The Cambridge N-Body Lectures

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Editorial Board

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Sverre J AarsethChristopher A ToutRosemary A Mardling (Eds)

The CambridgeN-Body Lectures

123

Sverre J AarsethUniversity of CambridgeInstitute of AstronomyMadingley RoadCambridge CB3 0HAUnited Kingdomsverreastcamacuk

Christopher A ToutUniversity of CambridgeInstitute of AstronomyMadingley RoadCambridge CB3 0HAUnited Kingdomcatastcamacuk

Rosemary A MardlingSchool of Mathematical SciencesMonash UniversityVictoria 3800Australiamardlingscimonasheduau

Aarseth S J et al (Eds) The Cambridge N-Body Lectures Lect Notes Phys 760(Springer Berlin Heidelberg 2008) DOI 101007978-1-4020-8431-7

The Royal Astronomical Society Series A series on Astronomy amp AstrophysicsGeophysics Solar and Solar-terrestrial Physics and Planetary Sciences

ISBN 978-1-4020-8430-0 e-ISBN 978-1-4020-8431-7

DOI 101007978-1-4020-8431-7

Lecture Notes in Physics ISSN 0075-8450

Library of Congress Control Number 2008929549

ccopy 2008 Springer-Verlag Berlin Heidelberg

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Preface

This book gives a comprehensive introduction to the tools required for directN -body simulations The contributors are all active researchers who writein detail on their own special fields in which they are leading internationalexperts It is their previous and current connections with the Cambridge Insti-tute of Astronomy as staff or visitors that gives rise to the title The materialis generally at a level suitable for a graduate student or postdoctoral workerentering the field

The book begins with a detailed description of the codes available forN -body simulations In a second chapter we find different mathematical for-mulations for special treatments of close encounters involving binaries ormultiple systems which have been implemented The concept of chaos andstability plays a fundamental role in celestial mechanics and is highlightedhere in a presentation of a new formalism for the three-body problem Theemphasis on collisional stellar dynamics enables the scope to be enlargedby including methods relevant for comparison purposes Modern star clus-ter simulations include additional astrophysical effects by modelling real starsinstead of point-masses Several contributions cover the basic theory and com-prehensive treatments of stellar evolution for single stars as well as binariesQuestions concerning initial conditions are also discussed in depth Furtherconnections with reality are established by an observational approach to dataanalysis of actual and simulated star clusters Finally important aspects ofhardware requirements are described with special reference to parallel andGRAPE-type computers The extensive chapters provide an essential frame-work for a variety of N -body simulations

During an extensive summer school on astrophysical N -body simulationsheld in Cambridge wwwcambodyorg the Royal Astronomical Society en-couraged us to edit a volume on the topic to be published in The Royal As-tronomical Society Series Subsequently we collected the tutorial lecture notesassembled in this volume We would like to take this opportunity to thankthe Royal Astronomical Society for sponsoring the school and the Institute ofAstronomy for provision of school facilities We are grateful to all the authors

VI Preface

who took time off from their busy schedules to deliver the manuscripts whichwere then checked for both style and scientific content by the editors Thiscollection of topics related to the gravitational N -body problem will proveuseful to both students and researchers in years to come

Cambridge Sverre J AarsethMay 2008 Christopher A Tout

Rosemary A Mardling

Contents

1 Direct N -Body CodesSverre J Aarseth 111 Introduction 112 Basic Features 213 Data Structure 314 N -Body Codes 415 Hermite Integration 616 AhmadndashCohen Neighbour Scheme 817 Time-Step Criteria 1018 Two-Body Regularization 1119 KS Decision-Making 13110 Hierarchical Systems 15111 Three-Body Regularization 17112 Wheel-Spoke Regularization 18113 Post-Newtonian Treatment 20114 Chain Regularization 21115 Astrophysical Procedures 23116 GRAPE Implementations 26117 Practical Aspects 28References 30

2 Regular Algorithms for the Few-Body ProblemSeppo Mikkola 3121 Introduction 3122 Hamiltonian Manipulations 3123 Coordinate Transformations 3324 KS-Chain(s) 3525 Algorithmic Regularization 3726 N -Body Algorithms 4427 AR-Chain 4528 Basic Algorithms for the Extrapolation Method 51

VIII Contents

29 Accuracy of the AR-Chain 56210 Conclusions 57References 58

3 Resonance Chaos and Stability The Three-Body Problemin AstrophysicsRosemary A Mardling 5931 Introduction 5932 Resonance in Nature 6133 The Mathematics of Resonance 6234 The Three-Body Problem 72References 95

4 FokkerndashPlanck Treatment of Collisional Stellar DynamicsMarc Freitag 9741 Introduction 9742 Boltzmann Equation 9843 FokkerndashPlanck Equation 10144 Orbit-Averaged FokkerndashPlanck Equation 10745 The FokkerndashPlanck Method in Use 113Acknowledgement 118References 118

5 Monte-Carlo Models of Collisional Stellar SystemsMarc Freitag 12351 Introduction 12352 Basic Principles 12453 Detailed Implementation 12654 Some Results and Possible Future Developments 145Acknowledgement 153References 153

6 Particle-Mesh Technique and SUPERBOX

Michael Fellhauer 15961 Introduction 15962 Particle-Mesh Technique 16063 Multi-Grid Structure of Superbox 166References 168

7 Dynamical FrictionMichael Fellhauer 17171 What is Dynamical Friction 17172 How to Quantify Dynamical Friction 17273 Dynamical Friction in Numerical Simulations 17574 Dynamical Friction of an Extended Object 177References 179

Contents IX

8 Initial Conditions for Star ClustersPavel Kroupa 18181 Introduction 18182 Initial 6D Conditions 20283 The Stellar IMF 22284 The Initial Binary Population 23885 Summary 253Acknowledgement 254References 254

9 Stellar EvolutionChristopher A Tout 26191 Observable Quantities 26192 Structural Equations 26493 Equation of State 26594 Radiation Transport 26895 Convection 27196 Energy Generation 27397 Boundary Conditions 27998 Evolutionary Tracks 27999 Stellar Evolution of Many Bodies 281References 282

10 N -Body Stellar EvolutionJarrod R Hurley 283101 Motivation 283102 Method and Early Approaches 284103 The SSE Package 286104 N -Body Implementation 289105 Some Results 293References 295

11 Binary StarsChristopher A Tout 297111 Orbits 298112 Tides 300113 Mass Transfer 302114 Period Evolution 307115 Actual Types 308References 318

12 N -Body Binary EvolutionJarrod R Hurley 321121 Introduction 321122 The BSE Package 321123 N -Body Implementation 325

X Contents

124 Binary Evolution Results 329References 331

13 The Workings of a Stellar Evolution CodeRoss Church 333131 Introduction 333132 Equations 333133 Variables and Functions 335134 Method of Solution 337135 The Structure of stars 339136 Problematic Phases of Evolution 340137 Robustness of Results 342References 345

14 Realistic N -Body Simulations of Globular ClustersA Dougal Mackey 347141 Introduction 347142 Realistic N -Body Modelling ndash Why and How 347143 Case Study Massive Star Clusters in the Magellanic Clouds 354144 Summary 375References 375

15 Parallelization Special Hardware and Post-NewtonianDynamics in Direct N-Body SimulationsRainer Spurzem Ingo Berentzen Peter Berczik David MerrittPau Amaro-Seoane Stefan Harfst and Alessia Gualandris 377151 Introduction 377152 Relativistic Dynamics of Black Holes in Galactic Nuclei 378153 Example of Application to Galactic Nuclei 380154 N -Body Algorithms and Parallelization 381155 Special Hardware GRAPE and GRACE Cluster 382156 Performance Tests 385157 Outlook and AhmadndashCohen Neighbour Scheme 386Acknowledgement 388References 388

A Educational N -Body WebsitesFrancesco Cancelliere Vicki Johnson and Sverre Aarseth 391A1 Introduction 391A2 wwwNBodyLaborg 391A3 wwwSverrecom 394A4 Educational Utility 396

References 397

Index 399

1

Direct N -Body Codes

Sverre J Aarseth

University of Cambridge Institute of Astronomy Madingley Road CambridgeCB3 0HA UKsverreastcamacuk

11 Introduction

The classical formulation of the gravitational N -body problem is deceptivelysimple Given initial values of N masses coordinates and velocities the taskis to calculate the future orbits Although the motions are in principle com-pletely determined by the underlying differential equations accurate solutionscan only be obtained by numerical methods Self-gravitating stellar systemsexperience highly complicated interactions which require efficient proceduresfor studying the long-term behaviour In this chapter we are concerned withdescribing aspects relating to direct summation codes that have been remark-ably successful This is the most intuitive approach and present-day technol-ogy allows surprisingly large systems to be considered for a direct attackAstronomers and mathematicians alike are interested in many aspects of dy-namical evolution ranging from highly idealized systems to star clusters wherecomplex astrophysical processes play an important role Hence the need formodelling such behaviour poses additional challenges for both the numericalanalyst and the code designer

In the present chapter we concentrate on describing some relevant proce-dures for star cluster simulation codes Such applications are mainly directedtowards studying large clusters However many techniques dealing with few-body dynamics have turned out to be useful here and their implementationwill therefore be discussed too At the same time the GRAPE special-purposesupercomputers are increasingly being used for large-N simulations Hence adiversity of tools are now employed in modern simulations and the practi-tioner needs to be versatile or part of a team This development has led tocomplicated codes which also require an effort in efficient utilization as well asinterpretation of the results It follows that designers of large N -body codesneed to pay attention to documentation as well as the programming itselfFinally bearing in mind the increasing complexity of challenging problemsposed by new observations further progress in software is needed to keeppace with the ongoing hardware developments

Aarseth SJ Direct N-Body Codes Lect Notes Phys 760 1ndash30 (2008)

DOI 101007978-1-4020-8431-7 1 ccopy Springer-Verlag Berlin Heidelberg 2008

2 S J Aarseth

12 Basic Features

Before delving more deeply into the underlying algorithms it is desirableto define units and introduce the data structure that forms the back-boneof a general N -body code From dimensional analysis we first constructfiducial velocity and time units by V lowast = 1 times 10minus5(GML

lowast)12 km sminus1T lowast = (Llowast3GM)12 s with G the gravitational constant and Llowast = 3times1018 cmas a convenient length unit Given the length scale or virial radius RV in pcand total mass NMS in M where MS is the average mass specified as in-put we can now write the corresponding values for a star cluster model asV lowast = 6557 times 10minus2(NMSRV)12 km sminus1and T lowast = 1494(R3

VNMS)12 MyrHence scaled (or internal) N -body units of distance velocity and time areconverted to corresponding astrophysical units (pc km sminus1 Myr) by r =RVr v = V lowastv t = T lowastt Finally individual masses in M are obtained fromm = MS m where MS is now redefined in terms of the scaled mean mass

As the next logical step on the road to an N -body simulation we considermatters relating to the initial data Let us assume that a complete set of initialconditions have been generated in the form mi ri vi for N particles wherethe masses coordinates and velocities can be in any units A standard clustermodel is essentially defined by NMS RV together with a suitable initialmass function (IMF) After assigning the individual data we evaluate thekinetic and potential energy K and U taking U lt 0 The velocities are scaledaccording to the virial theorem by taking vi = q vi where q = (QV|U |K)12

and QV is an input parameter (05 for overall equilibrium) Note that ingeneral the virial energy should be used however the additional terms arenot known ahead of the scaling We now introduce so-called standard unitsby adopting the scaling G = 1

summi = 1 E0 = minus025 where E0 is the

new total energy (lt 0) Here the energy condition is only applied for boundsystems (QV lt 1) otherwise the convention E0 = 025 is adopted The finalscaling is performed by ri = riS

12 vi = viS12 with S = E0(q2K + U)

These variables define a standard crossing time Tcr = 2radic

2T lowast MyrMany simulations include primordial binary stars for greater realism Be-

cause of their internal binding energies the above scaling cannot be imple-mented directly Instead the components of each binary are first combinedinto one object whereupon the reduced population of Ns single stars and Nb

binaries are subject to the standard scaling It then remains for the internaltwo-body elements such as semi-major axis eccentricity and relevant anglesto be assigned together with the mass ratio The choice of distributions is verywide but should be motivated by astrophysical considerations Of special in-terest here are the periods and mass ratios which may well be correlated forluminous stars (eg spectroscopic binaries) More complicated ways of pro-viding initial conditions with primordial binaries can readily be incorporatedThus for example a consistent set of initial conditions that do not requirescaling may be uploaded Such a data set might in fact be acceptable by awell-written code but this practice is not recommended

1 Direct N -Body Codes 3

13 Data Structure

The time has now come to introduce the data structure used in the CambridgeN -body codes Complications of describing the quantities in a stellar systemarise when some objects are no longer single stars In the first instance hardbinaries are treated by two-body regularization (Kustaanheime amp Stiefel 1965hereafter KS) Now a convenient description refers to the relative motion aswell as that of the centre of mass (cm) For the purposes of sequential pre-dictions and force summations it is natural to place the two KS componentsfirst in all relevant arrays followed by single stars with the cm last Thusgiven Np pairs the type of object can be distinguished by its location i inthe array compared to 2Np and N Likewise for long-lived triples where theinner binary of the hierarchy becomes the first member of the new KS pairand the outer component the second

The new arrangement necessitates the introduction of so-called ghost starswhich retain the quantities associated with the outer component except thatthe mass is temporarily set to zero In other words a ghost star is a dormantparticle without any gravitational effect since it now forms part of the tripleGeneralization to a quadruple consisting of two binaries forming a new KSfollows readily Note that in this case a ghost binary must be defined as wellas a ghost cm particle Higher-order systems of increasing complexity aredefined in an analogous manner The treatment of hierarchies continues as longas they are defined to be stable as will be discussed in subsequent sections

It now remains to introduce the final type of object in the form of acompact subsystem which is treated by chain regularization (Mikkola ampAarseth 1993) Briefly the idea here is to employ pairwise two-body regu-larization for the strongest interactions and include the other terms as per-turbations Such systems are invariably short-lived but the special treatmentis most conveniently carried out within the context of the standard data struc-ture At least two of the chain members are former components of a KS binaryand the initial membership may be three or four These systems are usuallycreated following a strong interaction between a binary and another singleparticle or binary Here one of the members is assigned to the role as the cmfor the subsystem while the others become ghosts

bull Single stars 2Np lt i le N Ni = ibull KS pairs 1 le i le 2Np ip = iicm minusNbull Cm particles i gt N N = N0 + Nk k = 2ip minus 1bull Stable triples KS + ghost Ncm = minusNk

bull Ghost particles Nghost = N2ipminus1 mghost = 0bull Stable quadruples KS + KS ghost Ncm = minusNk

bull Higher orders T + KS Ncm = minus (2N0 + Nk)bull Chain members 2Np lt icm le N Ncm = 0

The table summarizes the key features of the data structure In order to keeptrack of the identity of the particles we also assign a name to each denoted by

4 S J Aarseth

Ni This quantity is useful for distinguishing the type of object ie whethersingle binary or even chain cm Thus the name of a binary cm is definedby Ncm = N0 + Nk where N0 is the initial particle number and Nk is thename of the first KS component Likewise the cm of hierarchical systems ofdifferent levels are identified by Ncm lt 0 while Ni = 0 for a chain cm withi le N Note that an arbitrary number of binaries can be accommodated butonly one chain Given the location icm of any cm the corresponding KS pairindex is obtained from ip = icm minusN with the components at 2ip minus 1 2ip

A new KS pair is created by exchanging the individual particle componentswith the two first single-particle arrays and introducing the correspondingcm at N + Np after Np has been updated Conversely termination of aKS solution requires the former components to be placed in the first availablesingle-particle array (unless already in the correct location) and the cm to beeliminated The case of terminating a hierarchical system is more complicatedand will be considered later

There are many advantages of having a clearly defined and simple datastructure The analogy with molecules is striking and this also extends tointeractions since some objects may combine while others are disrupted inresponse to internal or external effects On the debit side all arrays of sizeN +Np must be in correct sequential order after each creation or destructionof an object Neighbour lists to be discussed later must also be updated con-sistently However the overheads still form a small fraction of the total CPUtime The same procedure applies when distant particles known as escapersare removed from the data set Again in the latter case the name identifiesthe type of object involved

14 N -Body Codes

A general N -body code consists of three main parts in the form of initial con-ditions integration and run-time data analysis of the results In the precedingsections we have discussed some relevant aspects dealing with the initial setupand data structure Before attacking the next stage it is useful to introducethe various algorithms that are used to advance the solutions Ideally differ-ent objects require a specially designed integration method in order to exploitthe characteristic features We start by considering single stars which usu-ally dominate by numbers and concentrate on the challenge of studying largesystems The first speed-up of such calculations can be obtained by assigningindividual time-steps according to the local conditions Since a Taylor seriesis used to describe the motion we are concerned with relative convergencewhere smooth orbits in low-density regions may have longer steps

From the N2 nature of the gravitational problem the calculation of theaccelerations requires an increasing fraction of the total effort Hence the sim-ple approach of direct summation for each integration step is too expensiveand restricts the type of problem for investigation A second efficiency feature

called a neighbour scheme (Ahmad amp Cohen 1973 hereafter AC) enables con-sistent solutions to be obtained while still employing direct summation Thebasic idea here is to introduce two time-scales for each particle where contri-butions from close neighbours are evaluated frequently by direct summationwhile the more distant forces are included (and recalculated) on a longer time-scale This two-polynomial scheme speeds up the calculation considerably atthe expense of extra programming Finally we also mention the modern wayto study large N and retain strict summation namely special-purpose com-puters known as GRAPE (Makino et al 1997)

Close encounters present another challenge that must be faced either inthe form of hyperbolic motion or as persistent binaries Although the time-steps of two interacting bodies can be reduced accordingly this may leadto significant accumulation of errors A more elegant way practised in theCambridge codes is to employ two-body regularization as mentioned aboveNow the programming requirements are quite formidable However the payoffis that such solutions can be used with confidence since the equations of motionare linear for weak perturbations

The next level of complexity arises when a regularized binary experiencesa strong interaction with another object A reliance on the two-body formu-lation makes for inefficient treatment during resonant interactions Compactsubsystems may instead be studied by three-body (Aarseth amp Zare 1974) orchain regularization (Mikkola amp Aarseth 1993) At present the former may beused if the external perturbations are small while the latter takes account ofperturbations and allows for up to six members Once again the programmingeffort is substantial but permits the study of extremely energetic interactions

One more special procedure remains to be discussed Although less spec-tacular the treatment of long-lived hierarchies requires careful decision-making A hierarchy is said to be stable if the orbital elements satisfy certainconditions The main property of a stable system is that the inner semi-majoraxis should be secularly constant in the presence of an outer bound perturberEssentially the outer pericentre needs to exceed the inner semi-major axis bya factor depending on the orbital parameters (Mardling amp Aarseth 1999)Once deemed to be stable the closest perturber is regularized with respectto the inner binary cm which is now treated as a point-mass However thespecial configuration is terminated on large external perturbations or if theouter eccentricity increases sufficiently to violate the stability criterion

The procedures outlined above constitute a veritable tool box for a widevariety of N -body simulations Efficient use of these tools requires a complexnetwork of decision-making Moreover it is desirable that the associated over-heads should only represent a small proportion of the total CPU effort Someof the relevant algorithms will be presented in later sections Suffice it for nowto state that this desirable requirement has been met as can be ascertainedby so-called run-time profiling

In the following we shall concentrate on the code nbody6 which combinesall of the above features and is suitable for studying realistic star clusters as

6 S J Aarseth

well as idealized systems on laptops and workstations However a section willbe devoted to GRAPE procedures With the above review as background wenow move to the next stage of presenting some of the main integration algo-rithms In each case further details are available elsewhere (Aarseth 2003)

15 Hermite Integration

Let us start by looking at the derivation of the Hermite scheme that hasproved so successful in modern simulations We expand Taylor series solutionfor the coordinates and velocities to fourth order in an interval Δt by

x1 = x0 + v0Δt+a0

2Δt2 +

a0

6Δt3 +

a(2)0

24Δt4 + α

a(3)0

120Δt5

v1 = v0 + a0Δt+a0

2Δt2 +

a(2)0

6Δt3 +

a(3)0

24Δt4 (11)

Here a represents the acceleration or force per unit mass which will alsobe referred to as force for convenience and α is an adjustable constant Thehigher-order Newmark implicit method (Newmark 1959) takes the form

x1 = x0 +12(v0 + v1)Δtminus α

10(a1 minus a0)Δt2 +

6αminus 5120

(a1 + a0)Δt3

v1 = v0 +12(a1 + a0)Δtminus 1

12(a1 minus a0)Δt2 (12)

As can be verified by substitution for v1 into the first equation with α = 1the standard Taylor series is recovered after some simplification

a1 = a0 + a0Δt+12a

(2)0 Δt2 +

16a

(3)0 Δt3

a1 = a0 + a(2)0 Δt+

12a

(3)0 Δt2 (13)

The subscripts 0 1 can be reversed hence the formulation is time-symmetricand consistent with the Hermite formulation It has been shown (Kokuboamp Makino 2004) that α = 76 is the optimal choice for the leading termin the error of the longitude of the periapse Moreover secular errors in theelements a and e are removed by using constant time-steps (in the absence ofencounters) for small eccentricities e le 01 This makes it an efficient schemefor planetesimal dynamics (see below) It has been found that energy errorsare improved by high-order prediction of the particle being advanced

It is also instructive to present a traditional formulation of standard Her-mite integration We first write a Taylor series for the force per unit mass Fand its explicit derivative F (1) for a given particle i (with index suppressed)to be advanced by a time interval t as

F = F 0 + F(1)0 t+

12F

(2)0 t2 +

16F

(3)0 t3

F (1) = F(1)0 + F

(2)0 t+

12F

(3)0 t2 (14)

After obtaining the initial values F 0 F(1)0 by summation the coordinates and

velocities of all particles are predicted to low order by

rj =[(

16F

(1)0 δtprimej +

12F 0

)

δtprimej + v0

]

δtprimej + r0

vj =(

12F

(1)0 δtprimej + F 0

)

δtprimej + v0 (15)

with δtprimej = t minus tj where tj is the time of the last force calculation New valuesF F (1) are now obtained in the usual way for the particle under considerationThis enables the higher derivatives to be constructed by inversion which yields

F(3)0 = [2(F 0 minus F ) + (F (1)

0 + F (1)) t]6t3

F(2)0 = [minus3(F 0 minus F ) minus (2F

(1)0 + F (1)) t]

2t2 (16)

Consequently the fourth-order corrector can be applied to the predicted so-lution of particle i by adding the contributions

Δri =124

F(2)0 Δt4 +

1120

F(3)0 Δt5

Δvi =16F

(2)0 Δt3 +

124

F(3)0 Δt4 (17)

Before proceeding we introduce so-called quantized time-steps according tothe rule

Δtn =(smax

2

)nminus1

(18)

where smax defines the maximum permitted value usually taken as unity withstandard scaling Hence every time-step Δti should correspond to some valueof n which entails a slight reduction from a provisional choice The reason forthis novel procedure is to reduce the overheads involved in the predictions ofall coordinates and velocities namely once per step Moreover this predictionis made by hardware when using GRAPE This procedure is referred to asa block-step scheme Thus it requires truncation of the natural step to thenearest value of n Moreover time-steps can only be increased by a factor of2 every other time to maintain synchronization of all ti + Δti

Here we also discuss a heliocentric formulation which has proved efficientfor planetesimal simulations (Kokubo Yoshinaga amp Makino 1998) In helio-centric coordinates the equation of motion for a mass-point mi is given by

ri = minusNsum

j=1 j =i

mj

[ri minus rj

|ri minus rj |3+

rj

r3j

]

minus M0 +mi

r3iri (19)

8 S J Aarseth

where M0 is the mass of the central star or dominant body If the total mass inplanetesimals is small (eg Saturnrsquos ring) the indirect terms may be neglected

In concise form the following algorithm describes the essential steps in-volved in the integration itself for a group of selected particles

bull Determine members due for updating at new time tbull Predict all r r to order Fbull Improve ri ri to order F (3) for the first memberbull Obtain F F due to planetesimalsbull Add optional gas drag or tidal dampingbull Include the dominant force and first derivativebull Apply the Hermite correctorbull Perform a second iteration by the two last stepsbull Specify provisional new time-step Δtibull Compare nearest neighbour step Δtnb = 01R2R middot Vbull Check for close encounter R lt Rcl R lt 0bull Complete the cycle for any other tj + Δtj = tbull Include optional boundary crossings

Some comments on this scheme are in order It is known as being time-symmetric Hermite of type P(EC)n (predict evaluate correct etc) The num-ber of iterations n is usually chosen as 2 but n = 3 may also be worth whileNote that for large N the expensive evaluation of the perturbations is not per-formed again because the two-body term dominates the errors On GRAPEthe procedure for identifying close encounters is implemented by using thenearest-neighbour facility which enables a suitable maximum time-step to bedefined In the alternative case of a standard calculation the closest parti-cle can readily be determined from the current neighbour list which wouldusually be small1 Typically a close encounter is defined by the distance Rclwhich signals switching the solution method to regularization (if desired)

16 AhmadndashCohen Neighbour Scheme

Most simulations aim for the largest systems that can be studied with a givenresource As already remarked this invariably means the use of some kind ofneighbour (or hybrid) procedure In the following we summarize the salientfeatures of the AC scheme since complete descriptions of the Hermite versionare already available (Makino amp Aarseth 1992 Aarseth 2003)

The basic idea is to split the total force acting on a particle into two partsformally represented by

F (t) =nsum

j=1

F j + F d(t) (110)

1A full-blown AC scheme might not satisfy the strict time-symmetry condition

where the first term contains the contributions from the n nearest neighboursand F d represents the distant members as well as any external effects Like-wise a similar equation can be written for the force derivative The basic ideais to perform direct summation over the neighbours at suitably chosen smallsteps and add the predicted contributions from the distant particles with fit-ting coefficients recalculated on a longer time-scale Δtd This leads to a gainin performance provided that N n and Δtd Δtn can be satisfied

The total force used for the integration is obtained on the time-scale Δtdwhen the neighbour list is also formed At intermediate times or so-calledirregular time-steps the total force and first derivative are evaluated by

F (t) = F n + F d(tminus t0) + F d(t0)F (t) = F n + F d (111)

where t0 is the time of the last regular force calculation For conveniencethe two time-steps are commensurate but this is not a formal requirementprovided the total force is evaluated at the nearest irregular time The deter-mination of time-steps for each force polynomial will be discussed in the nextsection

There are several possible strategies for neighbour selection Essentiallythe choice is between aiming for a constant value of n or adopt a more flexibleapproach depending on local conditions Given that particles in the halo havesmooth orbits as opposed to those in the core that are affected by stronginteractions it seems appropriate to employ a criterion depending on thedensity The neighbour radius itself is updated according to the relation

Rnews = Rold

s

(np

n

)13

(112)

Here the predicted neighbour number np is expressed in terms of the densitycontrast C prop nR3

s asnp = nmax(004C)12 (113)

subject to an upper limit Again the choice of nmax is a matter of taste but avalue near 2N12 has proved itself for large N In fact there are compensatingfactors affecting code performance such that smaller n requires more frequentupdating of the neighbours The neighbour selection is made during the totalforce calculation using |ri minus rj | lt Rs and is essentially free since all distancesare calculated in any case

The combination of two-force polynomials requires some care when thereis a change in the neighbour population In general there is a flux across theneighbour sphere which must be accounted for in the higher derivatives Todo this we evaluate the explicit derivatives F

(2)ij F

(3)ij from the corresponding

members j and add or subtract the corrections to the higher derivatives thatare kept separately However this extra cost may be avoided by performingthe energy check and result analysis at times commensurate with smax since

10 S J Aarseth

all the solutions are then known to highest order This is possible becauseonly predictions up to F

(1)i are used in the general integration

As regards performance the neighbour scheme is comparable to a single-force polynomial code for N 50 and speeds up as N14 Moreover a compar-ison with the GRAPE-6A (so-called micro-Grape) with the same host showsthe latter being faster by a factor of 11 for N = 25 000 Finally we emphasizethat neighbour lists are also very useful for identifying other close membersin connection with regularization and for estimating the density contrast

17 Time-Step Criteria

Any integration method based on individual time-steps tries to employ anappropriate criterion which optimizes the overall solution accuracy At thesimplest level are expressions of the type

Δt =α|r||v| Δt =

β|F ||F (1)|

(114)

where α and β are suitable dimensionless constants However such simpleforms invariably cause numerical problems mainly because close encountersare not detected in time for step reduction Since we are dealing with a Taylorseries for the force it is natural to look for a relative criterion involving higherderivatives The most convenient simple time-step can be constructed from

Δt =

(η|F ||F (2)|

)12

(115)

where η 002 would give reasonable behaviour For many years this relationwas used with success

The idea of relative convergence can be extended to take into account allthe force derivatives Consequently we write a general expression in the form

Δt =

(η(|F ||F (2)| + |F (1)|2)|F (1)||F (3)| + |F (2)|2

)12

(116)

This criterion has several useful properties Compared to (115) it gives a well-defined large value when the force is small as is the case near a tidal boundaryMoreover two bodies with different masses will tend to have similar time-stepsduring close encounters which facilitates decision-making In fact after thetruncation according to (18) the two steps are often identical but this cannotbe assumed It is worth emphasizing that a relative time-step criterion of theabove type is independent of the (non-zero) mass

From past experience it seems most efficient to assign slightly differentvalues for the dimensionless accuracy factors Hence in most practical work

regardless of N the respective values ηI = 002 ηR = 003 for the irregularand regular time-steps have been adopted For N 1000 typical time-stepratios of about 6 are seen this increases slowly as N is increased

In the case of planetesimal simulations special care is needed to ensuredetection of close encounters and physical collisions We therefore employ anadditional criterion based on the nearest neighbour

Δt =βR2

|R middot V | (117)

where β = 01 has proved sufficient The different strategies for GRAPE andconventional computers in this problem were commented on in a previoussection

For completeness we also include KS regularization in this discussion sinceit has relevance for the general time-step criterion Briefly for the unperturbedcase the equation governing the relative motion is given by

F u =12hu (118)

where h is the specific two-body energy and u the generalized coordinateswhich have the useful property u middot u = R Since h lt 0 for a binary we definethe constant time-step in terms of the frequency as

Δτ =ηu

(2|h|)12 (119)

with ηu = 02 for accurate solution (Mikkola amp Aarseth 1998) Substitutioninto (116) by carrying out explicit differentiation (with hprime = 0) simplifies tothe adopted form thereby giving some support for this apparently complicatedexpression Note that the basic time-step (119) is reduced appropriately inthe presence of significant perturbations

18 Two-Body Regularization

Regularization plays an important part in the codes under discussion In thefollowing we outline some of the main aspects of the KS method and describevarious relevant algorithms The latter can be divided into a purely localpart involved with studying the relative motion and a global part that formsan interface with the whole system Let us begin with a summary of thewell-known classical formulation (Kustaanheimo amp Stiefel 1965) for the 3Dtreatment which is described in more detail elsewhere (Aarseth 2003)

New coordinates in 4D are introduced by the condition

R = u21 + u2

2 + u23 + u2

4 (120)

12 S J Aarseth

As usual in regularization a time transformation is also needed and we choosethe simplest differential relation

dt = R dτ (121)

or tprime = R It turns out that the coordinate transformation

R = L(u)u (122)

is satisfied by the Levi-Civita matrix

L(u) =

⎡

⎣u1 minusu2 minusu3 u4

u2 u1 minusu4 minusu3

u3 u4 u1 u2

⎤

⎦ (123)

as can be verified by substitution into the equation for R For completenesswe also include the appropriate relations for the relative velocity Thus theregularized velocities are obtained by

uprime =12LT (u)R (124)

while the physical values are recovered from

R = 2L(u)uprimeR (125)

Starting from the perturbed two-body problem for mk and ml

R = minusmk +ml

R3R + P (126)

with P the tidal perturbation the equations of relative motion can be derivedThe complete set is given by

uprimeprime =12hu +

12RLT P

hprime = 2uprime middot LT P

tprime = u middot u (127)

where LT represents the transpose matrixThe 10 equations describing the relative motion in the presence of external

perturbations are regular in the sense that the solutions are well defined forR rarr 0 In order to describe the actual orbit in a stellar system we introducethe associated cm by

rcm =mk rk +mlrl

mk +ml (128)

Likewise the cm force is obtained from

rcm =mk P k +ml P l

mk +ml (129)

Hence the cm is added to the system of N particles as a fictitious memberto be advanced in time Individual coordinates are obtained by combining thetwo motions which yields

rk = rcm + μRmk

rl = rcm minus μRml (130)

where μ = mkml(mk +ml) is the reduced mass and similarly for the globalvelocities

Given the regularized time-step defined above the equations for therelative motion are advanced by an efficient Hermite method (Mikkola ampAarseth 1998) Although this formulation is fairly complicated the KS equa-tions can also be written in standard Hermite form by including the terms F

prime

u

and hprimeprimeImplementation of two-body regularization has many practical benefits

First the equations of motion take the form of a perturbed harmonic oscil-lator and are therefore regular This treatment permits a constant time-stepfor small perturbations while for direct integration Δt prop R32 which canbe troublesome when treating very eccentric binaries Moreover with lin-earized equations the accuracy per step is higher and only about 30 steps areneeded for an orbit Integration of relative motion also permits a faster forcecalculation because P prop 1R3 for tidal perturbation Finally on the creditside unperturbed two-body motion is justified in case there are no perturberswithin a distance d = λa(1 + e) with λ 100 Likewise if d gt λR the cmapproximation can be used in force calculations with binaries

The price to pay for all the advantages comes in the form of coordinate andvelocity transformations at the interface between relative and global motionHowever these operations are fast and do not involve the square root Asfor simulations using GRAPE there is a further cost due to differential forcecorrections since the hardware is based on point-mass interactions

Several optional features are worth mentioning For small perturbationsthe principle of adiabatic invariance can be used to slow down the motionby scaling the perturbation (Mikkola amp Aarseth 1996) So-called energy rec-tification improves the solutions of uuprime by scaling to the explicit value ofh which is integrated independently The availability of completely regulartwo-body elements like the semi-major axis (a) and eccentricity (e) can alsobe beneficial when employing averaged expressions to model secular evolutionof stable triples or tidal circularization (Mardling amp Aarseth 2001)

19 KS Decision-Making

A variety of algorithms are involved in the overall management of the regu-larization scheme Broadly speaking we may distinguish between aspects ofinitialization integration and termination and these will be covered in turn

14 S J Aarseth

The first question which presents itself is when to choose two particles forregularization treatment A close encounter is traditionally defined by the twomain parameters

Rcl =4 rh

N C13 Δtcl = β

(R3

cl

m

)12

(131)

where rh is the half-mass radius C is the central density contrast and β adimensionless constant determined by experimentation Thus a particle withtime-step Δtk lt Δtcl needs to have a close neighbour inside the distance RclFurther conditions of negative radial velocity and dominant two-body motionmust also be satisfied The latter is ensured by comparing the two-body termsdue to any other members identified in the close encounter search In the caseof GRAPE a list of particles with small time-steps is maintained and updatedduring the force calculation when the host computer is idle

The principle of initializing KS polynomials is the same as for single parti-cles except that time derivatives must also be obtained By employing explicitdifferentiation the latter terms are readily constructed from the available datainvolving u and its derivatives A conversion by Taylor series expansion forΔτ finally gives the time-step in physical units which is used for the schedul-ing of regularized solutions Thus any KS pair which needs to be advancedduring the next block-step is treated first

Initially and during the integration a consistent perturber list must alsobe available The perturber search is carried out after each apocentre passageRap = a(1+e) using the tidal limit approximation Particles inside a distance

rp =(

2mp

mbγmin

)13

a (1 + e) (132)

are selected from the neighbour list where mb is the mass of the binaryand γmin is a small dimensionless perturbation usually taken as 10minus6 Anextra procedure is included to increase the neighbour list for cm particles ifRs lt λa(1 + e)

A useful quantity for many purposes is the dimensionless relative pertur-bation defined by

γ =|P k minus P l|R2

mk +ml (133)

If evaluated in the apocentre region this dimensionless quantity is a measureof dominant two-body motion In general it is advantageous to initiate regu-larization if γ 01 but larger values are acceptable during the treatment

The KS integration itself begins with the prediction of u and uprime to high-est order u(5) while h is predicted to order h(2) As usual in the Hermitescheme perturbers are predicted to low order Transformations yield globalcoordinates and velocities rk rl rk rl which are needed for the force calcu-lation The physical perturbation P = P k minusP l and P can now be obtained

By virtue of the time transformation we have P prime = R P This enables thecorrector to be applied with new values uuprime to order u(5) and h to h(4)An iteration without recalculation of the perturbations improves the finalsolution

The conversion to physical time must also be carried out to highest orderTaylor series expansion yields the desired terms by successive explicit differ-entiation beginning with tprimeprime = 2u middot uprime and continued up to t(6) using knownterms This permits the corresponding physical time-step to be obtained by

Δt =6sum

k=1

1kt(k)0 Δτk (134)

Time inversion is required when calculating the force on single particles Givena physical interval δt this is achieved by expanding τ = 1R to sufficient orderNote that division by R is not dangerous here since the cm approximationis used for small values

Conditions for unperturbed motion have been alluded to above By carefulanalysis of the velocity distribution of nearby particles it is possible to extendthe analytical solution to many Kepler periods This is achieved by identifyingthe particles that provide the maximum force as well the smallest time ofminimum approach If there are no perturbers we estimate the minimumtime to reach the boundary γ γmin as well as the free fall time of thenearest particle Depending on the remaining time a number of unperturbedorbits may be adopted and the KS motion will remain dormant until the nexttime for checking Several extra conditions are also included in order to avoidpremature interactions inside the unperturbed boundary

Following the general exposition we now comment on the final stage of theKS cycle Termination of hard binaries is appropriate for strong perturbationsay γ ge 05 which would most likely result in switching to another dominantpair (temporary capture or so-called resonance) or chain regularization Forsofter binaries a smaller perturbation limit is called for After terminationstandard force polynomials are initialized for the two single particles

As a technical point except for collisions termination is delayed until theend of the block-step ie until the remaining interval δt = Tblock minus t fallsbelow the physical step Δt converted from Δτ A final iteration to the exactvalue can then readily be performed with Δτ obtained from τ τ and δt

110 Hierarchical Systems

Long-lived triples or even quadruples form an important constituent inN -body simulations Typically a triple is formed through a strong interac-tion between two hard binaries where the weakest binary is disrupted andone component is ejected The other component may then be captured intoan orbit around the inner binary because of energy and angular momentum

16 S J Aarseth

conservation Such systems may have long life-times and their treatment bydirect integration poses very severe numerical problems (or even code crash)by loss of accuracy as well as greater effort

Over the years there has been a quest for stability criteria which wouldallow the description of hierarchies to be simplified by assuming the innersemi-major axis to be constant permiting the cm approximation to be usedIn the absence of secular changes the outer component (a single particle oranother binary) may then be regularized with respect to the inner binary cmthereby speeding up the calculation by a large factor For this purpose we haveemployed a stability criterion that has been tested successfully for a limitedrange of parameters (Mardling amp Aarseth 1999 2001) A sharper stability cri-terion has been developed recently for the general three-body problem basedon first principles The underlying theory is discussed in Chap 3 togetherwith a practical algorithm that has been implemented in nbody46 Givenall the elements describing the inner and outer orbit this algorithm definesstability or otherwise for a hierarchical configuration instead of estimating thedistance from the stability boundary Consequently the stability test needsto be re-assessed during the subsequent evolution

The identification of a hierarchical candidate system involves checkingmany conditions In the first instance a search is initiated after each apocen-tre turning point provided the cm step is sufficiently small in other wordsif Δtcm lt Δtcl This condition implies that the new hierarchy is likely toform a hard outer binary However it should be stated that the same testis also performed for a new chain regularization which again involves stronginteractions After identifying the two most dominant neighbours the outertwo-body elements are constructed for the main perturber Among furtherconditions to be checked are the perturbation on the outer orbit as well asthe requirement of a new hard binary Moreover extra tests are performed ifthe outer component is another binary in which case a modified criterion isused depending on the ratio of semi-major axes

Acceptance of the stability condition entails a considerable programmingeffort in order to maintain a consistent data structure as discussed in anearlier section The relevant algorithmic steps are set out in the followingtable and are mostly self-explanatory

bull Increase the control index for decision-makingbull Save relevant masses mkml in a hierarchy tablebull Copy cm neighbour list for later correctionsbull Terminate KS solution and update Np and arraysbull Evaluate potential energy of components and old neighboursbull Record R = rk minus rl V = vk minus vl and h in the special tablebull Form binary cm in location of the primary j = 2Np + 1bull Define ghost (m = 0 x = 106) and initialize prediction variablesbull Obtain potential energy of inner cm body and neighboursbull Remove ghost from neighbour and perturber listsbull Initialize new KS for outer component in l = k + 1

bull Specify cm and ghost names Ncm = minusNk Nghost = Nl

bull Set pericentre stability limit in R0(Np) for termination testbull Update the internal and differential energy ΔE = μh0 + ΔΦ

Integration of hierarchical systems proceeds in the usual way except that thestability condition needs to be checked This is done at each apocentre turningpoint using the property Ncm lt 0 for identification One way in which thestability test may no longer apply is when the outer eccentricity increases dueto perturbations otherwise similar termination criteria are used as for hardbinaries For completeness we also give the algorithm dealing with the mainpoints of termination

bull Locate current position in the hierarchy table Ni = Ncm

bull Save cm neighbours for correction procedurebull Terminate the outer KS solution (k l) and update Np

bull Evaluate potential energy of cm wrt neighbours amp lbull Determine location of ghost Nj = Nghost j = 1 N +Np

bull Restore inner binary components from saved quantitiesbull Add l to neighbour lists containing first component kbull Initialize force polynomials for outer componentbull Copy basic KS variables h u uprime from the tablebull Re-activate inner binary as new KS solutionbull Obtain potential energy of inner components and perturbersbull Update internal energy for conservation ΔE = ΔΦ minus μhbull Reduce control index and compress tables (including escapers)

111 Three-Body Regularization

More than 30 years ago a break-through in regularization theory made it pos-sible to study the strong interactions of three particles (Aarseth amp Zare 1974)The basic idea is simple namely to employ two different KS solutions of m1

and m2 separately with respect to the so-called reference body m3 It is alsoinstructive to review this development because of its connection with the sub-sequent chain regularization mentioned above

In the following we summarize the key points of the formulation Theinitial conditions are first expressed in the local cm frame with coordinatesri and momenta pi Given the three respective distances R1 R2 R with Rthe distance between m1 and m2 and p3 = minus(p1 + p2) as the momentum ofm3 the basic Hamiltonian can be written as

H =2sum

k=1

12μk3

p2k +

1m3

pT1 middot p2 minus

m1m3

R1minus m2m3

R2minus m1m2

R (135)

with μk3 = mkm3(mk +m3) As can be seen the kinetic energy is expressedby the momenta of m1 and m2 together with a cross product which represents

18 S J Aarseth

the mutual interaction of m1 and m2 Likewise the potential energy is a sumof the three relevant terms Thus omitting any references to m2 reduces tothe familiar form of the two-body problem

In analogy with standard KS we introduce a coordinate transformation forthe distances R1 and R2 by

Q2k = Rk (k = 1 2) (136)

Several alternative time transformations are available Here we adopt the orig-inal choice which is the most intuitive but not necessarily the best giving thedifferential relation between physical and regularized time

dt = R1R2 dτ (137)

This enables a regularized Hamiltonian to be formed as Γlowast = R1R2 (H minusE0)where E0 is the initial energy By construct Γlowast should be zero along thesolution path Making use of the KS property p2

k = P 2k4Rk where P k now

is the regularized momentum the new Hamiltonian becomes

Γlowast =2sum

k=1

18μk3

Rl P2k +

116m3

P T1 A1 middot AT

2 P 2

minusm1m3R2 minusm2m3R1 minusm1m2R1R2

|R1 minus R2|minus E0R1R2 (138)

where l = 3 minus k For historical reasons Ai is taken as twice the transposeLevi-Civita matrix of (123) Finally the equations of motion are given by

dQk

dτ=

partΓlowast

partP k

dP k

dτ= minus partΓlowast

partQk

(139)

It can be seen from inspection of the Hamiltonian that the solutions are reg-ular for R1 rarr 0 or R2 rarr 0 Moreover the singular terms are numericallysmaller than the regular terms provided |R1 minus R2| gt max (R1 R2) Hence aswitch to another reference body can be made when R is no longer the largest(or second largest) distance which usually ensures a regular behaviour Fulldetails of the transformations can be found in the original publication

So far three-body regularization has only been used in unperturbed formwithin the N -body codes when chain regularization is not available whichis quite rare However it can be quite efficient as a stand-alone code forscattering experiments In particular the simplicity of decision-making as wellas the ability to achieve accurate results by a high-order integrator makes ita good choice for such problems (Aarseth amp Heggie 1976)

112 Wheel-Spoke Regularization

The recent interest in massive objects in the form of black holes has inspireda closer look at alternative regularization methods The so-called wheel-spoke

formulation is a direct generalization of three-body regularization to includemore members (Zare 1974) Such a configuration may be appropriate if thereference body dominates the mass in which case the need for switching isno longer an issue and leads to further simplification The scheme is outlinedhere in the expectation that it will prove a popular tool since its effectivenesshas been demonstrated recently (Aarseth 2007)

Let us consider a subsystem of n single particles of mass mi and a dominantbody of mass m0 where the initial conditions qi pi are expressed in the localcm frame Introducing relative coordinates qi with respect to m0 we writethe Hamiltonian as

H =nsum

i=1

p2i

2μi+

1m0

nsum

iltj

pTi middot pj minusm0

nsum

i=1

mi

Riminus

nsum

iltj

mimj

Rij (140)

where μi = mim0(mi + m0) and Ri = |qi| As can be seen this is a directgeneralization of (135) to n gt 2 where m0 plays the role of reference bodyThis implies that the technical treatment will also be similar However theoriginal time transformation is now replaced by the inverse Lagrangian energyas tprime = 1L since a multiple product would be cumbersome and might notwork for critical cases This choice has many advantages and would also besuitable for three-body regularization

The use of a fixed reference body albeit with dominant mass raises atechnical problem of dealing with close encounters between two light bodiesThus for small separations the last term of (140) may become arbitrarilylarge if Rij rarr 0 At present this difficulty is overcome by introducing a smallsoftening in these terms while still retaining the conservative nature of theHamiltonian It turns out that the powerful integrator (Bulirsch amp Stoer 1966)is able to handle quite small values of non-regularized distances so that theessential dynamics is preserved

The regularized coordinates and momenta Qi P i are obtained in the usualway Conversely the physical values are recovered from the inverse transfor-mations by

qi =12AT

i Qi pi =14AT

i P iRi (141)

For completeness we also give the full set of transformations to the final valuesin the local cm system corrected for a sign error

qi = q0 + qi q0 = minusnsum

i=1

miqi

nsum

i=0

mi

pi = pi (i = 1 n) p0 = minusnsum

i=1

pi (142)

The method presented here may also be used for more conventional calcula-tions involving comparable masses without the restriction of a fixed referencebody or softening This would be a simpler alternative to chain regularizationbut would at most be effective for four or five members

20 S J Aarseth

113 Post-Newtonian Treatment

The wheel-spoke formulation is particularly suited to studying a compact sub-system containing a massive object inside a star cluster Especially attractiveis the possibility of including relativistic terms in the most dominant two-body motion The corresponding post-Newtonian equation of motion can bewritten in the convenient form (Blanchet amp Iyer 2003 Mora amp Will 2004)

d2r

dt2=

mi +m0

r2

[(minus1 +A)

r

r+Bv

] (143)

where the dimensionless quantities A and B represent relativistic effects Herethe two-body term is contained in the regularized Hamiltonian with the re-maining contributions added as a perturbation

The coefficients A B can be expanded as functions of vc with c the speedof light Using the current notation this gives rise to the perturbing force

P GR =mim0

c2r2

[(

A1 +A2

c2+A52

c3

)r

r+(

B1 +B2

c2+B52

c3

)

v

]

(144)

Here the first-order precession is described by

A1 = 2(2 + η)mi +m0

rminus (1 + 3η)v2 +

32ηr2 B1 = 2(2 minus η)r (145)

with η = mim0(mi + m0)2 Next comes the second-order precession termsA2 B2 which are somewhat more complicated Of most interest is the energyloss by gravitational radiation represented by A52 B52

For energy conservation purposes an extra equation for the relativisticcontribution is integrated according to

ΔEGR =int

P GR middot v dt (146)

In order to carry out the treatment in regularized time the right-hand side isconverted into an expression analogous to hprime in (127) Also note that deriva-tive evaluations of the physical perturbation are not required for solution offirst-order equations The associated time-scale for shrinkage employed in thedecision-making is given by (Peters 1964)

τGR =5a4c5

64mim20

(1 minus e2)72

g(e) (147)

where g(e) is a known function and standard N -body units applyImplementation of the wheel-spoke scheme into a large N -body code

presents many interesting aspects To begin with a suitably compact sub-system is chosen from a binary containing the heavy body if there is at leastone close perturber inside Rcl The subsystem is initialized in the usual way

including transformations to KS-type variables Q P The perturber list isagain constructed according to (132) which now yields a smaller mass factorand hence requires less effort in coordinate prediction

Although the innermost binary is invariably long-lived the question ofmembership changes must be considered Decisions of addition or removal arebased on the central distance and radial velocity of perturbers or existingmembers respectively Simple criteria including a combination of an appro-priate perturbation (say γ gt 005) and distance (rp lt

sumRk) are used in

the former case while removal is controlled by R2 gt 2m0R and Rk gt RclIn analogy with the integration of KS binaries the cm force is obtained byvectorial summation over the components

The addition of post-Newtonian terms necessitates the introduction ofphysical units This is achieved by specifying the total mass and half-massradius as well as the speed of light From NMS and rh we have c = 3times105V lowastwith the velocity scaling factor V lowast expressed in km sminus1 This enables thecoalescence distance to be defined as three Schwarzschild radii by

rcoal =6(mi +m0)

c2 (148)

Alternatively a disruption distance may be defined for white dwarfs An ex-perimental scheme has been adopted where the different GR terms are acti-vated progressively depending on the value of the time-scale (147) Thus theradiation term is included first on the supposition that precession does notplay an important role during the early stages However due care must beexercised if the innermost binary is subject to Kozai cycles (Kozai 1962)

Simulations of centrally concentrated cluster models have been made witha GRAPE code for m0 = N12MS and N = 105 equal-mass stars Here theinnermost binary shrank by a significant factor and also developed very higheccentricity by the Kozai resonance In some cases the resulting pericentredistance was sufficiently small for stars with white dwarf radii to be affectedby further gravitational radiation shrinkage before disruption (Aarseth 2007)

114 Chain Regularization

This contribution would not be complete without a discussion of chain regu-larization which has proved to be a powerful tool in star cluster simulationsIn the following we shall review some of the essential features as well as themain algorithms since the relevant details can be found elsewhere (Mikkola ampAarseth 1993 Aarseth 2003)

The basic idea takes its cue from three-body regularization A system issuitable for special treatment if one hard binary has a close perturber in theform of a single particle or another binary Upon termination of the KS binarythe coordinates and momenta are expressed in the local cm frame Thus Nminus1

22 S J Aarseth

chain vectors connect the particles experiencing the strongest pair-wise forcesand are defined in terms of the coordinates qk by

Rk = qk+1 minus qk k = 1 N minus 1 (149)

In Hamiltonian theory the generating function

S =Nminus1sum

k=1

W k middot (qk+1 minus qk) (150)

connects the old momenta with the new ones by pk = partSpartq The relativephysical momenta W k can then be obtained by the recursion

W k = W kminus1 minus pk k = 2 N minus 2 (151)

with W 1 = minusp1 and W Nminus1 = minuspN due to the cm condition Substitutioninto a Hamiltonian of the type (140) yields

H =12

Nminus1sum

k=1

(1mk

+1

mk+1

)

W 2k minus

Nminus1sum

k=2

1mk

W kminus1 middot W k

minusNminus1sum

k=1

mkmk+1

Rkminus

Nsum

1leilejminus2

mimj

Rij (152)

where the first momentum term contains the reduced mass In spite of the sim-ilarity with (140) the formalism differs in some important respects mainlybecause there is no reference body

As stated earlier the inverse Lagrangian energy is a good choice for thetime transformation Multiplication by tprime = 1L gives the regularized Hamil-tonian Γlowast = tprime(H minusE0) which can be differentiated in the usual way to yieldthe equations of motion Note that for technical reasons the differentiation ofthe product tprimeH is done explicitly This procedure enables the term H minus E0

(which should be zero) to be retained for stabilizing the solutions It can beseen that the two-body solutions are regular for any individual Rk rarr 0 atseparate times As usual the KS relations can be used to recover the physicalvariables via the standard transformations

Rk = Lk Qk W k = Lk P k2Q2k (153)

from which the momenta pk are readily derivedThe implementation of chain regularization into an N -body code contains

many algorithms some of which will be described briefly Following initial-ization in the cm frame and evaluation of the total energy E0 the chainvectors must be constructed The selection of the corresponding chain indicespresents a considerable algorithmic challenge if (as may occur later) thereare more than four members (cf Mikkola amp Aarseth 1993) Thus the scheme

may not work efficiently if the chain vectors fail to connect the dominant two-body forces The canonical variables Q P are introduced as before and theintegration can begin after specifying a suitably small time-step

Several quantities are useful for the decision-making Among these are thecharacteristic external perturbation γch and gravitational radius Rgrav wherethe latter represents the effective size of the subsystem Thus a perturber isconsidered for chain membership if γch is significant provided certain otherconditions are fulfilled The perturber list is updated at appropriate timesby (132) with Rgrav replacing the apocentre distance Likewise an existingmember with positive radial velocity is a candidate for removal if we have

R2k gt

2sum

mk

Rk Rk gt 3Rgrav (154)

Here the former condition requires transformation to the local cm systemThe chain integration is continued as long as there are at least three memberswith re-initialization after any changes Note that the membership procedurealso allows for a hard binary to be added or removed

It turns out that the chain structure is a convenient tool for checking thedynamical state Thus any escaping single particle or binary can readily beidentified by considering the distances at the beginning and end of the chainif N gt 3 As in the case of two-body regularization the internal integration iscontinued up to the next block-step time This entails inverting the integralof Ldt for an upper limit to ensure that the block-step is not exceeded Notethat here we do not have a Taylor series expansion for the time derivatives

In general termination is carried out if max Rk gt 3Rcl for three par-ticles or two hard binaries Provisions are also included for termination of astable hierarchy followed by switching to the more efficient KS treatmentAs discussed previously one way in which this can occur is after a stronginteraction of two binaries Finally procedures for physical collisions or tidalcircularization are also included albeit with considerable programming effort

115 Astrophysical Procedures

A star cluster simulation code should include a wide range of astrophysicalprocesses for a realistic treatment In the following we touch briefly on someof the most relevant aspects of the Cambridge codes By now the additionof synthetic stellar evolution has enabled the introduction of many interest-ing features that pose numerical challenges The simulation of realistic starclusters requires an IMF containing a significant proportion of heavy stars asdiscussed in Sect 74 It has been known for a long time that a few heavy bod-ies exert an unduly large influence on the dynamics of stellar systems Such adistribution also leads to mass segregation on a short time-scale which maybe comparable to the main-sequence life-time for typical cluster parametersMass loss from evolving stars is therefore important for all but the youngest

24 S J Aarseth

clusters and its inclusion in a simulation code is essential for observationalinterpretation

Since the basic ingredients of the stellar evolution scheme are discussedat length in Chaps 10 and 12 we concentrate on some of the related algo-rithms here The primary quantities associated with each star are updatedat sufficiently frequent intervals for a smooth representation For dynamicalpurposes only the process of mass loss requires special treatment It is usu-ally confined to a small fraction of all stars The main procedures can besummarized under the following headings

bull Mass loss from single stars and binariesbull Roche-lobe mass transfer and common-envelope evolutionbull Magnetic braking and spin-orbit couplingbull Inspiralling of compact binariesbull Supernova explosions and neutron star kicksbull Physical collisions (KS or chain regularization)

In the case of significant mass loss Δm gt 01M force polynomials for thenearest neighbours are re-initialized in order to reduce discontinuity effectsLikewise appropriate corrections are made to ensure overall energy conserva-tion This entails knowledge of the potential since we assume that the ejectedmass escapes rapidly from the cluster When using GRAPE the cost of a fullN summation can be avoided in most cases (except small Δti and large Δm)by employing the available potential corrected for the net force contributionup to the current time

Δφ = minusvi middot (F i minus F tide)(tminus ti) (155)

Close binaries undergoing general mass loss on a slow time-scale also re-quire updating of their KS elements Consequently the orbital parametersare modified at constant eccentricity based on the adiabatic approximationMba = const A corresponding correction for the inner binary elements of ahierarchical triple can be carried out explicitly Here it is necessary to re-assessthe stability condition because the inner orbit expands more than the outerone

A realistic period distribution invariably includes binaries that experienceRoche-lobe mass-transfer after the primary leaves the main sequence Thisstage is initiated by tidal circularization or the formation of a circular binaryfollowing common envelope evolution Since the complicated astrophysicalmodelling is discussed in Chap 12 we limit our comments to some computa-tional aspects for completeness For practical reasons the continuous processof mass transfer is divided into an active and a coasting phase where thelatter is updated at frequent intervals The duration of the active phase isrestricted to the cm time-step for consistency with the dynamics After theinternal adjustment of the essentially circular orbit has been completed anysystem mass loss is corrected for in the same way as for single stars

Magnetic braking and inspiralling of compact binaries by gravitationalradiation are catered for both within the Roche process as well as for certainnon-interacting binaries In either case changes in the rotational spin of thecomponents are treated according to the recipes outlined in Chap 12 Wenote that these processes themselves do not involve any mass loss

Stars above about 8M undergo supernova explosions and eject a signifi-cant amount of mass during the transition to neutron stars In the absence ofa consensus on neutron star kicks we have adopted a Maxwellian distributionwith large dispersion hence practically all the neutron stars escape from thecluster Now the correction procedure includes the increased kinetic energyas well as the potential energy contribution of the expelled mass Since theejection of high-velocity members is also a feature of stellar systems contain-ing binaries we have implemented an algorithm for preventing discontinuouschanges in the neighbour force for large time-steps

The determination and implementation of collisions in chain regularizationrequire special care and have been discussed elsewhere in considerable detail(Aarseth 2003) For highly eccentric binaries the KS solution facilitates acheck on the pericentre distance provisionally identified by a negative productof the old and new radial velocity Rprime = 2u middot uprime and R lt a The outcome ofa collision depends on the stellar types so that a variety of remnants may beproduced (see Chap 12) Here we note that the device of ghost stars can beused when two stars are replaced by one non-zero mass

Tidal fields represent another important feature of star cluster simulationsTwo different types of external effects are catered for Most open clustersin the solar neighbourhood move in nearly circular orbits which admit alinearized tidal force to be included in the equations of motion This simplerepresentation gives rise to an energy integral and imposes a tidal boundarythat is useful for defining escape The tidal radius is given by

rtide =(

GM

4A(AminusB)

)13

(156)

where A and B are the classical rotation constants Traditionally stars outside2rtide are removed from the calculation since their subsequent effect on boundcluster members is negligible

The general case of 3D motion requires a full galactic model with explicitexpressions for the force and its derivative The equations of motion are nowmost conveniently expressed in a non-rotating coordinate system (Aarseth2003) It is still possible to have an approximate energy integral by monitoringthe accumulated work done by the perturbing force P i during each (regular)time-step Expanding the integrated contribution to third order in terms ofthe initial values and expressing the result at the end of the time-step weobtain

ΔEi = mi

(12WiΔt2i minus WiΔti

)

(157)

26 S J Aarseth

where Wi = vimiddotP i Knowledge of P i enables the second order to be included inthe expansion and the resulting conservation is satisfactory Although distantstars are usually removed from the active data structure using a nominal valueof the tidal radius their orbits in the galactic potential can still be integratedHopefully these recent code innovations will encourage more comprehensivestudies of eccentric globular cluster orbits and associated tidal tails

116 GRAPE Implementations

Since the use of GRAPE-type special-purpose computers is gaining morewidespread use it may be of interest to describe some of the proceduresin the simulation code nbody4 In particular it should be emphasized thatthe internal GRAPE data structure differs from the host in several importantrespects which calls for additional software

We take advantage of the work-sharing facility to speed up the calcula-tion by carrying out some operations on the host while GRAPE is busy Ingeneral for large N many particles are due to be advanced at the same timebut the number may also be quite small during episodes of strong multipleinteractions After prediction of the first 48 block members nblock the relevantprocedures can be summarized as follows

bull Begin force calculation for the first block-step membersbull Predict the next 48 members (if any) while GRAPE is busybull Predict rivi of cm and perturbed KS components (first time)bull Form a list of small time-steps (first time nblock le 32)bull Correct the previous block members and specify new time-stepsbull Copy the force and force derivatives from GRAPEbull Correct the last block members after repeating the abovebull Send all the corrected rivi and also F i F i to GRAPE

The scheduling of particles to be advanced is essentially the same as innbody6 However coordinate and velocity predictions on the host are nowrestricted to block-step members since a fast prediction of all particles arecarried out on the GRAPE hardware When these quantities are copied tothe corresponding GRAPE variables for data transfer an optional predictionto second order in the force derivative may be included for increased accuracyWith regularized binaries present the data structure on GRAPE consists ofsingle particles and the cm of each KS pair Consequently the force actingon a binary is in the first instance obtained by direct summation from 2Np +1to N +Np where a cm is treated as a single particle Differential force cor-rections are then applied for each binary perturber to be consistent with thecm force and likewise for any perturber forces These corrections involve sub-tracting the cm terms before adding the vectorial contributions due to thetwo components Any particles which are not on the block-step must there-fore be predicted on the host before these corrections are performed Note

that the subtraction procedure invariably introduces small errors due to thelower precision of the GRAPE hardware

Another aspect of the prediction strategy concerns the indirect terms inthe heliocentric formulation (19) Again the coordinates and velocities of anysignificant members for which tj +Δtj gt t need to be predicted first This canmost readily be achieved by maintaining a list of any important planetesimalperturber which is updated following changes in the data structure In orderto check energy conservation in the heliocentric case the expression for kineticenergy takes the form

K =12

Nsum

i=1

miv2i minus

12

(M0 minus

summi

)v2

0 (158)

where v0 = minussum

miviM0 is the velocity of the dominant body of mass M0

and the second sum in (158) refers to the heavy perturbersAs mentioned in Sect 15 the determination of a maximum time-step also

differs when using a GRAPE in connection with (19) We employ a specialfunction that supplies the index of the closest neighbour at no extra costduring the force evaluation The current relative coordinates and velocityRV define an appropriate time-step Δtnb = 01R2R middot V which may besmaller than the standard value Another point to note is that the directforce summation does not include the dominant body whose effect is addedin the iteration Since provisional values of F i F i for each member on theblock-step are supplied to GRAPE for scaling purposes it is necessary tosubtract the dominant contributions first On the other hand decisions on newregularizations or terminations are made during the time-step determinationand executed in the usual way at the end of the block-step

Procedures for wheel-spoke regularization have also been combined withthe GRAPE code nbody4 making a separate version nbody7 A new featurehere is how to recognize a compact subsystem suitable for special treatmentGiven the presence of a massive binary together with the conditions

R lt 2Rcl rp lt14Rcl (m0m)12

rp lt 0 (159)

with rp the distance to the closest perturber this system is initialized andadditional perturbers are selected as for chain regularization A list of neigh-bours is updated on the local crossing time from which significant perturbersare selected Frequent checks are made on membership changes of the sub-system taking care to avoid near-collisions in the overlap region although nodirect test is made at present2

The post-Newtonian algorithms discussed above have also been imple-mented Again these procedures are carried out on the host computer Several

2Interactions between subsystem members and perturbers are not softened hencethe use of an overall perturbation with respect to the cm only acts as a guide

28 S J Aarseth

models where the relativistic terms become important have been studied forcentrally concentrated systems with N = 105 equal-mass particles and onemassive black hole of mass m0 = 300 m (Aarseth 2007) A typical simulationover 100 time units and including GR coalescence can be done in a few daysExperience shows that the less powerful GRAPE-6A is well suited for thispurpose since for much of the time the host constitutes the computationalbottleneck especially during relativistic episodes Because the central sub-system is now advanced by the accurate but more expensive BulirschndashStoermethod the overall energy conservation is somewhat better than for standardcluster simulations

When using GRAPE all regularization procedures are treated in essen-tially the same way as in nbody6 Depending on the requirements there isa choice of chain regularization time-transformed leapfrog (see Chap 2) orwheel-spoke method for studying three different types of problems but onlyone scheme is chosen for a given calculation Some of these procedures aredistinguished by options and there are also different directories containingroutines of the same name In conclusion this GRAPE software package hasalready yielded some interesting results that open up new avenues for futureexploration

117 Practical Aspects

In the preceding sections we have described the main procedures of the codenbody6 and also nbody4 which is similar The actual use of these codesinvolves many additional considerations Here we attempt a general summaryof some practical features that play a key role

To begin with the code needs to be installed and tested This neces-sitates downloading the software and extracting the relevant files3 Certainparameters governing maximum array sizes should be checked otherwise the(generous) defaults will be adopted It is expected that the code will com-pile successfully on most conventional computers Likewise results of the testinput should be examined before any further work is attempted When try-ing out a new code it is of interest to evaluate the performance by so-calledprofiling as explained in the manual which can also be downloaded

A versatile code requires a number of input parameters especially if thereare many alternative procedures To facilitate explanation we distinguish be-tween different types of input In the first group are the particle number N maximum neighbour membership nmax as well as the number of primordialbinaries nbin The second set of parameters ηI ηR ηu are concerned with theintegration itself and are dimensionless ie the same for most problems

Initial conditions may be generated internally or uploaded from a file Inthe former case there is a choice of IMF distributions with upper and lower

3See httpwwwastcamacukresearchnbody

mass limits The main scaling parameters are the length unit RV in pc andmean mass MS in solar units as well as the virial theorem ratio QV discussedearlier The network of 40 options are defined in a table and allows a vari-ety of tasks to be considered However the choice must be consistent whichrequires due care All the close encounter parameters have been discussed inthe KS section Special input templates are also available for simulations withprimordial binaries or cluster orbits in a 3D galactic potential

An example of typical input parameters is given for illustration purposeswhere the main categories are placed together

bull N = 1000 nmax = 70 ηI = 002 ηR = 003bull S0 = 03 ΔT = 2 Tcrit = 100bull QE = 2 times 10minus5 RV = 2 MS = 05bull 1 2 5 7 14 16 20 23bull Δtcl = 10minus4 Rcl = 0001 ηu = 02 γmin = 10minus6

bull α = 23 m1 = 100 mN = 02

In the second line S0 is an initial guess for the neighbour sphere the outputinterval is ΔT and Tcrit gives the termination time Moreover the relativeenergy tolerance QE is used for automatic error control The line of optionscontains some useful suggestions but is by no means complete Finally theIMF is defined by the classical Salpeter exponent α together with the upperand lower mass limits in terms of the average mass More detailed informationon the full set of input parameters can be found in the manual Thus for exam-ple there are options for external perturbations or stellar evolution Takinginto account the wide range of available procedures the complete input file isquite compact in comparison with many other large codes

Presentation of results constitutes another challenge for code developmentIt also requires an effort by the practitioner to extract the available data in asuitable form Here we may distinguish between result summaries and detailedinformation To elucidate the possibilities the table summarizes some of themain optional procedures with a brief explanation

Procedure Explanation

Cluster core N2 algorithm for core radius and density centreLagrangian radii Percentile mass radii and half-mass radiusError control Automatic error check and restart from last timeEscape Removal of distant members and table updatesTime offset Rescaling of all global times for large valuesEvent counters Stellar types and remnant statisticsBinary analysis Regularized binary histograms and energy budgetBinary data bank Characteristic parameters for regularized binariesHR diagram Evolutionary state of single stars and binariesGeneral data bank Detailed snapshots for data analysis

30 S J Aarseth

Each of these procedures is activated by specifying a non-zero option asdefined in the manual There is also a facility for changing any option atlater times Many of the result summaries are self-explanatory and will notbe reviewed here Likewise the manual illustrates the principle of adding newvariables to the code while preserving the total size of the common blocks

We conclude by commenting on the way in which the total energy is ob-tained Thus rather than evaluating the kinetic and potential energies di-rectly the different contributions are derived consistently according to thecalculation method For example the binding energies of KS pairs are givenby

sumμihi where hi is predicted to highest order Monitoring the internal

energies of hierarchical systems and collisions events enable a conservationscheme to be maintained at high accuracy because dissipative processes arealso accounted for

References

Aarseth S J 2003 Gravitational N-Body Simulations Cambridge University PressCambridge 6 8 11 21 25

Aarseth S J 2007 MNRAS 378 285 19 21 28Aarseth S J Heggie D C 1976 AampA 53 259 18Aarseth S J Zare K 1974 Celes Mech 10 185 5 17Ahmad A Cohen L 1973 J Comput Phys 12 389 5Blanchet L Iyer B 2003 Class Quantum Grav 20 755 20Bulirsch R Stoer J 1966 Num Math 8 1 19Kokubo E Makino J 2004 PASJ 56 861 6Kokubo E Yoshinaga K Makino J 1998 MNRAS 297 1967 7Kozai Y 1962 AJ 67 591 21Kustaanheimo P Stiefel E 1965 J Reine Angew Math 218 204 11Makino J Aarseth S J 1992 PASJ 44 141 8Makino J Taiji M Ebisuzaki T Sugimoto D 1997 ApJ 480 432 5Mardling R A Aarseth S J 1999 in Steves B A Roy A E eds The

Dynamics of Small Bodies in the Solar System Kluwer Dordrecht p 385 5 16Mardling R Aarseth S 2001 MNRAS 321 398 13 16Mikkola S Aarseth S J 1993 Celes Mech Dyn Ast 57 439 3 5 21 22Mikkola S Aarseth S J 1996 Celes Mech Dyn Ast 64 197 13Mikkola S Aarseth S J 1998 New Astron 3 309 11 13Mora T Will C M 2004 Phys Rev D 69 104021 (gr-qc0312082) 20Newmark N M 1959 J Eng Mech 85 67 6Peters P C 1964 Phys Rev 136 B1224Zare K 1974 Celes Mech 10 207 19

2

Regular Algorithms for the Few-Body Problem

Seppo Mikkola

Tuorla Observatory University of Turku Finlandmikkolautufi

21 Introduction

In N -body simulations the most common strong interactions are due to closeencounters of just two bodies Most classical numerical integration methodslose precision for such situations due to the 1r2 singularity of the mutualforce of the two bodies In a close encounter the relative motion of the partici-pating bodies is so fast that for a brief moment the rest of the system can beconsidered frozen Consequently the most important feature of a regularizingalgorithm must be that it can handle reliably the perturbed two-body prob-lem There are two basically different types of methods available Coordinateand time transformations and algorithms that produce regular results withoutcoordinate transformation

The first coordinate-transformation method was that of Levi-Civita (1920)but the method works only in two dimensions Later Kustaanheimo amp Stiefel(1965) generalized this by applying a transformation (KS-transformation)from four dimensions to three dimensions (see also Aarseth 2003) More re-cently two versions of algorithmic regularization have been proposed Theseare the logarithmic Hamiltonian (LogH) suggested by Mikkola amp Tanikawa(1999a b) and independently by Preto amp Tremaine (1999)

A further development the Time Transformed Leapfrog (TTL) was pre-sented by Mikkola amp Aarseth (2002) Finally Mikkola amp Merritt (2006 2008)combined the LogH and TTL as well as a generalized midpoint method tomodify the algorithmic regularization such that it can handle the case ofvelocity dependent perturbations which are important in for example post-Newtonian dynamics (Soffel 1989)

22 Hamiltonian Manipulations

All known regularization methods require the introduction of a new indepen-dent variable Due to the importance of the Hamiltonian formalism this is

Mikkola S Regular Algorithms for the Few-Body Problem Lect Notes Phys 760 31ndash58

(2008)

32 S Mikkola

often done by transforming the Hamiltonian Let qqq and ppp be the coordinatesand momenta T = T (ppp) the kinetic energy and U = U(rrr t) the potentialThen H(pppqqq t) = T (ppp) minus U(qqq t) is the Hamiltonian If one defines a newindependent variable s by the differential equation

dt = g(p q t)ds (21)

the equations of motion can be derived from the extended phase space Hamil-tonian Γ (Poincarersquos transformation)

Γ = g(p q t)(H(p q t) +B) (22)

where B is the momentum of time and initially

B(0) = minusH(p(0) q(0) t0) (23)

Time is now a coordinate and one notes that the Poincare transformationmakes the new Hamiltonian Γ conservative since it does not depend explicitlyon the new independent variable Due to this and the choice of the initial valuefor B the numerical values are Γ = 0 and B = minusH (binding energy) alongthe trajectory

One often uses

Γ = (H +B)L or Γ = (H +B)U (24)

Here U is the potential energy and L = T +U the Lagrangian The equationsof motion take the form

tprime =partΓpartB

= g qprime =partΓpartp

= +gpartH

partp+partg

partp(H +B) (25)

Bprime = minuspartΓpartt

= minusg partHpartt

minus partg

partt(H +B) pprime = minuspartΓ

partq= minusg partH

partqminus partg

partq(H +B)

which is correct because H + B = 0 along the orbit However this does notmean that the latter terms can be dropped The reason for this will becomeclear in the example in Sect 23

Another way to manipulate the Hamiltonian is the use of the functionalHamiltonian (Preto amp Tremaine 1999)

Λ = f(T +B) minus f(U) (26)

where f(z) is any function that satisfies f prime(z) ge 0 A most interesting functionis f(z) = log(z) (Mikkola amp Tanikawa 1999a b Preto amp Tremaine 1999)which gives tprime = partΛpartB = 1(T + B) Along the correct trajectory we alsohave 1(T + B) = 1U and thus the time transformation is essentially thesame as g = 1U A special feature of the functional Hamiltonian is that itallows the use of the (symplectic) leapfrog algorithm because the equations ofmotion

2 Regular Algorithms for the Few-Body Problem 33

rrr =partΛpartppp

= f prime(T +B)partT

partppp ppp = minuspartΛ

partrrr= f prime(U)

partU

partrrr(27)

are such that the right-hand sides do not depend on variables on the left-handside

23 Coordinate Transformations

231 One-Dimensional Case

A simple example is provided by the one-dimensional two-body problem TheKeplerian Hamiltonian H = p22 minusMq may be transformed by the point-transformation q = Q2 p = P(2Q) into the form H = P 2(8Q2) minusMQ2Using g = q = Q2 one obtains

Γ = Q2

(P 2

8Q2minus M

Q2+B

)

=18P 2 +BQ2 minusM (28)

and the equations of motion are

Qprime =14P P prime = minus2BQ or Qprimeprime = minusB

2Q (29)

which is a harmonic oscillator because B = minusH = constantNote that had we dropped the (H +B) factored terms in (25) we would

have had

Qprime =14P P prime = minus2

(18P 2 minusM

)

Q or Qprimeprime = minus12

(18P 2 minusM

)

Q

(210)

which is singular (but still analytically regular due to energy conservationie because 1

8P2 minusM = BQ2)

232 Three-Dimensional Case KS-Transformation

The KS-transformations (Kustaanheimo amp Stiefel 1965) between the three-dimensional position and momentum rrr and ppp and the corresponding four-dimensional KS-variables QQQ and PPP may be written

rrr = QQQQ ppp = QPPP(2Q2) (211)

Here Q is the KS-matrix (Stiefel amp Scheifele 1971 p 24)

Q =

⎛

⎜⎜⎝

Q1 minusQ2 minusQ3 Q4

Q2 Q1 minusQ4 minusQ3

Q3 Q4 Q1 Q2

Q4 minusQ3 Q2 minusQ1

⎞

⎟⎟⎠ (212)

34 S Mikkola

Another way to write the transformation is

x = Q21minusQ2

2minusQ23+Q2

4 y = 2(Q1Q2minusQ3Q4) z = 2(Q1Q3+Q2Q4) (213)

Note that the fourth components of rrr and ppp that (211) produces are zerosdue to the structure and properties of the transformation

Due to increased number of variables the Qrsquos corresponding to given phys-ical coordinates are not unique However one may choose any solution forexample with rrr = (x y z)t r = |rrr| we calculate

u1 =radic

12 (r + |x|)

u2 = Y(2u1) (214)u3 = Z(2u1)u4 = 0

and the components of QQQ are

QQQ =

(u1 u2 u3 u4)t X ge 0(u2 u1 u4 u3)t X lt 0 (215)

(This algorithm is used to avoid round-off error)Initial values for the KS momenta are given by

PPP = 2Qtppp (216)

For the two-body problem H = 12ppp

2minusMr the time-transformed HamiltonianΓ in (22) takes the form

Γ =18PPP 2 minusM +BQQQ2 (217)

ie a harmonic oscillator in complete analogy with the one-dimensional caseWhen regularized by the KS-transformation the equations of motion for

a perturbed binaryrrr +Mrrrr3 = FFF (218)

take the explicit form

QQQprimeprime = minus12BQQQ+

12rQtFFF

Bprime = minus2QQQprime middot QtFFF (219)tprime = r = QQQ middotQQQ

Here FFF is the physical perturbation exerted by other particles (or any otherphysical effect) and

B =M

rminus ppp2

2is the two-body binding (Kepler-)energy Since the equations are regular theycan be solved with any reasonable numerical method

24 KS-Chain(s)

When the KS-transformation is applied in N -body systems one does notobtain a harmonic oscillator but close approaches can still be regularizedFirst one forms a chain of particles such that all the small critical distancesare included in the chain and then one applies the KS-transformation to thechain vectors For details of the chain selection procedure see Sect 271

Let a time-transformed multiparticle Hamiltonian be

Γ = (T minus U +B)(T + U)

whereT =

sum

ν

ppp2ν(2mν) U =

sum

iltj

mimjrij

Let us introduce new coordinates

XXXk = rrrikminus rrrjk

then we can use the generating function

S =sum

k

WWW k middotXXXk =sum

k

WWW k middot (rrrikminus rrrjk

) (220)

In terms of the new momenta WWW the old ones are

pppν =partS

partrrrν=sum

k

WWW k middot (δνikminus δνjk

) (221)

where the δrsquos are the Kronecker symbols Thus we have

T =12

sum

αβ

TαβWWWα middotWWW β (222)

U =sum

k

mikmjk

|XXXk|+

sum

iltj (ij) isinikjk

mimj

rij (223)

whereTαβ =

sum

ν

1mν

(δνiαminus δνjα

)(δνiβminus δνjβ

)

and the second potential energy termsum

iltj (ij) isinikjk

mimj

rij

contains all the distances rij = rij(X1X2 ) that are not included amongthe vectors XXXk

36 S Mikkola

After application of the KS transformation by (211) to every momentum-coordinate pair by

WWW XXX rarr PPP QQQ

one can obtain the regularized Hamiltonian

Γ(PPP QQQ) = (T minus U +B)(T + U)

and form the canonical equations of motion

PPP prime = minus partΓpartQQQ

(224)

tprime =partΓpartB

QQQprime =partΓpartPPP

(225)

Note that the number of new variables may exceed the number of the oldones This however is not a problem all the physical results remain correct(Heggie 1974)

The above formulation is completely general at least to the point thatall the well-known methods the Zare (1974) method in which all particlesare regularized with respect to a central body Heggiersquos global regularization(Heggie 1974) (in which all the interparticle vectors are taken as new variablesand collisions are regularized by the KS transformation) and the chain method(Mikkola amp Aarseth 1993) are included The vectors XXX of these methods areschematically illustrated in Fig 21

ndash2

0

2

4

6

8

10

ndash2 ndash15 ndash1 ndash05 0 05 1 15 2

C

H

Z

Fig 21 Regularized interactions (schematically) in Zare method (Z) globalmethod of Heggie (H) and chain method (C)

In fact one can regularize any interparticle vector Thus any kind ofbranching and looping chains can be handled This could be seen as an in-termediate form between the Heggie method and the chain However it isnot clear if such alternatives are actually more useful than the simple chainComprehensive instructions for use of the KS-chain can be found in Mikkolaamp Aarseth (1993) and Aarseth (2003)

25 Algorithmic Regularization

The algorithmic regularization contrary to KS regularization does not usecoordinate transformation but only a time transformation and a suitable al-gorithm that produces regular results despite the singularity in the force Thefirst such methods were invented in 1999 independently in two places (Mikkolaamp Tanikawa 1999a b Preto amp Tremaine 1999)

251 The Logarithmic Hamiltonian (LogH)

Let ppp be the momenta and qqq the coordinates T (ppp) the kinetic energy andU(qqq t) the force function Then the Hamiltonian in extended phase-space is

H = T +B minus U (226)

Here B is the momentum of time (which is now a coordinate t = partHpartB = 1)

If B(0) = minusH(0) then the function

Λ = log(T +B) minus log(U) (227)

can be used as a Hamiltonian in the extended phase space

DemonstrationThe equations of motion derivable from Λ read

pppprime = minuspartΛpartqqq

=partU

partqqqU Bprime = minuspartΛ

partt=

partU

parttU (228)

qqqprime =partΛpartppp

=partT

partpppTe tprime =

partΛpartB

= 1Te (229)

where Te = T + B and a prime denotes differentiation with respect to the(new) independent variable s

Since Λ does not depend explicitly on s the value of Λ is constantThus T +B = U due to choice of initial value for B Using this and dividingthe equations of motion by the equation for time (229) we get for the timederivatives

ppp =partU

partqqq B =

partU

parttand qqq =

partT

partppp (230)

ie the normal Hamiltonian equations

38 S Mikkola

LogH for Two bodies

To introduce the method we first consider the simple case of two-body motionH = ppp22 minusMr which gives

Λ = log(ppp22 +B) + log(r) (231)

after dropping log(M)Thus the time transformation is

dt = dspartΛpartB

=ds

(ppp22 +B) (232)

B remains constant B = minus(ppp22 minusMr) The new independent variable s is

s =int t

(ppp22 +B) dt =int t M

rdt (233)

ie a quantity proportional to the eccentric anomaly increment

With stepsize h and initial values ppp0 rrr0 t0 the leapfrog algorithm takesthe form (illustration in Fig 22)

ndash04

ndash02

0

02

04

ndash02 0 02 04 06 08 1

Fig 22 Illustration of the working of the algorithmic regularization in the caseof an elliptic two-body motion The points on the ellipse are the starting and endpoints in a leapfrog step while those outside the ellipse are the rrr 1

2-points

rrr 12

= rrr0 +h

2ppp0(

ppp20

2+B) (234)

ppp1 = ppp0 minus h rrr 12r21

2(235)

rrr1 = rrr 12

+h

2ppp1(

ppp21

2+B) (236)

t1 = t0 +h

2

[1

(ppp202 +B)

+1

(ppp212 +B)

]

(237)

This algorithm produces correct positions and momenta on the associatedKeplerian ellipse (Mikkola amp Tanikawa 1999a b Preto amp Tremaine 1999)however time is not correct and the method thus has phase errors Thisresult applies even for collision orbits where the eccentricity e = 1

Although the singularity when r rarr 0 is not removed one expects thealgorithm to be applicable for the N -body problem since the functions arenot evaluated precisely at r = 0

252 Time-Transformed Leapfrog (TTL)

Consider the general system

rrr = vvv vvv = FFF (rrr) (238)

where rrr and vvv are position and velocity vectors of arbitrary dimension Wenow introduce a time transformation

ds = Ω(rrr) dt (239)

where Ω(rrr) gt 0 is arbitraryIf W = Ω then one may write

rrrprime = vvvW tprime = 1W vvvprime = FFFΩ

where a prime means dds If W is obtained from the differential equation

W = vvv middot partΩpartrrr

or W prime = vvv middot partΩpartrrr

Ω (240)

instead of W = Ω directly we have⎛

⎜⎜⎝

rrrprime

tprime

vvvprime

W prime

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

vvvW1W

0000

⎞

⎟⎟⎠+

⎛

⎜⎜⎝

0000

FFF (rrr)Ω(rrr)vvv middot part ln(Ω)partrrr

⎞

⎟⎟⎠ (241)

This allows the Time-Transformed Leapfrog (TTL)

40 S Mikkola

rrr 12

= rrr0 +h

2vvv0

W0(242)

t 12

= t0 +h

21W0

(243)

vvv1 = vvv0 + hFFF (rrr 1

2)

Ω(rrr 12)

(244)

W1 = W0 + hvvv0 + vvv1

2Ω(rrr 12)middotpartΩ(rrr 1

2)

partrrr 12

(245)

rrr1 = rrr 12

+h

2vvv1

W1(246)

t1 = t 12

+h

21W1

(247)

A Simple Fortran Code for Two Bodies (LogH)

implicit real8 (a-hmo-z)

read(5)htmxmass read stepsize maximum time amp mass

read(5)xyzvxvyvz read initial coordsvels

c initializations

t=0

r=sqrt(xx+yy+zz) distance

vv=vxvx+vyvy+vzvz v-square

B=massr-vv2 binding-E

c

c Integration of the two-body motion

1 continue

dt=h(vxvx+vyvy+vzvz+2B) time increment

x=x+dtvx

y=y+dtvy

z=z+dtvz

t=t+dt

dtc=h(xx+yy+zz)

vx=vx-xdtc

vy=vy-ydtc

vz=vz-zdtc

dt=h(vxvx+vyvy+vzvz+2B) new time increment

x=x+dtvx

y=y+dtvy

z=z+dtvz

t=t+dt time has an O(h^3) error

c diagnostics time coords amp error

write(62)txyz

amp (B+(vxvx+vyvy+vzvz)2)-masssqrt(x2+y2+z2)

if(tltTmx)goto 1

2 format(1x1p5g124)

end

If one takesΩ = 1r (248)

the increment of W in one step is

ΔW = minush rrr

r3middot vvv1 + vvv0

2(249)

and

Δ12vvv2 =

12(vvv2

1 minus vvv20) =

12(vvv1 minus vvv0) middot (vvv1 + vvv0) = minush rrr

2

which means that for the unperturbed two-body problem this algorithm ismathematically equivalent to the LogH-method (more generally this is thecase if Ω = U) Numerically however this does not apply The reason is thatin case of a close approach W first increases then decreases fast This meansthat the increments are large numbers and there is considerable cancellationand possible round-off error Combined with the extrapolation method thisalternative leapfrog can be a powerful integrator for some systems

Remark Especially interesting is the fact that the method can be efficientfor potentials that differ from the Newtonian 1r behaviour at small distancesOne notes that both the LogH and TTL are useful for the soft potential

U prop 1radicr2 + ε2

which cannot be regularized with the KS-transformationRemark If Ω = 1r the (numerical) relation W = 1r remains valid after

every step and somewhat surprisingly this is true for any radial force fieldFFF = f(r)rrrr

A Simple Fortran Code for Two Bodies (TTL)

read(5)htincrtmxmass read steptincr maxtime mass

tnext=0

c initializations

t=0

E0=vv2-massr

W=massr

c

c Integration of two-body motion

42 S Mikkola

1 continue

dt=hW2 time increment

t=t+dt

x=x+dtvx

y=y+dtvy

z=z+dtvz

c

dtc=h(xx+yy+zz)

dw= -(xvx+yvy+zvz)dtc2

vx=vx-xdtc

vy=vy-ydtc

vz=vz-zdtc

W=W+dw-(xvx+yvy+zvz)dtc2

c

dt=hW2 new time increment

t=t+dt this has an O(h^3) error

x=x+dtvx

y=y+dtvy

z=z+dtvz

c diagnostics

if(tlttnext)goto 1

tnext=tnext+tincr

r=sqrt(xx+yy+zz)

err=-E0+(vxvx+vyvy+vzvz)2-massr

write(62)txyzerrr Wr-mass time coords amp error

if(tltTmx)goto 1

2 format(1x1p10g124)

end

253 A Simple LogH Algorithm for the Three-Body Problem

The three-body problem is still one of the most studied problems in few-bodydynamics Therefore it may be of interest to consider in more detail a simpleregular three-body algorithm This also serves as further illustration of theuse of the algorithmic regularization

Following Heggie (1974) we use the three interparticle vectors (see Fig 23)

XXX1 = rrr3 minus rrr2 XXX2 = rrr1 minus rrr3 XXX3 = rrr2 minus rrr1 (250)

as new coordinates Let the corresponding velocities be VVV k = XXXk then thekinetic and potential energies (in cm system) can be written

T =1

2M

sum

iltj

mimjVVV2kij

U =sum

iltj

mimj

|Xkij | (251)

where M =sum

k mk is the total mass and kij = 6 minus i minus j The equations ofmotion are

ndash04

ndash02

0

02

04

06

08

1

0 05 1 15 2

X1 X2

X3

m1m2

m3

Fig 23 Labelling of vectors in the three-body regularization

XXXk = VVV k VVV k = minusM XXXk

|XXXk|3+mk

sum

ν

XXXν

|XXXν |3 (252)

and after the application of the logarithmic Hamiltonian modification theyread

tprime = 1(T +B) XXX primek = XXXk(T +B) VVV prime

k = VVV kU (253)

which are suitable for the leapfrog algorithm given in (258) and (259) aswell as for Yoshidarsquos (1990) higher-order leapfrogs

The usage of the relative vectors instead of some inertial coordinates isadvantageous in attempting to avoid large round-off effects One could alsointegrate only two of the triangle sides obtaining the remaining one from theconditions sum

k

XXXk = 000sum

k

VVV k = 000

However this hardly reduces the computational effort required by the methodInstead one may occasionally compute the longest side and the correspondingvelocity from the above triangle conditions Note however that the sums ofthe sides are not only integrals of the exact solution but are also exactlyconserved by the leapfrog mapping

The transformation from the variables XXX to centre-of-mass coordinates rrrcan be done as

44 S Mikkola

rrr1 =(m3XXX2 minusm2XXX3)

M rrr2 =

(m1XXX3 minusm3XXX1)M

M

(254)and the velocities obey the same rule

26 N -Body Algorithms

In an N -body system the Logarithmic Hamiltonian (LogH)

Λ = ln(T +B) minus ln(U) (255)

gives the equations of motion

tprime =partΛpartB

= 1(T +B) rrrprimek = vvvk(T +B) vvvprimek = AAAkU (256)

where vvvk = ˙rrrk and AAAk = partUpartrrrk

mk are the velocity and acceleration corre-spondingly

It is important to note that the derivatives of coordinates only depend onvelocities and vice versa This makes a simple leapfrog algorithm possible (seebelow) The most important feature is that as discussed in Sect 251 theresulting leapfrog is exact for two-body motion except for a phase error andthus regularizes close approaches

The Time-Transformed Leapfrog (TTL) method is a generalization of thisidea (Mikkola amp Aarseth 2002) In the time transformation one chooses someother function Ω(rrr) in place of the potential U and defines an auxiliary quan-tity W by the differential equation W = Ω = partΩ

partrrr middot vvvThe resulting TTL equations read

tprime = 1W rrrprimek =1W

partT

partpppk

vvvprimek =1ΩAAAk W prime =

sum

k

partΩpartrrrk

middot vvvkΩ (257)

and these can also be used to construct a leapfrog-like mapping which forsuitable functions Ω are asymptotically exact for two-body motion near col-lision It can be shown that TTL is mathematically equivalent to LogH if onetakes Ω = U

261 LogH Leapfrog

First one computes the constant B = minusT + U from initial values The equa-tions of motion can be used to define the basic mappings XXX(s) and VVV (s)as

XXX(s) δt = s(T +B) t rarr t+ δt rrrk rarr rrrk + δt vvvk (258)

VVV (s) δt = sU ppp rarr pppk + δtAAAk

which can be evaluated in a sequence

XXX(h2)VVV (h)X(h2)

using always the most recent results as input for the next operation

262 TTL

Here one first evaluates the initial value of W = Ω then uses the leapfrogmappings

XXX(s) δt = sW t rarr t+ δt rrrk rarr rrrk + δt vvvk (259)

VVV (s) δt = sΩ δvvvk = δtAAAk W rarr W + δtsum

k

partΩpartrrrk

middot(

vvvk +12δvvvk

)

vvvk rarr vvvk + δvvvk (260)

to advance the coordinates and velocities using the operation sequence

XXX(h2)VVV (h)XXX(h2)

repeatedlyFor Ω one may use any suitable function but usually it is advantageous

to takeΩ =

sum

iltj

Ωij

rij

whereΩij = 1 or Ωij = mimj

the latter choice being recommended if the masses are comparableThe leapfrog alone is however in many cases not accurate enough The

accuracy can be improved eg by using the higher-order leapfrog algorithmsof Yoshida (1990) Alternatively one may use the extrapolation method(Bulirsch amp Stoer 1966 Press et al 1986)

27 AR-Chain

First of all it is necessary to emphazise the importance of the chain structurenot only in the KS-chain method but also when one uses one of the algorith-mic regularizations The reason is round-off errors If one uses centre-of-masscoordinates the relative coordinates of a distant close pair are differencesof large numbers and there is considerable cancellation of significant figuresleading to irrecoverable errors

This section discusses a new code that uses the chain structure and amixture of the LogH and TTL-methods

271 Finding and Updating the Chain

We begin by finding the shortest interparticle vector for the first part of thechain Next we search for the particle closest to one or the other end of thepresently known part of the chain This particle is added to the closest end

46 S Mikkola

1

2

3

4

5

6

7

8

9

10

middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot

middotmiddotmiddotmiddotmiddotmiddotmiddotmiddottimes times

times

lowast lowast

Fig 24 Illustration of the chain and the checking of switching conditions Distanceslike R57 are compared with the smaller of the two distances R56 and R67 (markedby ) Interparticle distances like R410 are compared with the smallest of those incontact with the considered distance (marked by times)

of the already existing chain This is repeated until all particles are includedin the chain The particles are then re-numbered along the chain as 1 2 Nfor ease of programming

After every integration step we check for the need of updating the chainFigure 24 illustrates the case of a 10-particle chain To avoid some potentialround-off problems it is advantageous to carry out the transformation fromthe old chain vectors XXXk to the new ones directly by expressing the new chainvectors as sums of the old ones

Let the actual ldquophysicalrdquo names of the chain particles 1 N (as definedabove) be I1 I2 IN and let us use the notation Iold

k and Inewk for the

names in the old and new chains Then we may write

rrrIoldk

=kminus1sum

ν=1

XXXoldν (261)

XXXnewμ = rrrInew

μ+1minus rrrInew

μ (262)

Thus we need to use the correspondence between the old and the new indicesto express the new chain vectors XXX in terms of the old ones One finds that ifk0 and k1 are two indices such that Iold

k0= Inew

μ and Ioldk1

= Inewμ+1 then

XXXnewμ =

Nminus1sum

ν=1

BμνXXXoldν (263)

where Bμν = +1 if(k1 gt ν amp k0 le ν) and Bμν = minus1 if(k1 le ν amp k0 gt ν)otherwise Bμν = 0

272 Transformations

After selecting the chain and renaming the particles as 1 2 N alongthe chain one can evaluate the initial values for the chain vectors andvelocities as

XXXk = rrrk+1 minus rrrk (264)VVV k = vvvk+1 minus vvvk (265)

where vvvk = ˙rrrk At the same time one may evaluate the centre-of-mass quan-tities

M =sum

k

mk (266)

rrrcm =sum

k

mkrrrkM (267)

vvvcm =sum

k

mkvvvkM (268)

The transformation back to rrrvvv can be done by simple summation

rrr1 = 000 (269)vvv1 = 000 (270)

rrrk+1 = rrrk +XXXk (271)vvvk+1 = vvvk + VVV k (272)

followed by reduction to the centre of mass

rrrcm =sum

k

mkrrrkM (273)

vvvcm =sum

k

mkvvvkM (274)

rrrk = rrrk minus rrrcm (275)vvvk = vvvk minus vvvcm (276)

However it is not always necessary to reduce the coordinates to the centre-of-mass system since accelerations only depend on the differences

273 Equations of Motion and the Leapfrog

The equations of motion read

XXXk = VVV k (277)˙VVV k = AAAk+1 minusAAAk (278)

48 S Mikkola

where the accelerations AAAk with possible external effects fffk are

AAAk = minussum

j =k

mjrrrjk

|rrrjk|3+ fffk (279)

and for j lt k

rrrjk =

⎧⎪⎨

⎪⎩

rrrk minus rrrj if k gt j + 2XXXj if k = j + 1XXXj +XXXj+1 if k = j + 2

(280)

