Results

AMUSE runs are at left, [TPZ] at right. The plots are of mass (as a fraction of initial) versus time in Gyr. The thick dashed line (indicated) on the [TPZ] plots is the N=32k run.

764 TAKAHASHI & PORTEGIES ZWART Vol. 535

FIG. 2.ÈComparison between Aa Fokker-Planck models (solid lines) and N-body models (dashed and dotted lines) for the evolution of the total mass.Thickness of the lines stand for largeness of N. The thickest solid line in each panel represents the Fokker-Planck model with the instantaneous escapecondition, which corresponds to the limit of N ] O. The other Fokker-Planck models are calculated with the crossing-time escape condition withlesc \ 2.5for Ðnite N. The dot-dashed lines represent the mass evolution expected when mass loss occurs only through stellar evolution (with no escapers). (a) Initialconditions are N \ 1K, 16K, and 32K, from right to left. Only for the N \ 1K N-body model, ^p/2 deviation from the mean of(W0, a, family) \ (3, 2.5, 1) ;10 N-body runs is shown (two dotted lines). (b) N \ 8K, 16K. (c) N \ 4K, 8K. (d)(W0, a, family) \ (1, 2.5, 1) ; (W0, a, family) \ (7, 2.5, 1) ; (W0, a, family) \(3, 2.5, 4) ; N \ 8K, 16K. (e) N \ 8K, 16K. ( f ) N \ 8K, 16K.(W0, a, family) \ (7, 1.5, 4) ; (W0, a, family) \ (1, 3.5, 3) ;

rate dM/dt may be divided into two parts :

dMdt

\AdMdtB

esc]AdM

dtB

se, (13)

where the Ðrst term in the right-hand side represents themass loss due to escapers from the tidal radius and thesecond term represents the mass loss due to stellar evolu-tion. Since the stellar evolution is the same in the Fokker-Planck and N-body models, it may be useful to separate thestellar evolution mass loss from the total mass loss in orderto further clarify the di†erence between the two models.Thus in Figure 4 we plot the mass-loss rate due to escapers

as well as the total mass-loss rate deÐned as follows :mesc mtot

mtot \ [trh0M

dMdt

, mesc \ [trh0M

AdMdtB

esc. (14)

When there is no stellar evolution, is usually much lessmescthan 1 (Lee & Goodman 1995). This is to be expected, sincein that case mass loss proceeds only on two-body relaxation

timescale. In the present models, since the stellar evolutionmass loss causes the shrink of the tidal radius and thusproduces more escapers, exceeds 1 sometimes.mescFigure 2 as well as Figures 3 and 4 show that the Fokker-Planck and N-body calculations agree fairly well over theentire range of initial conditions we investigated. Appar-ently the single value of is applicable. The relativelesc \ 2.5di†erence of the total mass at each time is generally lessthan 10%. However, for the cases of (W0, a, family) \(3, 2.5, 4) and (7, 1.5, 4), the di†erence is rather large. Thislarge di†erence seems to well correlate with large ([1),mescespecially in the N-body models. Also when the cluster isabout to dissolve (see panels a and b), becomes largermescthan 1 and the Fokker-Planck and N-body models deviate.

In the case of the N-body(W0, a, family) \ (3, 2.5, 4),models tend to lose mass more quickly than the Fokker-Planck models. The mass loss in the N-body models accel-erates rapidly, compared with the Fokker-Planck models,from about 1 Gyr for the 16K model and from about 2 Gyrfor the 8K model, and after some time the rate of mass loss

Models

King Models [5] of a star cluster:

Tides are simulated using truncation at the Jacobi radius (= the King tidal radius). For all displayed runs, N = 32,000. Runs with N = 1,000 and N = 8,000 were also conducted.

diffuse concentrated

Simulating Star Clusters with AMUSEAlfred J. Whitehead1, Stephen L.W. McMillan1, Simon Portegies Zwart2, Enrico Vesperini1

AcknowledgementsThanks to Arjen van Elteren for help with AMUSE interfaces. This work has been

supported by NASA grants NNX07AG95G and NNX08AH15G, and by NSF grant AST 0708299. AMUSE is developed at the Leiden Observatory, a faculty of Leiden

University. AMUSE is funded by a NOVA grant.

