Notes on the Subject:

Astrophysics and Cosmology

(Astrophysics part)

Valent Bosch i Ramon, vbosch@am.ub.es, Universitat de Barcelona

October 30, 2013

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Contents

1 Introduction 6

1.1 Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Environment and Star formation . . . . . . . . . . . . . . . . 7

1.3 Energy generation and transport . . . . . . . . . . . . . . . . 8

1.4 Stellar timescales . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Hydrostatic equilibrium . . . . . . . . . . . . . . . . . . . . . 9

1.6 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . 10

I The Interstellar Medium and Star Formation 10

2 The Interstellar Medium 11

2.1 The galactic structure . . . . . . . . . . . . . . . . . . . . . . 11

2.2 The content of the interstellar medium . . . . . . . . . . . . . 12

3 The Virial Theorem 13

3.1 Consequences of the virial theorem . . . . . . . . . . . . . . . 14

3.1.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.2 Gravitational contraction and energy release . . . . . . 15

3.1.3 Exercise: the stellar mean temperature . . . . . . . . . 15

4 Star Formation 16

4.1 Protostar formation . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 The Jeans criterion . . . . . . . . . . . . . . . . . . . . 16

4.1.2 Homologous collapse . . . . . . . . . . . . . . . . . . . 17

4.1.3 Fragmentation of collapsing clouds . . . . . . . . . . . 18

4.1.4 Exercise: the Eddington luminosity . . . . . . . . . . . 20

II Stellar Structure 20

5 Hydrostatic Equilibrium 20

5.1 Mass conservation equation . . . . . . . . . . . . . . . . . . . 20

5.2 Momentum conservation equation and hydrostatic equilibrium 23

5.2.1 The gravitational field . . . . . . . . . . . . . . . . . . 23

5.2.2 Hydrostatic equilibrium . . . . . . . . . . . . . . . . . 24

5.3 The Lagrangian formulation . . . . . . . . . . . . . . . . . . . 24

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6 Dynamical, Thermal and Nuclear timescales 25

6.1 The dynamical timescale . . . . . . . . . . . . . . . . . . . . . 256.2 The thermal timescale . . . . . . . . . . . . . . . . . . . . . . 266.3 The nuclear timescale . . . . . . . . . . . . . . . . . . . . . . 27

7 Mechanical Structure 28

7.1 The equation of state . . . . . . . . . . . . . . . . . . . . . . . 297.2 Exercise: an estimate of the central pressure and temperature 297.3 Exercise: a lower limit on the central pressure . . . . . . . . . 30

8 The Polytropic Star 30

8.1 The Polytropic gas equation . . . . . . . . . . . . . . . . . . . 308.2 Solutions of the polytropic gas equation . . . . . . . . . . . . 33

8.2.1 Exercise: the Eddingtons radiative equilibrium star . 33

9 The Thermal Structure 34

9.1 The energy conservation equation . . . . . . . . . . . . . . . . 34

10 Energy Transport 37

10.1 Energy transfer: photon diffusion . . . . . . . . . . . . . . . . 3710.1.1 Exercise: the thermal adjustment timescale . . . . . . 3810.1.2 The Rosseland mean opacity . . . . . . . . . . . . . . 39

10.2 Energy transfer: conduction . . . . . . . . . . . . . . . . . . . 4010.3 Energy transfer: convection . . . . . . . . . . . . . . . . . . . 40

10.3.1 The stability Schwarzschild criterion . . . . . . . . . . 4010.3.2 Convection energy luminosity . . . . . . . . . . . . . . 4210.3.3 Causes for the convective instability . . . . . . . . . . 4210.3.4 The mixing-length theory . . . . . . . . . . . . . . . . 43

10.4 The temperature gradient equation . . . . . . . . . . . . . . . 43

11 Energy Generation and Stellar Nucleosynthesis 44

11.1 Nuclear burning . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.2 Stellar nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . 46

11.2.1 The proton-proton chains . . . . . . . . . . . . . . . . 4611.2.2 The CNO cycle . . . . . . . . . . . . . . . . . . . . . . 4611.2.3 Elements heavier than iron . . . . . . . . . . . . . . . 48