For k gt j one uses the fact that rrrjk = minusrrrkj The use of XXXj and XXXj +XXXj+1

reduces the round-off effect significantly More generally one could also use

rrrkj =kminus1sum

ν=j

XXXν (281)

but for many bodies it is faster to use the above recipe (280) and the latteralternative seems not to improve the resultsThe kinetic energy is

T =12

sum

k

mkvvv2k (282)

and the potential energyU =

sum

iltj

mimj

|rrrij | (283)

which is evaluated along with the accelerations according to (280) We intro-duce further a time transformation function

Ω =sum

iltj

Ωij

|rrrij | (284)

where Ωij are some selected coefficients (to be discussed below)Now one may define the two time transformations

tprime = 1(α(T +B) + βω + γ) = 1(αU + βΩ + γ) (285)

where α β and γ are adjustable constants B = U minusT is the N -body bindingenergy and ω is defined by the differential equation

ω =sum

k

partΩpartrrrk

middot vvvk (286)

and the initial value ω(0) = Ω(0) The binding energy B changes according to

B = minussum

k

mkvvvk middot fffk (287)

The equations of motion that can be used to construct the leapfrog whichprovides algorithmic regularization are for time and coordinates respectively

tprime = 1(α(T +B) + βω + γ) (288)

rrrprimek = tprimevvvk (289)

and for velocities B and ω

τ prime = 1(αU + βΩ + γ) (290)

vvvprimek = τ primeAAAk (291)

Bprime = τ primesum

k

(minusmkvvvk middot fffk) (292)

ωprime = τ primesum

k

partΩpartrrrk

middot vvvk (293)

To account for the vvv-dependence of Bprime and ωprime one must follow Mikkola ampAarseth (2002) ie first the vvvk are advanced and then the average lt vvvk gt=(vvvk(0) + vvvk(h))2 is used to evaluate Bprime and ωprime

The leapfrog for the chain vectors XXXk and VVV k can be written most easilyin terms of the two mappings

XXX(s)

δt = s(α(T +B) + βω + γ) (294)

t = t+ δt (295)XXXk rarr XXXk + δtVVV k (296)

(297)

VVV (s)

δt = s(αU + βΩ + γ) (298)

VVV k rarr VVV k + δt(AAAk+1 minusAAAk) (299)

B rarr B + δtsum

k

(minusmk lt vvvk gt middotfffk) (2100)

ω rarr ω + δtsum

k

partΩpartrrrk

middot lt vvvk gt (2101)

where lt vvvk gt is the average of the initial and final vvvrsquos here Note that it isalso necessary to evaluate the individual velocities vvvk because the expressionfor Bprime and ωprime would otherwise (in terms of the chain vector velocities VVV k)become rather cumbersome

One leapfrog step can then be written simply as

50 S Mikkola

and a longer sequence of n steps reads

XXX(h2)[Πnminus1

ν=1 (VVV (h)XXX(h))]VVV (h)XXX(h2)

This is the formulation to be used with the extrapolation method when pro-ceeding over a total time interval of length nh

274 Alternative Time Transformations

If one takesΩj = mimj (2102)

then α = 0 β = 1 γ = 0 is mathematically equivalent to α = 1 β = γ = 0as was shown in Mikkola amp Aarseth (2002) However numerically these arenot equivalent and the LogH alternative is much more stable On the otherhand as noted above it is desirable to get stepsize shortening (and thusregularization) also for encounters of small bodies and thus some function Ωshould also be included

To increase the numerical stability for strong interactions of big bodiesand smooth the encounters of small bodies one may use α = 1 β = 0 and

Ωij =

m2 if mimj lt εm2

0 otherwise (2103)

where m2 =sum

iltj mimj(N(N minus 1)2) is the mean mass product and ε

an adjustable parameter (ε sim 10minus3 may be a good guess) It is sometimesadvantageous to integrate (286) for ω even if β = 0 This is because theintegrator (extrapolation method) is forced to use short steps where ω islarge thus giving higher precision when required

Remarks

1 If (α β γ) prop (1 0 0) the method is the logarithmic Hamiltonian method(LogH) of Mikkola amp Tanikawa (1999a)

2 If (α β γ) prop (0 1 0) the method is the transformed leapfrog (TTL)(Mikkola amp Aarseth 2002)

3 If (α β γ) prop (0 0 1) the method is the normal basic leapfrog4 Which combination of the numbers (α β γ) is best cannot be answered in

general For N -body systems with very large mass ratios one must haveβ = 0 but some small value is advantageous This is because low-massbodies do not contribute significantly to the energies and if β = 0 thestepsize is not reduced sufficiently during a close encounter

28 Basic Algorithms for the Extrapolation Method

281 Leapfrog

The extrapolation method (Gragg 1964 1965 Bulirsch amp Stoer 1966) whichextrapolates results from a simple basic integrator to zero stepsize is one ofthe most efficient methods to convert results of low-order basic integrators intohighly accurate final outcomes Often such an integrator can be convenientlychosen to be a composite integrator like the leapfrog Let the differentialequations to be

xxx = fff(yyy) yyy = ggg(xxx) (2104)

then one can construct the the simple leapfrog algorithm

xxx 12

= xxx0 +h

2fff(yyy0) (2105)

yyy1 = yyy0 + hggg(xxx 12) (2106)

xxx1 = xxx 12

+h

2fff(yyy1) (2107)

One notes that this is a slightly generalized formulation of the very basicleapfrog which is obtained if fff(yyy) = yyy In this case therefore xxx would be thecoordinate vector yyy the velocity vector and ggg(xxx) the acceleration

Let us introduce the two mappings (or ldquosubroutinesrdquo)

XXX(s) xxx rarr xxx+ sfff(yyy) (2108)

andYYY (s) yyy rarr yyy + sggg(xxx) (2109)

with which the above leapfrog can be symbolized as XXX(h2)YYY (h)XXX(h2)When we want to compute n steps of stepsize = hn we can write

XXX

(h

2n

)[

YYY

(h

n

)

XXX

(h

n

)]nminus1

YYY

(h

n

)

XXX

(h

2n

)

(2110)

This advances the system over the time interval hThe final results can now be considered to be a function of hn and thus

it is possible to extrapolate to zero stepsize Due to the time symmetry of theleapfrog the error has an (asymptotic) expansion of the form

a2(hn)2 + a4(hn)4 +

ie the expansion contains only even powers of h This makes the extrapolationprocess particularly efficient

52 S Mikkola

282 Midpoint Method

In addition to the leapfrog algorithm commonly used in connection withthe extrapolation method we have the so-called modified midpoint methodThis algorithm can also be formally written as a leapfrog Let the differentialequation be

zzz = fff(zzz) (2111)

and let us split this into two parts as

xxx = fff(yyy) yyy = fff(xxx) (2112)

If this pair of equations is solved using the initial conditions xxx(0) = yyy(0) =zzz(0) the solution is simply xxx(t) = yyy(t) = zzz(t) On the other hand (2112) isof the same form as (2104) except that ggg = fff and it is possible to constructthe leapfrog algorithm

xxx 12

= xxx0 +h

2fff(yyy0) (2113)

yyy1 = yyy0 + hfff(xxx 12) (2114)

xxx1 = xxx 12

+h

2fff(yyy1) (2115)

the results of which can also be used for extrapolation to zero stepsize Notethat it is the vector xxx that is extrapolated while here yyy is just an auxiliaryquantity If one defines the mapping

AAA(yyyxxx s) xxx rarr xxx+ sfff(yyy) (2116)

then similar to (2110) one can write for the results with stepsize = hn

AAA

(

yyyxxxh

2n

)[

AAA

(

xxxyyyh

n

)

AAA

(

yyyxxxh

n

)]nminus1

AAA

(

xxxyyyh

n

)

AAA

(

yyyxxxh

2n

)

(2117)where xxx = zzz(0) yyy = zzz(0) initially

283 Generalized Midpoint Method

Here we introduce a generalization of the well-known modified midpointmethod In this algorithm the basic approximation to advance the solutionis not just the evaluation of the derivative at the midpoints but any methodto approximate the solution Thus eg the algorithmic regularization by theleapfrog can be used even when there are additional forces depending on ve-locities This provides a regular basic algorithm which is made suitable forthe extrapolation method by means of the generalized midpoint method

The starting point in this algorithm (Mikkola amp Merritt 2006 2008) is thesame as in the previous (midpoint method) section ie the problem consideredis

zzz = fff(zzz) zzz(0) = zzz0 (2118)

and it is split into two as xxx = fff(yyy) yyy = fff(xxx) and the leapfrog-like algorithm(the modified midpoint method) is

xxx 12

= xxx0 +h

2fff(yyy0) yyy1 = yyy0 + hf(xxx 1

2) xxx1 = xxx 1

2+h

2fff(yyy1)

A new interpretation of the above can be obtained by first rewriting it in theform

xxx 12

= xxx0 +[

+h

2fff(yyy0)

]

(2119)

yyy 12

= yyy0 minus[

minush

2f(xxx 1

2)]

(2120)

yyy1 = yyy 12

+[

+h

2f(xxx 1

2)]

(2121)

xxx1 = xxx 12minus[

minush

2fff(yyy1)

]

(2122)

In (2119) the bracketed term is an (Euler-method) approximation to theincrement of xxx over the time interval h2 with the initial value yyy0 while in(2120) the initial value is xxx 1

2asymp xxx(h2) and the time interval is minush2 Finally

this increment is added ndash with a minus sign ndash to yyy0 to obtain an approximationfor yyy(h2) In the remaining formulae (2121) and (2122) the idea is the samebut the roles of xxx and yyy have been changed

A generalization of this follows readily Let d(zzz0Δt) be an increment forzzz such that

zzz(Δt) asymp zzz0 + d(zzz0Δt) (2123)

is an approximation to the solution of (2118) over a time interval Δt Onestep in the generalized midpoint method can now be written

xxx 12

= xxx0 + d(

yyy0+h

2

)

(2124)

yyy 12

= yyy0 minus d(

xxx 12minush

2

)

(2125)

yyy1 = yyy 12

+ d(

xxx 12+

h

2

)

(2126)

xxx1 = xxx 12minus d

(

yyy1minush

2

)

(2127)

or if we define the mapping (or ldquosubroutinerdquo)

AAA(xxxyyy h) xxx rarr xxx+ d(

yyy+h

2

)

(2128)

yyy rarr yyy minus d(

xxxminush

2

)

(2129)

54 S Mikkola

we can write the algorithm with many (n) steps as

1 Initialize yyy = xxx2 Repeat AAA(xxxyyy h)AAA(yyyxxx h) n times (2130)3 Take xxx as the final result

Thus one simply calls the subroutine AAA alternately with arguments (xxxyyy) and(yyyxxx) such that the sequence is time-symmetric (starts and stops with xxx in(2130))

This basic algorithm has the correct symmetry ndash because it was derivedfrom a leapfrog-like treatment and thus the Gragg-Bulirsch-Stoer extrapola-tion method can be used to obtain high accuracy

This generalized midpoint algorithm may be especially useful if oneemploys a special method well-suited to the particular problem at hand to ob-tain the increment ddd For the few-body problem with velocity-dependent ex-ternal perturbations such a method is the algorithmic regularization leapfrogThe external perturbation (with possible dependence on velocities) can beadded to the increment as

d rarr d + Δtfff(vvv ) (2131)

where fff is the external perturbation and vvv is the most recent velocity valueavailable Further on the leapfrog can be replaced by any other method thatis not necessarily time-symmetric since the algorithm generates the right kindof symmetry

284 Lyapunov Exponents

When the Lyapunov exponents (usually the largest one is sufficient) are re-quired the normal practice is that one derives the variational equations andthen programs the integration of those equations In practice there exists an-other simpler way to do the necessary programming

1 First one writes the code to integrate the basic problem It is a good ideato use rather simple program statements

2 One differentiates the resulting (and tested) code line by line adding thenecessary lines for evaluation of the variations

3 This is the simplest way to write the code for the variations since thereis no reason to consider the variational equations at all Instead one me-chanically differentiates every program statement thus getting the exactvariations of the algorithm

4 That is the best one can do

Perhaps the best way to clarify the above is to give a simple example Hereis a leapfrog algorithm for the harmonic oscillator First is shown the pureharmonic oscillator code then the version with variations The differentiatedlines that evaluate the variations are marked as ldquovarrdquo

c Leapfrog code for a harmonic oscillator

c-----------------------------------------------

implicit real8 (a-ho-z)

x=1

p=0

h=001d0

E0=(pp+xx)2

t=0

1 continue

x=x+h2p this is

p=p-hx a leapfrog

x=x+h2p step

t=t+h

c diagnostics

E=(pp+xx)2

write(6)txpE-E0

if(tlt100)goto 1 max time=100

end

c Differentiated leapfrog for harmonic oscillator

c----------------------------------------------

x=1

dx=1 var

p=0

dp=0 var

E0=(pp+xx)2

dE0=pdp+xdx var

t=0

h=001d0 stepsize

1 continue

x=x+h2p this is

dx=dx+h2dp var

p=p-hx a leapfrog

dp=dp-hdx var

x=x+h2p step

dx=dx+h2dp var

t=t+h

c diagnostics

E=(pp+xx)2

dE=pdp+xdx var (this should be constant)

write(6)txpE-E0dE-dE0

end

The harmonic oscillator example is almost trivial but explains anyway how thevariations can be obtained by differentiating the original code mechanicallywithout any need to consider the variational equations The same technique

56 S Mikkola

is useful for almost any algorithm however complicated One easy check toimplement for the the variations is based on the fact that the differentialsof constants of motion are also constants of motion Above there is only oneintegral the total energy The differential should thus remain (approximately)constant In the few-body problem this applies to the components of angularmomentum also Finally in terms of the variations δq the Lyapunov expo-nents (approximations for) can be obtained as

λ asymp ln(|δq|)t (2132)

when the time t is sufficiently largeIn time-transformed systems all the variables including the time t have

variations Often the results are wanted in the ldquophysicalrdquo system where time isthe independent variable One must thus eliminate the time-variation effectIf f is any function of the system variables and time the physical systemvariation Δf and the time-transformed system variation δf are related by

Δf = δf minus δt f (2133)

where f is the total time derivative of f

29 Accuracy of the AR-Chain

To demonstrate the ability of the AR-chain code to handle large mass ratioswe plot in Fig 25 the energy and angular momentum errors in a system witha wide range of masses (two masses m1 = m2 = 1 and the rest were assignedvalues 01 001 0001 10minus8 Due to the large range of masses the KS-chain

cannot integrate the motions in this system satisfactorily but AR-chain is fastand accurate

The system evolves by ejecting most of the small masses in the time intervalillustrated The energy errors in this example are shown in two ways theuppermost curve gives the relative error in energy computed as 1minusEE0 whilethe lowermost curve is the value of the logarithmic Hamiltonian (essentiallythe same as (E minus E0)U The absolute error of the angular momentum isalso illustrated in the figure Somewhat surprisingly the relative error of theenergy fluctuates considerably while the value of the logarithmic Hamiltonianevolves much more slowly The reason for this is that since the Hamiltonianis log((T minusEU)) the algorithm attempts to keep this quantity constant (andnot the energy E) In fact it is inevitable that integration errors give a smallnon-zero value for the logarithmic Hamiltonian log((T minus E)U) = ε fromwhich we can derive the energy error

δE = εU (2134)

assuming the logarithmic Hamiltonian remains constant Thus it is essentiallythe variation of the potential energy U that causes the fluctuation of theenergy error in the above figure We conclude that all the illustrated errorsare sufficiently small of the order of magnitude of round-off error effects

ndash4endash13

ndash2endash13

0

2endash13

4endash13

6endash13

8endash13

1endash12

12endash12

0 20 40 60 80 100 120

erro

rs

time

1ndashEE0

AM

log((TndashE ) U )

0

Fig 25 Errors in a 10-body problem integrated with the AR-chain code Thesystem consists of a heavy binary (component masses = 1 eccentricity e = 05) andthe other particles have masses 10minusn for n = 1 2 3 8 Uppermost curve relativeerror of energy (= 1 minus EE0) lowermost curve log((T minus E)U) which is the valueof the logarithmic Hamiltonian the thick curve (AM) absolute error in the angularmomentum

210 Conclusions

Experience has shown that generally the AR-chain is comparable in accuracywith the KS-chain in most practical problems (the one-dimensional N -bodyproblem being an exception) With the modified midpoint method AR-chain

is efficient also in problems with velocity-dependent external forces A furtheradvantage is the fact that contrary to KS-chain soft potentials can readilybe treated without any problem Also the differentiation of the algorithmsis sufficiently simple especially for the three-body algorithm discussed inSect 253 so that one can evaluate the Lyapunov exponents

In summary

1 KS-chain is the most efficient KS-regularized code but restricted to com-parable masses (say mass ratios of sim 104) A possible drawback for someproblems is that a soft potential cannot be used

2 LogH is a good alternative for comparable masses3 TTL can handle large mass ratios but may suffer from round-off errors4 AR-chain can handle large mass ratios and soft potential With the gen-

eralized midpoint method velocity-dependent external forces can also be

58 S Mikkola

included with no problem Consequently AR-chain is a good alternativeto the KS-chain and in many problems the best method

5 For all the algorithms discussed here use of the extrapolation method(Bulirsch amp Stoer 1966 Press et al 1986) is necessary to improve theleapfrog results to high accuracy

Finally it is necessary to stress that the codes discussed here are stand-alonefew-body codes requiring additional programming when implementing themfor large N -body systems1

References

Aarseth S J 2003 Gravitational N-Body Simulations Cambridge University PressCambridge 31 37

Bulirsch R Stoer J 1966 Num Math 8 1 45 51 58Gragg W B 1964 PhD thesis University of California Los Angeles 51Gragg W B 1965 SIAM J Numer Anal 2 384 51Heggie D C 1974 Celes Mech 10 217 36 42Kustaanheimo P Stiefel E 1965 J Reine Angew Math 218 204 31 33Levi-Civita T 1920 Acta Math 42 99 31Mikkola S Aarseth S J 1993 Celes Mech Dyn Astron 57 439 36 37Mikkola S Aarseth S 2002 Celes Mech Dyn Astron 84 343 31 44 49 50Mikkola S Merritt D 2006 MNRAS 372 219 31 52Mikkola S Merritt D 2008 AJ 135 2398 50Mikkola S Tanikawa K 1999a MNRAS 310 745 50Mikkola S Tanikawa K 1999b Celes Mech Dyn Astron 74 287 31 32 37 39Press W H Flannery B PTeukolsky S A Wetterling W T 1986 Numerical

Recipes Cambridge University Press Cambridge 45 58Preto M Tremaine S 1999 AJ 118 2532 31 32 37 39Soffel M H 1989 Relativity in Astrometry Celestial Mechanics and Geodesy

Springer-Berlin p 141 31Stiefel E L Scheifele G 1971 Linear and Regular Celestial Mechanics Springer

Berlin 33Yoshida H 1990 Phys Lett A 150 262 43 45Zare K 1974 Celes Mech 10 207 36

1Some source codes can be found on httpwwwcambodyorgcodesphp

3

Resonance Chaos and StabilityThe Three-Body Problem in Astrophysics

Rosemary A Mardling

School of Mathematical Sciences Monash University Victoria 3800 Australiamardlingscimonasheduau

31 Introduction

In his Oppenheimer lecture entitled ldquoGravity is cool or why our universe isas hospitable as it isrdquo Freeman Dyson discusses how time has two faces thequick violent face and the slow gentle face the face of the destroyer and theface of the preserver (Dyson 2000) He entirely attributes these two faces togravity and the ease with which gravitational energy can change irreversiblyinto other forms of energy The simplest system exhibiting these two faces isthat of three gravitating bodies for most configurations the slow gentle faceis the norm while for a very important subset violence is the order of the dayIn fact it is this violence resulting in one of the bodies being ejected from thesystem which is responsible for much of the structure we see in the universefrom planets to giant elliptical galaxies

The simplest example of a quiescent gravitating system is that of twobodies orbiting each other at a distance large enough that their potentialsare essentially those of point masses Their paths about the common centreof mass are simple ellipses and these paths do not change from orbit toorbit their shapes (eccentricities) are preserved as are their sizes (semi-majoraxes) and orientations in space (inclination and longitudes of periastron andascending nodes measured with respect to some reference set of axes seeFig 31) However add one more body to the system and this wealth ofsymmetry is lost at least to some extent In the simplest case if the binarycomponents have equal mass and the third body orbits the binary in the sameplane and is ldquosufficiently distantrdquo the original binary will simply rotate aboutits centre of mass this is apsidal motion Its eccentricity and semi-major axiswill not be affected and the third body will orbit the centre of mass of thebinary as if the latter were a single body with mass equal to the sum of thecomponent masses No net energy or angular momentum is exchanged betweenthe inner and outer orbits in this simple case If the inner binary componentshave different masses some angular momentum is exchanged between theorbits with the result that the eccentricities oscillate about some mean values

Mardling RA Resonance Chaos and Stability The Three-Body Problem in Astrophysics

Lect Notes Phys 760 59ndash96 (2008)

60 R A Mardling

2

i

k

jI

Ωω

f

line of nodes

pericentre

m

Fig 31 Orbital elements specifying the orientation and phase of a binary relativeto a fixed coordinate system ω is the argument of periastron Ω is the longitude ofthe ascending node I is the orbital inclination and f is the true anomaly the latterbeing one of several ways of specifying the orbital phase

This is most pronounced when one body is much more massive than the othertwo as is the case in a planetary system because very close stable systemscan exist

If the orbit of the third body is out of the plane of the binary in additionto apsidal motion both orbits will rock (nutate) up and down that is theirrelative inclination will oscillate about some mean value and the planes of theirorbits will rotate about the direction defined by the total angular momentumof the system (precession)1 No energy and very little angular momentum isexchanged between the orbits of such a system2 even though the eccentricityof the inner binary may oscillate substantially about some mean value aphenomenon called the Kozai effect (Kozai 1962)

These variations of the elements generally occur on time-scales much longerthan the component orbital periods and are referred to as secular variationsThey are characterized by zero energy exchange between the orbits whichmanifests itself in the constancy of the semi-major axes of both the innerand the outer orbits3 In contrast to this unstable systems defined as thosefor which one body eventually escapes to infinity necessarily must exchangeenergy between the orbits in order for this to occur If one makes a plot inthe parameter space of initial conditions associated with secular and unstablebehaviour one finds a very sharp boundary between the two

I was led to the study of stability in the three-body problem after dis-covering that the energy exchange process between the tides and the orbitin a close binary system can be chaotic (Mardling 1995ab) One day Sverre

1Note that apsidal motion is often mistakenly referred to as precession2Again except if the system is a very close planetary-like system3Except for stable resonant systems see later

3 Three-Body Stability 61

Aarseth was looking at my stability plots and commented that they remindedhim of some plots made by Peter Eggleton and Luda Kiseleva for three-bodyhierarchies (Eggleton amp Kiseleva 1995) He wondered whether or not the twoproblems might be linked It turns out that they are much of the analysispresented in this chapter can equally be applied to the binary-tides problem

Throughout this chapter I will refer to five intimately related works sub-mitted or in progress M1a (Mardling 2008a) and M1b discuss stability in thethree-body problem the former coplanar systems and the latter inclined M2discusses the resonant structure of eccentric planetary systems M3 (Mardling2008b) presents a simple formalism for studying the secular evolution of arbi-trary triple configurations4 while M4 presents a new formalism for studyingstrong three-body interactions

32 Resonance in Nature

The most familiar example of resonance in action is a parent pushing a childon a swing The only way to increase the amplitude of the swing consistentlyis to push it at its natural frequency But if you think about it the ldquonaturalfrequencyrdquo varies depending on the amplitude of the swing while it is prettymuch constant over the range of amplitudes tolerated by most children forthe intrepid child who prefers heights substantially more than that of theparentrsquos one needs to wait considerably longer for her to complete a full swingbefore she gets her next push This amplitude dependence of the frequency is acharacteristic of non-linear oscillators of which the pendulum is one exampleand we will see that it is fundamental to understanding stability in the three-body problem

Resonance is responsible for both structure and destruction in Nature andnot just via gravity It is Naturersquos way of moving energy around in bulk Forexample molecular structure depends on resonance between internal elec-tronic states the formation of carbon in stars via the triple-alpha processrelies on a resonant reaction between an alpha particle and a very short-livedberyllium nucleus leading to the formation of an excited state of the carbonnucleus even the Archimedes spiral of a sunflower relies on resonance for itsformation [see Reichl (1992) for a discussion of the golden mean as the ldquomostirrational numberrdquo] But when gravity is involved resonance plays a role onevery astrophysical scale through the dynamics of three-body instability

321 Three-Body Processes in Astrophysics

Three-body processes are at the heart of structure on all astrophysical scalesfrom planet formation via the accumulation of planetesimals to giant ellipticalgalaxies through the forced collisions of smaller galaxies Processes occurring

4Some animations of stable and unstable triples may be found athttpusersmonasheduau~ro

62 R A Mardling

in star clusters include binaryndashsingle star scattering in the cores of globularclusters a process largely responsible for the prevention of total core col-lapse (Aarseth 1971) the formation of X-ray binaries in globular cluster coresthrough binaryndashsingle and binaryndashbinary collisions (Hills 1976) the formationof massive stars that almost certainly occasionally (if not exclusively) formthrough collisions induced in small-N systems the building of intermediate-mass black holes through the so-called Kozai mechanism (Aarseth 2007) theformation of close binaries through the Kozai mechanism (Eggleton amp Kiseleva2001 Fabrycky amp Tremaine 2007) the stability or otherwise of planetary sys-tems in star clusters (Spurzem et al 2006) and hypervelocity stars originat-ing from galactic centre (Hills 1976) In addition many objects thought tobe binary stars are revealing themselves to be triple or higher-order config-urations (Tokovinin et al 2006) such systems may well be the remnants ofeven higher-order systems that have decayed since their birth in the natal starcluster (Reipurth amp Clarke 2001)

To understand all these processes it is necessary to understand how energyand angular momentum move around inside a triple and under what circum-stances a given configuration is stable The rest of this chapter is devoted tothis question through a study of resonance in the three-body problem

33 The Mathematics of Resonance

331 The Pendulum

Before we discuss resonance it is necessary to review the mechanics of apendulum As we will show pendulum-like behaviour is fundamental to anunderstanding of the three-body problem

The equation governing the motion of a pendulum of length l in a uniformgravitational field g is

φ+ ω20 sinφ = 0 (31)

where ω20 = gl Clearly for max(φ) 1 (31) reduces to the equation for

simple harmonic motion with natural frequency ω0 We will refer to ω0 as thesmall angle frequency and to the associated libration period the small anglelibration period Figure 32(a) plots φ against time the latter measured inunits of small angle libration periods for φ(0) = 0 and various values of φ(0)while Fig 31(b) plots solutions in phase-space that is φ against φ Solutionsthat oscillate between fixed values of φ lt π are referred to as libratory andthose for which φ is unbounded are called circulatory These two kinds ofmotion are separated in phase space by the separatrix the two branches ofwhich are indicated by the dashed curves in each panel Clearly the librationperiod increases from 2πω0 for small maximum φ equiv φm to infinity for φm =π Note in particular the so-called hyperbolic fixed points on the separatrix(φ φ) = (plusmnπ 0) in panel (b) these play a vital role in unstable triples as wewill demonstrate

Fig 32 Libration versus circulation of a pendulum Corresponding curves in (a)and (b) have the same colour The dashed curves correspond to the separatrixafter starting at φ(0) = 0 the system takes an infinite amount of time to reach theunstable equilibrium points (φ φ) = (plusmnπ 0) (also known as hyperbolic fixed points)

Equation (31) has an integral of the motion which we refer to as thependulum energy

E =12φ2 minus ω2

0(cosφ+ 1) (32)

where we have chosen the zero of E to correspond to the separatrix that isthe curve which passes through (φ φ) = (π 0) The equation for the separa-trix is therefore

φ = plusmn2ω0 cos(φ2) (33)

For systems with E lt 0 the libration period Tlib is given by

Tlib =int Tlib

0

dt = 4int φm

0

dφφ

=2radic

2ω0

int φm

0

dφradiccosφminus cosφm

(34)

where again φm is the maximum value of φ therefore corresponding to φ = 0Note that for φm 1 Tlib 2πω0

For systems with E gt 0 the circulation period Tcirc is given by

Tcirc = 2int π

0

dφφ

= 2int π

0

dφradicφ2

0 + 2ω20(cosφminus 1)

(35)

where φ0 is the value of φ corresponding to φ = 0 Note that for φ0 2ω0Tcirc 2πφ0

The libration and circulation frequencies ωlib equiv 2πTlib and ωcirc equiv2πTcirc respectively are plotted in Fig 33 Note the steep dependence ofωlib on φm near φm = π and ωcirc on φ0 near φ0 = 0 As we will now demon-strate it is this steep dependence which is responsible for chaos in weaklycoupled non-linear systems

64 R A Mardling

m

E Eω

ω

ωω

Fig 33 Amplitude dependence of pendulum libration and circulation frequenciesNote the extremely steep dependence of ωlib on φm near π ndash one of the secrets tounderstanding chaos in weakly interacting systems The dashed curves correspondto (a) the small angle frequency and (b) φ0 = 2ω0

332 Linear Versus Non-Linear Resonance

Consider a simple undamped spring with natural frequency ω which is forcedat the frequency Ω If φ is the displacement away from equilibrium then giventhe initial conditions φ(0) = φ(0) = 0 the solution to the equation of motion

φ+ ω2φ = A sin Ωt (36)

is

φ(t) =A

Ω2 minus ω2[(Ωω) sinωtminus sin Ωt] (37)

when Ω = ω and

φ(t) =A

2ω2[sinωtminus ωt cosωt] (38)

when Ω = ω These two types of solution are plotted in Fig 34(a) and (b)respectively In the first case a near-resonant value of Ω = 09ω produces thephenomenon called beating where the frequency of the envelope of the solutionis |Ω minus ω| The maximum value attained is approximately (Aω)|Ω minus ω|However when Ω = ω the envelope is given by φ(t) = plusmnAt2ω and thesolution grows without bound This is linear resonance

Unlike a simple spring whose natural oscillation frequency is indepen-dent of the amplitude the libration frequency of a pendulum is amplitude-dependent except when the libration angle is small Consider a pendulumwhich is forced at a constant frequency Ω and let its small angle frequency beω0 Its equation of motion is almost identical to (36) except that φ is replacedby sinφ

φ+ ω20 sinφ = A sin Ωt (39)

ωΩ ωΩ

π

π π

π

ωΩΩ ω

t t

tt

tt t

t

Fig 34 Forced linear spring vs forced pendulum Linear spring (a) beating withΩ ltsim ω and (b) linear resonance with Ω = ω Pendulum (c) and (d) Both solutionsexhibit beating but the system which is forced with a frequency less than the small-angle frequency attains a larger amplitude because as the amplitude increases thelibration frequency decreases moving it closer to the forcing frequency In contrastsystem (d) moves away from the forcing frequency from the start and therefore doesnot attain as large an amplitude For all four systems A = 01 and φ(0) = φ(0) = 0

Now there is no closed-form solution in fact this differential equation admitschaotic solutions In order to understand how such solutions arise (and ulti-mately to understand why the three-body problem admits chaotic solutions)consider solutions to (39) with the same initial conditions as for the forcedspring these are shown in Fig 34(c) and (d) Both solutions exhibit beatingbut the system which is forced with a frequency less than the small anglefrequency attains a larger amplitude because as the amplitude increases thelibration frequency decreases moving it closer to the forcing frequency (seeFig 33) In contrast system (b) moves away from the forcing frequency fromthe beginning

What happens if A is increased in (39) While doing this merely scalesthe amplitude for a linear spring the response is quite different for a forcedpendulum because the response frequency actually depends on the amplitudeFigure 35 shows solutions for various values of A equiv Aω2

0 for φ(0) = φ(0) = 0

66 R A Mardling

π π

ππt

tt t

t

t t

t

A A

Fig 35 Strong forcing of a pendulum All systems have Ω = 09 ω0 and φ(0) =φ(0) = 0 except for the dashed curves for which φ(0) = 10minus6 (a) A equiv Aω2

0 = 03libration Here the pendulum frequency drops further below the forcing frequencyand beating is less pronounced Note especially that the amplitude gets dangerouslyclose to π that is the separatrix (b) A = 10 circulation Safely past the separatrixthe system is sufficiently forced to simply circulate (c) A = 0305 and (d) A = 105chaos The system is forced sufficiently strongly to show a mixture of libration andcirculation The dashed curves illustrate the sensitivity of chaotic systems to initialconditions In fact both (a) and (b) are also chaotic but these systems do not comesufficiently close to the separatrix during this time interval Note that the valuesof A in (c) and (d) are only slightly different to those in (a) and (b) respectivelysuggesting that the time at which obvious divergence of nearby trajectories takesplace is statistical Note also that different scales have been used for each panel

and Ω = 09ω0 In (a) A = 03 the motion remains libratory over this timeinterval (E lt 0) but the amplitude comes close to π (maximum 26) In (b)A = 10 and the stronger forcing allows the system to be completely circu-latory with E gt 0 at all times shown Panels (c) and (d) exhibit sensitivityto initial conditions a diagnostic of chaos even though their values for A areonly slightly different to those in (a) and (b) This is demonstrated by plot-ting trajectories with the same initial conditions except for the initial valuesfor φ which differ by 10minus6 Note that for longer integration times (a) and

(b) also display similar sensitivity to initial conditions including a mixture oflibration and circulation

333 The Butterfly Effect Explained

When a system is near the separatrix a small difference in φ can correspondto at least an order of magnitude difference in the pendulum frequency ωlib

or ωcirc (see Fig 33) Since the libration amplitude depends sensitively onthe current value of ωlib relative to the forcing frequency [for example com-pare Fig 34(c) and (d)] such differences can eventually lead to a significantdivergence of initially nearby solutions as long as the system is not periodicor quasi-periodic (see below)5 A system that is sufficiently strongly forcedmay even cross the separatrix and begin to circulate this almost never hap-pens at the same time as a neighbouring trajectory because of the differencesin their pendulum frequencies at the time The situation is indicated by ar-rows in Fig 35(c) and (d) This behaviour is the essence of chaos in weaklyinteracting systems

Let us consider the situation more closely Given the values of φ and φat any time t one can define the instantaneous (or osculating) pendulumfrequency ω to be such that

ω(t) =

ωlib E lt 0minusωcirc E gt 0 (310)

where again ωlib = 2πTlib and ωcirc = 2πTcirc with Tlib and Tcirc defined in(34) and (35) These latter quantities depend on knowing φm and φ0 that isrespectively φ at φ = 0 for a librating system and φ at φ = 0 for a circulatingsystem The instantaneous values of these can be defined via the pendulumenergy E (which is now not conserved) Thus from (32)

φ2 minus ω20(1 + cosφ) = minusω2

0(1 + cosφm) (311)

and

φ2 minus ω20(1 + cosφ) = φ2

0 minus 2ω20 (312)

Note that defining the pendulum frequency to be negative when E gt 0 simplyensures that dωdt is continuous through ω = 0 that is for the purpose ofgraphical representation there is a smooth transition from libration to circu-lation More importantly it allows for a meaningful measure of the ldquodistancerdquobetween neighbouring trajectories (see discussion below)

Figure 36(b) plots ω(t) for the stable case shown in panel (a) of the samefigure for which A = 01 Ω = 09ω0 The pendulum frequency is clearly

5A system is N-fold quasi-periodic if it can be represented as the product of NFourier series with associated frequencies ωi i = 1 N such that the ωi are notcommensurate If the ωi are commensurate the system is periodic

68 R A Mardling

ω ωΩ Ω

π π

ππ

π π

ω ωω

ωω

π πω

t ttt

tt

t t

tt

A A

Fig 36 Exponential divergence of chaotic trajectories Panel (a) shows the evolu-tion of φ (in units of π) for two initially close trajectories (δφ(0) = 10minus6) for A = 01and Ωω0 = 09 No unstable behaviour is indicated and this is supported by panel(c) which plots the logarithm of the difference in the pendulum frequencies Panel(b) shows the evolution of the pendulum frequency ω(t) ((310)) for the systemwith φ(0) = 0 Points are plotted only when the forcing is zero that is when thependulum is ldquofreerdquo Since φ is quasi-periodic (in fact for this example it is actuallyperiodic because ω0 and Ω are commensurate) the pendulum frequencies come inand out of step over time and their differences therefore never build up Panels (d)(e) and (f) show the evolution of these quantities for the chaotic system A = 02 andΩω0 = 09 The initially close trajectories diverge strongly around t2π = 30 eventhough the system appears to be stable before then However it is clearly not evenquasi-periodic and panel (f) reveals that the trajectories are in fact exponentiallydiverging because |φ| comes close enough to π for ω1 to be significantly different toω2 at those times In particular notice how individual peaks in panel (f) correspondto minimum values of |ω(t)| The forcing is strong enough to allow the system tocross the separatrix and occasionally circulate Since φ is not periodic differencesin ω accumulate and remain O(|ω|)

periodic with minima corresponding to maximum forcing (notice in (a) howthe response ldquostretchesrdquo at maximum amplitude this is seen in more detailin Fig 34(c)) Panel (c) plots the logarithm of the difference between thependulum frequencies ω1 and ω2 of two initially close systems for which thedifference in φ(0) is again 10minus6 equiv ε The difference remains of the order orless than ε for the time shown here and for longer times grows linearly beforeturning over when |ω1 minus ω2| 001 This behaviour is common to quasi-periodic (and periodic) systems for which accumulation of differences in ω islimited to how out of phase the two systems become

In contrast the right-hand panels (d) (e) and (f) show φ(t) ω(t) andlog |ω1 minusω2| for the chaotic system A = 02 and Ω = 09ω0 Unlike the stablesystem this one is not periodic or quasi-periodic and the consequence is thatdifferences in ω do accumulate These differences are a maximum when |ω(t)|is a minimum because of its steep dependence on φ0 as φ0 rarr π and this canbe seen if one compares panels (e) and (f) Eventually |ω1 minus ω2| = O(|ω|)when one of the systems is sufficiently forced to start circulating Note thatsystem 1 first circulates at t2π 84

The slope of the curve in panel (f) indicates the time-scale τ on whichexponential trajectory divergence takes place This is normally associated withthe largest Lyapunov exponent λ which is related to τ such that λ sim 1τ

The following questions arise how strong does the forcing have to be (howlarge should A be) andor how close should the forcing frequency Ω be to ω0

in order that the system is not exclusively libratory Are all systems whichdo not circulate quasi-periodic or periodic (ie do all chaotic systems involvecirculation) These and other related questions have been studied extensivelyin the context of conservative Hamiltonian systems of which the general three-body problem is an example In fact the three-body problem (or simplifiedversions of it) motivated Poincare to invent the modern theory of dynamicalsystems and chaos (Barrow-Green 1997) and led to the famous KolmogorovndashArnolrsquodndashMoser or KAM theory of weakly interacting Hamiltonian systems(see below)

334 Pendulums the Three-Body Problemand Resonance Overlap

The previous examples demonstrate how springs and pendulums respond tofixed forcing How are these related to the three-body problem Most three-body configurations can be regarded as being composed of an ldquoinner binaryrdquoand an ldquoouter binaryrdquo the latter being composed of the inner binary and thethird body this is referred to as a three-body hierarchy (see Fig 38) Whena system is stable (or at least close to stable) these two binaries constitutea weakly interacting conservative system with each binary forcing the other

Figure 37 shows the evolution of the semi-major axis ai of the innerbinary of (a) a stable triple and (b) an unstable triple The behaviour ofthe stable system is very similar to the forced pendulum in Figs 34(c) and

70 R A Mardling

Fig 37 Evolution of the semi-major axis ai of the inner binary of a stable triple(a) and an unstable triple (b) The initial conditions are such that for both (a)and (b) the ratio of the outer periastron distance to the inner semi-major axis is36 and the inner binary is circular while the outer eccentricity is 03 and 05 for(a) and (b) respectively In (b) we also show the evolution of an almost identicalconfiguration for which the initial inner eccentricities differ by 10minus6

36(a) here the forcing is provided by the third body with outer periastronpassage occurring at 05 phase The chaotic system in (b) is reminiscent ofFig 36(d) in this case with a mixture of oscillation between two fixed values(ldquolibrationrdquo) and approximately steady increase or decrease (ldquocirculationrdquo)of ai In fact the inner and outer orbits exchange energy via an interactionpotential or disturbing function which can be written as an infinite series ofresonance angles each a linear combination of all the angles in the systemand each obeying a forced pendulum equation The forcing of each individualldquopendulumrdquo is provided by all the other ldquopendulumsrdquo and when the systemis stable the forcing is negligible (in fact exponentially small) For almost allstable systems the pendulum motions are circulatory with exponentially smallamplitudes however some stable systems exist in a resonant state in whichcase one resonance angle librates6 In order for stability to be maintainedthe forcing of such an angle must remain small in the sense discussed inthe previous section When the forcing is such that the pendulum librationamplitude (ie the single resonance angle that is librating) comes close to πthe system is unstable again in the same sense as discussed in the previoussection However here the forcing is provided by another ldquopendulumrdquo withalmost the same frequency ie by another resonance angle In order for theforcing to be sufficiently strong it turns out that such a resonance angle (ingeneral) must also be librating and we have the situation where the systemexists in two ldquoneighbouringrdquo resonant states this is referred to as resonanceoverlap Thus the diagnostic for instability is simply that two neighbouringresonances be librating this is the resonance overlap stability criterion

6In fact the stable resonant state actually consists of a superposition of resonanceangles (M2) but this is usually only important for extreme mass-ratio systems thathave stable low-order resonances

The reader is referred to the original paper by Walker amp Ford (1969) inwhich this idea is discussed in a clear and straightforward way while Chirikov(1979) provides a deeper and more extensive analysis The concept of res-onance in weakly interacting conservative systems originates in a theoremproposed and partially proved by Kolmogorov (1954) itself inspired by thework of Poincare (1993) This theorem was fully proved by Arnolrsquod (1963)and independently by Moser (1962) The three papers constitute the famousKolmogorovndashArnolrsquodndashMoser or KAM theorem which would provide a proofthat ldquostablerdquo triple systems are formally stable for all time were it not forthe fact that one of the assumptions made in the proof of the theorem isviolated The aim of the KAM theorem is to show that if one perturbs anintegrable Hamiltonian system sufficiently weakly7 then some of the KAMtori on which solutions were originally quasi-periodic will be only slightly dis-torted and quasi-periodicity will be preserved Although not a conservativeHamiltonian system we see this behaviour in going from the forced spring inFig 34(a) to the forced pendulum in panel (c) of the same figure a pendulumcan be regarded as a linear spring with a non-linear perturbation Howeverif the perturbation is too strong quasi-periodicity is lost and the motion be-comes unpredictable If the KAM theorem applied to the three-body problemit would prove that a large subset of configurations exists whose members re-main stable for all time (because they are stuck on KAM tori) But the catchis that one requires the characteristic frequencies of the decoupled system tobe non-commensurate and this is not the case because the apsidal motion andprecession frequencies are equal (in fact equal to zero)

So a formal proof of the ultimate stability of general three-body configura-tions remains elusive although it can be proved in some restricted cases forexample when the eccentricities and inclinations are small so that the seculartheory of Laplace applies and can be used as the underlying ldquounperturbedrdquosystem see Arnolrsquod (1978) p 414 We must therefore (at least for now) becontent with our observation that apparently stable systems seem to mimicquasi-periodic systems for which the KAM theorem does apply and proceedto use the tools of the theorem (in particular the resonance overlap stabilitycriterion) to predict albeit approximately the boundary between stable andunstable behaviour

7An integrable Hamiltonian system that is a function of N coordinate and Nmomentum variables is one which has N integrals of the motion For such systemsone can then find a coordinate transformation such that the new momenta are theintegrals themselves and the new coordinates qi i = 1 N are linear functions oftime qi(t) = ωit + Ci where the ωi are the characteristic frequencies of the systemand the Ci are constants If the ωi are not commensurate that is there exists nointegers ki such that

sumkiωi = 0 the solutions are restricted to and densely cover

so-called KAM tori and the motion is quasi-periodic If the ωi are commensuratethe motion is periodic

72 R A Mardling

34 The Three-Body Problem

The three-body problem is famously easy to formulate and impossible tosolve ndash at least analytically Newton is said to have suffered from sleeplessnessand headaches trying to find closed-form solutions after having had such aneasy time with the two-body problem After many attempts by the best math-ematicians of their time Poincare noticed that perturbation techniques un-avoidably involved singularities associated with resonances and concluded thatthe three-body problem has solutions that cannot be represented by conver-gent series

In order to appreciate fully the dynamics of the three-body problem webegin by reviewing some aspects of the two-body problem in particular itsintegrals of the motion These express various symmetries inherent in theequations of motion one (sometimes more) of which survives when a thirdbody is added and the system is stable (the total energy and linear and angularmomenta are still conserved)

341 Symmetries in the Two-Body Problem

The equations of motion of two bodies with masses m1 and m2 acting underthe influence of each otherrsquos gravity are

m1r1 =Gm1m2

r212r12 (313)

m2r2 = minusGm1m2

r212r12 (314)

where r12 = r2 minus r1 Equations (313) and (314) constitute a twelfth-ordersystem of differential equations However it has eight independent integrals ofthe motion and as is well known this restricts the motion to a simple curve inspace as we now show Three of the integrals of motion are the components ofthe total linear momentum P which one obtains by adding (313) and (314)together and integrating that is

m1r1 +m2r2 equiv P (315)

Dividing through by the masses subtracting (313) from (314) and definingr to be the position vector of m2 relative to m1 that is r equiv r12 we reducethe system to sixth order

r = minusGm12

r2r (316)

where r = |r| and m12 = m1 +m2 Taking the cross product of each side withμr and integrating we get another three integrals of the motion these are thecomponents of the total angular momentum J

μr times r equiv J (317)

where μ = m1m2m12 is the reduced mass of the system A seventh integralof the motion is the total energy this is obtained by taking the dot productof (316) with μr and integrating

12μr middot r minus Gm1m2

requiv E (318)

where we have used the chain rule

ddt

=part

partt+ r middot part

partr (319)

with partpartr equiv nabla The seven integrals reflect natural symmetries of isolatedconservative mechanical systems the conservation of energy and linear mo-mentum reflect the fact that the equations of motion are independent of theorigin of time and space respectively while the conservation of angular mo-mentum reflects the fact that the solution is independent of the orientationof the system For all these symmetries there is no external landmark whichcould be used to distinguish one system from another under such transforma-tions

What symmetry does the eighth integral correspond to It is well knownthat solutions to (313) and (314) are conic sections In particular thesecurves are fixed in space that is their orientation is invariant a fact peculiarto the two-body problem (see Goldstein (1980) p 104 for a discussion of this)This is normally expressed as the invariance of the RungendashLenz vector (alsocalled the Laplace vector) a vector which points in the direction of periastronand is defined by

e = r times (r times r)Gm12 minus r (320)

and whose magnitude is the orbital eccentricity e But this appears to addthree extra integrals in fact one can show that only one is independent of theother seven (Goldstein 1980)

The two-body problem has six degrees of freedom and hence one only needssix integrals of the motion in order that the system be completely integrable(in the sense discussed in the footnote on p 71) The fact that we have eightrestricts the motion to closed curves in the frame of reference of the centreof mass of the system Solution curves are the conic sections (see Goldstein(1980) for a method of solution)

The equations of motion of three bodies with masses m1 m2 and m3 actingunder the influence of each otherrsquos gravity are

74 R A Mardling

m1r1 =Gm1m2

r212r12 +

Gm1m3

r213r13 (321)

m2r2 = minusGm1m2

r212r12 +

Gm2m3

r223r23 (322)

m3r3 = minusGm1m3

r213r13 minus

Gm2m3

r223r23 (323)

where the vectors ri i = 1 2 3 are referred to the centre of mass of thesystem (see Fig 38) and rij = rj minus ri with rij = |rij | The differentialequations (321) (322) and (323) constitute an 18th-order system While itagain yields the seven integrals of total energy linear momentum and angularmomentum there is no analogue of the RungendashLenz integral Thus we are twointegrals short of a totally integrable system This fact results in the possibilityof the system admitting chaotic solutions that is solutions that are exquisitelysensitive to the initial conditions and are hence unpredictable In fact for somesystems with negative total energy it allows for infinite separation of one bodyfrom the other pair These are systems referred to as Lagrange unstable whichin general do not rely on the close approach of two of the bodies (such systemsare referred to as Hill unstable)

We thus ask the general question given a particular three-body configu-ration how can we determine whether or not it is (Lagrange) stable for alltime As discussed in Sect 334 there is no rigorous answer to this ques-tion However there is no doubt that there exists a sharp (albeit fractal-like)boundary in parameter space between unstable systems which decay on arelatively short time-scale and those which appear to remain intact (are sta-ble) indefinitely It is this boundary that is approximately delineated in thischapter using the so-called resonance overlap criterion which itself involvesidentifying internal resonances in the system In order to do this we begin byintroducing Jacobi or hierarchical coordinates r and R which together with

RC123

r3

C12

m1

m2

m3

r1

r2

r

Fig 38 Centre of mass coordinates ri and Jacobi coordinates r and R C12 is thecentre of mass of bodies 1 and 2 while C123 is the centre of mass of the whole system

conservation of linear momentum replace the centre-of-mass coordinates r1r2 and r3 (see Fig 38)

343 Equations of Motion in Jacobi Coordinates

Intuitively it seems reasonable that three-body configurations are more likelyto be stable the further one of the bodies (let us take this to be body 3) isseparated from the other two In fact a very distant third body will orbitthe other two as if they were almost a single body Thus we can conceiveof an ldquoinner binaryrdquo composed of bodies 1 and 2 and an ldquoouter binaryrdquocomposed of bodies (1+2) and body 3 Jacobi coordinates conveniently expressthis arrangement Just as for the two-body problem r is defined to be theposition vector of m2 relative to m1 that is r = r2 minus r1 while R is theposition vector of m3 relative to the centre of mass of m1 and m2 In factit turns out that R passes through the centre of mass of the system and assuch is in the same direction as r3 with R = (m123m12) r3 (Fig 38) wherem123 = m1 +m2 +m3 Using these definitions we can reduce the 18th-ordersystem (321) (322) and (323) to the 12th-order system

μir +Gm1m2

r2r =

partRpartr

(324)

μoR +Gm12m3

R2R =

partRpartR

(325)

where R = |R| μi = m1m2m12 and μo = m12m3m123 are the reducedmasses associated with the inner and outer orbits respectively and

R = minusGm12m3

R+

Gm2m3

|R minus α1r|+

Gm1m3

|R + α2r|(326)

is the disturbing function8 with αi = mim12 i = 1 2 As rR rarr 0 andorm3m12 rarr 0 R rarr 0 and the inner and outer orbits decouple In fact thedisturbing function contains all the information about how the inner andouter orbits exchange energy and angular momentum Since we are interestedin determining which configurations are unstable that is which allow theescape to infinity of one of the bodies and this necessarily generally involvesa substantial exchange of energy between the orbits our focus for the rest ofthis chapter will be on the disturbing function it contains all the secrets ofthe three-body problem

Before we proceed we need to define the orbital elements of the inner andouter binaries in terms of which the stability boundary will be expressed Using

8Note that as a quantity introduced to study the restricted three-body problemthe disturbing function has historically been defined to have units of energy per unitmass Here it has units of energy

76 R A Mardling

subscripts i and o to denote the inner and outer orbits respectively9 these arethe semi-major axes ai and ao the eccentricities ei and eo the orientationangles ωi Ωi Ii and ωo Ωo Io which are respectively the arguments ofperiastron the longitudes of the ascending node and the inclinations (seeFig 31) and the phase angles fi Mi λi εi and fo Mo λo εo which arerespectively the true anomaly the mean anomaly the mean longitude andthe mean longitude at epoch (Murray amp Dermott 2000) Note that longitudeangles are measured with respect to a fixed direction (which here we taketo be the i direction in Figs 31 and 39) we will use longitudes when weconstruct the resonance angle in the next section Thus rather than ωio wewill use the longitudes of periastron defined to be i = ωi + Ωi and similarlyfor o From Fig 31 we see that for inclined orbits this is a dog-leg angleThe phase angles fio Mio and λio equiv Mio + io are used to express theangular positions of the bodies in the two-body orbit the choice of whichdepends on the application (there are at least another two phase angles inuse the true longitude equiv f + and eccentric anomaly neither of which wewill use here) The mean longitude at epoch is the mean longitude at t = 0((345)) See Murray amp Dermott (2000) for a more detailed discussion of thevarious orbital elements

344 Spherical Harmonic Expansions

Since our aim is to determine which configurations are stable it is useful towrite the disturbing function in terms of the orbital elements of the inner andouter binaries To do this we somehow need to separate information aboutthe inner orbit from that of the outer orbit The form of the second and thirdterms in (326) suggest using a Legendre expansion

1|b minus a| =

infinsum

l=0

(al

bl+1

)

Pl(cos γ) (327)

where b = |b| a = |a| with a lt b Pl(cos γ) is a Legendre polynomial of degreel and cos γ = a middot b However for us this involves the angle between r andR information about the two orbits is still ldquotangledrdquo We can go one stepfurther and use something called the addition theorem (Jackson 1975) whichexpresses a Legendre polynomial of order l in terms of spherical harmonicsYlm whose arguments are the spherical polar coordinate angles of the vectorsr and R both referred to a fixed coordinate system (Fig 39)

Pl(cos γ) =4π

2l + 1

lsum

m=minusl

Ylm(θ ϕ)Y lowastlm(ΘΨ) (328)

9When no subscript is used the elements refer to any (or either) two-body orbit

Ψ

Θ

iC

k

m

3

2

12

θ

ϕ

Fig 39 Spherical polar angles associated with r (θ ϕ) and R (Θ Ψ) The origincorresponds to the centre of mass of m1 and m2 C12

Spherical harmonics are defined in terms of associated Legendre functionsPm

l (cos θ) and trigonometric functions (see Jackson (1975) for an extensivediscussion of their properties)

Ylm(θ ϕ) =

radic2l + 1

4π(l minusm)(l +m)

Pml (cos θ) eimϕ (329)

where the numerical coefficient is chosen so that the spherical harmonics havea particularly simple orthogonality relation

int 2π

0

int π

0

Ylm(θ ϕ)Y lowastlprimemprime(θ ϕ) sin θ dθ dϕ = δllprimeδmmprime (330)

Spherical harmonics are especially important in quantum mechanics Com-bining (327) and (328) the disturbing function (326) becomes

R = Gμim3

infinsum

l=2

lsum

m=minusl

(4π

2l + 1

)

Ml

(rl

Rl+1

)

Ylm(θ ϕ)Y lowastlm(Θ ψ) (331)

where

Ml =mlminus1

1 + (minus1)lmlminus12

mlminus112

(332)

Notice how the sum over l begins at l = 2 and not l = 0 this is because thel = 0 term is cancelled by the first term in (326) while the l = 1 term (thedipole term) is zero because M1 = 0 Thus the leading term is proportional

78 R A Mardling

to r2R3 so that R provides a perturbation to the inner and outer orbits forsmall rR The l = 2 contribution is called the quadrupole term while thel = 3 contribution is called the octopole term Notice also that M2 = 1 andthat when m1 = m2 Ml = 0 for l odd

Since the focus of classical treatments of the three-body problem has beenthe Solar System in which mass ratios eccentricities and inclinations are gen-erally small these elements have been used as expansion parameters Theso-called literal expansion (Murray amp Dermott 2000) involves Laplace coef-ficients which are functions of the ratio of semimajor axes and is valid fororbits which cross an example of which is the NeptunendashPluto pair Apart frombeing restricted to small eccentricities and inclinations it also assumes thatone of the participating orbits is not affected by the presence of the third bodythis is the restricted three-body problem The formulation presented here isinstead restricted by the condition rR lt 1 for the spherical harmonic ex-pansion (331) to be valid Note that it is similar to the (rather tedious tofollow) formulation of Kaula (1961) however the latter is also based on therestricted three-body problem

Our aim here is to identify internal resonances so that we can apply theresonance overlap criterion and determine stability boundaries The two mostfundamental frequencies in the system are the inner and outer orbital frequen-cies νi and νo respectively and these are the only frequencies present whenthe orbits are not coupled For example recall that the orientation of a two-body orbit remains fixed in space and this is expressed by the constancy ofthe RungendashLenz vector However when a third body is introduced this sym-metry is broken and the original orbit rotates in space in a manner similar toa spinning top acting under the applied torque of the Earth As discussed inthe Introduction the presence of a third body introduces four new frequen-cies (apsidal advance and precession of the inner and outer orbits) which areusually much slower than the orbital frequencies Resonances will in generalinvolve linear combinations of all six frequencies Our next task then is toexpress the disturbing function in terms of six angles associated with thesefrequencies and as discussed earlier these are chosen to be longitudes Themean longitudes λio are associated with νio while the angles associated withapsidal motion and precession are the longitudes of periastron io and thelongitudes of the ascending node Ωio respectively

For clarity and simplicity the rest of the chapter will assume coplanarmotion see M1a and M3 for the general analysis involving inclined systemsTaking the plane of the orbits to be the xndashy plane the polar angles are thenθ = Θ = π2 so that from (329)

Ylm(π2 ϕ) =

radic2l + 1

4π(l minusm)(l +m)

Pml (0) eimϕ equiv

radic2l + 1

4πclm eimϕ (333)

and similarly for Ylm(π2Ψ) Values for c2lm for some values of l and m arelisted in Table 31

Table 31 Spherical harmonic constants

l m c2lm

2 2 380 14

3 3 5161 316

Referring to Figs 31 and 39 and recalling that we are working in the plane(I = 0) we have ϕ = fi + ωi + Ωi = fi + i and Ψ = fo + o Substitutingthese together with (333) into (331) gives

R = Gμim3

infinsum

l=2

lsum

m=minusl

c2lm Ml

(rleimfi

)(eminusimfo

Rl+1

)

eim(iminuso) (334)

where we have collected together plane polar variables associated with eachorbit in the two pairs of large brackets For uncoupled orbits these are pe-riodic functions with frequencies νi and νo Since we are interested in weakinteraction between the orbits it makes sense to expand these expressions inFourier series in these frequencies Using the familiar two-body expressions

r =ai(1 minus e2i )

1 + ei cos fiand R =

ao(1 minus e2o)1 + eo cos fo

(335)

we have(r

ai

)l

eimfi =infinsum

nprime=minusinfins(lm)nprime (ei) einprimeMi (336)

and

eminusimfo

(Rao)l+1=

infinsum

n=minusinfinF (lm)

n (eo) eminusinMo (337)

where

s(lm)nprime (ei) =

12π

int π

minusπ

(r

ai

)l

eimfieminusinprimeMi dMi (338)

and

F (lm)n (eo) =

12π

int π

minusπ

eminusimfo

(Rao)l+1einMo dMo (339)

Note that the mean anomalies are related to the orbital frequencies by

Mi(t) = νit+Mi(0) and Mo(t) = νot+Mo(0) (340)

80 R A Mardling

n = 1

n = 2

n = 3n = 1

n = 2

n = 1

l m

ei

ei ei

ei

e is n

l m

l ml m

Fig 310 Fourier coefficients s(lm)

nprime (ei) for various values of l m and nprime =1 2 10(= n in figure) Dashed curves correspond to nprime = m The most impor-

tant coefficient for the stability analysis of similar-mass systems is s(22)1 (ei) (shown

in red (grey) note that it is negative for all values of ei)

The real eccentricity-dependent Fourier coefficients s(lm)nprime (ei) and f

(lm)n (eo) =

(1 minus eo)l+1F(lm)n (eo) are plotted in Figs 310 and 311 for some values of l

m n and nprime In Sect 347 we present approximations to the functions usedin our stability analysis Substituting (336) and (337) into the disturbingfunction (334) gives

R = Gμim3

infinsum

l=2

lsum

m=minusl2

infinsum

nprime=minusinfin

infinsum

n=minusinfinc2lmMl

(al

i

al+1o

)

s(lm)nprime (ei)F (lm)

n (eo)eiφmnnprime

= 2Gμim3

sum

L

ζmc2lm Ml

(al

i

al+1o

)

n (eo) cos (φmnnprime) (341)

where

φmnnprime = nprimeMi minus nMo +m(i minuso)= nprimeλi minus nλo + (mminus nprime)i minus (mminus n)o (342)

is called a resonance angle Here ζm = 12 if m = 0 and is 1 otherwise and

n = 2

l m l m

l ml m

n = 3

n = 2

n = 4n = 2

n = 3

eo eo

eoeo

e of n

Fig 311 Fourier coefficients f(lm)n (ei) = (1 minus eo)

l+1F(lm)n for various values of l

m and n = 2 10 Dashed curves correspond to n = m The most importantcoefficients for the stability analysis of similar-mass systems are f

(22)n

sum

L

equivinfinsum

l=2

lsum

m=mmin2

infinsum

nprime=minusinfin

infinsum

n=minusinfin (343)

where the sum over m is in steps of two for coplanar systems (M1a) andmmin = 0 or 1 if l is even or odd respectively

We now have the disturbing function expressed in terms of all the relevantorbital elements including the four angles λi λo i and o which appear inlinear combination in the resonance angle (for coplanar systems the ascendingnode longitudes do not appear explicitly)

345 Energy Transfer Between Orbits

The defining characteristic of (most) stable hierarchical systems is that(essentially) no net energy is exchanged between the orbits over one outerorbital period The usual way to show this is via orbit-averaging over the in-ner orbit This involves a time-average over one entire orbit assuming that allthe orbital variables except the inner orbital phase remain constant on thisshort time-scale The form of (341) makes this extremely easy to performbut first we need an expression for the rate of change of the orbital energyThe simplest way to obtain such an expression is to use Lagrangersquos planetaryequation for the rate of change of the semi-major axis

82 R A Mardling

Lagrangersquos Planetary Equations

Lagrangersquos planetary equations express the rates of change of all the elementsof a two-body orbit which is being perturbed by some external potentialNo assumption is made about the smallness of mass ratios (or any otherparameters) so that it is perfectly well applicable to the general three-bodyproblem the results of which are meaningful as long as the inner and outerorbits retain their identities The derivation of these equations can be foundin Brouwer amp Clements (1961) and is based on the method of variation ofparameters The parameters in this case are the orbital element which remainconstant when the orbit is unperturbed that is e a Ω I and ε = M(0)+The Lagrange equation relevant to us here is that for the rate of change ofthe semi-major axis For the inner and outer orbits of a triple this is

dai

dt=

2μiνiai

partRpartεi

anddao

dt=

2μoνoao

partRpartεo

(344)

respectively where R is given by (341) (recall that our disturbing functionhas dimensions of energy)

Now the usual definition of the mean longitude is

λ = M + = νt+M(0) + = νt+ ε (345)

But this assumes that the orbital frequency (and hence the semi-major axis byKeplerrsquos law and also the orbital energy) is constant something we certainlydo not wish to assume once we consider unstable systems A more generaldefinition is

λ =int t

0

ν(tprime) dtprime + εlowast (346)

where εlowast is a generalization of ε which takes into account the variation of ν(Brouwer amp Clements (1961) p 286 and Murray amp Dermott (2000) p 252 wedo not need the precise definition here) It turns out that using this definitionof λ one can replace εi and εo with λi and λo in (344) so that the rates ofchange of the semi-major axes become

dai

dt=

2μiνiai

partRpartλi

anddao

dt=

2μoνoao

partRpartλo

(347)

Writing the inner orbital energy Ei in terms of inner semimajor axis Ei =minusGm1m22ai the rate of change of Ei is then

1Ei

dEi

dt= minus 1

ai

dai

dt

= 4νi

(m3

m12

)sum

L

nprimeζmc2lmMl

(ai

ao

)l+1

n (eo) sin (φmnnprime)

equivsum

L

nprime Clmnnprime sin(φmnnprime) (348)

Performing a time-average over the inner orbit assuming all elements exceptλi are constant (including ai ie putting λi = νit+ εi) gives

lang1Ei

dEi

dt

rang

=sum

L

nprime Clmnnprime

Ti

int Ti

0

sinφmnnprimedt

=sum

L

nprime Clmnnprime sin (φmnnprime) δnprime0 = 0 (349)

where Ti = 2πνi is the outer orbital period A simpler way to look at this isto ask for the contributions to (348) which are not rapidly varying (ie termswhich do not depend on λi and λo) that is to retain only the ldquosecularrdquo (slowlyvarying) terms by putting nprime = n = 0 This automatically gives ltEiEigt= 0due to the factor nprime in (348) This simple approach also yields the secular ratesof change of the other orbital elements via the Lagrange equations (M3)

Resonance

How do we reconcile (349) with the fact that significant energy transfer isneeded for escape of one body to occur It seems that the assumption thatelements other than λi hardly change over an inner orbital period must bewrong in such cases In fact it is not so much that the other elements donot change much but rather that in some circumstances certain combinationsof angles vary slowly and this can result in significant energy transfer Forexample imagine a system for which the outer orbital period is almost exactlytwo times the inner orbital period that is

νi minus 2νo 0 (350)

Noting from (342) and (346) that

φmnnprime = nprimeνi minus nνo + [nprimeεi minus nεo + (mminus nprime)i minus (mminus n)o] nprimeνi minus nνo (351)

where the frequencies in square brackets are generally much smaller than theorbital frequencies (350) is simply φm21 0 for any m In practice it isterms with m = 2 which contribute the most to energy transfer because theseinvolve the quadrupole l = 2 terms (note the power of aiao in (348) andrecall that the summation over l begins at 2) A system for which (350) holdsis referred to as resonant for obvious reasons In fact except for systems forwhich m2m3 m1 eg starndashplanetndashplanet systems or intermediatemassiveblack holendashstarndashstar systems the so-called 21 resonance is unstable becauseadjacent resonances overlap and produce instability However there are nowseveral stable 21 planetary systems known One example is GJ 876 (Riveraet al 2005) whose orbital periods are 3034 days and 60935 days with massesm1 = 03M m2 = 062MJ and m3 = 193MJ where MJ is the mass of

84 R A Mardling

π0minusπλ λ ω

oi

i

i io

Fig 312 The 21 resonance in the GJ 876 planetary system (a) the evolution ofthe inner semi-major axis for max(νiνo) = 21 The small wiggles correspond to en-ergy exchange during periastron passage of the outer planet (two peaks per passagecorresponding to superior and inferior conjunction) (b) libration and circulationνiνo equiv σ vs the resonance angle φ221 for (from centre) σ = 2008 21 and 22

Jupiter This period ratio is such that νiνo = 2008 that is the system isvery close to exact resonance In order to demonstrate clearly the resonantvariation of ai Fig 312(a) plots its evolution for a slightly larger value of σ(σ = 21) while Fig 312(b) plots νiνo equiv σ vs the resonance angle φ221 forσ = 2008 (the innermost set of points) σ = 21 (the librating set of pointsforming a fuzzy circle) and σ = 22 (the circulating set of points) The factthat ai varies significantly in Fig 312(a) indicates that a substantial amountof energy is exchanged between the orbits (when the inner orbit shrinks theouter orbit expands due to conservation of energy) Resonant orbits are alsoassociated with libration of one or more resonance angles The width of aresonance is the ldquodistancerdquo from exact resonance to the separatrix calculatedat φmnnprime = 0 if this separatrix overlaps the separatrix of a neighbouringresonance we have instability Thus our task is to determine the width ofresonances and to ask for what orbital parameters are these wide enough tooverlap neighbouring resonances

Before we leave this section on energy exchange and resonance we quotea result from M4 which gives approximately the energy exchanged betweenthe inner and outer orbits over one outer orbital period (from apastron toapastron)

ΔEi

Ei I2

22 + 2 ei(0) I22 sin [φ(0)] (352)

where ei(0) is the inner eccentricity at t = 0 and

I22 =94

(m3

m12

)(ai

ao

)3

E22(eo σ) (353)

with an asymptotic expression for the ldquooverlap integralrdquo

E22(eo σ) = νieminusiσπ

int To

0

eminus2ifo

(Rao)3eiνit dt (354)

4radic

2π3

(1 minus e2o)34

e2oσ52eminusσξ(eo) (355)

(M1a) Here To is the outer orbital period and ξ(eo) = coshminus1(1eo)minusradic

1 minus e2oAlso

φ(0) = Mi(0) + σπ + 2(i minuso) φ2n1(0) (356)

that is φ(0) is approximately the value of the resonance angle φ2n1 when theouter body is at apastron (see (342)) exact equality holding when σ = nThe expression (355) includes only quadrupole l = 2 m = 2 terms and isobtained using an asymptotic method similar to that of Heggie (1975) whichgives the energy exchanged during the flyby of a binary by a third body Notethat limeorarr0 E22 = 0 for σ gt 2 is finite for σ = 2 and is not defined for σ lt 2and that limeorarr1(1 minus eo)3E22 is finite

The form of (355) shows that the amount of energy transferred duringone outer orbit of a bound triple is exponentially small except when σξ(eo) issmall This is consistent with the orbit-averaging result 〈EiEi〉 = 0 and itstrongly suggests that ldquostablerdquo systems are stable for all time although aspreviously discussed a proof is not yet available

346 A Pendulum Equation for the Resonance Angle

Figure 312(b) illustrates how a resonance angle librates when the orbital fre-quencies are near-commensurate This suggests that resonance angles shouldsatisfy a pendulum-like equation the ability to write down such an equationwould then give us the full machinery outlined in Sect 331 for pendulumsIn particular we could calculate the distance from exact resonance to theseparatrix that is the resonance width recall that we need this in order todetermine when neighbouring resonances overlap and hence when a system isunstable

Referring to (31) we see the second time derivative of the resonance angleis required Starting from (351)

φmnnprime = nprimeνi minus nνo (357)

where we have replaced the approximation symbol with equality we then have

Relating the rates of change of the orbital frequencies to the rates of changeof the semi-major axes

νi

νi= minus3

2ai

aiand

νo

νo= minus3

2ao

ao (359)

86 R A Mardling

we can again make use of Lagrangersquos planetary equation for the rate of changeof the semi-major axis (347) together with (348) and its equivalent for aoSubstituting these into (358) and assuming that the resonance is isolated (notforced) that is that the only significant terms in the summations are thosewith the same values of m n and nprime we get

φmnnprime = minusnprime2ν2oAmnnprime sin (φmnnprime) (360)

where

Amnnprime equiv minus6 ζm

infinsum

l=lmin2

c2lms(lm)nprime (ei)F (lm)

n (eo)

middot[M

(l)i σminus(2lminus4)3 +M (l)

o (nnprime)2σminus2l3]

minus6 ζm

infinsum

l=lmin2

n (eo) (nnprime)minus(2lminus4)3

middot[M

(l)i +M (l)

o (nnprime)23]

(361)

and we have put σ nnprime in the last step Here lmin = 2 if m is even andlmin = 3 if m is odd The dependence on the masses is solely through thefunctions

M(l)i = Ml

(m3

m12

)(m12

m123

)(l+1)3

and M (l)o = Ml

(m1m2

m212

)(m12

m123

)l3

(362)

Except for very low values of n corresponding to planetary-like problems itis usually adequate to include only the first term in the summation over l

Comparing (360) with (31) we have that the ldquosmall angle frequencyrdquoω0 is nprimeνo|Amnnprime |12 When Amnnprime gt 0 we have libration around zero andwhen Amnnprime lt 0 we have libration around π It turns out that for systems forwhich at least two of the masses are reasonably similar (this is quantified inSect 3410) the dominant resonances are those with m = 2 and nprime = 1 Usingthe notation introduced in M1a these are the [n 1](2) resonances Referringto Figs 310 and 311 and recalling that we only need include l = 2 whenm = 2 we see that s(22)1 (ei) lt 0 for all 0 le ei le 1 and that f (22)

n (eo) gt 0for 0 le eo le 1 so that A2n1 gt 1 for all n Thus libration is around zero forall resonances of interest here Putting nprime = 1 and m = 2 in (361) retainingonly the l = 2 term and setting φ2n1 equiv φn and A2n1 equiv An the resonances ofinterest to us are governed by

φn = minusν2oAn sinφn (363)

whereAn = minus9

2s(22)1 (ei)F (22)

n (eo)[M

(2)i +M (2)

o n23] (364)

with

M(2)i =

m3

m123and M (2)

o =(m1m2

m212

)(m12

m123

)23

(365)

and we have used c222 = 38 from Table 31 In Sect 345 p 84 we definedthe width of a resonance to be the distance from exact resonance and theseparatrix calculated at φmnnprime = 0 Equation (33) gives an expression forthe separatrix so that the width of a resonance is

Δφ = 2ω0 = 2νo

radicAn (366)

for the [n 1](2) resonances of interest here It is usually more convenient todefine the width of a resonance in terms of the change in σ Since

φn = νi minus n νo = νo(σ minus n) (367)

we can define the width of the [n 1](2) resonance to be

Δσn = 2radicAn (368)

We can associate an ldquoenergyrdquo En with the pendulum-like motion of a res-onance such that En lt 0 for libration and En gt 0 for circulation of φnFollowing (32) we then have

En =12φ2

n minus ν2oAn(cosφn + 1) (369)

It is useful to define a dimensionless version of this such that En = νoEnthat is

En =12[δσn]2 minusAn(cosφn + 1) (370)

where δσn = σ minus n is the ldquodistancerdquo from exact resonance corresponding toφn Note that δσn is a maximum when φn = 0 (for libration around φn = 0)We will use (370) in a simple algorithm to determine the stability of anygiven configuration (Sect 3410)

The form of (364) makes it relatively easy to see how resonance widthsdepend on the various parameters Before we make use of (368) to determinethe stability boundary it is necessary to discuss evaluation of the eccentricityfunctions s(22)1 (ei) and F

(22)n (eo)

88 R A Mardling

347 Eccentricity Functions

Since the eccentricity functions s(22)1 (ei) and F(22)n (eo) are integrals with no

closed form expressions (except for n = 0 see M3) it is of interest to findapproximations A simple Taylor expansion of the integrand of s(22)1 (ei) aboutei = 0 allows for the integral to be performed and if one expands up to O(e7i )allows for the function to be well represented for all ei le 1 This proceduregives

s(22)1 (ei) minus3ei +

138e3i +

5192

e5i minus2273072

e7i (371)

If εi is the difference between the exact and approximate expression |εi| lt0001 for ei lt 063 |εi| lt 001 for ei lt 079 and |εi| lt 01 for ei lt 1

While it is possible to find Taylor series approximations to F(22)n (eo) we

would need hundreds of these for a general stability algorithm since systemswith very high outer eccentricity can involve very high values of n (sinceσ = νiνo nnprime = n) Instead we make use of the asymptotic expression(354) to evaluate (339) Making the substitution Mo = νotminusπ in (354) (sincethe outer orbit starts at minusπ that is Mo(0) = minusπ) so that νit = σ(Mo + π)and νidt = σdMo the integral becomes

E22(eo σ) = σ

int π

minusπ

eminus2ifo

(Rao)3eiσMo dMo (372)

Comparing this with (339) we see that

F (22)n (eo) E22(eo n)2πn (373)

Thus we have the beautiful result that the resonance widths are exponentiallysmall when σξ(eo) is small consistent with the fact that an exponentially smallamount of energy is exchanged between the orbits in such circumstances

348 Induced Eccentricity and Secular Effects

The expression for the resonance width (368) together with (364) and (371)suggest that systems whose inner binary is circular have zero resonance widths(since s(22)1 (0) = 0) But this surely is not true Figure 313 plots the evolutionof the inner eccentricity for an equal mass three-body system whose initialeccentricities are ei(0) = 0 and eo(0) = 05 and for which (a) σ = 10 and(b) σ = 8 Both systems start at outer apastron and significant eccentricityis induced when they pass through outer periastron The formalism used toestimate the energy transferred between orbits (see Sect 345 and (352)) canalso be used to estimate the induced inner eccentricity This is given by

ei(To) =[ei(0)2 minus 2 ei(0) I22 sin[φ(0)] + I2

22

]12 (374)

σ σe i e i

Fig 313 Induced inner eccentricity of a circular binary (a) σ = 10 and (b)σ = 8 In both cases eo = 05 and the system is started at outer apastron withMi(0) = 0 and i minus o = 0 Both systems are chaotic but (a) is on the stabilityboundary while (b) is deep inside the unstable region The dashed lines correspondto the estimated induced eccentricity ((374)) following the first outer periastronpassage

where ei(0) and ei(To) are the inner eccentricity at initial and final outerapastron and I22 and φ(0) are given by (353) and (356) respectively Thedashed curves in Fig 313 indicate these estimates

It turns out that using ei(To) instead of ei(0) in the expression for the res-onance width quite accurately predicts the stability boundary when octopoleeffects are unimportant (see Fig 315)

Octopole Variations for Coplanar Systems

For systems with m1 = m2 secular octopole contributions to the disturbingfunction (terms with n = nprime = 0) can cause the inner eccentricity to varyconsiderably on time-scales of thousands of inner orbits (Murray amp Dermott2000 M3) This is especially important for close planetary systems Whilethe outer eccentricity also varies the main effect on the resonance widthscomes from the variation of s(22)1 (ei) which is a maximum at the maximumof the octopole cycle in ei Referring to this maximum as e(oct)

i it is givenapproximately by (Mardling 2007 M1a)

e(oct)i =

(1 + α)e(eq)

i α le 1ei(0) + 2e(eq)

i α gt 1(375)

where α = |1 minus ei(0)e(eq)i | and e

(eq)i is the ldquoequilibriumrdquo or ldquofixed pointrdquo

eccentricity which is the root of the eighth-order polynomialsum8

n=1 anxn in

[01] where the an are given by

a0 = minusB2

a1 = 2ABa2 = B2 + C2 minusA2

90 R A Mardling

a3 = minus2(AB + 4CD)a4 = A2 + 3C2 + 16D2

a5 = minus18CD

a6 =94C2 + 24D2

a7 = minus9CDa8 = 9D2 (376)

with

A =34

(m3

m12

)(ai

ao

)3

εminus3o

B =1564

(m3

m12

)(m1 minusm2

m12

)(ai

ao

)4

εminus5o

C =34

(m1m2

m212

)(ai

ao

)2

εminus4o

D =1564

(m1m2

m212

)(m1 minusm2

m12

)(ai

ao

)3 (1 + 4e2oeoε6o

)

(377)

and εo =radic

1 minus e2o In the limit ei 1 the equilibrium eccentricity reduces to

e(eq)i =

(54)eom3(m1 minusm2)(aiao)2σεminus1o

|m1m2 minusm12m3(aiao)εoσ| (378)

Note that even though (378) is not accurate away from the stability boundarywhere ei(To) is large it can be used to determine the boundary if ei(0) is smallbecause ei(To) tends to be small there in that case (see Fig 313)

349 Resonance Overlap and the Stability Boundary

The stability of any given coplanar configuration depends on the values of theeight parameters m2m1 m3m12 σ ei eo i minus o Mi(0) and Mo(0) Inorder to represent the stability boundary in two dimensions we need to fixthe values of six of these and vary the other two Here we choose to plot eo

against σ for i minuso = Mi(0) = 0 and Mo = minusπ and for a selection of massratios and ei(0)

For a given value of n and for fixed values of ei(0) m2m1 and m3m12the two boundaries of the [n 1](2) resonance are given by

σ(eo) = nplusmn Δσn(eo) = nplusmn 2 [An(eo)]12

(379)

12

librationof

nΔσ

φ

e o e o

Fig 314 (a) The [121](2) resonance (b) Resonance overlap This example cor-responds to m2m1 = m3m1 = 001 and ei(0) = 05 (see Fig 316) See text fordiscussion

where An(eo) is given by (364) Note that this assumes exact resonance occurswhen

φn = νi minus nνo = 0 (380)

that is when σ = νiνo = n however if iνo is significant it will shift exactresonance away from this (recall the precise expression (351) for φn see alsoFig 315) Figure 314(a) plots eo against σ for the [121](2) resonance for aparticular set of initial conditions with the shaded region corresponding tolibration of the resonance angle φ1210 while panel (b) shows the overlap ofthe resonances [n 1](2) n = 9 10 15 for the same initial conditions Thelower (green)-shaded regions in panel (b) formally correspond to stable libra-tion of the resonance angles φn while the unshaded regions correspond to sta-ble circulation for which the inner and outer orbits have constant semi-majoraxes The upper (red)-shaded region corresponds to the overlap of neighbour-ing resonances (as well as more distant resonances) so that a system withinitial conditions corresponding to any point in this region is predicted by theresonance overlap stability criterion to be unstable

How does this compare with direct numerical experiments Figure 315(a)shows a stability map for equal-mass configurations with initially circular in-ner binaries for various initial period ratios and outer eccentricities A dotcorresponding to the initial values of σ and eo is plotted if a direct numericalintegration of the three-body equations of motion results in an unstable sys-tem Rather than integrating the system until one of the bodies escapes twoalmost identical systems (the given system and its ldquoghostrdquo) are integrated inparallel and the difference in the inner semi-major axes at outer apastron ismonitored (because this variable is approximately constant for non-resonantsystems) Taking advantage of the sensitivity of a chaotic system to initial con-ditions this difference will grow in proportion to the initial difference between

10Even though (379) gives σ as a function of eo it seems more natural to plotthe resonance boundaries with σ as the independent variable

92 R A Mardling

Fig 315 Experimental vs theoretical stability boundary The position of each red(grey) dot in (σ minus eo) space corresponds to the initial conditions of an unstablesystem for which the masses are equal ei(0) = 0 and Mi(0) = 0 and Mo(0) = minusπThe black curves are the resonance boundaries given by (379) which terminate atpoints for which ei(To) = 1 Notice the structure of the distribution of dots near thesetermination points this reflects the process of exchange of m3 into the inner binary(consistent with ei(To) gt 0) Systems deemed stable (see text for how this decisionis made) are those for which exchange occurs rapidly While the resonance overlapstability criterion predicts the stability boundary fairly accurately some of the reddots fall inside single-resonance regions which ought to be stable according to thecriterion But the criterion assumes that when only one resonance angle is libratingthe forcing is negligible this clearly is not true at these points Also notice how thered dots trace the separatrix at the left-hand boundaries and in particular noticethe offset which is prominent for the 51 resonance this is analogous to spectral linesplitting by a magnetic field and is a result of the influence of i which has beenneglected in (379)

two systems (10minus7 in the inner eccentricity) for a stable system but will growexponentially for an unstable system as discussed in Sect 333 The actualstability boundary fairly accurately follows the points at which neighbour-ing resonances overlap however the stability criterion does not predict theunstable nature of some systems inside single-librating regions (correspond-ing to the green regions in Fig 314(b)) because it assumes that forcing isnegligible there

Figure 316 shows stability maps for a variety of initial conditions Eachmap has m1 = 1 Mi(0) = 0 and Mo(0) = minusπ and aligned periastra exceptfor panel (f) Consider the systems (a) (c) and (e) for which ei(0) = 0 andη = i minuso = 0 The librating regions for which there is no overlap with aneighbouring resonance are relatively free of unstable systems while those for

eie o

ei

eiei

ei

e oe o e o

e oe o

m m m m

mmmm

mi mi

Fig 316 Stability maps for a variety of initial conditions (m1 = 1) Notice howresonance shapes vary significantly from panel to panel but the resonance overlapstability criterion is still successful at predicting the stability boundary (except forthe single-librating regions) The dashed curve in the top left-hand corner of eachpanel corresponds to Rpai = 1 where Rp = ao(1 minus eo) is the outer periastrondistance (data were not collected beyond this curve) (a) planetary-like systemwith significant inner eccentricity (b) low-mass secondary with zero initial innereccentricity (c) Jupiter-like outer body orbiting an equal-mass eccentric binary(d) ldquobinaryrdquo consisting of a heavy body and an equal-mass binary (e) and (f)equal-mass system with ei(0) = 02 Here η = i minuso the two plots demonstratingthe effect of rotating the orbits relative to each other Notice that even resonances aremore stable than odd in (a) while the opposite is true in (b) (see text for discussion)

94 R A Mardling

odd resonances tend to be full down to near the resonance cross-over pointsThe reason for this is as follows Referring to (342) on p 80 we see (puttingnprime = 1 and m = 2) that for these initial conditions φn(0) = nπ Since librationis around zero (because An gt 0) a system starting at exact resonance thatis with σ = n will stay there if n is even because it is at the very centreof the resonance (see Fig 312 on p84) while if n is odd the system startsat the hyperbolic fixed point on the separatrix An odd-n system for whichσ = n (and is indicated on the stability map to be inside a resonance) actuallybegins outside the librating region recall that the definition of the resonanceboundary uses the value of the separatrix at φn = 0 However it will still bestrongly forced and its proximity to the separatrix will cause it to be unstableA more detailed analysis can be found in M1a

We should expect from this discussion that a system for which η = i minuso = 0 will exhibit different behaviour and this is indeed the case as panel(f) for which η = π2 reveals In this case φn = (n+ 1)π and we see that itis the even resonances that are now more unstable

The fact that ei(0) = 0 for the examples just discussed means that theinner orbit begins with a definite periastron direction What about whenei(0) = 0 Figure 315 as well as panels (b) and (d) in Fig 316 show thatpoints on the left-hand sides of the resonances tend to be unstable whilepoints on the right-hand side are stable up to where the resonances overlapWe interpret this as indicating that the induced periastron direction associ-ated with the induced eccentricity tends to be such that η(To) π4 so thatφn = (2n+ 1)π2

Another feature of Fig 316 worth noting is the patch of instability atthe lower-left corner of panel (a) This is common for low-order resonances inplanetary-like systems and actually corresponds to libration around π (this isdiscussed in detail in M2)

3410 A Simple Algorithm for Predicting Stability

For most applications one needs to know the stability characteristics of singlesystems Thus rather than give a formula for the stability boundary we endthis chapter by presenting an algorithm for testing the stability of individualconfigurations Note that it only holds for coplanar systems11 and is restrictedto systems for which the [n 1](2) resonances dominate These are such thateither both m2m1 gt 001 and m3m1 gt 001 or at least one of m2m1 gt 005or m3m1 gt 005 The algorithm is as follows

1 Identify which [n 1](2) resonance the system is near and calculate thedistance δσn from that resonance δσn = σminusn where n = σ (the nearestinteger for which n le σ)

11A Fortran routine for arbitrarily inclined systems is available from the author

σ σ

e o e o

Fig 317 Comparison of (a) experimental and (b) theoretical data for equal masscoplanar systems with ei(0) = 0

2 Take the associated resonance angle to be zero rather than the definition(342) (see discussion below) φn = 0

3 Calculate the induced eccentricity from (374) and (if m1 = m2) the maxi-mum octopole eccentricity from (375) Determine ei = max[ei(To) e

(oct)i ]

for use in s(22)1 (ei)

4 Calculate An from (364)5 Calculate En and En+1 from (370) and deem the system unstable if En lt 0

and En+1 lt 0

Figure 317 compares the experimental data shown in Fig 315 with datagenerated using the algorithm above A dot is plotted if a system is deemed tobe unstable The boundary structure is reproduced reasonably well althoughthe boundary itself should be slightly lower a result of the fact that theresonance overlap criterion does not recognize the unstable nature of pointsnear to but outside the separatrix This is also the reason for taking φn = 0for all initial conditions (recall the discussion in the previous section on oddand even resonances)

References

Aarseth S J 1971 ApampSS 13 324 62Aarseth S J 2007 MNRAS 378 285 62Arnolrsquod V I 1963 Russian Mathematical Surveys 18 9 71Arnolrsquod V I 1978 Mathematical Methods of Classical Mechanics Springer-Verlag

New York 71Barrow-Green J 1997 Poincare and the Three Body Problem (History of Mathe-

matics V 11) American Mathematical Society 69Brouwer D Clements G M 1961 Methods of Celestial Mechanics Academic Press

New York and London 82Chirikov B V 1979 Phys Rep 52 263 71

96 R A Mardling

Dyson F J 2000 Oppenheimer Lecture University of California Berkeleyhttpwwwhartford-hwpcomarchives20035html 59

Eggleton P Kiseleva L 1995 ApJ 455 640 61Eggleton P P Kiseleva-Eggleton L 2001 ApJ 562 1012 62Fabrycky D Tremaine S 2007 ApJ 669 1298 62Goldstein H 1980 Classical Mechanics Addison-Wesley Philippines 73Heggie D C 1975 MNRAS 173 729 85Hills J G 1976 MNRAS 175 1P 62Hills J G 1988 Nature 331 687Jackson J D 1975 Classical Electrodynamics Wiley New York 2nd ed 76 77Kaula W M 1961 Geophys J Roy Astr Soc 5 104 78Kolmogorov A N 1954 Dokl Akad Nauk 98 527 71Kozai Y 1962 AJ 67 591 60Mardling R A 1995a ApJ 450 722 60Mardling R A 1995b ApJ 450 732 60Mardling R A 2007 MNRAS 382 1768 89Mardling R A 2008a submitted to MNRAS 61Mardling R A 2008b submitted to MNRAS 61Moser J 1962 Nachr Akad Wiss Gottingen II Math Phys KD 1 1 71Murray C D Dermott S F 2000 Solar System Dynamics Cambridge Univ Press

Cambridge 76 78 82 89Poincare H 1993 New Methods of Celestial Mechanics (Vol 1) Goro D L ed

AIP New York I23 22 71Reichl L E 1992 The Transition to Chaos in Conservative Classical Systems

Quantum Manifestations Springer-Verlag New York 61Reipurth B amp Clarke C 2001 AJ 122 432 62Rivera E J et al 2005 ApJ 634 625 83Spurzem R Giersz M Heggie D C Lin D N C 2006 astro-ph0612757 62Tokovinin A Thomas S Sterzik M amp Udry S 2006 AampA 450 681 62Walker G H Ford J 1969 Physical Review 188 416 71

4

FokkerndashPlanck Treatment of Collisional StellarDynamics

Marc Freitag

University of Cambridge Institute of Astronomy Madingley Road CambridgeCB3 0HA UKmarcfreitaggmailcom

41 Introduction

In this chapter I explain how the evolution of an N -body system can be de-scribed using a formalism explicitly based on the distribution function in phasespace Such an approach can be contrasted with direct N -body simulations inwhich the trajectories of a large number of particles are integrated Becausetrajectories with close initial conditions diverge exponentially in gravitationalN -body systems (Goodman et al 1993 Hemsendorf amp Merritt 2002 andreferences therein) most results of N -body simulations must be interpretedstatistically It is therefore interesting to consider the simulation methods thattreat the gravitational system in an explicitly statistical way

Since the early 1980s the numerical solution of the FokkerndashPlanck (FP)equation has been the technique of choice for a statistical treatment of colli-sional systems such as globular clusters or dense galactic nuclei In its basicversion on which I focus this equation (combined with the Poisson equa-tion) describes the evolution of a stellar system in dynamical equilibrium butevolving slowly through the effects of two-body relaxation In this chapterI further restrict myself to spherically symmetric configurations with no netrotation as most researchers in the field have done to make the problemeasier to tackle As far as relaxation is concerned the Monte-Carlo numericalscheme presented in Chap 5 is essentially equivalent to solving the FP equa-tion using a particle-based representation of the distribution function insteadof tabulated data Therefore the assumptions and limitations inherent in theFP description of relaxation which are described in detail in this chapteralso apply to Monte-Carlo techniques

A note of caution is required here The dynamics of a gravitational N -bodysystem is highly non-linear with the possibility that small differences in theldquomicroscopicrdquo conditions (such as the existence and properties of a binarystar) can lead to rather large macroscopic differences in evolution The FPapproach does not provide a statistical description of the various macroscop-ically distinct possible evolutions When such divergences are expected to

Freitag M Fokkerndashplanck Treatment of Collisional Stellar Dynamics Lect Notes Phys 760

97ndash121 (2008)

98 M Freitag

occur such as in the process of collisional runaway or post-collapse core os-cillations (see Sect 45) the only way to capture them in a satisfying wayby means of FP simulations is probably by including some explicit stochasticprocess and repeat the simulation several times with different random se-quences (see Takahashi amp Inagaki (1991) for an example in the case of coreoscillations)

In the last decade or so FP codes have lost some ground to direct N -bodyand Monte-Carlo codes Indeed these particle-based methods make it easierto include a variety of physical effects thought to play an important rolein real systems and faster computers enable the use of higher and higherparticle numbers Nevertheless because FP computations are very fast andproduce data that are much smoother less memory-consuming and easierto manipulate than particle-based simulations they are an invaluable tool forexploring large volumes of parameter space They also help in gaining a betterunderstanding of ldquomacroscopicrdquo collisional stellar dynamics by providing adescription at a level more suitable than that of ldquomicroscopicrdquo point-massparticles attracting each other

In Sect 42 I present the Boltzmann equation which is at the heart ofthe statistical description of an N -body system In Sect 43 I give an outlineof the derivation of the main forms of the FP equation used to simulate theeffects of relaxation in spherical stellar systems Finally Sect 45 is a quickoverview of the applications of the FP approach in stellar dynamics with afocus on the additional physics that can be incorporated into that framework

42 Boltzmann Equation

421 Notation

The following notations are in use in this section Position and velocity in 3Dspace are denoted by

x = (x y z) = (x1 x2 x3)

andv = (vx vy vz) = (v1 v2 v3)

For a point in the 6D phase space I use the notation

w = (xv)

The gradient of a field u in 3D space is written

nablau equiv partu

partx=(partu

partxpartu

partypartu

partz

)

and the gradient in the 6D phase space is

nablau equiv partu

partw=(partu

partxpartu

partypartu

partzpartu

partvxpartu

partvypartu

partvz

)

4 FokkerndashPlanck Treatment 99

422 Collisionless System

In this section I follow mostly the treatment presented in Sect 41 of Binneyamp Tremaine (1987 hereafter BT87)

We consider a large number Nlowast of bodies moving under the influence ofa smooth gravitational potential Φ(x t) Here smooth means essentially thatΦ does not change much over distances of the order of (a few times) the av-erage inter-particle distance nminus13 where n is the particle number densityNo other forces affect the motion of these objects The potential Φ may bethe gravitational field created by these bodies themselves or an external fieldThe system of particles is described through the one-particle phase-space dis-tribution function (DF for short) f(xv t) A useful interpretation of f is asa probability density if it is normalised to 1 Then f(xv t)d3xd3v is theprobability of finding at time t any given particle within a volume of phasespace d3xd3v around the 6D phase-space point w = (xv) The mean numberof particles in this volume is Nlowastf(xv t)d3xd3v

From the knowledge of the initial conditions f0(xv) equiv f(xv t0) wewant to predict f(xv t) at some future time t gt t0 We define the velocityin the 6D phase-space

w = (x v) = (vminusnablaΦ) (41)

As long as Φ is sufficiently smooth the particles evolve in a smooth continuousway in the phase-space Therefore f must satisfy a continuity equation

partf

partt+ nabla middot (fw) =

partf

partt+

3sum

i=1

part(fvi)partxi

minus3sum

i=1

part(fpartxiΦ)partvi

= 0 (42)

This equation can be simplified using the fact that in the phase-space repre-sentation the xi and vi are independent variables (partvipartxj = 0) and that Φdoes not depend on the velocities so that partΦpartvi = 0 Therefore we have

partf

partt+

3sum

i=1

vipartf

partximinus

3sum

i=1

partxiΦpartf

partvi=

partf

partt+ v middot nablaf minus nablaΦ middot partf

partv= 0 (43)

This is the collisionless Boltzmann equation It can be written simply as

Dtf = 0 (44)

where Dt is a notation for the ldquoLagrangianrdquo or advective rate of change of f This equation means that if we follow the trajectory of a (real or imaginary)particle in the phase-space the number density around it does not change Inother words the flow in phase-space is incompressible

We note that there is an equation which is equivalent but more general(and of less practical use) for the distribution function in the Nlowast-particlephase-space in which a point represents all the positions and velocities of

100 M Freitag

the Nlowast bodies of the system It is Liouvillersquos theorem (BT87 Sect 82) Thecollisionless Boltzmann equation follows from Liouvillersquos theorem and the as-sumptions that the number of particles is very large and that there are no two-particle correlations In other words the probability of finding particle 1 at w1

and particle 2 at w2 is simply given by the product f(w1 t)f(w2 t)d6w1 d6w2

(BT87 Sect 83) While the first approximation is certainly valid in many as-trophysical situations such as galaxies and globular clusters (but see commentsbelow about multi-component systems) the second is violated by two-bodyeffects such as mutual deflections or the existence of small bound sub-groupsin particular binaries In fact as long as they do not interact closely withother objects and are themselves numerous enough binaries can in principlebe treated as just a special component for which a particle is really a bi-nary Two-particle effects such as deflection due to close encounters are calledcollisional effects and the FokkerndashPlanck treatment described below is anapproximate but manageable way to take them into account