References [1] S. Casertano & P. Hut, ApJ, 298, 80, 1985 [CW] D.F. Chernoff & M.D. Weinberg, ApJ, 351, 121, 1990 [2] E. Gaburov et al., New Astronomy, 14, 630, 2007 [3] V. Garcia, PhD Thesis, Université de Nice - Sophia Antipolis, 2008 [4] S. Harfst et al., New Astronomy, 12, 357, 2007 [5] J. Hurley et al., MNRAS, 315, 543, 2000 [6] King, I.R., AJ, 71, 64, 1966 [7] S. Portegies Zwart et al., New Astronomy, 14, 369, 2009 [TPZ] K. Takahashi & S. Portegies Zwart, ApJ, 535, 759, 2000

Introduction

Our goal is to demonstrate that the new AMUSE code framework is sufficiently developed to be useful for scientific work. This is done by attempting to reproduce well-known results from Chernoff & Weinberg (1990) [CW] and Takahashi & Portegies Zwart (2000) [TPZ] for clusters evolving under the combined influence of gravitational dynamics and stellar evolution.

The results survey the parameter space of King models in a highly idealized tidal field. We vary the concentration, mass function slope and tidal time scale. The parameters chosen specifically avoid core-collapse as the AMUSE multiple module is still in development.

1Drexel University, Philadelphia, PA, 19104, USA 2Leiden Observatory, Leiden University, Leiden, The Netherlands

AMUSEThe Astrophysical Multipurpose Science Environment (AMUSE) is a new code base growing out of the MUSE project [7]. The core idea behind AMUSE is that it links together codes specialized to a single physical problem in order to create a multi-physics simulation rather than combining all codes into a single monolithic program.

AMUSE uses MPI to allow each module to exist in its own process, possibly in parallel and on a different machine than the Python control script. The AMUSE framework provides an easy way to import new legacy codes.

phiGRAPE [4] provides N-Body dynamics using SAPPORO [2] for GPU acceleration. SSE [5] provides stellar evolution. knnCUDA [3] is used to compute densities (12th nearest neighbour) in a stand-alone code similar to [1], but separate from AMUSE. This code finds all nearest neighbours, regardlesss of distance. Further InformationPlease contact alf.whitehead@drexel.edu

Control Script

phiGRAPE Interface

SSE Interface

Density Interface

MPI MPI

phiGRAPE SSE

knnCUDA Library

GPU

GPU

more high-mass

stars

more low-mass

stars short tidaltimescalelong tidaltimescale

Stellar Evolution Model Comparison

This plot shows the same N=16,000 model (W0 = 3, α=-2.5, family=1) evolved using different stellar evolution models. AMUSE easily allows switching stellar evolution models in the same code. All curves used AMUSE, except for the Starlab comparison. The inset shows the population synthesis results for these models.

W0 = 3α = -2.5

family = 4

W0 = 7α = -2.5

family = 1

W0 = 7α = -1.5

family = 4

W0 = 3 or 7 α = -1.5 or -2.5 family = 1, 2, 3 or 4

764 TAKAHASHI & PORTEGIES ZWART Vol. 535

FIG. 2.ÈComparison between Aa Fokker-Planck models (solid lines) and N-body models (dashed and dotted lines) for the evolution of the total mass.Thickness of the lines stand for largeness of N. The thickest solid line in each panel represents the Fokker-Planck model with the instantaneous escapecondition, which corresponds to the limit of N ] O. The other Fokker-Planck models are calculated with the crossing-time escape condition withlesc \ 2.5for Ðnite N. The dot-dashed lines represent the mass evolution expected when mass loss occurs only through stellar evolution (with no escapers). (a) Initialconditions are N \ 1K, 16K, and 32K, from right to left. Only for the N \ 1K N-body model, ^p/2 deviation from the mean of(W0, a, family) \ (3, 2.5, 1) ;10 N-body runs is shown (two dotted lines). (b) N \ 8K, 16K. (c) N \ 4K, 8K. (d)(W0, a, family) \ (1, 2.5, 1) ; (W0, a, family) \ (7, 2.5, 1) ; (W0, a, family) \(3, 2.5, 4) ; N \ 8K, 16K. (e) N \ 8K, 16K. ( f ) N \ 8K, 16K.(W0, a, family) \ (7, 1.5, 4) ; (W0, a, family) \ (1, 3.5, 3) ;

rate dM/dt may be divided into two parts :

dMdt

\AdMdtB

esc]AdM

dtB

se, (13)

where the Ðrst term in the right-hand side represents themass loss due to escapers from the tidal radius and thesecond term represents the mass loss due to stellar evolu-tion. Since the stellar evolution is the same in the Fokker-Planck and N-body models, it may be useful to separate thestellar evolution mass loss from the total mass loss in orderto further clarify the di†erence between the two models.Thus in Figure 4 we plot the mass-loss rate due to escapers

as well as the total mass-loss rate deÐned as follows :mesc mtot

mtot \ [trh0M

dMdt

, mesc \ [trh0M

AdMdtB

esc. (14)