12 Stellar Atmospheres 49

12.1 Radiation transfer . . . . . . . . . . . . . . . . . . . . . . . . 4912.2 Deriving T () . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.3 Mechanical structure . . . . . . . . . . . . . . . . . . . . . . . 51

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12.4 Processes shaping spectroscopic lines . . . . . . . . . . . . . . 51

III Stellar Evolution 51

13 Pre-main Sequence Evolution 52

13.1 The Hayashi track . . . . . . . . . . . . . . . . . . . . . . . . 52

13.2 Evolution of a Sun-like star . . . . . . . . . . . . . . . . . . . 53

13.3 Lower and higher mass stars . . . . . . . . . . . . . . . . . . . 53

14 Main Sequence 54

14.1 The zero age main sequence . . . . . . . . . . . . . . . . . . . 54

14.2 Low-mass main sequence . . . . . . . . . . . . . . . . . . . . . 55

14.3 High-mass main sequence . . . . . . . . . . . . . . . . . . . . 57

15 Mass-loss in Low- and High-Mass Stars 57

15.1 Magnetic activity and mass-loss in low-mass stars . . . . . . . 57

15.1.1 Exercise: a model for the stellar wind . . . . . . . . . 59

15.2 Stellar winds in high-mass stars . . . . . . . . . . . . . . . . . 60

15.3 Mass-loss and stellar evolution . . . . . . . . . . . . . . . . . 61

16 Post-Main Sequence Stellar Evolution 61

16.1 The fate of stars with M < 8M . . . . . . . . . . . . . . . . 61

16.1.1 Initial stages . . . . . . . . . . . . . . . . . . . . . . . 61

16.1.2 From the He burning core on . . . . . . . . . . . . . . 62

16.1.3 Beyond the C-O-core below 8 M . . . . . . . . . . . 63

16.2 The fate of massive Stars and Supernovae . . . . . . . . . . . 65

16.2.1 Supernova remnants . . . . . . . . . . . . . . . . . . . 66

17 Calculations of the Stellar Evolution 67

17.1 The system of equations . . . . . . . . . . . . . . . . . . . . . 67

17.2 Simplifying assumptions . . . . . . . . . . . . . . . . . . . . . 68

17.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 69

17.4 The Vogt-Russell theorem . . . . . . . . . . . . . . . . . . . . 69

17.5 Solving the equations . . . . . . . . . . . . . . . . . . . . . . . 69

IV Outcomes of Stellar Evolution 70

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18 The Degenerate Gas 70

18.1 The Fermi energy . . . . . . . . . . . . . . . . . . . . . . . . . 7018.1.1 Exercise: the condition for degeneracy . . . . . . . . . 72

18.2 The equation of state of the degenerate gas . . . . . . . . . . 7318.3 The Chandrasekhar limit . . . . . . . . . . . . . . . . . . . . . 73

19 Compact Objects 74

19.1 White dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . 7419.2 Neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 7519.3 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

20 Binary Systems 79

21 Accretion and Relativistic Outflows 81

21.1 Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8121.2 Relativistic outflows . . . . . . . . . . . . . . . . . . . . . . . 82

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1 Introduction

Stars are very important objects in shaping the Universe. They processthe ambient medium converting it into heavier elements, produce light, andshape their environment in different ways, through their radiation, winds,gravity, and explosive events. Stellar formation, evolution, and end, giveform to our sky to a large extent, and only a proper study of these processeswill allow a proper understanding of the Universe as it is nowadays and wasin the past.

The goal of these notes is to give an overview of the way in which starswork. The focus is put on the structure of stars, and on a description of theirevolution from their formation to the point when they end up their lives byviolent processes and become compact objects. In this section a brief andsketchy description of the contents of these notes is provided. More detailedexplanations will be presented in the following sections.

1.1 Stars

Stars are balls of gas whose structure and evolution are mainly determinedby their own gravity and pressure, plus the thermonuclear processes thattake place in their interiors.