The Boltzmann equation is valid whether f is interpreted as a numbermass luminosity or probability density The distribution function f does notneed to represent a system of objects with identical physical properties (stel-lar masses radii etc) but may be used globally for a mixed populationAs long as all sub-populations share the same f0 or if we are not interestedin distinguishing between them and the system is collisionless a unique fis enough to describe the system and its evolution If there are different sub-populations with initially distinct distribution functions (as would be the casefor a globular cluster with primordial mass segregation) each population (in-dex α) can be assigned its own DF fα In the absence of collisional termsthe only coupling between the evolution of the various fα is through the factthat they move in the same global potential Φ to which each componentcontributes unless it is treated as a mass-less tracer Specifically Φ is ob-tained from the fαrsquos and a possible external potential Φext through the Poissonequation

Φ(x) = Φself + Φext with nabla2Φself = 4πGNcompsum

α=1

Mα

int

d3v fα(xv)︸︷︷︸

ρα

(45)

where Ncomp is the number of components and Mα the total mass in com-ponent α (with the normalisation

intd3v d3x fα = 1) In the following we will

generally assume a fully self-gravitating system Φ(x) = Φself Because the Boltzmann equation simply states conservation of the phase-

space density along physical trajectories it keeps the same form if anothercoordinate system is used instead of the Cartesian (x y z) as long as f stillrepresents the number density per unit volume of the (x y z vx vy vz) phase-space

423 Collision Terms

When particles are subject to forces other than those produced by the smoothΦ the convective derivative of f does not vanish anymore In particular ina real self-gravitating N -particle system the potential cannot be smooth onsmall scales Instead it exhibits some graininess ie short-term small-scalefluctuations Φreal = Φ + ΔΦgrainy Here I call relaxation the effects of thesefluctuations on the evolution of the system described by f Schematically theyare due to the fact that a given particle does not see the rest of the system asa smooth mass distribution but as a collection of point-masses Relaxationaleffects also known (somewhat confusingly) as collisional effects can there-fore be seen as particles influencing each other individually as opposed tocollectively To allow for these effects a right-hand collision term Γ has to beintroduced into the Boltzmann equation

Dtf = Γ [f ] (46)

We now develop an expression for Γ Let Ψ(wΔw)d6(Δw)dt be the probabil-ity that a particle at the phase-space position w is perturbed (through forcesnot derived from Φ) to w+Δw during dt In general Ψ is also a function of tbut I drop this dependence here to simplify notation Stars are scattered outof an element of phase space around w at a rate

Γminus = minusf(w)int

d6(Δw)Ψ(wΔw) (47)

while stars from other phase-space positions (wminusΔw) are scattering into thiselement at a rate

Γ+ =int

d6(Δw)f(w minus Δw)Ψ(w minus ΔwΔw) (48)

The collision term is thus Γ = Γ+ + Γminus and the Boltzmann equation withsuch a collision term is called the master equation

43 FokkerndashPlanck Equation

431 FokkerndashPlanck Equation in Position-Velocity Space

Theoretically the master equation is of very general applicability because veryfew simplifying assumptions have been made so far Unfortunately it is of lit-tle practical use unless some explicit expression for the transition probabilityΨ is known The FokkerndashPlanck treatment is based on the assumption that Ψis sufficiently smooth and that typical changes Δw are small We can then de-velop Ψ and f around w in a Taylor series to second order in Δw Specificallyin the term Γ+ we write

102 M Freitag

f(w minus Δw)Ψ(w minus ΔwΔw) = f(w)Ψ(wΔw) minus6sum

i=1

Δwipart

partwi[Ψ(wΔw)f(w)]

+12

6sum

ij=1

ΔwiΔwjpart2

partwipartwj[Ψ(wΔw)f(w)] + O((Δw)3)

(49)

Defining the diffusion coefficients (DCs)

〈Δwi〉 equivint

d6(Δw)ΔwiΨ(wΔw)

〈ΔwiΔwj〉 equivint

d6(Δw)ΔwiΔwjΨ(wΔw)(410)

and plugging the development (49) into the collision term of the master equa-tion we obtain the general FokkerndashPlanck (FP) equation

Dtf = minus6sum

i=1

part

partwi[f(w)〈Δwi〉] +

12

6sum

ij=1

part2

partwipartwj[f(w)〈ΔwiΔwj〉] (411)

Here 〈Δwi〉 is the mean change in wi per unit time due to collisional effectsThese diffusion coefficients are generally functions of w and t but I have notwritten these dependencies explicitly

Now in the case of stellar dynamics we identify the collisional changesΔw with the effect of Keplerian hyperbolic uncorrelated two-body encountersand assume that they occur instantaneously ie on a time-scale much shorterthan the dynamical time-scale tdyn equiv R

32cl (GMcl)minus12 where Mcl is the total

mass of the system and Rcl is some typical length scale such as the half-massradius In this local approximation we neglect the change in position and onlyconsider changes in velocity This means that Ψ(wΔw) = 0 if Δx = 0 andthe FokkerndashPlanck equation reads

Dtf = minus3sum

i=1

part

partvi[f(xv)〈Δvi〉] +

12

3sum

ij=1

part2

partvipartvj[f(xv)〈ΔviΔvj〉] (412)

432 Diffusion Coefficients and Approximations for Relaxation

Let us sketch the computation of the velocity diffusion coefficients In practicewe do not need to compute the transition probability Ψ Instead we use thefact that for instance 〈Δvi〉 is the mean rate of change of the component iof the velocity of a given particle (called the test particle) as it is perturbedby all other particles (the field particles) To carry out the computationswe have to adopt the following set of approximations usually referred to asldquoChandrasekhar theory of relaxationrdquo (Chandrasekhar 1943 1960 See forinstance Henon 1973 Saslaw 1985 Spitzer 1987 Binney amp Tremaine 1987Heggie amp Hut 2003)

1 Local approximation The collisional perturbations to the motion of thetest particle are assumed to take place on a scale much smaller than thesize of its orbit Formally this holds if perturbations from distant starswith a long time-scale are negligible

2 Small perturbations approximation We assume that on time-scales of or-der tdyn (or shorter) the ldquocollisionsrdquo produce only a small change in theorbital parameters of a particle for the diffusion coefficients this trans-lates into tdyn〈Δvi〉 v tdyn〈ΔviΔvj〉 v2 This is an extension of theFP approximation which will make it possible to average the FP equationover the orbit of stars Most importantly for the time being it justifiesthe assumption that perturbations are two-body effects only and that theyadd linearly In other words to this level of approximation the combinedeffect of two field particles on a test particle are the same as the sumof the effects of each taken independently In particular the interactionbetween both field particles can be neglected Hence we are only con-sidering the so-called two-body relaxation This simplification only holdsif perturbations from very close stars (leading to large changes in v) arenegligible

3 Homogeneity approximation This is sometimes considered part of the lo-cal approximation We assume that the cumulative effects of the pertur-bations on the test object are as if the properties of the field particles(density velocity distribution) were the same in the whole system andequal to what they are in the vicinity of the test object In other wordsthe local conditions are representative of the global ones This arguablylooks like an unjustified assumption given how heterogeneous stellar sys-tems are (for instance the density in a globular cluster or galactic nucleusdecreases by many orders of magnitude from the centre to the half-massradius) and the long-range unshielded nature of the gravitational forceWe will see as we proceed why it may be a reasonable simplification butwe note that it can only work if distant perturbations do not dominate

To sum up the standard theory of relaxation is based on the assumptionsthat relaxation can be reduced to the cumulative effects of a large number ofuncorrelated two-body encounters that can be treated like (local) Kepleriansmall-angle hyperbolic velocity deflections due to objects with a density andvelocity distribution identical to the local ones

All these approximations are shared by other explicitly statistical methodsused to follow the long-term evolution of stellar clusters such as the Monte-Carlo scheme (see Chap 5) and the gaseous model (Bettwieser amp Spurzem1986 Louis amp Spurzem 1991 Giersz amp Spurzem 1994 Spurzem amp Takahashi1995 Amaro-Seoane et al 2004) but some approximations can be improvedon In particular large velocity changes (due to close encounters) can beincluded (Goodman 1983a Freitag et al 2006a)

To compute the diffusion coefficients we start by looking at the hyperbolicKeplerian encounter between the test particle with velocity v and mass m and

104 M Freitag

a field particle with velocity vf and mass mf We only consider field particles ofa given mass possibly different from m Standard numerical methods basedon the FP equation require that the mass spectrum is discretised Hencewe assume there are Nf particles of mass mf described by the distributionfunction ff now with the normalisation

intd3xd3vff = Nf

Using the local approximation we can assume that the encounter takesplace in a vacuum In other words the orbits are straight lines at large sep-aration (ldquoinfinityrdquo) The relative velocity at infinity is vrel = v minus vf and thevelocity of the centre-of-mass (CM) of the pair vcm = μv + (1 minus μ)vf withμ = m(m + mf) If b is the impact parameter the effect of the encounter issimply to rotate the relative velocity by an angle

tan(θ

2

)

=b0b

with b0 =G (m+mf)

v2rel

(413)

The value b0 is the impact parameter leading to a deflection angle π2 (inthe CM frame) We decompose the change of velocity Δv into componentsparallel and perpendicular to the initial relative velocity vrel

Δvperp = 2(1minus μ)vrelb

b0

(

1 +b2

b20

)minus1

Δv = 2(1minus μ)vrel

(

1 +b2

b20

)minus1

(414)

We then transform from the reference frame aligned with vrel (dependent onvf) to the external frame to get the Δvirsquos The next step is to average overall (equally probable) possible orientations of the impact parameter vectoraround the direction of vrel This gives values of 〈Δvi〉 and 〈ΔviΔvj〉 forfixed vf and b Now we sum the effects of all the encounters with field starshaving this velocity The number density of such objects is ffd3vf (consideredindependent of the position owing to the homogeneity approximation) andthe rate of encounters with an impact parameter between b and b + db is2πbdbvrelffd3vf We have to integrate over all possible impact parametersThis involves the integrals

int bmax

0

Δvbdb = vrel(1 minus μ)b20 ln(1 + Λ2)

int bmax

0

(Δv)2bdb = 2v2rel(1 minus μ)2b20

(

1 minus 11 + Λ2

)

int bmax

0

(Δvperp)2bdb = 2v2rel(1 minus μ)2b20

(

ln(1 + Λ2) minus 1 +1

1 + Λ2

)

(415)

In these relations Λ = bmaxb0 where bmax is the ill-defined maximum impactparameter For a system that is not too centrally concentrated we can setb = Rcl In most cases Λ 1 so the integrals can be approximated by

int bmax

0

Δvbdb 2vrel(1 minus μ)b20 lnΛ

int bmax

0

(Δv)2bdb 0

int bmax

0

(Δvperp)2bdb 4v2rel(1 minus μ)2b20 lnΛ

(416)

Hence the cut-off bmax only enters the computation of the diffusion coefficientsthrough the multiplicative Coulomb logarithm lnΛ Due to the very weaklogarithmic dependency we can replace m and mf in b0 by the mean valueMclNlowast and vrel by the 1D velocity dispersion σv measured for example atthe half-mass radius unless σv is a very steep function of the position suchas around a massive black hole Further for a self-gravitating system in virialequilibrium σ2

v asymp GMclRcl so that Λ must be of order Nlowast Putting Λ = γcNlowastdirect N -body experiments indicate that γc asymp 01 for single-mass systemsand γc asymp 001 (with considerable uncertainty) if objects have a realistic massspectrum (See Henon 1975 for theoretical estimates and Giersz amp Heggie 19941996 amongst others for the determinations based on N -body simulations)

Although the above integrals are carried out from b = 0 remember that theFP approximation requires small changes in v This suggests that encounterswith b smaller than a few b0 (causing deflection angles not small comparedto π2) cannot be taken into account But truncating the integrations atbmin = a few b0 would just bring in terms smaller than those in (416) by afactor lnΛ This is reflected by the fact that the typical time-scale for anencounter within kb0 with k some numerical coefficient is

tla =[

nσvπ(kb0)2(

1 +2Gmkb0σv

)]minus1

asymp(nσvπ(kb0)2

)minus1 asymp σ3v

k G2m2n (417)

where n is the number density σv the velocity dispersion and m the (mean)mass of a particle For k asymp 1 this large-angle deflection time-scale is of orderlnΛ longer than the relaxation time (see (424)) However from these consid-erations it does not follow that large-angle deflection cannot play an impor-tant role in some circumstances while the standard two-body relaxation bydefinition leads to gradual changes in orbital properties a single large-angleencounter causes sudden orbit modifications which may have very differentconsequences This may produce ejections or lead to strong interactions be-tween stars and a central massive black hole in a galactic nucleus (Henon1960 Lin amp Tremaine 1980 Freitag et al 2006a See also Chap 5)

The contribution to the relaxation of encounters with b between b1 andb2 with b2 gt b1 b0 is proportional to ln(b1b2) This explains why thestructure of the stellar system at large distances from the test particle haslittle importance in practice The average inter-particle distance is

d equiv nminus13 =(m

ρ

)13

asymp(mR3

cl

Mcl

)13

= Nlowastminus13Rcl (418)

106 M Freitag

while b0 asymp Nlowastminus1Rcl So somewhat surprisingly about two thirds of the contri-

bution to two-body relaxation come from encounters with impact parameterssmaller than d This is why the homogeneity approximation is a good one

Carrying out the computation of the diffusion coefficients using (416) wearrive at

〈Δvi〉 = 4π lnΛG2mf(m+mf)parth(v)partvi

〈ΔviΔvj〉 = 4π lnΛG2m2f

part2g(v)partvipartvj

(419)

where h(v) and g(v) are the Rosenbluth potentials (Rosenbluth et al 1957)

h(v) =int

d3uff(u) |v minus u|minus1 and g(v) =int

d3uff(u) |v minus u| (420)

Recall that all these quantities have an implicit x-dependenceIf the velocity distribution is isotropic we can go further in the computa-

tion of the diffusion coefficients for the velocity We find (eg Spitzer 1987)

〈Δv〉 = minus4πλm2f

(

1 +m

mf

)

Elt2 (V )

〈Δvperp〉 = 0

〈(Δv)2〉 =8π3λm2

f v(Elt4 (v) + Egt

1 (v))

〈(Δvperp)2〉 =8π3λm2

f v(3Elt4 (v) minus Elt

4 (v) + 2Egt1 (v))

〈ΔvΔvperp〉 = 0

(421)

where λ equiv 4πG2 lnΛ

Eltn (v) =

int v

u=0

(u

v

)n

ff(u)du and Egtn (v) =

int infin

u=v

(u

v

)n

ff(u)du (422)

We see that the mass of the test particle m only appears in the coefficient〈Δv〉 for dynamical friction From this the diffusion coefficients for the energycan be computed using ΔE = vΔv + 1

2 (Δvperp)2 + 12 (Δv)2 which gives

〈ΔE〉 = 4πλm2f v

(

Egt1 (v) minus m

mfElt

2 (v))

〈(ΔE)2〉 =8π3λm2

f v3(Elt

4 (v) + Egt1 (v)

)

(423)

We can write Egtltn = ξgtlt

n nσminus3v where ξgtlt

n are dimensionless order-of-unity (and position-dependent) numbers n is the local number density offield stars and σv their local 1D velocity dispersion The time-scale trlx over

which the direction of the velocity of a typical star (with v = v equiv 312σv) haschanged completely due to relaxation can be estimated using (423) and thedefinition 〈(Δvperp)2〉vtrlx equiv σ2

v We find tminus1rlx asymp lnΛG2m2

f nσminus3v A conventional

definition of the local relaxation time is obtained by assuming that the velocitydistribution is isotropic and Maxwellian and using the mean stellar mass m(Spitzer 1987)

trlx equiv 0339σ3

v

lnΛG2m2n (424)

In the case of a system with objects of different masses the relaxational effectof a species α is proportional to nαm

2α rather than its density (eg Perets et al

2007) On the other hand dynamical friction corresponding to the secondnegative term for 〈ΔE〉 (see (423)) has a time-scale proportional to ρ = mnthe total mass density of the field irrespective of the individual masses of thestars (for more on dynamical friction see Chap 7)

This is as far as we can go without further restriction on the distributionfunction ff If there is a single species of particles ff = f and the FP equationconsisting of (412) with the above diffusion coefficients (419) together withthe Poisson equation determine the evolution of the DF in a self-containedway Unfortunately the FP equation is a very intricate integro-differentialequation which at this point cannot be solved in whole generality

Furthermore realistic stellar systems are composed of objects with a rangeof properties (in particular masses) We can assume that there is a discreteset of populations orbiting in their common total potential and influencingeach other through two-body relaxation Each component k is described byDF fk which follows an FP equation but the diffusion coefficients are now asum of contributions from each component

〈Δvi〉k = 4π lnΛG2

timesNcompsum

l=1

[

ml(mk +ml)part

partvi

(int

d3ufl(u) |v minus u|minus1

)]

(425)

44 Orbit-Averaged FokkerndashPlanck Equation

441 General Considerations

To go further and obtain more easily usable versions of the FP equation weneed to restrict ourselves to stellar systems that are spherically symmetric inall their properties1 The use of the FP equation to study the structure and

1This does not imply that the velocity distribution is isotropic meaning thatit is spherically symmetric in velocity space but that the local velocity distribu-tion depends only on the moduli of the components of the velocity parallel andperpendicular to the radius-vector

108 M Freitag

evolution of stellar clusters was pioneered by Henon (1961) who derived theFP equation for an isotropic (but multi-mass) cluster and found an analyt-ical self-similar solution for the single-mass case assuming the existence ofa central energy source The first numerical codes producing general time-dependent solutions were written by Cohn (1979 1980) and to this daymost of the work in this field is based on the formalism and numerical meth-ods developed by this author (but see Takahashi 1995 and references thereinfor a finite-element scheme to solve the FP equation based on a variationalprinciple)

The FP equation can also be used for systems with axial symmetry suchas globular clusters or galactic nuclei with global rotation but we will nottreat this approach here (see Goodman 1983b Einsel 1996 Einsel amp Spurzem1999 Kim et al 2002 2004 Fiestas 2006 Fiestas et al 2006 Kim et al 2008for this original line of research under active development)

We also assume that the stellar system is in (quasi-)dynamical equilibriumIn other words it evolves very little over dynamical timescales

∣∣∣ff

∣∣∣ tdyn

If evolution is only due to two-body relaxation and the system is fully self-gravitating this assumption holds provided Nlowast is sufficiently large because∣∣∣ff

∣∣∣ asymp trlx asymp Nlowast(lnΛ)minus1tdyn with lnΛ = ln(γcNlowast) asymp 5minus 15 For single-mass

systems with Nlowast 103 the distinction between dynamical and relaxationaleffects (or between the smooth and grainy parts of the potential) becomesblurred When stars have a broad mass spectrum a larger number of stars isrequired for a clear distinction between dynamical and relaxational regimes

From Jeansrsquo theorem (Jeans 1915 Merritt 1999) for a spherical system indynamical equilibrium the DF f can depend on the phase-space coordinates(xv) only through the (specific) orbital energy E and modulus of the angularmomentum J

f(xv) = F (E(xv) J(xv)) with E = φ(r) +12v2 J = r vt (426)

where r = |x| v = |v| in a system of reference centred on the cluster centre2

φ is the spherically symmetric smooth gravitational potential so that Φ(x) =φ(r) and vt is the modulus of the component of the velocity perpendicular tothe radius-vector x

442 Isotropic Spherical Cluster

We first consider the simpler case of a cluster with isotropic velocity dispersionwhere F depends on E only We also assume only one component LetN(E)dEbe the number of stars with energy between E and E+dE The transformationfrom F to N is found by integrating over the phase-space accessible to orbits

2I use the word ldquoclusterrdquo to designate all (spherically) symmetric stellar systemsincluding galactic nuclei

with energy between E and E + δE and then letting δE be an infinitesimalδE rarr dE

N(E)δE =int

[EE+δE]

d3xd3vF (E) = 16π2

int

r

dr r2[int

v

dv v2F (E)]

(427)

We bring F (E) out of the integrals because it is nearly constant in the in-tegration domain (by definition) We first realise the v-integration at fixedr which runs from v =

radic2(E minus φ(r)) to v + δv with δv δEv giving

intvdv v2

radic2(E minus φ(r))δE Finally remains the integration over r which

runs from 0 to rmax(E) defined such that φ(rmax) = E We neglect the smallpart of the integration domain with r between rmax(E) and rmax(E + δE)because its contribution is of higher order in δE Once we replace δE by dEwe find

N(E) = 16π2p(E)F (E) (428)

withp(E) =

int rmax

0

r2v dr =int rmax

0

r2radic

2(E minus φ(r))dr (429)

Note that the quantity p(E) is proportional to the radial orbital period aver-aged in J space (isotropised orbital period)

p(E) =12

int J2c (E)

0

d(J2)Porb(E J) with Porb(E J) = 2int rmax

rmin

drvr (430)

where Jc(E) is the angular momentum of a circular orbit of energy EWe could transform the FP equation in (xv)-space (412) into an equation

for the rate of change of N(E) but it is much simpler to start over fromscratch The collisional term of an FP equation for N(E) simply reads

dNdt

∣∣∣∣coll

= minus part

partE[ΔEN(E)] +

12part2

partE2

[(ΔE)2N(E)

] (431)

Here the computation of the diffusion coefficients involve averaging over thevolume of space accessible to a particle of energy E reflecting the transfor-mation from F (E) to N(E) (428) and (429)

ΔE = p(E)minus1

int rmax

0

r2v〈ΔE〉dr (432)

where 〈ΔE〉 is the local diffusion coefficient for the kinetic energy In otherwords the mean rate of change of 1

2v2 for a particle at position r with velocity

v =radic

2(E minus φ(r))The smooth potential φ may change slowly as a result of the relaxational

evolution of the cluster itself or because of an external influence In any casethis will induce a change in the energy not accounted for by the collisional

110 M Freitag

term (431) So if we write DtN(E) for the ldquoLagrangianrdquo rate of change ofdensity in energy space following the φ-induced change in E we obtain theright-hand side of the FP for N(E)

DtN(E) =partN

partt+partN

partE

dEdt

∣∣∣∣φ

=dNdt

∣∣∣∣coll

(433)

where dEdt|φ is the change in energy due to the evolution of the potentialIt can be shown that it is equal to the phase-space averaged value of partφpartt

dEdt

∣∣∣∣φ

= p(E)minus1

int rmax

0

partφ(r)partt

r2vdr (434)

We see that the FP equation for N(E) as well as its generalisation to theanisotropic case (see Sect 443) are orbit-averaged Again the condition forthis averaging to be valid is that the system evolves only very little over onedynamical time staying close to dynamical equilibrium

To solve numerically the FP equation it is usual to write it in a flux-conservation form

DtN(E) = minuspartFE

partEwith FE = mDEF minusDEE

partF

partE (435)

Using (423) it can be shown that the flux coefficients are

DE =16π3λmf

int E

φ(0)

dEprimep(Eprime)Ff(Eprime)

DEE =16π3λm2f

[

q(E)int 0

E

dEprimeFf(Eprime) +int E

φ(0)

dEprimeq(Eprime)Ff(Eprime)

]

(436)

where

q(E) =int E

φ(0)

dEprimep(Eprime) =13

int rmax

0

r2v3 dr (437)

Here q(E) is the volume of phase-space accessible to particles with energieslower than E and p(E) is the area of the hypersurface bounding this volumethat is p(E) = partqpartE (Goodman 1983a) q(E) is also proportional to theisotropised radial action

q(E) =14

int J2c (E)

0

d(J2)Q(E J) with Q(E J) = 2int rmax

rmin

dr vr (438)

We have used an index ldquofrdquo for ldquofieldrdquo to distinguish the mass and DF of thepopulation we follow (test stars) from the ldquofieldrdquo objects This distinction doesnot apply to a single-component system but makes it very easy to generaliseto a multi-component situation by summing over components to get the totalflux coefficient

DE =Ncompsum

l=1

DEl DEE =Ncompsum

l=1

DEEl (439)

where the flux coefficient for component l can be written by replacing thesubscript ldquofrdquo by ldquolrdquo in (436) (eg Murphy amp Cohn 1988)

We now explain schematically how the FP equation is used numerically tofollow the evolution of star clusters A more detailed description can be foundin for example Chernoff amp Weinberg (1990) In the most common schemepioneered by Cohn (1980) two types of steps are realised in alternation

1 Diffusion step The change in the distribution function F for a discrete timestep Δt is computed by use of the FP equation assuming the potential φis fixed setting DtN = partN

partt = partNpartt

∣∣coll

The FP equation written as a flux-conserving equation is discretised on an energy grid The flux coefficientsare computed using the DF(s) of the previous step this makes the equationslinear in the values of F on the grid points The finite-differentiation schemeis the implicit Chang amp Cooper (1970) algorithm which is first-order intime and energy

2 Poisson step Now the change of potential resulting from the modification inF is computed and F is modified to account for the term dEdt|φ assumingDtN = partN

partt + partNpartE

dEdt

∣∣φ

= 0 This is done implicitly by using the fact thatas long as the change in φ over Δt is very small the actions of each orbitare adiabatic invariants Hence during the Poisson step the distributionfunction expressed as a function of the actions does not change Usingthe isotropised radial action q(E) defined above F (q)dq = F (E)p(E)dEwith F (q) = F (E(q)) In other words the modified F (E) is obtained byrecomputing the relation q(E) in the modified potential In practice aniterative scheme is used to compute the modified potential determinedimplicitly by the modified DF through the relation

φ(r) = minus4πG[1r

int r

0

dss2ρ(s) +int infin

r

dssρ(s)]

(440)

with

ρ(r) = 4πmint Emax

φ(r)

dEradic

2(E minus φ(r))F (E) (441)

for one component The iteration is started with the values of φ ρ etccomputed before the previous diffusion step

443 Anisotropic Spherical Cluster

The anisotropic FP treatment was already used to study some aspects ofthe structure of globular clusters by Spitzer amp Shapiro (1972) This typeof approach was then applied to the distribution of stars around a mas-sive black hole (assuming φ = minusGMBHr where MBH is the mass of the

112 M Freitag

black hole) by Lightman amp Shapiro (1977) and Cohn amp Kulsrud (1978)Although the first self-consistent FP simulations by Cohn (1979) made useof an anisotropic code further work on such models was relatively limitedin comparison to the isotropic case because the Chang amp Cooper (1970)discretisation scheme which proved so useful for getting good energy con-servation when the DF depended only on E (and t) has no exact equiva-lent for the case of a 2D (E J) dependence Also in most circumstancesit seems that forcing isotropy does not affect the results much and allowsa substantial reduction in the computational burden Cohn (1985) first pre-sented results of anisotropic FP models based on an extension of the ChangndashCooper scheme Since then Takahashi (1995 1996 1997) and Drukier et al(1999) have developed FP codes for spherical clusters with anisotropic velocitydistributions

Let F (E(xv) J(xv))d3xd3v be the number of stars with position withina volume d3x around x and velocity within d3v around v Because of sphericalsymmetry we can write d3x = 4πr2dr and d3v = 4πvtdvtdvr We note thatF (E J) = 0 if J gt Jc(E) Let N(E J)dE dJ be the number of stars withenergy between E and E + dE and angular momentum between J and J +dJ To convert from F (E J) to N(E J) we follow a star with energy Eand angular momentum J on its orbit and integrate the volume of phase-space along the way We use the distance from the centre r as integrationvariable

N(E J)dE dJ = 4πint rmax(EJ)

rmin(EJ)

r2drVr(E J)dE dJ (442)

Here Vr(E J)dE dJ denotes the (infinitesimal) volume in v-space with energybetween E and E + dE and angular momentum between J and J + dJ for afixed r We have

Vr(E J)dE dJ = 4πvtdvtdvr = 4πvt

∥∥∥∥

partEpartvt

partEpartvr

partJpartvt

partJpartvr

∥∥∥∥

minus1

dE dJ = 4πvt

rvrdE dJ

(443)which leads to

N(E J) = 8πPorb(E J)J F (E J) (444)

In numerical applications it is convenient to use R equiv (JJc(E))2 as a variableinstead of J Then the density of particles per unit E and R is

N(ER) = 4πJc(E)2Porb(E J)F (E J) (445)

The FP equation for N(ER) in its flux-conserving form is a direct extensionof the isotropic one

DtN(ER) = minuspartFE

partEminus partFR

partR (446)

with

FE = mDEF minusDEEpartF

partEminusDER

partF

partR

FR = mDRF minusDRRpartF

partRminusDER

partF

partE

(447)

The expression for the flux coefficients are significantly longer than in theisotropic case they are given by Cohn (1979) for single-mass clusters and byTakahashi (1997) for the multi-mass case3 To my knowledge in all numericalsolutions of the anisotropic FP equation for stellar systems an isotropised DFis used in the computation of the diffusion and flux coefficients For instancefor DEE we use

DEE =32π3

3λm2

f

int rmax

rmin

drvr

[v2

int 0

E

dEprimeFf(Eprime r)

+ vminus1

int E

φ(r)

dEprimeFf(Eprime r) (2(φ(r) minus Eprime))32]

(448)

Here Ff is the isotropised DF

Ff(Eprime r) =1

Jmax

int Jmax

0

dJFf(Eprime J) (449)

where Jmax(E r) =radic

2r2(φ(r) minus E) is the maximum (scaled) angular mo-mentum that an orbit of energy E can have if it goes through radius r andRmax = (JmaxJc)2

45 The FokkerndashPlanck Method in Use

To conclude this chapter I present a quick and partial overview of the workcarried out in cluster and galactic nucleus modelling using the direct resolutionof the FokkerndashPlanck equation My goal here is to provide pointers to theliterature that will allow the reader a deeper exploration of this rich field

451 Relaxational Evolution

The only physics included in the FokkerndashPlanck formalism presented here isself-gravity (through use of the Poisson equation) and two-body relaxationThis is enough to study the evolution of stellar clusters (with no or few pri-mordial binaries) up to core collapse The case of a single-mass cluster was

3Beware that in the work of these authors E is the binding energy and hastherefore the opposite sign as here with corresponding sign changes to be trackedin the computation of the coefficients and E-derivatives

114 M Freitag

initially computed by Cohn (1979 1980) for a Plummer model and revisitedseveral times since to explore a variety of initial cluster structures (Wiyantoet al 1985 Quinlan 1996) or to investigate the core-collapse physics in greaterdetail using more sophisticated FokkerndashPlanck codes (Takahashi 1995 Drukieret al 1999) Clusters with stars of different masses are much more realisticand have been considered by several authors (eg Merritt 1983 Inagaki ampWiyanto 1984 Inagaki amp Saslaw 1985 Murphy amp Cohn 1988 Chernoff ampWeinberg 1990 Lee 1995 Takahashi 1997 Kim et al 1998)

In a multi-mass cluster with a realistic mass spectrum the evolution tocore collapse is driven by mass segregation FP simulations are the ideal toolto investigate how this process operates in the limit of a very large numberof stars They are quick and their results are not affected by any significantnumerical noise in contrast to particle-based methods such as direct N -bodyor Monte-Carlo codes In Fig 41 I show the evolution of the Lagrangian radiifor a cluster with stellar mass spectrum dNlowastdMlowast prop Mminus235

lowast covering therange 02ndash10M The simulation was performed using an FP code providedby HM Lee (eg Lee et al 1991) using 12 mass components The initialstructure is a Plummer model In Fig 42 I plot the evolution of the centralldquotemperaturerdquo for several mass components We see that energy equipartitionis approached at the centre only amongst the most massive stars (roughly inthe range 3ndash10M)

Using an energy grid of 200 elements such an FP run requires only 1ndash2 minof CPU time on a laptop computer For an anisotropic code that solves the FPequation in (E J) space the simulation runs for about 4 days on a desktopcomputer (G Drukier 2007 personal communication) When the mass spec-trum is discretised into a larger number of mass components the computingtime increases approximately linearly with the number of components Thecorresponding direct N -body simulation with 256 000 particles took about 40days using special-purpose GRAPE hardware (H Baumgardt 2005 personalcommunication) and a Monte-Carlo simulation using 106 particles took aboutone week on a desktop computer (see Chap 5)

452 Models with Additional Physics

In order to simulate more realistic and complex systems the FokkerndashPlanckdescription of two-body relaxation has been complemented by approximatetreatment of a large variety of other physical effects Here I give a list of theseeffects with references to some pioneering or otherwise notable FP works wherethey have been considered

bull Central massive black hole Assuming a quasi-stationary regime and afixed Keplerian potential Lightman amp Shapiro (1977) and Cohn amp Kulsrud(1978) used the FP formalism to determine the distribution of stars arounda massive black hole (MBH) and the rate of stellar disruptions by theMBH The treatment of the loss cone developed for these works was later

MMMM

M

Fig 41 Core collapse of a Plummer cluster model with 02ndash10M Salpeter massfunction dNlowastdMlowast prop Nminus235

lowast Results of an isotropic FokkerndashPlanck code providedby H M Lee in solid lines are compared to a direct Nbody4 simulation with 256 000particles in dashes (H Baumgardt 2005 personal communication) To show masssegregation the evolution of Lagrangian radii for mass fractions of 1 and 50 per centis plotted for stars with masses within five different bins (corresponding to 5 of the12 discrete mass components used for the FP simulation) The length unit is theN -body scale (see Chap 1) The time unit is the initial half-mass relaxation time(Spitzer 1987) To convert the dynamical time units of the N -body simulation to arelaxation time a value of γc = 0045 was used for the Coulomb logarithm Comparewith Fig 54

introduced in self-consistent FP codes to study the evolution of globularclusters hosting an intermediate-mass black hole or of dense galactic nu-clei (Cohn 1985 David et al 1987a b Murphy et al 1991) Simplified FPcodes assuming in particular a fixed potential have been used to investi-gate the segregation of stellar-mass black holes around a MBH (Hopman ampAlexander 2006 Alexander 2007 OrsquoLeary et al 2008) and the formationof a central cusp of dark matter (Merritt et al 2007a) Very recently aFP code which includes the gravity of the stars self-consistently was usedto study the shrinkage of a binary MBH (Merritt et al 2007b) and theevolution of small nuclear clusters (Merritt 2008)

bull Stellar evolution Mass loss due to stellar evolution can be included byreducing the stellar mass represented by a mass component as a functionof time (eg Lee 1987a Chernoff amp Weinberg 1990 Quinlan amp Shapiro1990 Murphy et al 1991)

bull Collisions Some FP simulations have included the effects of collisions re-sulting in mergers (Lee 1987a Quinlan amp Shapiro 1989 1990) or (partial)

116 M Freitag

mm

σ

Fig 42 Evolution of the central temperatures during the core collapse of amulti-mass cluster model The temperature of component i is defined as Ti equiv32(mi〈m〉)σ2

i (0) where mi is the mass of a star of component i σi(0) the central1D velocity dispersion of that component in N -body units and 〈m〉 the mean stellarmass The data come from the same FokkerndashPlanck simulation as in Fig 41 Thesolid lines are the temperatures for the same five mass components (highest to lowestmass from top to bottom) The dashed line represents the mass-weighted averagecentral temperature

disruptions (David et al 1987a b Murphy et al 1991) The FP approachhas also been used to follow the evolution of galaxy clusters taking intoaccount galaxy mergers and mass stripping due to encounters betweengalaxies (Merritt 1983 1984 1985 Takahashi et al 2002) Collisions canonly be treated in an averaged and highly approximate fashion in the FPformalism because the mass and orbital energy of collision products ofany mass have to be transferred to the predefined mass components Fur-thermore the effects of collisions on stellar evolution cannot be includedin any detailed way Finally in the case of collisional runaway which isthe growth of one or a few stars to very high mass by successive mergersmass components have to be introduced that contain a very small num-ber of stars (sometimes less than one) Nevertheless comparisons with theMonte-Carlo algorithm (Chap 5) where collisions can be treated moreaccurately generally show good agreement as far as the overall effects ofcollisions are concerned (Freitag amp Benz 2002 Freitag et al 2006b)

bull Binary stars In a cluster containing no binaries initially some will formnear the centre during core collapse when the density reaches sufficientlyhigh values either through dissipative two-body effects or through close

three-body interactions (eg Aarseth 1971 Heggie amp Hut 2003) Bothkinds of mechanism have been included in FP codes (Statler et al 1987Lee et al 1991 Takahashi amp Inagaki 1991 Lee amp Ostriker 1993 amongstothers) In most cases the binary population is not followed explicitlyInstead the formation hardening and ejection of binaries are simply in-cluded as an effective central source of heating able to stop and reversecore collapse Binary heating can result in gravothermal core oscillations inthe post-collapse evolution (Cohn et al 1989 Takahashi amp Inagaki 1991Breeden et al 1994) A more detailed treatment of binaries would necessi-tate to represent them by at least one additional component (Lee 1987bGao et al 1991) Only limited physical realism can be achieved because itis not practical to extend the phase space to include the internal propertiesof the binaries which include mass ratio semi-major axis and eccentricityThis limitation explains why to the best of my knowledge primordial bi-naries have only been included into the FP framework by Gao et al (1991)Furthermore in the case of dynamically formed binaries only a few areexpected to be present in the core at any given time (Goodman 1984Baumgardt et al 2002) making a description based on the distributionfunction inadequate

bull Large-angle scatterings Goodman (1983a) included the effects of close two-body encounters in FP simulations and concluded that they do not affectappreciably the core collapse process

bull Evaporation Assuming the cluster is on a circular orbit around a sphericalgalaxy (or in the equatorial plane of an axially symmetrical galaxy) theevaporation of stars in the steady tidal field can be approximated in aspherical FP code by an outer boundary condition For an isotropic for-mulation the condition is F (Et) = 0 with Et = minusGMclR

minus1t and Rt is the

tidal truncation radius which can be identified with the distance betweenthe centre of the cluster and the Lagrange point L1 or L2 (eg Chernoffamp Weinberg 1990) A more accurate condition can be used in anisotropicmodels by setting the DF to zero for orbits with an apocentre distancelarger than Rt (Takahashi et al 1997) Delayed evaporation can be sim-ulated to account for the fact that a star can spend a significant amountof time in the cluster even when its orbital parameters would allow it toreach the Lagrange points (Lee amp Ostriker 1987 Takahashi amp PortegiesZwart 2000)

bull Gravitational shocking In general as it orbits its host galaxy a globu-lar cluster can experience strongly varying external gravitational stressesMurali amp Weinberg (1997a) and Gnedin et al (1999) have included so-called disc and bulge shocking in their FP simulations which allowedthem to study the evolution of whole globular cluster systems (Gnedin ampOstriker 1997 Murali amp Weinberg 1997b c) Thank to a new integrationscheme shocking has been studied in anisotropic FP models (Shin et al2008)

118 M Freitag

bull Gas dynamics (David et al 1987a b) coupled the FP algorithm with aspherical gas dynamical code to predict what amount of the gas releasedby stars through evolution and collisions is accreted by a central MBH inAGN models However gas motion is likely to be highly non-spherical andto vary on time-scales much shorter than those for evolution of the stellarcluster (eg Williams et al 1999 Cuadra et al 2005)

FP simulations including several of the above physical processes have beenused to interpret observations of a few specific globular clusters M 15(Grabhorn et al 1992 Dull et al 1997) M 71 (Drukier et al 1992) NGC 6397(Drukier 1993 1995) and NGC 6624 (Grabhorn et al 1992) In the futureit seems likely that particle-based methods will be used to produce detailedmodels of observed clusters (see Giersz amp Heggie 2003 2007 and Hurley et al2005 for pioneering examples) These codes can deal realistically with stel-lar populations that are rare or otherwise problematic to simulate with FPmethods such as primordial binaries blue stragglers or X-ray binaries How-ever because they are so much faster FP codes can be an invaluable toolto carry out extensive parameter-space exploration and determine the initialconditions and physical parameters most likely to fit the observational dataDirect N -body or Monte-Carlo simulations can then be used using these inputparameters to obtain more detailed models

Acknowledgement

I am indebted to Gordon Drukier and Hyung Mok Lee who provided invaluablehelp in the preparation of my FokkerndashPlanck lecture and took the time toread and comment on a draft of this chapter I also thank Hyung Mok Lee formaking available his FokkerndashPlanck code and helping me to use it and HolgerBaumgardt for providing unpublished N -body data My work is supported bythe STFC rolling grant to the IoA

References

Aarseth S J 1971 ApampSS 13 324 117Alexander T 2007 in Livio M Koekemoer A M eds 2007 STScI Spring Sympo-

sium Black Holes (astro-ph07080688) 115Amaro-Seoane P Freitag M Spurzem R 2004 MNRAS 352 655 103Baumgardt H Hut P Heggie D C 2002 MNRAS 336 1069 117Bettwieser E Spurzem R 1986 AampA 161 102 103Binney J Tremaine S 1987 Galactic Dynamics Princeton Univ Press Princeton

NJ 99 102Breeden J L Cohn H N Hut P 1994 ApJ 421 195 117Chandrasekhar S 1943 Rev Mod Phys 15 1 102Chandrasekhar S 1960 Principles of Stellar Dynamics Dover enlarged

edition 102

Chang J S Cooper G 1970 J Comp Phys 6 1 111 112Chernoff D F Weinberg M D 1990 ApJ 351 121 111 114 115 117Cohn H 1979 ApJ 234 1036 108 112 113 114Cohn H 1980 ApJ 242 765 108 111 114Cohn H 1985 in Goodman J Hut P eds Proc IAU Symp 113 Dynamics of

Star Clusters Reidel Dordrecht p 161 112 115Cohn H Hut P Wise M 1989 ApJ 342 814 117Cohn H Kulsrud R M 1978 ApJ 226 1087 112 114Cuadra J Nayakshin S Springel V Di Matteo T 2005 MNRAS 360 L55 118David L P Durisen R H Cohn H N 1987a ApJ 313 556 115 116 118David L P Durisen R H Cohn H N 1987b ApJ 316 505Drukier G A 1993 MNRAS 265 773 118Drukier G A 1995 100 347 118Drukier G A Cohn H N Lugger P M Yong H 1999 ApJ 518 233 112 114Drukier G A Fahlman G G Richer H B 1992 ApJ 386 106 118Dull J D Cohn H N Lugger P M Murphy B W Seitzer P O Callanan P J

Rutten R G M Charles P A 1997 ApJ 481 267 118Einsel C Spurzem R 1999 MNRAS 302 81 108Einsel M 1996 PhD thesis Christian-Albrechts-Universitat zu Kiel 108Fiestas J 2006 PhD thesis Heidelberg University 108Fiestas J Spurzem R Kim E 2006 MNRAS 373 677 108Freitag M Amaro-Seoane P Kalogera V 2006a ApJ 649 91 103 105Freitag M Benz W 2002 AampA 394 345 116Freitag M Rasio F A Baumgardt H 2006b MNRAS 368 121 116Gao B Goodman J Cohn H Murphy B 1991 ApJ 370 567 117Giersz M Heggie D C 1994 MNRAS 268 257 105Giersz M Heggie D C 1996 MNRAS 279 1037 105Giersz M Heggie D C 2003 MNRAS 339 486 118Giersz M Heggie D C 2007 in Vesperini E Giersz M Sills A eds Dynami-

cal Evolution of Dense Stellar Systems Proceedings of IAU Symposium No 246(astro-ph07110523) 118

Giersz M Spurzem R 1994 MNRAS 269 241 103Gnedin O Y Lee H M Ostriker J P 1999 ApJ 522 935 117Gnedin O Y Ostriker J P 1997 ApJ 474 223 117Goodman J 1983a ApJ 270 700 103 110 117Goodman J 1983b PhD thesis Princeton University 108Goodman J 1984 ApJ 280 298 117Goodman J Heggie D C Hut P 1993 ApJ 415 715 97Grabhorn R P Cohn H N Lugger P M Murphy B W 1992 ApJ 392 86 118Heggie D Hut P 2003 The Gravitational Million-Body Problem Cambridge Univ

Press Cambridge 102 117Hemsendorf M Merritt D 2002 ApJ 580 606 97Henon M 1960 Annales drsquoAstrophysique 23 668 105Henon M 1961 Annales drsquoAstrophysique 24 369 108Henon M 1973 in Martinet L Mayor M eds Lectures of the 3rd Advanced Course

of the Swiss Society for Astronomy and Astrophysics Obs de Geneve Genevep 183 102

Henon M 1975 in Hayli A ed Proc IAU Symp 69 Dynamics of Stellar SystemsReidel Dordrecht p 133 105

120 M Freitag

Hopman C Alexander T 2006 ApJ Lett 645 L133 115Hurley J R Pols O R Aarseth S J Tout C A 2005 MNRAS 363 293 118Inagaki S Saslaw W C 1985 ApJ 292 339 114Inagaki S Wiyanto P 1984 PASJ 36 391 114Jeans J H 1915 MNRAS 76 70 108Kim E Einsel C Lee H M Spurzem R Lee M G 2002 MNRAS

334 310 108Kim E Lee H M Spurzem R 2004 MNRAS 351 220 108Kim E Yoon I Lee H M Spurzem R 2008 MNRAS 383 2 108Kim S S Lee H M Goodman J 1998 ApJ 495 786 114Lee H M 1987a ApJ 319 801 115Lee H M 1987b ApJ 319 772 117Lee H M 1995 MNRAS 272 605 114Lee H M Fahlman G G Richer H B 1991 ApJ 366 455 114 117Lee H M Ostriker J P 1987 ApJ 322 123 117Lee H M Ostriker J P 1993 ApJ 409 617 117Lightman A P Shapiro S L 1977 ApJ 211 244 112 114Lin D N C Tremaine S 1980 ApJ 242 789 105Louis P D Spurzem R 1991 MNRAS 251 408 103Merritt D 1983 ApJ 264 24 114 116Merritt D 1984 ApJ 276 26 116Merritt D 1985 ApJ 289 18 116Merritt D 1999 PASP 111 129 108Merritt D 2008 preprint (astro-ph08023186)Merritt D Harfst S Bertone G 2007a Phys Rev D 75 043517 115Merritt D Mikkola S Szell A 2007b ApJ 671 53Murali C Weinberg M D 1997a MNRAS 288 749 117Murali C Weinberg M D 1997b MNRAS 291 717 117Murali C Weinberg M D 1997c MNRAS 288 767Murphy B W Cohn H N 1988 MNRAS 232 835 111 114Murphy B W Cohn H N Durisen R H 1991 ApJ 370 60 115 116Perets H B Hopman C Alexander T 2007 ApJ 656 709 107OrsquoLeary R M Kocsis B Loeb A 2008 preprint (astro-ph08072638)Quinlan G D 1996 New Astronomy 1 255 114Quinlan G D Shapiro S L 1989 ApJ 343 725 115Quinlan G D Shapiro S L 1990 ApJ 356 483 115Rosenbluth M N MacDonald W M Judd D L 1957 Physical Review 107 1 106Saslaw W C 1985 Gravitational Physics of Stellar and Galactic Systems Cam-

bridge Univ Press Cambridge 102Shin J Kim S S Takahashi K 2008 MNRAS 386 L67Spitzer L 1987 Dynamical evolution of globular clusters Princeton Univ Press

Princeton NJ 102 106 107 115Spitzer L J Shapiro S L 1972 ApJ 173 529 111Spurzem R Takahashi K 1995 MNRAS 272 772 103Statler T S Ostriker J P Cohn H N 1987 ApJ 316 626 117Takahashi K 1995 PASJ 47 561 108 112 114Takahashi K 1996 PASJ 48 691 112Takahashi K 1997 PASJ 49 547 112 113 114Takahashi K Inagaki S 1991 PASJ 43 589 98 117

Takahashi K Lee H M Inagaki S 1997 MNRAS 292 331 117Takahashi K Portegies Zwart S F 2000 ApJ 535 759 117Takahashi K Sensui T Funato Y Makino J 2002 PASJ 54 5 116Williams R J R Baker A C Perry J J 1999 MNRAS 310 913 118Wiyanto P Kato S Inagaki S 1985 PASJ 37 715 114

5

Monte-Carlo Models of Collisional StellarSystems

Marc Freitag

51 Introduction

In this chapter I describe a fast approximate particle-based algorithm tocompute the long-term evolution of stellar clusters and galactic nuclei Itrelies on the assumptions of spherical symmetry of the stellar system dynam-ical equilibrium and local diffusive two-body relaxation It allows for velocityanisotropy an arbitrary stellar mass spectrum stellar evolution a centralmassive object collision between stars binary processes and two-body en-counters leading to large deflection angles Using one to ten million particlesa run extending over several relaxation times takes a few days to a few weeksto compute on a single-CPU personal computer and the CPU time scalesas tCPU prop Np lnNp where Np is the number of particle used Because eachphysical process is implemented with its explicit scaling the number of starssimulated can be (much) larger than Np making it possible to simulate galac-tic nuclei with (in particular) the correct rate of relaxation

The Monte-Carlo (MC) numerical scheme is intermediate both in termsof realism and computing time between FokkerndashPlanck or gas approaches anddirect N -body codes The former are very fast but based on a significantly ide-alised description of the stellar system the latter treat (Newtonian) gravity inan essentially assumption-free way but are extremely demanding in terms ofcomputing time (Binney amp Tremaine 1987 Sills et al 2003) The MC schemewas first introduced by Henon to follow the relaxational evolution of globularclusters (Henon 1971ab Henon 1973a Henon 1975) To my knowledge thereexist three independent codes based on Henonrsquos ideas in active developmentand use The first is the one written by M Giersz (Giersz 1998 2001 2006Giersz et al 2008) which implements many of the developments first intro-duced by Stodolkiewicz (1982 1986) Second is the code written by K Joshi(Joshi et al 2000 2001) and greatly improved and extended by A Gurkanand J Fregeau (see for instance Fregeau et al 2003 Gurkan et al 2004 2006Fregeau amp Rasio 2007) These codes have been applied to the study of globu-lar and young clusters Finally we developed a MC code specifically aimed at

Freitag M Monte-Carlo Models of Collisional Stellar Systems Lect Notes Phys 760

123ndash158 (2008)

124 M Freitag

the study of galactic nuclei containing a central massive black hole (Freitag ampBenz 2001c Freitag amp Benz 2002 Freitag et al 2006a Freitag et al 2006bc)The description of the method given here is based on this particular imple-mentation1

This chapter is organised as follows In Sect 52 the core principles andassumptions of the method are presented In Sect 53 I expose the innerworkings of the code in detail the basic algorithm which treats global self-gravity and two-body relaxation is the subject of Sect 531 while Sect 532covers the additional physical processes (collisions central object binariesstellar evolution etc) Finally in Sect 54 I show a few applications anddiscuss possible avenues for future developments of the method in the contextof research on star clusters (Sect 541) and on galactic nuclei (Sect 542)

52 Basic Principles

The MC code shares most of its underlying assumptions with the FokkerndashPlanck (FP) approach presented in Chap 4 Essentially Henonrsquos algorithmcan be seen as a particle-based method to solve the coupled FP and Pois-son equations for a stellar cluster using Monte-Carlo sampling to determinethe long-term effects of two-body relaxation An advantage of the MC ap-proach over FP integrations is that it can include a continuous stellar massspectrum and extra physical ingredients such as stellar evolution collisionsbinaries or a central massive black hole in a much more straightforward andrealistic way On the downside MC simulations require considerably morecomputing time Furthermore the MC results show numerical noise whilethose obtained with the FP codes are smooth and easier to analyse and ma-nipulate

The assumptions shared by both methods are the following

1 Dynamical equilibrium2 Spherical symmetry3 Diffusive relaxation4 Adequacy of representation with a one-particle distribution function

An isolated system is likely to attain dynamical equilibrium after an ini-tial phase of violent relaxation spanning a few dynamical times tdyn =radicR3

cl(GMcl) where Rcl is a characteristic length (such as the half-massradius) and Mcl the mass of the cluster The MC code developed by Spitzerand collaborators (Spitzer amp Hart 1971ab Spitzer amp Thuan 1972 Spitzer amp

1This code is available at httpwwwastcamacukresearchrepositoryfreitagMChtmGeneral information on the MC method and more references can be found onthe web pages created for the MODEST consortium (ldquoMOdeling DEnse STellarsystemsrdquo) at httpwwwmanybodyorgmodest (follow the link to the workinggroup on stellar-dynamics methods WG5)

5 Monte-Carlo Models 125

Shull 1975 Spitzer amp Mathieu 1980) allows for out-of-equilibrium situationsat the price of computing speed but the assumption of spherical symmetrystrongly limits the usefulness of this feature

In practice the strongest restriction is that of spherical symmetry Vio-lent relaxation generally leads to an equilibrium configuration with signifi-cant triaxiality (eg Aguilar amp Merritt 1990 Theis amp Spurzem 1999 Boily ampAthanassoula 2006) Although it is likely that two-body relaxation makes thesystem more symmetrical flattening owing to global rotation can persist overmany relaxation times (Einsel amp Spurzem 1999 Kim et al 2002 2004 Fiestaset al 2006) In galactic nuclei the interaction between the stars and a binarymassive black hole (eg Merritt amp Milosavljevic 2005) or a massive accre-tion disc (eg Subr et al 2004) cannot be studied accurately when sphericalsymmetry is assumed (see Sect 542)

The last two assumptions have been discussed in Chap 4 on FP methodsThey imply that correlations between particles beyond random two-bodyencounters are neglected but I stress that three- and four-body interactionsin the form of binary processes can be included in the MC approach with muchmore realism than permitted by the direct FP formalism (see Sect 532)

It should be noted at once that all these assumptions can only be validif the system under consideration contains a large number of stars In myexperience the MC approach is suitable if the number of particles Np satisfies

Np 3000mmax

〈m〉 (51)

where mmax and 〈m〉 are the maximum and mean stellar mass respectivelyIn Henonrsquos scheme the numerical realisation of the cluster is a set of spher-

ical shells with zero thickness each of which is given a mass M a radius Ra specific angular momentum J and a specific kinetic energy T These parti-cles can be interpreted as spherical layers of synchronised stars that share thesame stellar properties orbital parameters and orbital phase and experiencethe same processes (relaxation collision etc) at the same time

From the radii and masses of all particles the potential can be computedat any time or place and the orbital energies of all particles are straightfor-wardly deduced from their kinetic energies and positions Hence the set ofparticles can be regarded as a discretised representation of the distributionfunction (DF) f(xv) = F (E J) But whereas a functional or tabulated ex-pression of the DF (as implemented in direct FP methods) would require theintegration of the Poisson equation to yield the gravitational potential theMonte-Carlo realisation of the cluster provides it directly From this point ofview the Monte-Carlo method is closer to N -body philosophy than to directFP methods

The main difference between the MC code and a spherical 1D N -bodysimulation (eg Henon 1973b) is that the former does not explicitly followthe continuous orbital motion of particles which preserves E and J How-ever these orbital constants as well as other properties of the particles are

126 M Freitag

modified by collisional processes to be incorporated explicitly two-body relax-ation stellar collisions etc So the MC simulation proceeds through millionsto billions of steps each of them consisting of the selection of particles themodification of their properties to simulate the effects these physical processesand the selection of radial positions R on their new orbits

53 Detailed Implementation

531 Core Algorithm

This subsection is divided into four parts In the first I present the treatmentof relaxation and the overall structure of the code In the following partsI explain in detail some important aspects of the algorithm which are theselection of a pair of particles to evolve the representation of the gravitationalpotential and the determination of a new orbital position for updated particles

Two-Body Relaxation and General Organisation

The treatment of two-body relaxation is the backbone of Henon-type Monte-Carlo schemes It relies on the usual diffusive approximation developed byChandrasekhar and presented in Chap 4 I recall that the basic idea behindthe concept of relaxation is that the gravitational potential of a stellar systemcontaining a large number of bodies can be described as the sum of a dom-inating smooth contribution plus a small granular part that fluctuates oversmall scales and short times When only the smooth part is taken into accountthe DF of the cluster obeys the collisionless Boltzmann equation Howeverin the long run the fluctuating part makes E and J change slowly and theDF evolve The basic simplifying assumption underlying Chandrasekhar relax-ation theory is to treat the effects of the fluctuating part as the sum of multipleuncorrelated two-body hyperbolic gravitational encounters with small devia-tion angles Under these assumptions if a test star of mass m travels througha field of stars with homogeneous number density n which all have massmf and the same velocity after a time span δt its velocity in the referenceframe of the encounters will deviate from the initial direction by an angle θsuch that

〈θ〉δt = 0 and

langθ2rang

δt 8πn lnΛ

G2 (m+mf)2

v3rel

δt(52)

where vrel the relative velocity between the test star and the field stars andlnΛ ln(γcNlowast) for a self-gravitating cluster with the value of γc dependingon the mass spectrum (see Chap 4)

Henonrsquos method avoids the computational burden and some of the nec-essary simplifications connected with the numerical evaluation of diffusioncoefficients The repeated application of (52) to a given particle implicitlyamounts to a Monte-Carlo integration of the orbit-averaged diffusion coef-ficients provided the orbital positions and properties of field particles arecorrectly sampled Under the usual assumption that encounters are local thislatter constraint is obeyed if we take these properties to be those of the clos-est neighbouring particle Furthermore this allows us to actually modify thevelocities of both particles at a time each acting as a representative from thefield for the other Evolving particles in symmetrical pairs not only speeds upthe simulations by a factor 2 but also and more critically ensures strictconservation of energy

Therefore at the heart of the MC treatment of relaxation are super-encounters encounters between two neighbouring particles with a deflectionangle θSE devised to reproduce statistically the cumulative effects of the nu-merous physical deflections taking place in the real system over a time spanδt Using the indices 1 and 2 to designate the particles in a pair we see thatin order to reproduce the values of (52) for deflection angles correspondingto a time step δt we must set

θSE =π

2

radicδt

trlx 12

(53)

where

trlx 12 equiv π

32v3rel

lnΛG2 (m1 +m2)2n

(54)

is the pair relaxation time

With no other physical process than relaxation included a single step ina MC simulation consists of the following operations

1 Selection of a pair of adjacent particles to evolve This procedure alsodetermines the (local) value of the time step δt as explained below

2 Modification of the orbital properties (Ei and Ji) of the particles througha super-encounter This involves(a) estimation of the local density n entering trlx 12 in (54)(b) random orientation of the velocity vectors vi of the particles respecting

their angular momenta Ji = Ji and specific kinetic energy Ti = 12vi

2

(this sets the centre-of-mass [CM] and relative velocities vCM and vrelthe former defines the encounter CM frame while the latter allows θSE

to be determined through (53) and (54)(c) random orientation of the orbital plane in the CM frame around the

direction of the relative velocity (the angle θSE is known so computingthe post-encounter velocities in the CM frame is trivial) and

(d) transformation back to the cluster frame to obtain the modified Eprimei

and J primei

128 M Freitag

3 For each particle selection of a new position on the (EprimeiJ

primei)-orbit As a

particle is a spherical shell its position is simply its radius Ri This stepcomprises the update of the potential to take these new positions intoaccount

To compute the local density required in step 2a we build and maintaina radial Lagrangian grid the cells of which typically contain a few tens ofparticles each Frequent updates (each time a particle gets a new position R)and occasional rebuilds of the mesh introduce only a very slight computationaloverhead

Selection of a Pair of Particles and Determination of Time Step

For the sake of efficiency we wish to use time steps that reflect the largevariations of the relaxation time between the central and outer parts of astellar cluster The other constraint determining the selection procedure isthat particles in an interacting pair must have the same δt lest energy not beconserved2 But adjacent particles only form a pair momentarily and separateafter their interaction as each is attributed a new position This necessitatesthe use of local time steps ie δt should be a function of R alone instead ofbeing attached to particles

For the time steps to be sufficiently short we impose

δt(R) le fδttrlx(R) (55)

where trlx is a locally averaged relaxation time

trlx prop 〈v2〉 32

lnΛG2〈m〉2n (56)

and 0005 le fδt le 005 typically The time trlx is evaluated approximatelywith a sliding averaging procedure and tabulated from time to time to reflectthe slow evolution of the cluster

The members of a pair arrived at their present position at different timesbut have to leave it at the same time after a super-encounter Building onthe statistical nature of the scheme instead of trying to maintain a particleat radius R during exactly δt(R) we only require the expectation value forthe residence time at R to be δt(R) As explained by Henon (1973a) thisconstraint can be fulfilled if the probability for a pair at R to be selected isproportional to 1δt(R) This is realised in the following way

bull Because it would be difficult to define and use a selection probability Pselec

that is a function of the continuous variable R we define it to depend on2When collisions are included a shared δt also ensures that the probability for

particle i to collide with particle j equates the symmetrical quantity

the rank i of the pair (rank 1 designates the two particles that are closestto the centre rank 2 the second and third particles at increasing R and soon) For a given clusterrsquos state local relaxation times trlx are computed atthe radial position of every pair Rank-depending time steps are definedto obey inequality (55)

δt(i) le fδttrlx(R(i)) (57)

bull Normalised selection probabilities are computed by

Pselec(i) =δt

δt(i)with δt =

⎛

⎝Npminus1sum

j=1

1δt(j)

⎞

⎠

minus1

(58)

from which we derive a cumulative probability

Qselec(i) =isum

j=1

Pselec(j) (59)

bull At each evolution step another particle pair is randomly chosen accordingto Pselec To do this a random numberXrand is first generated with uniformprobability between 0 and 1 The pair rank is then determined by inversionof Qselec

i = Qminus1selec(Xrand) (510)

The binary tree (see Sect 531) is searched twice to find the id-numbersof the member particles the (momentary) ranks of which are i and i+ 1

bull The pair is evolved through a super-encounter as explained above for atime step δt(i)

bull After a large number of elementary steps δt(i) and Pselec(i) are re-computed to reflect the slight modification of the overall cluster structure

For the sake of efficiency we must choose for Qminus1selec a function that is quickly

evaluated while Pselec(j) must approximate 1trlx(R(i)) as closely as possibleto avoid unnecessarily long time steps A good compromise is to use a piecewiseconstant representation ie divide the cluster into some 50 radial slices anduse a constant Pselec in each This is illustrated in Fig 51 (with only 20 slicesfor clarity) Once the selection probabilities have been determined the valueδt relating them to the time step is set to δt = fδt max(Trel(i)Pselec(i)) so asto ensure that the constraint of (55) is satisfied everywhere

It must be stressed that the probabilities Pselec(i) and corresponding timesteps are computed in advance and are only updated (to reflect the evolutionof the structure) after each particle has been treated several times on averageOnce the pair of adjacent particles of rank i has been selected to be subject toa super-encounter the time step δt(i) is imposed and the encounter relaxation

130 M Freitag

Fig 51 Selection probabilities in a King W0 = 5 cluster model consisting of 10 000particles The inverse of the locally estimated relaxation time is compared to thepiecewise approximation used to set the probabilities in the MC code

time trlx 12 is determined by the particlesrsquo properties and the local density(54) This imposes the value of the deflection angle (53) In order to performa proper orbit averaging and sampling over the field particles θSE should besmall so that a given particle would have experienced a large number of super-encounters by the time its orbit has changed significantly Unfortunately thisis impossible to enforce strictly as the δt(i) values are based on an estimate ofthe typical local relaxation time while trlx 12 can happen to be much shorterUsing a sufficiently small value of fδt we can keep the fraction of encountersleading to large values of θSE to a low level

Representation of the Gravitational Potential

The smooth part of the potential of the cluster is simply approximated asthe sum of the contributions of the Np particles each of which is a sphericalinfinitely thin shell In other terms compared to the potential in a systemof Np point-masses we (implicitly) perform a complete smoothing over theangular variables Between particles of rank i and i+1 the (smooth) potentialfelt by a particle at radius R isin [Ri Ri+1] is simply

Φ(R) = minusAi

RminusBi with Ai =

iminus1sum

j=1

Mj and Bi =Npsum

j=i

Mj

Rj (511)

where Mj and Rj are the mass and radius of the particle of rank j Althoughwe do not smooth the density distribution in the radial direction tests showthat in practice this spherically symmetric potential does not introduce sig-nificant unwanted relaxation for Np 104 in simulations extending to an av-erage number of steps per particle of a few thousands (Henon 1971b Freitagamp Benz 2001c) However too small a time step parameter fδt can yield anartificially accelerated evolution owing to this numerical relaxation

At each step in the simulation two particles are selected undergo a super-encounter and are given new positions on their slightly modified orbits Toenforce exact energy conservation the Ai and Bi coefficients are updatedafter every such orbital displacement Doing so saves much trouble connectedwith a potential that lags behind the actual distribution of particlesrsquo radii (andmasses when stellar evolution or collisions are included) However performingpotential updates only after a large number of particle moves has advantagesof its own in particular the possibility of algorithm parallelisation (Joshi et al2000) but requires special measures to ensure satisfactory energy conservation(Stodolkiewicz 1982 Giersz 1998 Fregeau amp Rasio 2007)

The potential information is not represented by linear arrays (for the Ai

and Bi) but by a binary tree (Sedgewick 1988) This tree also contains rankinginformation It allows us to find a particle of a given rank compute the poten-tial at its position and update the potential data once the particle is movedto another radius in O(logNp) operations instead of O(Np) as would be thecase with simple arrays At any given time each particle is represented by anode in the tree Each node is connected to (at most) two sub-trees All thenodes in the left sub-tree of a given node correspond to particles with smallerradii and all the nodes in its right sub-tree to particles at larger radii Thespherical potential is represented by (floating-point) δAk and δBk coefficientsattached to nodes A third (integer) value δik allows the determination ofthe radial rank of any particle If we define LT k and RT k to be the sets ofnodes in the left and right sub-trees of node k these quantities are defined by

δik = 1 + number of nodes in LT k

δAk = Mk +sum

misinLT k

Mm and δBk =Mk

Rk+

sum

misinRT k

Mm

Rm

(512)

An example of binary tree is shown in Fig 52 After a large number ofspecified steps the binary tree is rebuilt from scratch to keep it well balanced

Selection of a New Orbital Position

In a spherical potential Φ(R) a star of specific orbital energy E and angularmomentum J spends during one complete radial oscillation a time dt =vminus1rad(R)dR in an infinitesimal interval of radius [RR + dR] with

132 M Freitag

Fig 52 Binary tree for a cluster of 50 particles The structure of the tree is shownafter many particles have been moved around since the tree was built The loweraxis shows the radius of each particle The tree keeps the particles sorted in radiusThe table on the right is the content of the three arrays used in the Fortran-77

code to implement the logical structure of the tree Arrays l son(k) and r son(k)

indicate the root nodes for the left and right sub-trees of node k Array father(k)

allows us to climb back to the root

v2rad = 2E minus 2Φ(R) minus J2

R2 (513)

Without knowledge of orbital phase the probability density of finding the starat R is thus

dPorb

dR=

2Porb

1vrad(R)

(514)

where

Porb = 2int Rapo

Rperi

dRvrad(R)

(515)

is the radial orbital periodSince dynamical equilibrium is assumed the knowledge of the explicit or-

bital motion R(t) is not necessary Instead once a particle is updated its posi-tion R is picked up at random but with the requirement of correct statisticalsampling This means that the fraction of time spent at R must follow (514)Let the sought-for probability of placing the particle at R isin [Rperi Rapo] befplac(R) equiv dPplacdR We have to compensate for the fact that if the particleis placed at R it will stay there for an average time δtPselec(R) The averageratio of times spent at two different radii R1 and R2 on the orbit is

langtstay(R1)tstay(R2)

rang

=fplac(R1)Pselec(R2)fplac(R2)Pselec(R1)

=vrad(R2)vrad(R1)

(516)

This imposes the relation

fplac(R) prop Pselec(R)vrad(R)

(517)

The numerical implementation of this probability law is complicated by thefact that vrad(R)minus1 is not known analytically and becomes infinite at the peri-centre and apocentre However vrad(R)minus1 can always be capped by theKeplerian value with the same J Rperi and Rapo allowing the use of anefficient rejection method (Press et al 1992 Sect 73) to pick up R accordingto (517)3

532 Additional Physics

Because it is based on particle representation it is relatively easy to add avariety of physical ingredients to the MC algorithm in order to improve therealism of the simulations or the domain of applicability of the methods

Collisions

Direct collisions are likely to occur in very dense stellar systems from youngclusters to core-collapsed globular clusters to nuclei of small galaxies (eg thevarious contributions in Shara 2002)

Let us consider a close approach between two stars with masses and radiim1 r1 and m2 r2 respectively The relative velocity at infinity is vrel and the

3This is the only significant improvement of the relaxation-only MC algorithmover the method described by Henon He also used a binary tree in the latest versionsof his code although he did not describe it in his articles

134 M Freitag

impact parameter b Neglecting tidal effects a collision requires the centres ofthe stars to come closer than dcoll = r1 + r2 Although neglected in our MCcode (because rare in galactic nuclei) tidal captures (Fabian et al 1975) canbe be considered using dcapt = η(r1 + r2) with η gt 1 a numerical coefficientdependent on the velocity masses and structures of the stars (eg Lee ampOstriker 1986 Kim amp Lee 1999) Treating the approach until physical contactas a point-mass problem (assuming hyperbolic trajectories) we obtain thelargest impact parameter leading to contact bmax and the cross section

Scoll 12 = πb2max = π(r1 + r2)2[

1 +(vlowast 12

vrel

)2]

(518)

where

v2lowast 12 =

2G(m1 +m2)r1 + r2

(519)

is the relative velocity the stars would have at contact on a parabolic orbit It isof the order a few 100 km sminus1 for main-sequence (MS) objects The second termin the bracket of (518) is the gravitational focusing which highly enhancesthe cross section over the geometrical value π(r1 +r2)2 as long as vrel lt vlowast 12So the collision rate for a star 1 travelling through a field of stars 2 withidentical masses sizes and velocities with number density n2 is simply

dNcoll

dt

∣∣∣∣12

= n2vrelScoll 12 equiv tminus1coll 12 (520)

which defines the collision time tcoll 12 If all stars have the same mass m andsize r a number density n and their velocities follow a Maxwellian distributionwith 1D dispersion σ2

v the average collision rate is (Binney amp Tremaine 1987)

tminus1coll = 16

radicπnσvr

2

(

1 +Gm

2σ2vr

)

(521)

Adding stellar collisions to the MC algorithm is relatively straightforwardthanks to the use of particles to represent the cluster (as opposed to DFs asdone in FP codes)

First the determination of time steps (and corresponding pair-selectionprobabilities) has to include in addition to (55) the following constraint

δt(R) le fδttcoll(R) (522)

with

tminus1coll = 16

radicπnσv〈r2〉

(

1 +G〈mr〉2σ2

v〈r2〉

)

(523)

where σ2v = 13〈v2〉m The notations 〈middot middot middot 〉 and 〈middot middot middot 〉m denote number- and

mass-weighted averaged quantities respectively4 The choice of quantities to4Note that (15) of Freitag amp Benz (2002) is slightly incorrect

average is such that we retrieve the correct value for the average collision ratein the limits σ2

v G〈m〉〈r〉minus1 and σ2v G〈m〉〈r〉minus1

Next when a pair is selected for update and once the local density andrelative velocity have been determined the pair collision time is computedusing (518) (519) and (520) but with n instead of n2 Hence the probabilityof collision between the pair during the time step δt is

Pcoll 12 = nvrelScoll 12 δt (524)

The use of n rather than n2 is of central importance This way the collisionprobabilities are symmetric as they should be Pcoll 12 = Pcoll 21 Further-more it would be impossible to estimate the local density of each populationparticularly because in MC codes as in N -body each particle can represent astar (or stars) with properties different from any other particle What makesthis simplification possible is that for a given particle the (local) probabilitythat the neighbouring particle is of type x (whatever the definition of a typeis) is simply nxn so the process of selecting the next particle as interactionpartner will statistically produce a rate of collisions with objects of type xproportional to nx because n rather than nx is used to compute the pair col-lision time Including the estimate of the collision time in the determinationof the time steps ensures that in a vast majority of cases Pcoll 12 fδt 1avoiding time steps during which more than one collision should have occurredIn the MC algorithm a collision between two particles has a statistical weightof NlowastNp This means that every star in the first particle collides with a starof the second particle and that all these collisions are identical so that the out-come can be represented by (at most) two particles corresponding to NlowastNp

collision products eachThen a random number Xrand with uniform deviate between 0 and 1 is

generated and a collision between the two particles has to be implemented ifXrand lt Pcoll 12 In low-velocity environments it is justified to assume thatcollisions result in mergers with negligible mass loss (Freitag et al 2006b)but this simplification breaks down in galactic nuclei where σv gt 100 km sminus1

(Freitag amp Benz 2002) We use prescriptions for the boundary between mergersand fly-bys and for the amount of mass and energy lost based on a large setof SPH simulations of collisions between MS stars (Freitag amp Benz 2005)The impact parameter is selected at random with uniform probability in b2

between 0 and b2max Because evolution on the MS is neglected a collision isentirely determined by the values of m1 m2 vrel and b and its outcome isdetermined using 4D interpolation and extrapolation from the SPH results(Freitag amp Benz 2002 Freitag et al 2006c) The properties of the particlesare updated from the post-collision values of m1 m2 and vrel

The particles are then placed at random radii on their new orbits accord-ing to (517) This concludes the step as two-body relaxation is not imple-mented when a collision is detected In highly collisional systems this canlead to an underestimate of relaxation effects and we have experimented witha modified scheme in which every second step is collisional and the others are

136 M Freitag

reserved for relaxation This makes the code approximately twice as slow butdoes not seem to affect the results significantly In case of a merger or if oneor both stars are completely disrupted (a rare outcome requiring velocities inexcess of about 5 vlowast 12) the number of particles in the simulation is reducedcorrespondingly

One major theoretical uncertainty still to be tackled when it comes to theeffects of collisions in stellar dynamics is how they affect stellar evolution Incase of mergers the problem is made particularly difficult by the very highrotation rate of the collision product (eg Sills et al 1997 2001 Lombardiet al 2002) In the face of this uncertainty we adopt a simple approach inwhich we set the effective age of the collision product based on its mass and theamount of core helium and assume no collisional mixing at all (see PortegiesZwart et al 1999 for another prescription)

While the hydrodynamics of collisions between two MS stars is now rela-tively well understood (Sills et al 2002 Freitag amp Benz 2005 Dale amp Davies2006 Trac et al 2007 and references therein) our knowledge about encountersfeaturing other stellar types is still very limited mostly because the physicsinvolved is more challenging Collisions between a giant and a more compactobject are probably more common than MSndashMS encounters at least in galac-tic nuclei where gravitational focusing is weaker but only a few authors haveattempted to model such events (Davies et al 1991 Rasio amp Shapiro 1991Bailey amp Davies 1999 Lombardi et al 2006) The main question mark con-cerns the evolution of the common envelope system resulting from the captureof the more compact star (see eg Taam amp Ricker 2006 and Chap 11) Colli-sions between a compact remnant and a MS (or giant) star have been studiednumerically in a larger number of papers (Regev amp Shara 1987 Benz et al1989 Rozyczka et al 1989 Davies et al 1992 Ruffert 1993 to mention afew) but clear and comprehensive predictions for their outcome are still miss-ing This is unfortunate because in our models for galactic nuclei collisionsbetween a MS star and a remnant occur at a rate comparable to collisions be-tween two MS stars (a few 10minus6 yrminus1 in a Milky-Way-like nucleus see Freitaget al 2006a) Finally in young dense clusters where mergers may contributeto the formation of massive stars (m gt 10M) or lead to the build-up of verymassive stars (m gt 100M eg Bally amp Zinnecker 2005 and Sect 541)collisions involving pre-MS objects are likely a type of event only simulatedvery recently (Laycock amp Sills 2005 Davies et al 2006)5

Central Massive Object

To study the structure and evolution of galactic nuclei with a central mas-sive black hole (MBH MBH 104 M) or globular clusters hosting an

5For more pointers to the literature on stellar collisions and tidaldisruptions by a massive black hole see the MODEST web pages athttpwwwmanybodyorgmodestWGwg4html

intermediate-mass black hole (IMBH 104 M MBH 102 M) or a verymassive star (Mlowast 200M) the effects of a central massive object have beenincluded in the MC code (Freitag 2000 Freitag amp Benz 2002 Freitag et al2006a Freitag et al 2006b) Here I concentrate on the case of an (I)MBH (seeFerrarese amp Ford 2005 for a review of the observational evidence for MBHs incentres of galaxies and Miller amp Colbert 2004 van der Marel 2004 for reviewson the possible existence of IMBHs)

Recall that the MC approach is only valid for spherical systems in dy-namical equilibrium and useful mostly if collisional effects such as two-bodyrelaxation produce noticeable evolution over the period of interest Galacticnuclei hosting MBH less massive than about 107 M are probably relaxedand therefore amenable to MC modelling Indeed assuming naively that theSgr Alowast cluster at the centre of our Galaxy is typical as far as the total stellarmass and density are concerned (Genzel et al 2003 Ghez et al 2005 Schodelet al 2007) and that we can scale to other galactic nuclei using the observedcorrelation between the mass of the MBH and the velocity dispersion of thehost spheroid σ in the form σ = σMW(MBH4times 106 M)1β with β asymp 4minus 5(Ferrarese amp Merritt 2000 Tremaine et al 2002) we can estimate the relax-ation time at the radius of influence (the limit of the region where the gravityof the MBH dominates) to be trlx(Rinfl) asymp 1010 yr (MBH4 times 106 M)(2minus3β)

All the key aspects of the interaction between the central MBH and itshost stellar system (ldquoclusterrdquo in short) are included in the MC code

Gravity of the MBH The contribution of the MBH is treated as a centralfixed point mass Newtonian gravity is assumed so the only modification incomputing the potential φ is to add MBH to the coefficients Ai in (511) TheMBH is allowed to grow by accretion of material from the stars or through anad hoc prescription to account for gas inflow Care is taken to make the timesteps significantly shorter than φ(dφdt)minus1 so as to ensure that the adiabaticeffects of the growth of the MBH on the cluster are accounted for (Young1980 Quinlan et al 1995) The MBH imposes very high stellar velocities inits vicinity causing stellar collisions to be more disruptive The gas emitted ina collision is assumed to accrete completely and immediately onto the MBH orto accumulate in an unresolved disc around the MBH if its growth is limitedby the Eddington rate

Tidal disruptions A star of mass Mlowast and radius Rlowast which comes withina distance Rtd = k Rlowast(MBHMlowast)13 of the MBH is torn apart by the tidalforces (eg Fulbright 1996 Diener et al 1997 Ayal et al 2000 Kobayashiet al 2004) Here k is a constant of order unity depending on the structureof the star In the present implementation we assume that the tidal disrup-tion is always complete and that a fixed fraction of the mass of the disruptedstar is accreted immediately usually 50 per cent as suggested by most hy-drodynamical simulations The rest is lost from the cluster These events arepredicted to trigger month- to year-long accretion flares in the UVX domain(Hills 1975 Rees 1988) some of which might have been detected already (see

138 M Freitag

Komossa 2005 for a review and Gezari et al 2006 Esquej et al 2007 for recentobservations)

In a spherical galactic nucleus in dynamical equilibrium the velocity vectorv of a star at distance R from the MBH has to point inside the loss conein direction to or away from the centre for its orbit to pass within Rtd Theaperture angle of the loss cone θLC is given by the relation

sin2(θLC) = 2(Rtd

vR

)2 [v2

2+GMBH

Rtd

(

1 minus Rtd

R

)

+ Φlowast(R) minus Φlowast(Rtd)]

2GMBHRtd

(vR)2asymp Rtd

R

(525)

where Φlowast(R) = Φ(R) + GMBHR is the cluster contribution to the gravita-tional potential The first approximation is valid as long as R Rtd whichis nearly always the case the second is an order-of-magnitude estimate validwithin the sphere of influence of the MBH where v2 asymp GMBHR

minus1Stars on loss-cone orbits are removed on an orbital time-scale In a spher-

ical potential it is generally assumed that loss-cone orbits are replenishedby two-body relaxation but orbital perturbations by resonant relaxation (seeSect 542) or deflections by massive objects such as molecular clouds (Peretset al 2007) may play an important role Barring such non-standard processestwo loss-cone regimes can be distinguished (Frank amp Rees 1976 Lightman ampShapiro 1977 Cohn amp Kulsrud 1978) (1) The loss cone is kept full and doesnot induce any significant anisotropy in the velocity distribution when relax-ation is strong enough to repopulate loss-cone orbits over an orbital timecorresponding to the condition θ2

LCtrlx Porb For stars in this regime whichtypically occurs at large distances the average time before tidal disruption isof order tdisrfull θminus2

LCPorb (when averaged over all directions of v) (2) Theloss cone is (nearly) empty in the opposite case θ2

LCtrlx Porb and corre-sponds to an absorbing region of phase space into which the stars diffuse Thedensity of stars on orbits close to but out of the loss cone is reduced In thisregime it takes on average tdisrempty trlx ln(θminus2

LC) for a star to be disruptedPlunges through the horizon The last stable parabolic orbit around a non-

spinning massive black hole corresponds to a (Newtonian) pericentre distanceRLSPO = 8GMBHc

minus2 Sufficiently dense stars such as compact remnants havea tidal disruption radius Rtd inside RLSPO (or even inside the horizon) mean-ing that such objects will be swallowed whole rather than be tidally disruptedand produce no accretion flare6 From the point of view of stellar dynamicsthis situation is identical to the case of tidal disruptions with the quantityRtd replaced by RLSPO

6In fact when RLSPO gt Rtd gt Rhor = 2GMBHcminus2 the star is disrupted before itdisappears through the horizon To my knowledge the detectability of such eventshas not been investigated

Inspirals by emission of gravitational waves Significant emission of grav-itational waves (GWs) occurs during very close encounters with the MBH(Peters amp Mathews 1963) For a compact massive stellar object on a veryeccentric orbit GW emission may dominate orbital evolution over two-bodyrelaxation yielding to progressive circularisation and shrinking of the semi-major axis (Peters 1964) until it plunges through the horizon of the MBH(or is tidally disrupted) For a 1ndash10M object orbiting a MBH with a massbetween 104 and 107 M the final months or years of inspiral should be de-tectable by the future spaceborn GW observatory LISA7 to distances of severalGpc Such extreme mass ratio inspirals (EMRIs) yield an unprecedented viewon the direct vicinity of MBHs The promise for physics and astrophysics is asexciting as the uncertainties about their physical rates and the challenges fordata analysis are high (see Amaro-Seoane et al 2007 for an extensive reviewof the various aspects of EMRI research)

I now explain in some detail how the loss-cone physics is implementedin the MC code This treatment is adequate only for the processes requir-ing a single passage within a well-defined critical distance of the MBH tobe successful such as tidal disruption plunges or non-repeating GW burstsemitted by stars on quasi-parabolic orbits (Hopman et al 2007) In contrastan EMRI is a progressive process that will only be successful (as a poten-tial source for LISA) if the stellar object experiences a very large number ofsuccessive dissipative close encounters with the MBHs (Alexander amp Hopman2003) The ability of the MC approach to deal with this situation is discussedin Amaro-Seoane et al (2007)

At the end of the step in which two particles have experienced an encounter(to simulate two-body relaxation) each particle is tested for entry into theloss cone J lt JLC where JLC = RV sin(θLC)

radic2GMBHRtd (525) A

complication arises because the time step δt used in the MC code is a frac-tion fδt = 10minus3 minus 10minus2 of the local relaxation time trlx(R) which is muchlarger than the critical timescale θ2

LCtrlx In other words the super-encounterdeflection angle θSE (53) is much larger than θLC This keeps the loss coneeffectively and artificially full However in contrast with direct N -body sim-ulations this is not due to the overall relaxation rate being too large whenNp lt Nlowast

To treat the empty loss-cone regime in the most accurate fashion we wouldneed to use time steps as short as the orbital period Unfortunately it is notpossible to give short time steps only to particles with eccentric orbits (andhence at risk of entering the loss cone) because the time step is a function ofthe positionR and cannot be attached to a particle Hence at least all particleswithin the critical radius defined by tdisrfull(Rcrit) = tdisrempty(Rcrit) wheret quantities are some local average would need to have much shorter timesteps which would slow down the code considerably Instead an approximate

7Laser Interferometer Space Antenna see httpwwwlisa-scienceorg

140 M Freitag

procedure is used to ensure that entry into the loss cone happens diffusivelywhen θ2

LCtrlx PorbAfter the super-encounter deflection angle θSE has been computed (53)

and before the particles in the pair are given their new energies angular mo-menta and positions we check each of them for entry into the loss cone inthe following manner First the orbital period is computed by integrating(515) using Chebyshev quadrature (Press et al 1992) We consider that dur-ing Porb δt the direction of the velocity of the particle would have changedby an rms angle θorb = (Porbδt)12θSE We then assume that the tip of thevelocity vector of the particle executes a random walk of NRW = δtPorb sub-steps of length θorb during δt The modulus of the velocity is kept constantEntry into the loss cone is tested at each of these sub-steps This random walkis executed in the reference frame of the super-encounter but independentlyfor each particle of the pair because they have different θorb and NRW If aparticle is found on a loss-cone orbit it is immediately removed and (part of)its mass is added to the MBH If the random walk never crosses into the losscone the particle is kept and in order to ensure exact energy conservationthe particle is given the velocity computed in the super-encounter not thatreached at the end of the random walk The random walk is a refinement of thesuper-encounter from a statistical point of view but because of its stochasticnature it cannot produce velocity vectors anti-parallel to each other for theparticles in a pair This means that energy in the reference frame of the cluster(as opposed to that of the pair) would not be conserved It might be possi-ble to improve this procedure by performing the random walk in the clusterreference frame and leaving the particle with the velocity attained at the endof it This would permit us to obtain the correct decrease of density on theorbits close to the loss cone

In the context of loss-cone physics I mention another type of Monte-Carlocode developed by Shapiro and collaborators at Cornell University (Shapiro1985 for a review and references) Their approach was essentially a hybridbetween that presented here entirely based on particles and with no explicitcomputation of diffusion coefficients and the direct FokkerndashPlanck integration(Chap 4) Instead of having particles interacting in pairs their density in the(E J) phase space was tabulated in order to compute diffusion coefficientsused to modify their orbital parameters during the next global step Withina global step each particle could be evolved independently of the others (andon its own time step) until the updated phase-space density (and potential)is recomputed This permitted to endow the particles in or close to the losscone with time steps as short as their orbital time Extending this scheme to amulti-mass situation seems feasible without explicit use of an augmented (andsparsely populated) (E JMlowast) phase space Unfortunately to my knowledgesuch a development was not attempted

Binary Stars

The MC code presented so far in this chapter only deals with the dynamicsand evolution of single stars This is a reasonable simplification as long as theoverall dynamics of galactic nuclei is concerned because in such environmentsmost binaries are very soft meaning that their internal orbital velocity is muchsmaller than velocity dispersion at least in the vicinity of a MBH where thedensity and interaction probability are the highest However binaries playa major role in the evolution of globular clusters where the hard ones actas an efficient central source of heat by being shrunk and eventually ejectedduring interactions with other stars (Aarseth 1974 Spitzer amp Mathieu 1980Gao et al 1991 Hut et al 1992 Heggie amp Hut 2003 Giersz 2006 Fregeauamp Rasio 2007 amongst many others) For a given stellar density binariesalso highly increase the rate of direct collision between stars (Portegies Zwartet al 1999 Portegies Zwart amp McMillan 2002 Portegies Zwart et al 2004Fregeau et al 2004) Beside their dynamical role binary interactions in denseclusters are also of high interest as a way to create a whole zoo of ldquostellarexoticardquo and phenomena including blue stragglers millisecond pulsars andmergers between compact stars as sources of supernovae gamma-ray burstsor gravitational waves (eg Hurley et al 2001 Davies 2002 Shara amp Hurley2002 Benacquista 2006 Grindlay et al 2006 OrsquoLeary et al 2007) Includingbinaries in models of galactic nuclei is also important to explain X-ray observa-tions at the Galactic centre (Muno et al 2005) hyper-velocity stars (eg Hills1988 Brown et al 2005) and as a possible channel to create extreme-massratio sources of gravitational waves for LISA (Miller et al 2005)

Here I put aside the very thorny question of binary evolution and howit might be affected by dynamics (see Chaps 11 and 12) and concentrateon the dynamical aspects Binaries have been included in MC simulationswith various levels of sophistication (Spitzer amp Mathieu 1980 Stodolkiewicz1985 1986 Giersz 1998 2001 2006 Giersz amp Spurzem 2000 2003 Fregeauet al 2003 Gurkan et al 2006 Fregeau amp Rasio 2007 Spurzem et al 2006)The approach of Fregeau amp Rasio (2007) is based on our own treatment ofcollisions and is the most direct and accurate one at least when each particlerepresents a single system (single star or binary) This treatment does notinclude formation of binaries through three-body interactions (see the worksof Stodolkiewicz and Giersz)

To include binaries in a MC code we first need to allow some of theparticles to represent binaries instead of single stars which requires extradata to keep track of the internal structure masses and evolutionary phase ofthe member stars semi-major axis abin and eccentricity ebin In the absence ofinteraction with another star or binary these parameters are updated by theuse of some binary evolution prescription Then similar to stellar collisionsincluding binary dynamics amounts to (1) determining the probability of abinary interaction Pbin between two neighbouring particle if at least one of

142 M Freitag

them is a binary (2) generating a random number Xrand and if Xrand lt Pbin(3) implementing a singlendashbinary or binaryndashbinary encounter

Steps (1) and (2) are the same as in the implementation of collision betweensingle stars Actually at this level binary interactions do not need to bedistinguished from stellar collisions We only need to give to binaries a radiusηabin where η gt 1 is a safety factor to ensure that all interactions that canperturb the binaries significantly are taken into account Fregeau amp Rasio(2007) chose η = 2 and checked that a value η = 4 (which could cause thetime steps to be about twice as short) do not lead to statistically differentresults as far as the overall evolution of the cluster and binary population isconcerned More complex forms of the criterion for the most distant encounterto be included have been used by other authors (eg Bacon et al 1996 Gierszamp Spurzem 2003) The simple rule described here based on proximity atthe closest approach (when each binary is treated as a point mass) shouldyield correct results if η is made sufficiently large but in studies of smallperturbations to binaries (or planetary systems) it may be less than optimalin the sense that large η values will yield small time steps Indeed for binarieswe have to substitute ηabin for r in (523) Roughly speaking with binariesat the hardndashsoft boundary (Gmbina

minus1bin σ2

v) the time step will be limited bybinary processes rather than by two-body relaxation if η gt lnΛ

Between interactions binaries are treated as unperturbed and their prop-erties are updated using binary evolution prescriptions Note that this is alsothe case in N -body codes unless another object comes within a distancedpert = γ

minus13min (2mpertmbin)13(1 + ebin)abin where mpert is the mass of the

perturber and γmin is the tidal perturbation parameter (Aarseth 2003 andChap 1) In most cases γmin is set to 10minus6 Hence in a similar-mass situa-tion (mpert asymp mbin) the N -body prescription corresponds to η asymp 100 minus 200in the MC collision formalism Whether this much more conservative condi-tion yields significantly different results in the evolution of the binaries andtheir host cluster has not been investigated in depth (see Giersz amp Spurzem2003 Spurzem et al 2006 for some discussion) Incidentally such researchmay open the possibility of a more approximate but much faster treatment ofbinary interactions in direct N -body codes

The most direct and accurate (but also time-consuming) way of imple-menting step (3) ie of determining the outcome of a binary encounter oc-curring in a MC simulation is to switch to a direct few-body integrator (seeChap 2 for algorithms) First the quantities not specified by the MC parti-cles have to be picked at random These are the orbital phase(s) and orienta-tion(s) and the impact parameter8 One difficulty arises with binaryndashbinaryencounters as they often result in the formation of a stable triple system As

8In principle we could keep track of the orbital phase of a binary between inter-actions However the MC method relies on the assumptions that strong interactionsare rare and that binaries are much smaller than any length scale in the cluster Thiseffectively randomises the orbital phase between interactions

mentioned by Giersz amp Spurzem (2003) and Fregeau amp Rasio (2007) it is inprinciple possible to have some particles representing triples (or higher-orderstable groups) in the MC framework with the appropriate book-keeping butthis has not been implemented so far Instead triple systems are forcefullybroken apart into a binary and a single star just unbound to the binary An-other type of outcome that may require special treatment is the formation ofa very wide soft binary with a size not much smaller than the typical sizeof the cluster Such pairs cannot be treated accurately in the MC formalismbut they are unlikely to survive the next interaction so they can be artifi-cially broken up without affecting the results Finally as mentioned above itis probably important to allow for direct collisions during binary interactionsOne source of uncertainty is the size of a merged star just after a collision It islikely to be several times the MS radius leading to a significant probability ofa triple or quadruple collision (Goodman amp Hernquist 1991 Lombardi et al2003 Fregeau et al 2004)

Once the outcome of a binaryndashsingle or binaryndashbinary interaction has beendetermined the products of the interaction are turned back into MC particlesrepresenting single or binary stars with the adequate internal and orbitalproperties and a position in the cluster is selected for each according to theprocedure presented in Sect 531

Integrating the few-body encounters in a cluster with a large fraction ofbinaries can account for a significant fraction of the computing time A muchfaster way to deal with binary dynamics is to use ldquorecipesrdquo which are fittingformulae for the cross section and outcome of interactions based on large pre-computed sets of scattering experiments (eg Heggie 1975 Hut 1993 Heggieet al 1996) However for stars of unequal masses the parameter space is toovast to be reliably covered by such recipes Even in the idealised case where allstars have the same mass for which comprehensive binary-interaction crosssections are available the use of such recipes rather than explicit few-bodyintegrations seems to yield quantitatively inaccurate results (Fregeau et al2003 Fregeau amp Rasio 2007)

Other Physical Ingredients

MC codes can include a few other physical processes that I describe moresuccinctly

Stellar evolution ndash Evolution of stars (single or binaries) can be taken intoaccount with various levels of refinement In our MC code a very simple pre-scription is used which assumes that a star of initial mass Mlowast spends a timetMS(Mlowast) on the MS without any evolution and abruptly turns into a compactremnant at the end of this period Thus the giant phase is neglected Therelation tMS(Mlowast) and the prescriptions for the nature and mass of the rem-nant are taken from stellar evolution models (Hurley et al 2000 Belczynski

144 M Freitag

et al 2002) To ensure that stellar evolution time-scales are resolved a sup-plementary constraint on the time step is introduced δti le fδtlowasttlowasti wheretlowasti is an estimate for the stellar evolution time-scale of stars at rank i andfδtlowast = 0025 typically In the present implementation tlowasti is simply the MSlifetime of the particle which has rank i at the moment the time steps arecomputed Because we use a piecewise constant representation of δt the timestep will generally be shorter than a fraction fδtlowast of the smallest local value oftMS Once a pair of particles is selected it is first checked for stellar evolutionand its masses and radii are updated if required before the super-encounter(or collision) is carried out Natal kicks can be given to newborn neutronsstars and black holes (Freitag et al 2006a)

This simplistic treatment can be improved by the use of detailed stel-lar evolution packages (Portegies Zwart amp Verbunt 1996 Portegies Zwart ampYungelson 1998 Hurley et al 2000 2001 See also Chaps 10 and 13) A diffi-culty to confront however is that this will involve shorter time-scales tlowast egto resolve the giant phase In general stars with short tlowast can be found any-where in a cluster imposing (unlike relaxation or collision) uniformly shorttime steps This could be prevented by using a time-stepping scheme for stel-lar evolution independent of the dynamical one For instance using a heapstructure (Press et al 1992) we could keep track of the next particle requir-ing update of its stellar parameters and realise this update when due withoutchanging the orbital parameters (except if a natal kick is imparted)

Large-angle scatterings ndash Gravitational encounters between stars of massm1 and m2 at a relative velocity vrel with an impact parameter smaller thana few b0 equiv G(m1 +m2)vminus2

rel lead to deflection angles too large to be accountedfor in the standard diffusive theory of relaxation On average a star willexperience an encounter with impact parameter smaller than fLAb0 (withfLA of order a few) over a time-scale

tLA [π(fLAb0)2nσ

]minus1 asymp lnΛf2LA

trlx (526)

The effects of large-angle scatterings on the overall evolution of a clusterare negligible in comparison with diffusive relaxation (Henon 1975 Goodman1983) However unlike the latter process they can produce velocity changesstrong enough to eject stars from an isolated cluster (Henon 1960 1969Goodman 1983) or more important from the region of influence around aMBH (Lin amp Tremaine 1980 Baumgardt et al 2004 OrsquoLeary amp Loeb 2008)Large-angle scatterings are easily included in MC simulations as a special caseof collision with a cross section π(fLAb0)2 (Freitag et al 2006a) but the timesteps will be limited by this (rare) process rather than by diffusive relaxationfor fLA 4

Tidal evaporation ndash Stellar clusters are subject to the tidal influence of theirhost galaxy Assuming spherical symmetry the MC code cannot deal with thegalactic field accurately but it is easy to include in an approximate way themost important effect which is the evaporation of stars from the cluster