When there is no stellar evolution, is usually much lessmescthan 1 (Lee & Goodman 1995). This is to be expected, sincein that case mass loss proceeds only on two-body relaxation

timescale. In the present models, since the stellar evolutionmass loss causes the shrink of the tidal radius and thusproduces more escapers, exceeds 1 sometimes.mescFigure 2 as well as Figures 3 and 4 show that the Fokker-Planck and N-body calculations agree fairly well over theentire range of initial conditions we investigated. Appar-ently the single value of is applicable. The relativelesc \ 2.5di†erence of the total mass at each time is generally lessthan 10%. However, for the cases of (W0, a, family) \(3, 2.5, 4) and (7, 1.5, 4), the di†erence is rather large. Thislarge di†erence seems to well correlate with large ([1),mescespecially in the N-body models. Also when the cluster isabout to dissolve (see panels a and b), becomes largermescthan 1 and the Fokker-Planck and N-body models deviate.

In the case of the N-body(W0, a, family) \ (3, 2.5, 4),models tend to lose mass more quickly than the Fokker-Planck models. The mass loss in the N-body models accel-erates rapidly, compared with the Fokker-Planck models,from about 1 Gyr for the 16K model and from about 2 Gyrfor the 8K model, and after some time the rate of mass loss

W0 = 3α = -2.5

family = 1

764 TAKAHASHI & PORTEGIES ZWART Vol. 535

FIG. 2.ÈComparison between Aa Fokker-Planck models (solid lines) and N-body models (dashed and dotted lines) for the evolution of the total mass.Thickness of the lines stand for largeness of N. The thickest solid line in each panel represents the Fokker-Planck model with the instantaneous escapecondition, which corresponds to the limit of N ] O. The other Fokker-Planck models are calculated with the crossing-time escape condition withlesc \ 2.5for Ðnite N. The dot-dashed lines represent the mass evolution expected when mass loss occurs only through stellar evolution (with no escapers). (a) Initialconditions are N \ 1K, 16K, and 32K, from right to left. Only for the N \ 1K N-body model, ^p/2 deviation from the mean of(W0, a, family) \ (3, 2.5, 1) ;10 N-body runs is shown (two dotted lines). (b) N \ 8K, 16K. (c) N \ 4K, 8K. (d)(W0, a, family) \ (1, 2.5, 1) ; (W0, a, family) \ (7, 2.5, 1) ; (W0, a, family) \(3, 2.5, 4) ; N \ 8K, 16K. (e) N \ 8K, 16K. ( f ) N \ 8K, 16K.(W0, a, family) \ (7, 1.5, 4) ; (W0, a, family) \ (1, 3.5, 3) ;

rate dM/dt may be divided into two parts :

dMdt

\AdMdtB

esc]AdM

dtB

se, (13)

where the Ðrst term in the right-hand side represents themass loss due to escapers from the tidal radius and thesecond term represents the mass loss due to stellar evolu-tion. Since the stellar evolution is the same in the Fokker-Planck and N-body models, it may be useful to separate thestellar evolution mass loss from the total mass loss in orderto further clarify the di†erence between the two models.Thus in Figure 4 we plot the mass-loss rate due to escapers

as well as the total mass-loss rate deÐned as follows :mesc mtot

mtot \ [trh0M

dMdt

, mesc \ [trh0M

AdMdtB

esc. (14)

When there is no stellar evolution, is usually much lessmescthan 1 (Lee & Goodman 1995). This is to be expected, sincein that case mass loss proceeds only on two-body relaxation

timescale. In the present models, since the stellar evolutionmass loss causes the shrink of the tidal radius and thusproduces more escapers, exceeds 1 sometimes.mescFigure 2 as well as Figures 3 and 4 show that the Fokker-Planck and N-body calculations agree fairly well over theentire range of initial conditions we investigated. Appar-ently the single value of is applicable. The relativelesc \ 2.5di†erence of the total mass at each time is generally lessthan 10%. However, for the cases of (W0, a, family) \(3, 2.5, 4) and (7, 1.5, 4), the di†erence is rather large. Thislarge di†erence seems to well correlate with large ([1),mescespecially in the N-body models. Also when the cluster isabout to dissolve (see panels a and b), becomes largermescthan 1 and the Fokker-Planck and N-body models deviate.