A star can escape from a cluster on a circular orbit of radius RG around aspherical host galaxy if its orbit allows it to reach the Lagrange point awayfrom or in the direction of the galaxy These locations are approximately ata distance RL = RG(Mcl(2MG))13 from the clusterrsquos centre where Mcl andMG are the masses of the cluster and a point-mass galaxy respectively In thespherical approximation we assume that a star escapes when its apocentredistance is larger than RL As the total mass of the cluster decreases the valueof RL is adjusted This can lead to more stars being lost if their apocentredistances happen to lie beyond the new RL value so we have to iterate untilconvergence is reached for the bound mass of the cluster Using such treatmentof tidal evaporation combined with a prescription for the orbital decay of thecluster owing to dynamical friction Gurkan amp Rasio (2005) have simulatedthe internal and orbital evolution of clusters at the Galactic centre

54 Some Results and Possible Future Developments

Monte-Carlo codes have been used in a variety of problems involving thecollisional evolution of globular clusters and galactic nuclei I do not attemptto review this variety of works but invite the reader to sample the referencescited in Sect 51 Here I limit myself to the quick presentation of a few typicalresults to give a flavour of the capabilities of the method

541 Young Clusters and Globular Clusters

In Figs 53 and 54 I show the evolution to core collapse of single-mass andmulti-mass Plummer models computed with the MC code described here withno other physics than two-body relaxation I compare with direct Nbody4

results (H Baumgardt 2005 personal communication) Provided the valueof γc needed to convert N -body time units (see Chap 1) to relaxation timeis adjusted in an ad hoc fashion very good agreement between the methodsis obtained for these cases We find γc 015 for the single-mass modeland γc 003 for Salpeter mass function (dNlowastdMlowast prop Mminus235

lowast ) extendingfrom 02 to 10 M in agreement with theoretical expectations and previousnumerical determinations (Henon 1975 Giersz amp Heggie 1994 1996 Freitaget al 2006c) We note that in N -body simulations core collapse is alwayshalted and reversed by the formation and hardening of binaries through closethree-body interactions (eg Aarseth 1971 Heggie amp Hut 2003) a process notincluded in the MC code When the mass function is extended to 120M theagreement between MC and N -body simulations is poorer but the time tocore collapse is found to be approximately the same in terms of relaxationtime namely a surprising 10ndash20 per cent of the initial central relaxation time(Spitzer 1987)

trc(0) equiv 0339σ3

v

lnΛG2〈m〉2n (527)

146 M Freitag

Fig 53 Core collapse of a single-mass cluster initialised as a Plummer modelThe results of the MC code using 250 000 particles in solid lines are comparedto a direct Nbody4 simulation using 64 000 particles in dashes (H Baumgardt2005 personal communication) Top panel evolution of radii of the Lagrangianspheres containing the indicated fraction of the mass Bottom panel evolution ofthe anisotropy parameter averaged over Lagrangian shells bounded by the indicatedmass fractions The length unit is the N -body scale (see Chap 1) The time unit isthe initial half-mass relaxation time (Spitzer 1987) To convert the dynamical timeunits of the N -body simulation to a relaxation time a value of γc = 015 was usedfor the Coulomb logarithm

MMMM

M

Fig 54 Core collapse of a Plummer cluster with 02ndash10 M Salpeter mass functionA MC code simulation with 106 particles in solid lines is compared to a directNbody4 simulation with 256 000 particles in dashes (H Baumgardt 2005 personalcommunication) To show mass segregation the evolution of Lagrangian radii isplotted for mass fractions of 1 and 50 per cent for stars with masses within fivedifferent bins To convert the dynamical time units of the N -body simulation to arelaxation time a value of γc = 003 was used for the Coulomb logarithm Comparewith Fig 41

where the quantities 〈m〉 n and σv are determined at the centre This isa result of great interest as it raises the possibility of triggering a phase ofrunaway collisions in young dense clusters (Quinlan amp Shapiro 1990 PortegiesZwart et al 1999 Portegies Zwart amp McMillan 2002 Gurkan et al 2004Portegies Zwart et al 2004 Freitag et al 2006bc)

A domain where MC simulations are bound to play a unique role in thenext few years is the evolution of large clusters with a high fraction of pri-mordial binaries This is one of the most challenging situations for directN -body codes because the evolution of regularised binaries cannot be com-puted on special-purpose GRAPE hardware At the time of writing the pub-lished N -body simulations tallying the largest number of binaries are those byHurley et al (2005) with 12 000 binaries amongst 36 000 stars and by PortegiesZwart et al (2007) with 13 107 binaries amongst 144 179 stars In contrastFregeau amp Rasio (2007) present tens of MC simulations for 105 particles somewith 100 per cent binaries and a few 3times105 particle cases with up to 15times105

binaries (see also Gurkan et al 2006) Although single and binary stellar evolu-tion were not included in these simulations they can be incorporated into MCcodes in the same way and with the same level of realism as in direct N -body

148 M Freitag

05

06

07

08

09

1M

bMb(0

) M

M(0

)

0 10 20 30 40 50 60t [t

rh]

01

1

r c rh

b rh

s [r N

B]

Fig 55 Evolution of a cluster containing 30 per cent of (hard) primordial binaries(J Fregeau 2007 personal communication) The cluster is set up as a Plummermodel of 105 particles with masses distributed according to a Salpeter IMF between02 and 12 M Stellar evolution is not simulated Top panel total cluster mass(dashed line) and mass in binaries (dot-dashes) normalised to the initial valuesBottom panel core radius (solid line) half-mass radius of single stars (dashes) andhalf-mass radius of binaries (dot-dashes) in N -body units Time is in units of theinitial half-mass relaxation time For more information on this work see Fregeau ampRasio (2007)

codes In Fig 55 I show the results from a simulation of a cluster with30 per cent primordial binaries ie Nbin(Nbin + Nsingle) = 03 (J Fregeau2007 personal communication) Binaries stabilise the core against collapse fora duration of tens of half-mass relaxation times corresponding to more thanthe Hubble time when applied to real globular clusters The quasi-equilibriumsize of the core maintained during this long phase of binary burning appearsto be too small to explain the observed core size of most non-collapsed Galac-tic clusters It is not yet clear whether this discrepancy is to be blamed onthe neglect of stellar evolution and other well-known physical effects (colli-sions non-stationary Galactic tides etc) or can only be resolved by assumingsome more exotic physics such as the presence of IMBHs in many clusters(Baumgardt et al 2005 Miocchi 2007 Trenti et al 2007) but it seems thatMC simulations are the ideal tool to investigate this issue

Monte-Carlo codes that treat the dynamics and evolution of single andbinary stars in great detail should be available very soon allowing the simula-tion of clusters containing up to 107 stars on a star-by-star basis with a highlevel of realism as long as the assumptions of spherical symmetry and dynam-ical equilibrium are justified I now mention a few strong motivations to tryand extend the realm of MC cluster simulations beyond these assumptions

bull Galactic tides The treatment of stellar evaporation from a cluster can beimproved significantly First stars have to find the narrow funnels aroundthe Lagrange points to exit the cluster (eg Fukushige amp Heggie 2000Ross 2004) Hence it takes a star several dynamical times to find theldquoexit doorrdquo even when some approximate necessary condition for the es-cape is reached such as an apocentre distance (in the spherical potential)larger than the distance to the Lagrange point Therefore a significantfraction of the stars in a cluster can be potential escapers (Fukushige ampHeggie 2000 Baumgardt 2001) Using (semi)analytical prescriptions fromthe cited studies one could take this effect into account in MC simula-tions by giving potential escapers a finite lifetime before they are actuallyremoved from the cluster (see Takahashi amp Portegies Zwart 2000 for asimilar approach applied to FokkerndashPlanck simulations) Other importanteffects of the galactic gravitational field absent from MC simulations (andmost other cluster simulations) come from its non-steadiness A cluster onan eccentric orbit experiences a stronger tidal stress at pericentre an ef-fect dubbed bulge shocking while compressive disc shocking happens whenthe cluster crosses the plane of the galactic disc (eg Spitzer 1987 Gnedinamp Ostriker 1997 Baumgardt amp Makino 2003 Dehnen et al 2004) Sucheffects can be included in MC codes using the same (semi)analytical pre-scriptions as in some FokkerndashPlanck integrations (Gnedin amp Ostriker 1997Gnedin et al 1999) Alternatively because shocking occurs on a time-scalemuch shorter than the relaxation time we could switch back and forth be-tween a fast non-collisional N -body algorithm (such as Superbox seeChap 6) to compute the effects of the shocks and a MC code to evolvethe cluster between shocks Another possibility would be a hybrid non-spherical MCN -body method suggested in the next point

bull Rotating clusters Observational evidence and theoretical models indicatethat clusters may be born with significant rotation possibly as a resultof the merger of two clusters (see references in Amaro-Seoane amp Freitag2006) The MC approach exposed here is not appropriate to study non-spherical systems but as already suggested by Henon (1971a) it might bepossible to develop a hybrid approach where a collisionless N -body codeis used for fast orbit sampling in a non-spherical geometry (by actual or-bital integration) and collisional effects are included explicitly in a MCfashion by realising super-encounters between neighbouring pairs A com-bination of the Self-Consistent Field N -body method with FokkerndashPlanckrelaxation terms was developed by S Sigurdsson to study the evolution

150 M Freitag

of globular clusters orbiting a galaxy (Johnston et al 1999) but to myknowledge no MCN -body hybrid has ever been developed Such a codewould also be of great interest in the study of galactic nuclei as mentionedin Sect 542

bull Primordial gas Observations show that when a cluster forms not morethan 30 per cent of the gas is eventually turned into stars (Lada 1999)In relatively small clusters the gas is expelled by the ionising radiationand winds of OB stars within the first 1ndash2 Myr In clusters with an escapevelocity larger than about sim 10 km sminus1 complete expulsion of the gasprobably only occurs when the first SN explodes (Kroupa et al 2001 Boilyamp Kroupa 2003ab Baumgardt amp Kroupa 2007 and references therein Seealso Sect 74) When still present in the cluster the gas dominates thegravitational potential Furthermore it can strongly affect the orbits andmass of stars as they accrete and slow down to conserve momentum thusshaping the mass function and producing strong segregation (Bonnell et al2001ab Bonnell amp Bate 2002) Such effects can be included in MC codesif the gas is treated as a smooth parametrised component However tofollow the reaction of the cluster to the fast gas expulsion we would haveto switch to a (collisionless) N -body code or Spitzer-type dynamical MCscheme because the Henon algorithm can only treat adiabatic potentialevolution

542 Galactic Nuclei

In addition to the study of globular and young clusters the MC code is also amethod of choice for the study of small galactic nuclei (Freitag 2001 Freitagamp Benz 2001ab 2002 Freitag 2003 Freitag et al 2006a) Massive black holes(MBHs) less massive than about 107 M are probably generally surroundedby a stellar nucleus with a relaxation time shorter than 1010 yr at the distancewhere the mass in stars is equal to the mass of the MBH (eg Lauer et al1998 Genzel et al 2003 Freitag et al 2006a Merritt amp Szell 2006) Althoughdirect N -body codes with GRAPE hardware can now be used to study someimportant aspects of the collisional evolution of galactic nuclei (Preto et al2004 Merritt amp Szell 2006 Merritt et al 2007b) they are still limited to 106 particles for this kind of application which falls short of the number ofstars in galactic nuclei

In Fig 56 I show the evolution of a small galactic nucleus computed withthe MC code described in this chapter In addition to two-body relaxation thephysics include the effects of a (growing) central MBH (tidal disruption directmergers for objects too compact to be disrupted) and stellar collisions Large-angle scatterings were found to be of secondary importance for such systemsand stellar evolution can be taken into account but this raises the questionof how much gas from stellar evolution will be accreted by the MBH (Freitaget al 2006a) For the model presented segregation of stellar-mass black holes

dd

Fig 56 Evolution of the model for a small galactic nucleus hosting a MBH witha mass of 35times 104 M with 21times 106 particles (model GN84 of Freitag et al 2006a)Top panel evolution of Lagrangian radii for the various stellar species (MS main-sequence WD white dwarfs NS neutron stars BH stellar black holes) The stellarpopulation has a fixed age of 10 Gyr Bottom panel accretion of stellar material bythe MBH For tidal disruptions 50 per cent of the mass of the star is accretedldquoMergersrdquo are events in which an object crosses the horizon whole Collisions be-tween MS stars are also taken into account with all the released gas being accretedby the MBH

152 M Freitag

to the centre occurs within some 50 Myr after which their swallowing by theMBH drives the expansion of the nucleus For models with parameters per-taining to the Milky Way nucleus mass segregation takes about 3ndash5 Gyr andonly little expansion occurs in a Hubble time The segregation of stellar blackholes is of key importance for the formation of EMRI sources for LISA (Hop-man amp Alexander 2006b Amaro-Seoane et al 2007 and references therein)

Simulations of galactic nuclei have not yet reached as high a level of realismas one might wish Several aspects of the physics are still laking including thefollowing elements

bull Binary stars Binary stars are probably not effective as a source of heat be-cause the ambient velocity dispersion is so high in galactic nuclei Howeverthis population is of interest in its own right as mentioned in Sect 532

bull Resonant relaxation Close to the MBH stars travel on approximately fixedKeplerian orbits exerting torques on each other causing the eccentricitiesto fluctuate randomly on a time-scale shorter than that of standard two-body relaxation (Rauch amp Tremaine 1996) This might affect moderatelythe rate of tidal disruptions (Rauch amp Ingalls 1998) and very significantlythat of EMRIs (Hopman amp Alexander 2006a) but being an intrinsicallynon-local effect it can probably only be included in an approximate fashionin MC models

bull Motion of the central MBH Direct N -body simulations have establishedthe importance of MBH wandering (eg Merritt et al 2007 and referencestherein) Because this is a dynamical non-spherical perturbation to theidealised cluster representation used in the MC approach it can only beincluded through ad hoc prescriptions determining for example the prob-ability for a star to be tidally disrupted It is not yet clear whether thewandering would affect the results appreciably and justify such modifica-tions to the MC code

bull Interplay between accretion disc and stars The orbits of stars repeatedlyimpacting a dense disc tend to align with it (eg Syer et al 1991 Subr et al2004 Miralda-Escude amp Kollmeier 2005) Stars may therefore be a majorcontributor to nuclear activity and the growth of SMBHs Testing this ideais challenging since what is required is a numerical scheme coupling stellardynamics for several millions of stars disc physics and some prescriptionfor the stellar and orbital evolution of the stars embedded in the disc Anon-spherical hybrid MCN -body code as suggested above could formthe backbone of this complex scheme

bull Binary massive black hole Galaxy mergers lead to the formation of massivebinaries the evolution and fate of which is still debated The key questionis whether interactions with stars and gas are efficient at shrinking thebinary to the point where it merges by the emission of gravitational waves(Begelman et al 1980 Merritt amp Milosavljevic 2005 Berczik et al 2006Merritt 2006 Sesana et al 2007 amongst others) If the binary insteadstalls for a very long time the next galactic merger can bring about a

highly dynamical three-body interaction involving MBHs likely to lead toa merger and the ejection of a single MBH (Hoffman amp Loeb 2007) If theparent galaxies are devoid of gas once its separation has become smallerthan about sim 4Gμσ2 where μ is the reduced mass and σ the stellar veloc-ity dispersion the MBH binary can only shrink by ejecting passing starsout of the nucleus These interactions also determine the evolution of theeccentricity which might play a key role in bringing the binary to coales-cence While only N -body methods can implement the non-symmetricalgeometry of this situation (eg Mikkola amp Aarseth 2002) they cannotinclude the gt 107 stars present in even a moderately small nucleus Anaxially symmetrical (hybrid) MC code would make it possible to simulatethe interaction of a massive binary with its host nucleus employing a real-istic mass ratio between the stars and the MBHs and hence the correctrate of relaxation into the loss cone for interaction with the massive binary

Acknowledgement

It is a pleasure to thank M Atakan Gurkan and John Fregeau for discussionsand comments on a draft of this chapter I also thank John Fregeau andHolger Baumgardt for providing unpublished simulation results My work issupported by the STFC rolling grant to the IoA

References

Aarseth S J 1971 ApampSS 13 324 145Aarseth S J 1974 AampA 35 237 141Aarseth S J 2003 Gravitational N-body Simulations Cambridge Univ Press

Cambridge 142Aguilar L A Merritt D 1990 ApJ 354 33 125Alexander T Hopman C 2003 ApJ Lett 590 L29 139Amaro-Seoane P Freitag M 2006 ApJ Lett 653 L53 149Amaro-Seoane P Gair J R Freitag M Miller M C Mandel I Cutler C J

Babak S 2007 Classical and Quantum Gravity 24 113 139 152Ayal S Livio M Piran T 2000 ApJ 545 772 137Bacon D Sigurdsson S Davies M B 1996 MNRAS 281 830 142Bailey V C Davies M B 1999 MNRAS 308 257 136Bally J Zinnecker H 2005 AJ 129 2281 136Baumgardt H 2001 MNRAS 325 1323 149Baumgardt H Kroupa P 2007 MNRAS 380 1589 150Baumgardt H Makino J 2003 MNRAS 340 227 149Baumgardt H Makino J Ebisuzaki T 2004 ApJ 613 1133 144Baumgardt H Makino J Hut P 2005 ApJ 620 238 148Begelman M C Blandford R D Rees M J 1980 Nature 287 307 152Belczynski K Kalogera V Bulik T 2002 ApJ 572 407 143

154 M Freitag

Benacquista M J 2006 Living Reviews in Relativity 9 2 141Benz W Hills J G Thielemann 1989 ApJ 342 986 136Berczik P Merritt D Spurzem R Bischof H-P 2006 ApJ Lett 642 L21 152Binney J Tremaine S 1987 Galactic Dynamics Princeton Univ Press

Princeton NJ 123 134Boily C M Athanassoula E 2006 MNRAS 369 608 125Boily C M Kroupa P 2003a MNRAS 338 665 150Boily C M Kroupa P 2003b MNRAS 338 673 150Bonnell I A Bate M R 2002 MNRAS 336 659 150Bonnell I A Bate M R Clarke C J Pringle J E 2001a MNRAS 323 785 150Bonnell I A Clarke C J Bate M R Pringle J E 2001b MNRAS 324 573 150Brown W R Geller M J Kenyon S J Kurtz M J 2005 ApJ Lett 622 L33 141Cohn H Kulsrud R M 1978 ApJ 226 1087 138Dale J E Davies M B 2006 MNRAS 366 1424 136Davies M B 2002 in van Leeuwen F Hughes J DPiotto G eds ASP Conf Ser

Vol 265 Omega Centauri A Unique Window into Astrophysics Astron SocPac San Francisco p 215 141

Davies M B Benz W Hills J G 1991 ApJ 381 449 136Davies M B Benz W Hills JG 1992 ApJ 401 246 136Davies M B Bate M R Bonnell I A Bailey V C Tout C A 2006 MNRAS

370 2038 136Dehnen W Odenkirchen M Grebel E K Rix H-W 2004 AJ 127 2753 149Diener P Frolov V P Khokhlov A M Novikov I D Pethick C J 1997 ApJ

479 164 137Einsel C Spurzem R 1999 MNRAS 302 81 125Esquej P Saxton R D Freyberg M J Read A M Altieri B Sanchez-Portal M

Hasinger G 2007 AampA 462 L49 138Fabian A C Pringle J E Rees M J 1975 MNRAS 172 15 134Ferrarese L Ford H 2005 Space Science Reviews 116 523 137Ferrarese L Merritt D 2000 ApJ Lett 539 L9 137Fiestas J Spurzem R Kim E 2006 MNRAS 373 677 125Frank J Rees M J 1976 MNRAS 176 633 138Fregeau J M Cheung P Portegies Zwart S F Rasio F A 2004 MNRAS 352 1 141 143Fregeau J M Gurkan M A Joshi K J Rasio F A 2003 ApJ 593 772 123 141 143Fregeau J M Rasio F A 2007 ApJ 658 1047 123 131 141 142 143 147 148Freitag M 2000 PhD thesis Universite de Geneve 137Freitag M 2001 Classical and Quantum Gravity 18 4033 150Freitag M 2003 ApJ Lett 583 L21 150Freitag M Amaro-Seoane P Kalogera V 2006a ApJ 649 91 124 136 137 144 150 151Freitag M Benz W 2001a in Deiters S Fuchs B Just R Spurzem R eds ASP

Conf Ser Vol 228 Dynamics of Star Clusters and the Milky Way Astron SocPac San Francisco p 428 150

Freitag M Benz W 2001b in Kaper L van den Heuvel E P J Woudt P AESO Astrophysics Symposia Black Holes in Binaries andGalactic Nuclei p 269 150

Freitag M Benz W 2001c AampA 375 711 124 131Freitag M Benz W 2002 AampA 394 345 124 134 135 137 150Freitag M Benz W 2005 MNRAS 358 1133 135 136Freitag M Gurkan M A Rasio F A 2006b MNRAS 368 141 124 135 137 147Freitag M Rasio F A Baumgardt H 2006c MNRAS 368 121 124 135 145 147

Fukushige T Heggie D C 2000 MNRAS 318 753 149Fulbright M S 1996 PhD thesis University of Arizona 137Gao B Goodman J Cohn H Murphy B 1991 ApJ 370 567 141Genzel R Schodel R Ott T Eisenhauer F Hofmann R Lehnert M Eckart A

Alexander T Sternberg A Lenzen R Clenet Y Lacombe F Rouan D RenziniA Tacconi-Garman L E 2003 ApJ 594 812 137 150

Gezari S Martin D C Milliard B Basa S Halpern J P Forster K FriedmanP G Morrissey P Neff S G Schiminovich D Seibert M Small T WyderT K 2006 ApJ Lett 653 L25 138

Ghez A M Salim S Hornstein S D Tanner A Lu J R Morris M BecklinE E Duchene G 2005 ApJ 620 744 137

Giersz M 1998 MNRAS 298 1239 123 131 141Giersz M 2001 MNRAS 324 218 123 141Giersz M 2006 MNRAS 371 484 123 141Giersz M Heggie D C 1994 MNRAS 268 257 145Giersz M Heggie D C 1996 MNRAS 279 1037 145Giersz M Heggie D C Hurley J R 2008 MNRAS 388 429Giersz M Spurzem R 2000 MNRAS 317 581 141Giersz M Spurzem R 2003 MNRAS 343 781 141 142 143Gnedin O Y Lee H M Ostriker J P 1999 ApJ 522 935 149Gnedin O Y Ostriker J P 1997 ApJ 474 223 149Goodman J 1983 ApJ 270 700 144Goodman J Hernquist L 1991 ApJ 378 637 143Grindlay J Portegies Zwart S McMillan S 2006 Nature Physics 2 116 141Gurkan M A Fregeau J M Rasio F A 2006 ApJ Lett 640 L39 123 141 147Gurkan M A Freitag M Rasio F A 2004 ApJ 604 123 147Gurkan M A Rasio F A 2005 ApJ 628 236 145Heggie D C 1975 MNRAS 173 729 143Heggie D Hut P 2003 The Gravitational Million-Body Problem A Multidisci-

plinary Approach to Star Cluster Dynamics CambridgeUniv Press Cambridge 141 145Heggie D C Hut P McMillan S L W 1996 ApJ 467 359 143Henon M 1960 Annales drsquoAstrophysique 23 668 144Henon M 1969 AampA 2 151 144Henon M 1971a ApampSS 14 151 123 149Henon M 1971b ApampSS 13 284 123 131Henon M 1973a in Martinet L Mayor M eds Lectures of the 3rd Advanced

Course of the Swiss Society for Astronomy and Astrophysics Obs de GeneveGeneve p 183 123 128

Henon M 1973b AampA 24 229 125Henon M 1975 in Hayli A ed Proc IAU Symp 69 Dynamics of Stellar Systems

Reidel Dordrecht p 133 123 144 145Hills J G 1975 Nature 254 295 137Hills J G 1988 Nature 331 687 141Hoffman L Loeb A 2007 MNRAS 334 153Hopman C Alexander T 2006a ApJ 645 1152 152Hopman C Alexander T 2006b ApJ Lett 645 L133 152Hopman C Freitag M Larson S L 2007 MNRAS 378 129 139Hurley J R Pols O R Aarseth S J Tout C A 2005 MNRAS 363 293 147Hurley J R Pols O R Tout C A 2000 MNRAS 315 543 143 144

156 M Freitag

Hurley J R Tout C A Aarseth S J Pols O R 2001 MNRAS 323 630 141 144Hut P 1993 ApJ 403 256 143Hut P McMillan S Goodman J Mateo M Phinney E S Pryor C Richer H B

Verbunt F Weinberg M 1992 PASP 104 981 141Johnston K V Sigurdsson S Hernquist L 1999 MNRAS 302 771 150Joshi K J Nave C P Rasio F A 2001 ApJ 550 691 123Joshi K J Rasio F A Portegies Zwart S 2000 ApJ 540 969 123 131Kim E Einsel C Lee H M Spurzem R Lee M G 2002 MNRAS 334 310 125Kim E Lee H M Spurzem R 2004 MNRAS 351 220 125Kim S S Lee H M 1999 AampA 347 123 134Kobayashi S Laguna P Phinney E S Meszaros P 2004 ApJ 615 855 137Komossa S 2005 in Merloni A Nayakshin S Sunyaev R A eds Growing Black

Holes Accretion in a Cosmological Context Springer Berlin p 269 138Kroupa P Aarseth S Hurley J 2001 MNRAS 321 699 150Lada E A 1999 in Lada C J Kylafis N D eds NATO ASIC Proc 540 The

Origin of Stars and Planetary Systems Kluwer Academic Publishers p 441 150Lauer T R Faber S M Ajhar E A Grillmair C J Scowen P A 1998 AJ 116

2263 150Laycock D Sills A 2005 ApJ 627 277 136Lee H M Ostriker J P 1986 ApJ 310 176 134Lightman A P Shapiro S L 1977 ApJ 211 244 138Lin D N C Tremaine S 1980 ApJ 242 789 144Lombardi Jr J C Proulx Z F Dooley K L Theriault E M Ivanova N Rasio

F A 2006 ApJ 640 441 136Lombardi J C Thrall A P Deneva J S Fleming S W Grabowski P E 2003

MNRAS 345 762 143Lombardi J C Warren J S Rasio F A Sills A Warren A R 2002 ApJ

568 939 136Merritt D 2006 ApJ 648 976 152Merritt D Berczik P Laun F 2007 AJ 133 553 152Merritt D Mikkola S Szell A 2007b ApJ 671 53 150Merritt D Milosavljevic M 2005 Living Reviews in Relativity 8 8 125 152Merritt D Szell A 2006 ApJ 648 890 150Mikkola S Aarseth S 2002 Celes Mech Dyn Ast 84 343 153Miller M C Colbert E J M 2004 International J Modern Phys D 13 1 137Miller M C Freitag M Hamilton D P Lauburg V M 2005 ApJ Lett

631 L117 141Miocchi P 2007 MNRAS 381 103 148Miralda-Escude J Kollmeier J A 2005 ApJ 619 30 152Muno M P Pfahl E Baganoff F K Brandt W N Ghez A Lu J Morris M R

2005 ApJ Lett 622 L113 141OrsquoLeary R M Loeb A 2008 MNRAS 383 86 144OrsquoLeary R M OrsquoShaughnessy R Rasio F A 2007 Phys Rev D 76 061504 141Perets H B Hopman C Alexander T 2007 ApJ 656 709 138Peters P C 1964 Phys Rev 136 1224 139Peters P C Mathews J 1963 Phys Rev 131 435 139Portegies Zwart S F Baumgardt H Hut P Makino J McMillan S L W 2004

Nature 428 724 141 147Portegies Zwart S F Makino J McMillan S L W Hut P 1999 AampA 348 117 136 141 147

Portegies Zwart S F McMillan S L W 2002 ApJ 576 899 141 147Portegies Zwart S F McMillan S L W Makino J 2007 MNRAS 374 95 147Portegies Zwart S F Verbunt F 1996 AampA 309 179 144Portegies Zwart S F Yungelson L R 1998 AampA 332 173 144Press W H Teukolsky S A Vetterling W T Flannery B P 1992 Numerical

Recipes in FORTRAN Cambridge Univ Press Cambridge 133 140 144Preto M Merritt D Spurzem R 2004 ApJ Lett 613 L109 150Quinlan G D Hernquist L Sigurdsson S 1995 ApJ 440 554 137Quinlan G D Shapiro S L 1990 ApJ 356 483 147Rasio F A Shapiro S L 1991 ApJ 377 559 136Rauch K P Ingalls B 1998 MNRAS 299 1231 152Rauch K P Tremaine S 1996 New Astronomy 1 149 152Rees M J 1988 Nature 333 523 137Regev O Shara M M 1987 MNRAS 227 967 136Ross S D 2004 PhD thesis Calif Inst Technology 149Rozyczka M Yorke H W Bodenheimer P Muller E Hashimoto M 1989 AampA

208 69 136Ruffert M 1993 AampA 280 141 136Schodel R Eckart A Alexander T Merritt D Genzel R Sternberg A Meyer

L Kul F Moultaka J Ott T Straubmeier C 2007 AampA 469 125 137Sedgewick R 1988 Algorithms Second Edition Addison-Wesley 131Sesana A Haardt F Madau P 2007 ApJ 660 546 152Shapiro S L 1985 in Goodman J Hut P eds Proc IAU Symp 113 Dynamics

of Star Clusters Reidel Dordrecht p 373 140Shara M ed 2002 ASP Conf Ser 263 Stellar Collisions amp Mergers and their

Consequences Astron Soc Pac San Francisco 133Shara M M Hurley J R 2002 ApJ 571 830 141Sills A Adams T Davies M B Bate M R 2002 MNRAS 332 49 136Sills A Deiters S Eggleton P Freitag M Giersz M Heggie D Hurley J Hut

P Ivanova N Klessen R S Kroupa P Lombardi J C McMillan S PortegiesZwart S F Zinnecker H 2003 New Astron 8 605 123

Sills A Faber J A Lombardi J C Rasio F A Warren A R 2001 ApJ 548323 136

Sills A Lombardi J C Bailyn C D Demarque P Rasio F A Shapiro S L1997 ApJ 487 290 136

Spitzer L 1987 Dynamical Evolution of Globular Clusters Princeton Univ PressPrinceton NJ 145 146 149

Spitzer L J Hart M H 1971a ApJ 164 399 124Spitzer L J Hart M H 1971b ApJ 166 483 124Spitzer L J Thuan T X 1972 ApJ 175 31 124Spitzer L Mathieu R D 1980 ApJ 241 618 125 141Spitzer L Shull J M 1975 ApJ 201 773 124Spurzem R Giersz M Heggie D C Lin D N C 2006 preprint (astro-

ph0612757) 141 142Stodolkiewicz J S 1982 Acta Astron 32 63 123 131Stodolkiewicz J S 1985 in Goodman J Hut P eds Proc IAU Symp 113 Dy-

namics of Star Clusters Reidel Dordrecht p 361 141Stodolkiewicz J S 1986 Acta Astron 36 19 123 141Subr L Karas V Hure J-M 2004 MNRAS 354 1177 125 152

158 M Freitag

Syer D Clarke C J Rees M J 1991 MNRAS 250 505 152Taam R E Ricker P M 2006 preprint (astro-ph0611043) 136Takahashi K Portegies Zwart S F 2000 ApJ 535 759 149Theis C Spurzem R 1999 AampA 341 361 125Trac H Sills A Pen U-L 2007 MNRAS 337 136Tremaine S Gebhardt K Bender R Bower G Dressler A Faber S M Filippenko

A V Green R Grillmair C Ho L C Kormendy J Lauer T R MagorrianJ Pinkney J Richstone D 2002 ApJ 574 740 137

Trenti M Ardi E Mineshige S Hut P 2007 MNRAS 374 857 148van der Marel R P 2004 in Ho L ed Coevolution of Black Holes and Galaxies

from the Carnegie Observatories Centennial Symposia Cambridge Univ PressCambridge p 37 137

Young P J 1980 ApJ 242 1232 137

6

Particle-Mesh Technique and Superbox

Michael Fellhauer

University of Cambridge Institute of Astronomy Madingley Road CambridgeCB3 0HA UKmadfastcamacuk

61 Introduction

Many problems in astronomy ranging from celestial mechanics via stellar dy-namics to cosmology require the solution of Newtonrsquos laws

F = a middotm = mdv

dt(61)

v =dr

dt (62)

where F is the gravitational force of all other (N minus 1) masses

F j =Nsum

i=1i =j

Gmjmi

r3ijrij (63)

acting on mass j (index ij denotes the vectors connecting particle i and j)While there is an analytical solution for the two-body system systems

involving three or more masses do not have an analytical solution Thus com-puter simulations of the time-evolution of multi-body systems are very com-mon in astronomy

The tools used for these purposes are diverse and widely range from high-precision integrators for the dynamics of the planetary systems to programmesusing up to a billion particles to investigate the structure formation in theuniverse This article focuses on the particle-mesh technique and a programmeto simulate galaxies called Superbox

The particle-mesh (PM) technique is explained in Sect 62 Then themulti-grid structure of Superbox is described in Sect 63

Fellhauer M Particle-Mesh Technique and SUPERBOX Lect Notes Phys 760 159ndash169 (2008)

160 M Fellhauer

62 Particle-Mesh Technique

621 Overview

In the particle-mesh technique the density of the particles is sampled on agrid covering the simulation area and then Poissonrsquos equation

nabla2Φ = 4πG (64)

is solved on the grid-based density using a suitable Greenrsquos function to derivethe grid-based gravitational potential Particles are integrated using the forcesderived from this grid-based potential

The first step is to locate the grid-point of each particle according toits position and derive a grid of densities This density-grid is Fourier-transformed via the Fast Fourier Transform(FFT) algorithm This requiresthat the number of grid-cells per dimension is a power of 2 The Fourier-transformed density-grid is multiplied cell-by-cell with a suitable alreadyFourier-transformed Greenrsquos function Then these values are back-transformedwhich results in a grid of potential values From these potential values theforces of each particle are derived via discrete differentiation Finally the par-ticle velocities and positions are integrated forward in time

A flow-chart of a standard PM-code is shown in Fig 61

read input data

forward FFT of Greenrsquos Function

start timeminusstep loop

derive gridminusbased density array

forward FFT of density array

cellminusbyminuscell multiplication with Greenrsquos Fkt

backward FFT to derive potential array

start particle loop

differentiate potential to get force

integrate velocities

integrate positions

collect output data

write final data

Fig 61 Flow-chart of a standard PM-code

6 Particle-Mesh Technique and Superbox 161

622 Suitable Greenrsquos Function

The usual geometry of the grid in a particle-mesh code is Cartesian and cu-bic Therefore the standard Greenrsquos function which describes the distancesbetween cells looks like

Hijk =1

radici2 + j2 + k2

i j k = 0 n

H000 =1ξ (65)

This formula implies that the length of one grid-cell is unity n is the numberof grid-cells per dimension and has to be a power of 2

The value for H000 has to be chosen carefully It describes the strength ofthe force between particles in the same cell including the non-physical lsquoself-gravityrsquo of the particle acting on itself In the one-dimensional case analyticalstudies by D Pfenniger showed a value of ξ = 34 gives the best results interms of energy conservation Numerical experiments showed that this is alsotrue in the three-dimensional case

Nevertheless in the case of very low particle numbers per cell this valuecould lead to spurious self-accelerations and a value that excludes the forcesof particles from the same cell would be more suitable In the Superbox

differentiation scheme the value to exclude self-gravity is ξ = 1 In a latersection we discuss why one should avoid low particle-per-cell ratios if possible

Finally it can be stated that the grid-array of the Greenrsquos function hasto be set up and Fourier-transformed only once at the beginning of eachsimulation and can then be used throughout the whole simulation

623 Deriving the Density-Grid

The actual positions and velocities of each particle (x y z vx vy vz) are storedin the particle array From the actual positions the grid-cell in which eachparticle is located is derived via

ix = nearest integer(enh middot x) + n2 (66)

ix denotes the grid-cell number in the x-direction enh is a numerical factorthat stretches or compresses the physical extension of the x-direction of thesimulation area to allow the grid-cell length to be unity The grid-cell numbersin the y- and z-direction are derived accordingly

There are two possibilities to assign the mass of the particle to the density-grid covering the simulation area One is called nearest-grid-point scheme andassigns the whole mass of the particle to the grid-cell that the particle is inA second more advanced procedure is called cloud-in-cell scheme and assignsa radius of half a cell length to each particle The mass of the particle is nowdistributed to the cells this extended particle is in according to the actual

162 M Fellhauer

1

2

3

n

1 3

ix

iy +mass

grid of densities

1

2

3

4

5

N

x z vx vy vz

array of particles

ix = nint(enhx + n2)

iy = nint(enhy + n2)

y

n2

Fig 62 Deriving the density-grid from the particle positions The z-dimensionis omitted for clarity In the NGP scheme the total mass is placed in one cell inthe CIC scheme contributions of the mass are distributed in neighbouring cells also(denoted by the circle)

deviation of the particle position from the centre of the cell In Fig 62 thisassignment is shown for two dimensions

The CIC scheme allows for a much smoother distribution of the densitiesbut does not allow for sub-cell-length resolution This has to be added via di-rect summation of the forces of neighbouring particles within a certain sphereof influence A code that employs direct summation in the vicinity of eachparticle is usually called P3M-code (particle-particle particle-mesh) The CICscheme also allows for a smooth and high accuracy derivation of the forces(this will be discussed in a sub-section below)

Superbox still uses the lsquoold-fashionedrsquo NGP-scheme which results in amuch faster assignment of the densities and allows for sub-cell-length resolu-tion if H000 = 1 To reach the high accuracy we later apply a higher-orderdifferentiation scheme to obtain the forces

624 The FFT-Algorithm

Poissonrsquos equation is solved for the density-grid to get the grid-based potentialΦijk which becomes

Φijk = Gnminus1sum

abc=0

abc middotHaminusibminusjcminusk i j k = 0 nminus 1 (67)

where n denotes the number of grid-cells per dimension (n3 = Ngc totalnumber of grid-cells) and Hijk is the Greenrsquos function To avoid this N2

gc pro-cedure the discrete Fast Fourier Transform (FFT) is used for which n = 2kk gt 0 being an integer The stationary Greenrsquos function is Fourier-transformed

once at the beginning of the calculation and only the density array is trans-formed at each time-step

abc =nminus1sum

ijk=0

ijk middot exp(

minusradicminus1

2πn

(ai+ bj + ck))

Habc =nminus1sum

ijk=0

Hijk middot exp(

minusradicminus1

2πn

(ai+ bj + ck))

(68)

The two resulting arrays are multiplied cell by cell and transformed back toget the grid-based potential

Φijk =G

n3

nminus1sum

abc=0

abc middot Habc middot exp(radic

minus12πn

(ai+ bj + ck))

(69)

The FFT-algorithm gives the exact solution of the grid-based potential for aperiodic system For the exact solution of an isolated system which is whatsimulators are interested in the size of the density array has to be doubled(2n) filling all inactive grid cells with zero density and extending the Greenrsquosfunction in the empty regions in the following way (also shown in Fig 63)

H2nminusijk = H2nminusi2nminusjk = H2nminusij2nminusk = H2nminusi2nminusj2nminusk

= Hi2nminusjk = Hi2nminusj2nminusk = Hij2nminusk = Hijk (610)

This provides the isolated solution of the potential in the simulated area be-tween i j k = 0 and n minus 1 In the inactive part the results are unphysicalTo keep the data size as small as possible only a 2n times 2n times n-array is usedfor transforming the densities and a (n+ 1)times (n+ 1)times (n+ 1)-array is usedfor the Greenrsquos function For a detailed discussion see Eastwood amp Brownrigg(1978) and also Hockney amp Eastwood (1981)

The FFT-routine incorporated in Superbox is a simple one-dimensionalFFT and is taken from Werner amp Schabach (1979) and Teukolsky et al (1992)It is fast and makes the code portable and not machine-specific The low-storage algorithm for extending the FFT to three dimensions to obtain the3-D potential is taken from Hohl (1970) The performance of Superbox canbe increased by incorporating machine-optimised FFT routines

A detailed description of the low-storage FFT algorithm used in Super-

box can be found in the manual available directly from the author (Fellhauer2006)

625 Derivation of the Forces

After the FFT procedure has been completed one has a grid-based potentialof the simulation area From this potential the forces acting on each particleare derived via discrete numerical differentiation of the potential

164 M Fellhauer

simulated object

active simulation area

empty ghost region empty ghost region

empty ghost region

1 3 n n+1 2n

gridminusarray rho not existent as array

not existent as arraynot existent as array

2

Fig 63 Virtual extension of the simulation area to provide isolated solution(z-direction omitted)

As with the mass assignment of the density array the forces are also cal-culated differently depending on whether a NGP or CIC scheme is used ANGP scheme only uses the force calculated for the grid-cell the particle isin while in a CIC scheme forces of the neighbouring cells are used with thesame weights the mass was distributed to interpolate the force to the particleposition

For simplicity the force derivation of the different schemes is given in a1D case

NGP a(xi + dx) =partΦpartx

∣∣∣∣i

(611)

SUPERBOX a(xi + dx) =partΦpartx

∣∣∣∣i

+part2Φpartx2

∣∣∣∣i

middot dxΔx

(612)

CIC a(xi + dx) =partΦpartx

∣∣∣∣i

middot Δxminus dxΔx

+partΦpartx

∣∣∣∣i+1

middot dxΔx

(613)

where a denotes the acceleration xi is the position of the cell with index i theparticle is located in and dx is the deviation of the particle from the centreof the cell As one can see the standard NGP scheme does not account forthe deviation of the particle from the centre of the cell The acceleration isa step function from cell to cell and is not steady at all The CIC schemeaccounts for this deviation and the acceleration of the particle is a weightedmean from the cell the particle is in and the neighbouring cell Superbox hasa non-standard force calculation scheme which is definitely NGP in nature(only the force for the cell i is used) but accounts for the deviation by usingthe next term of a Taylor series of the acceleration around the cell i Thesteadiness of the force is not guaranteed when crossing the cell boundaries at

an arbitrary angle but anisotropies of the force are suppressed The full 3Dexpression for the acceleration in Superbox is

aijkx(dxdydz) =partΦpartx

∣∣∣∣ijk

+part2Φpartx2

∣∣∣∣ijk

dx+part2Φpartxparty

∣∣∣∣ijk

dy +part2Φpartxpartz

∣∣∣∣ijk

dz

(614)

The partial derivatives are replaced in the code by second-order central dif-ferentiation quotients and now the 3D expression for the acceleration in thex-direction reads

aijkx(dxdydz) =Φi+1jk minus Φiminus1jk

2Δx

+Φi+1jk + Φiminus1jk minus 2 middot Φijk

(Δx)2middot dx

+Φi+1j+1k minus Φiminus1j+1k + Φiminus1jminus1k minus Φi+1jminus1k

4ΔxΔymiddot dy

+Φi+1jk+1 minus Φiminus1jk+1 + Φiminus1jkminus1 minus Φi+1jkminus1

4ΔxΔzmiddot dz (615)

Note that generally Δx = Δy = Δz = 1 ie the cell-length is assumed to beequal along the three axes and unity i j k are the cell indices of the particlein the three Cartesian coordinates The accelerations in y- and z-direction arecalculated analogously

626 Integrating the Particles

The orbits of the particles are integrated forward in time using the leapfrogscheme For example for the x-components of the velocity vx and positionx vectors of particle l

vn+12xl = v

nminus12xl + an

xl middot Δt

xn+1l = xn

l + vn+12xl middot Δt (616)

where n denotes the nth time step and Δt is the length of the integrationstep

Superbox uses a fixed global time step ie the time step is the same forall particles and does not vary in time

The leapfrog integrator together with the fixed time step is very fast (nodecision-making necessary) and is accurate enough for a grid-based code It isin principle time-reversible and has very good energy conservation propertiesconsidering its simplicity

166 M Fellhauer

63 Multi-Grid Structure of SUPERBOX

A detailed description of the code is also found in Fellhauer et al (2000) Foreach galaxy five grids with three different resolutions are used This is madepossible by invoking the additivity of the potential (Fig 64)

The five grids are as follows

bull Grid 1 is the high-resolution grid that resolves the centre of the galaxy Ithas a length of 2timesRcore in one dimension In evaluating the densities allparticles of the galaxy within r le Rcore are stored in this grid

bull Grid 2 has an intermediate resolution to resolve the galaxy as a wholeThe length is 2 times Rout but only particles with r le Rcore are stored hereie the same particles as are also stored in grid 1

bull Grid 3 has the same size and resolution as grid 2 but it contains onlyparticles with Rcore lt r le Rout

bull Grid 4 has the size of the whole simulation area (ie lsquolocal universersquo with2 times Rsystem) and has the lowest resolution It is fixed Only particles ofthe galaxy with r le Rout are stored in grid 4

RoutRout

Grid 4 Grid 5

Rout Rout

Rcore

RcoreRcore

Rcore

Rsystem

RsystemRsystem

Rsystem

Grid

1

Grid

2

Grid 1 + 2 Grid 3

Fig 64 The five grids of Superbox In each panel solid lines highlight the relevantgrid Particles are counted in the shaded areas of the grids The lengths of the arrowsare (N2)minus2 grid-cells (see text) In the bottom left panel the grids of a hypotheticalsecond galaxy are also shown as dotted lines

bull Grid 5 has the same size and resolution as grid 4 This grid treats theescaping particles of a galaxy and contains all particles with r gt Rout

Grids 1 to 3 are focused on a common centre of the galaxy and move with itthrough the lsquolocal universersquo as detailed below All grids have the same numberof cells per dimension n for all galaxies The boundary condition requiringtwo empty cells with = 0 at each boundary is open and non-periodic thusproviding an isolated system This however means that only nminus 4 active cellsper dimension are used

To keep the memory requirement low all galaxies are treated consecutivelyin the same grid-arrays whereby the particles belonging to different galaxiescan have different masses Each of the five grids has its associated potentialΦi i = 1 2 5 computed by the PM technique from the particles of onegalaxy located as described above The accelerations are obtained additivelyfrom the five potentials of each galaxy in turn in the following way

Φ(r) = [θ(Rcore minus r) middot Φ1 + θ(r minusRcore) middot Φ2 + Φ3] middot θ(Rout minus r)+ θ(r minusRout) middot Φ4 + Φ5

Φ(Rcore) = Φ1 + Φ3 + Φ5

Φ(Rout) = Φ2 + Φ3 + Φ5 (617)

where θ(ξ) = 1 for ξ gt 0 and θ(ξ) = 0 otherwise This means

bull For a particle in the range r le Rcore the potentials of grids 1 3 and 5 areused to calculate the acceleration

bull For a particle with Rcore lt r le Rout the potentials of grids 2 3 and 5 arecombined

bull And finally if r gt Rout the acceleration is calculated from the potentialsof grids 4 and 5

bull Any particle with r gt Rsystem is removed from the computation

Due to the additivity of the potential (and hence its derivatives the accel-erations) the velocity changes originating from the potentials of each of thegalaxies can be separately updated and accumulated in the first of the leapfrogformulae (616) The final result does not depend on the order by which thegalaxies are taken into account and it could be computed even in parallel ifa final accumulation takes place After all velocity changes have been appliedto all galaxies the positions of the particles are finally updated

As long as the galaxies are well separated they feel only the low-resolutionpotentials of the outer grids But as the galaxies approach each other theirhigh-resolution grids overlap leading to a high-resolution force calculationduring the interaction

631 Grid Tracking

Two alternative schemes to position and track the inner and middle grids canbe used The most useful scheme centres the grids on the density maximum

168 M Fellhauer

of each galaxy at each step The position of the density maximum is found byconstructing a sphere of neighbours centred on the densest region in whichthe centre of mass is computed This is performed iteratively The other optionis to centre the grids during run-time on the position of the centre of mass ofeach galaxy using all its particles remaining in the computation

632 Edge-Effects

It can be seen in Fig 64 that only spherical regions of the cubic grids containparticles (except for grid 5) Particles with eccentric orbits can cross the borderof two grids thus being subject to forces resolved differently No interpolationof the forces is done at the grid boundaries This keeps the code fast andslim but the grid sizes have to be chosen properly in advance to minimise theboundary discontinuities It leads to some additional but negligible relaxationeffects because the derived total potential has insignificant discontinuities atthe grid boundaries (Wassmer 1992) The best way to avoid these edge-effectsis to place the grid boundaries at lsquoplacesrsquo where the slope of the potential isnot steep

633 Choice of Parameters

Finally we make some comments on the right choice of parameters In princi-ple Superbox works with all sets of parameters but the outcome might beunphysical The user has to check if the choice makes sense or not There area few rules that help to ensure that the simulation is not unrealistic Firstone should check if there are enough particles for the given resolution As arule-by-thumb one can divide the number of particles by the total number ofcells of one grid If the mean number of particles per cell amounts to a fewthen one is on the safe side (conservative lt N gtasymp 10minus15) Second one shouldcheck the time-step Particles should not travel much more than one grid-cellper time step otherwise one again loses resolution Another rule-by-thumb istake the shortest crossing-time of all objects and divide it by 10 (conservative50ndash70) This ensures that this object stays stable It is also not useful to havelarge resolution steps between the grid levels At least one should avoid themin all places of interest

References

Eastwood J W Brownrigg D R K 1978 J Comput Phys 32 24 163Fellhauer M 2006 Superbox manual madfastcamacuk 163Fellhauer M Kroupa P Baumgardt H Bien R Boily C M Spurzem R Wassmer

N 2000 NewA 5 305 166Hockney R W Eastwood J W 1981 Computer Simulations Using Particles

McGraw-Hill 163

Hohl F 1970 NASA Technical Report R-343 163Teukolsky S A Vetterling W T Flannery B P 1992 Numerical Recipes in

Fortran Cambridge University Press Cambridge 163Wassmer N 1992 Diploma thesis University Heidelberg 168Werner H Schabach R 1979 Praktische Mathematik II Springer 163

7

Dynamical Friction

Michael Fellhauer

71 What is Dynamical Friction

Dynamical friction is as the name says a deceleration of massive objects Itoccurs whenever a massive object travels through another extended objectThis behaviour makes dynamical friction one of the most important effects instellar dynamics

It occurs on all kinds of length-scales and objects from the sinking to thecentre of massive stars inside a star cluster leading to mass segregation viasinking of star clusters and dwarf galaxies inside the host galaxy to collisionsof massive galaxies

Dynamical friction is a pure gravitational interaction between the massiveobject (M) and the multitude of lighter stars (m) of the extended object it istravelling through (see Fig 71 left panel) In the rest-frame of the moving ob-ject M the lighter stars are oncoming from the front and get deflected behindthe object (see Fig 71 middle panel) These many gravitational interactionssum up to an effective deceleration of the object while some of the deflectedlighter particles m build up a wake behind M (see Fig 71 right panel) Thiswake can be measured and may induce an extra drag on the moving objectbut the drag is neglected in the determination of the standard description ofdynamical friction It is dynamical friction which causes the wake and not thewake being responsible for the dynamical friction

Mv

Mwake

Fig 71 Dynamical friction as a cartoon

Fellhauer M Dynamical Friction Lect Notes Phys 760 171ndash179 (2008)

172 M Fellhauer

Hence dynamical friction causes a deceleration of the object M and there-fore if it was on a stable orbit before causes a shrinking of this orbit andsinking to the centre in response to the deceleration If the object is initiallyon an eccentric orbit dynamical friction acts in a way that the orbit gets moreand more circular

72 How to Quantify Dynamical Friction

Dynamical friction was first quantified by Chandrasekhar (1943) In this sec-tion the classical way to derive the dynamical friction formula will be followed(see for example Binney amp Tremaine 1987 chapter 71)

Before the multitude of encounters can be treated one has to focus on asingle encounter The geometry of this encounter is shown in the left panel ofFig 72 Defining r = xm minus xM as the separation vector between m and Mand V = r one gets the relative velocity change

ΔV = Δvm minus ΔvM (71)

Because this two-body system is conservative one can apply momentum con-servation which leads to

mΔvm +MΔvM = 0 (72)

Combining these two equations and eliminating Δvm gives ΔvM as a functionof ΔV

ΔvM = minus(

m

m+M

)

ΔV (73)

In the right panel of Fig 72 we show the hyperbolic geometry of the Keplerproblem in the frame of the reduced particle mass travelling in the combinedpotential due to both particles (m + M) The conserved angular momentum

m

M

xM

xm

r

vm

vMV0

V0

ψ ψ0

θb r

Fig 72 Left Geometry of a single encounter Right The motion of the reducedparticle during a hyperbolic encounter V 0 = V (t = minusinfin) is the initial velocity b isthe impact parameter and θ is the deflection angle

7 Dynamical Friction 173

(per unit mass) in this system is L = bV0 = r2Ψ From the analytical solu-tion of the Kepler problem we know the equation that relates radius r andazimuthal angle Ψ

1r

= C cos(Ψ minus Ψ0) +G(m+M)

b2V 20

(74)

where C and Ψ0 are constants defined by the initial conditions If (74) isdifferentiated with respect to time one gets

drdt

= Cr2Ψ sin(Ψ minus Ψ0) = CbV0 sin(Ψ minus Ψ0) (75)

Evaluating (74) and (75) at t = minusinfin one obtains

0 = C cos(Ψ0) +G(m+M)

b2V 20

(76)

minusV0 = CbV0 sin(minusΨ0) (77)

Using these two equations to eliminate C leads to

tan(Ψ0) = minus bV 20

G(m+M) (78)

The point of closest approach is reached when Ψ = Ψ0 and since the orbitis symmetrical about this point the deflection angle is θ = 2Ψ0 minus π Byconservation of energy the length of the relative velocity vector is the samebefore and after the encounter and has the value V0 Hence the componentsΔV and ΔV perp of ΔV are given by

=2bV 3

0

G(m+M)

[

1 +b2V 4

0

G2(m+M)2

]minus1

(79)

∣∣ΔV

∣∣ = V0 [1 minus cos(θ)] = V0(1 + cos(2Ψ0)) =

2V0

1 + tan2(Ψ0)

= 2V0

[

1 +b2V 4

0

G2(m+M)2

]minus1

(710)

ΔV always points in the direction opposite to V 0 Using (73) one finallygets

|ΔvMperp| =2mbV 3

0

G(m+M)2

[

1 +b2V 4

0

G2(m+M)2

]minus1

(711)

∣∣ΔvM

∣∣ =

2mV0

m+M

[

1 +b2V 4

0

G2(m+M)2

]minus1

(712)

174 M Fellhauer

Hence by (73) ΔvM always points in the same direction as V 0Let us now imagine that M travels through an infinite homogeneous ldquosea

of particlesrdquo Then there are as many deflections from ldquoaboverdquo as from ldquobe-lowrdquo or from ldquorightrdquo or ldquoleftrdquo and the changes in ΔvMperp sum up to zeroFurthermore one has to invoke the ldquoJeans swindlerdquo to neglect the gravita-tional potential of the ldquosea of particlesrdquo so the motion of each particle isdetermined only by M The changes in ΔvM are all parallel to V 0 and forma non-zero resultant ie the mass M suffers a steady deceleration which issaid to be dynamical friction

To determine the deceleration one now has to integrate over all possibleimpact parameters b and velocities vm The number density of particles mwith velocity distribution f(v) in the velocity-space element d3vm at impactparameters between b and b+ db is

2πbdbtimes V0 times f(vm)d3vm (713)

Hence the net rate of change of vM is

dvM

dt

∣∣∣∣vm

= V 0f(vm)d3vm

int bmax

0

2mV0

m+M

[

1 +b2V 4

0

G2(m+M)2

]minus1

2πbdb

(714)

with bmax the largest impact parameter to be considered Performing theintegration over all b one finds

dvM

dt

∣∣∣∣vm

= 2π ln(1 + Λ2)G2m(m+M)f(vm)vm minus vM

|vm minus vM |3d3vm (715)

with

Λ =bmaxV

20

G(m+M)=

bmax

bmin (716)

Usually Λ is very large and so one can assume that 12 ln(1 + Λ2) asymp ln(Λ)

which is called the Coulomb logarithm Furthermore one replaces V0 by thetypical speed vtyp Equation (715) states that particles that have velocity vm

exert a force on M that acts parallel to vmminusvM and is inversely proportionalto the square of this vector The problem to integrate over all velocities vm isequivalent to finding the gravitational field at the point with position vectorin velocity space vM which is generated by the ldquomass densityrdquo ρ(vm) =4π ln(Λ)Gm(m + M)f(vm) If the particles move isotropically the densitydistribution is spherical and according to Newtonrsquos first and second theoremthe total acceleration of M is equal to Gv2

M times the total ldquomassrdquo at vm ltvM Hence

dvM

dt= minus16π2 ln(Λ)G2m(m+M)

int vM

0f(vm)v2

mdvm

v3M

vM (717)

ie only particles m with velocities slower than M contribute to the force thatalways opposes the motion of M and this equation is henceforth called theChandrasekhar dynamical friction formula

If f(vm) is Maxwellian with dispersion σ then

f =n0

(2πσ2)32exp

(

minus v2

2σ2

)

(718)

and introducing ρ = n0m as the background density one can perform theintegration which gives

dvM

dt= minus4π ln(Λ)G2ρM

v3M

[

erf(X) minus 2Xradicπ

exp(minusX2)]

vM (719)

with X = vMradic

2σ This formula holds for M mWith this formula one can derive some useful relations If keeping ln Λ

constant we can determine the time a star cluster or dwarf galaxy needs tospiral into the centre of its host system

tfric =117D2

0vcirc

ln(Λ)GM=

264 times 1011

ln(Λ)

(D0

2 kpc

)2 ( vcirc

250 km sminus1

)(106 MM

)

yr

(720)

Furthermore McMillan amp Portegies Zwart (2003) derived a formula for thesinking rate if the background is a mass distribution following a power law ofthe form M(D) = A middotDα Then the distance D of an object to the centre ofthe host system vs time is given by

D(t) = D0

[

1 minus α(α+ 3)α+ 1

radicG

ADα+30

χM ln(Λ)t

]23+α

(721)

with

χ = erf(X) minus 2Xradicπ

exp(minusX2) (722)

where X = vMradic

2σEven though one might think that the derivation of Chandrasekharrsquos for-

mula has too many vague definitions and approximations in it it has beenshown that it is a really powerful tool to describe dynamical friction in allkinds of environments

73 Dynamical Friction in Numerical Simulations

Especially in numerical simulations the validity of Chandrasekharrsquos formulahas been verified throughout the decades Still some words of caution haveto be added In the previous section it was shown that Λ = bmaxbmin withbmintheo = G(m + M)v2

M in the extreme case of a point mass being a verysmall quantity (eg for a 106 M black hole with a velocity of 50 km sminus1 gives

176 M Fellhauer

bmin asymp 2 pc) For extended objects like a star cluster bmin is of the order ofthe size of the cluster

However even if one uses a point mass to determine dynamical friction it isnot easy to reach the correct result All standard N -body codes are resolution-limited Even if one does not introduce softening and uses a direct summationN -body code the limitation gets introduced through the finite particle num-ber In a study how dynamical friction is influenced by the resolution of thesimulation code (ie the softening length used) Spinnato et al (2003) showedthat with a given softening length ε (or in the case of a particle-mesh codethe cell-length )

bmineff asymp bmintheo + ε (or ) (723)

This is shown as the actual sinking curve for two choices of resolution in aparticle-mesh code in the left panel of Fig 73 and for all choices of ε as thederived ln(Λ) in the comparison to a direct summation N-body codes a treecode and a particle-mesh code in the right panel

In this study ln(Λ) was assumed to be constant during the whole simula-tion time independently of the actual distance D to the centre of the back-ground Fitting bmax of a constant ln(Λ) to the data resulted in bmax = kD0

with k asymp 05In another study Fellhauer amp Lin (2007) used the same approach but fitted

ln(Λ) at many small time-slices during the sinking process and determinedbmin as function of the resolution and bmax as function of the distance D asshown in Fig 74

ln Λ = ln(bmax) minus ln(bmin)= ln(kprime middotD(t)) + bmineff (724)

The values for bmineff were in very good agreement with (723) for the differentresolutions Superbox the particle-mesh code used in this study has threelevels of grid-resolutions While the point-mass starts inside the medium res-olution it crosses the grid-boundary to the high-resolution area when D lt 1

77

6

5

4

30 05 1 15 2 25

6

5

4

3

2

1

00 5 10 15 20 25 30

InΛ

N = 80 000 PP dataPP fit

tree datatree fit

PM dataPM fit

N = 2 000 000

I asymp 23ε0

εIε0

I asymp 10ε0

I asymp 5ε0

12

1

08

06

04

02

00 100 300 400

t500 600200

RR

0

Fig 73 Influence of the resolution on the dynamical friction of a point mass

D

InΛ

Fig 74 ln(Λ) as a function of the distance to the centre of the background Alsovisible is the change in resolution for D lt 1 which leads to a smaller value of bmin

and a larger value of ln(Λ) ln(Λ) is decreasing with decreasing distance Fittingcurves assume bmax prop D (724)

in the above simulation The values for kprime differ from the value k found in theprevious study and also seem to be dependent on the resolution

74 Dynamical Friction of an Extended Object

In the previous section the dependence of ln(Λ) on environment was investi-gated which was possible because the studies involved the sinking of a pointmass with constant mass In many cases of dynamical friction the sinking ob-ject is extended and due to tidal forces acting on it the mass is not constantThis section investigates which mass one has to insert into the dynamicalfriction formulae like (719) and (721)

The initial mass and orbit of the extended object (it could be a star clusteror a dwarf galaxy) is the same as the one of the point-mass of the previoussection We use again (721) to fit now the combined quantity Mcl ln(Λ) Forthe left panel this quantity is converted into ln(Λ) in the following two ways

ln Λ(t)crosses = (Mcl ln Λ)(t)Mbound(t = 0) (725)ln Λ(t)triminuspods = (Mcl ln Λ)(t)Mbound(t) (726)

The curves show that either way does not give the correct answer If themass is kept constant and the initial mass is inserted the data points fallbelow the reference line of the point-mass case This disparity is expectedsince an extended object should have a larger bmin than that of a point-mass

178 M Fellhauer

potential For t lt 30 or D gt 1 the difference between these two simulationsis less than 20 per cent However it can also be seen that the deviation fromthe fitting line grows with time especially at t gt 30 (or equivalently as Ddecreases below 1) This growing difference is due to the loss of mass fromthe stellar cluster This divergence shows that a constant Mcl approximationdoes not adequately represent the results of the simulation If one inserts thebound mass as responsible for the dynamical friction the measured values aresystematically above the fitting line that represents the cluster with a point-mass potential However using the above argument that an extended objectshould have a larger bmin than that of a point-mass potential the tri-podsmeasured from this simulation would be systematically below the fitting lineif the bound stars adequately account for all the mass that contributes to thedynamical friction This disparity is a first hint that more particles may takepart in the dynamical friction than just the bound stars In the later stagesof the evolution these values of ln Λ increase quite dramatically which is aclear sign that Mcl is underestimated

In the right panel of Fig 75 the bound mass of the object as a function oftime (solid line) is plotted In the same figure crosses and squares representthe mass of the cluster taking part in the dynamical friction process if the sameln Λ as that derived for a point-mass is assumed Then one solves for Mcl with(721) For the crosses the actual values from the point-mass simulation isapplied while the data-points of the squares are derived using the smoothed

InΛ

tD

Mcl

Fig 75 Dynamical friction on an extended object Left Fitting Mcl ln(Λ) to thesinking curve in small time-slices like in Fig 74 and deriving ln(Λ) according to(725) amp (726) Right Using the values of ln(Λ) derived from the point-mass case todetermine Mcl the mass responsible for dynamical friction (yellow squares using thefitting formulae black crosses with error-bars using the actual values of the point-mass simulation) (Red) solid line shows the bound mass of the object long dashed(green) line the bound mass plus the unbound mass in a ring around the centre ofthe background with size of the object (Red) short dashed line is the rule-by-thumbbound mass plus half of the unbound mass

fitting curve for ln Λ(D) from (724) (Since it has already been shown that themagnitude of ln Λ(D) for a cluster with a Plummer potential is smaller thanthat for a point mass the actual total mass that contributes to the dynamicalfriction is slightly larger than both the values represented by the crosses andthe squares) Even though the uncertainties are large the data points showthat the total mass responsible which contribute to the effect of dynamicalfriction is systematically above the bound mass in the bound mass curve

In addition to the bound mass the lost mass of the cluster which is locatedin a ring of the cluster dimension around the galaxy at the same distanceis calculated and only the particles with the same velocity signature as thecluster are counted Adding this mass to the bound mass is shown as theshort dashed line in the right panel of Fig 75 This mass estimate seems tofit the data much better This value is not easy to access and surely has to bereplaced by a more elaborate formulation of dynamical friction ie assigningweights to all unbound particles with respect to their position and velocityto the cluster Thus applying a simple rule-by-thumb by adding half of theunbound mass to the bound mass (shown as long dashed line in the rightpanel of Fig 75) fits the data nicely taking into account that the ldquoactualrdquoln Λ of an extended object should be smaller than the one of a point mass iethe data points have to be regarded as lower limits Even though this simpleestimate has no physical explanation and breaks down during the very finalstages of the dissolution of the cluster it gives an easy accessible estimate ofthe dynamical friction of an extended object suffering from mass-loss

References

Binney J Tremaine S 1987 Galactic Dynamics Princeton Univ PressPrinceton NJ 172

Chandrasekhar S 1943 ApJ 97 255 172Fellhauer M Lin D N C 2006 MNRAS 375 604 176McMillan S L Portegies Zwart S F 2003 ApJ 596 314 175Spinnato P F Fellhauer M Portegies Zwart S F 2003 MNRAS 344 22 176

8

Initial Conditions for Star Clusters

Pavel Kroupa

Argelander-Institut fur Astronomie Auf dem Hugel 71 D-53121 Bonn Germanypavelastrouni-bonnde

81 Introduction

Most stars form in dense star clusters deeply embedded in residual gas Thepopulations of these objects range from small groups of stars with about adozen binaries within a volume with a typical radius of r asymp 03 pc throughto objects formed in extreme star bursts containing N asymp 108 stars withinr asymp 36 pc Star clusters or more generally dense stellar systems must there-fore be seen as the fundamental building blocks of galaxies Differentiationof the term star cluster from a spheroidal dwarf galaxy becomes blurred nearN asymp 106 Both are mostly pressure-supported that is random stellar motionsdominate any bulk streaming motions such as rotation The physical processesthat drive the formation evolution and dissolution of star clusters have a deepimpact on the appearance of galaxies This impact has many manifestationsranging from the properties of stellar populations such as the binary frac-tion and the number of type Ia and type II supernovae through the velocitystructure in galactic discs such as the agendashvelocity dispersion relation to theexistence of stellar halos around galaxies tidal streams and the survival andproperties of tidal dwarf galaxies the existence of which challenge current cos-mological perspectives Apart from this cosmological relevance dense stellarsystems provide unique laboratories in which to test stellar evolution theorygravitational dynamics the interplay between stellar evolution and dynamicalprocesses and the physics of stellar birth and stellar feedback processes duringformation

Star clusters and other pressure-supported stellar systems in the skymerely offer snap-shots from which we can glean incomplete information Be-cause there is no analytical solution to the equations of motion for more thantwo stars these differential equations need to be integrated numerically Thusin order to gain an understanding of these objects in terms of the above is-sues a researcher needs to resort to numerical experiments in order to testvarious hypotheses as to the possible physical initial conditions (to test star-formation theory) or the outcome (to quantify stellar populations in galaxies

Kroupa P Initial Conditions for Star Clusters Lect Notes Phys 760 181ndash259 (2008)

182 P Kroupa

for example) The initialisation of a pressure-supported stellar system is suchthat the initial object is relevant for the real physical Universe and is thereforea problem of some fundamental importance

Here empirical constraints on the initial conditions of star clusters arediscussed and some problems to which star clusters are relevant are raisedSection 82 contains information to set up a realistic computer model of a starcluster including models of embedded clusters The initial mass distribution ofstars is discussed in Sect 83 and Sect 84 delves into the initial distributionfunctions of multiple stars A brief summary is provided in Sect 85

811 Embedded Clusters

In this section an outline is given of some astrophysical aspects of dense stellarsystems in order to help differentiate probable evolutionary effects from initialconditions A simple example clarifies the meaning of this An observer maysee two young populations with comparable ages (to within 1Myr say) Theyhave similar observed masses but different sizes and a somewhat differentstellar content and different binary fractions Do they signify two differentinitial conditions derived from star-formation or can both be traced back toa t = 0 configuration which is the same

Preliminaries

Assume we observe a very young population of N stars with an age τage andthat we have a rough estimate of its half-mass radius rh and embedded stellarmass Mecl1 The average mass is

m =Mecl

N (81)

Also assume we can estimate the star-formation efficiency (SFE) ε within afew rh For this object

ε =Mecl

Mecl +Mgas (82)

where Mgas is the gas left over from the star-formation process The tidalradius of the embedded cluster can be estimated from the Jacobi limit((Eq (7-84) in Binney amp Tremaine 1987) as determined by the host galaxywhen any contributions by surrounding molecular clouds are ignored

rtid =(Mecl +Mgas

3Mgal

) 13

D (83)

1Throughout all masses m M etc are in units of M unless noted otherwiseldquoEmbedded stellar massrdquo refers to the man in stars at the time before residual gasexpulsion and when star-formation has ceased

8 Initial Conditions for Star Clusters 183

where Mgal is the mass of the spherically distributed galaxy within the dis-tance D of the cluster from the centre of the galaxy This radius is a roughestimate of that distance from the cluster at which stellar motions begin tobe significantly influenced by the host galaxy

The following quantities that allow us to judge the formal dynamical stateof the system the formal crossing time of the stars through the object canbe defined as

tcr equiv2 rhσ

(84)

where2

σ =radicGMecl

ε rh(85)

is up to a factor of order unity the three-dimensional velocity dispersion of thestars in the embedded cluster Note that these equations serve to estimate thepossible amount of mixing of the population If τage lt tcr the object cannotbe mixed and we are seeing it close to its initial state It takes a few tcr for adynamical system out of dynamical equilibrium to return back to it This isnot to be mistaken for a relaxation process

Once the stars orbit within the object they exchange orbital energythrough weak gravitational encounters and rare strong encounters The sys-tem evolves towards a state of energy equipartition The energy equipartitiontime-scale tms between massive and average stars (Spitzer 1987 p 74) whichis an estimate of the time massive stars need to sink to the centre of the systemthrough dynamical friction on the lighter stars is

tms =m

mmaxtrelax (86)

Here mmax is the massive-star mass and the characteristic two-body relax-ation time (eg Eq (4ndash9) in Binney amp Tremaine 1987) is

trelax = 01N

lnNtcr (87)

This formula refers to a pure N -body system without embedded gas A roughestimate of trelaxemb for an embedded cluster can be found in Eq (8) of Adamsamp Myers (2001) The above (87) is a measure for the time a star needs tochange its orbit significantly from its initial trajectory We often estimate itby calculating the amount of time that is required to change the velocity vof a star by an amount Δv asymp v

Thus if for example τage gt tcr and τage lt trelax the system is probablymixed and close to dynamical equilibrium but it is not yet relaxed That isit has not had sufficient time for the stars to exchange a significant amountof orbital energy Such a cluster may have erased its sub-structures

2As an aside note that G = 00045 pc3M Myr2 and that 1 km sminus1 =102 pcMyr

184 P Kroupa

Fragmentation and Size

The very early stages of cluster evolution on a scale of a few parsecs aredominated by gravitational fragmentation of a turbulent magnetised contract-ing molecular cloud core (Clarke Bonnell amp Hillenbrand 2000 Mac Low ampKlessen 2004 Tilley amp Pudritz 2007) Gas-dynamical simulations show theformation of contracting filaments which fragment into denser cloud coresthat form subclusters of accreting protostars As soon as the protostars ra-diate or lose mass with sufficient energy and momentum to affect the cloudcore these computations become expensive because radiative transport anddeposition of momentum and mechanical energy by non-isotropic outflows aredifficult to handle with present computational means (Stamatellos et al 2007Dale Ercolano amp Clarke 2007)

Observations of the very early stages at times less than a few hundreds ofthousands of years suggest that protoclusters have a hierarchical protostellardistribution a number of subclusters with radii less than 02 pc and separatedin velocity space are often seen embedded within a region less than a pcacross (Testi et al 2000) Many of these subclusters may merge to form amore massive embedded cluster (Scally amp Clarke 2002 Fellhauer amp Kroupa2005) It is unclear though if subclusters typically merge before residual gas

blow-out or if the residual gas is removed before the sub-clumps can interactsignificantly nor is it clear if there is a systematic mass dependence of anysuch possible behaviour

Mass Segregation

Whether or not star clusters or subclusters form mass-segregated remains anopen issue Mass segregation at birth is a natural expectation because proto-stars near the density maximum of the cluster have more material to accreteFor these the ambient gas is at a higher pressure allowing protostars to ac-crete longer before feedback termination stops further substantial gas inflowand the coagulation of protostars is more likely there (Zinnecker amp Yorke2007 Bonnell Larson amp Zinnecker 2007) Initially mass-segregated subclus-ters preserve mass segregation upon merging (McMillan Vesperini amp Porte-gies Zwart 2007) However for mmmax = 05100 and N le 5 times 103 stars itfollows from (86) that

tms le tcr (88)

That is a 100M star sinks to the cluster centre within roughly a crossingtime (see Table 81 below for typical values of tcr)

Currently we cannot say conclusively if mass segregation is a birth phe-nomenon (eg Gouliermis et al 2004) or whether the more massive starsform anywhere throughout the protocluster volume Star clusters that havealready blown out their gas at ages of one to a few million years are typicallymass-segregated (eg R136 Orion Nebula Cluster)

Table 81 Notes the Y in the O stars column indicates that the maximum stellarmass in the cluster surpasses 8 M (Fig 81) The average stellar mass is taken tobe m = 04 M in all clusters A star-formation efficiency of ε = 03 is assumed Thecrossing time tcr is (84) The pre-supernova gas evacuation time-scale is τgas =rvth where vth = 10 km sminus1 is the approximate sound velocity of the ionised gasand τgas = 005 Myr for r = 05 pc while τgas = 01 Myr for r = 1 pc

MeclM N O stars tcrMyr τgastcr tcrMyr τgastcr(rh = 05 pc 05 pc 1 pc 1 pc)

40 100 N 09 ndash 26 ndash100 250 YN 06 008 16 02500 1250 Y 03 02 07 01103 25 times 103 Y 02 025 05 02104 25 times 104 Y 006 08 02 05105 25 times 105 Y 002 25 005 2106 25 times 106 Y 0006 83 002 5

To affirm natal mass segregation would impact positively on the notionthat massive stars (more than about 10M) only form in rich clusters andnegatively on the suggestion that they can also form in isolation For recentwork on this topic see Li Klessen amp Mac Low (2003) and Parker amp Goodwin(2007)

Feedback Termination

The observationally estimated SFE (82) is (Lada amp Lada 2003)

02 le ε le 04 (89)

which implies that the physics dominating the star-formation process on scalesless than a few parsecs is stellar feedback Within this volume the pre-clustercloud core contracts under self-gravity and so forms stars ever more vigorouslyuntil feedback energy suffices to halt the process (feedback termination)

Dynamical State at Feedback Termination

Each protostar needs about tps asymp 105 yr to accumulate about 95 of itsmass (Wuchterl amp Tscharnuter 2003) The protostars form throughout thepre-cluster volume as the protocluster cloud core contracts The overall pre-cluster cloud-core contraction until feedback termination takes (84 85)

tclform asymp few times 2radicG

(Mecl

ε

)minus 12

r32h (810)

(a few times the crossing time) which is about the time over which the clusterforms Once a protostar condenses out of the hydro-dynamical flow it becomes

186 P Kroupa

a ballistic particle moving in the time-evolving cluster potential Because manygenerations of protostars can form over the cluster-formation time-scale andif the crossing time through the cluster is a few times shorter than tclform thevery young cluster is mostly in virial equilibrium when star-formation stopswhen any residual gas has been lost3 It is noteworthy that for rh = 1pc

tps ge tclform forMecl

εge 1049 M (811)

(the protostar-formation time formally surpasses the cluster formation time)which is near the turnover mass in the old-star cluster mass function (egBaumgardt 1998)

A critical parameter is thus the ratio

τ =tclformtcr

(812)

If it is less than unity protostars condense from the gas and cannot reachvirial equilibrium in the potential before the residual gas is removed Suchembedded clusters may be kinematically cold if the pre-cluster cloud core wascontracting or hot if the pre-cluster cloud core was pressure confined becausethe young stars do not feel the gas pressure

In those cases where τ gt 1 the embedded cluster is approximately in virialequilibrium because generations of protostars that drop out of the hydrody-namic flow have time to orbit the potential The pre-gas-expulsion stellarvelocity dispersion in the embedded cluster (85) may reach σ = 40pc Myrminus1

if Mecl = 1055 M which is the case for ε rh lt 1 pc This is easily achievedbecause the radius of one-Myr old clusters is r05 asymp 08 pc with no dependenceon mass Some observationally explored cases are discussed by Kroupa (2005)Notably using K-band number counts Gutermuth et al (2005) appear tofind evidence for expansion after gas removal

Interestingly recent Spitzer results suggest a scaling of the characteristicprojected radius R with mass4

Mecl prop R2 (813)

(Allen et al 2007) so the question of how compact embedded clusters formand whether there is a massndashradius relation needs further clarification Notethough that such a scaling is obtained for a stellar population that expandsfreely with a velocity given by the velocity dispersion in the embedded cluster(85)

3A brief transition time ttr tclform exists during which the star-formation ratedecreases in the cluster while the gas is being blown out However for the purposeof the present discussion this time may be neglected

4Throughout this text projected radii are denoted by R while the 3D radiusis r

r(t) asymp ro + σ t rArr Mecl =1G

(r(t) minus ro

t

)2

(814)

where ro le 1 pc is the birth radius of the cluster Is the observed scaling thena result of expansion from a compact birth configuration after gas expulsionIf so it would require a more massive system to be dynamically older whichis at least qualitatively in-line with the dynamical time-scales decreasing withmass Note also that the observed scaling (813) cannot carry through toMecl ge 104 M because the resulting objects would not resemble clusters

There are two broad camps suggesting on one hand that molecular cloudsand star clusters form on a free-fall time-scale (Elmegreen 2000 Hartmann2003 Elmegreen 2007) and on the other hand that many free-fall times areneeded (Krumholz amp Tan 2007) The former implies τ asymp 1 while the latterimplies τ gt 1

Thus currently unclear issues concerning the initialisation ofN -body mod-els of embedded clusters are the ratio τ and whether a massndashradius relationexists for embedded clusters before the development of HII regions To makeprogress I assume for now that the embedded clusters are in virial equilibriumat feedback termination (τ gt 1) and that they form highly concentrated withr le 1 pc independently of mass

The Mass of the Most Massive Star

Young clusters show a well-defined correlation between the mass of the mostmassive star mmax and the stellar mass of the embedded cluster Mecl Thisappears to saturate at mmaxlowast asymp 150M (Weidner amp Kroupa 2004 2006)This is shown in Fig 81 This correlation may indicate feedback terminationof star-formation within the protocluster volume coupled to the most mas-sive stars forming latest or turning-on at the final stage of cluster formation(Elmegreen 1983)

The evidence for a universal upper mass cutoff near

mmaxlowast asymp 150M (815)

(Weidner amp Kroupa 2004 Figer 2005 Oey amp Clarke 2005 Koen 2006Maız Apellaniz et al 2007 Zinnecker amp Yorke 2007) seems to be rather wellestablished in populations with metallicities ranging from the LMC (Z asymp0008) to the super-solar Galactic centre (Z ge 002) so that the stellar massfunction (MF) simply stops at that mass This mass needs to be understoodtheoretically (see discussion by Kroupa amp Weidner 2005 Zinnecker amp Yorke2007) It is probably a result of stellar structure stability but may be near

80M as predicted by theory if the most massive stars reside in near-equalcomponent-mass binary systems (Kroupa amp Weidner 2005) It may also bethat the calculated stellar masses are significantly overestimated (MartinsSchaerer amp Hillier 2005)

188 P Kroupa

Fig 81 The maximum stellar mass mmax as a function of the stellar mass ofthe embedded cluster Mecl (Weidner private communication an updated versionof the data presented by Weidner amp Kroupa 2006) The solid triangle is an SPHmodel of star-cluster formation by Bonnell Bate amp Vine (2003) while the solidcurve stems from stating that there is exactly one most massive star in the cluster1 =

int 150

mmaxξ(m) dm with the condition Mecl =

int mmax008

m ξ(m) dm where ξ(m) isthe stellar IMF The solution can only be obtained numerically but an easy-to-usewell-fitting function has been derived by Pflamm-Altenburg Weidner amp Kroupa(2007)

The Cluster Core of Massive Stars

Irrespective of whether the massive stars (more than about 10M) form at thecluster centre or whether they segregate there owing to energy equipartition(86) they ultimately form a compact sub-population that is dynamicallyhighly unstable Massive stars are ejected from such cores very efficiently ona core-crossing time-scale and for example the well-studied Orion Nebulacluster (ONC) has probably already shot out 70 of its stars more massivethan 5M (Pflamm-Altenburg amp Kroupa 2006) The properties of O andB runaway stars have been used by Clarke amp Pringle (1992) to deduce thetypical birth configuration of massive stars They find them to form in binarieswith similar-mass components in compact small-N groups devoid of low-massstars Among others the core of the Orion Nebula Cluster (ONC) is just sucha system

The Star-Formation History in a Cluster

The detailed star-formation history in a cluster contains information aboutthe events that build up the cluster Intriguing is the recent evidence for someclusters that while the bulk of the stars have ages that differ by less thana few 105 yr a small fraction of older stars are often encountered (Palla ampStahler 2000 for the ONC Sacco et al 2007 for the σ Orionis cluster) Thismay be interpreted to mean that clusters form over about 10 Myr with afinal highly accelerated phase in support of the notion that turbulence of amagnetised gas determines the early cloud-contraction phase (Krumholz ampTan 2007)

A different interpretation would be that as a pre-cluster cloud core con-tracts on a free-fall time-scale it traps surrounding field stars which thenbecome formal cluster members Most clusters form in regions of a galaxythat has seen previous star-formation The velocity dispersion of the previ-ous stellar generation such as an expanding OB association is usually ratherlow around a few km sminus1 to 10 km sminus1 The deepening potential of a newlycontracting pre-cluster cloud core is able to capture some of the precedinggeneration of stars so that these older stars become formal cluster membersalthough they did not form in the cluster Pflamm-Altenburg amp Kroupa(2007) study this problem for the ONC and show that the age spread re-ported by Palla et al (2007) can be accounted for in this way This suggeststhat the star-formation history of the ONC may in fact not have started about10 Myr ago supporting the argument by Elmegreen (2000) Elmegreen (2007)and Hartmann (2003) that clusters form on a time-scale comparable to thecrossing time of the pre-cluster cloud core Additionally the sample of clus-ter stars may be contaminated by enhanced fore- and back-ground densitiesof field stars by focussing of stellar orbits during cluster formation (Pflamm-Altenburg amp Kroupa 2007)

For very massive clusters such as ω Cen Fellhauer Kroupa amp Evans(2006) show that the potential is sufficiently deep that the pre-cluster cloudcore may capture the field stars of a previously existing dwarf galaxy Up to30 or more of the stars in ω Cen may be captured field stars This wouldexplain an age spread of a few Gyr in the cluster and is consistent with thenotion that ω Cen formed in a dwarf galaxy that was captured by the MilkyWay The attractive aspect of this scenario is that ω Cen need not have beenlocated at the centre of the incoming dwarf galaxy as a nucleus but withinits disc because it opens a larger range of allowed orbital parameters for theputative dwarf galaxy moving about the Milky Way The currently preferredscenario in which ω Cen was the nucleus of the dwarf galaxy implies thatthe galaxy was completely stripped while falling into the Milky Way leavingonly its nucleus on its current retrograde orbit (Zhao 2004) The new scenarioallows the dwarf galaxy to be absorbed into the bulge of the Milky Way withω Cen being stripped from it on its way in

190 P Kroupa

Another possibility for obtaining an age spread of a few Gyr in a massivecluster such as ω Cen is gas accretion from a co-moving inter-stellar medium(Pflamm-Altenburg amp Kroupa 2008) This could only have worked for ω Cenbefore it became unbound from its mother galaxy though That is the clustermust have spent about 2ndash3Gyr in its mother galaxy before it was capturedby the Milky Way

This demonstrates beautifully how an improved understanding of dynam-ical processes on scales of a fewpc impinges on problems related to the forma-tion of galaxies and cosmology (through the sub-structure problem) Finallythe increasingly well-documented evidence for stellar populations in massiveclusters with different metallicities and ages and in some cases even significantHe enrichment may also suggest secondary star-formation occurring from ma-terial that has been pre-enriched from a previous generation of stars in thecluster Different IMFs need to be invoked for the populations of different ages(see Piotto 2008 for a review)

Expulsion of Residual Gas

When the most massive stars are O stars they destroy the protocluster neb-ula and quench further star-formation by first ionising most of it (feedbacktermination) The ionised gas at a temperature near 104 K and in seriousover-pressure pushes out and escapes the confines of the cluster volume atthe sound speed (near 10 km sminus1) or faster if the winds blow off O stars withvelocities of thousands of km sminus1 and impart sufficient momentum

There are two analytically tractable regimes of behaviour instantaneousgas removal and slow gas expulsion over many crossing times

bull First consider instantaneous gas expulsion τgas = 0 The binding energyof the object of mass M and radius r is

Eclbind = minusGM2

r+

12M σ2 lt 0 (816)

Before gas expulsion M = Minit = Mgas +Mecl rarr M and

σ2init =

GMinit

rinitminusrarr σ (817)

After instantaneous gas expulsion Mafter = Mecl rarr M but σafter =σinit rarr σ and the new binding energy is

Eclbindafter = minusGM2after

rinit+

12Mafter σ

2init (818)

But the cluster relaxes into a new equilibrium so that by the scalar virialtheorem5

5The scalar virial theorem states that 2 K + W = 0 rArr E = K + W = (12) W where K W are the kinetic and potential energy and E is the total energy of thesystem

Eclbindafter = minus12GMafter

rafter (819)

and on equating these two expressions for the final energy and using (817)we find that

rafterrinit

=Mecl

Mecl minusMgas (820)

Thus as Mgas rarr Mecl then ε rarr 05 from above rafter rarr infin Thismeans that as the SFE approaches 50 from above the cluster unbindsitself But by (89) this result would imply either (see Kroupa Aarseth ampHurley 2001 and references therein)ndash all clusters with OB stars (and thus τgas tcr) do not survive gas

expulsion orndash the clusters expel their gas slowly τgas tcr This may be the case if

surviving clusters such as the Pleiades or Hyades formed without OBstars

bull Now consider slow gas removal τgas tcr τgas rarr infin By (820) and theassumption that an infinitesimal mass of gas is removed instantaneously

rinit minus δr

rinit=

Minit minus δMgas

Minit minus δMgas minus δMgas (821)

For infinitesimal steps and for convenience dM lt 0 but dr gt 0

r minus drr

=M + dMM + 2dM

(822)

Re-arranging this we find

drr

=dMM

(

1 minus 2dMM

)

(823)

so that

drr

=dMM

rArr lnrafterrinit

= lnMinit

Mafter (824)

upon integration of the differential equation Thus

rafterrinit

=Mecl +Mgas

Mecl=

1ε (825)

and for example for a SFE of 20 the cluster expands by a factor of 5rafter = 5 rinit without dissolving

Table 81 gives an overview of the type of behaviour one might expect forclusters with increasing number of stars N and stellar mass Mecl for twocharacteristic radii of the embedded stellar distribution rh It can be seen thatthe gas-evacuation time-scale becomes longer than the crossing time through

192 P Kroupa

the cluster for Mecl ge 105 M Such clusters would thus undergo adiabaticexpansion as a result of gas blow out Less-massive clusters are more likelyto undergo an evolution that is highly dynamic and that can be described asan explosion (the cluster pops) For clusters without O and massive B starsnebula disruption probably occurs on the cluster-formation time-scale of abouta million years and the evolution is again adiabatic A simple calculation ofthe amount of energy deposited by an O star into its surrounding cluster-nebula suggests it is larger than the nebula binding energy (Kroupa 2005)This however only gives at best a rough estimate of the rapidity with whichgas can be expelled An inhomogeneous distribution of gas leads to the gasremoval preferentially along channels and asymmetrically so that the overallgas-excavation process is highly non-uniform and variable (Dale et al 2005)

The reaction of clusters to gas expulsion is best studied numerically withN -body codes Pioneering experiments were performed by Tutukov (1978) andthen Lada Margulis amp Dearborn (1984) Goodwin (1997ab 1998) studied gasexpulsion by supernovae from young globular clusters Figure 82 shows theevolution of an ONC-type initial cluster with a stellar mass Mecl asymp 4000Mand a canonical IMF (8124) and stellar evolution a 100 initial binary popu-lation (Sect 842) in a solar-neighbourhood tidal field ε = 13 and sphericalgas blow-out on a thermal time-scale (vth = 10 km sminus1) The figure demon-strates that the evolution is far more complex than the simple analytical esti-mates above suggest and in fact a substantial Pleiades-type cluster emergesafter losing about two-thirds of its initial stellar population (see also p 195)Subsequent theoretical work based on an iterative scheme according to whichthe mass of unbound stars at each radius is removed successively shows that

Fig 82 The evolution of 5 10 20 50 of the Lagrangian radius and the coreradius (Rc = rc thick lower curve) of the ONC-type cluster discussed in the textThe gas mass is shown as the dashed line The cluster spends 06 Myr in an embeddedphase before the gas is blown out on a thermal time-scale The tidal radius (83) isshown by the upper thick solid curve (Kroupa Aarseth amp Hurley 2001)

the survival of a cluster depends not only on ε τgastcr and rtid but also on thedetailed shape of the stellar distribution function (Boily amp Kroupa 2003) Forinstantaneous gas removal ε asymp 03 is a lower limit for the SFE below whichclusters cannot survive rapid gas blow-out This is significantly smaller thanthe critical value of ε = 05 below which the stellar system becomes formallyunbound (820) However if clusters form as complexes of subclusters eachof which pop in this way then overall cluster survival is enhanced to evensmaller values of ε asymp 02 (Fellhauer amp Kroupa 2005)

Whether clusters pop and what fraction of stars remain in a post-gas expul-sion cluster depend critically on the ratio between the gas-removal time-scaleand the cluster crossing time This ratio thus mostly defines which clusters suc-cumb to infant mortality and which clusters merely suffer cluster infant weightloss The well-studied observational cases do indicate that the removal of mostof the residual gas does occur within a cluster-dynamical time τgastcr le 1Examples noted (Kroupa 2005) are the ONC and R136 in the LMC both ofwhich have significant super-virial velocity dispersions Other examples arethe Treasure-Chest cluster and the very young star-bursting clusters in themassively interacting Antennae galaxy that appear to have HII regions ex-panding at velocities so that the cluster volume may be evacuated within acluster dynamical time However improved empirical constraints are needed todevelop further an understanding of cluster survival Such observations wouldbest be the velocities of stars in very young star clusters as they should showa radially expanding stellar population

Indeed Bastian amp Goodwin (2006) note that many young clusters havethe radial-density profile signature expected if they are expanding rapidlyThis supports the notion of fast gas blow out For example the 05ndash2Myrold ONC which is known to be super-virial with a virial mass about twicethe observed mass (Hillenbrand amp Hartmann 1998) has already expelled itsresidual gas and is expanding rapidly It has therefore probably lost its outerstars (Kroupa Aarseth amp Hurley 2001) The super-virial state of young clus-ters makes measurements of their mass-to-light ratio a bad estimate of thestellar mass within them (Goodwin amp Bastian 2006) and rapid dynamicalmass-segregation likewise makes naive measurements of the ML ratio wrong(Boily et al 2005 Fleck et al 2006) Goodwin amp Bastian (2006) and de Grijsamp Parmentier (2007) find the dynamical mass-to-light ratios of young clustersto be too large strongly implying they are in the process of expanding aftergas expulsion

Weidner et al (2007) attempted to measure infant weight loss with asample of young but exposed Galactic clusters They applied the maximal-star-mass to cluster mass relation from above to estimate the birth mass ofthe clusters The uncertainties are large but the data firmly suggest that thetypical cluster loses at least about 50 of its stars

194 P Kroupa

Binary Stars

Most stars form as binaries with as far as can be stated today universal orbitaldistribution functions (Sect 84) Once a binary system is born in a denseenvironment it is perturbed This changes its eccentricity and semi-majoraxis Or it undergoes a relatively strong encounter that disrupts the binary orhardens it perhaps with exchanged companions The initial binary populationtherefore evolves on a cluster crossing time-scale and most soft binaries aredisrupted It has been shown that the properties of the Galactic field binarypopulation can be explained in terms of the binary properties observed for veryyoung populations if these go through a dense cluster environment (dynamicalpopulation synthesis Kroupa 1995d) A dense cluster environment hardensexisting binaries (p 240) This increases the SN Ia rate in a galaxy withmany dense clusters (Shara amp Hurley 2002)

Binaries are significant energy sources (see also Sect 84) A hard binarythat interacts via a resonance with a cluster field star occasionally ejects onestar with a terminal velocity vej σ The ejected star either leaves the clus-ter causing cluster expansion so that σ drops or it shares some of its kineticenergy with the other cluster field stars through gravitational encounters caus-ing cluster expansion Binaries in a cluster core can thus halt and reverse corecollapse (Meylan amp Heggie 1997 Heggie amp Hut 2003)

Mass Loss from Evolving Stars

An old globular cluster with a turn-off mass near 08M has lost 30 of themass that remained in it after gas expulsion by stellar evolution (Baumgardtamp Makino 2003) Because the mass loss is most rapid during the earliest timesafter the cluster returned to virial equilibrium once the gas was expelled thecluster expands further during this time This is nicely seen in the Lagrangianradii of realistic cluster-formation models (Kroupa Aarseth amp Hurley 2001)

812 Some Implications for the Astrophysics of Galaxies

In general the above have a multitude of implications for galactic and stellarastrophysics

1 The heaviest-starndashstar-cluster-mass correlation constrains feedback modelsof star cluster formation (Elmegreen 1983) It also implies that the sumof all IMFs in all young clusters in a galaxy the integrated galaxy initialmass function (IGIMF) is steeper than the invariant stellar IMF observedin star clusters This has important effects on the massndashmetallicity rela-tion of galaxies (Koeppen Weidner amp Kroupa 2007) Additionally star-formation rates (SFRs) of dwarf galaxies can be underestimated by up tothree orders of magnitude because Hα-dark star-formation becomes possible(Pflamm-Altenburg Weidner amp Kroupa 2007) This indeed constitutes an

important example of how sub-pc processes influence the physics on cos-mological scales

2 The deduction that type-II clusters probably pop (p 190) implies thatyoung clusters will appear to an observer to be super-virial ie to havea dynamical mass larger than their luminous mass (Bastian amp Goodwin2006 de Grijs amp Parmentier 2007)

3 It further implies that galactic fields can be heated and may also lead togalactic thick discs and stellar halos around dwarf galaxies (Kroupa 2002b)

4 The variation of the gas expulsion time-scale among clusters of differenttype implies that the star-cluster mass function (CMF) is re-shaped rapidlyon a time-scale of a few tens of Myr (Kroupa amp Boily 2002)

5 Associated with this re-shaping of the CMF is the natural production ofpopulation II stellar halos during cosmologically early star-formation bursts(Kroupa amp Boily 2002 Parmentier amp Gilmore 2007 Baumgardt Kroupaamp Parmentier 2008)

6 The properties of the binary-star population observed in Galactic fields areshaped by dynamical encounters in star clusters before the stars leave theircluster (Sect 84)

Points 2ndash5 are considered in more detail in the rest of Sect 81

Stellar Associations Open Clusters and Moving Groups

As one of the important implications of point 2 a cluster in the age range1ndash50Myr has an unphysical ML ratio because it is out of dynamical equilib-rium rather than because it has an abnormal stellar IMF (Bastian amp Goodwin2006 de Grijs amp Parmentier 2007)

Another implication is that a Pleiades-like open cluster would have beenborn in a very dense ONC-type configuration and that as it evolves a moving-group-I is established during the first few dozen Myr This comprises roughlytwo-thirds of the initial stellar population and the cluster is expanding witha velocity dispersion that is a function of the pre-gas-expulsion configura-tion (Kroupa Aarseth amp Hurley 2001) These computations were amongthe first to demonstrate with high-precision N -body modelling that the re-distribution of energy within the cluster during the embedded phase and dur-ing the expansion phase leads to the formation of a substantial remnant clusterdespite the inclusion of all physical effects that are disadvantageous for thisto happen (explosive gas expulsion low SFE ε = 033 galactic tidal field andmass loss from stellar evolution and an initial binary-star fraction of 100see Fig 82) Thus expanding OB associations may be related to star-clusterbirth and many OB associations ought to have remnant star clusters as nuclei(see also Clark et al 2005)

As the cluster expands becoming part of an OB association the radiationfrom its massive stars produce expanding HII regions that may trigger furtherstar-formation in the vicinity (eg Gouliermis Quanz amp Henning 2007)

196 P Kroupa

A moving-group-II establishes later ndash the classical moving group made upof stars that slowly diffuse or evaporate out of the readjusted cluster remnantwith relative kinetic energy close to zero The velocity dispersion of moving-group-I is thus comparable to the pre-gas-expulsion velocity dispersion of thecluster while moving-group-II has a velocity dispersion close to zero

The Velocity Dispersion of Galactic-Field Populationsand Galactic Thick Discs

Thus the moving-group-I would be populated by stars that carry the initialkinematic state of the birth configuration into the field of a galaxy Each gen-eration of star clusters would according to this picture produce overlappingmoving-groups-I (and II) and the overall velocity dispersion of the new fieldpopulation can be estimated by adding the squared velocities for all expandingpopulations This involves an integral over the embedded-cluster mass func-tion ξecl(Mecl) which describes the distribution of the stellar mass content ofclusters when they are born Because the embedded cluster mass function isknown to be a power-law this integral can be calculated for a first estimate(Kroupa 2002b 2005) The result is that for reasonable upper cluster masslimits in the integral Mecl le 105 M the observed agendashvelocity dispersionrelation of Galactic field stars can be reproduced

This idea can thus explain the much debated energy deficit namely thatthe observed kinematic heating of field stars with age could not until nowbe explained by the diffusion of orbits in the Galactic disc as a result of scat-tering by molecular clouds spiral arms and the bar (Jenkins 1992) Becausethe velocity-dispersion for Galactic-field stars increases with stellar age thisnotion can also be used to map the star-formation history of the Milky Waydisc by resorting to the observed correlation between the star-formation ratein a galaxy and the maximum star-cluster mass born in the population ofyoung clusters (Weidner Kroupa amp Larsen 2004)

An interesting possibility emerges concerning the origin of thick discs Ifthe star-formation rate was sufficiently high about 11 Gyr ago star clustersin the disc with masses up to 1055 M would have been born If they poppeda thick disc with a velocity dispersion near 40 km sminus1 would result naturally(Kroupa 2002b) This notion for the origin of thick discs appears to be qual-itatively supported by the observations of Elmegreen Elmegreen amp Sheets(2004) who find galactic discs at a red shift between 05 and 2 to show massivestar-forming clumps

Structuring the Initial Cluster Mass Function

Another potentially important implication from this picture of the evolution ofyoung clusters is that if the ratio of the gas expulsion time to the crossing timeor the SFE varies with initial (embedded) cluster mass an initially featurelesspower-law mass function of embedded clusters rapidly evolves to one with

peaks dips and turnovers at cluster masses that characterise changes in thebroad physics involved

As an example Adams (2000) and Kroupa amp Boily (2002) assumed thatthe function

Micl = fst(Mecl)Mecl (826)

exists where Mecl is as above and Micl is the classical initial cluster massand fst = fst(Mecl) According to Kroupa amp Boily (2002) the classical initialcluster mass is that mass which is inferred by standard N -body computationswithout gas expulsion (in effect this assumes ε = 1 which is however unphys-ical) Thus for example for the Pleiades Mcl asymp 1000M at the present time(age about 100 Myr) A classical initial model would place the initial clustermass near Micl asymp 1500M by standard N -body calculations to quantify thesecular evaporation of stars from an initially bound and relaxed cluster (Porte-gies Zwart et al 2001) If however the SFE was 33 and the gas-expulsiontime-scale were comparable to or shorter than the cluster dynamical timethe Pleiades would have been born in a compact configuration resemblingthe ONC and with a mass of embedded stars of Mecl asymp 4000M (KroupaAarseth amp Hurley 2001) Thus fst(4000M) = 038 (= 15004000)

By postulating that there exist three basic types of embedded clusters(Kroupa amp Boily 2002) namely

Type I clusters without O stars (Mecl le 1025 M eg Taurus-Auriga pre-main sequence stellar groups ρ Oph)

Type II clusters with a few O stars (1025 le MeclM le 1055 eg theONC)

Type III clusters with many O stars and with a velocity dispersion compara-ble to or higher than the sound velocity of ionized gas (Mecl ge 1055 M)

it can be argued that fst asymp 05 for type I fst lt 05 for type II and fst asymp 05for type III The reason for the high fst values for types I and III is thatgas expulsion from these clusters may last longer than the cluster dynamicaltime because there is no sufficient ionizing radiation for type I clusters orthe potential well is too deep for the ionized gas to leave (type III clusters)The evolution is therefore adiabatic ((825) above) Type II clusters undergoa disruptive evolution and witness a high infant mortality rate (Lada amp Lada2003) They are the pre-cursors of OB associations and Galactic clusters Thisbroad categorisation has easy-to-understand implications for the star-clustermass function

Under these conditions and an assumed functional form for fst = fst(Mecl)the power-law embedded cluster mass function transforms into a cluster massfunction with a turnover near 105 M and a sharp peak near 103 M (Kroupaamp Boily 2002) This form is strongly reminiscent of the initial globular clustermass function which is inferred by for example Vesperini (1998 2001)Parmentier amp Gilmore (2005) and Baumgardt (1998) to be required for a

198 P Kroupa

match with the evolved cluster mass function that is seen to have a universalturnover near 105 M By the reasoning given above this ldquoinitialrdquo CMF ishowever unphysical being a power-law instead

This analytical formulation of the problem has been verified nicely withN -body simulations combined with a realistic treatment of residual gas expul-sion by Baumgardt Kroupa amp Parmentier (2008) who show the Milky Wayglobular cluster mass function to emerge from a power-law embedded-clustermass function Parmentier et al (2008) expand on this by studying the ef-fect that different assumptions on the physics of gas removal have on shapingthe star-cluster mass function within about 50 Myr The general ansatz thatresidual gas expulsion plays a dominant role in early cluster evolution maythus solve the long-standing problem that the deduced initial cluster massfunction needs to have this turnover while the observed mass functions ofyoung clusters are featureless power-law distributions

The Origin of Population II Stellar Halos

The above view implies naturally that a major field-star component is gen-erated whenever a population of star clusters forms About 12Gyr ago theMilky Way began its assembly by an initial burst of star-formation throughouta volume spanning about 10 kpc in radius In this volume the star-formationrate must have reached 10M yrminus1 so that star clusters with masses up toasymp 106 M formed (Weidner Kroupa amp Larsen 2004) probably in a chaoticturbulent early interstellar medium The vast majority of embedded clus-ters suffered infant weight loss or mortality The surviving long-lived clus-ters evolved to globular clusters The so-generated field population is thespheroidal population-II halo which has the same chemical properties as thesurviving (globular) star clusters apart from enrichment effects evident inthe most massive clusters All of these characteristics emerge naturally inthe above model as pointed out by Kroupa amp Boily (2002) Parmentier ampGilmore (2007) and most recently by Baumgardt Kroupa amp Parmentier(2008)

813 Long-Term or Classical Cluster Evolution

The long-term evolution of star clusters that survive infant weight loss andthe mass loss from evolving stars is characterised by three physical processesthe drive of the self-gravitating system towards energy equipartition stellarevolution processes and the heating or forcing of the system through externaltides One emphasis of star-cluster work in this context is to test the theoryof stellar evolution and to investigate the interrelation of stellar astrophysicswith stellar dynamics The stellar-evolution and the dynamical-evolution time-scales are comparable The reader is directed to Meylan amp Heggie (1997) andHeggie amp Hut (2003) for further details

Tidal Tails

Tidal tails contain the stars evaporating from long-lived star clusters (themoving-group-II above) The typical S-shaped structure of tidal tails close tothe cluster are easily understood stars that leave the cluster with a slightlyhigher galactic velocity than the cluster are on slightly outward-directed galac-tic orbits and therefore fall behind the cluster as the angular velocity aboutthe galactic centre decreases with distance The outward-directed trailing armdevelops Stars that leave the cluster with slower galactic velocities than thecluster fall towards the galaxy and overtake the cluster

Given that energy equipartition leads to a filtering in energy space of thestars that escape at a particular time one expects a gradient in the stellarmass function progressing along a tidal tail towards the cluster so that themass function becomes flatter richer in more massive stars This effect isdifficult to detect but for example the long tidal tails found emanating fromPal 5 (Odenkirchen et al 2003) may show evidence for it

As emphasised by Odenkirchen et al (2003) tidal tails have another veryinteresting use they probe the gravitational potential of the Milky Way ifthe differential motions along the tidal tail can be measured They are thusimportant future tests of gravitational physics

Death and Hierarchical Multiple Stellar Systems

Nothing lasts forever and star clusters that survive initial relaxation to virialequilibrium after residual gas expulsion and mass loss from stellar evolutionultimately cease to exist after all member stars evaporate to leave a binary ora long-lived hierarchical multiple system composed of near-equal mass com-ponents (de la Fuente Marcos 1997 1998) Note that these need not be singlestars These cluster remnants are interesting because they may account formost of the hierarchical multiple stellar systems in the Galactic field (Good-win amp Kroupa 2005) with the implication that these are not a product ofstar-formation but rather of star-cluster dynamics

814 What is a Galaxy

Star clusters dwarf-spheroidal (dSph) and dwarf-elliptical (dE) galaxies aswell as galactic bulges and giant elliptical (E) galaxies are all stellar-dynamicalsystems that are supported by random stellar motions ie they are pressure-supported But why is one class of these pressure-supported systems referredto as star clusters while the others are galaxies Is there some fundamentalphysical difference between these two classes of systems

Considering the radius as a function of mass we notice that systems withM le 106 M do not show a massndashradius relation (MRR) and have r asymp 4 pcMore massive objects however show a well-defined MRR In fact Dabring-hausen Hilker amp Kroupa (2008) find that massive compact objects (MCOs)

200 P Kroupa

which have 106 le MM le 108 lie on the MRR of giant E galaxies (about1013 M) down to normal E galaxies (1011 M) as is evident in Fig 83

Rpc = 10minus315

(M

M

)060plusmn002

(827)

Noteworthy is that systems with M ge 106 M also exhibit complex stel-lar populations while less massive systems have single-age single-metallicitypopulations Remarkably Pflamm-Altenburg amp Kroupa (2008) show that astellar system with M ge 106 M and a radius as observed for globular clus-ters can accrete gas from a co-moving warm inter-stellar medium and mayre-start star-formation The median two-body relaxation time is longer thana Hubble time for M ge 3 times 106 M and only for these systems is there evi-dence for a slight increase in the dynamical mass-to-light ratio Intriguingly(ML)V asymp 2 for M lt 106 M while (ML)V asymp 5 for M gt 106 M with apossible decrease for M gt 108 M (Fig 84) Finally the average stellar den-sity maximises at M = 106 M with about 3 times 103 Mpc3 (DabringhausenHilker amp Kroupa 2008)

Thus

Fig 83 Massndashradius data plotted against the dynamical mass of pressure-supported stellar systems (Dabringhausen Hilker amp Kroupa 2008) MCOs aremassive compact objects (also referred to as ultra compact dwarf galaxies) Thesolid and dashed lines refer to (827) while the dash-dotted line is a fit to dSph anddE galaxies

Fig 84 Dynamical ML values in dependence of the V-band luminosity ofpressure-supported stellar systems (Dabringhausen Hilker amp Kroupa 2008) MCOsare massive compact objects (also referred to as ultra compact dwarf galaxies)

bull the mass 106 M appears to be specialbull stellar populations become complex above this massbull evidence for some dark matter only appears in systems that have a median

two-body relaxation time longer than a Hubble timebull dSph galaxies are the only stellar-dynamical systems with 10 lt (ML)V lt

1000 and as such are total outliers andbull 106 M is a lower accretion limit for massive star clusters immersed in a

warm inter-stellar medium

M asymp 106 M therefore appears to be a critical mass scale so that less-massive objects show characteristics of star clusters that are described wellby Newtonian dynamics while more massive objects show behaviour moretypical of galaxies Defining a galaxy as a stellar-dynamical object which hasa median two-body relaxation time longer than a Hubble time ie essentiallya system with a smooth potential may be an objective and useful way todefine a galaxy (Kroupa 1998) Why only smooth systems show evidencefor dark matter remains at best a striking coincidence at worst it may besymptomatic of a problem in understanding dynamics in such systems

202 P Kroupa

82 Initial 6D Conditions

The previous section gave an outline of some of the issues at stake in therealm of pressure-supported stellar systems In order to attack these and otherproblems we need to know how to set up such systems in the computerIndeed as much as analytical solutions may be preferred the mathematicaland physical complexities of dense stellar systems leave no alternatives otherthan to resort to full-scale numerical integration of the 6N coupled first-order differential equations that describe the motion of the system through6N -dimensional phase space There are three related questions to ponderGiven a well-developed cluster how is one to set it up in order to evolve itforward in time How does a cluster form and how does the formation processaffect its later properties How do we describe a realistic stellar population(IMF binaries) Each of these questions is dealt with in the following sections

821 6D Structure of Classical Clusters

Because the state of a star cluster is never known exactly it is necessary toperform numerical experiments with conditions that are statistically consis-tent with the cluster snap-shot To ensure meaningful statistical results forsystems with few stars say N lt 5000 many numerical renditions of the sameobject are thus necessary For example systems with N = 100 stars evolveerratically and numerical experiments are required to map out the range ofpossible states at a particular time the range of half-mass radii at an age of20 Myr in 1000 numerical experiments of a cluster initially with N = 100 starsand with an initial half-mass radius r05 = 05 pc can be compared with anactually observed object for testing consistency with the initial conditionsExcellent recent examples of this approach can be found in Hurley et al(2005) and Portegies Zwart McMillan amp Makino (2007) with a recent reviewavailable by Hut et al (2007) and two text books have been written dealingwith computational and more general aspects of the physics of dense stellarsystems (Aarseth 2003 Heggie amp Hut 2003)

The six-dimensional structure of a pressure-supported stellar system attime t is conveniently described by the phase-space distribution functionf(rv t) where r and v are the phase-space variables and

dN = f(rv t) d3x d3v (828)

is the number of stars in 6D phase-space volume element d3x d3v In the case ofa steady state the Jeans theorem (Binney amp Tremaine 1987 their Sect 44)allows us to express f in terms of the integrals of motion ie the energyand angular momentum The phase-space distribution function can then bewritten as

f = f(rv) = f(εe l) (829)

whereεe =

12v2 + Φ(r) (830)

is the specific energy of a star and

l = |r times v| (831)

is the specific orbital angular momentum of a star The Poisson equation is

nabla2Φ(r) = 4πGρm(r) = 4π Gint

allspace

mf d3v (832)

or in spherical symmetry

1r2

ddr

(

r2dΦdr

)

= 4πGint

allspace

fm

(12v2 + Φ |r times v|

)

d3v (833)

where fm is the phase-space mass-density of all matter and is equal to mffor a system with equal-mass stars Most pressure-supported systems have anear-spherical shape and so in most numerical work it is convenient to assumespherical symmetry

For convenience it is useful to introduce the relative potential6

Ψ equiv minusΦ + Φ0 (834)

and the relative energy

E equiv minusεe + Φ0 = Ψ minus 12v2 (835)

where Φ0 is a constant so that f gt 0 for E gt 0 and f = 0 for E le 0The Poisson equation becomes nabla2Ψ = minus4π Gρm subject to the boundarycondition Ψ rarr Φ0 as r rarr infin

One important property of stellar systems is the anisotropy of their velocitydistribution function We define the anisotropy parameter

β(r) equiv 1 minus v2θ

v2r

(836)

where v2θ v

2r are the mean squared tangential and radial velocities at a par-

ticular location r respectively It follows that systems with β = 0 everywherehave an isotropic velocity distribution function

If f only depends on the energy the mean squared radial and tangentialvelocities are respectively

v2r =

1ρ

int

all vel

v2r f

[

Ψ minus 12(v2

r + v2θ + v2

φ

)]

dvr dvθ dvφ (837)

6The following discussion is based on Binney amp Tremaine (1987)

204 P Kroupa

and

v2θ =

1ρ

int

all vel

v2θ f

[

Ψ minus 12(v2

r + v2θ + v2

φ

)]

dvr dvθ dvφ (838)

If the labels θ and r are exchanged in (838) it can be seen that one arrives at(837) Equations (837) and (838) are thus identical apart from the labellingThus if f = f(E) β = 0 and the velocity distribution function is isotropic

If f depends on the energy and the orbital angular momentum of the stars(|l| = |r times v|) then the mean squared radial and tangential velocities arerespectively

v2r =

1ρ

int

all vel

v2r f

[

Ψ minus 12(v2

r + v2θ + v2

φ

) rradicv2

θ + v2φ

]

dvr dvθ dvφ (839)

and

v2θ =

1ρ

int

all vel

v2θ f

[

Ψ minus 12(v2

r + v2θ + v2

φ

) rradicv2

θ + v2φ

]

dvr dvθ dvφ (840)

If the labels θ and r are exchanged in (840) it can be seen that this time onedoes not arrive at (839) Thus if f = f(E l) then β = 0 and the velocity dis-tribution function is not isotropic This serves to demonstrate an elementarybut useful property of the phase-space distribution function

A very useful series of distribution functions can be arrived at from thesimple form

fm(E) =F Enminus 3

2 E gt 00 E le 0

(841)

The mass density

ρm(r) = 4π Fint radic

2 Ψ

0

(

Ψ minus 12v2

)nminus 32

v2 dv (842)

where the upper integration bound is given by the escape condition E =Ψ minus (12)v2 = 0 Substituting v2 = 2Ψ cos2θ for some θ leads to

ρm(r) =cn Ψn Ψ gt 0

0 Ψ le 0 (843)

For cn to be finite n gt 12 ie homogeneous (n = 0) systems are excludedThe LanendashEmden equation follows from the spherically symmetric Poisson

equation after introducing dimensionless variables s = rb ψ = ΨΨ0 whereb = (4π GΨnminus1

0 cn)minus12 and Ψ0 = Ψ(0)

1s2

dds

(

s2dψds

)

=minusψn ψ gt 0

0 ψ le 0 (844)

H Lane and R Emden worked with this equation in the context of self-gravitating polytropic gas spheres which have an equation of state

p = K ργm (845)

where K is a constant and p the pressure It can be shown that γ = 1 + 1nThat is the density distribution of a stellar polytrope of index n is the sameas that of a polytropic gas sphere with index γ

The natural boundary conditions to be imposed on (844) are at s = 0

1 ψ = 1 because Ψ(0) = Ψ0 and2 dψds = 0 because the gravitational force must vanish at the centre

Analytical solutions to the LanendashEmden equation are possible only for afew values of n and we remember that a homogeneous (n = 0) stellar densitydistribution has already been excluded as a viable solutions of the generalpower-law phase-space distribution function

The Plummer Model

A particularly useful case is

ψ =1

radic1 + 1

3 s2 (846)

It follows immediately that this is a solution of the LanendashEmden equation forn = 5 and it also satisfies the two boundary conditions above and so consti-tutes a physically sensible potential By integrating the Poisson equation itcan be shown that the total mass of this distribution function is finite

Minfin =radic

3 Ψ0 bG (847)

although the density distribution has no boundary The distribution functionis

fm(E) =

F(Ψ minus 1

2 v2) 7

2 v2 lt 2Ψ0 v2 ge 2Ψ

(848)

with the relative potential

Ψ =Ψ0radic

1 + 13

(rb

)2(849)

and density lawρm =

ρm0(1 + 1

3

(rb

)2) 5

2(850)

with the above total mass This density distribution is known as the Plummermodel named after Plummer (1911) who showed that the density distributionthat results from this model provides a reasonable and in particular verysimple analytical description of globular clusters The Plummer model is in

206 P Kroupa

fact a work-horse for many applications in stellar dynamics because many ofits properties such as the projected velocity dispersion profile can be calculatedanalytically Such formulae are useful for checking numerical codes used to setup models of stellar systems

Properties of the Plummer Model

Some useful analytical results can be derived for the Plummer density law(see also Heggie amp Hut 2003 their p 73 for another compilation) For thePlummer law of mass Mecl the mass-density profile (850) can be written as

ρm(r) =3Mecl

4π r3pl

1[

1 +(

rrpl

)2] 5

2 (851)

where rpl is the Plummer scale length The central number density is thus

ρc =3N

4π r3pl

(852)

The mass within radius r follows from M(r) = 4πint r

0ρm(rprime) rprime

2drprime

M(r) = Mecl

(r

rpl

)3

[

1 +(

rrpl

)2] 3

2 (853)

Thus

rpl contains 354 of the mass2 rpl contain 7165 rpl contain 943 and10 rpl contain 985 of the total mass

For the half-mass radius we have

rh = (223 minus 1)minus

12 rpl asymp 1305 rpl (854)

The projected surface mass density ΣM (R) = 2intinfin0

ρm(r) dz where R isthe projected radial distance from the cluster centre and Z is the integrationvariable along the line-of-sight (r2 = R2 + Z2) is

Σρ(R) =Mecl

π r2pl

1[

1 +(

Rrpl

)2]2 (855)

We assume there is no mass segregation so that the mass-to-light ratio Υ equiv(ML) measured in some photometric system is independent of radius Theintegrated light within projected radius R is

I(R) = (1Υ )int R

0

Σρ(Rprime) 2π Rprime dRprime (856)

I(R) =Mecl r

2pl

Υ

[1r2pl

minus 1R2 + r2pl

]

(857)

Thus rpl is the half-light radius of the projected star cluster I(rpl) =05 I(infin)

In the above equations ρ(r) = ρm(r)m N(r) = M(r)m and Σn =Σρm are respectively the stellar number density the number of stars withinradius r and the projected surface number density profile if there is no masssegregation within the cluster Thus the average stellar mass m is constant

The velocity dispersion can be calculated at any radius from the Jeansequation (8120) For an isotropic velocity distribution (σ2

θ = σ2φ = σ2

r) suchas the Plummer model the Jeans equation yields

σ2r(r) =

1ρ(r)

int infin

r

ρ(rprime)GM(rprime)

r2drprime (858)

because dφ(r)dr = GM(r)r2 and the integration bounds have been chosento make use of the vanishing ρm(r) as r rarr infin Note that the above equationis also valid if M(r) consists of more than one spherical component such as adistinct core plus an extended halo Combining (851) (853) and (858) weare led to

σ2(r) =(GMecl

2 rpl

)1

[

1 +(

rrpl

)2] 1

2 (859)

where σ(r) is the three-dimensional velocity dispersion of the Plummer sphereat radius r σ2(r) =

sumk=rθφ σ

3k(r) or σ2(r) = 3σ2

1D(r) because isotropy isassumed

A star with mass m positioned at r and with speed v =(sum3

k=1 v2k

)12

can escape from the cluster if it has a total energy ebind = ekin + epot =05mv2 + mφ(r) ge 0 so that v ge vesc(r) So the escape speed at radiusr is vesc(r) =

radic2 |φ(r)| The potential at r is given by the mass within r

plus the potential contributed by the surrounding matter It is calculated byintegrating the contributions from each radial mass shell

φ(r) = minus[

GM(r)r

+int infin

r

G1rprime

ρ(rprime) 4π rprime2drprime

]

= minus(GMecl

rpl

)1

[1 + (rrpl)2]12

(860)

so that

vesc(r) =(

2GMecl

rpl

)12 1

[1 + (rrpl)2]14

(861)

208 P Kroupa

The circular speed vc of a star moving on a circular orbit at a distancer from the cluster centre is obtained from centrifugal acceleration v2

cr =dφ(r)dr = GM(r)r2

v2c =

(GMecl

rpl

)(rrpl)

2

[1 + (rrpl)2]32 (862)

In many but not all instances of interest the initial cluster model is chosento be in the state of virial equilibrium That is the kinetic and potentialenergies of each star balance so that the whole cluster is stationary Thescalar virial theorem

2K +W = 0 (863)

where K and W are the total kinetic and potential energy of the cluster7

K =12

int infin

0

ρ(r)σ2(r) 4πr2dr

=3π64

GM2ecl

rpl for the Plummer sphere (864)

W =12

int infin

0

φ(r) ρ(r) 4πr2dr

= minus3π32

GM2ecl

rplfor the Plummer sphere (865)

The total or binding energy of the cluster Etot = W +K is

Etot = minusK =12W (866)

The characteristic three-dimensional velocity dispersion of a cluster can bedefined as σ2

cl equiv 2KMecl so that

σ2cl =

3π32

GMecl

rpl (867)

equiv GMecl

rgrav (868)

equiv s2(GMecl

2 rh

)

(869)

which introduces the gravitational radius of the cluster rgrav equiv GM2ecl|W |

For the Plummer sphere rgrav = (323π)rpl = 34 rpl and the structure factor

s =(

6 times 1305π32

) 12

asymp 088 (870)7Equation (32514) on p 295 of Gradshteyn amp Ryzhik (1980) is useful to solve

the integrals for the Plummer sphere

We define the virial ratio by

Q =K

|W | (871)

so that a cluster can initially be in three possible states

Q

⎧⎪⎨

⎪⎩

= 12 virial equilibrium

gt 12 expanding

lt 12 collapsing

(872)

Note that if initially Q lt 12 the value Q = 12 will be reached temporarilyduring collapse after which Q increases further until the cluster settles invirial equilibrium after this violent relaxation phase (Binney amp Tremaine 1987p 271)

The characteristic crossing time through the Plummer cluster

tcr equiv2 rpl

σ1Dcl (873)

=(

128πG

) 12

Mminus 1

2ecl r

32pl (874)

with the characteristic one-dimensional velocity dispersion σ1Dcl = σclradic

3Observationally the core radius is that radius where the projected surface

density falls to half its central value For a real cluster it is much easier todetermine than the other characteristic radii For the Plummer sphere

Rcore =(radic

2 minus 1) 1

2rpl = 064 rpl (875)

from (855) with the assumption that the mass-to-light ratio Υ is indepen-dent of radius For a King model

Rkingcore =

(9

4πGσ2

ρm(0)

) 12

(876)

is the King radius From (859) σ2(0) = GMecl(2 rpl) and from (851)ρm(0) = 3Mecl(4π r3pl) so that

rpl =(

64πG

σ(0)2

ρm(0)

) 12

= 082 Rkingcore (877)

The Singular Isothermal Model

Another useful set of distribution functions can be arrived at by consideringn = infin The LanendashEmden equation is not well defined in this limit but for a

210 P Kroupa

polytropic gas sphere (845) implies γ rarr 1 as n rarr infin Thus p = K ρm which isthe equation of state of an isothermal ideal gas with K = kB Tmp where kB

is Boltzmannrsquos constant T the temperature and mP the mass of a gas particleFrom the equation of hydrostatic support dpdr = minusρm(GM(r)r2) whereM(r) is the mass within r the following equation can be derived

ddr

(

r2d ln ρm

dr

)

= minusGmp

kB T4π r2 ρm (878)

For a distribution function (our ansatz)

fm(E) =ρm1

(2π σ2)32e

Eσ2 (879)

where σ2 is a new quantity related to a velocity dispersion and E = Ψminus v22one obtains from ρm =

intfm(E) 4π v2 dv

Ψ(r) = ln(ρm(r)ρm1

)

σ2 (880)

From the Poisson equation it then follows that

σ = const =kB T

mp(881)

for consistency with (878)Therefore the structure of an isothermal self-gravitating sphere of ideal

gas is identical to the structure of a collisionless system of stars whose phase-space mass-density distribution function is given by (879) Note that f(E) isnon-zero at all E (cf Kingrsquos models below)

The number-distribution function of velocities is F (v) =intall x

f(E) d3x ie

F (v) = F0 eminus v2

2 σ2 (882)

This is the MaxwellndashBoltzmann distribution which results from the kinetictheory of atoms in a gas at temperature T that are allowed to bounce offeach other elastically This exact correspondence between a stellar-dynamicalsystem and a gaseous polytrope holds only for an isothermal case (n = infin)

The total number of stars in the system is Ntot = Ntot

intinfin0

F (v) 4π v2 dvand the number of stars in the speed interval v to v + dv is

dN = F (v) 4π v2 dv = Ntot1

(2πσ2)32eminus

v2

2 σ2 4π v2 dv (883)

which is the MaxwellndashBoltzmann distribution of speeds The mean-squarespeed of stars at a point in the isothermal sphere is

v2 =4π

intinfin0

σ2 F (v) dv4π

intinfin0

F (v) dv= 3σ2 (884)

and the 1D velocity dispersion is σ1D = σα = σ where α = r θ φ x y z To obtain the radial mass-density of this model the ansatz ρm = C rminusb

together with the Poisson equation (878) implies

ρm(r) =σ2

2πG1r2 (885)

That is a singular isothermal sphere

The Isothermal Model

The above model has a singularity at the origin This is unphysical In order toremove this problem it is possible to force the central density to be finite Tothis end new dimensionless variables are introduced ρm equiv ρmρm0 r equiv rr0The density ρm is the finite central density while r0 = RKing

core is the King radius(876) at which the projected density falls to 05013 (ie about half) its centralvalue The radius r0 is also sometimes called the core radius (but see furtherbelow for King models on p 211) The Poisson equation (878) then becomes

ddr

(

r2d ln ρm

dr

)

= minus9 ρm r2 (886)

This differential equation must be solved numerically for ρm(r) subject to theboundary conditions (as before)

ρm(r = 0) = 1dρm

dr

∣∣∣∣∣r=0

= 0 (887)

The solution is the isothermal sphereBy imposing physical reality (central non-singularity) on our mathematical

ansatz we end up with a density profile that cannot be arrived at analyticallybut only numerically The isothermal density sphere must be tabulated in thecomputer with entries such as

rr0 log10

(ρ

ρ0

)

and log10

(Σ

r0 ρ0

)

(888)

where Σ is the projected density (Binney amp Tremaine 1987 for example seetheir Table 41 and Fig 47 of) The circular velocity vc(r) = GM(r)r of theisothermal sphere is obtained by integrating Poissonrsquos equation (878) fromr = 0 to r = rprime with r2(d ln ρmdr) = minus(Gσ2)M(r) and

v2c (r) = minusσ2 d ln ρm(r)

d ln r (889)

212 P Kroupa

Numerical solution of differential (886) shows that vc rarrradic

2σ (constant) forlarge r

The isothermal sphere is a useful model for describing elliptical galaxieswithin a few core radii and disc galaxies because of the constant rotationcurve However combining the two equations for v2

c above one finds thatM(r) asymp (2σ2G) r for large r ie the isothermal sphere has an infinite massas it is not bounded

The Lowered Isothermal or King Model

We have thus seen that the class of models with n = infin contain as the simplestcase the singular isothermal sphere By forcing the central density to be finitewe are led to the isothermal sphere which however has an infinite mass Thefinal model considered here within this class is the lowered isothermal modelor the King model8 which forces not only a finite central density but alsoa cutoff in radius These have a distribution function similar to that of theisothermal model except for a cutoff in energy

fm(E) =

ρm1

(2 π σ2)32

(e

Eσ2 minus 1

) E gt 0

0 E le 0(890)

The density distribution becomes

ρm = ρm1

[

eΨσ2 erf

(radicΨσ

)

minusradic

4Ψπ σ2

(

1 +2Ψ3σ2

)]

(891)

with integration only to E = 0 as before The Poisson (878) becomes

ddr

(

r2d ln ρm

dr

)

= minus4πGρm1 r2

[

eΨσ2 erf

(radicΨσ

)

minusradic

4Ψπ σ2

(

1 +2Ψ3σ2

)]

(892)

Again this differential equation must be solved numerically for Ψ(r) subjectto the boundary conditions

Ψ(0)dΨdr

|r=0 = 0 (893)

The density vanishes at r = rtid (the tidal radius) where Ψ(r = rtid) = 0also A King model is thus limited in mass and has a finite central density

8Note that King (1962) suggested a three-parameter (mass core radius and cut-offtidal radius) empirical projected (2D) density law that fits globular clustersvery well These do not have information on the velocity structure of the clustersThe King (non-analytical) 6D models which are solutions of the Jeans equation((8120) below) and discussed here are published by King (1966)

Fig 85 The King concentration parameter W0 as a function of c (cf with Fig 4ndash10of Binney amp Tremaine 1987) This figure has been produced by Andreas Kupper

but the parameter σ is not the velocity dispersion It is rather related to thedepth of the potential via the concentration parameter

Wo equiv Ψ(0)σ2

(894)

The concentration is defined as

c equiv log10

(rtidro

)

(895)

For globular clusters 3 lt Wo lt 9 075 lt c lt 175 and the relation betweenWo and c is plotted in Fig 85 Note also that the true core radius defined asΣ(Rc) = (12)Σ(0) where Σ(R) is the projected density profile and R is theprojected radius is unequal in general to the King radius r0 (876) Finallyit should be emphasised that it is not physical to use an arbitrary rtid Thetidal radius must always match the value dictated by the cluster mass andthe host galaxy (eg (83))

822 Comparison Plummer vs King Models

The above discussion has served to show how various popular models can befollowed through from a power-law distribution function (841) with differentindices n The Plummer model (p 205) and the King model (p 212) are par-ticularly useful for describing star clusters The Plummer model is determinedby two parameters the mass M and the scale radius rh asymp 1305 rpl TheKing model requires three parameters M a scale radius rh and a concen-tration parameter W0 or c Which subset of parameters yield models that aresimilar in terms of the overall density profile

214 P Kroupa

Fig 86 Comparison of a King model (solid curve) with a Plummer model (dashedcurve) Both have the same mass and that Plummer model is sought which min-imises the unweighted reduced chi-squared between the two models The upper panelshows a high-concentration King model with c = 255 and W0 = 11 and the best-fitPlummer model has rPlummer

h = 0366 rKingh (rh equiv rh) as stated in the panel The

lower panel compares the two best matching models for the case of an intermediate-concentration King model This figure was produced by Andreas Kupper

To answer this the mass is set to be constant King models with differentW0 and rh are computed and Plummer models are sought which minimisethe reduced chi-squared value between the two density profiles Figure 86shows two examples of best-matching density profiles and Fig 87 revealsthe family of Plummer profiles that best match King models with differentconcentrations Note that a good match between the two is only obtained forintermediate-concentration King models (25 le W0 le 75)

823 Discretisation

To set up a computer model of a stellar system withN particles (eg stars) thedistribution functions need to be sampled N times The relevant distribution

Fig 87 The ratio rPlummerh rKing

h (rh equiv rh) for the best-matching Plummer andKing models (Fig 86) are plotted as a function of the King concentration param-eter W0 The uncertainties are unweighted reduced chi-squared values between thetwo density profiles It is evident that there are no well-matching Plummer modelsfor low- (c lt 25) and high-concentration (c gt 75) King models This figure wasproduced by Andreas Kupper

functions are the phase-space distribution function the stellar initial massfunction and the three distribution functions governing the properties of bi-nary stars (periods mass-ratios and eccentricities)

Assume the distribution function depends on the variable ζmin le ζ le ζmax

(eg stellar mass m) There are various ways of sampling from a distributionfunction (Press et al 1992) but the most efficient way is to use a generatingfunction if one exists Consider the probability X(ζ) of encountering a valuefor the variable in the range ζmin to ζ

X(ζ) =int ζ

ζmin

p(ζ prime) dζ prime (896)

with X(ζmin) = 0 le X(ζ) le X(ζmax) = 1 and p(ζ) is the distribution func-tion normalised so that the latter equal sign holds (X = 1) p(ζ) is the prob-ability density The inverse of (896) ζ(X) is the generating function It is aone-to-one map of the uniform distribution X isin [0 1] to ζ isin [ζmin ζmax]If an analytical inverse does not exist it can be found numerically in astraightforward manner for example by constructing a table of X ζ andthen interpolating this table to obtain a ζ for a given X

Example The Power-Law Stellar Mass Function

As an example consider the distribution function

ξ(m) = kmminusα α = 235 05 le m

Mle 150 (897)

216 P Kroupa

The probability density is p(m) = kp mminusα and

int 150

05p(m) dm = 1 rArr kp =

053 Thus

X(m) =int m

05

p(m) dm = kp1501minusα minus 051minusα

1 minus α(898)

and the generating function for stellar masses becomes

m(X) =[

X1 minus α

kp+ 051minusα

] 11minusα

(899)

It is easy to programme this into an algorithm Obtain a random variate Xfrom a random number generator and use the above generating function toget a corresponding mass m Repeat N times

Generating a Plummer Model

Perhaps the most useful and simplest model of a bound stellar system is thePlummer model (p 205) It is worth introducing the discretisation of thismodel in some detail because analytical formulae go a long way which isimportant for testing codes A condensed form of this material is available inAarseth Henon and Wielen (1974)

The mass within radius r is (rpl = b here)

M(r) =int r

0

ρm(rprime) 4π rprime2drprime = Mcl

(rrpl)3

[1 + (rrpl)

2] 3

2 (8100)

A number uniformly distributed between zero and one can then be defined

X1(r) =M(r)Mcl

=ζ3

[1 + ζ2] (8101)

where ζ equiv rrpl and X1(r = infin) = 1 This function can be inverted toyield the generating function for particle distances distributed according to aPlummer density law

ζ(X1) =(X

minus 23

1 minus 1)minus 1

2 (8102)

The coordinates of the particles x y z r2 = (ζ rpl)2 = x2 + y2 + z2 can beobtained as follows For a given particle we already have r For all possiblex and y z has a uniform distribution p(z) = const = 1(2 r) over the rangeminusr le z le +r Thus for a second random variate between zero and one

X2(z) =int z

minusr

p(zprime) dzprime =12 r

(z + r) (8103)

with X2(+r) = 1 The generating function for z becomes

z(X2) = 2 r X2 minus r (8104)

Having obtained r and z x and y can be arrived at as follows noting theequation for a circle r2 minus z2 = x2 + y2 Choose a random angle θ which isuniformly distributed over the range 0 le θ le 2π Thus p(θ) = 1(2π) andthe third random variate becomes

X3(θ) =int θ

0

12π

dθprime =θ

2π (8105)

The corresponding generating function is

θ(X3) = 2πX3 (8106)

Finally

x =(r2 minus z2

) 12 cosθ and y =

(r2 minus z2

) 12 sinθ (8107)

The velocity for each particle cannot be obtained as simply as the positionsIn order for the initial stellar system to be in virial equilibrium the potentialand kinetic energy need to balance according to the scalar virial theoremThis is ensured by forcing the velocity distribution function to be that of thePlummer model

fm(εe) =

(24

radic2

2 π3r2pl

(G Mcl)5

)(minusεe)

72 εe le 0

0 εe gt 0(8108)

whereεe(r v) = Φ(r) + (12) v2 (8109)

is the specific energy per star and

Φ(r) = minusGMcl

rpl

(

1 +(

r

rpl

)2)minus 1

2

(8110)

is the potential Now the Plummer distribution function can be expressed interms of r and v

f(r v) = fo

(

minusΦ(r) minus 12v2

) 72

(8111)

for a normalisation constant fo and dropping the mass subscript because weassume the positions and velocities do not depend on particle mass With theescape speed at distance r from the Plummer centre vesc(r) =

radicminus2Φ(r) equiv

vζ it follows that

f(r v) = fo

(12vesc

)7 (1 minus ζ2

) 72 (8112)

218 P Kroupa

The number of particles with speeds in the interval v to v + dv is

dN = f(r v) 4π v2 dv equiv g(v) dv (8113)

Thus

g(v) = 16π fo

(12vesc(r)

)9 (1 minus ζ2(r)

) 72 ζ2(r) (8114)

that isg(ζ) = go ζ

2(r)(1 minus ζ2(r)

) 72 (8115)

for a normalisation constant go determined by demanding that

X4(ζ = 1) = 1 =int 1

0

g(ζ prime) dζ prime (8116)

for a fourth random number variate X4(ζ) =int ζ

0g(ζ prime) dζ prime It follows that

X4(ζ) =12(5 ζ3 minus 3 ζ5

) (8117)

This cannot be inverted to obtain an analytical generating function for ζ =ζ(X4) Therefore numerical methods need to be used to solve (8117) Forexample one way to obtain ζ for a given random variate X4 is to find theroot of the equation 0 = (12) (5 ζ3 minus3 ζ5)minusX4 or one can use the Neumannrejection method (Press et al 1992)

The following procedure can be implemented to calculate the velocity vec-tor of a particle for which r and ζ are already known from above Computevesc(r) so that v = ζ vesc Each speed v is then split into its componentsvx vy vz assuming velocity isotropy using the same algorithm as above forx y z

vz(X5) = (2X5 minus 1) v θ(X6) = 2πX6 (8118)

vx =radicv2 minus v2

z cosθ vy =radicv2 minus v2

z sinθ (8119)

Note that a rotating Plummer model can be generated by simply switchingthe signs of vx and vy so that all particles have the same direction of motionin the x y plane

As an aside an efficient numerical method to set up triaxial ellipsoidswith or without an embedded rotating disc is described by Boily Kroupa ampPenarrubia-Garrido (2001)

Generating an Arbitrary Spherical Non-Rotating Model

In most cases an analytical density distribution is not known (eg theKing models above) Such numerical models can nevertheless be discretisedstraightforwardly as follows Assume that the density distribution ρ(r) isknown Compute M(r) and Mcl Define X(r) = M(r)Mcl as above We thus

have a numerical grid of numbers r M(r) X(r) For a given random variateX isin [0 1] interpolate r from this grid Compute x y z as above

If the distribution function of speeds is too complex to yield an analyticalgenerating function X(ζ) for the speeds ζ we can resort to the followingprocedure One of the Jeans equations for a spherical system is

ddr

(ρ(r)σr(r)2

)+ρ(r)r

[2σ2

r(r) minus(σθ(r)2 + σφ(r)2

)]= minusρ(r) dΦ(r)

dr

(8120)For velocity isotropy σ2

r = σ2θ = σ2

φ this reduces to

d(ρ σ2

r

)

dr= minusρ dΦ

dr (8121)

Integrating this by making use of ρ rarr 0 as r rarr infin and remembering thatdΦdr = minusGMr2

σ2r(r) =

1ρ(r)

int infin

r

rprime2drprime (8122)

For each particle at distance r a one-dimensional velocity dispersion σr(r) isthus obtained Choosing randomly from a Gaussian distribution with disper-sion σi i = r θ φ x y z then gives the velocity components (eg vx vy vz)for this particle

Rotating Models

Star clusters are probably born with some rotation because the pre-clustercloud core is likely to have contracted from a cloud region with differentialmotions that do not cancel How large this initial angular momentum contentof an embedded cluster is remains uncertain because the dominant motionsare random and chaotic owing to the turbulent velocity field of the gas Oncethe star-formation process is quenched as a result of gas blow-out (Sect 811)the cluster expands This must imply substantial reduction in the rotationalvelocity A case in point is ω Cen which has been found to rotate with a peakvelocity of about 7 km sminus1 (Pancino et al 2007 and references therein)

A setup for rotating cluster models is easily made for instance by increas-ing the tangential velocities of stars by a certain factor A systematic studyof relaxation-driven angular momentum re-distribution within star clustershas become available through the work of the group of Rainer Spurzem andHyung-Mok Lee and the interested reader is directed to that body of work(Kim et al 2008 and references therein) One important outcome of thiswork is that core collapse is substantially accelerated in rotating models Theprimary reason for this is that increased rotational support reduces the role ofsupport through random velocities for the same cluster dimension Thus therelative stellar velocities decrease and the stars exchange momentum and en-ergy more efficiently enhancing two-body relaxation and thence the approachtowards energy equipartition

220 P Kroupa

824 Cluster Birth and Young Clusters

Some astrophysical issues related to the initial conditions of star clusters havebeen raised in Sect 811 In order to address most of these issues numericalexperiments are required The very initial phase the first 05Myr can onlybe treated through gas-dynamical computations that however lack the nu-merical resolution for the high-precision stellar-dynamical integrations whichare the essence of collisional dynamics during the gas-free phase of a clusterrsquoslife This gas-free stage sets in with the blow out of residual gas at an age ofabout 05ndash15Myr The time 05ndash15Myr is dominated by the physics of stel-lar feedback and radiation transport in the residual gas as well as energy andmomentum transfer to it through stellar outflows The gas-dynamical com-putations cannot treat all the physical details of the processes acting duringthis critical time which also include early stellar-dynamical processes such asmass segregation and binaryndashbinary encounters

One successful procedure to investigate the dominant macroscopic physicalprocesses of these stellar-dynamical processes gas blow-out and the ensuingcluster expansion through to the long-term evolution of the remnant clusteris to approximate the residual gas component as a time-varying potential inwhich the young stellar population is trapped The pioneering work usingthis approach has been performed by Lada Margulis amp Dearborn (1984)whereby the earlier numerical work by Tutukov (1978) on open clusters andlater N -body computations by Goodwin (1997ab 1998) on globular clustersmust also be mentioned in this context

The physical key quantities that govern the emergence of embedded clus-ters from their clouds and their subsequent appearance are (BaumgardtKroupa amp Parmentier 2008 Sect 811)

bull sub-structuringbull initial mass segregationbull the dynamical state at feedback termination (dynamical equilibrium col-

lapsing or expanding)bull the star-formation efficiency εbull the ratio of the gas-expulsion time-scale to the stellar crossing time through

the embedded cluster τgastcross andbull the ratio of the embedded-cluster half-mass radius to its tidal radius rhrt

It becomes rather apparent that the physical processes governing theemergence of star clusters from their natal clouds is terribly messy and theresearch-field is clearly observationally driven Observations have shown thatstar clusters suffer substantial infant weight loss and probably about 90 of allclusters disperse altogether (infant mortality) This result is consistent withthe observational insight that clusters form in a compact configuration witha low star-formation efficiency (02 le ε le 04) and that residual-gas blow-outoccurs on a time-scale comparable or even faster than an embedded-clustercrossing time-scale (Kroupa 2005) Theoretical work can give a reasonable

description of these empirical findings by combining some of the above pa-rameters such as an effective star-formation efficiency as a measure of theamount of gas removed for a cluster of a given stellar mass if this cluster werein dynamical equilibrium at feedback termination and that the gas and starswere distributed according to the same radial density function with the samescaling radius

Embedded Clusters One way to parameterise an embedded cluster is to setup a Plummer model in which the stellar positions follow a density law withthe parameters Mecl and rpl and the residual gas is a time-varying Plummerpotential initially with the parameters Mgas and rpl ie modelled with thesame radial density law The effective star-formation efficiency is then given by(82) Stellar velocities must then be calculated from a Plummer law with totalmass Mecl +Mgas following the recipes of Sect 823 The gas can be removedby evolving Mgas or rpl For example Kroupa Aarseth amp Hurley (2001) andBaumgardt Kroupa amp Parmentier (2008) assumed the gas mass decreasesexponentially after an embedded phase lasting about 05Myr during whichthe cluster is allowed to evolve in dynamical equilibrium Bastian amp Goodwin(2006) as another example do not include a gas potential but take the initialvelocities of stars to be 1

radicε times larger vembedded = (1

radicε) vno gas to model

the effect of instantaneous gas removal Many variations of these assumptionsare possible and Adams (2000) for example investigated the fraction of starsleft in a cluster remnant if the radial scale length of the gas is different to thatof the stars ie for a radially dependent star-formation efficiency ε(r)

Subclustering Initial subclustering has been barely studied Scally amp Clarke(2002) considered the degree of sub-structuring of the ONC allowed by its

current morphology while Fellhauer amp Kroupa (2005) computed the evolutionof massive star-cluster complexes assuming each member cluster in the com-plex undergoes its own individual gas-expulsion process McMillan Vesperiniamp Portegies Zwart (2007) showed that initially mass-segregated subclustersretain mass segregation upon merging This is an interesting mechanism foraccelerating dynamical mass segregation because it occurs faster in smaller-Nsystems which have a shorter relaxation time

The simplest initial conditions for such numerical experiments are to set upthe star-cluster complex (or protoONC-type cluster for example) as a Plum-mer model where each particle is a smaller subcluster Each subcluster is alsoa Plummer model embedded in a gas potential given as a Plummer modelThe gas-expulsion process from each subcluster can be treated as above

Mass Segregation and Gas Blow-Out The problem of how initially mass-segregated clusters react to gas blow-out has not been studied at all in thepast This is due partially to the lack of convenient algorithms to set up mass-segregated clusters that are in dynamical equilibrium and which do not gointo core collapse as soon as the N -body integration begins An interesting

222 P Kroupa

consequence here is that gas blow-out will unbind mostly the low-mass starswhile the massive stars are retained These however evolve rapidly so thatthe mass lost from the remnant cluster owing to the evolution of the massivestars can become destructive enhancing infant mortality

Ladislav Subr has developed a numerically efficient method to set up ini-tially mass-segregated clusters close to core-collapse based on a novel conceptthat uses the potentials of subsets of stars ordered by their mass (Subr Kroupaamp Baumgardt 2008)9 An alternative algorithm based on ordering the starsby increasing mass and increasing total energy that leads to total mass seg-regation and also to a model that is not in core collapse and which thereforeevolves towards core collapse has been developed by Baumgardt Kroupa ampde Marchi (2008) An application concerning the effect on the observed stellarmass function in globular clusters shows that gas expulsion leads to bottom-light stellar mass functions in clusters with a low concentration consistentwith observational data (Marks Kroupa amp Baumgardt 2008)

83 The Stellar IMF

The stellar initial mass function (IMF) ξ(m) dm where m is the stellar massis the parent distribution function of the masses of stars formed in one eventHere the number of stars in the mass interval m to m+ dm is

dN = ξ(m) dm (8123)

The IMF is strictly speaking an abstract theoretical construct because anyobserved system of N stars merely constitutes a particular representation ofthis universal distribution function if such a function exists (Elmegreen 1997Maız Apellaniz amp Ubeda 2005) The probable existence of a unique ξ(m) canbe inferred from the observations of an ensemble of systems each consisting ofN stars (eg Massey 2003) If after corrections for (a) stellar evolution (b)unknown multiple stellar systems and (c) stellar-dynamical biases the indi-vidual distributions of stellar masses are similar within the expected statisticalscatter we (the community) deduce that the hypothesis that the stellar massdistributions are not the same can be excluded That is we make the case fora universal standard or canonical stellar IMF within the physical conditionsprobed by the relevant physical parameters (metallicity density mass) of thepopulations at hand

Related overviews of the IMF can be found in Kroupa (2002a) Chabrier(2003) Bonnell Larson amp Zinnecker (2007) Kroupa (2007a) and a review

with an emphasis on the metal-rich problem is available in Kroupa (2007b)Zinnecker amp Yorke (2007) provide an in-depth review of the formation anddistribution of massive stars Elmegreen (2007) discusses the possibility thatstar-formation occurs in different modes with different IMFs

9The C-language software package plumix may be downloaded from the websitehttpwwwastrouni-bonnde~webaiubenglishdownloadsphp

831 The Canonical or Standard Form of the Stellar IMF

The canonical stellar IMF is a two-part-power law (8128) The only structurefound with confidence so far is the change of index from the SalpeterMasseyvalue to a smaller one near 05M

10

ξ(m) prop mminusαi i = 1 2(8124)

α1 = 13 plusmn 03 008 le mM le 05α2 = 23 plusmn 05 05 le mM le mmax

where mmax le mmaxlowast asymp 150M follows from Fig 81 Brown dwarfs havebeen found to form a separate population with α0 asymp 03plusmn 05 (8129) (Thiesamp Kroupa 2007)

It has been corrected for bias through unresolved multiple stellar systemsin the low-mass (m lt 1M) regime (Kroupa Gilmore amp Tout 1991) by amulti-dimensional optimisation technique The general outline of this tech-nique is as follows (Kroupa Tout amp Gilmore 1993) First the correct form ofthe stellar-massndashluminosity relation is extracted using observed stellar bina-ries and theoretical constraints on the location amplitude and shape of theminimum of its derivative dmdMV near m = 03MMV asymp 12MI asymp 9 incombination with the observed shape of the nearby and deep Galactic-fieldstellar luminosity function (LF)

Ψ(MV ) = minus(

dmdMV

)minus1

ξ(m) (8125)

where dN = Ψ(MV ) dMV is the number of stars in the magnitude inter-val MV to MV + dMV Once the semi-empirical massndashluminosity relation ofstars which is an excellent fit to the most recent observational constraints byDelfosse et al (2000) is established a model of the Galactic field is calculatedwith the assumption that a parameterised form for the MF and different val-ues for the scale-height of the Galactic disc and different binary fractions init Measurement uncertainties and age and metallicity spreads must also beconsidered in the theoretical stellar population Optimisation in this multi-parameter space (MF parameters scale-height and binary population) againstobservational data leads to the canonical stellar MF for m lt 1M

One important result from this work is the finding that the LF of main-sequence stars has a universal sharp peak near MV asymp 12MI asymp 9 It resultsfrom changes in the internal constitution of stars that drive a non-linearity inthe stellar massndashluminosity relation A consistency check is then performedas follows The above MF is used to create young populations of binary sys-tems (Sect 842) that are born in modest star clusters consisting of a fewhundred stars Their dissolution into the Galactic field is computed with an

10The uncertainties in αi are estimated from the alpha-plot (Sect 832) as shownin Fig 5 of Kroupa (2002b) to be about 95 confidence limits

224 P Kroupa

Fig 88 The Galactic field population that results from disrupted star clustersunification of both the nearby (solid histogram) and deep (filled circles) LFs withone parent MF (8124) The theoretical nearby LF (dashed line) is the LF of allindividual stars while the solid curve is a theoretical LF with a mixture of about50 per cent unresolved binaries and single stars from a clustered star-formationmode According to this model all stars are formed as binaries in modest clusterswhich disperse to the field The resulting Galactic field population has a binaryfraction and a mass-ratio distribution as observed The dotted curve is the initialsystem LF (100 binaries) (Kroupa 1995ab) Note the peak in both theoreticalLFs It stems from the extremum in the derivative of the stellar-massndashluminosityrelation in the mass range 02ndash04 M (Kroupa 2002b)

N -body code and the resulting theoretical field is compared to the observedLFs (Fig 88) Further confirmation of the form of the canonical IMF comesfrom independent sources most notably by Reid Gizis amp Hawley (2002) andalso Chabrier (2003)

In the high-mass regime Massey (2003) reports the same slope or in-dex α3 = 23 plusmn 01 for m ge 10M in many OB associations and star clus-ters in the Milky Way and the Large and Small Magellanic clouds (LMCSMC respectively) It is therefore suggested to refer to α2 = α3 = 23 as theSalpeterMassey slope or index given the pioneering work of Salpeter (1955)who derived this value for stars with masses 04ndash10M

Multiplicity corrections await publication once we learn more about howthe components are distributed in massive stars (cf Preibisch et al 1999Zinnecker 2003) Weidner amp Kroupa (private communication) are in the pro-cess of performing a very detailed study of the influence of unresolved binaryand higher-order multiple stars on determinations of the high-mass IMF

Contrary to the SalpeterMassey index (α = 23) Scalo (1986) foundαMWdisc asymp 27 (m ge 1M) from a very thorough analysis of OB star countsin the Milky Way disc Similarly the star-count analysis of Reid Gizis ampHawley (2002) leads to 25 le αMWdisc le 28 and Tinsley (1980) Kennicutt(1983) (his extended Miller-Scalo IMF) Portinari Sommer-Larsen amp Tantalo(2004) and Romano et al (2005) find 25 le αMWdisc le 27 That αMWdisc gt α2

follows naturally is shown in Sect 834Below the hydrogen-burning limit (see also Sect 833) there is substantial

evidence that the IMF flattens further to α0 asymp 03 plusmn 05 (Martın et al 2000Chabrier 2003 Moraux et al 2004) Therefore the canonical IMF most likelyhas a peak at 008M Brown dwarfs however comprise only a few per cent ofthe mass of a population and are therefore dynamically irrelevant (Table 82)The logarithmic form of the canonical IMF

ξL(m) = log10 mξ(m) (8126)

which gives the number of stars in log10 m-intervals also has a peak near008M However the system IMF (of stellar single and multiple systemscombined to system masses) has a maximum in the mass range 04ndash06M(Kroupa et al 2003)

The above canonical or standard form has been derived from detailedconsiderations of Galactic field star counts and so represents an average IMFFor low-mass stars it is a mixture of stellar populations spanning a largerange of ages (0ndash10 Gyr) and metallicities ([FeH]ge minus1) For the massivestars it constitutes a mixture of different metallicities ([FeH]ge minus15) andstar-forming conditions (OB associations to very dense star-burst clustersR136 in the LMC) Therefore it can be taken as a canonical form and theaim is to test the

IMF universality hypothesis that the canonical IMF constitutes theparent distribution of all stellar populations

Negation of this hypothesis would imply a variable IMF Note that the work ofMassey (2003) has already established the IMF to be invariable for m ge 10Mand for densities ρ le 105 stars pcminus3 and metallicity Z ge 0002

Finally Table 82 compiles some numbers that are useful for simple insightsinto stellar populations

832 Universality of the IMF Resolved Populations

The strongest test of the IMF universality hypothesis (p 225) is obtainedby studying populations that can be resolved into individual stars Because wealso seek co-eval populations with stars at the same distance and with the samemetallicity to minimise uncertainties star clusters and stellar associationswould seem to be the test objects of choice But before contemplating suchwork some lessons from stellar dynamics are useful

226 P Kroupa

Table 82 The number fraction ηN = 100int m2

m1ξ(m) dm

int mu

mlξ(m) dm and the

mass fraction ηM = 100int m2

m1m ξ(m) dmMcl Mcl =

int mu

mlm ξ(m) dm in per cent of

BDs or main-sequence stars in the mass interval m1 to m2 and the stellar con-tribution ρst to the Oort limit and to the Galactic-disc surface mass-densityΣst = 2 hρst near to the Sun with ml = 001 M mu = 120 M and theGalactic-disc scale-height h = 250 pc (m lt 1 M Kroupa Tout amp Gilmore 1993)and h = 90 pc (m gt 1 M Scalo 1986) Results are shown for the canonical IMF(8124) for the high-mass-star IMF approximately corrected for unresolved compan-ions (α3 = 27 m gt 1 M) and for the present-day mass function (PDMF α3 = 45Scalo 1986 Kroupa Tout amp Gilmore 1993) which describes the distribution of stellarmasses now populating the Galactic disc For gas in the disc Σgas = 13plusmn3 Mpc2

and remnants Σrem asymp 3 Mpc2 (Weidemann 1990) The average stellar mass ism =

int mu

mlm ξ(m) dm

int mu

mlξ(m) dm Ncl is the number of stars that have to form in

a star cluster so that the most massive star in the population has the mass mmaxThe mass of this population is Mcl and the condition is

intinfinmmax

ξ(m) dm = 1 withint mmax001

ξ(m) dm = Ncl minus 1 ΔMclMcl is the fraction of mass lost from the clusterdue to stellar evolution if we assume that for m ge 8 M all neutron stars and blackholes are kicked out by asymmetrical supernova explosions but that white dwarfs areretained (Weidemann et al 1992) and have masses mWD = 0084 mini + 0444 [M]This is a linear fit to the data of Weidemann (2000 their Table 3) for progenitormasses 1 le mM le 7 and mWD = 05 M for 07 le mM lt 1 The evolutiontime for a star of mass mto to reach the turn-off age is available in Fig 20 of Kroupa(2007a)

Mass range ηN ηM ρst Σst

[M] [] [] [Mpc3] [Mpc2]α3 α3 α3 α3

23 27 45 23 27 45 45 45

001ndash008 372 377 386 41 54 74 32 times 10minus3 160008ndash05 478 485 497 266 352 482 21 times 10minus2 10505ndash1 89 91 93 161 213 292 13 times 10minus2 641ndash8 57 46 24 324 303 151 65 times 10minus3 128ndash120 04 01 00 208 78 01 36 times 10minus5 65 times 10minus3

mM = 038 029 022 ρsttot = 0043 Σst

tot = 196

α3 = 23 α3 = 27 ΔMclMcl

mmax Ncl Mcl Ncl Mcl mto [][M] [M] [M] [M] α3 = 23 α3 = 27

1 16 29 21 38 80 32 078 245 74 725 195 60 49 11

20 806 269 3442 967 40 75 1840 1984 703 11 times 104 2302 20 13 4760 3361 1225 22 times 104 6428 8 22 8080 4885 1812 36 times 104 11 times 104 3 32 15

100 6528 2451 53 times 104 15 times 104 1 44 29120 8274 3136 72 times 104 21 times 104 07 47 33

Star Clusters and Associations

To access a pristine population one would consider observing star-clustersthat are younger than a few Myr However such objects carry rather seriousdisadvantages The pre-mainsequence stellar evolution tracks are unreliable(Baraffe et al 2002 Wuchterl amp Tscharnuter 2003) so that the derived massesare uncertain by at least a factor of about two Remaining gas and dust leadto patchy obscuration Very young clusters evolve rapidly The dynamicalcrossing time is given by (84) where the cluster radii are typically rh lt1 pc and for pre-cluster cloud-core masses Mgas+stars gt 103 M the velocitydispersion σcl gt 2 km sminus1 so that tcr lt 1Myr

The inner regions of populous clusters have tcr asymp 01Myr and thus signifi-cant mixing and relaxation occurs there by the time the residual gas has beenexpelled by any winds and photo-ionising radiation from massive stars Thisis the case in clusters with N ge few times 100 stars (Table 81)

Massive stars (m gt 8M) are either formed at the cluster centre or getthere through dynamical mass segregation ie energy equipartition (Bonnellet al 2007) The latter process is very rapid ((86) p 184) and can occurwithin 1Myr A cluster core of massive stars is therefore either primordial orforms rapidly because of energy equipartition in the cluster and it is dynam-ically highly unstable decaying within a few tcr core The ONC for exampleshould not be hosting a Trapezium because it is extremely unstable The im-plication for the IMF is that the ONC and other similar clusters and the OBassociations which stem from them must be very depleted in their massivestar content (Pflamm-Altenburg amp Kroupa 2006)

Important for measuring the IMF are corrections for the typically highmultiplicity fraction of the very young population However these are veryuncertain because the binary population is in a state of change (Fig 814below) The determination of an IMF relies on the assumption that all starsin a very young cluster formed together However trapping and focussing ofolder field or OB association stars by the forming cluster has been found tobe possible (Sect 811)

Thus be it at the low-mass end or the high-mass end the stellar massfunction seen in very young clusters cannot be the true IMF Statistical cor-rections for the above effects need to be applied and comprehensive N -bodymodelling is required

Old open clusters in which most stars are on or near the main sequenceare no better stellar samples They are dynamically highly evolved becausethey have left their previous concentrated and gas-rich state and so they con-tain only a small fraction of the stars originally born in the cluster (Kroupaamp Boily 2002 Weidner et al 2007 Baumgardt amp Kroupa 2007) The binaryfraction is typically high and comparable to the Galactic field but does de-pend on the initial density and the age of the cluster as does the mass-ratiodistribution of companions So simple corrections cannot be applied equallyfor all old clusters The massive stars have died and secular evolution begins

228 P Kroupa

to affect the remaining stellar population (after gas expulsion) through energyequipartition Baumgardt amp Makino (2003) have quantified the changes ofthe MF for clusters of various masses and on different Galactic orbits Nearthe half-mass radius the local MF resembles the global MF in the clusterbut the global MF is already significantly depleted of its lower-mass stars byabout 20 of the cluster disruption time

Given that we are never likely to learn the exact dynamical history ofa particular cluster it follows that we can never ascertain the IMF for anyindividual cluster This can be summarised concisely with the following con-jecture

Cluster IMF Conjecture The IMF cannot be extracted for any indi-vidual star cluster

Justification For clusters younger than about 05Myr star-formation hasnot ceased and the IMF is therefore not yet assembled and the clustercores consisting of massive stars have already dynamically ejected members(Pflamm-Altenburg amp Kroupa 2006) For clusters with an age between 05and a few Myr the expulsion of residual gas has lead to loss of stars (KroupaAarseth amp Hurley 2001) Older clusters are either still losing stars owing toresidual gas expulsion or are evolving secularly through evaporation driven byenergy equipartition (Baumgardt amp Makino 2003) Furthermore the birthsample is likely to be contaminated by captured stars (Fellhauer Kroupa ampEvans 2006 Pflamm-Altenburg amp Kroupa 2007) There exists no time whenall stars are assembled in an observationally accessible volume (ie a starcluster)

Note that the Cluster IMF Conjecture implies that individual clus-ters cannot be used to make deductions on the similarity or not of their IMFsunless a complete dynamical history of each cluster is available Notwith-standing this pessimistic conjecture it is nevertheless necessary to observeand study star clusters of any age Combined with thorough and realisticN -body modelling the data do lead to essential statistical constraints on theIMF universality hypothesis Such an approach is discussed in the nextsection

The Alpha Plot

Scalo (1998) conveniently summarised a large part of the available observa-tional constraints on the IMF of resolved stellar populations with the alphaplot as used by Kroupa (2001 2002b) for explicit tests of the IMF univer-

sality hypothesis given the cluster IMF conjecture One example ispresented in Fig 89 which demonstrates that the observed scatter in α(m)can be readily understood as being due to Poisson uncertainties (see alsoElmegreen 1997 1999) and dynamical effects as well as arising from biasesthrough unresolved multiple stars Furthermore there is no evident systematicchange of α at a given m with metallicity or density of the star-forming cloud

Fig 89 The alpha plot The power-law index α is measured over stellar mass-ranges and plotted at the mid-point of the respective mass range The power-lawindices are measured on the mass function of system masses where stars not inbinaries are counted individually Open circles are the observations from open clus-ters and associations of the Milky Way and the Large and Small Magellanic cloudscollated mostly by Scalo (1998) The open stars (crosses) are theoretical star clus-ters observed in the computer at an age of 3 (0) Myr and within a radius of 32 pcfrom the cluster centre The 5 clusters have 3000 stars in 1500 binaries initially andthe assumed IMF is the canonical one The theoretical data nicely show a similarspread to the observational ones note the binary-star-induced depression of α1 inthe mass range 01ndash05 M The IMF universality hypothesis can therefore notbe discarded given the observed data Models are from Kroupa (2001)

More exotic populations such as the Galactic bulge have also been found tohave a low-mass MF indistinguishable from the canonical form (eg Zoccaliet al 2000) Thus the IMF universality hypothesis cannot be falsifiedfor known resolved stellar populations

Very Ancient andor Metal-Poor Resolved Populations

Witnesses of the early formation phase of the Milky Way are its globular clus-ters Such 104ndash106 M clusters formed with individual star-formation ratesof 01ndash1M yrminus1 and densities of about 5 times 103ndash105 M pcminus3 These are rel-atively high values when compared with the current star-formation activityin the Milky Way disc For example a 5 times 103 M Galactic cluster formingin 1Myr corresponds to a star-formation rate of 0005M yrminus1 The alphaplot however does not support any significant systematic difference betweenthe IMF of stars formed in globular clusters and present-day low-mass star-formation For massive stars it can be argued that the mass in stars moremassive than 8M cannot have been larger than about half the cluster massbecause otherwise the globular clusters would not be as compact as theyare today This constrains the IMF to have been close to the canonical IMF(Kroupa 2001)

230 P Kroupa

A particularly exotic star-formation mode is thought to have occurred indwarf-spheroidal (dSph) satellite galaxies The Milky Way has about 19 suchsatellites at distances from 50 to 250 kpc (Metz amp Kroupa 2007) These objectshave stellar masses and ages comparable to those of globular clusters butare 10ndash100 times larger and are thought to have 10ndash1000 times more mass indark matter than in stars They also show evidence for complex star-formationactivity and metal-enrichment histories and must therefore have formed underrather exotic conditions Nevertheless the MFs in two of these satellites arefound to be indistinguishable from those of globular clusters in the mass range05ndash09M So again there is consistency with the canonical IMF (Grillmairet al 1998 Feltzing Gilmore amp Wyse 1999)

The work of Yasui et al (2006) and Yasui et al (2008) have been pushingstudies of the IMF in young star clusters to the outer metal-poor regionsof the Galactic disc They find the IMF to be indistinguishable within theuncertainties from the canonical IMF

The Galactic Bulge and Centre

For low-mass stars the Galactic bulge has been shown to have a MF indistin-guishable from the canonical form (Zoccali et al 2000) However abundancepatterns of bulge stars suggest the IMF was top-heavy (Ballero Kroupa ampMatteucci 2007) This may be a result of extreme star-burst conditions pre-vailing in the formation of the bulge (Zoccali et al 2006)

Even closer to the Galactic centre models of the HertzsprungndashRusselldiagram of the stellar population within 1 pc of Sgr Alowast suggest the IMF wasalways top-heavy there (Maness et al 2007) Perhaps this is the long-soughtafter evidence for a variation of the IMF under very extreme conditions in thiscase a strong tidal field and higher temperatures (but note Fig 810 below)

Extreme Star Bursts

As noted on p 199 objects with a mass M ge 106 M have an increased MLratio If such objects form in 1 Myr their star-formation rates SFRge 1Myrand they probably contain more than 104 O stars packed within a regionspanning at most a few parsecs given their observed present-day massndashradiusrelation Such a star-formation environment is presently outside the reachof theoretical investigation However it is conceivable that the higher MLratios of such objects may be due to a non-canonical IMF One possibilityis that the IMF is bottom-heavy as a result of intense photo-destruction ofaccretion envelopes of intermediate to low-mass stars (Mieske amp Kroupa 2008)Another possibility is that the IMF becomes top-heavy leaving many stellarremnants that inflate the ML ratio (Dabringhausen amp Kroupa 2008) Workis in progress to achieve observational constraints on these two possibilities

Fig 810 The observed mass function of the Arches cluster near the Galacticcentre by Kim et al (2006) shown as the thin histogram is confronted with the the-oretical MF for this object calculated with the SPH technique by Klessen Spaansamp Jappsen (2007) marked as the hatched histogram The latter has a down-turn(bold steps near 1007) incompatible with the observations This rules out a the-oretical understanding of the stellar mass spectrum because one counter-examplesuffices to bring-down a theory One possible reason for the theoretical failure maybe the assumed turbulence driving For details of the figure see Kim et al (2006)

Population III The Primordial IMF

Most theoretical workers agree that the primordial IMF ought to be top-heavy because the ambient temperatures were much higher and the lack ofmetals did not allow gas clouds to cool and to fragment into sufficiently smallcores (Larson 1998) The existence of extremely metal-poor low-mass starswith chemical peculiarities is interpreted to mean that low-mass stars couldform under extremely metal-poor conditions but that their formation wassuppressed in comparison to later star-formation (Tumlinson 2007) Modelsof the formation of stellar populations during cosmological structure formationsuggest that low-mass population-III stars should be found within the Galactichalo if they formed Their absence to-date would imply a primordial IMFdepleted in low-mass stars (Brook et al 2007)

Thus the last three sub-sections hint at physical environments in whichthe IMF universality hypothesis may be violated

232 P Kroupa

833 Very Low-Mass Stars (VLMSs) and Brown Dwarfs (BDs)

The origin of BDs and some VLMSs is being debated fiercely One campbelieves these objects to form as stars because the star-formation processdoes not know where the hydrogen burning mass limit is (eg Eisloffel ampSteinacker 2008) The other camp believes that BDs cannot form exactly likestars through continued accretion because the conditions required for thisto occur in molecular clouds are far too rare (eg Reipurth amp Clarke 2001Goodwin amp Whitworth 2007)

If BDs and VLMSs form like stars they should follow the same pairingrules In particular BDs and G dwarfs would pair in the same manner ieaccording to the same mathematical rules as M dwarfs and G dwarfs Kroupaet al (2003) tested this hypothesis by constructing N -body models of Taurus-Auriga-like groups and Orion-Nebula-like clusters finding that it leads tofar too many starndashBD and BDndashBD binaries with the wrong semi-major axisdistribution Instead starndashBD binaries are very rare (Grether amp Lineweaver2006) while BDndashBD binaries are rarer than stellar binaries (BDs have a 15binary fraction as opposed to 50 for stars) and BDs have a semi-majoraxis distribution significantly narrower than that of starndashstar binaries Thehypothesis of a star-like origin of BDs must therefore be discarded BDs andsome VLMSs form a separate population which is however linked to that ofthe stars

Thies amp Kroupa (2007) re-addressed this problem with a detailed analysisof the underlying MF of stars and BDs given observed MFs of four popu-lations Taurus Trapezium IC348 and the Pleiades By correcting for unre-solved binaries in all four populations and taking into account the differentpairing rules of stellar and VLMS and BD binaries they discovered a signifi-cant discontinuity of the MF BDs and VLMSs therefore form a truly separatepopulation from that of the stars It can be described by a single power-lawMF (8129) which implies that about one BD forms per five stars in all fourpopulations

This strong correlation between the number of stars and BDs and thesimilarity of the BD MF in the four populations implies that the formationof BDs is closely related to the formation of stars Indeed the truncation ofthe binary binding energy distribution of BDs at a high energy suggests thatenergetic processes must be operating in the production of BDs as discussedby Thies amp Kroupa (2007) Two such possible mechanisms are embryo ejection(Reipurth amp Clarke 2001) and disc fragmentation (Goodwin amp Whitworth2007)

834 Composite Populations The IGIMF

The vast majority of all stars form in embedded clusters and so the correct wayto proceed to calculate a galaxy-wide stellar IMF is to add up all the IMFs ofall star clusters born in one star-formation epoch Such epochs may be iden-tified with the Zoccali et al (2006) star-burst events that create the Galactic

bulge In disc galaxies they may be related to the time-scale of transformingthe interstellar matter to star clusters along spiral arms Addition of the clus-ters born in one epoch gives the integrated galactic initial mass function theIGIMF (Kroupa amp Weidner 2003)

IGIMF definition The IGIMF is the IMF of a composite populationwhich is the integral over a complete ensemble of simple stellar populations

Note that a simple population has a mono-metallicity and a mono-age distri-bution and is therefore a star cluster Age and metallicity distributions emergefor massive populations with Mcl ge 106 M (eg ω Cen) This indicates thatsuch objects which also have relaxation times comparable to or longer thana Hubble time are not simple (Sect 814) A complete ensemble is a statis-tically complete representation of the initial cluster mass function (ICMF) inthe sense that the actual mass function of Ncl clusters lies within the expectedstatistical variation of the ICMF

IGIMF conjecture The IGIMF is steeper than the canonical IMF if theIMF universality hypothesis holds

Justification Weidner amp Kroupa (2006) calculate that the IGIMF issteeper than the canonical IMF for m ge 1M if the IMF universality

hypothesis holds The steepening becomes negligible if the power-law massfunction of embedded star clusters

ξecl(Mecl) prop Mminusβecl (8127)

is flatter than β = 18It may be argued that IGIMF = IMF (eg Elmegreen 2006) because

when a star cluster is born its stars are randomly sampled from the IMF upto the most massive star possible On the other hand the physically motivatedansatz of Weidner amp Kroupa (2005 2006) to take the mass of a cluster as theconstraint and to include the observed correlation between the maximal starmass and the cluster mass (Fig 81) yields an IGIMF which is equal to thecanonical IMF for m le 15M but which is systematically steeper above thismass By incorporating the observed maximal-cluster-mass vs star-formationrate of galaxies Meclmax = Meclmax(SFR) for the youngest clusters (Wei-dner Kroupa amp Larsen 2004) it follows for m ge 15M that low-surface-brightness (LSB) galaxies ought to have very steep IGIMFs while normal orLlowast galaxies have Scalo-type IGIMFs ie αIGIMF = αMWdisc gt α2 (Sect 831)follows naturally This systematic shift of αIGIMF (m ge 15M) with galaxytype implies that less massive galaxies have a significantly suppressed super-nova II rate per low-mass star They also show a slower chemical enrichmentso that the observed metallicityndashgalaxy-mass relation can be nicely accounted

234 P Kroupa

for (Koeppen Weidner amp Kroupa 2007) Another very important implica-tion is that the SFRndashHα-luminosity relation for galaxies flattens so that theSFR becomes greater by up to three orders of magnitude for dwarf galax-ies than the value calculated from the standard (linear) Kennicutt relation(Pflamm-Altenburg Weidner amp Kroupa 2007)

Strikingly the IGIMF variation has now been directly measured byHoversten amp Glazebrook (2008) using galaxies in the Sloan Digital Sky Sur-vey Lee et al (2004) have indeed found LSBs to have bottom-heavy IMFswhile Portinari Sommer-Larsen amp Tantalo (2004) and Romano et al (2005)find the Milky Way disc to have a an IMF steeper than Salpeterrsquos for massivestars which is in comparison with Lee et al (2004) much flatter than theIMF of LSBs as required by the IGIMF conjecture

835 Origin of the IMF Theory vs Observations

General physical concepts such as coalescence of protostellar cores mass-dependent focussing of gas accretion on to protostars stellar feedback andfragmentation of molecular clouds lead to predictions of systematic varia-tions of the IMF with changes of the physical conditions of star-formation(Murray amp Lin 1996 Elmegreen 2004) (But see Casuso amp Beckman 2007 fora simple cloud coagulationdispersal model that leads to an invariant massdistribution) Thus the thermal Jeans mass of a molecular cloud decreaseswith temperature and increasing density This implies that for higher metallic-ity (stronger cooling) and density the IMF should shift on average to smallerstellar masses (eg Larson 1998 Bonnell et al 2007) The entirely differentnotion that stars regulate their own masses through a balance between feed-back and accretion also implies smaller stellar masses for higher metallicitydue in part to more dust and thus more efficient radiation pressure on thegas through the dust grains Also a higher metallicity allows more efficientcooling and thus a lower gas temperature a lower sound speed and thereforea lower accretion rate (Adams amp Fatuzzo 1996 Adams amp Laughlin 1996)As discussed above a systematic IMF variation with physical conditions hasnot been detected Thus theoretical reasoning even at its most elementarylevel fails to account for the observations

A dramatic case in point has emerged recently Klessen Spaans amp Jappsen(2007) report state-of-the art calculations of star-formation under physicalconditions as found in molecular clouds near the Sun and they are able toreproduce the canonical IMF Applying the same computational technologyto the conditions near the Galactic centre they obtain a theoretical IMF inagreement with the previously reported apparent decline of the stellar MF inthe Arches cluster below about 6M Kim et al (2006) published their obser-vations of the Arches cluster on the astrophysics preprint archive shortly afterKlessen Spaans amp Jappsen (2007) and performed N -body calculations of thedynamical evolution of this young cluster revising our knowledge significantlyIn contradiction to the theoretical prediction they find that the MF continues

to increase down to their 50 completeness limit (13M) with a power-lawexponent only slightly shallower than the canonical MasseySalpeter valueonce mass-segregation has been corrected for This situation is demonstratedin Fig 810 It therefore emerges that there does not seem to exist any solidtheoretical understanding of the IMF

Observations of cloud cores appear to suggest that the canonical IMF isalready frozen in at the pre-stellar cloud-core level (Motte Andre amp Neri 1998Motte et al 2001) Nutter amp Ward-Thompson (2007) and Alves Lombardiamp Lada (2007) find however the pre-stellar cloud cores are distributed ac-cording to the same shape as the canonical IMF but shifted to larger massesby a factor of about three or more This is taken to perhaps mean a star-formation efficiency per star of 30 or less independently of stellar mass Theinterpretation of such observations in view of multiple star-formation in eachcloud-core is being studied by Goodwin et al (2008) while Krumholz (2008)outlines current theoretical understanding of how massive stars form out ofmassive pre-stellar cores

836 Conclusions IMF

The IMF universality hypothesis the cluster IMF conjecture andthe IGIMF conjecture have been stated In addition we may make thefollowing assertions

1 The stellar luminosity function has a pronounced maximum at MV asymp 12MI asymp 9 which is universal and well understood as a result of stellarphysics Thus by counting stars in the sky we can look into their interiors

2 Unresolved multiple systems must be accounted for when the MFs ofdifferent stellar populations are compared

3 BDs and some VLMSs form a separate population that correlates withthe stellar content There is a discontinuity in the MF near the starBDmass transition

4 The canonical IMF (8124) fits the star counts in the solar neighbourhoodand all resolved stellar populations available to-date Recent data at theGalactic centre suggest a top-heavy IMF perhaps hinting at a possiblevariation with conditions (tidal shear temperature)

5 Simple stellar populations are found in individual star clusters with Mcl

le 106 M These have the canonical IMF6 Composite populations describe entire galaxies They are a result of many

epochs of star-cluster formation and are described by the IGIMF Con-

jecture7 The IGIMF above about 1M is steep for LSB galaxies and flattens to the

Scalo slope (αIGIMF asymp 27) for Llowast disc galaxies This is nicely consistentwith the IMF universality hypothesis in the context of the IGIMF

conjecture

236 P Kroupa

8 Therefore the IMF universality hypothesis cannot be excluded de-spite the cluster IMF conjecture for conditions ρ le 105 stars pcminus3Z ge 0002 and non-extreme tidal fields

9 Modern star-formation computations and elementary theory give wrongresults concerning the variation and shape of the stellar IMF as well asthe stellar multiplicity (Goodwin amp Kroupa 2005)

10 The stellar IMF appears to be frozen-in at the pre-stellar cloud-core stageSo it is probably a result of the processes that lead to the formation ofself-gravitating molecular clouds

837 Discretisation

As discussed above a theoretically motivated form of the IMF that passesobservational tests does not exist Star-formation theory gets the rough shapeof the IMF right There are fewer massive stars than low-mass stars How-ever other than this it fails to make any reliable predictions whatsoever asto how the IMF should look in detail under different physical conditions Inparticular the overall change of the IMF with metallicity density or temper-ature predicted by theory is not evident An empirical multi-power-law formdescription of the IMF is therefore perfectly adequate and has important ad-vantages over other formulations A general formulation of the stellar IMF interms of multiple power-law segments is

ξ(m) = k

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(m

mH

)minusα0

mlow le m le mH(

mmH

)minusα1

mH le m le m0(

m0mH

)minusα1(

mm0

)minusα2

m0 le m le m1(

m0mH

)minusα1(

m1m0

)minusα2(

mm1

)minusα3

m1 le m le mmax

(8128)

where mmax le mmaxlowast asymp 150M depends on the stellar mass of the embeddedcluster (Fig 81) The empirically determined stellar IMF is a two-part-form(8124) with a third power-law for BDs whereby BDs and VLMSs form aseparate population from that of the stars (p 232)

ξBD prop mminusα0 α0 asymp 03 (8129)

(Martın et al 2000 Chabrier 2003 Moraux Bouvier amp Clarke 2004) and

ξBD(0075M) asymp 025 plusmn 005 ξ(0075M)

(Thies amp Kroupa 2007) where ξ is the canonical stellar IMF (8124) Thisimplies that about one BD forms per five stars

One advantage of the power-law formulation is that analytical generat-ing functions and other quantities can be readily derived Another importantadvantage is that with a multi-power-law form different parts of the IMF

can be varied in numerical experiments without affecting the other parts Apractical numerical formulation of the IMF is prescribed in Pflamm-Altenburgamp Kroupa (2006) Thus for example the canonical two-part power-law IMFcan be changed by adding a third power-law above 1M and making the IMFtop-heavy (αmgt1 M lt α2) without affecting the shape of the late-type stel-lar luminosity function as evident in Fig 88 The KTG93 (Kroupa Tout ampGilmore 1993) IMF is such a three-part power-law form relevant to the overallyoung population in the Milky Way disc This is top-light (αmgt1 M gt α2Kroupa amp Weidner 2003)

A log-normal formulation does not offer these advantages and requirespower-law tails above about 1M and for brown dwarfs for consistency withthe observations discussed above However while not as mathematically con-venient the popular Chabrier log-normal plus power-law IMF (Table 1 ofChabrier 2003) formulation leads to an indistinguishable stellar mass distri-bution to the two-part power-law IMF (Fig 811) Various analytical formsfor the IMF are compiled in Table 3 of Kroupa (2007a)

A generating function for the two-part power-law form of the canonicalIMF (8124) can be written down by following the steps taken in Sect 823The corresponding probability density is

p1 = kp1 mminusα1 008 le m le 05M (8130)p2 = kp2 mminusα2 05 lt m le mmax

where kpi are normalisation constants ensuring continuity at 05M andint 05

008

p1 dm+int mmax

05

p2 dm = 1 (8131)

N

M

Fig 811 Comparison between the popular Chabrier IMF (log-normal plus power-law extension above 1 M dashed curve Table 1 in Chabrier 2003) with the canon-ical two-part power-law IMF (solid line (8124)) The figure is from DabringhausenHilker amp Kroupa (2008)

238 P Kroupa

whereby mmax follows from Fig 81 Defining

X prime1 =

int 05

008

p1(m) dm (8132)

it follows that

X1(m) =int m

008

p1(m) dm if m le 05M (8133)

orX2(m) = X prime

1 +int m

05

p2(m) dm if m gt 05M (8134)

The generating function for stellar masses follows from inversion of the abovetwo equations Xi(m) The procedure is then to choose a random variate X isin[0 1] and to select the generating function m(X1 = X) if 0 le X le X1 orm(X2 = X) if X1 lt X le 1

This algorithm is readily generalised to any number of power-law segments(8128) such as including a third segment for brown dwarfs and allowing theIMF to be discontinuous near 008M (Thies amp Kroupa 2007) Such a formhas been incorporated into the Nbody467 programmes but hitherto with-out the discontinuity However Jan Pflamm-Altenburg has developed a morepowerful and general method of generating stellar masses (or any other quan-tities) given an arbitrary distribution function (Pflamm-Altenburg amp Kroupa2006)11

84 The Initial Binary Population

It has already been demonstrated that corrections for unresolved multiplestars are of much importance to derive correctly the shape of the stellar MFgiven an observed LF (Fig 88) Binary stars are also of significant importancefor the dynamics of star clusters because a binary has intrinsic dynamicaldegrees of freedom that a single star does not possess A binary can thereforeexchange energy and angular momentum with the cluster Indeed binariesare very significant energy sources as for example a binary composed of two1M main-sequence stars and with a semi-major axis of 01AU has a bindingenergy comparable to that of a 1000M cluster of size 1 pc Such a binarycan interact with cluster-field star accelerating them to higher velocities andthereby heating the cluster

The dynamical properties describing a multiple system are

bull the period P (in days throughout this text) or semi-major axis a (in AU)bull the system mass msys = m1 +m2

11The C-language software package libimf can be downloaded from the websitehttpwwwastrouni-bonnde~webaiubenglishdownloadsphp

bull the mass ratio q equiv m2m1

le 1 where m1m2 are respectively the primaryand secondary-star masses and

bull the eccentricity e = (rapo minus rperi)(rapo + rperi) where rapo rperi are re-spectively the apocentric and pericentric distances

Given a snapshot of a binary the above quantities can be computed fromthe relative position rrel and velocity vrel vectors and the masses of the twocompanion stars by first calculating the binding energy

Eb =12μ v2

rel minusGm1 m2

rrel= minusGm1 m2

2 arArr a (8135)

where μ = m1 m2 (m1 + m2) is the reduced mass From Keplerrsquos third lawwe have

msys =a3

AU

P 2yr

rArr P = Pyr times 36525 days (8136)

where Pyr is the period in years and aAU is in AU Finally the instantaneouseccentricity can be calculated using

e =

[(1 minus rrel

a

)2

+(rrel middot vrel)

2

aGmsys

] 12

(8137)

which can be derived from the orbital angular momentum too

L = μvrel times rrel (8138)

with

L =[

G

msysa (1 minus e2)

] 12

m1 m2 (8139)

The relative equation of motion is

d2rrel

dt2= minusGmsys

r3relrrel + apert(t) (8140)

where apert(t) is the time-dependent perturbation from other cluster membersIt follows that the orbital elements of a binary in a cluster are functions oftime P = P (t) and e = e(t) Also q = q(t) during strong encounters whenpartners are exchanged Because most stars form in embedded clusters thebinary-star properties of a given population cannot be taken to represent theinitial or primordial values

The following conjecture can be proposed

Dynamical population synthesis conjecture if initial binary popu-lations are invariant a dynamical birth configuration of a stellar populationcan be inferred from its observed binary population This birth configura-tion is not unique however but defines a class of dynamically equivalentsolutions

240 P Kroupa

The proof is simple Set up initially identical binary populations in clusterswith different radii and masses and calculate the dynamical evolution with anN -body programme For a given snapshot of a population there is a scalablestarting configuration in terms of size and mass (Kroupa 1995cd)

Binaries can absorb energy and thus cool a cluster They can also heata cluster There are two extreme regimes that can be understood with aGedanken experiment Define

Ebin equiv minusEb gt 0(8141)Ek equiv (12)mσ2 asymp (1N) times kinetic energy of cluster

Soft binaries have Ebin Ek while hard binaries have Ebin Ek A usefulequation in this context is the relation between the orbital period and circularvelocity of the reduced particle

log10 P [days] = 6986 + log10 msys[M] minus 3 log10 vorb[km sminus1] (8142)

Consider now the case of a soft binary a reduced-mass particle withvorb σ By the principle of energy equipartition vorb rarr σ (85) as timeprogresses This implies a uarr P uarr A hard binary has vorb σ Invoking en-ergy equipartition we see that vorb darr a darr P darr Furthermore the amount ofenergy needed to ionise a soft binary is negligible compared to the amountof energy required to ionise a hard binary And the cross section for sufferingan encounter scales with the semi-major axis This implies that a soft binarybecomes ever more likely to suffer an additional encounter as its semi-majoraxis increases Therefore it is much more probable for soft binaries to be dis-rupted rapidly than for hard binaries to do so Thus follows (Heggie 1975Hills 1975) a law

HeggiendashHills law soft binaries soften and cool a cluster while hard bi-naries harden and heat a cluster

Numerical scattering experiments by Hills (1975) have shown that harden-ing of binaries often involves partner exchanges Heggie (1975) derived theabove law analytically Binaries in the energy range 10minus2 Ek le Ebin le 102 Ek33minus1 σ le vorb le 33σ cannot be treated analytically owing to the complexresonances that are created between the binary and the incoming star or bi-nary It is these binaries that may be important for the early cluster evolutiondepending on its velocity dispersion σ = σ(Mecl) Cooling of a cluster is en-ergetically not significant but has been seen for the first time by Kroupa Petramp McCaughrean (1999)

Figure 812 shows the broad evolution of the initial period distributionin a star cluster At any time binaries near the hardsoft boundary withenergies Ebin asymp Ek and periods P asymp Pth (vorb = σ) (85) the thermal periodare most active in the energy exchange between the cluster field and thebinary population The cluster expands as a result of binary heating and

Fig 812 Illustration of the evolution of the distribution of binary star periods ina cluster (lP = log10 P ) A binary has orbital period Pth when σ3D (= σ) equals itscircular orbital velocity (8142) The initial or birth distribution (8164) evolves tothe form seen at time t gt tt

mass segregation and the hardsoft boundary Pth shifts to longer periodsMeanwhile binaries with P gt Pth continue to be disrupted while Pth keepsshifting to longer periods This process ends when

Pth ge Pcut (8143)

which is the cutoff or maximum period in the surviving period distributionAt this critical time tt further cluster expansion is slowed because the popu-lation of heating sources the binaries with P asymp Pth is significantly reducedThe details strongly depend on the initial value of Pth which determinesthe amount of binding energy in soft binaries which can cool the cluster ifsignificant enough

After the critical time tt the expanded cluster reaches a temporary stateof thermal equilibrium with the remaining binary population Further evolu-tion of the binary population occurs with a significantly reduced rate deter-mined by the velocity dispersion in the cluster the cross section given by thesemi-major axis of the binaries and their number density and that of singlestars in the cluster The evolution of the binary star population during thisslow phase usually involves partner exchanges and unstable but also long-lived hierarchical systems The IMF is critically important for this stage asthe initial number of massive stars determines the cluster density at t ge 5Myrowing to mass loss from evolving stars Further binary depletion occurs oncethe cluster goes into core-collapse and the kinetic energy in the core rises

242 P Kroupa

841 Frequency of Binaries and Higher-Order Multiples

The emphasis here is on late-type binary stars because higher-order multiplesare rare as observed The information on the multiplicity of massive stars isvery limited (Goodwin et al 2007) We define respectively the number ofsingle stars binaries triples quadruples etc by the numbers

(Nsing Nbin Ntrip Nquad ) = (S B T Q ) (8144)

and the multiplicity fraction by

fmult =Nmult

Nsys=

B + T + Q +

S + B + T + Q + (8145)

and the binary fraction is

fbin =B

Nsys (8146)

In the Galactic field Duquennoy amp Mayor (1991) derive from a decade-long survey for G-dwarf primary stars GNmult = (573841) and for M-dwarfsFischer amp Marcy (1992) find MNmult = (583371) Thus

Gfmult = 043 Gfbin = 038 (8147)Mfmult = 041 Mfbin = 033 (8148)

It follows that most stars are indeed binaryAfter correcting for incompleteness

Gfbin = 053 plusmn 008 (8149)

Kfbin = 045 plusmn 007 (8150)Mfbin = 042 plusmn 009 (8151)

where the K-dwarf data have been published by Mayor et al (1992) It followsthat

Gfbin asympK fbin asympM fbin asymp 05 asymp ftot (8152)

in the Galactic field perhaps with a slight decrease towards lower masses Incontrast for brown dwarfs BDfbin asymp 015 starsfbin (Thies amp Kroupa 2007and references therein)

An interesting problem arises because 1Myr old stars have fTTauri asymp 1(eg Duchene 1999) Given the above information the following conjecturecan be stated

Binary-star conjecture nearly all stars form in binary systems

Justification if a substantial fraction of stars were to form in higher-ordermultiple systems or as small-N systems the typical properties of these at

birth imply their decay within typically 104 to 105 yr leaving a predomi-nantly single-stellar population However the majority of 106 yr old stars areobserved to be in binary systems (Goodwin amp Kroupa 2005)

Higher-order multiple systems do exist and can only be hierarchical toguarantee stability Such systems are multiple stars which are stable overmany orbital times and are usually tight binaries orbited by an outer tertiarycompanion or two tight binaries in orbit about each other Stability issues arediscussed in detail in Chap 3 based on a theoretical development from firstprinciples In particular a new stability criterion for the general three-bodyproblem is derived in terms of all the orbital parameters For comparablemasses long-term stability is typically ensured for a ratio of the outer peri-centre to the inner semi-major axis of about 4 If the stability condition is notfulfilled higher-order multiple systems usually decay on a time-scale relatingto the orbital parameters Star cluster remnants (or dead star clusters) maybe the origin of most hierarchical higher-order multiple stellar systems in thefield (p 199)

842 The Initial Binary Population ndash Late-Type Stars

The initial binary population is described by distribution functions that are asfundamental for a stellar population as the IMF There are four distributionfunctions that define the initial dynamical state of a population

1 the IMF ξ(m)2 the distribution of periods (or semi-major axis) df = fP (logP ) d logP 3 the distribution of mass-ratios df = fq(q) dq and4 the distribution of eccentricities df = fe(e) de

where df is the fraction of systems between f and f +df Thus for exampleGflog P (log10 P = 45) = 011 ie of all G-dwarfs in the sky 11 have acompanion with a period in the range 4ndash5 d (Fig 816)

These distribution functions have been measured for late-type stars in theGalactic field and in star-forming regions (Fig 813) According to Duquennoyamp Mayor (1991) and Fischer amp Marcy (1992) both G-dwarf and M-dwarfbinary systems in the Galactic field have period distribution functions thatare well described by log-normal functions

fP (log10 P ) = ftot

(1

σlog10 P

radic2π

)

e

[

minus 12

(log10 Pminuslog10 P )2

σ2log10 P

]

(8153)

with log10 P asymp 48 and σlog10 P asymp 23 andintall P

flog10 P (log10 P ) d log10 P =ftot asymp 05 K-dwarfs appear to have an indistinguishable period distribution

From Fig 813 it follows that the pre-mainsequence binary fraction islarger than that of main-sequence stars (see also Duchene 1999) Is this anevolutionary effect

244 P Kroupa

fP

P

Fig 813 Measured period-distribution functions for G-dwarfs in the Galacticfield (histogram Duquennoy amp Mayor 1991) K-dwarfs (open circles Mayor et al1992) and M-dwarfs (asterisks Fischer amp Marcy 1992) About 1-Myr-old T Tauribinary data (open squares partially from the TaurusndashAuriga stellar groups) are acompilation from various sources (see Fig 10 in Kroupa Aarseth amp Hurley 2001)In all cases the area under the distribution is ftot