In the case of the N-body(W0, a, family) \ (3, 2.5, 4),models tend to lose mass more quickly than the Fokker-Planck models. The mass loss in the N-body models accel-erates rapidly, compared with the Fokker-Planck models,from about 1 Gyr for the 16K model and from about 2 Gyrfor the 8K model, and after some time the rate of mass loss

100 2 4 6 8

1

0

0.2

0.4

0.6

0.8

764 TAKAHASHI & PORTEGIES ZWART Vol. 535

FIG. 2.ÈComparison between Aa Fokker-Planck models (solid lines) and N-body models (dashed and dotted lines) for the evolution of the total mass.Thickness of the lines stand for largeness of N. The thickest solid line in each panel represents the Fokker-Planck model with the instantaneous escapecondition, which corresponds to the limit of N ] O. The other Fokker-Planck models are calculated with the crossing-time escape condition withlesc \ 2.5for Ðnite N. The dot-dashed lines represent the mass evolution expected when mass loss occurs only through stellar evolution (with no escapers). (a) Initialconditions are N \ 1K, 16K, and 32K, from right to left. Only for the N \ 1K N-body model, ^p/2 deviation from the mean of(W0, a, family) \ (3, 2.5, 1) ;10 N-body runs is shown (two dotted lines). (b) N \ 8K, 16K. (c) N \ 4K, 8K. (d)(W0, a, family) \ (1, 2.5, 1) ; (W0, a, family) \ (7, 2.5, 1) ; (W0, a, family) \(3, 2.5, 4) ; N \ 8K, 16K. (e) N \ 8K, 16K. ( f ) N \ 8K, 16K.(W0, a, family) \ (7, 1.5, 4) ; (W0, a, family) \ (1, 3.5, 3) ;

rate dM/dt may be divided into two parts :

dMdt

\AdMdtB

esc]AdM

dtB

se, (13)

where the Ðrst term in the right-hand side represents themass loss due to escapers from the tidal radius and thesecond term represents the mass loss due to stellar evolu-tion. Since the stellar evolution is the same in the Fokker-Planck and N-body models, it may be useful to separate thestellar evolution mass loss from the total mass loss in orderto further clarify the di†erence between the two models.Thus in Figure 4 we plot the mass-loss rate due to escapers

as well as the total mass-loss rate deÐned as follows :mesc mtot

mtot \ [trh0M

dMdt

, mesc \ [trh0M

AdMdtB

esc. (14)

When there is no stellar evolution, is usually much lessmescthan 1 (Lee & Goodman 1995). This is to be expected, sincein that case mass loss proceeds only on two-body relaxation

timescale. In the present models, since the stellar evolutionmass loss causes the shrink of the tidal radius and thusproduces more escapers, exceeds 1 sometimes.mescFigure 2 as well as Figures 3 and 4 show that the Fokker-Planck and N-body calculations agree fairly well over theentire range of initial conditions we investigated. Appar-ently the single value of is applicable. The relativelesc \ 2.5di†erence of the total mass at each time is generally lessthan 10%. However, for the cases of (W0, a, family) \(3, 2.5, 4) and (7, 1.5, 4), the di†erence is rather large. Thislarge di†erence seems to well correlate with large ([1),mescespecially in the N-body models. Also when the cluster isabout to dissolve (see panels a and b), becomes largermescthan 1 and the Fokker-Planck and N-body models deviate.

In the case of the N-body(W0, a, family) \ (3, 2.5, 4),models tend to lose mass more quickly than the Fokker-Planck models. The mass loss in the N-body models accel-erates rapidly, compared with the Fokker-Planck models,from about 1 Gyr for the 16K model and from about 2 Gyrfor the 8K model, and after some time the rate of mass loss

100 2 4 6 8

1

0

0.2

0.4

0.6

0.8

Conclusions• Our AMUSE runs are in good agreement with [TPZ] and [CW], apart from small differences due the different stellar evolution models used, validating the use of AMUSE as a research tool.• The modular structure of AMUSE facilitates comparison of physics modules and enables exploration of assumptions and approximations that is difficult or impossible with other simulation codes.• Specifically, AMUSE allows direct comparison of the effect of differing stellar evolution models. The choice of model can change the computed lifetime of a cluster near disruption by up to ~25%.• For the adopted parameters, AMUSE outperforms Starlab's kira by a factor of ~2.

100 2 4 6 8

1

0

0.2

0.4

0.6

0.8

N = 1,000

N = 32,000

100 2 4 6 8

1

0

0.2

0.4

0.6

0.8

100 1 2 3 4 5 6 7 8 9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SSEEFT89 CW

Starlab / SeBa

100 1 2 3 4 5 6 7 8 9

1

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

SSE

CWEFT89