Further Duquennoy amp Mayor (1991) derived the mass-ratio and eccen-tricity distributions for G-dwarfs in the Galactic field The mass-ratio dis-tribution of G-dwarf primaries is not consistent with random sampling fromthe canonical IMF (8124) as the number of observed low-mass companionsis underrepresented (Kroupa 1995c) In contrast the pre-mainsequence mass-ratio distribution is consistent within the uncertainties with random sam-pling from the canonical IMF for q ge 02 (Woitas Leinert amp Koehler 2001)The eccentricity distribution of Galactic-field G-dwarfs is found to be ther-mal for log10 P ge 3 while it is bell shaped with a maximum near e = 025for log10 P le 3 Not much is known about the eccentricity distribution ofpre-mainsequence binaries but numerical experiments show that fe does notevolve much in dense clusters ie the thermal distribution must be initial(Kroupa 1995d)

The thermal eccentricity distribution

fe(e) = 2 e (8154)

follows from a uniform binding-energy distribution (all energies are equallypopulated) as follows The orbital angular momentum of a binary is

L2 =G

msys

Gm1 m2

2Ebin

(1 minus e2

)(m1 m2)

2 (8155)

from which follows

e =(

1 minus 2Ebin L2 msys

G2 (m1 m2)2

) 12

(8156)

Differentiation leads to

dedEbin

=[

minusL2 msys

G2 (m1 m2)2

]

eminus1 prop eminus1 (8157)

The number of binaries with eccentricities in the range e e + de is the samenumber of binaries with binding energy in the range Ebin Ebin + dEbin (thesame sample of binaries)

f(e) de = f(Ebin) dEbin prop f(Ebin) ede (8158)

But int 1

0

f(e) de = 1 (8159)

That is

f(Ebin)int 1

0

ede prop f(Ebin)12e2|10 = const (8160)

Sof(Ebin) = const rArr f(e) de = 2 ede (8161)

Thus f(e) = 2 e is a thermalized distribution All energies are equally oc-cupied so f(Ebin) = const N -body experiments have demonstrated that theperiod distribution function must span the observed range of periods at birthbecause dynamical encounters in dense clusters cannot widen an initially nar-row distribution (Kroupa amp Burkert 2001) There are thus three discrepanciesbetween main-sequence and pre-mainsequence late-type stellar binaries

1 the binary fraction is higher for the latter2 the period distribution function is different and3 the mass-ratio distribution is consistent with random paring for the latter

while it is deficient in low-mass companions in the former for G-dwarfprimaries

Can these be unified That is are there unique initial flog P fq and fe con-sistent with the pre-mainsequence data that can be evolved to the observedmain-sequence distributions

This question can be solved by framing the following ansatz Assume theorbital-parameter distribution function for binaries with primaries of mass m1

can be separatedD(logP e q m1) = flog P fe fq (8162)

The stellar-dynamical operator ΩNrh can now be introduced so that theinitial distribution function is transformed to the final (Galactic-field) one

Dfin(logP e q m1) = ΩNrh [Din(logP e q m1)] (8163)

246 P Kroupa

This operator provides a dynamical environment equivalent to that of a starcluster with N stars and a half-mass radius rh (see also the Dynamical Pop-ulation Synthesis Conjecture p 239) Kroupa (1995c) and Kroupa (1995d)indeed show this to be the case for a cluster with N = 200 binaries andrh = 077 pc and derive the initial distribution function Din for late-typebinary systems that fulfils the above requirement and also has a simple gener-ating function (see below) It is noteworthy that such a cluster is very similarto the typical cluster from which most field stars probably originate The fullsolution for Ω so that the Galactic field is reproduced from forming and dis-solving star clusters requires full-scale inverse dynamical population synthesisfor the Galactic field

Thus by the dynamical population synthesis conjecture (p 239)the above ansatz with ΩNrh leads to one solution of the inverse dynamicalpopulation synthesis problem (the 200 binary rh = 08 pc cluster Fig 814 iemost stars in the Galactic field stem from clusters dynamically similar to thisone) provided the birth (or primordial) distribution functions for logP e qare

flog Pbirth = ηlogP minus logPmin

δ + (logP minus logPmin)2 (8164)

This distribution function has a generating function (Sect 823)

logP (X) =[δ(e

2 Xη minus 1

)] 12

+ logPmin (8165)

The solution obtained by Kroupa (1995d) has

η = 25 δ = 45 logPmin = 1 (8166)

so that logPmax = 843 sinceint log Pmax

log Pminflog P d logP = ftot = 1 is a require-

ment for stars at birth Intriguingly similar distributions can be arrived atsemi-empirically if we assume isolated formation of binary stars in a turbulentmolecular cloud (Fisher 2004)

The birth-eccentricity distribution is thermal (8154) while the birth mass-ratio distribution is generated from random pairing from the canonical IMFHowever in order to reproduce (1) the observed data in the eccentricityndashperiod diagram (2) the observed eccentricity distribution and (3) the observedmass-ratio distribution for short-period (logP le 3) systems a correlation ofthe parameters needs to be introduced through eigenevolution Eigenevolu-tion is the sum of all dissipative physical processes that transfer mass energyand angular momentum between the companions when they are still veryyoung and accreting

A formulation that is quite successful in reproducing the overall observedcorrelations between logP e q for short-period systems has been derived fromtidal circularisation theory (Kroupa 1995d) The most effective orbital dissi-pation occurs when the binary is at periastron

Fig 814 Evolution of ftot the total binary fraction for stellar mass 01 lemiM le 11 i = 1 2 with time for the four star-cluster models initially withN = 200 binaries computed by Kroupa (1995c) in the search for the existence of anΩrhN The initial half-mass radius of the clusters is denoted in this text as rh Notethat the rh = 08 pc cluster yields the correct ftot asymp 05 for the Galactic field Theperiod-distribution function and the mass-ratio distribution function that emergefrom this cluster also fit the observed Galactic-field distribution Some binary starsform by three-body encounters in clusters that initially consist only of single starsand the proportion of such binaries is shown for the single-star clusters (with ini-tially N = 400 stars) Such dynamically formed binaries are very rare and so ftot

remains negligible

rperi = (1 minus e)P23yr (m1 +m2)

13 (8167)

where Pyr = P36525 is the period in years Let the binary be born witheccentricity ebirth then the system evolves approximately according to (Gold-man amp Mazeh 1994) as

1e

dedt

= minusρprime rArr log10ein = minusρ+ log10ebirth (8168)

where 1ρprime is the tidal circularisation time-scale ein is the initial eccentricityand

ρ =int Δt

0

ρprime dt =(λRrperi

)χ

(8169)

where R is the Solar radius in AU λ χ are tidal circularisation parame-ters and rperi (in AU) is assumed to be constant because the dissipational

248 P Kroupa

force only acts tangentially at periastron Note that a large λ implies thattidal dissipation is effective for large separations of the companions (eg theyare puffed-up pre-mainsequence structures) and a small χ implies the dissi-pation is soft ie weakly varying with the separation of the companions Inthis integral Δt le 105 yr is the time-scale within which pre-mainsequenceeigenevolution completes The initial period becomes from (8167)

Pin = Pbirth

(mtotbirth

mtotin

) 12(

1 minus ebirth

1 minus ein

) 32

(8170)

Kroupa (1995d) assumed the companions merge if ain le 10R in which casem1 +m2 rarr m

In order to reproduce the observed mass-ratio distribution given randompairing at birth and to also reproduce the fact that short-period binaries tendto have similar-mass companions Kroupa (1995d) implemented a feeding al-gorithm according to which the secondary star accretes high angular momen-tum gas from the circumbinary accretion disc or material so that its massincreases while the primary mass remains constant Thus after generating thetwo birth masses randomly from the canonical IMF the initial mass-ratio is

qin = qbirth + (1 minus qbirth) ρlowast (8171)

where

ρlowast =ρ ρ le 11 ρ gt 1 (8172)

The above is a very simple algorithm which nevertheless reproduces theessence of orbital dissipation so that the correlations between the orbital pa-rameters for short-period systems are well accounted for The best parametersfor the evolution

birth rarr initial λ = 28 χ = 075 (8173)

Figure 815 shows an example of the overall model in terms of theeccentricityndashperiod diagram Figures 816 and 817 demonstrate that it nicelyaccounts for the period and mass-ratio distribution data respectively

Note that initial distributions are derived from birth distributions Thisis to be understood in terms of these initial distributions being the initiali-sation of N -body experiments while the birth distributions are more relatedto the theoretical distribution of orbital parameters before dissipational andaccretion processes have had a major effect on them The birth distributionsare however mostly an algorithmic concept Once the N -body integrationis finished eg when the cluster is dissolved the remaining binaries can beevolved to the main-sequence distributions by applying the same eigenevolu-tion algorithm above but with parameters

after Nbody integration rarr mainsequence λms = 247 χms = 8 (8174)

Fig 815 Eccentricityndashperiod after pre-mainsequence eigenevolution (λ = 28 χ =075) at t = 0 (upper panel) for masses 01 le miM le 11 and after cluster dis-integration (bottom panel note Tage means days) Systems with semi-major axesa le 10 R have been merged Binaries are only observed to have e log P below theenvelope described by Duquennoy amp Mayor (1991) The region above is forbiddenbecause pre-mainsequence dissipation depopulates it within 105 yr However dy-namical encounters can repopulate the eigenevolution region so that systems withforbidden parameters can be found but are short-lived Some of these are indicatedas open circles Eigenevolution (tidal circularisation) on the main sequence withλms = 247 and χms = 8 applied to the data in the lower panel depopulates theeigenevolution region and circularises all orbits with periods less than about 12 dThe dashed lines are constant periastron distances (8167) for rperi = λ R andmsys = 22 064 and 02 M (in increasing thickness) Horizontal and vertical cutsthrough this diagram produce eccentricity and period distribution functions andmass-ratio distributions that fit the observations (Kroupa 1995d)

250 P Kroupa

t

Fig 816 The period distribution functions (IPF (8164) with (8166) and forstellar masses 01 le miM le 11) The dashed histogram is derived from IPF withthe eigenevolution and feeding algorithms and represents the binary population at anage of about 105 yr The solid histogram follows from the dashed one after evolving acluster with initially N = 200 binaries and rh = 08 pc The agreement of the dashedhistogram with the observational pre-mainsequence data (as in Fig 813) and of thesolid histogram with the observed main sequence (Galactic field) data (also as inFig 813) is good A full model of the Galactic field late-type binary population hasbeen arrived at which unifies all available but apparently discordant observationaldata (see also Figs 814 815 and 817) nothing that the longest-period TTauribinary population is expected to show some disruption

The need for λms lt λ and χms gt χ to ensure for example the tidal circular-isation period of 12 days for G dwarfs (Duquennoy amp Mayor 1991) is nicelyqualitatively consistent with the shrinking of pre-mainsequence stars and theemergence of radiative cores that essentially reduce the coupling between thestellar surface where the dissipational forces are most effective and the cen-tre of the star The reader is also directed to Mardling amp Aarseth (2001) whointroduce a model of tidal circularisation to the N -body code Finally theabove work and the application to the ONC and Pleiades (Kroupa Aarsethamp Hurley 2001) suggests the following hypothesis

Initial binary universality hypothesis the initial period (8166) ec-centricity (8154) and mass-ratio (random pairing from canonical IMF)distributions constitute the parent distribution of all late-type stellarpopulations

Can this hypothesis be rejected

Fig 817 The mass-ratio distribution for stars with 01 le mM le 11 is thesolid histogram whereas the initial mass-ratio distribution (random pairing fromthe canonical IMF after eigenevolution and feeding at an age of about 105 yr) isshown as the dashed histogram The solid histogram follows from the dashed oneafter evolving a cluster with initially N = 200 binaries and rh = 08 pc The obser-vational data (solid dots Reid amp Gizis 1997) have been obtained after removing WDcompanions and scaling to the model This solar neighbourhood 8 pc sample is notcomplete and may be biased towards q = 1 systems (Henry et al 1997) Neverthe-less the agreement between model (solid histogram) and the data is striking A fullmodel of the Galactic field binary population has been arrived at which unifies allavailable but apparently discordant observational data (see also Figs 814 815816)

843 The Initial Binary Population ndash Massive Stars

The above semi-empirical distribution functions have been formulated for late-type stars (primary mass m1 le 1M) It is for these that we have the bestobservations It is not clear yet if they are also applicable to massive binaries

An approach taken by Clarke amp Pringle (1992) is to constrain the binaryproperties of OB stars by assuming that runaway OB stars are ejected fromstar-forming regions About 10ndash25 of all O stars are runaway stars whileabout 2 of B stars are runaways This approach leads to the result thatmassive stars must form in small-N groups of binaries that are biased to-wards unit mass ratio This is a potentially powerful approach but it can onlyconstrain the properties of OB binaries when they are ejected This occursafter many dynamical encounters in the cluster core which typically lead tothe mass-ratio evolving towards unity as the binaries harden The true birth

252 P Kroupa

properties of massive binaries therefore remain obscure and we need to resortto N -body experiments to test various hypotheses given the observations Onesuch hypothesis could be for example to assume massive stars form in bina-ries with birth pairing properties as for low-mass stars (Sect 842) ie mostmassive primaries would have a low-mass companion and to investigate if thishypothesis leads to the observed number of runaway massive stars throughdynamical mass segregation to the cluster core and partner exchanges throughdynamical encounters there between the massive stars

Apart from the fraction of runaway stars direct surveys have lead to someinsights into the binary properties of the observed massive stars Thus forexample Baines et al (2006) report a very high (f asymp 07 plusmn 01) binary frac-tion among Herbig AeBe stars with a binary fraction that increases withincreasing primary mass Furthermore they find that the circumbinary discsand the binary orbits appear to be coplanar This supports a fragmentationorigin rather than collisions or capture as the origin of massive binaries MostO stars are believed to exist as short-period binaries with q asymp 1 (Garcıa ampMermilliod 2001) at least in rich clusters On the other hand small-q appearto be favoured in smaller clusters such as the ONC consistent with randompairing (Preibisch et al 1999) Kouwenhoven et al (2005) report that the Aand late-type B binaries in the Scorpius OB2 association have a mass-ratiodistribution inconsistent with random pairing The lower limit on the binaryfraction is 052 while Kouwenhoven et al (2007) update this to a binary frac-tion of 72 They also find that the semi-major axis distribution containstoo many close pairs compared to a Duquennoy amp Mayor (1991) log-normaldistribution These are important constraints but again they are derived forbinaries in an OB association which is an expanded version of a dense starcluster (Sect 812) and therefore hosts a dynamically evolved population

Given the above results perhaps the massive binaries in the ONC repre-sent the primordial population whereas in rich clusters and in OB associa-tions the population has already evolved dynamically through hardening andcompanion exchanges (fq rising towards q = 1) This possibility needs to beinvestigated with high-precision N -body computations of young star clustersThe first simplest hypothesis to test would be to extend the pairing rules ofSect 842 to all stellar masses perform many (because of the small number ofmassive stars) N -body renditions of the same basic pre-gas expulsion clusterand to quantify the properties of the emerging stellar population at variousdynamical times (Kroupa 2001)

Another approach would be to constrain a and m2 for a given m1 ge 5Mso that

Ebin asymp Ek (8175)

(8141) Or we can test the initial massive-star population given by

a ltrc

N13OB

(8176)

which follows from stating that the density of a massive binary 2times3(a3 4π)be larger than the cluster-core density NOB 3(r3c 4π) So far none of thesepossibilities have been tested apart from the Initial Binary Universality

Hypothesis (p 250) extension to massive stars (Kroupa 2001)

85 Summary

The above material gives an outline of how to set up an initial birth orprimordial stellar population so that it resembles observed stellar populationsIn Sect 842 a subtle differentiation was made between initial and birthpopulations in the sense that an initial population is derived from a birthpopulation through processes that act too rapidly to be treated by an N -body integration

An N -body stellar system is generated for numerical experiments by speci-fying its 3D structure and velocity field (Sect 82) the mass distribution of itspopulation (Sect 83) and the properties of its binary population (Sect 84)Given the distribution functions discussed here and the existing numericalresults based on these it is surprising how universal the stellar and binarypopulation turns out to be at birth A dependence of the IMF or the birthbinary properties on the physical properties of star-forming clouds cannot bedetected conclusively In fact the theoretical proposition that there should bea dependency can be rejected except possibly (i) in the extreme tidal fieldenvironment at the Galactic centre or (ii) in the extreme protostellar den-sity environment of ultra-compact dwarf galaxies or (iii) for extreme physicalenvironments (pp 230ndash231)

The unified picture that has emerged concerning the origin of stellar pop-ulations is that stars form according to a universal IMF and mostly in binarysystems They form in very dense clusters which expel their residual gas andrapidly evolve to T- or OB-associations If the latter are massive enoughthe dense embedded clusters evolve to populous OB associations that maybe expanding rapidly and contain cluster remnants which may reach glob-ular cluster masses and beyond in intense star-bursts This unified pictureexplains naturally the high infant weight loss and infant mortality of clustersthe binary properties of field stars possibly thick discs of galaxies and theexistence of population II stellar halos around galaxies that have old globularcluster systems

Many open questions remain Why is the star-formation product so univer-sal within current constraints How are massive stars distributed in binariesDo they form at the centres of their clusters Why is the cluster mass ofabout 106 M special And which star cluster population is a full solutionto the inverse dynamical population synthesis problem (p 246) Many moreobservations are required These must not only be of topical high red-shiftstar-burst systems but also of the more mundane low red-shift and prefer-ably local star-forming objects globular and open star clusters

254 P Kroupa

Acknowledgement

It is a pleasure to thank Sverre Aarseth for organising a splendid and much tobe remembered Cambridge N -body school in the Summer of 2006 and alsoChristopher Tout for editing and proof-reading this chapter I am indebtedto Jan Pflamm-Altenburg who read parts of this manuscript carefully to An-dreas Kupper for producing the Plummer vs King model comparisons and forcarefully reading the whole text and to Joerg Dabringhausen who suppliedfigures from his work

References

Aarseth S J 2003 Gravitational N-Body Simulations Cambridge Univ Press Cam-bridge 202

Aarseth S J Henon M Wielen R 1974 AampA 37 183 216Adams F C 2000 ApJ 542 964 197 221Adams F C Fatuzzo M 1996 ApJ 464 256 234Adams F C Laughlin G 1996 ApJ 468 586 234Adams F C Myers P C 2001 ApJ 553 744 183Allen L Megeath S T Gutermuth R Myers P C et al 2007 in Reipurth

B Jewitt D Keil K eds Protostars and Planets V University Arizona PressTucson p 361 186

Alves J Lombardi M Lada C J 2007 AampA 462 L17 235Baines D Oudmaijer R D Porter J M Pozzo M 2006 MNRAS 367 737 252Ballero S Kroupa P Matteucci F 2007 MNRAS 467 117 230Baraffe I Chabrier G Allard F Hauschildt P H 2002 AampA 382 563 227Bastian N Goodwin S P 2006 MNRAS 369 L9 193 195 221Baumgardt H 1998 AampA 330 480 197Baumgardt H Kroupa P 2007 MNRAS 380 1589 227Baumgardt H Kroupa P de Marchi G 2008 MNRAS in press (astroph

08060622) 222Baumgardt H Kroupa P Parmentier G 2008 MNRAS 384 1231 195 198 220 221Baumgardt H Makino J 2003 MNRAS 340 227 194 228Binney J Tremaine S 1987 Galactic Dynamics Princeton Univ Press Princeton

NJ 182 183 202 203 209 211 213Boily C M Kroupa P 2003 MNRAS 338 673 193Boily C M Kroupa P Penarrubia-Garrido J 2001 New Astron 6 27 218Boily C M Lancon A Deiters S Heggie D C 2005 ApJ 620 L27 193Bonnell I A Bate M R Vine S G 2003 MNRAS 343 413 188Bonnell I A Larson R B Zinnecker H 2007 in Reipurth B Jewitt D Keil K

eds Protostars and Planets V University Arizona Press Tucson p 149 184 222 227 234Brook C B Kawata D Scannapieco E Martel H Gibson B K 2007 ApJ 661

10 231Casuso E Beckman J E 2007 ApJ 656 897 234Chabrier G 2003 PASP 115 763 222 224 225 236 237Clark P C Bonnell I A Zinnecker H Bate M R 2005 MNRAS 359 809 195

Clarke C J Bonnell I A Hillenbrand L A 2000 in Mannings V Boss A PRussell S S eds Protostars and Planets IV University Arizona Press Tucsonp 151 184

Clarke C J Pringle J E 1992 MNRAS 255 423 188 251Dabringhausen J Hilker M Kroupa P 2008 MNRAS 386 864 199 200 201 237Dabringhausen J Kroupa P Baumgardt H 2008 MNRAS submitted 230Dale J E Bonnell I A Clarke C J Bate M R 2005 MNRAS 358 291 192Dale J E Ercolano B Clarke C J 2007 MNRAS 1056 184de Grijs R Parmentier G 2007 Chinese J Astron Astrophys 7 155 193 195de la Fuente Marcos R 1997 AampA 322 764 199de la Fuente Marcos R 1998 AampA 333 L27 199Delfosse X Forveille T Segransan D Beuzit J-L Udry S Perrier C Mayor M

2000 AampA 364 217 223Duchene G 1999 AampA 341 547 242 243Duquennoy A Mayor M 1991 AampA 248 485 242 243 244 249 250 252Eisloffel J Steinacker J 2008 in The Formation of Low-Mass-Protostars and Proto-

Brown Dwarfs (in press astro-ph0701525) ASP conf series Vol 384 p 359ed Gerard von Belle 232

Elmegreen B G 1983 MNRAS 203 1011 187 194Elmegreen B G 1997 ApJ 486 944 222 228Elmegreen B G 1999 ApJ 515 323 228Elmegreen B G 2000 ApJ 530 277 187 189Elmegreen B G 2004 MNRAS 354 367 234Elmegreen B G 2006 ApJ 648 572Elmegreen B G 2007 ApJ 668 1064 187 189 222Elmegreen D M Elmegreen B G Sheets C M 2004 ApJ 603 74 196Fellhauer M Kroupa P 2005 ApJ 630 879 184 193 221Fellhauer M Kroupa P Evans N W 2006 MNRAS 372 338 189 228Feltzing S Gilmore G Wyse R F G 1999 ApJ 516 L17 230Figer D F 2005 Nature 434 192 187Fischer D A Marcy G W 1992 ApJ 396 178 242 243 244Fisher R T 2004 ApJ 600 769 246Fleck J-J Boily C M Lancon A Deiters S 2006 MNRAS 369 1392 193Garcıa B Mermilliod J C 2001 AampA 368 122 252Goldman I Mazeh T 1994 ApJ 429 362 247Goodwin S P 1997a MNRAS 284 785 192 220Goodwin S P 1997b MNRAS 286 669 192 220Goodwin S P 1998 MNRAS 294 47 192 220Goodwin S P Bastian N 2006 MNRAS 373 752 193Goodwin S P Kroupa P 2005 AampA 439 565 199 236 243Goodwin S P Kroupa P Goodman A Burkert A 2007 in Reipurth B Jewitt

D Keil K eds Protostars and Planets V University Arizona Press Tucsonp 133 242

Goodwin S P Nutter D Kroupa P Ward-Thompson D Whitworth A P 2008AampA 477 823 235

Goodwin S P Whitworth A 2007 AampA 466 943 232Gouliermis D Keller S C Kontizas M Kontizas E Bellas-Velidis I 2004 AampA

416 137 184Gouliermis D A Quanz S P Henning T 2007 ApJ 665 306 195

256 P Kroupa

Gradshteyn I S Ryzhik I M 1980 Table of Integrals Series and Products Aca-demic Press New York 208

Grether D Lineweaver C H 2006 ApJ 640 1051 232Grillmair C J et al 1998 AJ 115 144 230Gutermuth R A Megeath S T Pipher J L Williams J P Allen L E Myers

P C Raines S N 2005 ApJ 632 397 186Hartmann L 2003 ApJ 585 398 187 189Heggie D C 1975 MNRAS 173 729 240Heggie D Hut P 2003 The Gravitational Million-Body Problem Cambridge Univ

Press Cambridge 194 198 202 206Henry T J Ianna P A Kirkpatrick J D Jahreiss H 1997 AJ 114 388 251Hillenbrand L A Hartmann L W 1998 ApJ 492 540 193Hills J G 1975 AJ 80 809 240Hoversten E A Glazebrook K 2008 ApJ 675 163 234Hurley J R Pols O R Aarseth S J Tout C A 2005 MNRAS 363 293 202Hut P Mineshige S Heggie D C Makino J 2007 Progress Theor Phys 118

187 202Jenkins A 1992 MNRAS 257 620 196Kennicutt R C 1983 ApJ 272 54 225Kim S S Figer D F Kudritzki R P Najarro F 2006 ApJ 653 L113 231 234Kim E Yoon I Lee H M Spurzem R 2008 MNRAS 383 2 219King I R 1962 AJ 67 471 212King I R 1966 AJ 71 64 212Klessen R S Spaans M Jappsen A-K 2007 MNRAS 374 L29 231 234Koen C 2006 MNRAS 365 590 187Koeppen J Weidner C Kroupa P 2007 MNRAS 375 673 194 234Kouwenhoven M B N Brown A G A Portegies Zwart S F Kaper L 2007

AampA 474 77 252Kouwenhoven M B N Brown A G A Zinnecker H Kaper L Portegies Zwart

S F 2005 AampA 430 137 252Kroupa P 1995a ApJ 453 350 224Kroupa P 1995b ApJ 453 358 224Kroupa P 1995c MNRAS 277 1491 240 244 246 247Kroupa P 1995d MNRAS 277 1507 194 240 244 246 248 249Kroupa P 1998 MNRAS 300 200 201Kroupa P 2000 New Astron 4 615Kroupa P 2001 MNRAS 322 231 228 229 252 253Kroupa P 2002a Science 295 82 222Kroupa P 2002b MNRAS 330 707 195 196 223 224 228Kroupa P 2005 in Turon C OrsquoFlaherty K S Perryman M A C eds Proc Gaia

Symp Vol 576 The Three-Dimensional Universe with Gaia ESA PublicationsDivision Noordwijk p 629 (astro-ph0412069) 186 192 193 196 220

Kroupa P 2007a in Valls-Gabaud D Chavez M eds Resolved Stellar Populations(in press astro-ph0703124) 222 226 237

Kroupa P 2007b in Israelian G Meynet G eds The Metal Rich Universe Cam-bridge Univ Press Cambridge (astro-ph0703282) 222

Kroupa P Aarseth S J Hurley J 2001 MNRAS 321 699 191 192 193 194 195 197 221 228Kroupa P Boily C M 2002 MNRAS 336 1188 195 197 198 227Kroupa P Bouvier J Duchene G Moraux E 2003 MNRAS 346 354 225 232

Kroupa P Burkert A 2001 ApJ 555 945 245Kroupa P Gilmore G Tout C A 1991 MNRAS 251 293 223Kroupa P Petr M G McCaughrean M J 1999 New Astron 4 495 240Kroupa P Tout C A Gilmore G 1993 MNRAS 262 545 223 226 237Kroupa P Weidner C 2003 ApJ 598 1076 233 237Kroupa P Weidner C 2005 in Cesaroni R Felli M Churchwell E Walmsley

M eds Proc IAU Symp 227 Massive Star Birth A Crossroads of AstrophysicsCambridge Univ Press Cambridge p 423 187

Krumholz M R 2008 in Knapen J Mahoney T Vazdekis A eds PathwaysThrough an Eclectic Universe (astro-ph07063702) ASP conference series vol390 235

Krumholz M R Tan J C 2007 ApJ 654 304 187 189Lada C J Lada E A 2003 ARAampA 41 57 185 197Lada C J Margulis M Dearborn D 1984 ApJ 285 141 192 220Larson R B 1998 MNRAS 301 569 231 234Lee H-C Gibson B K Flynn C Kawata D Beasley M A 2004 MNRAS 353

113 234Li Y Klessen R S Mac Low M-M 2003 ApJ 592 975 185Mac Low M-M Klessen R S 2004 Rev Mod Phys 76 125 184Maız Apellaniz J Ubeda L 2005 ApJ 629 873 222Maız Apellaniz J Walborn N R Morrell N I Niemela V S Nelan E P 2007

ApJ 660 1480 187Maness H et al 2007 ApJ 669 1024 230Mardling R A Aarseth S J 2001 MNRAS 321 398 250Marks M Kroupa P Baumgardt H 2008 MNRAS 386 2047 222Martın E L Brandner W Bouvier J Luhman K L Stauffer J Basri G Zapatero

Osorio M R Barrado y Navascues D 2000 ApJ 543 299 225 236Martins F Schaerer D Hillier D J 2005 AampA 436 1049 187Massey P 2003 ARAampA 41 15 222 224 225Mayor M Duquennoy A Halbwachs J-L Mermilliod J-C 1992 in McAlister

H A Hartkopf W I eds ASP Conf Ser Vol 32 Complementary Approachesto Double and Multiple Star Research Astron Soc Pacific San Francisco p 73 242 244

McMillan S L W Vesperini E Portegies Zwart S F 2007 ApJ655 L45 184 221Metz M Kroupa P 2007 MNRAS 376 387 230Meylan G Heggie D C 1997 AampAR 8 1 194 198Mieske S Kroupa P 2008 ApJ 677 276 230Moraux E Bouvier J Clarke C 2004 in Combes F Barret D Contini T Mey-

nadier F Pagani L eds SF2A-2004 Semaine de lrsquoAstrophysique Francaise EdP-Sciences Conference Series p 251 225 236

Motte F Andre P Neri R 1998 AampA 336 150 235Motte F Andre P Ward-Thompson D Bontemps S 2001 AampA 372 L41 235Murray S D Lin D N C 1996 ApJ 467 728 234Nutter D Ward-Thompson D 2007 MNRAS 374 1413 235Odenkirchen M et al 2003 AJ 126 2385 199Oey M S Clarke C J 2005 ApJ 620 L43 187Palla F Randich S Pavlenko Y V Flaccomio E Pallavicini R 2007 ApJ 659

L41 189Palla F Stahler S W 2000 ApJ 540 255 189Pancino E Galfo A Ferraro F R Bellazzini M 2007 ApJ 661 L155 219

258 P Kroupa

Parker R J Goodwin S P 2007 MNRAS 380 1271 185Parmentier G Gilmore G 2005 MNRAS 363 326 197Parmentier G Gilmore G 2007 MNRAS 377 352 195 198Parmentier G Goodwin S Kroupa P Baumgardt H 2008 ApJ 678 347 198Pflamm-Altenburg J Kroupa P 2006 MNRAS 373 295 188 227 228 237 238Pflamm-Altenburg J Kroupa P 2007 MNRAS 375 855 189 228Pflamm-Altenburg J Kroupa P 2008 MNRAS submitted 190 200Pflamm-Altenburg J Weidner C Kroupa P 2007 ApJ 671 1550 188 194 234Piotto G 2008 in Cassisi S Salaris M XXI Century Challenges for Stellar Evo-

lution Mem d Soc Astron It Vol 792 (arXiv08013175) 190Plummer H C 1911 MNRAS 71 460 205Portegies Zwart S F McMillan S L W Hut P Makino J 2001 MNRAS 321

199 197Portegies Zwart S F McMillan S L W Makino J 2007 MNRAS 374 95 202Portinari L Sommer-Larsen J Tantalo R 2004 MNRAS 347 691 225 234Preibisch T Balega Y Hofmann K Weigelt G Zinnecker H 1999 New Astron

4 531 224 252Press W H Teukolsky S A Vetterling W T Flannery B P 1992 Numerical

Recipes Cambridge Univ Press Cambridge 2nd ed 215 218Reid I N Gizis J E 1997 AJ 113 2246 251Reid I N Gizis J E Hawley S L 2002 AJ 124 2721 224 225Reipurth B Clarke C 2001 AJ 122 432 232Romano D Chiappini C Matteucci F Tosi M 2005 AampA 430 491 225 234Sacco G G Randich S Franciosini E Pallavicini R Palla F 2007 AampA 462

L23 189Salpeter E E 1955 ApJ 121 161 224Scally A Clarke C 2002 MNRAS 334 156 184 221Scalo J M 1986 Fundamentals Cosmic Phys 11 1 225 226Scalo J 1998 in Gilmore G Howell D eds ASP Conf Ser Vol 142 The Stellar

Initial Mass Function (38th Herstmonceux Conference) Astron Soc Pac SanFrancisco p 201 228 229

Shara M M Hurley J R 2002 ApJ 571 830 194Spitzer L 1987 Dynamical Evolution of Globular Clusters Princeton Univ Press

Princeton NJ 183Stamatellos D Whitworth A P Bisbas T Goodwin S 2007 AampA 475 37 184Subr L Kroupa P Baumgardt H 2008 MNRAS 385 1673 222Testi L Sargent A I Olmi L Onello J S 2000 ApJ 540 L53 184Thies I Kroupa P 2007 ApJ 671 767 223 232 236 238 242Tilley D A Pudritz R E 2007 MNRAS 382 73 184Tinsley B M 1980 Fundamentals Cosmic Phys 5 287 225Tumlinson J 2007 ApJ 665 1361 231Tutukov A V 1978 AampA 70 57 192 220Vesperini E 1998 MNRAS 299 1019 197Vesperini E 2001 MNRAS 322 247 197Weidemann V 1990 ARAampA 28 103 226Weidemann V 2000 AampA 363 647 226Weidemann V Jordan S Iben I J Casertano S 1992 AJ 104 1876 226Weidner C Kroupa P 2004 MNRAS 348 187 187Weidner C Kroupa P 2005 ApJ 625 754 233

Weidner C Kroupa P 2006 MNRAS 365 1333 187 188 233Weidner C Kroupa P Larsen S S 2004 MNRAS 350 1503 196 198 233Weidner C Kroupa P Nurnberger D E A Sterzik M F 2007 MNRAS 376

1879 193 227Woitas J Leinert C Koehler R 2001 AampA 376 982 244Wuchterl G Tscharnuter W M 2003 AampA 398 1081 185 227Yasui C Kobayashi N Tokunaga A T Saito M Tokoku C 2008 (astro-

ph08010204) 230Yasui C Kobayashi N Tokunaga A T Terada H Saito M 2006 ApJ 649

753 in Formation and Evolution of Galaxy Disks ASP Conf series in press edsJ G Funes EM Cossini (astro-ph08010204) 230

Zhao H 2004 MNRAS 351 891 189Zinnecker H 2003 in van der Hucht K Herrero A Esteban C eds Proc IAU

Symp 212 A Massive Star Odyssey From Main Sequence to Supernova AstronSoc Pacific San Francisco p 80 224

Zinnecker H Yorke H W 2007 ARAampA 45 481 184 187 222Zoccali M et al 2006 AampA 457 L1 230 232Zoccali M Cassisi S Frogel J A Gould A Ortolani S Renzini A Rich R M

Stephens A W 2000 ApJ 530 418 229 230

9

Stellar Evolution

Christopher A Tout

University of Cambridge Institute of Astronomy Madingley Road CambridgeCB3 0HA Englandcatastcamacuk

The bodies in any N -body system can change The most changeable bodiesare stars In order to fully model the evolution of a cluster of stars we need toknow how they interact with their environment particularly how much massthey lose and how they interact with each other Is their evolution affectedby a companion or close encounter In this chapter we describe the physicsand the mathematical formulation that we use to describe it If we could wewould evolve each star in every detail (Church Tout amp Aarseth 2007) butup to now in practice we have had to approximate the detailed evolution byempirical models (Hurley Tout amp Pols 2000) As the number of bodies we canmodel increases with increasing computing power it becomes more reasonableto include the full evolution (Chap 13) So let us examine the physics of stars

91 Observable Quantities

When we look at stars in the night sky they have two immediately discernibleproperties they vary in brightness and colour The brightness is assessedin terms of magnitudes Historically and we are going back to the ancientGreeks here stars fall into six magnitude classes The brightest stars areof first magnitude and the faintest stars visible to the naked eye are sixthmagnitude though these are rarely visible amongst todayrsquos city lights Theeye measures brightness logarithmically so that a star of magnitude 50 turnsout to be one hundred times fainter than a star of magnitude 10 Modernphotometry can measure the magnitude of stars extremely accurately and indifferent wavelength ranges But these magnitudes are only apparent A starcan vary in brightness for two reasons First it may be brighter because it isintrinsically more luminous Alternatively it might just be brighter becauseit is close to us Indeed Herschel (1783) hoped that all stars were of similarintrinsic luminosity so that he might map the Galaxy by taking variations inbrightness to indicate variations in distance Today the distances to nearbystars can be determined by accurate trigonometric parallaxes The motion of

Tout CA Stellar Evolution Lect Notes Phys 760 261ndash282 (2008)

262 C A Tout

the star is measured against the background of distant apparently immovablestars and galaxies as the Earth moves around its orbit Once the distance isknown an absolute magnitude can be calculated from the observed apparentmagnitude and from this we get an estimate of the luminosity of the star

The second observable quantity is a starrsquos colour Some stars appear redderwhile others are distinctly blue The colour of a star is related to its surfacetemperature The apparent surface or photosphere of a star represents thelocus of points at which the majority of photons were last emitted or scat-tered before they began their journey through space to the Earth Typicallythe spectrum of radiation emitted by a star is close to that of a black bodyThe hotter the black body the bluer is the peak in its spectrum Thus bluestars are hot while red stars are relatively cool Another way of determiningthe surface temperature of a star is to look at the dark lines in its spectrumThese generally occur at wavelengths where an atomic transition of an elec-tron makes the absorption of a photon particularly favourable Historicallyspectra where classified by the strength of their hydrogen lines Those withthe strongest hydrogen lines are of type A while those with the weakest areof type M Hydrogen ionizes at about 10 000K and it is stars of this temper-ature that have the most prominent hydrogen lines As the temperature risesfewer and fewer atoms have bound electrons and the lines disappear from thespectra As the temperature falls the electrons around the hydrogen nucleibecome more and more energetically confined to the ground-state orbits Thisin turn leads to fewer hydrogen lines in the spectra However lines from moreweakly bound electrons and bands owing to molecular rotations and vibra-tions become more prominent So it is easy to distinguish the very hot O starsfrom the relatively very cool M stars The sequence of spectral types from thehottest to the coolest normal stars follows

O B A F G K M

Once we know the temperature and nature of a starrsquos atmosphere we canrelate its absolute magnitude to a bolometric luminosity This bolometricluminosity L is the total energy radiated by the star per unit time

In the early years of the twentieth century Russell (1913) who workedpartly in Cambridge at the time and the Danish astronomer and chemistHertzsprung (1905) examined the correlations of these two quantities witheach other The resulting HertzsprungndashRussell (HR) diagram has become themajor tool for describing the evolution of stars over their lifetimes Ratherthan populating the whole of such a diagram we find that most of the starslie on a band running from hot bright stars to cool faint stars (Fig 91) Thisis the main sequence Because the radiation from stars is very close to a blackbody the temperature of the photosphere is close to the effective temperaturegiven by

L = 4πσR2T 4eff (91)

where σ is the Stefan-Boltzmann constant and R is the radius of the photo-sphere This means that the loci of stars of constant radius are straight lines

9 Stellar Evolution 263

log T

log

L

Main Sequence

Red

Gia

nts

AG

B S

tars

Super Giants

White Dwarfs

Horizontal Branch

Fig 91 A schematic HertzsprungndashRussell diagram showing the position of starsin a surface temperature ndash luminosity or colour magnitude diagram Temperatureincreases from right to left along the horizontal axis Colour changes from blue tored from left to right Most stars like the Sun lie along the main sequence but otherdistinct groups of stars are visible particularly in such diagrams of clusters

of slope minus4 in the HR diagram so that stars at the top left of the main se-quence are blue supergiants while those at the bottom right are red dwarfsIn a diagram of the brightest stars another region to the right and nearlyvertically upwards from the main sequences is prominent These are the redgiants In diagrams of globular clusters this giant branch splits into two dis-tinct parts the normal red giants and the asymptotic giant branch (AGB)We shall see later how these are populated by stars in quite distinct evolution-ary phases In HR diagrams of nearby stars the fainter but relatively commonwhite dwarfs appear in a band below the main sequence Also discernible asseparate though not so distinct regions are the supergiants from blue to redacross the very top of the diagram and the subgiants between the main se-quence and the true red giants Globular clusters have the advantage that the

264 C A Tout

stars all lie at approximately the same distance so that relatively though notabsolutely the errors associated with distance measurements are significantlyreduced Today some very beautiful HR diagrams of globular clusters can beplotted with the data obtained with large telescopes (Pancino et al 2000) andthese reveal all sorts of detail Of particular interest is the horizontal branch atrelatively constant luminosity extending from red to blue across from the redgiants The structure and population of this feature vary considerably fromcluster to cluster and contain clues to the age and initial chemical compositionof the constituent stars The Sun itself lies right in the middle of the mostpopulated part of the main sequence so that we can deduce that it is typicalof the majority of stars In the next sections we shall investigate the physicsand the mathematical models that have allowed us to unravel the life of a staras it moves about the HR diagram from the main sequence to the red giantbranch perhaps to the horizontal branch or back to the subgiant area thenon to the AGB and finally to a white dwarf if the star has lost enough massto avoid a supernova explosion

92 Structural Equations

The structure of a star can be described in essence by four differential equa-tions Two of these that describe the variation of mass and pressure withradius can be called the structural equations They are the subject of thissection Supplemented with an equation of state these two are the basic build-ing blocks of a stellar model When the equation of state depends on twophysical state variables we must add an equation to describe the variation oftemperature through the star and another to incorporate energy-generatingprocesses to complete the set

The first equation is easily derived by considering a thin shell of mass δmand thickness δr at radius r in the star (Fig 92) The mass in the shell is justits volume multiplied by the local density ρ(r) and when we take the limit asδr tends to zero we obtain

dmdr

= 4πr2ρ (92)

the mass equationThe mass interior to this shell exerts on it an attractive radial force of

magnitude δmg = 4πr2ρgδr where g(r) = Gmr2 is the local gravitationalacceleration and m(r) is the mass inside radius r This must be balanced bythe differences in the pressure force on either side of the shell 4πr2[P (r+δr)minusP (r)] Again taking the limit as δr tends to zero we obtain

dPdr

= minusGmρ

r2 (93)

This is the equation of hydrostatic equilibrium Equations (92) and (93) arespecial spherically symmetric cases of the more general equations of mass

r+δ rr

m(r)p(r)

p(r+ r)δ

δm = 4π r2ρδr

δmg

Fig 92 The structure of a fluid sphere The mass enclosed by a spherical surfaceof radius r is m(r) A shell of mass δm of thickness δr at this radius is supportedagainst gravity by a pressure gradient

conservation and the Euler momentum equation of fluid dynamics when thevelocity in the fluid is everywhere zero

If we can write P explicitly as a function of ρ only we can obtain a fullsolution to the structure of the star The simplest boundary conditions toapply are at r = 0

m(0) = 0 rArr dPdr

= 0 (94)

and at r = Rm(R) = M ρ(R) = 0 (95)

where M is the total mass of the star It turns out that the equation of stateof very degenerate matter takes just such a form and white dwarfs can bemodelled immediately (Chandrasekhar 1939)

93 Equation of State

In practice pressure does not depend only on density Figure 93 illustratesthe various contributions to the pressure as temperature and density varyTypically the state of stellar material depends on its composition plus anytwo state variables In general there are many contributions to the equation ofstate but for most normal stars the fluid behaves very similarly to an ideal gasfor which the pressure may be written as a function of density temperatureT and mean molecular weight μ

P = ρRT

μ (96)

266 C A Tout

TK

M M

M

PP

PI

C

p gp r

XZ

Fig 93 Contributions to the equation of state as a function of temperature anddensity The thick solid lines are the run of temperature and density through zero-agemain-sequence stars of masses 01 03 1 10 and 100 M Their centres are towardsthe top right of the figure A dashed line marks where gas and radiation pressureare equal with increasing PrPg to the left A second dashed line indicates where theelectron chemical potential ψ = 0 To the right of this line material becomes moreand more degenerate The shaded regions represent the range over which ionisationof H He and He+ and dissociation of molecular hydrogen take place Thin solid linesindicate the effects of pressure ionisation and dotted lines corrections to account forplasma effects Dot-dashed lines indicate when the fluid can be considered a plasmaand when it begins to crystallize into the solid state

where R is the gas constant per unit mass The mean molecular weight is thereciprocal of the number of particles each of which contributes to the pressureequally at a given temperature per atomic mass unit Thus neutral hydrogencontributes one particle for each mass unit and has μ = 1 while fully ionizedhydrogen contributes two particles an electron and a proton for each massunit and so has μ = 12 Fully ionised helium contributes two electrons andand a helium nucleus made up of two protons and two neutrons for its four

mass units and so has μ = 43 Anything heavier than hydrogen and heliumis designated a metal and when fully ionized contributes approximately halfas many particles as its atomic mass because the nucleus typically consists ofequal numbers of protons and neutrons and each positively charged proton isbalanced by an electron Thus metals have μ asymp 2 For a fully ionized mixtureadding the numbers and masses we find

1μ

= 2X +34Y +

12Z (97)

where X is the mass fraction of hydrogen Y is that of helium and Z that ofall metals and X + Y + Z = 1 In the deep interiors of stars temperaturesare such that all atoms are ionized but as the temperature falls electronsrecombine with their nuclei to form atoms in various ionization states Themost strongly bound electrons recombine at the highest temperatures Thusin the Sun hydrogen recombines between about 10 000 and 20 000K whileiron is still not completely ionized at 100 000K

An important consequence of (97) is that the equation of state changesas nuclear reactions convert one element to another This is one of the driv-ing forces behind stellar evolution and is responsible for the Sun graduallyexpanding and brightening with time

At high temperatures the pressure exerted by energetic photons becomescomparable with that exerted by the particles and we must include a term

Pr =13aT 4 (98)

where a is the radiation constantAt high densities electrons contribute a degeneracy pressure This arises

because free electrons must occupy a discrete set of momentum states andas the volume to which an electron is confined is reduced the energies of itsavailable states increase Thus squeezing an electron gas increases the mo-menta of the electrons and this requires energy So work must be done andthe gas exerts a force against compression The contribution to this degener-acy pressure Pe becomes important when the electron chemical potential ψbecomes positive It is already becoming important in the core of the Sun andlower-mass main-sequence stars but comes into its own in the white dwarfswhere it provides sufficient support against gravity even when the gas is coldAlthough we might expect a cold gas to consist of neutral atoms this is notthe case at very high densities because the nuclei are so close to one anothermuch nearer than the radius of an atom that the electrons are not boundto a particular nucleus but behave as a free gas similar to those in metal-lic elements at room temperature This effect of pressure ionization is alsoimportant to some extent in the Sun

There are various other corrections to the pressure Pc that must be in-cluded such as plasma effects at high densities and eventually liquefactionand crystallization to the solid state as density increases and temperaturefalls

268 C A Tout

94 Radiation Transport

When temperature is important for the equation of state we require twofurther equations to describe the star The first is for the temperature gra-dient This depends on the rate at which energy can be transported fromwhere it is generated usually at the hot centre through the star One of thethree processes dominates energy transport under different conditions Radi-ation or the diffusion of photons dominates in the central parts of the SunConduction or the diffusion of particles is prevalent in degenerate materialConvection or energy transport by bulk fluid motion operates when the tem-perature gradient becomes too large for stable radiative transfer This is thecase in the outer layers of the Sun

In radiative regions we can estimate the temperature gradient by consid-ering two surfaces of different temperatures separated by a distance λ thedistance that a photon moves between interactions with the matter and overwhich it maintains memory of the conditions when it last interacted (Fig 94)Deep in the star everything is in local thermodynamic equilibrium so thata surface at temperature T emits energy as a blackbody providing a fluxof energy per unit area of F = σT 4 where the Stefan Boltzmann constant

= σ(T + δT )4

F = σT4

F + δF

λ

Fig 94 Radiation diffuses through the star The interior of the star is locallyin thermodynamic equilibrium so that the radiation flux emitted by any surfacedepends on the temperature of that surface Photons travel until they are absorbedor scattered typically a mean free path length from where they were emitted or lastscattered In this way heat diffuses from hotter to cooler regions

σ = ac4 Consider two such surfaces one at temperature T and one at T+δT In our spherically symmetric star the surfaces are spheres of area 4πr2 and Tusually decreases as r increases We call the net energy flow through a sphereof radius r the local luminosity Lr and we have

Lr = 4πr2δF (99)

whereδF asymp minus4σT 3δT (910)

is the difference between the inward flux from the surface at temperatureT+δT and the outward from the surface at T The difference in temperature isjust the temperature gradient multiplied by the distance between the surfaces

δT = λdTdr

(911)

So we haveLr asymp 16πσr2λT 3 dT

dr (912)

The typical distance travelled by a photon between interactions its meanfree path depends on the opacity of the material Opacity is defined as theeffective cross-section per unit mass seen by a photon The probability ofinteraction of a photon passing along a cylinder (Fig 95) of cross-sectionequal to κ times the mass in the cylinder and length λ is unity Thus formaterial of density ρ

ρκλ = 1 (913)

Combining this with (912) we find

dTdr

=minusκρLr

4πacr2T 3 (914)

λ

ρ κ

Fig 95 The relation between mean free path and opacity A photon is likely tobe absorbed or scattered once within a cylinder of height λ and cross-sectional areaκm aligned with its motion which contains one target of mass m

270 C A Tout

This is not quite correct because we have not taken proper account of the factthat the radiation field from a point on a surface is isotropic and not directedtowards the other surface With somewhat more effort we should obtain

dTdr

=minus3κρLr

16πacr2T 3 (915)

which is the equation of radiative transferThe detailed calculation of opacity is a long and complex procedure

Figure 96 illustrates how it varies with temperature and density in stellarmaterial At high temperatures all material is ionized and the only sourceof opacity is scattering by electrons This is independent of temperature anddensity until at very high temperatures when relativistic effects become im-portant At intermediate temperatures atomic processes where electrons aremoved from one state to another by absorption of a photon dominate Thestates may be either bound or free and a dependence

κ prop ρTminus35 (916)

κ

T K

Z

Fig 96 Opacity as a function of temperature for various stellar densities

emerges Just above 10 000K the opacity drops rapidly with decreasing tem-perature as hydrogen recombines and fewer and fewer photons have sufficientenergy to change the electronic states At lower temperatures it begins to riseagain as Hminus ions and various molecules become important sources but thecalculation becomes even more complex

Conductivity can be described in a similar way with electrons replacingthe photons as the energy carriers Usually the mean free path of electrons ismuch shorter than that of photons so that their effective opacity is much largerand radiation transport dominates However in degenerate material electronsare not easily scattered because they must scatter into an empty momentumstate but all neighbouring momentum states are already occupied The meanfree path becomes very large and the fluid is effectively superconducting Inpractice this means that degenerate regions of stars are close to isothermal

95 Convection

The process of convection is sufficiently important to warrant a separate dis-cussion Fluid is convectively unstable when the temperature gradient is suchthat a packet of material displaced vertically parallel to the direction of grav-ity continues to rise or fall Suppose we displace a blob of material by a smalldistance δz upwards in the star (Fig 97) the density of the material out-side the blob changes according to the ambient gradient Let the new densitywithin the blob be ρprime Then the blob continues to rise if it is now less densethan its surroundings

T p ρ

g

δ z

Tpρ

T p ρ T+ zδ dT

dz

Fig 97 The convective instability A blob of fluid displaced upwards continues torise if its density is less than that of its surroundings when it has reached pressureequilibrium adiabatically

272 C A Tout

ρprime lt ρ+ δzdρdz

(917)

and is convectively unstableThe sound speed in the fluid is generally short so that the blob quickly

reaches pressure equilibrium with its surroundings and

P prime = P + δzdPdz

(918)

Initially the displaced blob has had no time to exchange heat with its sur-roundings so that its density changes adiabatically at constant entropy s Wecan then write

ρprime minus ρ = δρs =(part log ρpart logP

)

s

ρ

P

dPdz

δz (919)

The adiabatic change in density with pressure can be found from the equationof state and is written as

1Γ1

=(part log ρpart logP

)

s

(920)

From the structure of the star we also have

dρdz

δz =(

d log ρd logP

)ρ

P

dPdz

δz (921)

and we define Γ by1Γ

=(

d log ρd logP

)ρ

P (922)

the density exponent with respect to pressure in the surrounding materialBecause P must always fall as z increases in order to maintain hydrostaticequilibrium dPdz lt 0 always and so the fluid is unstable to convection if

1Γlt

1Γ1

(923)

the Schwarzschild criterionBy considering the ideal gas equation of state we can see that Γ is large

when the temperature gradient in the star is large Thus just as in a boilingkettle convection is driven when there is a strong heat source that would drivea very large temperature gradient Convection is also induced by a small valueof Γ1 This occurs in ionization regions where the number of particles and sothe pressure increases over a small temperature range

In unstable regions efficient turbulent mixing of the fluid takes place andthis leads to an adiabatically stratified region of constant entropy

Γ asymp Γ1 (924)

So in convective regions we write the temperature gradient as

dTdr

= nablaaT

P

dPdr

+ ΔnablaT (925)

where ΔnablaT is the superadiabatic gradient It is one of the least certain fea-tures of stellar evolution but is usually calculated by mixing length theory(Bohm-Vitense 1958) Throughout most of a convective region it is small andnot important but at the outer edge of the solar convection zone it becomesrelatively large and determines the adiabat on which the whole convective zonelies It can be calibrated by ensuring that the radius of a model of the Sunfits the measured radius but there is no guarantee that the same calibrationor even the same theory can be applied to other stars

There are further complications that have yet to be fully satisfactorilyaddressed Convective overshooting might occur at Schwarzschild boundariesbecause although the acceleration of a blob goes to zero at the edge of aconvective region its velocity does not However the deceleration of a blobthat crosses a boundary is generally extremely fast and any overshooting quitenegligible Even so the concept is still popular because there is much evidencefor composition mixing in radiative regions that does not have an establishedcause Semiconvection occurs when there is a composition gradient Convec-tion may be stable according to the Schwarzschild criterion if no materialis mixed across the boundary but unstable if it is There is an equilibriumwhen just enough material mixes to maintain stability What is uncertain isthe timescale on which this equilibrium is attained Varying it significantlychanges some evolutionary phases and in particular the size of the burnt coreat the end of helium burning (Dewi Stancliffe amp Tout private communica-tion)

96 Energy Generation

The luminosity of a star is created by various sources of energy The changein luminosity from radius r to r+ δr is the total energy generated by materialin the shell of mass δm between the two radii (Fig 98) Thus for an energygeneration rate per unit mass of ε

dLr

dr= 4πr2ρε (926)

This is a simple equation but a great deal of complexity is hidden within therate ε which depends on the state of the fluid particularly its temperatureand its composition

There are three major contributions First as a star contracts the fluidreleases gravitational energy This is the dominant source of luminosity duringstar formation when a gas cloud collapses to form the star and before its coreis hot enough to ignite hydrogen fusion It is occasionally important later in

274 C A Tout

r+δ rr

Lr

Lr+ r = Lr + Lδ rδ

2rδL 4 rε δρπ r=

Fig 98 Luminosity variation The local luminosity Lr of a star is the energyflux outwards through the sphere of radius r within the star Luminosity increasesbetween r and r+δr when there is energy generation in the shell of mass δm betweenthese spheres

the evolution too when contraction can release energy at a comparable rateto nuclear burning For an ideal gas the contribution is

εgrav = minusCVTpart

partt

(

loge

P

ργ

)

(927)

where γ = CPCV is the ratio of the specific heat at constant pressure CP

to the specific heat at constant volume CV This term is negative when thestar is expanding but it generally does not dominate nuclear energy sourcesIt also introduces stellar evolution via the time derivative

Secondly energy is generated by nuclear reactions and the discussion ofthese will compose the major part of this section Thirdly at very high tem-peratures and densities neutrino loss processes become important Reversibleweak reactions release two energetic neutrinos both of which escape from thestar because the matter cross-section to neutrinos is very small Their meanfree path is much greater than the radius of the star The contribution εν isalways negative

961 Nuclear Burning

One 4He nucleus is less massive than four protons and two electrons This isbecause the magnitude of the binding energy per nucleon is larger in helium-4It is more stable In general the binding energy of a nucleus

EB = (Zmp + [Aminus Z]mn minusmnuc)c2 (928)

Fig 99 Binding energy per nucleon for nuclides of atomic mass A The moststable isotope is plotted for each atomic number Up to the iron group elementsaround 56Fe the binding energy per nucleon increases and energy is usually releasedin nuclear reactions that create heavier stable nuclei For higher mass nuclei theenergy per nucleon decreases with A Energy is required to create these nuclei fromless massive ones

for a nucleus of mass mnuc containing Z protons of mass mp and A minus Zneutrons of mass mn This is zero for a hydrogen nucleus which is just asingle proton Z = A = 1 Figure 99 shows the binding energy per nucleonEBA as a function of atomic number A This average binding energy tends torise up to iron-56 and then falls again There are notable peaks of stability athelium-4 carbon-12 and oxygen-16 When any of these are formed from lessstable nuclei the binding energy is released As A increases beyond 56 thebinding energy per nucleon falls again so that it is not energetically favourableto fuse lower-mass isotopes to form higher-mass ones

962 Hydrogen Burning

The energy released when converting four protons to one helium-4 nucleus is2673MeV However the actual energy available to the star depends on thereaction pathway Energy is released in three forms high-energy gamma rayskinetic energy of the reacting particles and neutrinos The first two forms arethermalized locally but once again the neutrinos can escape from the starand carry off their energy At relatively low temperatures as in the Sun thereaction proceeds via the protonndashproton chain The first and slowest reaction

276 C A Tout

is the combination of two protons to form a deuterium nucleus

1H + 1H rarr 2H + e+ + ν (929)

The neutrino escapes with an energy of 026MeV while the positron annihi-lates with an electron

e+ + eminus rarr γ (930)

to leave an energetic gamma ray Another proton can then react with thedeuterium nucleus

1H + 2H rarr 3He + γ (931)

and two of these 3He nuclei can then combine

3He + 3He rarr 4He + 2 1H + γ (932)

The actual energy released to the stellar material is 2620MeV because twoneutrinos are lost for each 4He nucleus created This is the ppI chain At highertemperatures the ppII and ppIII chains which involve lithium beryllium andboron also operate but each of these loses more energy in neutrinos

Above a temperature of 2 times 107 K hydrogen burns faster via a catalyticcycle the CNO cycle

12C(p γ)13N( e+ν)13C(p γ)14N(p γ) (933)

15O( e+ν)15N(p α)12C (934)

with a rare branch when 15N captures a proton before it decays

15N(p γ)16O(p γ)17F( e+ν)17O(p α)14N (935)

The component of the cycle 12C(p γ)13N represents

12C + 1H rarr 13N + γ (936)

etc The neutrino losses are greater than those in the ppI chain so that thetotal energy available per 4He nucleus created is reduced to 238MeV The coretemperature of main-sequence stars increases with their mass and the CNOcycle begins to dominate at about 15M Hydrogen burns faster but lessefficiently because of the greater neutrino losses

963 Reaction Rates

Quite a complicated mixture of theory and experiment is required to estimatereaction rates and details may be found in Clayton (1968) Charged-particlereactions can only occur at all because the most energetic nuclei in the tail ofthe Maxwellian distribution are able to quantum-mechanically tunnel throughthe Coulomb barrier Once they reach the nucleus the bound states tend to

be of much lower energy and they face being reflected unless they can entera similar energy resonant state All these lead to very strong temperaturedependences for nuclear reactions The energy generation rate of the pp chainat 107 K

εpp prop ρT 46 (937)

and for the CNO cycle at 2 times 107 K

εCNO prop ρT 14 (938)

In most cases these temperature dependences lead to thermostatic control ofthe reactions If energy production were to rise the star would expand inresponse and the temperature would fall As a result hydrogen burning takesplace at a temperature much too low for helium burning which in turn takesplace at a temperature much too low for carbon burning so that stars use upone fuel at a particular radius at a time before igniting the next

As mentioned before as nuclear reactions change the composition of thematerial the star evolves because the equation of state is changed The opac-ities and the energy generation rates which depend on the state also changeOnce a star has begun nuclear burning it is these composition changes thatdrive evolution

964 Helium Burning

Above 108 K with hydrogen long gone helium can fuse to carbon First two4He nuclei react and form the unstable 8Belowast

4He + 4He 8Belowast (939)

This is a resonant state but unlike the deuterium nucleus formed in the ppchain there is no stable state of 8Be to which it can decay Indeed there isno stable nucleus of atomic mass 8 at all The 8Belowast nucleus has no choice butto split up into two 4He nuclei again with a half life of 3 times 10minus16 s Thoughshort this is long enough for a third α-particle to collide if the temperature ishigh enough Interestingly there is a resonant state of 12C not very differentfrom that of the colliding nuclei This reaction too is reversible but now thereis a stable state into which the 12Clowast nucleus can decay by the emission of twophotons to conserve spin and complete the process

8Belowast + 4He 12Clowast rarr 12C + γ + γ (940)

The first two reactions are endothermic Formation of an 8Belowast nucleusrequires 0092MeV and formation of the 12Clowast requires a further 0285MeVBut when this decays to the stable 12C the photons extract 765MeV Thetotal energy liberated by the whole process is therefore 727MeV 0606MeVper nucleon or about one tenth of that released during hydrogen burning Theenergy generation rate

278 C A Tout

ε3α prop ρ2T 40 (941)

This is perhaps the most extreme sensitivity to temperature found in natureand in the Sun it will lead to a thermonuclear runaway when it ignites in thedegenerate helium ash in the core

At temperatures required to run this triple-α reaction it is easy to addanother helium nucleus

12C + 4He rarr 16O + γ (942)

and in many cases helium burning produces more oxygen than carbon

965 Later Burning Stages

Hydrogen and helium burning account for most of the energy production in astarrsquos life but stars more massive than about 8M can go on to ignite carbonat T asymp 5 times 108 K

12C + 12C rarr

⎧⎪⎨

⎪⎩

20Ne + 4He23Na + 1H23Mg + n rare

(943)

The next major phase is neon burning by photodisintegration Temperaturesof about 109 K are sufficient to provide energetic photons capable of ejecting anα-particle from a neon nucleus At these temperatures the α-particle can read-ily combine with another neon nucleus and produce more stable magnesium

γ + 20Ne 16O + 4He (944)20Ne + 4He rarr 24Mg + γ (945)

At 2 times 109 K oxygen can burn to produce a variety of products includingsilicon

16O + 16O rarr 28Si + 4He + γ (946)

then at 3 times 109 K photons are energetic enough to break up the silicon

γ + 28Si 24Mg + 4He (947)

This is followed by a series of α captures and photodisintegrations that culmi-nate in the iron group elements The actual combination of isotopes dependson the nuclear statistical equilibrium which is controlled by the number ofprotons and neutrons present When numbers are about equal the dominantproduct is 56Ni which is the power source of most supernovae as it decays to56Fe via 56Co

97 Boundary Conditions

We now have the set of four equations of stellar structure together with thetime dependence that drives stellar evolution We discussed boundary con-ditions in Sect 92 We want the surface of a star to be what we see whenwe look at it This is the surface from which the photons that reach us areemitted Photons escape freely when the optical depth

τ =int infin

r

κρdr asymp 1 (948)

More carefully we can use a thin grey atmosphere with the Eddington Closureapproximation (Woolley amp Stibbs 1953) Then at τ = 23

Lr = 4πR2σT 4 (949)

and with hydrostatic equilibrium

P asymp 23g

κ (950)

With yet more sophistication we can make a full model of the radiative trans-fer in the atmosphere and fit it to the stellar interior Unfortunately the so-lution to this is sufficiently complex to consume as much time as a full stellarevolution sequence and so tends not to be used unless absolutely necessary

98 Evolutionary Tracks

Figure 910 shows the path followed in the HR diagram for stars of 1 5and 32M as they evolve from the zero-age main sequence when no hydrogenhas yet been converted to helium They have been evolved with the CambridgeSTARS code that is described in more detail in Chap 13 There details of howto obtain and run the program can be found so that the reader can reproducethis and similar diagrams On the ZAMS our 5M star has a radius of 265Rand a luminosity of 540L It is burning hydrogen to helium via the CNOcycle in its core Because of the relatively strong temperature dependence ofthe CNO reactions the burning mostly occurs right at the centre but thetemperature gradient drives convection out to 12M and the whole of thiscore is burnt The core shrinks in both mass and radius as burning proceedsso that only the inner 053M is completely converted to helium Just beforethis after 824 times 107 yr when the starrsquos luminosity has reached 900L andits radius grown to 535R the fraction of hydrogen at the centre by masshas dropped to 005 At this point it is more energetically favourable for thewhole star to contract This is the hook in the HR diagram at the end ofthe main sequence Shortly afterwards (23 times 106 yr later) central hydrogenis exhausted completely and burning moves to a shell surrounding the core

280 C A Tout

T K

LL

Fig 910 Model tracks in the HertzsprungndashRussell diagram from Pols et al (1995)

After another 39 times 106 yr this core has grown so large (to about 06M)that it can no longer support itself with gas pressure It starts to contractgradually forcing the nuclei and electrons together but the core does not getvery degenerate at this stage It does however rapidly contract and the starmoves over to the giant branch in the relatively short time of 84 times 105 yrAs the core contracts the envelope expands Though no one has yet explainedsimply why it expands we do appear to include all the relevant physics becauseour models expand A star is complex and behaves in very non-linear waysso it is often not easy to predict what will happen or even to explain whyit has An important result of the expansion is that the surface temperaturedescends and convection sets in reaching right down to parts of the stellar corethat have previously been processed Once established on the giant branchthe helium core grows as hydrogen burns outwards It contracts in radius asit does so and heats up This raises the temperature at the burning shell sothat the reactions run faster and the luminosity rises The star makes its firstascent of the giant branch

The core growing in mass but contracting in radius continues to heat upuntil at 12times108 K it is hot enough for helium to ignite Once again the heliumburning drives convection in the core which this time grows as the burning

proceeds Eventually helium fuel is exhausted in the core too and heliumburning moves to a shell that starts to follow the hydrogen-burning shell outthrough the envelope During core helium burning our star had settled backto a lower luminosity shrunk and lost its deep convective envelope It nowmoves back over to the giant branch but only slowly resumes the same risingtrack so we call this AGB

At this point we should note that the production of elements in stars isnot on its own enough to ensure their availability when a new generation ofstars and planets condense The processed material must actually be somehowdriven off into the interstellar medium at a velocity that exceeds the escapevelocity of the star Indeed stars leave behind remnants that might be whitedwarfs neutron stars or black holes depending on mass and these remnantsswallow a substantial part of the processed core in the most common stars

In comparison two significant differences characterise the evolution of a1M star First the central temperature on the main sequence is lower sothat hydrogen burning proceeds via the pp chain rather than the CNO cycleThen the lower core temperature on the giant branch means that the corebecomes very degenerate before it reaches the temperature at which heliumcan ignite Because the degenerate equation of state does not respond to therising temperature as the reaction generates energy it is not thermostaticallycontrolled in the normal way This is coupled with the incredible temperaturesensitivity so that a thermonuclear runaway ensues during which the energyproduction reaches the luminosity of a small galaxy But it lasts only a fewseconds before the degeneracy is raised and the star drops rapidly down thegiant branch to begin stable core helium burning The energy produced isabsorbed by the starrsquos envelope and is hardly noticed at its surface Fromthen it evolves much like the 5M Once high on the AGB it is mass lossthat controls the evolution of these stars A very strong dusty wind eventuallyremoves all the hydrogen envelope and exposes the burning shells These cooland extinguish leaving a white dwarf that rapidly falls in luminosity to belowthe main sequence and then cools from left to right across the diagram

The 32M star on the other hand goes on to ignite carbon in its corewhich is processed all the way to iron When the iron core reaches theChandrasekhar mass of 144M the maximum that can be supported byelectron degeneracy it collapses to a tiny neutron star The energy releasedblows the entire envelope off in a spectacularly bright supernova

99 Stellar Evolution of Many Bodies

In Chap 10 Jarrod Hurley describes how single-star stellar evolution can beincorporated in N -body calculations It is important to know how the massesof the stars change both by mass loss in stellar winds and any sudden massloss in a supernova because this affects the dynamics of the cluster As thebodies interact dynamics can also influence the stellar evolution This is most

282 C A Tout

apparent when stars are in or form close binary systems These form the topicof Chap11

References

Alexander D R Ferguson J W 1994 ApJ 437 879Bohm-Vitense E 1958 Z Astrophys 46 108 273Chandrasekhar S 1939 An Introduction to the Study of Stellar Structure Chicago

Univ Press Chicago 265Church R P Tout C A Aarseth S J 2007 private communication 261Clayton D D 1968 Principles of Stellar Evolution and Nucleosynthesis Chicago

Univ Press Chicago 276Herschel W 1783 Phil Trans R Soc 73 247 261Hertzsprung E 1905 Z Wissenschaftliche Photographie 3 422 262Hurley J R Tout C A Pols O R 2000 MNRAS 315 543 261Iglesias C A Rogers F J Wilson B G 1992 ApJ 397 717Pancino E Ferraro F R Bellazzini M Piotto G Zoccali M 2000 ApJ 534 L83 264Pols O R Tout C A Eggleton P P Han Z 1995 MNRAS 274 964 280Russell H N 1913 Obs 36 324 262Woolley R v d R Stibbs D W N 1953 The Outer Layers of a Star Clarendon

Press Oxford 279

10

N -Body Stellar Evolution

Jarrod R Hurley

Centre for Astrophysics and Supercomputing Swinburne University of TechnologyPO Box 218 VIC 3122 Australiajhurleyswineduau

101 Motivation

The advent of the Hubble Space Telescope (HST) with its ability to peer deepinside the globular clusters (GCs) of our Galaxy and resolve individual stars(Paresce et al 1991) provided reason enough to include stellar evolution incluster models We only have to look at the beautiful images of stars in the coreof say Omega Centauri1 (Carson Cool amp Grindlay 2000) to be motivated toproduce colour-magnitude diagrams (CMDs) from simulations to match thoseemanating from HST There are also a number of questions relating to stellarpopulations in star clusters that require a combination of stellar evolutionand stellar dynamics for investigation For example population gradients areobserved which indicate a central concentration of blue stragglers (BSs) aswell as a central depletion of red giants (Yanny et al 1994) A possible expla-nation is that close encounters between stars in the dense core of a GC leadsto enhanced production of BSs in collisions (or mergers) of main-sequencestars Encounters are also then expected to enhance the stripping of the en-velopes of giant stars to produce blue horizontal branch stars or white dwarfs(WDs) The situation is not straightforward though as evidenced by the clas-sic second-parameter pair of GCs M3 and M13 (Ferraro et al 1997) Herewe have two clusters of the same mass density metallicity and (apparently)age but with dramatic differences in their blue straggler and blue horizon-tal branch star populations Also HST is not alone in exposing the cores ofstar clusters ndash the Chandra X-ray Telescope has provided a wealth of comple-mentary information on objects such as millisecond pulsars and cataclysmicvariables (Grindlay et al 2001ab)

Aside from a desire to produce models to match observations of stellarpopulations in star clusters there is a more basic need for stellar evolution inN -body models Here we are talking specifically about mass loss from starsas they evolve This can have a dramatic effect on the lifetime and structure

1httphubblesiteorgnewscenterarchivereleases200133imagea

Hurley JR N-Body Stellar Evolution Lect Notes Phys 760 283ndash296 (2008)

284 J R Hurley

of a star cluster Put simply mass lost from stars in stellar winds is expectedto escape from a cluster and therefore weakens its potential The cluster thenexpands which leads to a temporary increase in the loss of stars across thetidal boundary This weakening of the potential leaves the cluster more ex-posed to the possibility of disruption if for example the cluster encountersa giant molecular cloud or orbits through the Galactic disc In the long-termstellar evolution mass loss affects the timescale for two-body relaxation andcore-collapse (the reader is directed to Meylan amp Heggie 1997 for an overviewof the processes involved in cluster evolution) Thus stellar and cluster evolu-tion are intertwined and an accurate description of the former in concert withthe dynamics is required

102 Method and Early Approaches

To meet the needs described above there are a minimum set of variablesthat a stellar evolution algorithm must be able to provide within the N -bodycode In order to detect and enact collisions between stars the stellar radius isrequired for each star in the model To produce CMDs requires the luminosityand effective temperature (or radius) of each star The mass of each star isrequired and information on the mass and radius of the core is important fordetermining the nature of stellar remnants as well as the outcomes of collisions(and the inclusion of binary evolution) Therefore the algorithm must beable to account for the mass size and appearance of the N -body stars as thecluster evolves Ideally it should be able to do this with metallicity as a freeparameter The open clusters of the Galaxy typically contain stars of close tosolar metallicity while the GCs are metal-poor (see Meylan amp Heggie 1997)and in comparison the star clusters of the Large Magellanic Cloud exhibita wide range of metallicity (Mackey amp Gilmore 2003) This is an importantdistinction to make because the evolution timescale and appearance of a stardepends critically on its composition as well as its mass

When deciding on an appropriate stellar evolution method there are threeapproaches from which to choose (i) a detailed evolution code (ii) look-uptables and (iii) fitted functions An overriding concern is that the stellar evo-lution algorithm should not impede the progress of the N -body calculationsthe algorithm must be robust and provide rapid updating of the necessaryvariables for all possible stages of evolution The robustness requirement hasalways been a stumbling block for using a detailed evolution code to providestellar evolution because the codes are liable to break down at critical stagesin the evolution However steps have recently been taken to overcome thisshortcoming and live stellar evolution in N -body simulations is now an excit-ing possibility Computational constraints make this method more relevant tothe large-N regime At present look-up tables constructed from the output ofa series of detailed evolution calculations represent a more reliable approach

10 N -Body Stellar Evolution 285

These require interpolation and the associated data files can be very largeif a fine grid in mass is used to ensure accuracy especially when a range ofmetallicities is also considered This would not be of much concern today butit was in the early to mid-1990s when including stellar evolution in dynam-ics codes was under serious consideration and computing memory was at apremium As a result the third approach ndash a set of functions approximatingthe detailed dataset ndash has proven to be the most popular to date This isthe most time-consuming approach to set up but the reward is a relativelycompact algorithm that lends itself well to the requirements of an N -bodycode

One drawback of the fitted function approach is that much of the informa-tion provided by a detailed stellar evolution code is discarded and not availableto the dynamics code This could be important for example in the case ofstellar collisions where the outcome of the collision and nature of the collisionproduct depends on the internal density profiles of the colliding stars This iscircumvented somewhat by also predetermining the collision outcomes basedon prior calculations (see Hurley Tout amp Pols 2002) Another potential prob-lem with this approach to stellar evolution is that if the detailed models onwhich the functions are based become outdated for any reason it is non-trivialto generate a new set of functions Nevertheless the fitted function approachis the method of choice in the codes nbody4 and nbody6 and has provensuccessful to date

An early approach to combining stellar and dynamical evolution was pro-vided by the Fokker-Planck models of Chernoff amp Weinberg (1990) This wasa two-step method based on an expression for the main-sequence lifetime ofa star as a function of stellar mass and a WD initial-final mass relation ieat the end of the main sequence a star would lose mass instantaneously andbecome a white dwarf Even earlier attempts had employed simple schemes todescribe mass loss in supernovae events (eg Wielen 1968) see Aarseth (2003)for an overview In their population synthesis work Eggleton Fitchett amp Tout(1989) provided a more sophisticated algorithm that described the luminosityradius and core mass of the stars for a range of evolution phases This treat-ment was included in nbody4 in 1994 and is still adopted by other dynamicscodes Improvements were made to this algorithm by Tout et al (1997) specif-ically for use in nbody4 The next major development in the fitted functionapproach came with the creation of the Single Star Evolution (SSE) packageby Hurley Pols amp Tout (2000) This was based on an updated set of de-tailed stellar models that included convective overshooting and for the firsttime metallicity was a free parameter ndash all previous algorithms were solarmetallicity only It also included an expanded range of evolution phases amore detailed description of the evolution within each phase and an updatedmass-loss algorithm SSE currently provides stellar evolution in nbody4 andnbody6 and is outlined below A general introduction to stellar evolutiontheory has been presented in Chap 9

286 J R Hurley

103 The SSE Package

The goal here is to provide an overview of the method used to construct theSSE package and to discuss some aspects relevant to inclusion in an N -bodycode A full description of the SSE package is given in Hurley Pols amp Tout(2000)

The basic idea of the algorithm is to break the evolution of a star intoa series of evolution phases These are listed in Table 101 Each phase hasan associated index kstar which identifies the stellar type2 The phases fallinto three groupings normal nuclear burning evolution kstar isin [1 6] nakedhelium star evolution kstar isin [7 9] and remnant evolution kstar isin [10 14]

All stars are assumed to be born on the zero-age main sequence (ZAMS)where core hydrogen burning is initiated Stars then move through a seriesof phases as they evolve although a particular star may not experience allphases For example a 1M star stays on the main sequence (kstar= 1)for about 11Gyr before quickly passing through the Hertzsprung gap phase(kstar= 2) as hydrogen burning commences in a shell surrounding the he-lium core It then ascends the giant branch (kstar= 3) until helium is igniteddegenerately in the core and the core helium flash brings the star to the core-helium burning or horizontal branch phase (kstar= 4) This is as far asa 1M star would get within the age of the Galaxy If for some reason thestar was stripped of its envelope while on the giant branch as a result ofa collision or close binary evolution it would become a helium white dwarf(kstar= 10) Otherwise given enough time it would eventually evolve to be-come a WD comprised primarily of carbon and oxygen (kstar= 11) A 5M

Table 101 Evolution phases identified in SSE and the assigned kstar index

kstar Evolution phase kstar Evolution phase

1 main sequence 10 helium white dwarf2 Hertzsprung gap 11 carbon oxygen white dwarf3 first giant branch 12 oxygen neon white dwarf4 core helium burning 13 neutron star5 early asymptotic giant branch 14 black hole6 thermally pulsing AGB7 helium main sequence8 helium Hertzsprung gap9 helium giant branch

2There is an additional phase (kstar= 0) not listed which is used to denotelow-mass main-sequence stars with mass less than 07 M This is carried over fromTout et al (1997) and distinguishes stars with deeply or fully convective envelopeswhich respond differently to mass changes during binary evolution (see Chaps 11and 12)

star evolves through phases 1 rarr 6 before becoming a 1M carbon oxygenwhite dwarf (kstar= 11) This takes about 100Myr The asymptotic giantbranch (AGB) is divided into two separate phases by the the onset of seconddredge-up or more generally the time at which the growing carbon oxygencore reaches the helium core in mass On the early AGB (kstar= 5) lumi-nosity is dominated by a helium-burning shell At the onset of the thermallypulsing AGB (kstar= 6) a hydrogen shell source is ignited and subsequentlyprovides the bulk of the luminosity Thermal pulses that reduce the growth ofthe core mass are modelled during this phase Stars of approximately 8Mignite carbon on the AGB and evolve to become oxygen neon white dwarfs(kstar= 12) More massive stars (10ndash25M) evolve to become neutron stars(kstar= 13) and even more massive stars become black holes (kstar= 14)A 20M star for example evolves through phases 1 rarr 2 rarr 4 rarr 5 rarr 13in approximately 10Myr In this case central helium burning is ignited dur-ing phase 2 so that phase 3 is skipped Furthermore a 25M star sheds itsenvelope during phase 4 and thus becomes a naked helium main-sequencestar (kstar= 7) rather than reach the AGB It then evolves onto the heliumHertzsprung gap (kstar= 8) and giant branch (kstar= 9) before becominga black hole Transitions from 12 rarr 13 and 13 rarr 14 are also possible throughmass accretion in a close binary (see Chap 12 and Hurley Tout amp Pols 2002for details) Note that the quoted evolution times and landmark masses arefor solar metallicity and vary for different composition

The SSE package comprises a set of analytical evolution functions thatprovide quantities such as the luminosity radius and core mass for a starwhich evolves through the phases mentioned above Input variables are themass M metallicity Z and age of the star The method used in constructingSSE was to first find functions to fit the end-points of the various evolutionaryphases as well as the timescales Then the behaviour within each phase wasfitted A starting point was the set of formulae provided by Tout et al (1996)to describe the ZAMS luminosity and radius as a function of M and Z Thiswas then extended to fit aspects of the evolution such as the luminosityand radius at the end of the main sequence with rational polynomials thatare continuous and differentiable where possible For example the formula todescribe the time taken for a star to evolve from the ZAMS to the base of thegiant branch is

tBGB =a1 + a2M

4 + a3M55 +M7

a4M2 + a5M7 (101)

where the coefficients an are functions of Z Data to create the functions forthe standard nuclear burning phases was taken from the detailed models ofPols et al (1998) The models cover a range in mass from 01 to 50M and arange of metallicity from 00001 to 003 with Z 002 being solar The result-ing functions are accurate to within 5 of the detailed stellar models over allphases of the evolution The errors introduced by this approach are less thanthe intrinsic errors of the detailed models themselves owing to uncertainties

288 J R Hurley

in the input physics Note that the functions can be safely extrapolated up to100M but for greater mass SSE evolves the star using timescales and quan-tities for a 100M star Extrapolation outside of the Z range of the modelsis not recommended

The functions for the naked helium star phases were fitted to models pro-duced by Onno Pols (see Dewi et al 2002 for some details) The luminosityevolution of white dwarfs in SSE was initially modelled according to standardcooling theory but has subsequently been expanded to reflect better currentwhite dwarf models (see Hurley amp Shara 2003 for details) Radii for whitedwarfs come from Eq (17) of Tout et al (1997) and mass-dependent lumi-nosities and radii are also assigned to neutron stars and black holes (see Hurleyet al 2000) Another change to SSE subsequent to Hurley Pols amp Tout (2000)is the adoption of the prescription suggested by Belczynski Kalogera amp Bulik(2002) for calculating the masses of neutron stars and black holes Relatedto this the default maximum mass for a neutron star is now assumed to be30M rather than 18M as suggested in Hurley Pols amp Tout (2000) ndash thisis an adjustable input parameter

The models of Pols et al (1998) neglect mass loss from the surface of astar owing to a stellar wind However the SSE package supplements thesemodels by including a prescription for mass loss in a simple subroutine formthat can easily be altered or added to This prescription is drawn from arange of current mass-loss theories available in the literature It is applicableto all nuclear burning evolution phases (kstar isin [1 9]) and includes standardReimersrsquo mass loss (Kudritzki amp Reimers 1978) for giants pulsation-drivenwinds for AGB stars and a Wolf-Rayet like mass loss for helium stars Thereader is referred to Sect 7 of Hurley Pols amp Tout (2000) for full detailsTo achieve a smooth transition from the Pols et al (1998) models (withoutmass loss) to the beginning of remnant evolution SSE employs perturbationfunctions that alter the radius and luminosity of a star as the envelope be-comes small in mass SSE also follows the spin evolution of a star and includesmagnetic braking

The SSE package can be obtained by contacting the author or fromhttpastronomyswineduaujhurleybsedloadhtml (where the asso-ciated binary evolution package is also available) It provides a rapid and reli-able method for evolving stars and is therefore well suited for use in populationsynthesis and dynamics codes The bulk of the SSE functions are contained ina subroutine called zfuncsf and before any of these are used the subroutinezcnstsf must be called to set all the Z-dependent coefficients (this in turnrequires the zdatah data file) The routine hrdiagf determines which evo-lution stage a star is currently at and calculates the appropriate propertiessuch as luminosity radius and core mass It must be preceded by a call tostarf which sets the timescales for the evolution phases (as a function ofM and Z) as well as various landmark luminosities Other associated routinesare mlwindf which calculates the current mass-loss rate mrenvf which

Table 102 Subroutines in nbody4 and nbody6 associated with stellar evolution

SSE routines Related routines

hrdiagf fcorrf ( larr mdot )magbrkf hrplotflowast ( larr output )mlwindf instarflowast ( larr start )mrenvf kickf ( larr fcorr )starf mdotflowast ( larr intgrt )zcnstsf mixflowast ( larr cmbody )zdatah trdotflowast ( larr instarmdot )zfuncsf

corerdf cmbodyf

gntagef dataf

mturnf

routines marked with lowast call hrdiag directly

calculates the mass and radius of the convective envelope (if one exists) andmagbrkf which determines the rate of angular momentum change owingto magnetic braking These are the main SSE routines They are listed inthe left-hand column of Table 102 along with some further routines that arementioned in the next section

104 N -Body Implementation

The core SSE routines as described in the previous section are included inthe N -body codes in their entirety That is to say they operate indepen-dently of the structure of the N -body codes ndash if any of these routines areupdated in the SSE package they can simply be copied into nbody4 andnbody6 without any further concern This also means that a routine such ashrdiagf could be swapped for any other routine that sets the stellar param-eters provided that the current interface or subroutine arguments are thesame The SSE subroutines that are involved in the N -body codes are shownin Table 102 Also shown are all nbody46 subroutines that either interactwith these routines directly or are associated with the stellar evolution pro-cedure in some way Note that the subroutines that call hrdiagf have beenhighlighted and it was also considered instructive to identify from where innbody46 these routines were called (as shown in the parentheses on the farright)

Within Table 102 there exist some grey areas For example trdotf is ac-tually a SSE routine that calculates the appropriate stellar evolution timestepfor a star based on its type and the restriction that the radius should notchange by more than 10 in a single timestep This is listed in the right-hand

290 J R Hurley

column of Table 102 as an N -body routine because it contains additional linesof code specific to nbody46 The same goes for kickf which is a SSE rou-tine that sets the velocity kick for newly born neutron stars and black holesSome subsidiary SSE routines are utilised by nbody46 and these are alsolisted in Table 102 (on the left-hand side below the dividing line) The rou-tine corerdf contains a function to calculate the core radius of a star and isrendered somewhat obsolete by the combination of hrdiagf and zfuncsfHowever it is still used in nbody46 for convenience The routine mturnfprovides an estimate of the turn-off mass of a star cluster the most massivestar that currently resides on the main sequence based on the current timeand the SSE function that calculates the main-sequence lifetimes of the starsIt is not a routine that is essential to the evolution algorithm On the otherhand the SSE routine gntagef is an essential component of a stellar evolu-tiondynamics interface but its use is more relevant in a discussion of binaryevolution Given a stellar type current mass and core mass of a star thisroutine calculates an appropriate age and initial mass Thus it is essentiallyan inverse of hrdiagf and is used to set the parameters of stars produced inmergers and collisions

Before proceeding to give an overview of the nbody46 stellar evolutionalgorithm it is first pertinent to describe the associated stellar variables Eachstar has an initial mass body0 a current mass body a radius radius a lu-minosity zlmsty spin angular momentum spin and a stellar type kstarThese are all common arrays of size NMAX where NMAX is set in paramsh andmust be greater than N to accommodate binaries A star of index i has quan-tities saved at the ith position of these arrays eg body0(i) Other quan-tities such as the core mass are not stored and are obtained from hrdiagfas required The need to keep track of both the current and initial massesis driven by the stellar evolution algorithm In both SSE and its predeces-sor (Tout et al 1997) it was recognised that the evolution timescales andlandmark luminosities depend on the initial mass whereas the stellar radiusis more correctly a function of the current mass Note that both body0 andbody are in dimensionless N -body units and the scale-factor ZMBAR (or equiv-alently SMU) is used to convert to solar masses Similarly radius is convertedto solar radii using SU and spin is converted to units of MR2

yrminus1 usingSPNFAC

To allow stars to have different update frequencies each star has an associ-ated stellar evolution update time specified by the tev array This recognisesthat massive stars and the stars in advanced evolution stages such as onthe AGB require more frequent updates than say low-mass main-sequencestars or white dwarfs Thus it would not be computationally efficient to havethe update frequency of all stars dictated by the most rapidly evolving star atthe time A second update variable tev0(i) is also utilised This denotes thetime at which star i was last updated as opposed to tev(i) which representsthe next required update time and the two are used to compute the amountof mass lost between updates Also associated with the time-keeping for each

star is a quantity called epoch This is a product of the SSE package and isused to calculate the effective stellar evolution age of a star ie if tphys isthe current physical time in Myr the stellar evolution age of star i is tphys ndashepoch(i) To illustrate the need for such a variable consider a star that hasjust lost its envelope on the AGB and evolved to become a white dwarf Theluminosity evolution of a white dwarf is calculated from a cooling law thatis a function of the time elapsed since the birth of the white dwarf So theevolution algorithm needs to know when the white dwarf was born This iscommunicated by setting epoch (i) = tphys when the star leaves the AGBThe epoch variable is also used to reset the stellar evolution clock of starsthat lose (or gain) mass during certain phases of evolution (see Hurley Polsamp Tout 2000 Hurley Tout amp Pols 2002 for more details on the use of epoch)Note that the units of epoch are Myr whereas tev and tev0 are in N -bodyunits and the scale-factor TSTAR is required to convert to N -body times tophysical units of Myr

The next step is to be aware of N -body input variables that are relevant tostellar evolution These are read by the routine dataf and are the maximumstellar mass body1 the minimum stellar mass bodyn the metallicity zmetan offset parameter for the stellar evolution time epoch0 and the time in-terval between writing stellar evolutionndashrelated output dtplot Also relatedare the input options kz(19) and kz(20) (actually read in inputf) Settingkz(19) = 3 is necessary to activate stellar evolution according to SSE If this isindicated dataf calls zcnstsf with zmet to set the metallicity dependentcoefficients This only needs to be done once as it is assumed that all starsare of the same composition However if a restart is required then zcnstsfis called once more but from the main routine (nbody4f or nbody6f) Thevalue of kz(20) affects the choice of initial mass function Options includethe distribution of masses derived by Kroupa Tout amp Gilmore (1993) fromstars in the solar neighborhood (kz(20) = 5) and a power-law mass func-tion (kz(20) = 0) If the latter is indicated the exponent alpha is also re-quired from the input file The stellar masses ie body(i) for i = 1 N are required to lie between the bounds of bodyn and body1 and are set indataf according to kz(20) ndash it is also possible to read these from a file usingkz(22)

After reading the input file and generating the stellar masses the N -bodystellar evolution algorithm starts by initialising the stellar variables for eachof the N stars The routine instarf is responsible for this process For eachstar i it sets body0(i) = body(i) kstar(i) = 1 or 0 and epoch(i) = 00before calling the starf and hrdiagf combination to set radius(i) andzlmsty(i) The spin angular momentum spin(i) is also set using the SSEpackage (see Hurley Pols amp Tout 2000) For the stellar evolution update timesthe routine sets tev0(i) = 00 and tev(i) are initialised by a call to trdotffor each star Note that it is possible to start the stars at an advanced evolutionstage by setting the input parameter epoch0 to some negative value (see theusage of epoch above) In this case epoch(i) = epoch0

292 J R Hurley

Subsequent to initialisation stellar evolution is controlled by the mdotfsubroutine Frequent updates are performed in step with the dynamical in-tegration by means of a variable TMDOT the minimum of tev(i) for alli = 1 N At the end of each integration step (in intgrtf) a check is madeto determine if the new time exceeds TMDOT If it does then mdotf is calledin order to update each star that has tev(i) less than the current time (morethan one star may be due) Within mdotf the stellar variables for star i areupdated to an age of tev(i) lowast TSTAR minus epoch(i) by calling the starf andhrdiagf combination The mass-loss rate m for the star is obtained by a callto mlwindf which gives m and the actual mass lost in the interval tev0(i)rarr tev(i) is

Δm = m (tev(i) minus tev0(i)) lowast TSTAR lowast 1 times 106ZMBAR (102)

in N -body units If non-zero this correction is applied to body(i) to updatethe stellar mass If kstar(i) le 2 or kstar(i) = 7 then body0(i) is reset to beequal to body(i) and epoch(i) is updated to reflect the change in mass Notethat epoch(i) is also updated when the stellar type changes Also if massloss occurs the spin angular momentum of the star is adjusted accordingly ndasha call to magbrkf makes any further adjustments resulting from magneticbraking In the case of Δm gt 0 the routine fcorrf is called to perform forceand energy corrections for the mass loss If a new neutron star or black hole isdetected this routine calls kickf to generate the velocity kick arising fromthe supernova event and deals with the ramifications of the velocity changeIf the mass loss is substantial (Δm lowast ZMBAR gt 01) or a velocity kick hasoccurred it is also necessary to initialise new force polynomials for the starand its neighbours This is performed in mdotf (with calls to the appropriatesubroutines) The update procedure in mdotf for star i is then completedby setting tev0(i) = tev(i) and calling trdotf to set a new tev(i) Beforeleaving mdotf and after dealing with each star that is due TMDOT is updatedto the new minimum in the tev array

Output of the stellar evolution variables is performed by the routinehrplotf which is called from outputf at intervals of dtplot Note thatdtplot must be greater than or equal to deltat ndash the time interval in N -bodyunits for major output ndash and ideally the two input variables should commen-surate A call to hrplotf creates a snapshot of the model stars at the cur-rent time This involves two output files fort83 contains a line for eachsingle star and fort82 contains a line for each binary These files providethe necessary information for generating descriptions of the model in theform of colour-magnitude diagrams radial profiles and mass functions forexample

The possibility of stellar collisions has been mentioned and the N -bodycodes allow for such events Direct hyperbolic collisions between stars arerare in the cluster simulations for which nbody4 and nbody6 have typicallybeen used Rather two stars in a close gravitational encounter more likely

form a binary and this may be followed by a merging of the two stars Assuch a discussion of how these events are dealt with falls more naturallyunder the banner of binary evolution and will be described in Chap 12Here it suffices to say that collisions of all types (eccentric parabolic orhyperbolic) are processed by the routine cmbodyf which calls mixf iftwo stars are to merge The routine mixf determines the nature of themerger product and initialises its stellar variables through calls to gntagefand hrdiagf (see also Hurley Tout amp Pols 2002 for more details of thisprocedure)

The interested reader may find Hurley et al (2001) and Aarseth (2003p 279) useful for additional discussions regarding the implementation of stel-lar evolution in N -body codes To complement these discussions the materialin this section is rounded off by making the user aware of SSE parameters thatare hardwired so to speak into various N -body routines For example theparameter η appears in the Reimers mass-loss formula in mlwindf ndash in thestand-alone SSE package this is an input parameter but in nbody46 it isset in the header of the subroutine The same goes for the maximum neutronstar mass which is set in the header of hrdiagf rather than appearing asan input variable There may be occasions when the user would wish to varythese parameters and this requires an edit of the relevant file and recompilingthe code

105 Some Results

The stellar evolution capability in nbody4 and nbody6 has been used togood effect to produce realistic models of star clusters (Baumgardt amp Makino2003 for example) The results of such endeavours are presented in Chap 14Given that the option to use metallicity as a free parameter is a unique featurethat SSE has added to the N -body codes this section briefly highlights someresults relating to the models of varying metallicity

In Hurley et al (2004) a series of nbody4 simulations was presented inorder to investigate the effect of metallicity on the evolution of open clustersEach simulation started with 30 000 single stars Figure 101 shows CMDsnapshots at four times for one of these simulations at solar metallicity Thiswas constructed using the fort83 output file Note that stellar evolutionnot only affects the distribution of stars in the nuclear burning phases as thecluster evolves but also affects the locus of the white dwarf stars To illustratehow metallicity affects the CMD appearance Fig 102 shows the snapshotsof four models at the same age but with different metallicities

The models of Hurley et al (2004) showed that clusters with low-Z starsexperienced more mass loss from stellar evolution over the first 5 000Myrof evolution compared to clusters of solar metallicity This lead to increasedexpansion of the cluster and a decreased stellar mass range with a knock-on

294 J R Hurley

Fig 101 Colour-magnitude diagram showing four N -body isochrones Dataare taken from a Z = 002 NBODY4 simulation that started with 30 000 sin-gle stars Shown are stars in the simulation at 500Myr (diamond symbols)1 000Myr ( symbols) 4 000 Myr (+ symbols) and 9 000 Myr (star symbols)Stars in the upper-right of the diagram are in normal nuclear burning phasesof evolution (kstar le 6) and stars in the lower-left are white dwarfs Thereare no naked helium stars present Any neutron stars or black holes are notshown The luminosity and effective temperature provided for each star bySSE have been converted to magnitude and colour with the bolometric correc-tions given by the models of Kurucz (1992) and in the case of white dwarfsBergeron Wesemael amp Beauchamp (1995)

effect of a delay in the onset of core-collapse and binary formation Overallthis means that low-Z clusters have extended lifetimes Models with low-Zalso produced many more double-WD binaries This is a result of shortermain-sequence lifetimes and greater AGB core-masses producing more WDsand more massive WDs in comparison to high-Z models of the same ageThis is a direct illustration of the interaction between stellar and dynamicalevolution within the star cluster environment (see Hurley et al 2004 for moredetails)

The focus so far has been on models of single stars ndash in Chap 11 we shallbegin to discuss the intricacies of binary evolution This will be followed inChap 12 by details of the binary evolution algorithm used in nbody4 andnbody6

Fig 102 Colour-magnitude diagram showing N -body isochrones at 4 000 Myrfor simulations of different metallicity Shown are stars with Z = 003 ( symbols)Z = 002 (+ symbols) Z = 0001 (diamond symbols) and Z = 00001 (star symbols)Data are from NBODY4 simulations begun with 30 000 single stars The simulationsare described in Hurley et al (2004) Only stars with kstar le 6 are shown

References

Baumgardt H Makino J 2003 MNRAS 340 227 293Belczynski K Kalogera V Bulik T 2002 ApJ 572 407 288Bergeron P Wesemael F Beauchamp A 1995 PASP 107 1047 294Carson J E Cool A M Grindlay J E 2000 ApJ 532 461 283Chernoff D F Weinberg M D 1990 ApJ 351 121 285Dewi J D M Pols O R Savonije G J van den Heuvel E P J 2002 MNRAS

331 1027 288Eggleton P P Fitchett M Tout C A 1989 ApJ 347 998 285Ferraro F R Paltrinieri B Fusi Pecci F Cacciari C Dorman B Rood R T

1997 ApJ 484 L145 283Grindlay J E Heinke C Edmonds P D Murray S S 2001a Science 292 2290 283Grindlay J E Heinke C O Edmonds P D Murray S S Cool A M 2001b

ApJ 563 L53 283Hurley J R Pols O R Tout C A 2000 MNRAS 315 543 285 286 288 291Hurley J R Shara M M 2003 ApJ 589 179 288Hurley J R Tout C A Aarseth S J Pols O R 2001 MNRAS 323 630 293Hurley J R Tout C A Aarseth S J Pols O R 2004 MNRAS 355 1207 293 294 295

296 J R Hurley

Hurley J R Tout C A Pols O R 2002 MNRAS 329 897 285 287 291 293Kroupa P Tout C A Gilmore G 1993 MNRAS 262 545 291Kudritzki R P Reimers D 1978 AampA 70 227 288Kurucz R L 1992 in Barbuy B Renzini A eds Proc IAU Symp 149 The Stellar

Populations of Galaxies Kluwer Dordrecht p 225 294Mackey A D Gilmore G F 2003 MNRAS 338 85 284Meylan G Heggie D C 1997 AampARv 8 1 284Paresce F Meylan G Shara M Baxter D Greenfield P 1991 Nature

352 297 283Pols O R Schroder K -P Hurley J R Tout C A Eggleton P P 1998 MNRAS

298 525 287 288Tout C A Aarseth S J Pols O R Eggleton P P 1997 MNRAS 291 732 285 286 288 290Tout C A Pols O R Eggleton P P Han Z 1996 MNRAS 281 257 287Wielen R 1968 Bull Astron 3 127 285Yanny B Guhathakurta P Schneider D P Bahcall J N 1994 AJ

435 L59 283

11

Binary Stars

Christopher A Tout

In clusters there are both primordial binary stars and binaries created bydynamical interactions Occasionally a new binary system can be formed(Fabian Pringle amp Rees 1975) but more often new systems are the resultof exchanges In Chap 12 Hurley describes an algorithm for including theinteraction of the components of a binary star in N -body simulations In thischapter we investigate the underlying physics and note that though we havea good qualitative idea of what goes on there is still much to be determinedfully quantitatively

Double stars have been known since ancient times and were referred toin written records as early as Ptolemy But the concept of a binary star as agravitationally bound entity did not exist before the late eighteenth centuryThe Revd John Michell (Michell 1767) showed statistically that not all doublestars could be chance superpositions on the sky He concluded that ldquo[DoubleStars] were brought together by their mutual gravitation or some other lawor by the appointment of the Creatorrdquo This statement sums up well ourunderstanding of the formation of binary stars the physics of which stilleludes us At the time Herschel disagreed with Michellrsquos deductions becausehe wanted to use stars as standard candles to map the structure of the MilkyWay If the two very different components of a double star were actually at thesame distance such a mapping would be impossible He eventually acquiescedand himself introduced the term ldquobinary starrdquo in 1803 (Herschel 1803)

Binary stars are common and consequently perhaps represent the normalformation mode The ratio of single (or unresolved) systems to binary to tripleor higher multiple systems is two to five to two to one As a requirement forthe dynamical stability of a system higher multiples must be hierarchicaland can be considered as a sequence of binary stars within binary stars Forinstance a quadruple system (Fig 111) can take essentially two forms Eitherthey are two pairs of stars both orbiting one another or a very close binaryin a wider orbit with a third star and then this triple system in a yet widerorbit with the fourth star Typically the separations of nested pairs must bea factor of four or more smaller for long-term stability Though no multiple

Tout C A Binary Stars Lect Notes Phys 760 297ndash319 (2008)

298 C A Tout

aaa

aa

AA ABA ABB B

AA AB BA BB

aa ab

aaaa gt 4 aa gt 4

a gt 4 4

Fig 111 Two possible configurations of quadruple systems with long lifetimesThe convention of labelling the stars in a binary as A and B is extended through thehierarchy Separations of binary systems within the hierarchy must be typically afactor four or more larger when moving from one level up to the next for long-termsurvival

system is indefinitely stable many can be expected to survive the current ageof the Universe (Chap 3)

111 Orbits

The orbits of binary stars (Fig 112) obey a form of Keplerrsquos laws generalisedto the case where both stars have similar masses First the orbits are conicsections and bound orbits are ellipses The diagram shows the semi-major axisa the semi-minor axis b and the semi-latus rectum l These are related to theeccentricity e by

l = a(1 minus e2) (111)

and

e2 = 1 minus b2

a2 (112)

A general point on the ellipse is given parametrically by

r =l

1 + e cos θ (113)

where r is the distance from the primary focus F and θ is the angle from thesemi-major axis to the line joining the F to P Secondly the line connectingthe two bodies sweeps out equal areas in equal times If one body is consideredfixed at F while the other orbits at P this is equivalent to

11 Binary Stars 299

a

Fprime

b lr

P

F

θ

Fig 112 Stars in a bound binary follow elliptical orbits One star is at the focusF and the other orbits at P around the ellipse

12r2θ =

πa2(1 minus e2)12

P (114)

where the numerator is the area of the ellipse and the denominator P is theperiod of the binary the time taken for a complete orbit This follows fromthe conservation of angular momentum Third the period and separation arerelated by

(P

2π

)2

=a3

G(M1 +M2) (115)

where G is Newtonrsquos gravitational constant and M1 and M2 are the massesof the two stars

Each of these laws is a consequence of Newtonrsquos laws of motion and hislaw of gravity Both stars orbit the centre of mass in ellipses and both feel acentrally directed force so angular momentum is conserved Again with r theinstantaneous separation we have

r2θ = h =MJ

M1M2= const (116)

where M = M1 + M2 is the total mass J is the total angular momentum ofthe system and h is the specific angular momentum per unit reduced massSolving the equations of motion we find that

l =h2

GM(117)

300 C A Tout

so that conservation of angular momentum fixes the semi-latus rectum of theorbit Similarly we find the total energy kinetic plus potential to be

E = minusGM1M2

2a(118)

so that the energy determines the semi-major axis and thence the period ofthe system

112 Tides

Though angular momentum can be lost in stellar winds and gravitational ra-diation let us first consider the case when the total orbital angular momentumis conserved Because the stars are luminous they can radiate orbital energy ifit is converted to heat by tides or any other process We may write the energyin terms of the angular momentum and eccentricity as

E = minusGM1M2

2h2GM(1 minus e2) (119)

from which we can see that(partE

parte

)

J

prop 2e and(part2E

parte2

)

J

gt 0 at e = 0 (1110)

Thus a circular orbit is the most stable configuration for a given angularmomentum

1121 Tidal Forces

So far we have considered both stars as point masses This is a good approx-imation when they are well separated but when they are closer the finite sizeof the stars becomes important and tidal interactions and eventually masstransfer occur between the two Let us assume that star 2 is sufficiently smallto still be considered a point mass and let star 1 have a radius R (Fig 113)The potential at a point P a distance r from the centre of star 1 along a lineat an angle θ to the line joining the centres of the two stars and a distance rprime

from star 2 owing to star 2 can be expanded as

Φ2 = minusGM2

rprime=

minusGM2radica2 + r2 minus 2ar cos θ

= minusGM2

a

infinsum

n=0

( r

a

)n

Pn(cos θ) (1111)

where Pn is the nth Legendre polynomial The force on material in star 1 isminusnablaΦ2 The n = 1 term balances the overall orbital motion Of most interestfor the evolution of the system is the n = 2 term because it is the largestthat leads to both transfer of angular momentum between star 1 and the

11 Binary Stars 301

1M 2M

PR r

a gtgt R

r

δ

Ω

ω

Fig 113 The tidal potential of star 2 distorts star 1 If as here the star is spinningfaster than the orbit (Ω gt ω) viscosity drags the tidal bulges ahead of the orbit anddissipates energy The force between star 2 and the two bulges provides a torquethat transfers angular momentum from star 1 to the orbit

orbit and dissipation of energy Star 1 is distorted as illustrated by the dashedcurve in Fig 113 If the star is not rotating synchronously with the orbit thedistortion is dragged around it If the star is spinning more slowly viscosityleads to a lag of angle minusδ so that the tidal bulges lag behind the line joiningthe stars If the star is spinning faster than the orbit the companion lagsbehind the bulges The gravitational force between the two bulges and star 2provides a torque that tends to synchronize the stellar spin and the orbit Atthe same time energy dissipation circularises the orbit Tides also align thespin axes with the orbital axis (Hut 1981)

The synchronous state is not always stable (Hut 1980) Transfer of angularmomentum from the orbit to a star increases both the spin of the star andthe orbital angular velocity because the orbital angular momentum

Jorb prop a2ω prop ωminus13 (1112)

in a circular orbit with angular velocity ω If there is insufficient total angularmomentum in the system the stars end up spiralling together This is theexpected fate of contact binary stars and some planetary systems though theprocess can take a very long time (Rasio Tout amp Livio 1996)

For a typical system in which the extended star star 1 is convective withmass ratio q = M1M2 separation a and radius of the largest star (star 1here) R the circularisation time

τcirc asymp2q2

1 + q

( a

R

)8

yr (1113)

We shall see in the next section that much more drastic interaction beginswhen R = RL asymp 1

3a At this point τcirc asymp 2000 yr Even when R = 12RL

τcirc asymp 6times 105 yr which is still much less than the nuclear timescale for stellar

302 C A Tout

evolution that ranges from 1010 yr for a 1M star on the main sequence to106 yr for a massive giant Synchronization times are even shorter

τsync asymp q2( a

R

)6

yr (1114)

or 300 yr for R asymp RL and 2 times 104 yr for R asymp 12RL

113 Mass Transfer

When the two stars are very close and R asymp a we can no longer ignore thehigher terms in the expansion of the tidal potential We shall begin the analysisagain and make use of the fact that by the time the radius of either star getslarge enough tides will have already circularised the orbit and synchronizedthe spin of the star We can therefore work in a frame rotating at Ω asillustrated in Fig 114 Let all the material be stationary except for a testparticle at P Then in an inertial frame the velocity of P is

v = r + Ω times r (1115)

and its acceleration is

a = r + 2Ω times r + Ω times (Ω times r) (1116)

where the first term may be familiar as the Coriolis force and the second as thecentrifugal We can then apply the Euler momentum equation in the inertialframe

ρa = minusnablaP minus ρnablaφG (1117)

2M1M

2M a_____

Ma

1 =

a

P

x

y

C of M

r

Fig 114 Coordinates rotating with the binary system centred on its centre ofmass with the z-axis perpendicular to the orbital plane

11 Binary Stars 303

where ρ is the density P is the pressure

nabla2φG = 4πGρ (1118)

and φG is the gravitational potential In corotation r and r vanish and aligningthe z-axis with Ω we may write

Ω times (Ω times r) = minusnablaφΩ (1119)

withφΩ = minus1

2Ω2s2 (1120)

where s is the distance from the z-axis Thus the Euler equation reduces to

1ρnablaP + nablaΦ = 0 (1121)

with Φ = φG +φΩ So surfaces of constant pressure are surfaces of constant ΦIn particular the surface of the star if defined as P = 0 is a surface of constantΦ Stars are centrally condensed so to a good approximation φG is just thegravitational potential of two point masses at the centres of the stars and inCartesian coordinates with star 1 at the origin and star 2 at (a 0 0) we find

Φ =minusGM1radic

x2 + y2 + z2+

minusGM2radic(xminus a)2 + y2 + z2

minus 12GM

a3

[(

xminus a

1 + q

)2

+ y2

]

(1122)

which is just a function of the mass ratio q = M1M2 GM and a Moreover ifwe scale all lengths by the separation x rarr xa the shape of the equipotentialsurfaces is a function of q only We plot them for q = 2 in Fig 115 Corotatingmaterial in hydrostatic equilibrium fills up to an equipotential surface Thuswhen the radii are small compared to a the surface equipotentials are spheresFar from the binary surfaces are again spheres Of interest to us are the twoinnermost critical surfaces on which the lines meet at stationary Lagrangianpoints Moving outwards from the centres of the stars the first meeting atthe inner Lagrangian point L1 determines when material is more attractedto its companion than to the star itself The second opens to the right at theL2 point and determines the maximum size of a joint star or contact binaryaround the two orbiting masses The three other stationary points are alsoshown but are not of interest to us now because beyond the surface throughthe L2 point there is nothing to keep the material corotating and (31) is nolonger valid

Figure 116 shows the value of the potential along the x-axis and illustrateshow stars fill their equipotential surfaces to form three different classes ofbinary star In a wide binary system both stars have radii small compared tothe separation and the system is said to be detached As either star grows

304 C A Tout

x a

y a

q

Fig 115 Equipotential lines in the xminusy plane Solid lines pass through Lagrangianpoints where nablaΦ = 0

it is gradually distorted until it fills the critical potential surface that crossesat the inner Lagrangian L1 point between the two stars This equipotentialaround the star is its Roche lobe If the star grows any larger material at L1 ismore attracted to its companion than to itself and the material can flow fromit to the other star This is known as Roche lobe overflow and the systemis said to be semi-detached Algols (Sect 1151) and cataclysmic variablestars (Sect 1153) are in this state If the second star expands so that it toowould overfill its Roche lobe the two stars can exist in equilibrium in contactSuch systems appear to be common but do not last long Material and heatare transferred between the two until the mass ratio becomes large and tidalinstability shrinks the orbit and merges the two stars

Even the surface through the L1 point is almost spherical When the massratio q = 1 the difference in extent between the x and z directions is only5 of the diameter and this rises to only 10 when q = 10 We define theRoche lobe radius RL to be the radius of a sphere with the same volume asthe Roche lobe

VL =43R3

L (1123)

11 Binary Stars 305

Fig 116 The potential along the x-axis in Fig 115 Three binary star configura-tions are shown

The volume can be evaluated numerically and various simple fits to RL havebeen deduced Eggleton (1983) fitted the Roche lobe radius of star 1 by

RL

a=

049q23

06q23 + loge(1 + q13) (1124)

This is accurate to better than 1 over the whole range 0 lt q lt infin It is thepreferred form for numerical work but for analytic work a formula deducedby Paczynski (1971)

RL

a= 0462

(M1

M

) 13

(1125)

which is accurate to better than 3 for 0 lt q lt 08 is much more usefulThe rate of flow through the L1 point is a rapidly rising function of the

amount by which the star overfills its Roche lobe ΔR = R minus RL So as longas the rate at which the star expands or the Roche lobe shrinks is longcompared with the dynamical timescale on which hydrostatic equilibrium is

306 C A Tout

regained we can expect the mass transfer rate to adjust to maintain

R asymp RL and R asymp RL (1126)

If this timescale is much less we can expect ΔR and consequently M toincrease on a dynamical timescale We consider the consequences of such un-stable mass transfer in Sect 1154 but first we examine under what conditionsmass transfer is stable

1131 Stability of Mass Transfer

To examine the stability of mass transfer we follow Webbink (1985) and definethree derivatives of radii with respect to the mass of the lobe-filling star Thefirst is the rate of change of the Roche lobe radius RL for conservative masstransfer in which the angular momentum of the system J and the total massM are conserved Any material lost by star 1 is accreted by star 2 so that

ζL =(part logRL1

part logM1

)

MJ

(1127)

This can be approximated by ζL = 213q minus 167 (Eggleton 2006) and we seethat it is positive for M1 gt 078M2 so that in this case the Roche lobeshrinks in response to mass transfer from star 1 to star 2 and otherwise itexpands The initial response of the star to mass loss is adiabatic as it regainshydrostatic equilibrium and loses thermal equilibrium in the process So wedefine a second derivate at constant entropy s and composition of each isotopeXi throughout the star

ζad =(part logR1

part logM1

)

sXi

(1128)

For stars with radiative envelopes ζad gt 0 so they shrink on mass loss whilefor stars with convective envelopes ζad lt 0 and they expand on mass loss Ona thermal timescale the star regains full equilibrium at its new mass but stillwith constant composition A third derivative

ζeq =(part logR1

part logM1

)

Xi

(1129)

describes the rate of change of radius with mass in equilibrium For main-sequence stars ζeq gt 0 typically while for red giants and stars crossing theHertzsprung gap ζeq lt 0

The rate at which mass transfer proceeds depends on the relative valuesof these derivatives If ζL gt ζad then the Roche lobe shrinks faster than theradius of the star in direct response to mass transfer So ΔR increases andconsequently M increases rapidly There is positive feedback and the masstransfer is unstable

11 Binary Stars 307

∣∣∣∣M1

M1

∣∣∣∣ rarr τdyn asymp 10 minus 100 yr (1130)

and mass transfer proceeds on a dynamical timescale Star 2 often cannotaccrete the material at such a high rate Instead it expands itself and thetransferred material ends up in a common envelope around the two stars Weshall discuss this in detail in Sect 1154 This is typically the outcome when agiant fills its Roche lobe when in orbit with a less massive companion becausethe giant expands while its Roche lobe is shrinking Positive feedback drivesthe mass transfer up to the dynamical rate

If ζL lt ζad but ζL gt ζeq then the star shrinks in its immediate responseto mass transfer but then expands on its thermal timescale τth and

∣∣∣∣M1

M1

∣∣∣∣ rarr τth asymp 105 minus 106 yr (1131)

Mass transfer proceeds on a thermal timescale This is the case when a sub-giant in the Hertzsprung gap with a radiative or thin convective envelope fillsits Roche lobe

If both ζad gt ζL and ζeq gt ζL the star shrinks in response to mass transferand does not expand again to fill its Roche lobe until driven to either by itsown nuclear evolution or until some angular momentum loss mechanism causesthe orbit to shrink sufficiently Either

∣∣∣∣M1

M1

∣∣∣∣ rarr τnuc asymp 107 minus 109 yr (1132)

the case for main-sequence stars or red giants in present-day Algols (seeSect 1151) or ∣

∣∣∣M1

M1

∣∣∣∣ rarr τJ (1133)

the timescale on which angular momentum is lost from the system This isthe case for cataclysmic variables that form the subject of Sect 1153

114 Period Evolution

When the angular momentum of the component stars is negligible comparedto that of their orbit we can derive simple formulae for how the orbit evolveswith mass loss and mass transfer We allow a wind from star 1 that escapesfrom the system and mass transfer from star 1 to star 2 so that minusM1 is themass loss rate from star 1 M2 is the rate of accretion by star 2 the masstransfer rate and minusM is the rate of mass loss from the system the wind fromstar 1 Then

minus M1 = minusM + M2 (1134)

308 C A Tout

with M and M1 le 0 and M2 ge 0 The wind from star 1 carries off angularmomentum intrinsic to the orbit of the star so that the rate of change ofangular momentum of the orbit is

J = Ma21Ω (1135)

We recall thatJ =

M1M2

Ma2Ω (1136)

so that we can differentiate log J to find

J

J=

M1

M1+M2

M2minus M

M+ 2

a

a+

ΩΩ

=M

M1M2

(M2

M

)2

M =M2

M1

M

M (1137)

from (1135) Differentiating Keplerrsquos third law we find

2P

P= minus2

ΩΩ

= 3a

aminus M

M(1138)

and combining these gives us

M2

M1

M

M=

M1

M1+M2

M2minus 1

3M

M+

13P

P (1139)

When there is no mass transfer but mass loss in a wind M2 = 0 and M1 = Mso that

P

P= minus2

M

M (1140)

We can integrate this to give P 2M = const or with (1138) aM = constThe period and separation increase as mass is lost Indeed as the Sun losesmass so the planets of the solar system will drift further away from it

When there is mass transfer but no mass lost from the system M = 0 andJ = 0 so that

P

P= minus3

M1

M1minus 3

M2

M2 (1141)

This can be integrated to give P (M1M2)3 = const or a(M1M2)2 = constThe period and separation decrease while mass is transferred from the moremassive to the less massive component reach minima when the masses areequal and then increase as mass is transferred from the less massive to themore massive component

115 Actual Types

We have described the basic physics of binary stars and their interactionsCoupling this with stellar evolution leads to a veritable zoo of different typesof binary star as described by Eggleton (1985) Observations do overlap with

11 Binary Stars 309

what we expect but often require the introduction of new physical processessuch as common envelope evolution (Sect 1154) that are not fully under-stood We shall illustrate with just three examples The Algols as the proto-types the cataclysmic variables as those studied in most detail and the type Iasupernovae that have recently been used as standard candles to measure thestructure and evolution of the Universe

1151 Algols

As one of the brightest stars in the northern hemisphere Algol or β Perseihas been known for a long time It is an eclipsing SB2 and so yields a greatdeal of information about its current state Its variability was first definitelyrecorded by Montanari (1671) in Bologna but the name Algol suggests that itmay have been recognised much earlier Algol is derived from the Arabian AlGhul which has been variously translated as demon or changing spirit (Kopal1959) However Allen (1899) felt it is more likely that the name is derivedfrom Ptolemy who referred to it as the brightest star in the Gorgonrsquos heada constellation recognised by the Greeks at the time and indeed generallyuntil quite recently (Goodricke 1783) The Hebrews called it Rosh-ha-Satanor Satanrsquos head and the Chinese Tseih She or the piled up corpses Whetherthese names reflect the variability or not must be left to our imaginationsbecause no actual record has been found

Its eclipses were not noted for over a century until John Goodricke (1783)sent a short letter to the Royal Society describing how he had spotted aperiodicity in the light variations of Algol He and his friend Edward Pigotthad by then already obtained a fairly accurate estimate of the period of 2 daysand 21 h Goodricke in a short paragraph at the end of his letter went on tosuggest that the cause of the variation might be either a dark object orbitingand eclipsing the star or a dark spot on its surface Confirmation of his firsthypothesis did not come for yet another century when Vogel (1890) observedradial velocity shifts in the spectrum of Algol and found the positions ofminimum light to correspond to the conjunctions of the eclipse model

Observations improved with time giving better photometric and spectro-scopic measurements of Algol and a number of similar systems It seems thatit had been apparent that something was not quite right with Algol for sometime before Hoyle (1955) recorded what he described as the Algol ParadoxFrom the shapes of the eclipses it was clear that the fainter star was largerSuch a situation was thought not to be possible according to the theory ofstellar evolution If both stars were on the main sequence then the brighterwould be larger In fact the fainter could only be larger if it had evolvedoff the main sequence and indeed Parenago (1950) had already claimed thatthe fainter components of Algols were in many cases sub-giants Hoyle arguedthat although it would be possible to pick the two stars from the H-R di-agram one on the main sequence and the other a much older sub-giant allreasonable theories of the formation of binary stars suggested that the twocomponents would have formed at the same time and would be of the same

310 C A Tout

age now Thus he had identified the paradox without the need to introducethe masses of the stars directly and went on to explain it successfully in termsof the initially brighter star evolving to such a size that its fainter companiongobbled up matter from its surface This companion could then move up themain sequence and become the brighter of the two In clusters such stars couldlater appear as blue stragglers (Sandage 1953)

At the same time Crawford (1955) was also solving the same paradoxthough more specifically in terms of the limitations placed on the mass ratiosby the spectroscopically determined mass functions and the assumption thatthe brighter component does in fact lie on the main sequence Struve (1948)had already pointed out that these mass functions are low Crawford alsointroduced the concept of the giant filling its Roche lobe In fact Walter(1931) had pointed out that the cool stars in Algols are close to the limit ofdynamical stability but this had gone largely unnoticed

This semi-detached nature of Algols provided mutual support for the hy-pothesis formulated by Struve (1949) that the existence of gaseous streamsbetween the two stars in Algols could account for an asymmetry in the ra-dial velocity curve Although the photometric light curve of U Cephei showedsymmetric eclipses the radial velocity curve is asymmetric Struve explainedthis in terms of the spectrum of a gaseous stream moving faster than thetwo stars superimposed on the symmetric curve of the star Evidence hadalso been provided by Wood (1950) who had found that binaries with periodfluctuations almost always have one star filling its Roche lobe

With a fairly definite theory and the dawn of numerical stellar evolu-tion the stage was set for the construction of theoretical models of thesesemi-detached systems The first step was taken by Morton (1960) who con-centrating on the initially more massive star examined the process of masstransfer He pointed out that since all observed Algols have the sub-giantcomponent already less massive the initial rate of mass transfer must havebeen much faster than that taking place now It must have been sufficientlyfast to make it unusual to observe a system in a state where the primary isstill the more massive

1152 Critical Mass Ratio

A simple calculation reveals why Let the mass-losing giant be star 1 Its radius

R1 asymp f(L)Mminus0271 (1142)

where f is a function of its luminosty L which does not vary much with massloss The fully convective giant envelope is isentropic so that ζad asymp ζeq andfor timescales short compared with the nuclear evolution timescale on whichL varies

R1

R1= minus027

M1

M1 (1143)

11 Binary Stars 311

For stable mass transfer we must have negative feedback

R lt RL when R1 = RL (1144)

because otherwise the process of mass transfer would mean that the staroverfills its Roche lobe even more and the rate of overflow would increase

We can differentiate formula (1125) which recall is valid for q lt 08 tofind

RL

RL=

13M1

M1minus 1

3M

M+a

a (1145)

Then assuming conservative mass transfer (M = 0 and J = 0) we require

minus 027M1

M1lt

(1

3M1minus 2(M2 minusM1)

M1M2

)

M1 (1146)

But M1 lt 0 so

M1 lt 07M2 or q lt qcrit = 07 (1147)

Over the decade following Mortonrsquos work detailed models were made bymany independent workers Paczynski (1966) Kippenhahn amp Wiegert (1967)and Plavec et al (1968) all confirmed Mortonrsquos results Kippenhahn andWeigert introduced the nomenclature of case A to indicate mass transfer be-fore the exhaustion of central hydrogen burning and case B for mass transferafterwards when the star has evolved off the main sequence In all of thesemodels conservative mass transfer (all the matter lost by the primary beingaccreted by the secondary) was assumed but Paczynski amp Ziolkowski (1967)showed that the resulting Algol systems are more realistic if half the mass lostby the primary is actually lost from the system In order to avoid dynamicalmass transfer all Algols must have begun mass transfer before the most mas-sive star has evolved on to the giant branch unless it has suffered sufficientmass loss that q lt qcrit asymp 07 and the Roche lobe expands faster than the star(Tout amp Eggleton 1988)

1153 Cataclysmic Variables

Cataclysmic variables are very close binary stars in which the primary com-ponent is a white dwarf which is accreting material transferred from itsRoche-lobe-filling companion Figure 117 illustrates the basic componentsThe companion to the white dwarf is always less massive often substantiallyand is typically a low-mass main-sequence star for which the Roche-fillingstate dictates an orbital period of a few hours and a separation of about asolar radius In a very few systems the secondary star can be slightly evolvedFor example GK Per the widest system classified as a cataclysmic variablehas an orbital period of 47 h and its white dwarf has a subgiant companion

312 C A Tout

Cataclysmic Variable Star

Accretion Disc

White Dwarf

Hot Spot

Accretion Stream

Secondary Star

Fig 117 A schematic diagram of a cataclysmic variable with the major observablecomponents marked According to general practice the accreting white dwarf isstar 1 and the Roche lobe filling companion is star 2

The nuclear or in some cases mass-loss timescales of evolved companionscan be relatively short and their nature is therefore fundamentally differentfrom those systems with unevolved low-mass secondaries Most importantlythe mass transfer rates are higher These systems particularly those with verylarge red or supergiant secondaries are classified as symbiotic stars At theother extreme the companion can be another white dwarf of lower mass thanthe primary AM CVn is the prototype of this class of cataclysmic variablesand has a period of 89min

In addition to the two stars a third component an accretion disc is impor-tant and often dominates the light from the cataclysmic variable It is formedbecause the material overflowing from the companion at the inner Lagrangianpoint L1 has too much angular momentum to fall directly on to the whitedwarf Viscous dissipation allows the slow infall of the majority of the matterthrough the disc while angular momentum is carried outwards until it canbe tidally returned to the orbit Many cataclysmic variables are observation-ally very clean systems in which the light variations and spectra of each ofthe three main components can be separated out Often the signature of thehigh-velocity accretion stream and the hot spot where it impacts the edge ofthe disc can also be identified An excellent detailed and very readable reviewof the observations from early times forms a substantial part of the book byWarner (1995) to which the interested reader is encouraged to turn

11 Binary Stars 313

Two instabilities gave cataclysmic variables their name and were respon-sible for their early observation The first is the classical nova Hydrogen-richmaterial transferred to the white dwarf from its companion builds up in adegenerate layer on the surface When the base of this layer becomes denseenough and hot enough the hydrogen ignites in a thermal nuclear runawaythat leads to a large increase in brightness and probably the ejection of mostof the accreted material The second is an instability in the accretion discUnder some conditions material can accumulate in the disc and fall throughin bursts The quasiperiodic increase in brightness of the disc makes thesevisible as dwarf novae There are yet other systems that have never displayedeither of these phenomena and others that are dominated by magnetic fields

Typically the nuclear timescale on which the donor star evolves τN gt1011 yr so that evolution cannot be the driving force behind the mass transferRather this is direct angular momentum loss In the closest systems typicallythose with P lt 3 h it is achieved in gravitational radiation (Peters amp Mathews1963) at a fractional rate

JGR

J= minus32G3

5c5M1M2(M1 +M2)

a4 (1148)

In longer period systems this is too weak and the most likely mechanism is aprocess of magnetic braking (Fig 118) A very mild wind carrying off massat |M | lt |M1| the mass transfer rate can be dragged round by the star outto large distances beyond the Alfven radius RA at which the magnetic energydensity equals the specific kinetic energy in the wind

Dead Zone

Wind Zone

Magnetic Field anchored to Star

Fig 118 A very weak wind can be dragged around by a magnetic field linked toa star In dead zones the wind cannot escape but where it can open the field linesit carries of substantial angular momentum because it effectively corotates with thestar to the Alfven radius RA

314 C A Tout

12v2w =

B2

2μ0 (1149)

where vw is the wind velocity B is the magnetic field strength and μ0 is thevacuum permeability The combined angular momentum loss rate in the windand owing to magnetic torques is

J = MR2AΩ (1150)

where Ω is the spin angular velocity of the star effectively as if the wind werecorotating to RA (Mestel amp Spruit 1987) This can be very effective whenRA R which is usually the case when |M | is small It is most probablymagnetic braking that is responsible for bringing cataclysmic variables intothe semidetached state in the first place

1154 Common Envelope Evolution

The white dwarfs in cataclysmic variables must have originally formed as thecores of giants which must have had room to grow to 100 or even 1000R be-fore interaction However their orbital separation is now only a few solar radiiThe generally accepted route by which a binary reduces its period is common-envelope evolution (Paczynski 1976) Following dynamical mass transfer fromthe giant the pair becomes a common-envelope system (Fig 119) in whichthe degenerate core of the original giant and the relatively dense red dwarf areorbiting within the low-density envelope of the giant that now engulfs bothstars From here on what happens is as much fantasy as fact By some fric-tional process the two cores are supposed to spiral together towards the centreof the envelope During this process the orbital energy released is transferredto the envelope which it drives away in a strong wind Because the orbitalenergy of the cores and the binding energy of the envelope are of the sameorder it can be envisaged that in some cases the balance is just such that theentire envelope is blown away when the cores reach a separation of a few solarradii If more energy is transferred the envelope is lost while the orbit is stillquite wide If less energy is transferred the cores coalesce before the envelopeis lost In practice coalescence most likely occurs when the red dwarf reachesa depth in the envelope where it has comparable density with the envelope orwhen it is tidally disrupted by the white dwarf

Webbink (1984) defined a parameter αCE to be the fraction of the or-bital energy released during the spiralling-in which goes into driving awaythe envelope Knowing αCE and the binding energy of the envelope we cancalculate the final orbital separation from the initial Note that the bindingenergy of the envelope is calculated differently by different authors The mostsignificant discrepancy is whether we use the binding energy of the single-stargiant envelope before the common envelope forms (Webbink 1984) or thatof the common envelope itself on the assumption that it has swollen up tothe size of the orbit (Iben amp Tutukov 1984) The value of αCE is expected

11 Binary Stars 315

Cores Spiral Together

Envelope Lost Coalescence

Magnetic BrakingGravitational

Radiation

Rapidly Spinning GiantClose Binary in Planetary Nebula

Normal Giant

10 yr4

Cataclysmic Variable

Fig 119 Common-envelope evolution After dynamical mass transfer from a gianta common envelope enshrouds the relatively dense companion and the core of theoriginal giant These two spiral together as their orbital energy is transferred tothe envelope until either the entire envelope is lost or they coalesce In the formercase a close white-dwarf and main-sequence binary is left initially as the core of aplanetary nebula Magnetic braking or gravitational radiation may shrink the orbitand create a cataclysmic variable Coalescence results in a rapidly rotating giantwhich will very quickly spin down by magnetic braking

to be less than one because at least part of the released energy should beradiated away However population synthesis models that recreate sufficientnumbers of cataclysmic variables and other close systems such as X-ray bi-naries and the progenitors of SNe Ia indicate that large values of αCE are

316 C A Tout

required Typically about three times the energy released seems to be needed(Hurley Tout amp Pols 2002)

Sources of energy other than the orbital energy are available but it is notyet established exactly how they might be tapped There is always ongoingnuclear burning around the giantrsquos core and indeed this energy is importantif it is assumed that the common envelope expands to fill the orbit as it formsand so is included surreptitiously in the formalism of Iben amp Tutukov but notin that of Webbink In general this requires that the timescale for common-envelope evolution be comparable with or longer than the thermal timescaleof the envelope so that the nuclearly generated energy is comparable withthe envelope binding energy It also requires an efficient means of convertingthis nuclear luminosity to the kinetic energy of mass loss and avoid radiationHan Podsiadlowski amp Eggleton (1994) include the ionization energy in thebinding energy of the envelope This greatly reduces what is required but tosuch an extent that the envelopes of many normal AGB star models are un-bound It is also difficult to see how this energy can be tapped in an envelopethat is hot enough to remain fully ionised Yet another source has been iden-tified by Ivanova amp Podsiadlowski (2001) During the formation of a commonenvelope a stream of hydrogen-rich material can penetrate to hot hydrogen-exhausted regions where rapid non-equilibrium burning takes place Indeedin their models often enough energy is released to destroy the envelope beforeany spiralling of the cores has begun

1155 Type Ia Supernovae

Luminous SNe Ia are amongst the brightest objects in the Universe and theiruse as standard candles by cosmologists has elevated the need to understandtheir progenitors The major energy source of SNe Ia is the decay of 56Nito 56Fe and the total energy released in a SN Ia is consistent with the de-cay of approximately a solar mass of 56Ni These facts strongly implicate thethermonuclear explosion of a white dwarf though the actual explosion mech-anism is not fully understood (Hillebrandt amp Niemeyer 2000) White dwarfsmay be divided into three major types (i) helium white dwarfs composedalmost entirely of helium form as the degenerate cores of low-mass red gi-ants which lose their hydrogen envelope before helium can ignite (ii) car-bonoxygen white dwarfs composed of about 20 carbon and 80 oxygenform as the cores of asymptotic giant branch stars or naked helium burningstars that lose their envelopes before carbon ignition and (iii) oxygenneonwhite dwarfs composed of heavier combinations of elements form from gi-ants that ignite carbon in their cores but still lose their envelopes before thedegenerate centre collapses to a neutron star

In binary systems mass transfer can increase the mass of a white dwarfClose to the Chandrasekhar mass (MCh asymp 144M) degeneracy pressure canno longer support the star that collapses releasing its gravitational energy TheONe white dwarfs lose enough energy in neutrinos and collapse sufficiently

11 Binary Stars 317

before oxygen ignites to avoid explosion (accretion induced collapse AIC)The CO white dwarfs on the other hand reach temperatures early enoughduring collapse (at a mass of 138M) for carbon fusion to set off a ther-monuclear runaway under degenerate conditions and release enough energyto create a SN Ia Accreting He white dwarfs reach sufficiently high temper-atures to ignite helium at M asymp 07M MCh (Woosley Taam amp Weaver1986) An explosion under these conditions is expected to be quite unlike aSN Ia

The process is further complicated by the nature of the accreting materialIf it is hydrogen-rich accumulation of a layer of only 10minus4 M or so leads toignition of hydrogen burning sufficiently violent to eject most if not all of ormore than the accreted layer in the novae outbursts of cataclysmic variablesThe white dwarf mass does not significantly increase and ignition of its in-terior is avoided However if the accretion rate is high M gt 10minus7 M yrminus1hydrogen can burn as it is accreted bypassing novae explosions (Paczynskiamp Zytkow 1978) and allowing the white dwarf mass to grow Though if it isnot much larger than this M gt 3 times 10minus7 M yrminus1 hydrogen cannot burnfast enough and accreted material builds up a giant-like envelope around thecore and burning shell that rapidly leads to more drastic interaction withthe companion and the end of the mass transfer episode Rates in the nar-row range for steady burning are found only when the companion is in theshort-lived phase of thermal-timescale expansion as it evolves from the end ofthe main sequence to the base of the giant branch Super-soft X-ray sources(Kahabka amp van den Heuvel 1997) are probably in such a state but withoutinvoking some special feedback mechanism such as disc winds (Hachisu Katoamp Nomoto 1996) cannot be expected to remain in it for very long and whitedwarf masses very rarely increase sufficiently to explode as SNe Ia

At first sight a more promising scenario might be mass transfer fromone white dwarf to another In a very close binary orbit gravitational radia-tion can drive two white dwarfs together until the less massive fills its Rochelobe If both white dwarfs are CO and their combined mass exceeds MChenough mass could be transferred to set off a SN Ia However if the massratio MdonorMaccretor exceeds 0628 mass transfer is dynamically unstablebecause a white dwarf expands as it loses mass Based on the calculations atsomewhat lower steady accretion rates Nomoto amp Iben (1985) have claimedthat the ensuing rapid accretion of material allows carbon to burn in mild shellflashes converts the white dwarf to ONe and ultimately leads to AIC and nota SN Ia They found a limit of one fifth of the Eddington accretion rate wasnecessary to avoid igniting carbon non-degenerately The Eddington accretionrate is that rate at which the outward radiation pressure that results from theenergy released as the material falls into the potential well of the star balancesthe gravitational attraction on an atom Even for stable mass transfer drivenby gravitational radiation this is exceeded Recently Martin Tout amp Lesaffre(2005) have found that the accretion limit for steady accretion is more liketwo-fifths of the Eddington rate and further that short periods of accretion at

318 C A Tout

much higher rates can be tolerated They showed that a 11M white dwarfcould accrete all the material from a companion white dwarf of 03M at thefull rate driven by gravitational radiation and still ignite degenerately at thecentre However there is no simple way to create a 03M CO white dwarfand accretion of helium rich material can lead to similar to but more extremeexplosions than novae We are still searching for the progenitors of SNe Iafrom among the diverse binary systems in the stellar zoo

References

Allen R H 1899 Star Names and Their Meanings Stechert New York 309Crawford J A 1955 ApJ 121 71 310Eggleton P P 1983 ApJ 268 368 305Eggleton P P 1985 in Pringle J E Wade R A eds Interacting Binary Stars

Cambridge Univ Press Cambridge p 21 308Eggleton P P 2006 Evolutionary Processes in Binary and Multiple Stars Cam-

bridge Univ Press Cambridge 306Fabian A C Pringle J E Rees M J 1975 MNRAS 172 15 297Goodricke J J 1783 Phil Trans R Soc London 73 474 309Hachisu I Kato M Nomoto K 1996 ApJ 470 L97 317Han Z Podsiadlowski P Eggleton P P 1994 MNRAS 270 121 316Herschel W 1803 Phil Trans R Soc London 93 339 297Hillebrandt W Niemeyer J C 2000 ARAampA 38 191 316Hoyle F 1955 Frontiers of Astronomy Heinemann London 309Hurley J R Tout C A Pols O R 2002 MNRAS 329 897 316Hut P 1980 AampA 92 167 301Hut P 1981 AampA 99 126 301Iben I Jr Tutukov A V 1984 ApJS 54 335 314Ivanova N Podsiadlowski P 2001 in Podsiadlowski P Rappaport S King A R

DrsquoAntona F Burderi L eds ASP Conf Ser Vol 229 Evolution of Binary andMultiple Star Systems Astron Soc Pac San Fransisco p 261 316

Kahabka P van den Heuvel E P J 1997 ARAampA 35 69 317Kippenhahn R Wiegert A 1967 Z Astrophys 65 251 311Kopal Z 1959 Close Binary Systems Chapman and Hall London 309Martin R G Tout C A Lesaffre P 2005 MNRAS 373 263 317Mestel L Spruit H C 1987 MNRAS 226 57 314Michell J 1767 Phil Trans R Soc London 57 234 297Montanari 1671 Prose di Signori Academici Gelati di Bologna (see Kopal 1959

p 12) 309Morton D C 1960 ApJ 132 146 310Nomoto K Iben I Jr 1985 ApJ 297 531 317Paczynski B 1966 AampA 16 231 311Paczynski B 1971 ARAampA 9 183 305Paczynski B 1976 in Eggleton P P Mitton S Whelan J eds Proc IAU

Symp 73 Structure and Evolution of Close Binary Systems Reidel Dordrechtp 75 314

Paczynski B Ziolkowski J 1967 AampA 17 7 311

11 Binary Stars 319

Paczynski B Zytkow A N 1978 ApJ 222 604 317Parenago P P 1950 Astron Zh 27 41 309Peters P C Mathews J 1963 Phys Rev 131 435 313Plavec M Kriz S Harmenec P Horn J 1968 Bull Astr Inst Czech 19 24 311Rasio F A Tout C A Livio M 1996 MNRAS 470 1187 301Sandage A R 1953 AJ 58 61 310Struve O 1948 Ann Astrophys 11 117 310Struve O 1949 MNRAS 109 487 310Tout C A Eggleton P P 1988 MNRAS 231 823 311Vogel N C 1890 Astron Nachr 123 289 309Walter K 1931 Konigsberg Veroff 2 (see Kopal 1959 p 545) 310Warner B 1995 Cataclysmic Variable Stars Cambridge Univ Press Cambridge 312Webbink R F 1984 ApJ 277 355 314Webbink R F 1985 in Pringle J E Wade R A eds Interacting Binary Stars

Cambridge Univ Press Cambridge p 39 306Wood F B 1950 ApJ 112 196 310Woosley S E Taam R E Weaver T A 1986 ApJ 301 601 317

12

N -Body Binary Evolution

Jarrod R Hurley

121 Introduction

It has long been recognized that binary stars represent a significant and im-portant population within a star cluster and are present from the time offormation (Hut et al 1992) As such binary stars have been included in N -body models of star cluster evolution for quite some time (Heggie amp Aarseth1992 for example) However these early models focused only on the dynami-cal evolution of binaries ndash orbital changes resulting from encounters with othercluster stars It was not until the emergence of rapid binary evolution algo-rithms (also called population synthesis codes Tout et al 1997 Yungelsonet al 1995) that facets of internal binary evolution such as mass-transfer werefollowed in N -body codes This chapter provides a description of how binaryevolution is treated in nbody4 and nbody6 and what is included in the al-gorithm It follows closely on from the overview of N -body stellar evolutiongiven in Chap 10 and the theory of binary stars presented in Chap 11 soit is strongly suggested that these are read beforehand The material in thischapter does not deal with dynamical considerations such as the transfor-mation of the two-body orbital elements to regularized variables for a moreaccurate treatment of close encounters the integration of hierarchical sub-systems and gravitational perturbations of binary orbits These are coveredin Chaps 1 and 3 as well as comprehensively in Aarseth (2003)

122 The BSE Package

The modelling of binary evolution in nbody4 and nbody6 follows closelythe Binary Star Evolution (BSE) algorithm presented in Hurley Tout amp Pols(2002) Before discussing the implementation of this algorithm in the N -bodycodes it will first be useful to give an overview of what it entails This willalso serve to give the reader some insight into how a prescription-based bi-nary evolution code operates BSE is the binary evolution analogue of the

Hurley JR N-Body Binary Evolution Lect Notes Phys 760 321ndash332 (2008)

322 J R Hurley

Single Star Evolution (SSE) package described in Chap 10 The SSE packageis fully incorporated within BSE and provides the underlying stellar evolutionof the binary stars as the orbital characteristics are evolved Throughout thedescription of the binary evolution algorithm given below references will bemade to SSE subroutines as listed in Table 102 of Chap 10

The first step in the evolution algorithm is to initialize the binary Thisrequires setting the masses of the two stars (which we will call M1 and M2)an orbital separation (or equivalently an orbital period) and an eccentricityIn the next section there will be some discussion of how these parameters canbe chosen from appropriate distribution functions but for now it is assumedthey are simply set to arbitrary values For the purposes of stellar evolutionthe metallicity Z is also required and it is generally assumed that this isthe same for the two stars Normally the evolution begins with both stars onthe zero-age main-sequence (ZAMS) and a separation such that the binary isdetached However beginning with evolved stars andor a semi-detached stateis possible A final consideration for the initialization phase is the spins orrotation rates of the stars Unless otherwise specified each star begins with aZAMS spin set by SSE according to the ZAMS stellar mass (this is based on afit to rotational data of observed main-sequence stars as described in HurleyPols amp Tout 2000) Other options such as starting the stars in co-rotationwith the orbit ie tidally locked are available

For the purposes of the algorithm the evolution of a binary is separatedinto two distinct phases

1 detached evolution if neither star is filling its Roche lobe2 roche evolution if one or both of the stars are filling their Roche lobes

The Roche-lobe radius is calculated using the expression given by Eggleton(1983) which depends on the mass-ratio of the stars and the orbital sepa-ration If the radius of a star exceeds its Roche-lobe radius it is deemed tobe filling its Roche lobe In its most basic form the algorithm can be seen asmoving the binary forward in time within the detached phase (according tosome chosen timestep) until one of the stars fills its Roche lobe and is thereforestarting to transfer mass to the companion star The evolution then switchesto the roche phase which deals with all facets of the evolution associated withmass transfer including contact and common-envelope evolution This mayonce again involve moving the binary forward through a series of timestepsor the outcome may be decided immediately Switching between the detachedand roche phases is permitted as is the possibility of following the evolutionof a single star after a merger event

Each iteration during a timestep Δt within the detached phase includesthe following steps (taken in turn)

bull calculate the stellar wind mass-loss rate from each star (via a call tomlwindf) and determine if any of this material is accreted by the com-panion

12 N -Body Binary Evolution 323

bull calculate the rate of change of the orbital angular momentum and eccen-tricity owing to stellar wind mass loss and accretion

bull calculate the rate of change of orbital angular momentum and eccentricityowing to gravitational radiation (only effective for separations less than10R)

bull calculate the change in the intrinsic spin of each star owing to mass changesand magnetic braking

bull calculate the rate of change of the spin of each star and the orbital eccen-tricity owing to tidal interactions between the stars and the orbital motion(spin-orbit coupling)

bull restrict Δt if necessary to ensure that the relative changes in stellar massspin angular momentum owing to magnetic braking and orbital angularmomentum owing to tides are less than 1 3 and 2 respectively

bull update the mass of each star and for main-sequence (MS) and sub-giantstars adjust the epoch parameter if necessary (see Chap 10 and HurleyTout amp Pols 2002 for usage)

bull update the intrinsic spin of each star with a check to ensure that the stardoes not exceed its break-up speed

bull update the orbital parameters (angular momentum separation period andeccentricity)

bull advance the time by Δtbull evolve each star to the current time using calls to starf and hrdiagf

in order to update the stellar parameters (stellar type radius core-massetc)

bull if a supernova has occurred call kickf and adjust the orbital parame-ters accordingly including a check that the orbit is still bound (the nextiteration is done with Δt = 0)

bull check if either star now fills its Roche lobe and switch to the roche phaseif this is true (if the Roche-lobe radius exceeds the stellar radius by morethan 1 the algorithm interpolates backwards until this condition is metbefore switching)

bull for an eccentric binary check if a collision is expected at periastron andswitch to the roche phase if this is true

bull choose a new Δt from the minimum of the current recommended stellarevolution timestep for each star (based on the stellar type and a require-ment that the radius changes by less than 10 see Chap 10)

bull start the next iteration

Note that if a single star emerges from the roche phase after a coales-cencemerger of the binary stars this new star will be evolved within thedetached phase Likewise if the binary becomes unbound the evolution oftwo single stars can be followed in the detached phase with the irrelevantsteps such as tidal evolution and mass accretion skipped

The general steps involved with each iteration of the roche phase are asfollows

324 J R Hurley

bull calculate the dynamical timescale for the primary star (the star filling itsRoche lobe)

bull determine if mass-transfer occurs on a dynamical timescale (dependenton the stellar types and the mass-ratio) and if this is true determine theinstantaneous outcome ndash either a single star or a post-common-envelopebinary ndash and switch back to the detached phase

bull otherwise the mass-transfer occurs on a nuclear or thermal timescale andthe algorithm proceeds by first calculating the amount of mass transferredfrom the primary per orbital period

bull determine what fraction of the mass transferred from the primary willbe accreted by the companion star ndash this depends on the nature of thecompanion star as well as the mass-transfer rate and includes intricaciessuch as novae eruptions

bull set Δt (based on a relative mass loss from the primary of 05)bull calculate the change in orbital angular momentum owing to mass loss

from the system during the mass-transfer (any mass not accreted by thecompanion) and adjust the spin angular momentum of each star owing tomass-transfer

bull calculate mass loss and accretion owing to stellar winds as for the detachedphase

bull calculate any changes to the orbital angular momentum and stellar spinsowing to stellar-wind mass changes magnetic braking gravitational radi-ation andor tidal interaction as for the detached phase

bull update the stellar spinsbull update the mass of each star and for the companion check for special cases

(such as the mass of a carbonndashoxygen white dwarf reaching the Chan-drasekhar mass which results in a type Ia supernova and a return to thedetached phase with only the primary remaining to evolve)

bull update the orbital parametersbull advance the time by Δt and evolve both stars to the current timebull if a supernova has occurred call kickf and if the binary has become

unbound return to the detached phasebull test whether or not the primary still fills its Roche lobe (return to the

detached phase if it does not)bull test if the companion fills its Roche lobe ie a contact binary (merge the

two stars and return to the detached phase to evolve the merger productif true)

bull start the next iteration of the roche phase

Details of the calculations and decision-making involved in each step of thealgorithm can be found in Hurley Tout amp Pols (2002) In most cases theseare based on expressions and theory sourced from the literature For examplethe equations that parameterize tidal evolution are taken from Hut (1981)with additions from Zahn (1977) and Campbell (1984) for tides raised onradiative and degenerate stars respectively Prescribed outcomes are derived

from the most accepted theory or models available at the time For examplemodels suggest that white dwarfs (WDs) composed primarily of oxygen andneon that reach the Chandrasekhar mass by accreting oxygen-rich materialwill collapse to form a neutron star (Nomoto amp Kondo 1991) Thereforethis is the outcome currently adopted in BSE If the theory changes or newmodels emerge suggesting a different outcome the algorithm is updated toreflect this Updates to the BSE algorithm since its publication in HurleyTout amp Pols (2002) include the addition of an expression to calculate if anaccretion disk is present during Roche-lobe overflow (as given by Ulrich ampBurger 1976) The disk itself is not modelled within BSE but its presence isaccounted for when making changes to the orbital angular momentum Futureupdates might include an extension of the Roche-lobe treatment to includenon-circular theory along the lines of Sepinsky et al (2007)

As with the SSE package BSE can be obtained by downloading it fromhttpastronomyswineduaujhurleybsedloadhtml or by contactingthe author Within this package the steps describing the detached and rochephases are contained in the evolv2f subroutine The package also containsa subroutine comenvf to deal with common-envelope evolution this is calledfrom evolv2f during the roche phase if the mass-transfer is deemed to bedynamical and the primary is a giant-like star If the binary evolves intocontact (both stars filling their Roche lobes) the two stars are merged and thesubroutine mixf is called to determine the outcome after complete mixingAn additional routine gntagef is included to calculate the parameters of thenew star that results from such a merger or from coalescence during common-envelope evolution

Parameterized binary evolution naturally involves a number of input pa-rameters that reflect uncertainties in the underlying theory These can affectthe evolution and outcomes An example in BSE is the common-envelope pa-rameter α which determines the efficiency with which energy is transferredfrom the orbit to the envelope surrounding the two stellar cores as they spi-ral towards each other Other parameters affect aspects of the evolution suchas mass accretion from a stellar wind mass ejected in a nova explosion andthe change in orbital angular momentum when mass is lost from the binarysystem during mass-transfer These features will be returned to in the nextsection and full descriptions can be found in Hurley Tout amp Pols (2002)

To evolve a population of binaries using the BSE population synthesis algo-rithm is a straightforward process It simply involves taking each binary inturn evolving it to the desired physical time (such as the age of the Galaxy)and recording the outcome Thus only one call to evolv2f is required foreach binary In an N -body code it is not so straightforward as the binaryevolution must be performed in step with the dynamical evolution of the star

326 J R Hurley

cluster If the mass of a binary changes owing to mass transfer this mustbe communicated to the dynamical interface of the code with minimal delayso that the gravitational force calculations remain accurate Conversely dy-namical interactions between a binary and cluster stars can lead to perturba-tions that alter the orbital parameters of the binary including disassociationwith consequences for the binary evolution outcomes Binary evolution withinthree- and four-body sub-systems must also be accounted for (see Chap 3)as well as the possible existence of non-primordial binaries that form duringthe cluster evolution The binary evolution treatment must also interface withthe regularization methods that are used to follow accurately the dynamicalevolution of binaries sub-systems and close encounters (see Aarseth 2003)

In nbody46 the tasks performed in the BSE subroutine evolv2f aresplit with the detached phases implemented in mdotf and the roche phasescontained in the rochef subroutine Stars in a binary have their individualtev values (time of next stellar evolution update) set equal (to the minimumof the two) so that they will be evolved together within mdotf This al-lows corrections to the spin and orbital angular momentum owing to stellarwind mass changes to be performed as the stars are evolved Gravitationalradiation for short-period detached binaries is taken care of by the subroutinegrradf from mdotf Similarly tidal interactions within circular binaries areaccounted for by bsetidf ndash tidal circularization of eccentric binaries is dealtwith elsewhere as part of the two-body regularization process (see below)The subroutine brakef is then used by mdotf in order to update the bind-ing energy of the binary and re-scale the associated two-body regularizationvariables after any orbital changes

Decision-making for binaries is aided by assigning the centre-of-mass par-ticle for each binary its own tev0 and tev values Here tev is the expectedtime of the next mass-transfer update the next call to rochef for the binaryFor detached binaries this will be the time when one of the component starshas evolved to fill its Roche lobe and is estimated by the subroutine trflowf(called from mdotf each time a stellar evolution update is performed for thecomponent stars) For a semi-detached binary in an ongoing Roche-lobe over-flow phase this will be set in rochef (see below) The binary tev values areincluded in setting TMDOT (the smallest tev) and if mdotf is called owing totev(i) being less than the current time where i represents a centre-of-massparticle1 the evolution update switches to rochef (called from mdotf)

The subroutine rochef includes all of the processes outlined in the rochephase of the BSE algorithm with a few N -body related additions First asmentioned above a steady mass-transfer phase must now be dealt with ina piece-wise fashion so that the binary evolution time does not get too farahead of the dynamical time This is put into place using the tev and tev0

1For a system of N stars and NBIN binaries the centre-of-mass particle for binaryj sits at position i = N + j in the various arrays The component stars sit at(2 times j)minus1 and 2timesj while the single stars occupy the (2 times NBIN)+1 to N positions

variables each call to rochef evolves the binary from tev0(i) rarr tev(i)unless something happens within the interval such as a merger Before exitingrochef the routine sets tev0(i) = tev(i) and updates tev(i) If rochefsignals termination because the primary star no longer fills its Roche lobethis is done with a call to trflowf Otherwise tev(i) is set to the currenttime plus some multiple of the current mass-transfer timestep (as describedin the previous section) This multiplication factor is in the range of 10ndash50depending on whether or not the binary has a nearby perturber The updateof tev(i) also takes into account any major stellar evolution changes on thehorizon for the component stars such as an impending supernova explosion

Analogous to the stellar type index used to describe the evolution stateof individual stars there is also a kstar index for the binary centre-of-massparticle that describes the current state of each binary This takes on valuessuch as 0 for a standard eccentric binary minus2 for a circularizing binary and 10for a circular binary The first time that a binary enters rochef the kstarindex is set to 11 and when the binary next becomes detached it is set to12 Subsequently kstar is increased by one each time a binary switches froma detached to a roche phase and vice-versa such that kstar(i) = 16 wouldindicate that binary i minus N is currently detached but has previously evolvedthrough three distinct roche phases

Another addition to the N -body version of the roche process is the subrou-tine coalf which is called from rochef when mass-transfer has ended incoalescence of the two stars This routine takes care of the associated N -bodybook-keeping such as removing the second star and the centre-of-mass particlefrom the relevant arrays and performing the necessary force corrections

Unlike isolated binary evolution the cluster environment provides for theformation of non-standard binary configurations through dynamical interac-tions An example would be an eccentric binary that emerges from a four-bodyhierarchy with one of the stars filling its Roche lobe If such a binary entersrochef it is currently dealt with by first calculating the tidal circularizationtimescale and if this is less than 10Myr calling bsetidf to circularize thebinary before proceeding with the mass-transfer process

Some of the subroutines associated with the roche phase are also utilizedvia an nbody46 subroutine cmbodyf This is called from various parts ofthe N -body code when a hyperbolic collision or a collision at periastron in aneccentric (and non-Roche-lobe filling) binary is detected If one or both of thestars involved in the collision is a sub-giant or giant cmbodyf calls expelfwhich in turn calls comenvf to determine the outcome via common-envelopeevolution Otherwise the two stars are merged directly with mixf whichdetermines the outcome If this results in the formation of a new giant starthe BSE routine gntagef is used to set the appropriate age and initial massto match the core-mass and mass of the star (this routine is also used bycomenvf and rochef when needed)

The main difference between the treatment of binary evolution within BSEand that of the N -body codes relates to how tidal interactions for eccentric

328 J R Hurley

binaries are dealt with Mardling amp Aarseth (2001) have developed algorithmsthat combine tidal circularization neatly with the two-body regularizationmethod for following the orbital evolution of binaries These algorithms alsocope with N -body complications such as the orbit of an eccentric binary be-coming chaotic owing to perturbations The subroutines involved are tcircfand spiralf (as well as some subsidiary routines) There is also a relatedsubroutine synchf which models tidal synchronization The underlying the-ory for tides in the Mardling amp Aarseth (2001) algorithm is Hut (1981) as itis in BSE so the two treatments are consistent However the option to modeltidal circularization within nbody46 using the BSE algorithm may be addedin the future for the sake of completeness

Subroutines in nbody46 that are directly related to binary evolution aresummarized in Table 121 The only one not yet mentioned above is rlfwhich contains the Eggleton (1983) function for calculating the Roche-loberadius of a star

An important facet of binary evolution is setting the initial parameters ndashfor a population of binaries this is critical in determining the range of outcomesthat are possible In the case of a star cluster the relative number of tightlybound binaries is an important factor in how the cluster itself will evolve Thefirst step towards initializing a population of primordial binaries in nbody46

is to decide how many are to be included This is set by the parameter NBIN0read from the input file in the dataf subroutine If NBIN0 is non-zero thesubroutine binpopf generates the parameters of the NBIN0 binaries Thisinvolves a number of choices that are controlled by a line of input variablesread from the input file in binpopf These include SEMI0 ECC0 RATIO RANGEand ICIRC Both SEMI0 and RANGE affect the semi-major axes of the binariesif RANGE is negative the log-normal distribution from Eggleton Fitchett ampTout (1989) is used with a peak at SEMI0 (in AU) if RANGE is positive auniform logarithmic distribution is used with a maximum of SEMI0 (in N -body units) and covering RANGE orders of magnitude and if RANGE = 0 SEMI0is the semi-major axis of all binaries The input variable ECC0 determines theeccentricity distribution (constant or thermal distribution) and RATIO controls

Table 121 Subroutines in nbody4 and nbody6 associated with binary evolution

BSE-related Other

bsetidf brakef

comenvf cmbodyf

gntagef coalf

grradf expelf

mdotf tcircf

mixf trflowf

rlf spiralf

rochef synchf

how the masses of the two stars are assigned from the binary mass (see alsoimff) If the variable ICIRC is non-zero pre-MS eigen-evolution of the orbitalparameters is invoked (Kroupa 1995)

There are also a number of input options that affect binary evolution andrelated diagnostic output The option kz(34) must be set non-zero for binaryevolution (Roche-lobe mass-transfer and tides) to occur If kz(34) = 1 tidalsynchronization of circular binaries is performed using synchf otherwise it isperformed using bsetidf The option kz(6) controls the level of diagnosticoutput for regularized binaries and kz(8) affects output relating to primordialbinaries To date input parameters in BSE that affect particular aspects of thebinary evolution algorithm are not included as input variables in nbody46Instead they are hardwired into the various subroutines where they are usedFor example the common-envelope efficiency parameter mentioned in theprevious section is set in the header of comenvf while a number of parametersare set in rochef ndash the fraction of accreted mass that is ejected from thesurface of a WD in a nova explosion (EPSNOV) the Eddington-luminosity factor(EDDFAC) and the stellar-wind velocity factor (BETA) to name a few

This completes the overview of how binary evolution is treated in nbody4

and nbody6 It is by no means a comprehensive description but should givethe interested user enough information to get started More details can befound in Aarseth (2003) and Hurley et al (2001)

124 Binary Evolution Results

The colour-magnitude diagram (CMD) of a binary-rich nbody4 simulationis shown in Fig 121 This simulation started with 28 000 stars and a 40primordial binary fraction The initial separations (or equivalently orbital pe-riods) of the binaries were drawn from the Eggleton Fitchett amp Tout (1989)distribution with a peak at 10AU and a maximum of 100AU The modelshown is at an age of 4 000Myr when the binary fraction is still at about40 ndash preservation of the primordial binary fraction is a common featureof star cluster evolution noted in Hurley Aarseth amp Shara (2007) Howeveras the cluster evolution progresses it becomes increasingly likely that a sig-nificant component of the binary population will be non-primordial For themodel in Fig 121 about 20 of the binaries are non-primordial and these areprimarily the result of exchange interactions The exact proportion of binariesformed by dynamical processes depends on factors such as the fraction of bi-naries in relatively wide orbits the cluster density and the stage of evolutionFigure 121 can be compared to the CMD at 4 000Myr shown in Fig 101of Chap 10 for a simulation starting with 30 000 stars and 0 binaries Theeffects of binary evolution on the locus of points in the CMD is clearly seenand the result is much closer to the reality presented by the observations ofopen clusters (Fan et al 1996 for example)

330 J R Hurley

Fig 121 Colour-magnitude diagram after 4 000Myr of evolution for a Z = 002nbody4 simulation that started with 12 000 single stars and 8 000 binaries At4 000Myr there are 3 382 single stars and 2 360 binaries in the model cluster Eachbinary is shown as a single point ie unresolved The luminosity and effective tem-perature provided for each star by SSEBSE have been converted to magnitude andcolour using the bolometric corrections given by the models of Kurucz (1992) andin the case of white dwarfs Bergeron Wesemael amp Beauchamp (1995)

Some features to note in Fig 121 include the broadening of the MS owingto the presence of MSndashMS binaries with the upper edge defined by the equal-mass binaries Similar behaviour can be seen for the WD sequence owing toWDndashWD binaries Points below the MS but distinct from the WD sequenceare MSndashWD binaries These evolve away from the WD sequence and towardsthe MS as the WD cools and the MS star comes to dominate the colourThe points that form an extension of the MS hotter and bluer than the MSturn-off represent blue stragglers (BSs) These are MS stars that have longercentral hydrogen-burning lifetimes than expected for their mass That is to sayif these stars were born in the cluster with their current mass (or higher) theywould already have evolved away from the MS to become giants or WDs Theirpresence is explained by obtaining their current mass either through steadymass-transfer in a short-period binary or as the result of a merger of twoMS stars Either way they are a product of binary evolution In Hurley et al(2005) nbody4 models were used to demonstrate how the combination of thecluster environment and close binary evolution could explain the number and

N

rp R

Fig 122 Distribution of periastron Rp = a (1 minus e) where a is the semi-major axisand e the eccentricity for the 8 000 primordial binaries in the NBODY4 simulationdescribed in Fig 121 (solid line) the binaries remaining in this simulation after4 000Myr (dashed line) and the primordial binaries evolved to the same age usingBSE only (dotted line) Each distribution is normalized to a maximum of unity

nature of the BSs observed in the old open cluster M67 This included theproduction of BSs in eccentric binaries which cannot be explained by binaryevolution alone

Figure 122 shows the periastron distribution for the binaries in thenbody4 simulation of Fig 121 and compares this to the primordial dis-tribution as well as the distribution obtained when the binaries are evolvedto the same age using BSE only We see from comparing the latter two distri-butions that binary evolution steadily removes binaries with short periastrondistances However the nbody4 distribution shows that a star cluster is ef-fective in replenishing the relative numbers of interacting binaries This isdone at the expense of the wide binaries which are broken up in encounterswith other cluster members In closing it is noted that binary evolution isimportant for proper accounting of the orbital properties of the binary popu-lations of star clusters especially as the presence of binaries and in particulartightly bound binaries can critically affect properties such as the structureand lifetime of a star cluster

References

Aarseth S J 2003 Gravitational N-Body Simulations Cambridge Univ PressCambridge 321 326 329

Bergeron P Wesemael F Beauchamp A 1995 PASP 107 1047 330

332 J R Hurley

Campbell C G 1984 MNRAS 207 433 324Eggleton P P 1983 ApJ 268 368 322 328Eggleton P P Fitchett M Tout C A 1989 ApJ 347 998 328 329Fan X et al 1996 AJ 112 628 329Heggie D C Aarseth S J 1992 MNRAS 257 513 321Hurley J R Pols O R Tout C A 2000 MNRAS 315 543 322Hurley J R Tout C A Aarseth S J Pols O R 2001 MNRAS 323 630 329Hurley J R Tout C A Pols O R 2002 MNRAS 329 897 321 323 324 325Hurley J R Pols O R Aarseth S J Tout C A 2005 MNRAS 363 293 330Hurley J R Aarseth S J Shara M M 2007 ApJ 665 707 329Hut P 1981 AampA 99 126 324 328Hut P McMillan S Goodman J Mateo M Phinney E S Pryor C Richer H B

Verbunt F Weinberg M 1992 PASP 104 981 321Kroupa P 1995 MNRAS 277 1507 329Kurucz R L 1992 in Barbuy B Renzini A eds Proc IAU Symp 149 The Stellar

Populations of Galaxies Kluwer Dordrecht p 225 330Mardling R A Aarseth S J 2001 MNRAS 321 398 328Nomoto K Kondo Y 1991 ApJ 367 L19 325Sepinsky J F Willems B Kalogera V Rasio F A 2007 ApJ 667 1170 325Tout C A Aarseth S J Pols O R Eggleton P P 1997 MNRAS 291 732 321Ulrich R K Burger H L 1976 ApJ 206 509Yungelson L Livio M Tutukov A Kenyon S J 1995 ApJ 447 656 321Zahn J-P 1977 AampA 57 383 324

13

The Workings of a Stellar Evolution Code

Ross Church12

1 University of Cambridge Institute of Astronomy Madingley RoadCambridge CB3 0HA UKrpc25srcfucamorg

2 Centre for Stellar and Planetary Astrophysics Monash UniversityPO Box 28M Clayton Victoria 3800 Australia

131 Introduction

Models of stellar clusters link the theoretical gravitational N -body problemto the study of real astrophysical systems Such models require a descriptionof the stars contained within the cluster Stars are interesting objects in theirown right and the study of stellar evolution is important across astronomyfrom the formation of exotic objects such as X-ray binaries and gamma-raybursts to measuring the ages of galaxies

The physical processes important for stellar evolution theory as well asqualitative results are discussed elsewhere in this book Here the technicalproblem of computing the structure and evolution of the stars is consideredHow can we solve the set of differential equations that describe the interiorof a star to obtain a model of its physical properties A brief mention will bemade of some of the uncertainties in stellar physics and how they affect theresults obtained

The stellar evolution code used as an example in this text is stars the Cam-bridge Stellar Evolution Code Written originally by Peter Eggleton (1971)it is widely used by astronomers working in the field of stellar evolutionIt has the advantage of being relatively concise and simple in its construc-tion owing mainly to the elegant treatment of meshpoint placement and con-vective mixing The code itself can be downloaded from httpwwwastcamacukresearchstars

132 Equations

In order that a star can be modelled efficiently for its entire lifetime whichgreatly exceeds its dynamical timescale simplifying physical assumptionsmust be made The star is usually taken to be spherically symmetric and in hy-drostatic equilibrium This reduces the problem to a single spatial dimensionbut necessitates that the process of convection be treated empirically These

Church R The Workings of a Stellar Evolution Code Lect Notes Phys 760 333ndash345 (2008)

334 R Church

assumptions lead to the four equations of stellar structure A detailed deriva-tion of these equations can be found in any standard text on stellar struc-ture and evolution for example Schwarzschild (1965) Cox amp Giuli (1968)Kippenhahn amp Weigert (1994) Prialnik (2000) as well as Chap 9 of thisbook In summary the equations are

dmdr

= 4πr2ρ (131)

dPdr

= minusGmρ

r2 (132)

dTdr

=

⎧⎪⎪⎨

⎪⎪⎩

minus 3κρL16πacr2T 3

(radiative regions)

nablaaT

P

dPdr

+ ΔnablaT (convective regions)

(133)

dLdr

= 4πr2ρε (134)

where m is the mass within radius r of the centre of the star P the pressureρ the density L the luminosity T the temperature and κ the Rosseland meanopacity The adiabatic temperature gradient nablaa is calculated from the equa-tion of state of the star whilst the superadiabatic temperature gradient ΔnablaTis obtained from mixing length theory The energy liberation rate per unitmass ε contains contributions from gravitational expansion and contractionnuclear reactions and neutrino emission

In addition to the equations of stellar structure it is necessary to takeinto account composition changes owing to nuclear burning and mixing Theprocess of mixing can be modelled as diffusion with an appropriate coefficientThis leads to a set of equations for the evolution of the chemical composition

partXi

partt=

mi

ρ

⎛

⎝sum

j

Rji minussum

k

Rik

⎞

⎠minus part

partr

(

σ2 partXi

partr

)

(135)

where Rij is the rate of conversion of element i into element j per unit volumemi is the atomic mass of element i and σ is a diffusion coefficient usuallyobtained from mixing length theory

The central boundary conditions of a stellar model are straightforward at thecentre m = 0 r = 0 and L = 0 although in practice stars does not use acentral meshpoint The surface of the star is placed where the temperatureequals the effective temperature given by

L = 4πR2σT 4eff (136)

for a star of luminosity L and radius R

13 Stellar Evolution Code 335

The Eddington closure approximation together with a thin grey atmo-sphere is used to obtain the gas pressure at the surface

Pg =23g

κ

(

1 minus L

LEdd

)

(137)

where LEdd is the limiting Eddington luminosity and g the surface gravityThe total mass of the star equal to the value of m at the outermost meshpointchanges according to

dMdt

= minusW (138)

where W is the stellar wind There is no general theory of stellar winds anda number of empirically determined formulae are commonly used During themain-sequence phase all but the most massive stars are assumed to lose nomass the solar wind being evolutionarily negligible For the red-giant phasethe formula of Kudritzki amp Reimers (1978) is commonly used whereas on theasymptotic giant branch (AGB) the formulae of Blocker (1995) and Vassiliadisamp Wood (1993) are popular

133 Variables and Functions

A stellar model is defined in terms of a set of independent variables1 Thephysical variables used in stars are log T logm log r L log f X1H X4HeX12C X16O and X20Ne The first four are standard physical quantities definedabove Note that the luminosity can be negative and hence its logarithm can-not be used The quantity f is a function of the electron degeneracy parameterψ and is explained in Sect 1332 The composition of the star is measured bythe mass fractions Xi of various isotopes Because the mass fractions must sumto unity these numbers also determine the mass fraction of another isotope14N All other compositions are assumed to either be constant for exampleiron or zero

1331 The Mesh

Whilst a real star is continuous a computer can only hold a finite quantityof data and hence the star must be discretised on to a mesh of points Theplacement of these points is crucial to the functionality of the code Areasof interest in a star must be sufficiently resolved in particular the burning

1Speaking in a strictly mathematical sense there are only two independent vari-ables in the problem m and t All the other variables are dependent on these im-plicitly through the equations listed in Sect 132 To explicitly define a model of astar however one needs values of all the 11 variables and it is possible to vary thesevariables independently the resulting model may not however represent a physicalstar Hence it is reasonable to refer to them as independent variables

336 R Church

shells in giants and ionisation zones in the envelope The use of too manymeshpoints however increases the memory requirements and slows the codedown An Eulerian mesh of points at constant radii performs poorly becausethe stellar radius can change by several orders of magnitude over the starrsquos life-time Meshpoints placed at constant mass co-ordinates to form a Lagrangianmesh work better but then the points must be moved as the evolution pro-ceeds to keep interesting parts of the star well resolved A unique feature ofstars is that the mesh is positioned automatically by the equation solvingpackage A further equation is solved by the code to make the gradient withrespect to the meshpoint number of a function Q constant throughout thestar The function is chosen to cause points to be placed in regions of physicalsignificance The form usually adopted is

Q = c4 log(P ) + c5 log(P + c9P + c1

)

+ c2 log(P + c10P + c1

)

+ c7 log(

T

T + c11

)

+ log(

c6M23

c6M23 +m23

)

+ c3 log(r2

c8+ 1

)

(139)

where the constants ci are chosen by the user Because

C =dQdk

=dQdm

dmdk

(1310)

is constant the mass resolution which is inversely proportional to dmdkis largest where Q varies most quickly with mass Given appropriate val-ues of the coefficients the second and third terms have the effect of drivingmeshpoints into the hydrogen and helium burning shells This substantiallyimproves numerical stability during thermal pulses on the AGB

1332 The Equation of State

It is necessary to have an equation of state for the material that makes up astar A common approach is to use a set of tables for different temperaturesdensities etc stars conversely utilises the semi-analytic equation of statedescribed by Pols et al (1995) Contributions to the Helmholtz free energyfrom radiation ions and electrons are considered along with some non-idealeffects The Fermi-Dirac integral over the momentum states of the electronis simplified by working with the quantities f and g chosen so that a powerseries therein has the correct asymptotic form for limiting values of ψ and T The quantities f and g are defined by

ψ = 2radic

1 + f + logradic

1 + f minus 1radic1 + f + 1

(1311)

andg =

kT

mec2

radic1 + f (1312)

Full details of the series can be found in Eggleton Faulkner amp Flannery (1973)Although most of the equation of state is calculated in real time there

are still a few tabulated quantities The opacities are too complicated to becalculated analytically likewise the nuclear reaction and neutrino loss ratesThese are included as tables of numerical values bicubic spline interpolationis used within the opacity tables

134 Method of Solution

The Henyey Forbes amp Gould (1964) relaxation method solves the equationsof stellar structure and evolution by making small changes to the structureobtained at the previous timestep and adjusting the resulting model until itsolves the equations This use of information from a previous timestep greatlyimproves the speed of calculations over a simple shooting method and is usedin almost every modern stellar evolution code

If the subscript i is allowed to run over the set of Ne equations at Np

meshpoints and the subscript j over the Nv variables at Np meshpoints bybringing all the terms on to one side of the equations of the solved code canbe written implicitly as

Ei(vj) = 0 (1313)

Then for a complete stellar model vj the degree to which it does not satisfythe equations is

δEi = Ei(v) (1314)

The model from the previous timestep is used as an initial guess for v Bynumerical differentiation of each equation with respect to each variable onecan obtain

Aij =partEi

partvj (1315)

Most of the entries in A vanish Because the equations are either first orsecond order spatially an element in A depends only on values within theadjacent one or two meshpoints hence A is block-diagonal This enables it tobe economically inverted and corrections to the variables are calculated as

δvj = Aminus1ji δEi (1316)

This process is iterated in a manner analogous to the Newton-Raphson methoduntil the convergence criterion is met It is required that the average change inδvj in a single iteration is less than a user-supplied constant In practice thisprocedure is sometimes slightly modified to improve stability of the solutionmethod Only part of the correction is applied under some circumstancesto prevent the solution being overshot This is equivalent to reducing themagnitudes of the eigenvalues of the iteration matrix It is also usually betterto use vj + δvj from the previous timestep as a first guess rather than vj thatis to start the iteration with the changes applied at the previous timestep

338 R Church

1341 Timesteps

The timestep δτi that the code uses is determined by an ad-hoc formula

δτi = δτiminus1 timesΔ

sumjk |δXjk|

(1317)

where δτiminus1 is the previous timestep δXjk the change in variable j at mesh-point k and Δ is a user-supplied constant The sum is evaluated over thevariables omitting the luminosity because this fluctuates too much to be use-ful A larger value of Δ allows the variables to change more in a single timestepand hence larger timesteps to be taken Because the change at a single mesh-point is independent of the number of meshpoints it is necessary to scale Δlinearly with the number of meshpoints different values are appropriate todifferent phases of evolution In the standard case of 199 meshpoints Δ = 5provides adequate results

If the iterative process fails to find a set of values for the variables thatsatisfy the equations with sufficient accuracy a model is deemed to have notconverged The code reverts to the previous model and the timestep is reducedby a factor of 08 Multiple reductions in timestep are possible for a systemthat is failing to converge but when the timestep has fallen below 1 of itsfirst tried value the code stops attempting to converge

A graph of the variation of the timestep with model number during theevolution of a 1M star is shown in Fig 131 One can see that it has a

1

100

104

106

108

Δt

yr

0 500 1000 1500 2000Model number

Fig 131 Variation of the timestep during the evolution of a 1 M star The modelnumber is plotted on the abscissa this increments by unity for each converged stellarmodel The ordinate shows the timestep in years This model was run with 199meshpoints and Δ = 5 throughout

very large dynamic range there is a difference of approximately 109 betweenthe shortest and the longest timesteps The initial peak in the timestep andthat around model number 600 are the main sequence and horizontal branchrespectively The timestep increases substantially again towards the end ofthe run as the star descends the white dwarf cooling track The discontinuityaround model 500 represents pseudo-evolution through the helium flash (seeSect 1361) and the period of short timesteps from model 1000 onward onthe post-AGB

135 The Structure of STARS

stars comprises 20 subprocedures which can be divided up into four groupsthe solution package physics package the flow control routines and the initialsetup routines as well as a few vestigial routines The solution package consistsof the following procedures

bull solver which solves the implicit matrix equation (see (1313))bull difrns which differentiates the equations to be solvedbull elimin8 which carries out some matrix manipulations andbull divide which implements matrix inversion

The physics package contains

bull equns1 which calculates the values of the difference equations and theirboundary conditions

bull funcs1 which calculates various quantities from the principal variablesmostly for use in equns1

bull statef which evaluates the equation of state at a given meshpointbull statel which decides whether it is necessary to call statefbull fdirac which evaluates Fermi-Dirac integralsbull pressi which approximates pressure ionisationbull opacty which does spline interpolation within the opacity tables andbull nucrat which calculates nuclear reaction rates

The flow control routines are

bull main which provides the main integration loop and basic flow controlbull printa which determines the next timestep updates the matrix controls

input and output and does sundry minor tasks for which there is no obviousalternative location and

bull printb which writes most of the output files

Finally the initial setup routines are

bull opspln which sets up the opacity tablesbull spline which calculates spline coefficientsbull remesh which attempts to remesh the model to a different grid

340 R Church

main

nucrat

consts

opspln

remesh

compos

printb

spline

statel

statef

fdirac

pressi

opacty

equns1 difrns

elimin8

divide

printa solver

funcs1

SolutionPackage

PhysicsPackage

Initial setup

Fig 132 A schematic illustration of the operation of the stars code Arrowsindicate the direction in which one subroutine calls another The division of thecode into sections with different functionality is shown

bull consts which sets up physical constants andbull compos which sets small or negative compositions to zero

The interaction of the first three groups of these subroutines can be seenin Fig 132 Note that the physics package is called via funcs1 from severalplaces in the rest of the code

136 Problematic Phases of Evolution

The iterative procedure that stars uses to converge a model is not guaranteedto arrive at a solution Usually the desired solution is sufficiently close to thestarting model that it does so but in some situations this is not the caseProblematic phases of evolution are mostly those where the structure of thestar is changing quickly As well as requiring small timesteps such phasesof evolution often cause the mesh to move rapidly through the model Theadvection terms in the equations that are included to deal with movementof the mesh are then large in magnitude but opposite in sign This causesnumerical problems

Phases of evolution that routinely cause problems are the helium flashthermal pulses on the AGB the post-AGB degenerate carbon ignition insuper-AGB stars heavy element burning subsequent to neon ignition and thevery late stages of white dwarf evolution Brief notes on how these problemscan be tackled are given below

1361 The Helium Flash

In stars of M 23M the core is degenerated at the time of helium ignitionThe increased temperature owing to helium burning does not cause expansionand thermonuclear runaway occurs (Schwarzschild amp Harm 1962) This isthe helium flash To circumvent these problems one can use an empiricalprocedure to construct approximate post-flash models with stable core heliumburning A star of mass M 3M that has evolved successfully throughnon-degenerate core helium ignition is taken and matter removed from theenvelope until the desired mass is reached The hydrogen burning shell isallowed to burn outwards with helium consumption disabled in order to obtainthe correct core mass The envelope compositions are reset to their pre-flashvalues and normal evolution is resumed Whilst not physically rigorous thisprocess provides models that can be used to study subsequent evolution

1362 The AGB

Evolution through thermal pulses on the AGB using stars is possible butonly with a modified version of the code and considerable effort (StancliffeTout amp Pols 2004) An easier though less accurate approach is to avoidmodelling the pulses A relatively low resolution of 199 meshpoints per modeland a comparatively large value of the timestep control parameter Δ = 5suppress thermal pulses on the AGB Their exclusion changes the compositionof material ejected in stellar winds and for the more massive AGB stars themass of the core and hence the final white dwarf mass

1363 Late Stages of Intermediate-Mass and High-Mass Stars

The problems in the late stages of the lives of intermediate-mass and high-massstars are more tricky to deal with Degenerate carbon ignition in lower-masssuper-AGB stars and the post-AGB cannot be avoided as thermal pulses andthe helium flash can Stars that ignite carbon mildly degenerately probablygo on to form oxygen-neon white dwarfs although the most massive amongstthem may end their lives as neutron stars The post-AGB is the final stageof evolution of AGB stars and it is reasonable to assume that once a starreaches this point it forms a white dwarf

Heavy element burning is only really of interest for the calculation of pre-supernova models Very little stellar evolution significant for N -body calcula-tions takes place after the ignition of neon and it is reasonable to terminate

342 R Church

a starrsquos evolution at this point Likewise problems in the evolution of whitedwarfs mostly occur at times comparable with the Hubble time In any casethe bulk properties of the star change very little after this point

137 Robustness of Results

The theory of stellar structure and evolution contains substantial uncertaintyIn particular some of the input physics is not well determined Convection is athree-dimensional process and the one-dimensional mixing length theory usedto approximate it cannot be entirely accurate Mixing length theory containsa free parameter α related to the length scale of convective plumes Its valueis usually obtained by fitting a solar model but there is no reason why itshould not vary between stars of different masses or in different evolutionaryphases There is substantial evidence that for many stars the amount of mix-ing predicted by the Schwarzschild criterion is insufficient and that processesthat cause extra mixing occur in stars Some candidates for these are stellarrotation convective overshooting and internal gravity waves Nuclear reactionrates even some of those most important to the structure of a stellar modelare substantially uncertain For example the rate of the 14N(p γ) reactionthat forms the slowest step in the CNO cycle is uncertain to approximatelya factor of 2 (Herwig Austin amp Lattanzio 2006) There is no general theoryof stellar mass loss so it is necessary to use empirically measured values ofquestionable accuracy There are also uncertainties in the opacity of stellarmaterial and in models of stellar atmospheres

To illustrate briefly the effects of two of these uncertainties a set of stellarmodels with varying input physics are presented here Models of masses 1M2M 4M 8M and 16M have been calculated varying two uncertainphysical parameters In one set of models extra mixing was added accordingto the prescription of Schroder Pols amp Eggleton (1997) In the the other therate of the 14N(p γ) reaction was doubled This is the slowest step in the CNOcycle and hence determines how fast hydrogen burning occurs according tothat process

1371 HR Diagrams

The effects on the HR diagram of changing the input physics are largestin the case of the 4M and 8M stars HR diagrams for these two starsare presented in Fig 133 It can be seen that changing the degree of extramixing has a dramatic effect on the position of the blue loop (horizontalbranch) in the HR diagram The increased mixing draws more hydrogen intothe core increasing its size and hence the luminosity of the star There is alsoa slight but much less pronounced difference when the CNO burning rate ischanged

2 2

2 4

2 6

2 8

3

log 1

0(L

L)

log 1

0(L

L)

3 63 844 2

StdMixingCNO rate

3 5

4

log 1

0(L

L)

log 1

0(L

L)

3 63 844 24 4

log10(T K)log10(T

StdMixingCNO rate

Fig 133 HR diagrams for stellar models of mass 4 M (top panel) and 8 M(bottom panel) The thick solid line is the standard model the dashed line the modelwith extra mixing and the dotted line the model with the enhanced CNO burningrate

1372 Stellar Lifetimes

The effect of increased mixing and the enhanced CNO rate on main-sequencelifetimes is shown in Fig 134 Stars spend the majority of their lives onthe main sequence and hence this time is a useful measure It also has theadvantage of being better defined than the total stellar lifetime

The main effect that can be seen is that models more massive than the Sunwith extra mixing have substantially increased lifetimes This is because theirconvective cores are enlarged by the extra mixing The cores have more fuelto burn and hence the main sequence is prolonged As the 1M model has aradiative core it is unaffected by changing the degree of convective mixing

344 R Church

minus10

0

10

20

30

Per

cent

age

chan

gein

lifet

ime

Per

cent

age

chan

gein

lifet

ime

1 2 5 10 20

MMM

106

107

108

109

1010

Mai

nse

quen

celif

etim

e(y

r)M

ain

sequ

ence

lifet

ime

(yr)

Fig 134 The effect of enhanced mixing and increased CNO reaction rate onthe main-sequence lifetimes of stellar models The top panel shows the lifetimes ofthe standard stellar models as a function of their masses The lower panel shows thepercentage change in the main-sequence lifetime with respect to the standard modelwhen the input physics is changed The crosses represent the calculations with extramixing the squares those with an enhanced CNO reaction rate

The effect of increasing the CNO rate on the main-sequence lifetime isconsiderably counter-intuitive For the stars in which the CNO cycle is thedominant reaction on the main sequence the lifetime increases slightly whereasfor the 1M model where it is not the dominant reaction it decreases slightlyThe reason for the increase in lifetime is that the structure of the modeldepends on the conditions in the core If the CNO rate is doubled from thestandard value too much energy is generated in the core of the star for thestructure that it supports As a result the star expands and the core becomescooler and less dense until equilibrium is regained At the new equilibriumpoint the structure is such that a lower energy flux is needed to support thestar Hence hydrogen burns more slowly and the star lives longer In the

1M model the dominant reaction rate is the pp chain and hence the changein the CNO rate does not have the same structural effect on the model Thesmall amount of CNO burning that does take place however is increased andhence the main-sequence lifetime is reduced This effect demonstrates anotherimportant point about stellar evolution it is a highly non-linear process andsimple assumptions about the behaviour of stars that are not supported bydetailed calculations often turn out to be incorrect

References

Blocker T 1995 AampA 297 727 335Cox J P Giuli R T 1968 Principles of Stellar Structure Gordon and Breach 334Eggleton P P 1971 MNRAS 151 351 333Eggleton P P Faulkner J Flannery B P 1973 AampA 23 325 337Henyey L G Forbes J E Gould N L 1964 ApJ 139 306 337Herwig F Austin S M Lattanzio J C 2006 Phys Rev C 73 025802 342Kippenhahn R Weigert A 1994 Stellar Structure and Evolution Springer-Verlag 334Kudritzki R P Reimers D 1978 AampA 70 227 335Pols O R Tout C A Eggleton P P Han Z 1995 MNRAS 274 964 336Prialnik D 2000 An Introduction to the Theory of Stellar Structure and Evolution

Cambridge Univ Press Cambridge 334Schroder K-P Pols O R Eggleton P P 1997 MNRAS 285 696 342Schwarzschild M 1965 Structure and Evolution of the Stars Dover Publication 334Schwarzschild M Harm R 1962 ApJ 136 158 341Stancliffe R J Tout C A Pols O R 2004 MNRAS 352 984 341Vassiliadis E Wood P R 1993 ApJ 413 641 335

14

Realistic N -Body Simulations of GlobularClusters

A Dougal Mackey

Institute for Astronomy University of Edinburgh Royal Observatory BlackfordHill Edinburgh EH9 3HJ UKdmyroeacuk

141 Introduction

This chapter is an introduction to realistic N -body modelling of globular clus-ters ndash specifically why it might be desired to conduct such models and whatconstitutes their key ingredients Detailed consideration is also given to theanalysis of data from such simulations and how it is increasingly becomingmore important to perform simulated observations in order to derive quan-tities that are directly comparable with real-world measurements The mostsalient points from this general discussion are illustrated via an extensive casestudy concerning N -body modelling of massive stellar clusters in the Largeand Small Magellanic Clouds

142 Realistic N -Body Modelling ndash Why and How

N -body modelling has long been an important tool for exploring the evolu-tion of star clusters All major phases of cluster evolution from early massloss through to core collapse gravothermal oscillations and tidal disruptionhave been investigated with N -body simulations (as well as other types ofmodelling) and they have played a large part in forming our current under-standing of cluster evolutionary processes (see eg the review by Meylan ampHeggie 1997) Even so due to the massive computational workload involvedwith the direct accurate integration of a large number of particles over verylong time-scales historically N has been restricted to relatively small values(a few thousand or with major effort a few tens of thousand) In additionmuch of the complexity of real clusters (such as the processes involved withstellar evolution binary star evolution stellar collisions time-varying tidalfields and so on) has often by necessity been neglected These two factorshave meant that the investigation of globular cluster evolution with N -bodymodelling has generally involved the extrapolation of results to larger N and

Mackey AD Realistic N-Body Simulations of Globular Clusters Lect Notes Phys 760

347ndash376 (2008)

348 A D Mackey

approximations due to incomplete implementation of the complicated inter-play between various internal and external evolutionary processes

In the last decade however and particularly within the last few yearsthere have been two major advances that have propelled the field of clusterN -body modelling into a new era The first of these is the advent of specialpurpose hardware most recently the GRAPE-6 machines (Makino et al 2003Fukushige Makino amp Kawai 2005) to accelerate the direct N2 summation ofgravitational forces These have greatly reduced the computational bottleneckassociated with large N and simulations covering a Hubble time of evolutionwith N sim 105 ndash that is at the lower end of the globular cluster mass function ndashare now within reach

The second advance concerns the sophistication of the N -body codes them-selves Several of the major codes such as Aarsethrsquos nbody4 (Chap 1 seealso Aarseth 2003)1 and the starlab software environment2 have now pro-gressed to the stage where most if not all of the major internal and externalevolutionary processes in a star cluster have successfully been incorporatedSuch processes include single-star and binary star evolution stellar collisionsand the formation and destruction of hierarchical systems and arbitrary ex-ternal tidal fields The sophistication of available N -body codes combinedwith the integrating power of special purpose hardware means that directrealistic simulations of massive stellar clusters are now possible

This aim of this chapter is to present an overview of realistic N -body mod-elling of globular clusters In particular we will discuss in what situations itis desirable to invest the time and effort to run and analyse a realistic N -bodymodel and examine the most important aspects of the N -body codes whichallow such realism Since many (if not all) of the latter have been covered insignificant detail elsewhere in this series of lectures we will spend most of ourtime examining the processes involved with reducing the large amounts of datathat come out of a realistic simulation and in particular discuss the conceptof ldquosimulated observationsrdquo which is becoming increasingly prominent Sincethis constitutes some very general discussion much of it from an observerrsquosperspective the best way to illustrate the most important points is via a spe-cific case study ndash we examine recent direct realistic N -body modelling of theevolution of massive stellar clusters in the Magellanic Clouds

1421 Why Run a Realistic N-Body Model

There are a number of advantages to running large-scale realistic N -bodymodels First unlike with many methods used to model star cluster evolu-tion a sophisticated N -body code includes all the important physics with aminimum of simplifying assumptions Hence for example if one is interestedin investigating the long-term evolution of hierarchical systems within a stel-lar cluster in a realistic N -body simulation it is possible to integrate directly

1Available for download from httpwwwastcamacukresearchnbody2See httpwwwidsiasedu~starlab

14 Realistic N -Body Simulations of Globular Clusters 349

the orbits of all stars ndash no gravitational softening or similar modificationsare required

Similarly because all the important physics is being included in a self-consistent manner (eg the stars and binaries are evolving in step with thecluster evolution) one can be reasonably confident that the complex inter-play between various evolutionary processes in a cluster is being accountedfor Even though star clusters are generally considered to be relatively simpleastrophysical systems in that they are often approximately spherically sym-metric and consist of stars with a uniform single age and metallicity theyare in fact complicated objects and it is often extremely difficult to isolate (orpredict) the effects of individual physical processes in a cluster

For example consider the production of blue stragglers in a globular clus-ter It is generally accepted that there are a number of channels leading to theformation of such objects ndash for example Roche-lobe mass transfer in a binarystar or the coalescence of a highly eccentric binary star after a strong inter-action It is complicated to determine the relative importance of formationchannels in a star cluster and the resulting properties of the blue stragglersbecause much interplay between competing processes occurs For example thestructural and dynamical state of the cluster plays an integral role in definingthe collision (strong interaction) rate between individual members Howeverthe state of the cluster is strongly affected by the stellar evolution within thecluster by related parameters such as the initial mass function the metallic-ity and so on and by the properties of the external tidal field In additionthe properties of any binary stars in the cluster are strongly affected by boththe structural and the dynamical state of the cluster as well as the stellarevolution of the individual members of the binary (especially if processes suchas mass transfer occur) In certain cases (such as during deep core collapse)the binaries themselves can in turn affect the cluster structure and dynamicsGiven all this if one wishes to investigate the production and properties ofblue stragglers in a cluster a realistic N -body simulation offers a very powerfulmeans of accounting for (and following) this complicated interplay

A third advantage to running realistic N -body simulations is that withpresent technology one is now able to directly compare simulations with realclusters for realistic N up to that corresponding to low-mass globular clustersEven for higher-mass clusters it is almost always possible to choose an Nwhich corresponds within an order of magnitude We are therefore now movinginto the regime where many of the scaling-with-N issues which have beennecessary to account for in the past when applying the results of N -bodysimulations to the evolution of real clusters (eg Aarseth amp Heggie 1998)are circumvented In addition with such large N fluctuations in the globalevolution of the N -body model are reduced to the point where they are notsignificant For small-N models it has been standard practice to average theresults of a number of simulations to reduce such fluctuations the amplitudesof which increase with decreasing N (eg Giersz amp Heggie 1994 Wilkinson

350 A D Mackey

et al 2003) For large-N models it is becoming increasingly clear that thisprocess is not necessary (eg Hurley et al 2005 Mackey et al 2007 2008a)

Finally given both the fact that processes such as stellar evolution aremodelled along with the gravitational interactions between particles and thatwe often do not have to worry about extrapolating our results to larger N it is possible to apply sophisticated techniques to the analysis of realisticN -body simulations More specifically it is possible to realistically simulateobservations of N -body models This aspect is especially important if oneis trying to compare an N -body simulation with a real system (which willinevitably have properties defined through observation) or if one is trying tomake predictions about the properties of a real system (which will have tobe tested observationally) This concept is discussed in more detail below inSect 1423 and examples are given in Sect 143

Even taking into account the above advantages it is important to under-stand that it will not always be necessary to invest the time and effort inrunning a large-scale realistic N -body model One should always considercarefully what question is under investigation and how best to answer it Ifthe physics can be sufficiently well modelled with small-N clusters or withoutneeding to include degrees of sophistication such as stellar evolution or sim-ulated observations then running less complicated models will naturally bepreferable (and almost certainly far quicker and more efficient) than investingin a direct realistic N -body simulation

1422 Key Ingredients in a Realistic N-Body Model

There are two main ingredients in setting up and running a realistic N -bodymodel ndash the N -body code itself and the generation of initial conditions

N-Body Codes

It is worth considering briefly the major components of a realistic N -bodycode As noted earlier there are a number of such codes publicly availableProminent examples are nbody4 (for use with the GRAPE-6 special pur-pose hardware) nbody6 (for use without GRAPE-6) and nbody6++ (aparallelised version of nbody6) and the starlab environment Here we willconsider the code nbody4 and note that much of the discussion also appliesto the other codes Since most of the following is covered in great detail byother contributions to this lecture series we will not delve too deeply intothe computational details Nonetheless it is important to understand whatprimary ingredients make up a realistic N -body code

These main components can be divided into three different groups theintegration routines the stellar evolution routines and the binary evolutionroutines Let us consider these in order In nbody4 the equations of mo-tion are integrated using the fourth-order Hermite scheme (Makino 1991) in

combination with a GRAPE-6 An external tidal field is incorporated by inte-grating the equations of motion in an accelerating but non-rotating referenceframe centred on the clusterrsquos centre-of-mass (see eg Wilkinson et al 2003and references therein for more details) The integration proceeds using theN -body units of Heggie amp Mathieu (1986) which are converted to physicalunits for output using a length scale generally set at the beginning of a run viacomparison to a real cluster (see Sect 1432) A close multiple system (suchas a hard binary) is treated as a combined centre-of-mass object in the Her-mite integration while the detailed orbits of the individual components of themultiple system are integrated separately using state-of-the-art two-body orchain regularization schemes as applicable (Mikkola amp Aarseth 1993 1998)The point of two-body regularization is that binary star orbits and partic-ularly perturbed binary motion can be followed at high accuracy withoutresorting to the introduction of gravitational softening Chain regularizationextends this possibility to close encounters between more than two stars (suchas in a binaryndashbinary interaction)

Stellar evolution in nbody4 is incorporated by means of the analytical for-mulae of Hurley Pols amp Tout (2000) who derived them from detailed stellarevolution models following stars from the zero-age main sequence throughto remnant phases (such as white dwarfs neutron stars and black holes)Each star is initially assigned a mass (the formulae cover the mass range01ndash100 M) and a single metallicity for the cluster may be selected in therange Z = 00001ndash003 The stellar evolution is calculated in step with the dy-namical integration and includes a mass-loss prescription such that evolvingstars lose gas through winds and supernova explosions This gas is instanta-neously removed from the cluster which is a reasonable approximation sinceoutflow speeds are generally large compared to the cluster escape velocity Animportant consequence of the introduction of stellar evolution is that eachstar possesses a finite radius (as opposed to being a point mass) which variesas its evolution progresses This is vital when considering close encountersbetween stars including effects such as tidal capture Furthermore the stellarevolution parameters calculated in the routines in nbody4 (such as luminos-ity and effective temperature) may be used to derive absolute magnitudes andcolours although this is not done within the code itself This allows simulatedobservations of the model cluster to be made if necessary

Binary star evolution is calculated in a similar manner to single-star evo-lution following the analytical prescription of Hurley Tout amp Pols (2002) andallowing for such phases as the tidal circularization of orbits mass transfercommon-envelope evolution and mergers Algorithms such as stability testswhich allow the consideration of triples and higher-order hierarchical systemsare also implemented within the code Details of the tidal evolution and sta-bility routines are discussed in Chap 3 and Mardling amp Aarseth (2001) Aswith the single-star evolution binary star evolution is calculated in step withthe overall dynamical integration

352 A D Mackey

Initial Conditions

Generating high-quality initial conditions is of paramount importance whenrunning a realistic N -body model Generally the reason for wanting to run arealistic N -body simulation will be to directly model one or more real clustersIn such cases the initial conditions are defined naturally by the clusters underconsideration although it may be necessary to infer them (for example if thereal clusters are dynamically evolved) In addition since the initial conditionsfor the real clusters are almost certainly defined (or at least constrained) byobservational measurements it may well be necessary to implement simulatedobservations in order to confirm the generated initial conditions in the N -bodymodel are as accurate as possible (see eg Sect 1437)

There is a significant number of variables to consider when setting initialconditions and the parameter space can therefore be very large For exampleconsider the following (non-exhaustive) list

bull What is the initial cluster structure The central density core andorhalf-mass radius tidal limit and the radial density profile all need to beappropriate to the problem under consideration

bull What is the initial dynamical state Should the cluster be starting in virialequilibrium or is some other state more appropriate

bull What is the most appropriate initial mass function (IMF)bull What is the most appropriate range of stellar massesbull What is the total cluster mass Mtotbull Mtot the IMF and the stellar mass range allow N to be calculated Is this

number realistic to model in a reasonable time-framebull What is the cluster metallicitybull Should there be any primordial mass segregation in the clusterbull Are there any primordial binaries in the cluster If so then what should the

overall binary fraction be and how should they be distributed spatiallybull What properties do any primordial binaries have What are the distribu-

tions for the mass ratio semi-major axis and orbital ellipticitybull What is the external tidal fieldbull Are any special modifications to the code required For example to incor-

porate specific stellar evolution or a new external tidal field etc

It is also important to consider practicalities for a given simulation like itsrequired duration (this will be constrained by the real systems being mod-elled) how frequently data should be produced during the run (this will beconstrained by the temporal resolution required to investigate properly allquestions under consideration) and whether the resulting disk space require-ments can be met

1423 Data Analysis Simulated Observations

There is a number of reasons why one may be running a large-scale real-istic N -body simulation For example the aim may be to directly model

one specific cluster (see eg Hurley et al 2005) to try and understand theglobal properties of a system of clusters (see eg Mackey et al 2007) or toinvestigate a more general question like the effect of cluster metallicity onstructural evolution (see eg Hurley et al 2004) In most (if not all) suchcases the problems under investigation will be defined by the observationsof real systems Furthermore any results from the simulations may lead topredictions for real systems that will require observational verification Forthese reasons it is necessary to treat the analysis of data from a realisticN -body simulation with some degree of sophistication Specifically the mostuseful results are likely to be obtained by simulating observations of the modelcluster(s)

This will not constitute all of the data analysis for a given simulationIt is still necessary to perform more traditional analysis to understand theaspects of the global or specific evolution of a model cluster Nonetheless ifone wishes to obtain measurements from an N -body simulation which areto be compared directly with observational measurements of real systemsconsiderable care must be taken that the derived quantities are indeed directlycomparable If this is not the case significant error can result as highlightedin Sect 1436 The most straightforward means by which it can be ensuredthat directly comparable quantities are obtained is by closely reproducing theoriginal observational analysis on the N -body model

In undertaking such a process the most important thing is to adopt an ob-serverrsquos perspective In particular it is vital to be aware of the circumstancesand limitations of the genuine observations and make sure that these are ap-plied to the simulated observations It should be clearly understood exactlywhat was observed in a cluster (eg maybe just red giant branch stars) whatquantities were actually measured and what processes were used to obtainthese measured parameters Detailed examples of this methodology are set outin Sects 1434 1436 and 1437 For a theoretician or N -body modeller ac-customed to being able to consider any aspect of a simulated cluster at will itis often surprising how crude many genuine observations are Detailed observ-ing in a globular cluster can be a very difficult feat which has only recentlybecome fairly routine due to the arrival of extremely high-quality telescopesand instruments such as the Hubble Space Telescope (HST) particularlyits associated cameras (WFPC2 ACS etc) and the Very Large Telescope(VLT) particularly its spectrographs (UVES FLAMES) and adaptive opticsinstruments (eg NACO) Even so the process of obtaining simulated ob-servational measurements from a realistic N -body run will invariably involvedegrading the data significantly because star cluster observations generallyonly measure a small fraction of the stars in a cluster

Simulated observations serve a number of functions in addition to theiruse in the primary analysis of the results from an N -body simulation Asdiscussed above in Sect 1422 in many situations the initial conditions fora realistic N -body model will be defined or constrained by the observationsof a genuine system or systems In such cases simulated observations of the

354 A D Mackey

initial state of the model N -body cluster can be used to verify the validity ofthe adopted initial conditions and can often be used to fine-tune these initialconditions Examples of this are provided in Sects 1433 and 1437

Furthermore simulated observations of an N -body cluster can provideimportant information about the quality of the real set of observations theyare designed to reproduce Since it is possible to do ldquoperfectrdquo observationson an N -body model and thus gauge the true state of the model at anyparticular time by then degrading the observational quality to that of the realmeasurements one can investigate how accurately those real measurementsquantify that state and search for any biases that may have been introducedSubsequently it may be possible to use further simulated observations toexamine the modifications that could be made to the real observations ordata reduction procedure in order to improve their quality An example ofsuch a process is presented in Sect 1436

Similarly if one has calculated a realistic N -body model that makes somekind of prediction about a quantity which can potentially be observed in aglobular cluster it is important to examine whether it is feasible to search forthat signature with presently available facilities Simulated observations inwhich the capabilities of a given telescope andor instrument are incorporatedcan provide such information and also allow one to assess the complexity ofsuch observations along with the time allocation requirements for them to becarried out

Conducted with due care and attention simulated observations of realisticN -body models can be an extremely powerful tool for both modellers andobservers

143 Case Study Massive Star Clustersin the Magellanic Clouds

The above discussion is quite general and many of the points are best il-lustrated via a specific case study For the remainder of this chapter we willtherefore examine recent work concerning the evolution of globular clustersin the Large and Small Magellanic Clouds (LMC and SMC respectively)(Mackey et al 2007 2008a)

Before proceeding to this however it is worth noting that another excellentexample of realistic N -body modelling with a different focus to the case studyconsidered below is the recent lsquowork concerningrsquo the old Galactic open clusterM67 by Hurley et al (2005) in which they investigate the evolution of thecluster structure and mass loss along with formation mechanisms and proper-ties of blue stragglers evolution of the cluster colourndashmagnitude diagram andvarious stellar populations and modification of the cluster luminosity func-tion due to external tidal forces Some aspects of this work are discussed inChap 12

1431 Observational Background The RadiusndashAge Trend

The star cluster systems belonging to the LMC and SMC (which are two closecompanion galaxies of the Milky Way) are of fundamental importance in starcluster astronomy particularly the field of star cluster evolution While theGalactic system provides the nearest globular cluster ensemble from an obser-vational point of view these objects are not ideal for studying cluster evolutionbecause of their uniform ancient nature (ages sim 10 minus 13 Gyr) Therefore wecan determine very well the end-points of massive star cluster evolution butmust infer the complete long-term development that brought them to theseobserved states

In contrast the LMC and SMC possess extensive systems of star clusterswith masses comparable to the Galactic globulars but crucially of all ages106 le τ le 1010 yr These systems are hence the nearest places we can observedirect snapshots of cluster development over the last Hubble time

Elson and her collaborators were among the first to consider the struc-tural evolution of massive star clusters in the LMC (Elson Fall amp Freeman1987 Elson Freeman amp Lauer 1989 Elson 1991 1992) They measured radialbrightness profiles and derived structural parameters for a sample of clusterscovering a wide range of ages to search for evolutionary trends The moststriking relationship they discovered concerns the sizes of the cluster cores3

The spread in core radius was observed to be a strongly increasing function ofage in that the youngest clusters possessed compact cores with rc sim 1minus2 pcwhile the oldest clusters exhibited a range 0 le rc le 6 pc (cf Fig 141) Theydid not observe any significant trend between cluster mass and radius Theradius-age trend provided intriguing evidence that our understanding of mas-sive star cluster evolution may be incomplete since quasi-equilibrium modelsof star cluster evolution do not predict large-scale core expansion over thecluster lifetime (see eg Meylan amp Heggie 1997)

The advent of the Hubble Space Telescope has allowed this problem tobe re-addressed observationally in significantly more detail than was possiblewith ground-based facilities HST imaging can resolve LMC and SMC starclusters (at distances of sim 50 and sim 60 kpc respectively) even in their innercores so that star counts may be conducted to very small projected radiiand very accurate surface densitybrightness profiles constructed Work withHST observations using the Wide Field Planetary Camera 2 (WFPC2) andAdvanced Camera for Surveys (ACS) has recently been conducted (Mackey ampGilmore 2003ab Mackey et al 2008b) These authors have a combined sampleconsisting of 84 LMC and 23 SMC clusters covering the full age range andwith masses generally comparable to those of the Galactic globular clustersFor the interested reader full details of the data reduction construction ofsurface brightness profiles and measurement of structural parameters may befound in Mackey amp Gilmore (2003a) and Mackey et al (2008b)

3As parametrised by the observational core radius rc defined in this case as theradius at which the surface brightness is half its central value

356 A D Mackey

Fig 141 Core-radius versus age for massive stellar clusters in the Large and SmallMagellanic Clouds This figure includes all clusters from the HSTWFPC2 measure-ments of Mackey amp Gilmore (2003ab) as well as the HSTACS measurements ofMackey et al (2008b)

The resulting core-radius versus age diagram is shown in Fig 141 Thisrepresents the most up-to-date information available regarding the radius-agetrend in the LMC and SMC cluster systems The upper envelope is very welldefined for all ages up to a few Gyr At older times than this the full range ofcore radii observed for massive stellar clusters is allowed In fact the situationis even more dramatic than appreciated in earlier studies Several of the oldestclusters in the sample lie off the top of the diagram the Reticulum cluster inthe LMC with age τ sim 12minus 13 Gyr and rc sim 148 pc and Lindsay 1 and 113in the SMC with τ sim 9 Gyr and rc sim 164 pc and τ sim 5 Gyr and rc sim 11pc respectively Hence the range for the oldest clusters is 0 le rc le 17 pc

It is interesting to note that the observed distribution of core radii forthe oldest clusters is quite consistent with that observed for Galactic globularclusters Indeed if only globular clusters in the remote outer Milky Way haloare considered (where destructive tidal processes particularly affecting diffuseclusters are minimized) the distributions match very closely indeed (Mackeyet al 2008a) It is worth emphasizing however that the radius-age relation-ship cannot be inferred solely from the observations of the Galactic globularclusters ndash the full trend is only evident when the age spectrum present in theLMC and SMC cluster systems is exploited

1432 Realistic N-Body Modelling of Magellanic Cloud Clusters

The key question resulting from these observations concerns the origin of theradius-age trend This is important for our understanding of star cluster evo-lution ndash since standard models never predict an order-of-magnitude expansionof the cluster core radius over the cluster lifetime these models are possiblyincomplete

There exist a number of interpretations of the radius-age diagram Themost straightforward (which we consider here) postulates that massive starclusters (or at least the long-lived variety) are always formed as compact ob-jects and that some for an as-yet unidentified reason expand for the durationof their lives while the remainder do not In this case we are searching for adynamical explanation of the trend ndash a problem ideally suited to large-scalerealistic N -body modelling

A number of possible dynamical mechanisms for the radius-age trend havepreviously been proposed and investigated however none can fully explainthe observed distribution of clusters For example a strongly varying intra-cluster IMF (Elson et al 1989) or binary star fraction (Wilkinson et al 2003)have been ruled out as viable explanations as have the effects of a temporallyvarying tidal field such as that which a cluster on a highly elliptical orbitmight feel (Wilkinson et al 2003) In the present case study we considerthe effects of a population of stellar-mass black holes (BHs) Usually suchobjects are assumed to receive a large velocity kick at formation in a supernovaexplosion which means they rapidly escape from their cluster Therefore weconsider here the effects if a star cluster can somehow retain a fraction of theseBHs Large-scale realistic N -body modelling has been conducted to investigatethis question using the nbody4 code (Mackey et al 2007 2008a)

As discussed in more general terms earlier in this chapter there are twokey aspects to conducting realistic N -body simulations The first is to developmodel clusters that have properties as similar as possible to those observedfor the real LMC and SMC clusters The second concerns the data analysisSince we are trying to reproduce an observationally defined trend we mustobtain measurements from the simulations that are directly comparable to themeasurements which were determined for the real clusters The most logicalway to do this is to perform simulated observations of the simulated clustersin just the manner that the genuine observations were conducted This willbe discussed in more detail in Sect 1434 below

Returning then to the question of setting up realistic models we mustfirst identify the key characteristics of the youngest LMC and SMC clustersThese are summarized in Fig 142 All the observed young LMC and SMCclusters have profiles with cores (rather than cusps) ndash even the ultra-compactcluster R136 exhibits a small core (see eg the detailed discussion in Mackeyamp Gilmore 2003a and the references therein) The radial brightness profiles ofthe youngest clusters are well fit by models of the form (Elson Fall amp Freeman1987 EFF models hereinafter)

358 A D Mackey

Fig 142 Properties of the youngest massive clusters observed in the LMC andSMC Structural data are taken from Mackey amp Gilmore (2003ab) while the centraldensity and total mass estimates are taken from McLaughlin amp van der Marel (2005)

μ(r) = μ0

(

1 +r2pa2

)minusγ2

(141)

where rp is the projected radius (ie the radius on the sky) μ0 is the centralsurface brightness γ determines the power-law slope of the fall-off in surfacebrightness at large radii and a is the scale length It is straightforward toshow that this latter parameter is related to the core-radius by

rc = a(22γ minus 1)12 (142)

Typical values for these structural parameters in young LMC and SMC clus-ters are rc le 2 pc and γ sim 26 Excluding R136 the young LMC and SMCclusters generally have central densities in the range 16 le log ρ0 le 30 andtotal masses in the range 4 le logMtot le 5 R136 is the youngest cluster in thesample sim 3 Myr and also has the greatest central density with log ρ0 asymp 48

Given these observational constraints we generate model clusters in virialequilibrium according to an EFF profile with γ = 3 ndash this is the member ofthe EFF family of models closest to γ sim 26 which possesses analytic expres-sions for the radial dependence of the enclosed mass and isotropic velocity

dispersion Full details of the generation procedure may be found in Mackeyet al (2008a)

Using the IMF of Kroupa (2001) we assign a range of masses to the starsin a model cluster according to the multiple-part power law

ξ(m) prop mminusαi (143)

where ξ(m)dm is the number of single stars falling in the mass interval m tom+ dm and the exponents αi are

α0 = +03 plusmn 07 001 le mM lt 008α1 = +13 plusmn 05 008 le mM lt 050 (144)α2 = +23 plusmn 03 050 le mM lt 100α3 = +23 plusmn 07 100 le mM

Kroupa (2001) derived his IMF from a large compilation of measurementsfrom young stellar clusters including many in the LMC This is in contrastwith many other widely used IMFs ndash the Kroupa (2001) IMF is therefore themost suitable for the present N -body modelling

We impose a stellar mass range 01ndash100M for our model clusters Thelower mass limit is set by the lowest mass stars for which stellar evolutionroutines are incorporated in nbody4 while the upper limit is consistent withthe observations of very young massive star clusters Note that the lower masslimit means that in practice only the exponents α1ndashα3 in the IMF describedabove are utilized

Selection of the IMF described above along with the requirement thatour model clusters have masses typical of those of young LMC and SMCclusters (Fig 142) allows the total number of stars in each given model tobe assigned For all present simulations N sim 105 stars which gives typicalinitial total cluster masses of Mtot sim 56 000M (ie logMtot sim 475)

In the interest of maintaining a high degree of realism in the simulationsmodel clusters are evolved in a weak external tidal field rather than in iso-lation This external field is incorporated by imposing the gravitational po-tential of a point-mass LMC with Mg = 9 times 109M and placing the clusterson circular orbits of galactocentric radius Rg = 6 kpc Adopting a point-massLMC is a significant over-simplification however as described by Wilkinsonet al (2003) the gradient of this potential is within a factor of 2 of that in theLMC mass model of van der Marel et al (2002) at the assigned orbital radiusIn any case the relatively weak tidal field of the LMC does not significantlyaffect the core-radius evolution of its massive stellar clusters (Wilkinson et al2003)

Incorporating a tidal field in the N -body modelling serves two importantpurposes First it allows the gradual evaporation of stars from a simulatedcluster to be modelled in a self-consistent fashion so that the rates of evapo-ration between different models with the same external potential and escape

360 A D Mackey

criterion may be easily compared Second it lets us impose a natural scalingbetween N -body units in which the integration is computed and physicalunits which we use to compare the model cluster to observational results Inparticular the length scaling controls the physical density of the cluster andhence the physical time-scale on which internal dynamical processes occurThe tidal radius rt of a star cluster (mass Mcl) on a circular orbit of radiusRg in the external point-mass potential of a point-mass galaxy (mass Mg)may be estimated from the relationship (King 1962)

rt = Rg

(Mcl

3Mg

) 13

(145)

The initial tidal radius of the cluster estimated via (145) is used to determinethe length-scale conversion It is important to check that this results in clusterdensities consistent with those observed for young LMC and SMC clusters ndashwe quantify this more carefully below

Since we wish to examine the dynamical effects of populations of stellar-mass black holes on star cluster evolution it is important to consider how suchobjects may be incorporated naturally into our N -body simulations The mostunambiguous method is to generate black holes from the supernova explosionsof the most massive stars in the cluster nbody4 includes such formation in itsstellar evolution routines however we added small modifications so that theprogenitors masses and natal kicks of the generated BHs could be controlledTo ensure a sizeable population of BHs we form one whenever a star with aninitial mass greater than 20M explodes For a cluster with N = 105 starsand a Kroupa (2001) IMF with an upper mass limit of 100M this results inNBH = 198 BHs When a BH is formed we assign it a mass randomly selectedfrom a uniform distribution in the range 7 lt MBH lt 13M so that the meanmass is 10M This process is again undoubtably a simplification howeverthe mass characteristics of the progenitors and BHs are reasonably consistentwith theoretical expectations (see eg Zhang Woosley amp Heger 2007) as wellas observational evidence (see eg Casares 2006)

The natal kicks which the BHs are given are very important A large kick(a few hundred km sminus1) is usually used for both black holes and neutron starsThis generally means no BHs are retained in a typical cluster which mighthave an escape velocity of 10ndash20 km sminus1 In order to control the retentionfraction we modified nbody4 so that the natal kicks given to generated BHscould be easily controlled and varied from run to run

It is also important to specify the metallicity of the model clusters sincethis parameter strongly affects the stellar evolution and hence the mass lossat early times in the N -body simulations (see eg Hurley et al 2004) Inthe present example we select solar metallicity (Z = 002) to be consistentwith observations of young clusters in the Magellanic Clouds However it isimportant to be aware that since there is a strong age-metallicity relationin both Clouds there is a metallicity gradient across the radius-age diagram

(ie the oldest clusters are also very metal poor) In any ensemble of N -body runs seeking to explain the radius-age trend the significance of thisfact should be investigated (although we do not consider it any further in thepresent example)

One additional key aspect of young LMC and SMC clusters is that thosewhich have been observed in detail generally exhibit some degree of masssegregation ndash that is the most massive stars in a given cluster are preferentiallylocated near the centre of that cluster For example mass segregation hasbeen observed in the LMC clusters NGC 1805 and NGC 1818 (de Grijs et al2002ab) and R136 (Malumuth amp Heap 1994 Brandl et al 1996 Hunter et al1995 1996) as well as the SMC cluster NGC 330 (Sirianni et al 2002) It doesnot necessarily follow from these observations that mass segregation occurs inall young LMC and SMC clusters and nor is it clear whether the segregationis primordial or dynamical in the clusters where it has been found howevermass segregation is clearly an important factor which we must consider inour models

In order to produce mass-segregated clusters in a self-consistent fashion(ie close to virial equilibrium with all members having appropriate velocities)a cluster is first generated as described above (with no mass segregation) Wethen implement a mass-truncation setting all stars in the cluster with massesgreater than 8M to have mass 8M Next the cluster is evolved dynami-cally using nbody4 but with the stellar evolution routines turned off Hencethe cluster begins to dynamically relax and mass segregate The degree ofprimordial mass segregation is controlled by the length of time for which thecluster is ldquopre-evolvedrdquo The truncation limit of 8M is selected so that thepre-evolution can extend for a reasonable period (a few hundred Myr) with-out the most massive stars sinking to the cluster centre forming a collapsedcore and ejecting each other through close interactions Once the desired pre-evolution time is reached the simulation is halted the mass-truncated starsreplaced with their original masses and the resulting cluster taken as theinput for the simulation proper

The truncation and replacement process introduces some small inconsis-tencies in the velocities of some stars once the simulation proper is startedHowever these are small and are erased by dynamical processes within a fewcrossing times In addition during the pre-evolution phase some stars escapefrom the cluster This process is very gradual however and even clusters withlong pre-evolution times (several hundred Myr) only lose a few per cent of theirmass Since the scaling of all models is set by (145) which varies as the cuberoot of the cluster mass the differences in scaling between non-segregated andprimordially segregated clusters are tiny

It is important to check whether this artificial mass segregation processproduces clusters that have properties comparable to the observed mass-segregated young LMC and SMC clusters We do this by comparing simulatedobservations of the model clusters with the genuine cluster observations Thisis considered in more detail in the next section and in Sect 1437

362 A D Mackey

1433 Summary of N-Body Runs

With the initial conditions specified as described above four N -body simula-tions are required to address the question under consideration ndash namely thedynamical effects of a population of stellar-mass black hole remnants on mas-sive star cluster evolution ndash at a basic level The parameter space of interest isspanned by two types of clusters ndash those with no primordial mass segregationand those with a strong degree of primordial mass segregation In each ofthese types we consider evolution with no black holes (that is where the na-tal kick is large so the retention fraction is zero) and a significant populationof black holes (that is where the natal kick is zero so the retention fractionis unity)

These four runs cover the extreme limits of the parameter space we aimto investigate and hence are expected to cover the extreme limits of clus-ter evolutionary behaviour Subsequent to their completion it is sensible tocheck this is indeed the case by adding further runs which sample interme-diate regions of the parameter space (eg a cluster with only moderate masssegregation or a black hole retention fraction around 05) Although such runshave been carried out we will not consider them in any detail here

The properties of the four N -body runs are listed in Table 141 Notethat for Runs 3 and 4 ldquostrong mass segregationrdquo is rather difficult to de-fine numerically however a pre-evolution duration of sim 450 Myr is adequateto reproduce observational results of mass segregation in young Magellaniccloud clusters This aspect is discussed in more detail in Sect 1437 belowEach model is run until late times (Tmax gt 10 Gyr) which match the ages ofthe oldest Magellanic Cloud globular clusters Each such run took approxi-mately 2 weeks of full-time calculation on the GRAPE-6 at the Institute ofAstronomy in Cambridge The first week takes any given run to an age ofsim 15 Gyr after which time the computation becomes rather swifter mainlydue to decreasing particle number and much less demanding stellar evolutioncalculations

We selected data for output every 15 Myr at ages less than 100 Myr andevery 15 Myr thereafter This allowed close examination of the early phases

Table 141 Details of N -body runs and initial conditions Each cluster begins withN0 stars with masses summing to Mtot and initial central density ρ0 Initial clusterstructure is ldquoobservedrdquo to obtain rc and γ Each model is evolved until Tmax

Name N0 log Mtot log ρ0 rc γ Initial mass Black hole Tmax

(M) (M pcminus3) (pc) segregation kicks (Myr)

Run 1 100 881 4746 231 190 296 None Large 16 996Run 2 100 881 4746 231 190 296 None Zero 10 668Run 3 95 315 4728 458 025 233 452 Myr Large 11 274Run 4 95 315 4728 458 025 233 452 Myr Zero 10 000

of cluster evolution and suitable resolution at all times to consider in detailthe development and evolution of any black hole populations Typically eachsim 10 Gyr N sim 105 star run takes up sim 10 Gb of space on disk This can bereduced considerably by compressing the output for storage and backup

For each run we measured the initial cluster mass central density andthe structural parameters rc and γ ndash these are all listed in Table 141 Thestructural parameters were derived from simulated observations as discussedin Sect 1434 below It is worth re-emphasizing how closely these correspondto the observed quantities for the youngest massive clusters in the Magellanicclouds This can be seen explicitly by comparing the values listed in Table 141with the plots in Fig 142 In addition the evolution of the central density(ρ0) over the first tens of Myr for Runs 1 and 3 is plotted in Fig 143

The model clusters with no primordial mass segregation have rc sim 19 pcγ sim 3 and log ρ0 sim 23 These clusters therefore appear very similar to anumber of Magellanic Cloud clusters with ages of sim 20 Myr In contrast theheavily mass-segregated model clusters have much smaller cores and highercentral densities with rc sim 03 pc and log ρ0 sim 48 They also have flatterpower-law fall-offs with γ sim 23 In this respect they look very similar tothe very compact massive young LMC cluster R136 which has an age ofsim 3 minus 4 Myr

Fig 143 Early evolution of the central density ρ0 for Runs 1 and 3 (solid lines)compared with the observations for young LMC clusters (points) Run 1 has noprimordial mass segregation while Run 3 is heavily segregated Run 3 looks verysimilar to R136 at early times but by a few tens of Myr looks more like otherobserved young LMC and SMC clusters and indeed rather similar to Run 1

364 A D Mackey

1434 Simulated Observations of Core Radius Evolution

As described in Sect 142 a key advantage of running realistic N -body simu-lations is that they allow the opportunity to conduct simulated observationson the models In particular this is a vital ingredient if the problem underinvestigation is defined observationally If this is the case it is essential toensure that whatever measurements obtained from the N -body modelling aredirectly comparable to those determined observationally

In our present case study we are investigating the origin of the radius-age trend in the LMC and SMC star cluster systems This trend is definedobservationally through measurements of cluster core radii To determinewhether our N -body simulations have been successful in reproducing the trendor not a directly comparable parameter must be obtained from them Themost unambiguous method of achieving this is by passing the N -body datathrough as similar a process as possible to that which generated the observedmeasurements

The first step is to identify and account for the limitations of the clusterobservations In any given LMC or SMC cluster in the sample displayed inFig 141 only a fraction of the stars in the cluster were imaged and usedto produce the brightness profiles from which core-radius measurements weremade There are two primary reasons for this First the HST field of view(whether it be with WFPC2 or ACS) is not large enough to cover the fullspatial extent of an LMC or SMC cluster The core is imaged but the radialprofile is cut off typically at sim 20 pc much less than the nominal tidal radiusof roughly sim 40ndash50 pc

Second the exposure times are too short to see the faintest stars in thecluster and too long to allow accurate measurement of the brightest starsThis point is illustrated in Fig 144 The displayed colour-magnitude diagram(CMD) is from ACS imaging of 47 Tuc a bright Galactic globular cluster Themain sequence is clearly visible as is the turn-off The image exposure timeswere not long enough to measure stars fainter than sim6 mag below the turn-offA large fraction of the stars in 47 Tuc are fainter than this (for example nowhite dwarfs were observed) but would not be included in any star counts usedto construct a brightness profile from these observations At the bright end thedata are cut off just above the sub-giant branch Brighter stars (ie all the redgiant branch and horizontal branch stars) do appear on the images howeverthe exposure times were long enough that these objects were saturated on theCCD That is the pixels imaging these stars have received too many photonsand the signal has overflowed into neighbouring areas Accurate photometrycannot be done above a certain level of saturation hence the bright cut-offlimit on the CMD in Fig 144 None of the saturated stars would be countedin a radial brightness profile either

Exactly similar processes apply to the LMC and SMC clusters we aretrying to model Each has a bright and faint cut-off determined by the expo-sure times of the imaging These are illustrated in Fig 144 for the complete

m

m m

Fig 144 Left Colour-magnitude diagram of the Galactic globular cluster 47 Tucfrom HSTACS imaging The measured signal-to-noise ratios for the detected starsare indicated in several places The bright and faint cut-offs are evident RightBright and faint stellar detection limits on the HSTWFPC2 and ACS images ofLMC and SMC clusters used for the measurements presented in Fig 141 LMCclusters are blue circles while SMC objects are magenta triangles Filled symbolsrepresent the WFPC2 imaging described in Mackey amp Gilmore (2003ab) while opensymbols are the ACS imaging from Mackey et al (2008b) Clusters are split intofour age bins shown with solid vertical lines Within each bin the mean bright andfaint detection limits are marked by dashed lines while the approximate maximumscatter about each mean is marked by a pair of dotted lines

sample The clusters are split into four age bins delineated on the plot withsolid vertical lines Within each of these the mean bright and faint detectionlimits are marked with dashed lines and the approximate maximum scatterabout these means with dotted lines From this figure it is clear that the brightand faint limits and hence the portion of the mass function sampled by theobservations vary systematically with cluster age This is due to the fact thatobservations of star clusters in the LMC and SMC are commonly aimed attargeting stars near the main-sequence turn-off Consequently the requiredexposure time increases with cluster age meaning that both the brighter andthe fainter detection limits decrease with age

To observe our model clusters we pass the N -body data at each out-put time through a measurement pipeline essentially identical to that usedto obtain structural quantities for the real LMC and SMC cluster sample(full details of the observational pipeline may be found in Mackey amp Gilmore2003a) At a given output time the luminosity and effective temperature ofeach star in the cluster is first converted to magnitude and colour using thebolometric corrections of Kurucz (1992) (see also eg Hurley et al 2005)We also convert the position and velocity of each star to physical units usingthe appropriate length-scale and velocity factor (see Sect 1432) With this

366 A D Mackey

completed we next impose the bright and faint detection limits appropriateto the output time (these are the dashed mean limits in Fig 144) This leavesan ensemble of stars with which to construct a surface brightness profile Weproject the three-dimensional position of each star onto a plane (to mimic theobservation of a cluster projected onto the sky) construct annuli of a givenwidth about the cluster centre and calculate the surface brightness in eachannulus For consistency with the observational pipeline we use a variety ofannulus widths so that both the bright inner core and the fainter outer regionsof the cluster are well measured Measurements are truncated at a radius com-mensurate with that imposed by the HST field of view as discussed aboveWe next fit an EFF model to the resulting surface brightness profile and fromthis model derive the structural parameters in particular the core radius Toreduce noise we repeat this process for each of the three orthogonal planarprojections at each output time and average the results

1435 Results from the Simulations

In this chapter we are primarily concerned with investigating the processesinvolved in running realistic N -body simulations and analysing the resultingdata illustrated through the examination of a case study Therefore we willnot delve deeply into the results of the four N -body runs themselves (the inter-ested reader is referred to Mackey et al (2008a) for full details) Nonethelessit is interesting to take a moment to consider these results in the context ofthe radius-age trend described in Sect 1431

Because we have taken care to construct models where N is sufficientlylarge that no scaling with N is necessary to interpret the output and be-cause we have taken care to obtain measurements closely mimicking the realobservations it is legitimate to directly plot the core-radius evolution of ourN -body models over Fig 141 This is shown in Fig 145 for Runs 1 and 2and Fig 146 for Runs 3 and 4

The simplest model is Run 1 which is not primordially mass-segregatedand in which black holes formed in supernova explosions receive a large natalkick ejecting them almost immediately from the cluster The retention frac-tion is thus zero As could be expected the evolution follows the standardpath expected for an ordinary globular cluster (see eg Meylan amp Heggie1997) There is an initial phase of violent relaxation and mass loss due to stel-lar evolution which lasts for the first sim100 Myr This phase is hardly reflectedin the core-radius evolution because as there is no primordial mass segrega-tion the mass loss is distributed widely over the cluster The remainder ofthe cluster evolution consists of a slow contraction of the core as dynamicalmass segregation is established and the cluster moves towards core collapsewhich happens near the end of the run at sim15 Gyr

Run 2 is identical except for the fact that natal black hole kicks are setto be zero so that the retention fraction is one This results in a populationof 198 stellar mass black holes within the cluster Initially the core radius

Fig 145 Core-radius evolution of N -body Runs 1 and 2 Both runs have noprimordial mass segregation and start from identical initial conditions The onlydifference between them is the retention fraction of stellar-mass black holes (zero andone respectively) Run 1 evolves exactly as expected with the main trend being aslow contraction in rc as the cluster relaxes and moves towards core collapse In starkcontrast Run 2 evolves very similarly up to a point after which strong expansionin the core radius is observed The presence of 198 stellar-mass black holes in thiscluster thus leads to strikingly different core radius evolution

evolution appears identical to that of Run 1 The mass loss phase passes andrelaxation processes set in However starting at about sim500 Myr the coreradius of Run 2 begins to expand dramatically This is due to the dynamicalinfluence of the black holes These objects because they are dark are notincluded in the core-radius measurements (they fall far below the faint cut-offon the CMD) All we can see is how the stars which are included in the profilecalculations are affected After their formation and a few tens of Myr of stellarevolution within the cluster the black holes are by far the most massive clustermembers They therefore sink rapidly to the cluster centre via dynamical masssegregation and after a few hundred Myr form a compact black hole coreThe densities within this core are such that close encounters between BHsare frequent and soon black hole binaries are formed Encounters betweenbinary BHs and single BHs and between binary BHs and other binary BHsscatter single BHs out of the core which then sink back in again via masssegregation Since an individual BH may undergo this process a number oftimes significant energy is transferred to the core stars through the repeatedmass segregation In addition in very strong encounters BHs are ejected from

368 A D Mackey

Fig 146 Core-radius evolution of N -body Runs 3 and 4 Both runs have strongprimordial mass segregation and start from identical initial conditions The onlydifference between them is the retention fraction of stellar-mass black holes (zeroand one respectively) Compared with Runs 1 and 2 there is strong early expan-sion due to the concentrated central mass loss Subsequently Run 3 without blackholes begins to mass segregate and contract whereas Run 4 undergoes continuedexpansion due to the dynamical effect of its black hole population

the cluster By the end of the run only about 30 of the original populationremains This ejection process serves as an additional heating mechanism

In contrast to Runs 1 and 2 the primordially mass-segregated Runs 3 and4 expand dramatically at early times Given that these two runs have the sameIMF as Runs 1 and 2 this early expansion must be a direct result of theirdifferent initial structure Unlike in Runs 1 and 2 where the early mass lossfrom stellar evolution is spread throughout the cluster in Runs 3 and 4 it isheavily concentrated in the core which in turn reacts with strong expansionThis expansion lasts for the first sim250 Myr by which time the highest massstars in the cluster have completed their evolution and the stellar mass-lossrate has been significantly reduced After this point the evolution follows verysimilar paths to those for Runs 1 and 2 The model with no black hole retention(Run 3) gradually begins to dynamically relax and mass segregation sets incausing slow contraction Because this cluster expanded at early times it isless dense than Run 1 and hence its relaxation time is longer Thus it doesnot reach a state of core collapse by the end of the simulation In contrast Run4 where the black hole retention fraction is unity undergoes core expansionfor the full duration of its evolution As in Run 2 the BHs segregate to the

centre of the cluster and form a compact core by sim 500 Myr This initiatesBHndashBH interactions inducing the expansion Because of its additional earlyexpansion Run 4 reaches larger core radii than Run 2 at late times evolvingoff the top of the figure to rc sim 12 pc

This set of four runs hence demonstrates that we can cover all regions ofthe observed cluster distribution in the radius-age plane simply by varying twobasic parameters within the ranges constrained by observation ndash the degree ofinitial mass segregation in a cluster and the retention fraction of stellar massblack holes Additional runs have been performed which demonstrate thatas should be expected models with intermediate degrees of mass segregationor an intermediate BH retention fraction evolve somewhere between the fourextremes modelled in the present example

1436 More Detail on Simulated Observations of rc

As well as directly addressing the question of the origin of the radius-age trendconducting simulated observations of the four N -body models described abovealso allows us to investigate the quality of the data reduction carried out onthe original observational data

For example when we examined the bright and faint saturation limitspresent in the imaging and constructed Fig 144 it became clear that theselimits vary systematically with cluster age We were able to implement thisvariation in our simulated observations of the N -body clusters and henceaccount for any systematic effect on the measurement of the core radiusHowever it also became clear that at any given age there is considerablescatter in the bright and faint limits between clusters ndash something we didnot account for in the simulated observations This raises the question as towhether this cluster-to-cluster variation at similar ages introduces significantscatter into the observed distribution of clusters on the radius-age diagramFurthermore if it does is it possible to reduce this scatter by re-analysing theobservational data and artificially imposing uniform bright and faint limits ata given age

To investigate these questions we re-calculated the core-radius evolutionof the N -body clusters using simulated observations with new bright and faintlimits implemented in place of the mean limits previously adopted In thesenew calculations we used the ldquomaximum scatterrdquo limits marked in Fig 144 ndashin one set we used the brightest pair of limits at any given age and in a secondset we used the faintest pair of limits at any given age The resulting evolutionis plotted in Fig 147 along with the evolution derived using the mean brightand faint limits

It can be seen from this figure that in all four runs for the majority ofthe evolution the selected bright and faint limits make little difference to thecalculated core radius at least at the level of the cluster-to-cluster scatterdetermined to be present in the Magellanic Cloud cluster observations How-ever in the case where a cluster is heavily mass-segregated and where it still

370 A D Mackey

Fig 147 Core-radius evolution derived from the simulated observations with threedifferent sets of saturation and faint limits implemented as indicated in Fig 144The black lines represent rc calculated using the mean limits as in Figs 145and 146 while the magenta lines represent rc calculated using the brightest max-imum scatter limits and the green lines represent rc calculated using the faintestmaximum scatter limits Agreement between the three is excellent except in thecase where a cluster is mass-segregated and young (so that it still possesses massiveluminous stars)

possesses massive luminous stars the adopted bright and faint limits make asignificant difference to the measured core radius

This result is readily understood In any given cluster since we constructbrightness profiles rather than simple stellar density profiles the presence ofany luminous stars strongly weights the resulting structural calculations Inparticular when mass segregation is present the most luminous stars are pre-disposed to lie near the cluster centre resulting in a small core radius Henceif the saturation limit is varied in the observations of such a cluster differentnumbers of luminous stars will be included in the calculation resulting in astrong variation in the measured rc This is clearly evident for Runs 3 and4 at early times in Fig 147 and suggests that the cluster-to-cluster scatterin saturation limits present in the observational data for the youngest clus-ters may have introduced significant scatter in the positions of clusters in the

radius-age diagram for ages up to sim200 Myr It would therefore be worthwhilere-reducing the observational data for clusters younger than this limit arti-ficially imposing uniform bright and faint detection limits With this donea major source of scatter in the positions of the youngest clusters on theradius-age diagram would be removed

This example shows that while genuine cluster observations define sim-ulated observations to be carried out on any N -body modelling of theseclusters additional simulated observations of the N -body models can leadto improvements in the genuine cluster observations in an iterative processThis illustrates one of the key advantages to running direct realistic N -bodysimulations and implementing a sophisticated data reduction procedure

One additional aspect worth a brief investigation is a comparison betweenthe measured core-radius (now using the mean bright and faint limits again)and the core-radius computed internally by nbody4 which one might betempted to use rather than proceeding down the more complicated and time-consuming path of implementing simulated observations

The core-radius calculated by nbody4 is more correctly termed the den-sity radius (rd) and is based on a quantity described by Casertano amp Hut(1985) so that rd is defined as the density-weighted average of the distanceof each star from the density centre of the cluster (see eg Aarseth 2003)The local density at each star is computed from the mass within the spherecontaining the six nearest neighbours This parameter was designed to behavein a similar manner to the observational core radius however as we will see itcan be strongly biased by particles that would not be included in any genuineobservation aimed at deriving the structural parameters of a cluster

In Fig 148 comparison between the observational core radius as calcu-lated above in Sect 1434 and the density radius computed by nbody4 ispresented for each of the four runs For Runs 1 and 3 where black holes arenot retained the agreement between the two radii is generally satisfactoryalthough there is a significant tendency for the density radius to be largerthan the observational core radius In comparison for Runs 2 and 4 whereblack holes are retained the agreement is very poor indeed with no correla-tion between the behaviour of the two radii The reason for this is simple ndashblack holes are included in the computation of rd but not included in thecomputation of rc (since they are dark particles) Hence for Runs 2 and 4 rd

is effectively tracing only the evolution of the black hole sub-system ratherthan the distribution of the luminous matter

Based on this result it is clear why one should be very careful aboutselecting measurements that are directly comparable to any observations beingmodelled If two disparate quantities are compared the potential for seriousmistakes exists In the above example if the density radius from nbody4 hadbeen taken as a proxy for the observational core-radius instead of makinguse of the simulated observations method the dramatic expansion evident inFigs 145 and 146 may not have been noticed and an ultimately successfulexplanation for the radius-age trend possibly not investigated any further

372 A D Mackey

Fig 148 Comparison between the evolution of the core radius rc derived fromsimulated observations and the density radius rd implemented in nbody4 for eachof the four N -body runs In each plot the upper panel shows the evolution of thetwo radii (rc in magenta rd in blue) while the lower panel shows the evolution ofthe ratio rcrd A ratio of unity is marked with a dashed line In runs with blackhole populations the density radius is a poor match to the observational core radius

1437 Simulated Observations of the Initial Mass Segregation

As a final example it is worth investigating the fact that we can use de-tailed simulated observations to examine the quality of the initial conditionswe constructed in Sect 1432 especially for the primordially mass-segregatedmodels We have already demonstrated that these model clusters closely re-semble the youngest massive LMC and SMC clusters in terms of their basicstructural parameters central densities and masses However we would like toverify that the method used to primordially segregate these clusters producesmass segregation similar to that observed in genuine objects Ideally we wouldalso like to integrate stellar velocities into the initial conditions (so that wecan see whether the assumption of virial equilibrium is valid) however unfor-tunately suitably detailed internal velocity measurements for young massiveMagellanic Cloud clusters do not yet exist

Nonetheless detailed observations of the radial dependence of the massfunction in such clusters do exist In particular there are three studies thatare very useful to us ndash that of Hunter and collaborators for R136 (Hunter et al1995 1996) that of de Grijs and collaborators for NGC 1805 and NGC 1818(de Grijs et al 2002ab) and that of Sirianni and collaborators for NGC 330(Sirianni et al 2002) R136 in the LMC is the youngest of these four clusters(sim3 Myr) followed by NGC 1805 (sim10 Myr) and NGC 1818 (sim20 Myr) bothalso in the LMC and finally NGC 330 (sim30 Myr) in the SMC This age range

allows us to closely trace the evolution of the primordially mass-segregatedmodels by comparing simulated observations to genuine observations reportedin the relevant papers

Consider first R136 and the work of Hunter et al (1995 1996) who usedHSTWFPC2 observations of this cluster to measure the slope of the massfunction as a function of projected radius Their results are reproduced inFig 149 Note that in their work the mass function is represented by a func-tion ζ(m) which is the number of single stars per logarithmic mass interval asopposed to the mass function ξ(m) defined in (143) It is straightforward to

Γ

Fig 149 Mass and luminosity function slopes as a function of projected radius forvarious young LMC and SMC clusters compared with the results from simulatedobservations of N -body Run 3 Upper left Mass function slope Γ as a function ofradius in R136 in the LMC from Hunter et al (1995 1996) Upper right Luminosityfunction slopes β as a function of projected radius for NGC 1805 and NGC 1818 inthe LMC from de Grijs et al (2002b) Lower Mass function slope Γ for NGC 330in the SMC from Sirianni et al (2002)

374 A D Mackey

demonstrate that if a function ξ(m) has an exponent minusα then the functionζ(m) also a power law has exponent Γ = minusα + 1 Hence the exponentα3 = 23 in the Kroupa (2001) IMF in (145) becomes Γ = 13 if the massfunction is represented by ζ(m) rather than ξ(m)

Hunter et al (1996) found some flattening of the mass function slope withincreasingly small radius in R136 Using their annulus widths together withthe specific bright and faint detection limits they list for each annulus we di-rectly simulated their measurements on N -body Run 3 at an output time of3 Myr As usual it is vital to this process that the annulus widths and brightand faint limits per annulus are exactly reproduced so that directly compa-rable mass function slopes are derived Radii in arcseconds were obtained byapplying an LMC distance modulus of 185 which defines a scale of 4116arcsec per parsec The N -body results are plotted on the relevant panel inFig 149 and clearly closely match the results of Hunter et al (1996) Thegreatest deviation occurs in the innermost part of the cluster where severecrowding prevented Hunter et al (1996) from obtaining a secure measure-ment It is also worth noting that the overall mass function slope agrees wellThis value is flatter than the input value (ie flatter than Γ = minus13 whichis the slope in the mass ranges under consideration) because we are onlyconsidering the innermost 15 arcsec of Run 3 to match the radial extent ofthe genuine R136 measurements In the outer regions of the N -body clusterthe mass function slope is somewhat steeper than the input slope so thatin the entire cluster we obtain Γ = minus13 Observations of R136 extending tolarge projected radii would presumably also find a steeper mass function slopein its outer regions

We followed a similar procedure to reproduce the observations of de Grijset al (2002b) for NGC 1805 and NGC 1818 (in this case we used an inter-mediate output time from Run 3 of 15 Myr) and the observations of Sirianniet al (2002) for NGC 330 (we used an output time from Run 3 of 30 Myr)In each case we adopted the annulus widths and annulus-specific detectionlimits listed by the authors Note that in the case of NGC 1805 and NGC1818 the slope β of the luminosity function (rather than the mass function)is measured This is easily reproduced by using the brightnesses of the N -bodystars rather than their masses

Our N -body measurements are plotted on the relevant panels in Fig 149In all cases agreement is close The largest deviation comes in the outer re-gions of NGC 330 where Sirianni et al (2002) note that their measurementsare uncertain due to field star contamination (which is not present in theN -body models and which is not straightforward to include in simulated ob-servations) The fact that this more detailed testing of our initial conditionsmatches well the best available observations of young LMC and SMC clusterssuggests we have managed to set up sufficiently realistic clusters and vali-dates the procedure we used to generate primordial mass segregation in theN -body models Once even more detailed observations of young Magellanic

cloud clusters are available (say velocity profiles for example) these will beable to be incorporated into the initial conditions in a very similar manner

144 Summary

Realistic large-scale N -body modelling of low-mass globular clusters suchas those found in the LMC and SMC is now feasible and routinely carriedout This is mainly due to the advent of special purpose hardware combinedwith the ever-increasing sophistication of leading N -body codes which nowincorporate all the major physical processes that occur in star clusters Directmodelling of typical mass globular clusters is still an order of magnitude out ofreach (this is the so-called million body problem) however within a few yearsthis goal should be reached The next generation GRAPE machine will shortlybe in production (GRAPE-DR) and it is expected that this will provide therequired order of magnitude leap Furthermore exciting new code develop-ments are taking place For example Church (PhD dissertation Universityof Cambridge) includes live stellar evolution in an N -body code (as opposedto stellar evolution calculated from analytic formulae) Borch Spurzem ampHurley (2007) are associating spectral libraries with evolving stars in N -bodyclusters These will allow new levels of sophistication and realism in both themodels themselves and the types of simulated observations it will be possibleto carry out

This chapter has provided an introduction to what is presently possiblewithin the field of realistic N -body simulations and a general description ofvarious aspects of the philosophy and methodology required for successfulsimulations and data analysis A detailed example has demonstrated how theinteraction between observation and modelling is essential throughout theprocess of applying realistic large-scale N -body simulations to real systems

References

Aarseth S J Heggie D C 1998 MNRAS 297 794 349Aarseth S J 2003 Gravitational N -Body Simulations Cambridge Univ Press

Cambridge 348 371Brandl B et al 1996 ApJ 466 254 361Borch A Spurzem R Hurley J 2007 328 662 375Casares J 2006 in Karas V Matt G eds Proc IAU Symp 238 Black Holes

From Stars to Galaxies Cambridge Univ Press Cambridge p 3 360Casertano S Hut P 1985 ApJ 298 80 371de Grijs R Johnson R A Gilmore G F Frayn C M 2002a MNRAS 331 228 361 372de Grijs R Gilmore G F Johnson R A Mackey AD 2002b MNRAS 331 245 361 372 373 3Elson R A W 1991 ApJS 76 185 355Elson R A W 1992 MNRAS 256 515 355Elson R A W Fall S M Freeman K C 1987 ApJ 323 54 355 357

376 A D Mackey

Elson R A W Freeman K C Lauer T R 1989 ApJ 347 L69 355 357Fukushige T Makino J Kawai A 2005 PASJ 57 1009 348Giersz M Heggie D C 1994 MNRAS 268 257 349Heggie D C Mathieu R D 1986 in Hut P McMillan S eds Lecture Notes

in Physics Vol 267 The Use of Supercomputers in Stellar Dynamics Springer-Verlag Berlin p 233 351

Hunter D A Shaya E J Holtzman J A Light R M 1995 ApJ 448 179 361 372 373Hunter D A OrsquoNeil Jr E J Lynds R Shaya E J Groth E J Holtzman J A

1996 459 L27 361 372 373 374Hurley J R Pols O R Tout C A 2000 MNRAS 315 543 351Hurley J R Tout C A Pols O R 2002 MNRAS 329 897 351Hurley J R Tout C A Aarseth S J Pols O R 2004 MNRAS 355 1207 353 360Hurley J R Pols O R Aarseth S J Tout C A 2005 MNRAS 363 293 350 353 354 365King I R 1962 AJ 67 471 360Kroupa P 2001 MNRAS 322 231 359 360 374Kurucz R L 1992 in Barbuy B Renzini A eds Proc IAU Symp 149 The Stellar

Populations of Galaxies Kluwer Dordrecht p 225 365Mackey A D Gilmore G F 2003a MNRAS 338 85 355 356 357 358 365Mackey A D Gilmore G F 2003b MNRAS 338 120 355 356 358 365Mackey A D Wilkinson M I Davies M B Gilmore G F 2007 MNRAS 379

L40 350 353 354 357Mackey A D Wilkinson M I Davies M B Gilmore G F 2008a MNRAS in

press 350 354 356 357 359 366Mackey A D et al 2008b in prep 355 356 365Makino J 1991 ApJ 369 200 350Makino J Fukushige T Koga M Namura K 2003 PASJ 55 1163 348Malumuth E M Heap S R 1994 AJ 107 1054 361Mardling R A Aarseth S J 2001 MNRAS 321 398 351McLaughlin D E van der Marel R P 2005 ApJS 161 304 358Meylan G Heggie D C 1997 AampAR 8 1 347 355 366Mikkola S Aarseth S J 1993 Celest Mech Dyn Astron 57 439 351Mikkola S Aarseth S J 1998 New Astron 3 309 351Sirianni M Nota A De Marchi G Leitherer C Clampin M 2002 ApJ 579 275

361 372 373 374van der Marel R P Alves D R Hardy E Suntzeff N B 2002 AJ 124 2639 359Wilkinson M I Hurley J R Mackey A D Gilmore G F Tout C A 2003

MNRAS 343 1025 349 351 357 359Zhang W Woosley S E Heger A 2008 ApJ 679 639 360

15

Parallelization Special Hardwareand Post-Newtonian Dynamicsin Direct N -Body Simulations

Rainer Spurzem15 Ingo Berentzen15 Peter Berczik15 David Merritt2Pau Amaro-Seoane3 Stefan Harfst425 and Alessia Gualandris24

1Astronomisches Rechen-Institut Zentr Astron Univ Heidelberg (ZAH)Monchhofstrasse 12-14 69120 Heidelberg Germany2College of Science Dept of Physics Rochester Instute of Technology 85 LombMemorial Drive Rochester NY 14623-5603 USA3Max-Planck Institut fur Gravitationsphysik (Albert-Einstein-Institut) AmMuhlenberg 1 D-14476 Potsdam Germany4Astronomical Institute Anton Pannekoek and Section Computational ScienceUniversity of Amsterdam The Netherlands5The Rhine Stellar Dynamical Networkspurzemariuni-heidelbergde

151 Introduction

The formation and evolution of supermassive black hole (SMBH) binaries dur-ing and after galaxy mergers is an important ingredient for our understandingof galaxy formation and evolution in a cosmological context eg for predic-tions of cosmic star formation histories or of SMBH demographics (to predictevents that emit gravitational waves) If galaxies merge in the course of theirevolution there should be either many binary or even multiple black holes orwe have to find out what happens to black hole multiples in galactic nucleieg whether they come sufficiently close to merge resulting from emission ofgravitational waves or whether they eject each other in gravitational slingshotinteractions

According to the standard theory the subsequent evolution of the blackholes is divided in three successive stages (Begelman Blandford amp Rees 1980)1 Dynamical friction causes a transfer of the black holesrsquo kinetic energy tothe surrounding field stars and the black holes spiral to the centre where theyform a binary 2 While hardening the effect of dynamical friction reduces andthe evolution is dominated by superelastic scattering processes that is theinteraction with field stars closely encountering or intersecting the binariesrsquoorbit thereby increasing the binding energy 3 Finally the black holes coalescethrough the emission of gravitational radiation potentially detectable by theplanned space-based gravitational wave antennae LISA For a more detailed

Spurzem R et al Parallelization Special Hardware and Post-Newtonian Dynamics in

Direct N-Body Simulations Lect Notes Phys 760 377ndash389 (2008)

378 R Spurzem et al

account of the state of research in this field see Milosavljevic amp Merritt (20012003) Makino amp Funato (2004) Berczik Merritt amp Spurzem (2005) In ourcontext the problem will be used as an example where relativistic dynamicsbecomes important during the evolution of an otherwise classical NewtonianN -body system

152 Relativistic Dynamics of Black Holesin Galactic Nuclei

Relativistic stellar dynamics is of paramount importance for the study of anumber of subjects For instance if we want to have a better understanding ofwhat the constraints on alternatives to supermassive black holes are in orderto explore the possibility of ruling out stellar clusters one must do detailedanalysis of the dynamics of relativistic clusters Furthermore the dynamicsof compact objects around an SMBH or multiple SMBHs in galactic nucleirequires the inclusion of relativistic effects Our current work deals with theevolution of two SMBHs in bound orbit and looks at the phase when theyget close enough to each other that relativistic corrections to Newtonian dy-namics become important which ultimately leads to gravitational radiationlosses and coalescence

Efforts to understand the dynamical evolution of a stellar cluster inwhich relativistic effects may be important have already been made by Lee(1987) Quinlan amp Shapiro (1989 1990) and Lee (1993) In the earlier work1PN and 2PN terms were neglected (Lee 1993) and the orbit-averaged for-malism (Peters 1964) used We describe here a method to deal with deviationsfrom Newtonian dynamics more rigorously than in most existing literature(but compare Mikkola amp Merritt (2007) Aarseth (2007) which are on thesame level of PN accuracy) We modified the nbody6++ code to allow forpost-Newtonian (PN ) effects of two particles getting very close to each otherimplementing the 1PN 2PN and 25PN corrections fully from Soffel (1989)and Kupi Amaro-Seoane amp Spurzem (2006)

Relativistic corrections to the Newtonian forces are expressed by expand-ing the relative acceleration between two bodies in a power series of 1c inthe following way (Damour amp Dereulle 1987 Soffel 1989)

a = a0︸︷︷︸Newt

+ cminus2a2︸︷︷︸1PN

︸︷︷︸periastron shift

+ cminus5a5︸︷︷︸25PN︸︷︷︸

grav rad

+O(cminus6) (151)

where a is the acceleration of particle 1 a0 = minusGm2nr2 is the Newtonian ac-

celeration G is the gravitation constant m1 and m2 are the masses of the twoparticles r is the distance of the particles n is the unit vector pointing fromparticle 2 to particle 1 and the 1PN 2PN and 25PN are post-Newtoniancorrections to the standard acceleration responsible for the pericentre shift

15 Parallelization and Post-Newtonian Dynamics 379

(1PN 2PN ) and the quadrupole gravitational radiation (25PN ) corre-spondingly as shown in (151) The expressions for the accelerations are

a2 =Gm2

r2middot

n

[

minusv21 minus 2v2

2 + 4v1v2 +32(nv2)2 + 5

(Gm1

r

)

+ 4(Gm2

r

)]

+(v1 minus v2) [4nv1 minus 3nv2]

(152)

a4 =Gm2

r2middot

n

[

minus2v42 + 4v2

2(v1v2) minus 2(v1v2)2 +

3

2v21(nv2)

2

+9

2v22(nv2)

2 minus 6(v1v2)(nv2)2 minus 15

8(nv2)

4

+Gm2

rmiddot(

4v22 minus 8v1v2 + 2(nv1)

2 minus 4(nv1)(nv2) minus 6(nv2)2)

+Gm1

rmiddot(

minus15

4v21 +

5

4v22 minus 5

2v1v2 +

39

2(nv1)

2 minus 39(nv1)(nv2) +17

2(nv2)

2)]

+(v1 minus v2)

[

v21(nv2) + 4v2

2(nv1) minus 5v22(nv2) minus 4(v1v2)(nv1)

+4(v1v2)(nv2) minus 6(nv1)(nv2)2 +

9

2(nv2)

3

+Gm1

rmiddot(

minus63

4nv1 +

55

4nv2

)

+Gm2

rmiddot(

minus 2nv1 minus 2nv2

)]

+G3m2

r4middot n

[

minus57

4m2

1 minus 9m22 minus 69

2m1m2

]

(153)

a5 =45G2m1m2

r3

(v1 minus v2)[

minus(v1 minus v2)2 + 2

(Gm1

r

)

minus 8(Gm2

r

)]

+n(nv1 minus nv2)[

3(v1 minus v2)2 minus 6

(Gm1

r

)

+523

(Gm2

r

)]

(154)

In the last expressions v1 and v2 are the velocities of the particles For sim-plification we have denoted the vector product of two vectors x1 and x2as x1x2 The basis of direct nbody4 and nbody6++ codes relies on an im-proved Hermite integration scheme (Makino amp Aarseth 1992 Aarseth 1999)for which we need not only the accelerations but also their time derivativesThese derivatives are not included here for succinctness We include our cor-rection terms in the KS regularisation scheme (Kustanheimo amp Stiefel 1965)as perturbations similarly to what is done to account for passing stars influ-encing a KS pair Note that formally the perturbing force in the KS equationsdoes not need to be small compared to the two-body force (Mikkola 1997)If the internal KS time step is properly adjusted the method works even forrelativistic terms becoming comparable to the Newtonian force component

380 R Spurzem et al

153 Example of Application to Galactic Nuclei

In Fig 151 the importance of relativistic post-Newtonian dynamics for theseparation of the binary black holes in our simulations is seen The curvedeviates from the Newtonian results when gravitational radiation losses setin and causes a sudden coalescence (1a rarr infin) at a finite time Gravitationalradiation losses are enhanced by the high eccentricity of the SMBH binary Itis interesting to note that the inclusion or exclusion of the conservative 1PNand 2PN terms changes the coalescence time considerably Details of theseresults will be published in a larger parameter study (Berentzen et al 2008in preparation) Note that Aarseth (2003a) presents two models very similarto those discussed here which agree qualitatively with our work regarding therelativistic merger time and the eccentricity of the SMBH binary

Once the SMBH binary starts to lose binding energy dramatically due togravitational radiation its orbital period drops from a few thousand yearsto less than a year very quickly (time-scale much shorter than the dynami-cal time-scale in the galactic centre which defines our time unit) Then theSMBH binary will enter the LISA band ie its gravitational radiation will bedetectable by LISA The Laser Interferometer Space Antenna is a system ofthree space probes with laser interferometers to measure gravitational wavessee eg httplisaesaint Once the SMBH binary decouples from therest of the system we just follow its relativistic two-body evolution starting

001

01

1

10

100

1000

10000

100000

0 50 100 150 200

1a

e

time

PN25PN

Fig 151 Effect of post-Newtonian (PN) relativistic corrections on the dynamicsof black hole binaries in galactic nuclei Plotted are inverse semi-major axis andeccentricity as a function of time The solid line uses the full set of PN correctionswhile the dashed line has been obtained by artificially only using the dissipative25PN terms Note that the coalescence time in the latter case has changed sig-nificantly Further details will be published elsewhere (Berentzen et al 2008 inpreparation)

with exactly the orbital parameters (including eccentricity) as they were ex-tracted from the N -body model It is then possible to predict the gravitationalradiation of the SMBH binary relative to the LISA sensitivity curve (Pretoet al 2008 in preparation) For some values of the eccentricity our simu-lated SMBH binaries indeed enter the LISA sensitivity regime for a circularorbit the n = 2 harmonic of the gravitational radiation is dominant whilefor eccentric orbits higher harmonics are stronger (Peters amp Mathews 1963Peters 1964)

154 N -Body Algorithms and Parallelization

Numerical algorithms for solving the gravitational N -body problem (Aarseth2003) have evolved along two main lines in recent years Direct-summationcodes compute the complete set of N2 interparticle forces at each time stepThese codes are designed for systems in which the finite-N graininess of thepotential is important or in which binary- or multiple-star systems form anduntil recently were limited by their O(N2) scaling to moderate (N lt 105)particle numbers The best-known examples are the NBODY series of codes(Aarseth 1999) and the Starlab environment developed by McMillan Hutand collaborators (eg Portegies Zwart et al 2001)

A second class of N -body algorithms replaces the direct summation offorces from distant particles by an approximation scheme Examples are theBarnesndashHut tree code (Barnes amp Hut 1986) which reduces the number offorce calculations by subdividing particles into an oct-tree and fast multipolealgorithms that represent the large-scale potential via a truncated basis-setexpansion (van Albada amp van Gorkom 1977 Greengard amp Rokhlin 1987) Suchalgorithms have a milder O(N logN) or even O(N) scaling for the force calcu-lations and can handle much larger particle numbers although their accuracyare substantially lower than that of the direct-summation codes (Spurzem1999) The efficiency of both sorts of algorithm can be considerably increasedby the use of individual time steps for advancing particle positions (Aarseth2003)

A natural way to increase both the speed and the particle number inan N -body simulation is to parallelize (Dubinski 1996 Pearce amp Couchman1997) Parallelization on general-purpose supercomputers is difficult howeverbecause the calculation cost is often dominated by a small number of particlesin a single dense region eg the nucleus of a simulated galaxy Communicationlatency becomes the bottleneck the time to communicate particle positionsbetween processors can exceed the time spent computing the forces The bestsuch schemes use systolic algorithms (in which the particles are successivelypassed around a ring of processors) coupled with non-blocking communica-tion between the processors to reduce the latency (Makino 2002 DorbandHemsendorf amp Merritt 2003)

382 R Spurzem et al

A major breakthrough in direct-summation N -body simulations came inthe late 1990s with the development of the GRAPE series of special-purposecomputers (Makino amp Taiji 1998) which achieve spectacular speed-ups byimplementing the entire force calculation in hardware and placing many forcepipelines on a single chip The GRAPE-6 in its standard implementation (32chips 192 pipelines) can achieve sustained speeds of about 1 Tflops at a costof just sim $50 K In a standard setup the GRAPE-6 is attached to a singlehost workstation in much the same way that a floating-point or graphicsaccelerator card is used Advancement of particle positions [O(N)] is carriedout on the host computer while coordinate and velocity predictions and inter-particle forces [O(N2)] are computed on the GRAPE More recently ldquomini-GRAPEsrdquo (GRAPE-6A) (Fukushige Makino amp Kawai 2005) have becomeavailable which are designed to be incorporated into the nodes of a parallelcomputer The mini-GRAPEs have four processor chips on a single PCI cardand deliver a theoretical peak performance of sim 131 Gflops for systems of upto 128 K particles at a cost of about $6 K By incorporating mini-GRAPEsinto a cluster both large (106) particle numbers and high (1Tflops) speedscan be achieved

In the following we describe the performance of direct-summation N -bodyalgorithms on two computer clusters that incorporate GRAPE hardware

155 Special Hardware GRAPE and GRACE Cluster

The GRAPE-6A board (Fig 152 top panel) is a standard PCI short cardon which a processor an interface unit and a power supply are integratedThe processor is a module consisting of four GRAPE-6 processor chips eightSSRAM chips and one FPGA chip The processor chips each contain six forcecalculation pipelines a predictor pipeline a memory interface a control unitand IO ports (Makino et al 2003) The SSRAM chips store the particledata The four GRAPE chips can calculate forces their time derivatives andthe scalar gravitational potential simultaneously for a maximum of 48 par-ticles at a time this limit is set by the number of pipelines (six force cal-culation pipelines each of which serves as eight virtual multiple pipelines)There is also a facility to calculate neighbour lists from predefined neigh-bour search radii this feature is not used in the algorithms presented belowThe forces computed by the processor chips are summed in an FPGA chipand sent to the host computer A maximum of 131 072 (217) particles canbe held in the GRAPE-6A memory The peak speed of the GRAPE-6A is1313 Gflops (when computing forces and their derivatives) and 875 Gflops(forces only) assuming 57 and 38 floating-point operations respectively perforce calculation (Fukushige Makino amp Kawai 2005) The interface to thehost computer is via a standard 32-bit33 MHz PCI bus The FPGA chip (Al-tera EP1K100FC256) realizes a 4-input 1-output reduction when transferringdata from the GRAPE-6 processor chip to the host computer The complete

Fig 152 Top interior of a node showing a GRAPE-6A card (note the large blackfan) and an Infiniband card Bottom the GRACE cluster at ARI The head nodeand the 14Tbyte raid array are visible on the central rack The other four racks holda total of 32 compute nodes each equipped with a GRAPE-6A card and MPRACEcards

384 R Spurzem et al

GRAPE-6A unit is roughly 11 cm times 19 cm times 7 cm in size Note that 58 cm ofthe height is taken up by a rather bulky combination of cooling body and fanwhich may block other slots on the main board Possible ways to deal withthis include the use of even taller boxes for the nodes (eg 5U) together witha PCI riser of up to 6 cm which would allow the use of slots for interface cardsbeneath the GRAPE fan or the adoption of the more recent flatter designssuch as that of the GRAPE6-BL series The reader interested in more technicaldetails should seek information from the GRAPE (httpastrograpeorg)and Hamamatsu Metrix (httpwwwmetrixcojp) websites

A computer cluster incorporating GRAPE-6A boards became fully op-erational at the Rochester Institute of Technology (RIT) in February 2005This cluster named ldquogravitySimulatorrdquo consists of 32 compute nodes plusone head node each containing dual 3 GHz-Xeon processors In addition to astandard Gbit-ethernet the nodes are connected via a low-latency Infinibandnetwork with a transfer rate of 10 Gbits The typical latency for an Infini-band network is of the order of 10minus6 seconds or a factor sim 100 better thanthe Gbit-Ethernet A total of 14 Tbyte of disc space is available on a level5 RAID array The disc space is equivalent to 25 times 105 N -body data setseach with 106 particles The discs are accessed via a fast Ultra320 SCSI hostadapter from the head node or via NFS from the compute nodes which inaddition are each fitted with an 80 Gbyte hard disc Each compute node alsocontains a GRAPE-6A PCI card (Fig 152 top panel) The total theoreticalpeak performance is approximately 4 Tflops if the GRAPE boards are fullyutilized Total cost was about $ 450 000 roughly half of which was used topurchase the GRAPE boards

Some special considerations were required in order to incorporate theGRAPE cards into the cluster Since our GRAPE-6Arsquos use the relativelyold PCI interface standard (32 bit33 MHz) only one motherboard was avail-able the SuperMicro X5DPL-iGM that could accept both the GRAPE-6Aand the Infiniband card (A newer version of the GRAPE-6A which uses thefaster PCI-X technology is now available) The PC case itself has to be tallenough (4U) to accept the GRAPE-6A card and must also allow good air flowfor cooling since the GRAPE card is a substantial heat source The clusterhas a total power consumption of 17 kW when the GRAPEs are fully loadedCluster cooling was achieved at minimal cost by redirecting the air condition-ing from a large room toward the air-intake side of the cluster Temperaturesmeasured in the PC case and at the two CPUs remain below 30C and 50Crespectively

A similar cluster called ldquoGRACErdquo (GRAPE + MPRACE) has been in-stalled in the Astronomisches Rechen-Institut (ARI) at the University ofHeidelberg (Fig 152 bottom panel) There are two major differences be-tween the RIT and ARI clusters (1) Each node of the ARI cluster incorpo-rates a reconfigurable FPGA card (called ldquoMPRACErdquo) in addition to to theGRAPE board MPRACE is optimized to compute neighbour forces and othernon-Newtonian forces between particles in order to accelerate calculations of

molecular dynamics smoothed-particle hydrodynamics etc (2) The newermain board SuperMicro X6DAE-G2 was used which supports Pentium Xeonchips with 64-bit technology (EM64T) and the PCIe (PCI express) bus Thismade it possible to use dual-port Infiniband interconnects via the PCI ex-press Infiniband times8 host interface card used in the times16 Infiniband slot of theboard (it has another times4 Infiniband slot which is reserved for the MPRACE-2 Infiniband card) As discussed below the use of the PCIe bus substantiallyreduces communication overhead The benchmark results presented here forthe ARI cluster were obtained from algorithms that do not access the FPGAcards

156 Performance Tests

Initial conditions for the performance tests were produced by generatingMonte-Carlo positions and velocities from self-consistent models of stellarsystems Each of these systems is spherical and is completely described bya steady-state phase-space distribution function f(E) and its self-consistentpotential Ψ(r) where E = v22+Ψ is the particle energy and r is the distancefrom the centre The models were a Plummer sphere two King models withdifferent concentrations and two Dehnen models (Dehnen 1993) with differentcentral density slopes The Plummer model has a low central concentrationand a finite central density it does not represent any class of stellar systemaccurately but is a common test case King models are defined by a singledimensionless parameter W0 characterizing the central concentration (eg ra-tio of central to mean density) we used W0 = 9 and W0 = 12 which areappropriate for globular star clusters Dehnen models have a divergent innerdensity profile ρ prop rminusγ We took γ = 05 and γ = 15 which correspond ap-proximately to the inner density profiles of bright and faint elliptical galaxies

In what follows we adopt standard N -body units G = M = minus4E = 1where G is the gravitational constant M the total mass and E the total energyof the system In some of the models the initial time step for some particleswas smaller than the minimum time step tmin set to 2minus23 These models werethen rescaled to change the minimum time step to a large enough value Sincethe rescaling does not influence the performance results we will present allresults in the standard N -body units

We realized each of the five models with 11 different particle numbersN = 2k k = [10 11 20] ie N = [1K 2K 1M]1 We also testedPlummer models with N = 2M and N = 4M the latter value is the maximumN -value allowed by filling the memory of all 32 GRAPE cards Thus a totalof 57 test models were used in the timing runs

Two-body relaxation ie exchange of energy between particles due togravitational scattering induces a slow change in the characteristics of the

1Henceforth we use K to denote a factor of 210 = 1024 and M to denote a factorof 220 = 1 048 576

386 R Spurzem et al

models In order to minimize the effects of these changes on the timing runswe integrated the models for only one time unit The standard softening εwas set to zero for the Plummer models and to 10minus4 for the Dehnen and Kingmodels For the time step parameters used see Harfst et al (2007)

We analyzed the performance of the hybrid scheme as a function of particlenumber and also as a function of number of nodes using p = 1 2 4 8 16and 32 nodes The compute time w for a total of almost 350 test runs wasmeasured using MPI Wtime() The timing was started after all particles hadfinished their initial time step and ended when the model had been evolvedfor one time unit No data evaluation was made during the timing interval

The top panel of Fig 153 shows wallclock times wNp from all integrationson the ARI cluster For any p the clock time increases with N roughly as N2

for large N However when N is small communication dominates the totalclock time and w increases with increasing number of processors This be-haviour changes as N is increased for N gt 10K (the precise value depends onthe model) the clock time is found to be a decreasing function of p indicatingthat the total time is dominated by force computations

The speedup for selected test runs is shown in the bottom panel ofFig 153 The speedup s is defined as

sN p =wN 1

wN p (155)

The ideal speedup (optimal load distribution zero communication and la-tency) is sNp = p For particle numbers N ge 128K the wallclock time wN1

on one processor is undefined asN exceeds the memory of the GRAPE card Inthat case we used wN1 = w128 K1(N128K)2 assuming a simple N2-scalingIn general the speedup for any given particle number is roughly proportionalto p for small p then reaches a maximum before reducing at large p Thenumber of processors at maximum speedup is ldquooptimumrdquo in the sense thatit provides the fastest possible integration of a given problem The optimump is roughly the value at which the sum of the communication and latencytimes equals the force computation time in the zero-latency case popt prop N(Dorband Hemsendorf amp Merritt 2003) Figure 153 (bottom panel) showsthat for N ge 128K popt ge 32 for all the tested models The reader interestedin more details is referred to Harfst et al (2007)

157 Outlook and AhmadndashCohen Neighbour Scheme

At present there exist only the relatively simple parallel N -body code de-scribed above and in Harfst et al (2007) which uses GRAPE special hard-ware in parallel but always computes full forces for every particle at everystep This code sometimes dubbed p-GRAPE (sources are freely available seelink in the cited paper) also does not include any special few-body treatments(regularisations) as in the N -body codes of Aarseth (1999 2003)

1 2 4 8 16 32Number of Processors - p

1

10

Spe

edup

ideal speedup

N = 8kN = 32kN = 128kN = 512kN = 1024kN = 2048k

Plummer (ARI)

103 104 105 106

Number of Particles - N

100

101

102

103

104

105W

allc

lock

tim

e [s

]one day

one hour

one minute

Processors p = 1Processors p = 2Processors p = 4Processors p = 8Processors p = 16Processors p = 32

Plummer (ARI)

Fig 153 Top wallclock time w versus particle number N for different numbersof processors p Bottom speedup s versus processor number p for different N Boththe plots show the results obtained for a Plummer model on the ARI cluster

388 R Spurzem et al

There is the already mentioned parallel N -body code nbody6++ whichincludes all regularizations and the use of the Ahmad-Cohen neighbour scheme(Ahmad amp Cohen 1973) as in the standard nbody6 code However the pub-licly provided source code (ftpftpariuni-heidelbergdepubstaffspurzemnb6mpi) is not yet able to make parallel use of special hardwareIt parallelizes very efficiently over the regular and irregular force loops (cfSpurzem 1999 Khalisi et al 2003) but current work is in progress on animplementation of nbody6++ for special-purpose hardware (such as GRAPEMPRACE or graphical processing units GPU) as well as on an efficient paralleltreatment of many regularized perturbed binaries (see first results in Maalejet al 2005) New results in these topics will be published early at the wiki ofnbody6++ developers and users at httpnb6mpipbwikicom Last butnot least a nice visualization interface specially developed for nbody6++ ishosted by FZ Julich see httpwwwfz-juelichdejscxnbody

Similar to the GRAPE development nearly two decades ago the recentintroduction of GPUs and other new hardware devices (such as FPGA orMPRACE cards in the GRACE project

httpwwwariuni-heidelbergdegrace) is inspiring a new interestin improving and developing efficient N -body algorithms It is expected thatvery soon the use of most advanced special hardware and software (such asnbody6 and nbody6++) will not mutually exclude each other any more

Acknowledgement

Computing time at NIC Julich on the IBM Jump is acknowledged Finan-cial support comes partly from Volkswagenstiftung (I80 041-043) GermanScience Foundation (DFG) via SFB439 at the University of Heidelberg andSchwerpunktprogramm 1177 (Project ID Sp 34517-1) lsquoBlack Holes Witnessesof Cosmic Historyrsquo It is a pleasure to acknowledge many enlightening discus-sions with and support by Sverre Aarseth and very useful interactions aboutrelativistic dynamics with A Gopakumar and G Schafer

References

Aarseth S J 1999 PASP 111 1333 379 381 386Aarseth S J 2003a ApSS 285 367 380Aarseth S J 2003 Gravitational N -Body Simulations Cambridge University Press

Cambridge 381 386Aarseth S J 2007 MNRAS 378 285 378Ahmad A Cohen L 1973 J Comput Phys 12 349Barnes J Hut P 1986 Nature 324 446 381Begelman M C Blandford R D Rees M J 1980 Nature 287 307 377Berczik P Merritt D Spurzem R 2005 ApJ 633 680 378

Berczik P Merritt D Spurzem R Bischof H-P 2006 ApJ 642 L21Berentzen I Preto M Berczik P Merritt D Spurzem R 2008 to be submitted 380Damour T Dereulle N 1987 Phys Lett 87 81 378Dehnen W 1993 MNRAS 265 250 385Dorband E N Hemsendorf M Merritt D 2003 J Comput Phys 185 484 381 386Dubinski J 1996 New Astron 1 133 381Fukushige T Makino J Kawai A 2005 PASJ 57 1009 382Greengard L Rokhlin V 1987 J Comput Phys 73 325 381Harfst S Gualandris A Merritt D Spurzem R Portegies Zwart S Berczik P

2007 New Astron 12 357 386Khalisi E Omarov C T Spurzem R Giersz M Lin D N C 2003 in Krause E

Jaeger W Resch M eds Performance Computing in Science and EngineeringSpringer Verlag p 71 388

Kupi G Amaro-Seoane P Spurzem R 2006 MNRAS 371 L45 378Kustaanheimo P Stiefel E Journ fur die reine und angew Math 1965 218 204Lagoute C Longaretti P -Y 1996 AampA 308 441Lee H M 1987 ApJ 319 801 378Lee M H 1993 ApJ 418 147 378Maalej K P Boily C David R Spurzem R 2005 in Casoli F Contini T

Hameury J M Pagani L eds SF2A-2005 Semaine de lrsquoAstrophysique Fran-caise EdP-Sciences Conference Series p 629 388

Makino J 2002 New Astron 7 373 381Makino J Aarseth S J 1992 PASJ 44 141 379Makino J Fukushige T Koga M Namura K 2003 PASJ 55 1163 382Makino J Funato Y 2004 ApJ 602 93 378Makino J Taiji M 1998 Scientific Simulations with Special-Purpose Computers mdash

the GRAPE systems Wiley 382Mikkola S 1997 Celes Mech Dyn Ast 68 87 379Mikkola S Merritt D 2007 ArXiv e-prints 709 arXiv07093367 378Milosavljevic M Merritt D 2001 ApJ 563 34 378Milosavljevic M Merritt D 2003 ApJ 596 860 378Pearce F R Couchman H M P 1997 New Astron 2 411 381Peters P C 1964 Phys Rev 136 B1224 378 381Peters P C Mathews J 1963 Phys Rev 131 435 381Portegies Zwart S F McMillan S L W HutP Makino J 2001 MNRAS 321 199 381Preto M Berentzen I Berczik P Spurzem R 2008 in preparation 381Quinlan G D Shapiro S L 1989 ApJ 343 725 378Quinlan G D Shapiro S L 1990 ApJ 356 483 378Soffel M H 1989 Relativity in Astrometry Celestial Mechanics and Geodesy

Springer-Verlag 378Spurzem R 1999 J Comput Applied Math 109 407 381 388van Albada T S van Gorkom J H 1977 AampA 54 121 381

A

Educational N -Body Websites

Francesco Cancelliere1 Vicki Johnson2 and Sverre Aarseth3

1 Free University Brussels Pleinlaan 2 B-1050 Brussels Belgiumfcancellvubacbe

2 Interconnect Technologies LLC POB 1517 Placitas NM 87043 USAvljinterconnectcom

3 University of Cambridge Institute of Astronomy Madingley Road CambridgeCB3 0HA UKsverreastcamacuk

A1 Introduction

The 2006 Cambridge N -body School introduced participants to educationalwebsites for N -body simulations wwwSverrecom and wwwNBodyLaborgThese websites run versions of the freely available open-source NBODY4TRIPLE and CHAIN codes (Aarseth 2003) that have been adapted for theweb The websites provide guidance and documentation They support simu-lations of small N (3 and 4 bodies) on both sites and higher N (up to 15000)on NBodyLaborg Numerical results graphics and animations are displayedNBodyLaborg supports NBODY4 running on a GRAPE-6A hardware acceler-ator and demonstrates its accuracy and speed The websites were developedwith different approaches NBodyLaborg runs N -body codes on the serverside and Sverrecom uses Java to run locally

The websites were recommended as homework before the N -body Schooland practical demonstrations were given during the School Use of these sitesby participants also continued afterwards Such web-based tools can be a use-ful and convenient part of the curriculum for teaching N -body simulationsand also serve as test-beds for prospective buyers of GRAPE hardware accel-erators for large simulations This Appendix describes the websites and theireducational utility

A2 wwwNBodyLaborg

NBodyLaborg (Johnson amp Aarseth 2006) is a laboratory where we can exper-iment with small N -body simulations with a desktop GRAPE-6A supercom-puter (Fukushige Makino amp Kawai 2005 Makino amp Taiji 1998) The NBODY4TRIPLE and CHAIN codes are adapted for the web from the current versions

Cancelliere F et al Educational N-Body Websites Lect Notes Phys 760 391ndash396 (2008)

392 F Cancelliere et al

of the UnixFORTRAN codes 1 and simulations are run on the server sidePlots and 3D animations are created from the simulation output

NBodyLab was initially developed in 2002 to augment an undergradu-ate astrophysics course Prior to upgrading to NBODY4 NBodyLab was usedfor homework assignments an undergraduate senior thesis on tidal shockingof globular clusters small system studies of Ursa Major Hyades Collinder70 the solar system Halleyrsquos comet and a masterrsquos thesis on N -body sim-ulations and HR diagrams of nearby stars (Johnson amp Ates 2004) Incor-porating NBODY4 has significantly improved the sitersquos N -body simulationcapabilities

Examples of NBODY4 simulations that can be run on NBodyLaborg include

bull single Plummer sphere cluster model (N = 1000)bull single Plummer sphere cluster with 200 additional primordial binariesbull two Plummer models in orbitbull massive perturber and planetesimal diskbull evolution of a dominant binary andbull upload specialized initial conditions

Input parameters are entered via forms (NBODY4 concise style or simplified)

The presentation of NBodyLaborg at the N -body School included dis-cussion of the sitersquos goals parameter limits and an overview of the mainfeatures of NBODY4 such as GRAPE acceleration for direct integration reg-ularization of close encounters and stellar evolution with mass loss and colli-sions The main NBODY46 input parameters were introduced including modeloptions choices for binaries stellar evolution and mass loss initial massfunction scaling and chain regularization NBODY4 and NBODY6 were com-pared It should be noted that NBODY6 uses a neighbour scheme to speedup the integration Output data analysis and output quantities were dis-cussed along with plots and stellar evolution features such as the time

1downloads at httpwwwastcamacukresearchnbody

Educational N -Body Websites 393

dependence of the half-mass radius and core radius in N -body units as wellas the HR diagram for the initial and final population of single stars (see nextfigures)

Animations of model evolution can be viewed in 3D with a Java applet

TRIPLE and CHAIN with regularization are used for small N simulations onNBodyLaborg Examples of three-body simulations with 3D animations in-clude

bull figure-8 periodic orbit and perturbations (Heggie 2000)bull idealized triple system and perturbationsbull Pythagorean problem and perturbations andbull criss-cross periodic orbit and perturbations (Moore 1993)

and examples of four-body simulations with 3D animations include

bull great circle unstable orbit andbull symmetrical exchange for two binaries

Examples of graphics for the three-body figure-8 stable orbit and with per-turbations are displayed in the following figures

A manual for running simulations with NBODY4 and NBODY6 was preparedfor the Cambridge N -body School (Aarseth amp Johnson 2006) It covers pa-rameter selection suggested simulations astrophysical and N -body units in-tegration methods the relationship between NBODY4 and NBODY6 and othertopics Sample runs are interpreted and annotated

A3 wwwSverrecom

This interactive website was made available in 2005 to support movies ofthe three-body problem where the initial conditions are specified online Inthe summer of 2006 a second similar presentation was implemented for thefour-body problem The main technical difference is that a three-body regu-larization method (Aarseth amp Zare 1974) is used for the former while N = 4is handled by chain regularization (Mikkola amp Aarseth 1993) which can alsodeal with N = 3 after one body escapes The calculations are done in realtime by a Java applet or Java application that can be downloaded In spiteof considerable loss in programming efficiency owing to the use of Java in-stead of FORTRAN the viewing time is sufficiently short even at the highesttime-step resolution

Online simulations can be instructive and also great fun For practicalconvenience only 2D calculations are performed A number of useful featuresare available such as a scale factor for magnification smoothness index tovary the viewing time maximum run time (otherwise until escape) a facilityfor play pause or reset and also for displaying the orbits at the end Thescreen shots show initial and final configurations for the two movie versionswith the interactive initial conditions specified in appropriate boxes The basicFORTRAN codes without the interactive part as well as TRIPLE and CHAINcan be downloaded from the URL specified above

Some examples of interesting initial conditions are provided as templatesand shown above together with the final orbits Users are encouraged toexperiment by exploring the large parameter space It can be seen that verysmall changes in the initial conditions may produce widely different behaviourowing to the chaotic nature of the problem Although most solutions shouldbe accurate complex interplays of long duration are notoriously difficult and

even small errors are subject to exponential growth which may lead to thewrong outcome However since close two-body encounters are treated very ac-curately with regularization the result of the strong interactions themselvesis reliable

A4 Educational Utility

For undergraduate and graduate astronomy and physics courses and specialadvanced programs such as the N -body School web-based tools can be auseful part of the curriculum The primary educational utility of the websitesdiscussed here is their ease-of-use Documentation is available for beginnersand experienced users and initial values are given for interesting examplesRuns can be made with a click of a button and no compilation and additionalgraphical displays are produced which are not supported in the standardcode versions Specially constructed initial conditions can also be uploadedto satisfy individual requirements for GRAPE simulations The websites havealso been used by researchers writing their own N -body codes for comparingresults and testing (eg for stellar evolution)

The websites enable and encourage migration from simulations via thewebsites to in-depth runs code development and research on personal work-stations After becoming acquainted with the program functionality userscan download the freely available open-source software and run NBODY46TRIPLE and CHAIN with NBODY4 also available in an emulator version withoutGRAPE hardware Discussion of the programs in the book (Aarseth 2003)and documentation on the websites facilitate online use and local computing

Simulations on the websites have been made by users world-wide About300 simulations per month were run on wwwNBodyLaborg in the last halfof 2006 and the guide Introduction to Running NBODY46 Simulations wasdownloaded about 100 times per month Following the Cambridge N -bodySchool NBodyLaborg was used in late 2006 in assigned exercises for studentsof a Stellar Dynamics course at the University of Bonn In 2007 a three-bodysimulation code with relativistic effects was added The development of thesewebsites has led to improvements in the N -body codes and documentationSuggestions for other features and new educational uses are welcomed

References

Aarseth S J 2003 Gravitational N -Body Simulations Cambridge UnivPress Cambridge

Aarseth S J Johnson V L 2006 posted on NBodyLaborgAarseth S J Zare K 1974 Celes Mech 10 185Fukushige T Makino J Kawai A 2005 PASJ 57 1009Heggie D C 2000 MNRAS 318 L61Johnson V L Aarseth S J 2006 in C Gabriel C Arviset D Ponz

E Solano eds ADASS XV ASP Conf Ser 351 165Johnson V L Ates A 2004 in P Shopbell M Britton R Ebert eds

ADASS XIV ASP Conf Ser 347 524Makino J Taiji M 1998 Scientific Simulations with Special-Purpose

Computers the GRAPE System John Wiley amp SonsMikkola S Aarseth S J 1993 Celes Mech Dyn Ast 57 439Moore C 1993 Phys Rev Lett 70 3675

front-matterpdf
fulltextpdf
- 1 Direct N-Body Codes
- - Sverre J Aarseth
  - - Introduction
    - Basic Features
    - Data Structure
    - N-Body Codes
    - Hermite Integration
    - Ahmad--Cohen Neighbour Scheme
    - Time-Step Criteria
    - Two-Body Regularization
    - KS Decision-Making
    - Hierarchical Systems
    - Three-Body Regularization
    - Wheel-Spoke Regularization
    - Post-Newtonian Treatment
    - Chain Regularization
    - Astrophysical Procedures
    - GRAPE Implementations
    - Practical Aspects
    - References
    - - fulltext_2pdf
      - 2 Regular Algorithms for the Few-Body Problem
        
        Seppo Mikkola
        
        Introduction
        
        Hamiltonian Manipulations
        
        Coordinate Transformations
        
        KS-Chain(s)
        
        Algorithmic Regularization
        
        N-Body Algorithms
        
        AR-Chain
        
        Basic Algorithms for the Extrapolation Method
        
        Accuracy of the AR-Chain
        
        Conclusions
        
        References
        
        fulltext_3pdf
        
        3 Resonance Chaos and Stability The Three-Body Problem in Astrophysics
        
        Rosemary A Mardling
        
        Introduction
        
        Resonance in Nature
        
        The Mathematics of Resonance
        
        The Three-Body Problem
        
        References
        
        fulltext_4pdf
        
        4 Fokker--Planck Treatment of Collisional Stellar Dynamics
        
        Marc Freitag
        
        Introduction
        
        Boltzmann Equation
        
        Fokker--Planck Equation
        
        Orbit-Averaged Fokker--Planck Equation
        
        The Fokker--Planck Method in Use
        
        Acknowledgement
        
        References
        
        fulltext_5pdf
        
        5 Monte-Carlo Models of Collisional Stellar Systems
        
        Marc Freitag
        
        Introduction
        
        Basic Principles
        
        Detailed Implementation
        
        Some Results and Possible Future Developments
        
        Acknowledgement
        
        References
        
        fulltext_6pdf
        
        6 Particle-Mesh Technique and S710UPERBOX
        
        Michael Fellhauer
        
        Introduction
        
        Particle-Mesh Technique
        
        Multi-Grid Structure of Superbox
        
        References
        
        fulltext_7pdf
        
        7 Dynamical Friction
        
        Michael Fellhauer
        
        What is Dynamical Friction
        
        How to Quantify Dynamical Friction
        
        Dynamical Friction in Numerical Simulations
        
        Dynamical Friction of an Extended Object
        
        References
        
        fulltext_8pdf
        
        8 Initial Conditions for Star Clusters
        
        Pavel Kroupa
        
        Introduction
        
        Initial 6D Conditions
        
        The Stellar IMF
        
        The Initial Binary Population
        
        Summary
        
        Acknowledgement
        
        References
        
        fulltext_9pdf
        
        9 Stellar Evolution
        
        Christopher A Tout
        
        Observable Quantities
        
        Structural Equations
        
        Equation of State
        
        Radiation Transport
        
        Convection
        
        Energy Generation
        
        Boundary Conditions
        
        Evolutionary Tracks
        
        Stellar Evolution of Many Bodies
        
        References
        
        fulltext_10pdf
        
        10 N-Body Stellar Evolution
        
        Jarrod R Hurley
        
        Motivation
        
        Method and Early Approaches
        
        The SSE Package
        
        N-Body Implementation
        
        Some Results
        
        References
        
        fulltext_11pdf
        
        11 Binary Stars
        
        Christopher A Tout
        
        Orbits
        
        Tides
        
        Mass Transfer
        
        Period Evolution
        
        Actual Types
        
        References
        
        fulltext_12pdf
        
        12 N-Body Binary Evolution
        
        Jarrod R Hurley
        
        Introduction
        
        The BSE Package
        
        
        Binary Evolution Results
        
        References
        
        fulltext_13pdf
        
        13 The Workings of a Stellar Evolution Code
        
        Ross Church
        
        Introduction
        
        Equations
        
        Variables and Functions
        
        Method of Solution
        
        The Structure of stars
        
        Problematic Phases of Evolution
        
        Robustness of Results
        
        References
        
        fulltext_14pdf
        
        14 Realistic N-Body Simulations of Globular Clusters
        
        A Dougal Mackey
        
        Introduction
        
        Realistic N-Body Modelling -- Why and How
        
        Case Study Massive Star Clusters in the Magellanic Clouds
        
        Summary
        
        References
        
        fulltext_15pdf
        
        15 Parallelization Special Hardware and Post-Newtonian Dynamics in Direct N-Body Simulations
        
        Rainer Spurzem Ingo Berentzen Peter Berczik David Merritt Pau Amaro-Seoane Stefan Harfst and Alessia Gualandris
        
        Introduction
        
        Relativistic Dynamics of Black Holes in Galactic Nuclei
        
        Example of Application to Galactic Nuclei
        
        N-Body Algorithms and Parallelization
        
        Special Hardware GRAPE and GRACE Cluster
        
        Performance Tests
        
        Outlook and Ahmad--Cohen Neighbour Scheme
        
        Acknowledgement
        
        References
        
        fulltext_16pdf
        
        A Educational N-Body Websites
        
        Francesco Cancelliere Vicki Johnson and Sverre Aarseth
        
        Introduction
        
        wwwNBodyLaborg
        
        wwwSverrecom
        
        Educational Utility
        
        References
        
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 SUO 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 SVE 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 ENU 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 gtgtgtgt setdistillerparamsltlt HWResolution [2400 2400] PageSize [595276 841890]gtgt setpagedevice