Stars can be studied through the light they produce and reaches us. Thislight is generated in the stellar atmosphere, and brings primarily surfaceinformation thanks to its spectral features. Stellar global properties canbe also derived, like the surface gravity and the rotational velocity. Thestar luminosity and surface temperature also help to characterize the global(inner) stellar properties. For the case of the Sun, finer details can be resolvedgiven its proximity. Finally, time and morphological data are also relevant forthe study of the star properties, like its magnetic activity, stellar seismology,the stellar outermost layers or winds, etc.

The evolution of stars is very slow and cannot be followed for individualcases excepting violent events, but since there are many stars with apparentlysimilar properties but different ages, one can get a quite complete view ofthe stellar evolution. The main differences between stars are mostly relatedto mass and to a lesser extent to chemical composition, but also to rotation,magnetic fields, binarity, etc (not considered here).

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1.2 Environment and Star formation

Our galaxy is a complex structure composed of a central bulge, a bar crossingthe bulge, and a disc around the centre that consists of several spiral arms.The bulge, the bar and the disc contain about 1011 stars. In addition tostars, about a 50% of the disc volume is composed by atomic cold hydrogenplus some helium and heavier elements, another 50% consists of ionizeddiluted and hot hydrogen, and 1% corresponds to cold and dense molecularhydrogen and dust, the latter consisting of grains made of complex moleculescontaining C, Si, etc. The galactic halo, which contains very diluted and hotgas, surrounds the other mentioned galactic components. The galactic massand structure is complemented also by dark matter of yet unknown nature.This dark matter is required to explain the rotation velocity profile of theGalaxy, since it violates Keplers law if a dark component is not accounted.

Stars are born in the cold and dense cores of the molecular clouds thatpopulate the inner regions of the galactic disc, mainly concentrated in thespiral arms and the galactic centre. These stars may be products of starformation in the early Universe and therefore contain little amount of metals(elements heavier than He), the so-called Population II stars. Stars maybe also younger and more metallic (Population I), as our sun and all the(relatively) young stars. An initial Population of metal-less stars has beenhypothesized: the Population III stars. These would have been the first starsformed, with no other elements beside those originated in the Universe initialnucleosynthesis.

The formation of stars requires of a gravitational bound plus an unbal-ance between attractive and centrifugal (in a generic sense) forces. An atomof hydrogen is considered bound to the surrounding material, say forming acloud of mass Mc, mean molecular weight c, temperature Tc, density c,and radius Rc, if approximately the atom kinetic energy is smaller than itspotential energy (see Sect- 4.1.1). This determines a radius, the Jeans radius

RJ

15 k Tc4GcmHc

, (1)

within which the material is detached from the surrounding medium, andcan suffer gravitational instabilities that lead to its gravitational collapse.Initially, this collapse is radiatively efficient and in free fall and is sloweddown later on due to radiation capture. Density inhomogeneities can producefragmentation and generate smaller structures. The collapsing cloud keepscontracting with timescale KH (see Sect. 1.4) until the conditions at its core

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are suitable for nuclear burning, which marks the beginning of the mainsequence.

Roughly, the stellar mass range goes from 0.08M, limit determinedby electron gas degeneration preventing further contraction and the ignitionof efficient nuclear burning, to 100M, beyond which radiation pressureblocks the accretion of material.

1.3 Energy generation and transport

Stars are considered black bodies for simplicity, to which one assigns a fixedradius and effective temperature, from which their luminosity and continuumspectrum can be obtained. Stars keep hot for most of their lives through theburning of hydrogen and its conversion into helium, although sometimesgravitational energy can be important when contraction takes place.

The transport of energy takes often place through diffusion, in whichphotons are absorbed and re-emitted when they interact with bound or freeelectrons of the medium. The mean free path is extremely short, with thephoton following a random walk. The radiation is almost isotropic withinthe star until it reaches the stellar external layers, when the flux outwardsgrows.

Under strong temperature gradients and/or large opacities, the net fluxoutwards fueling the surface emission cannot be sustained only by radia-tion diffusion. Too much radiation momentum and energy is deposited inthe material through matter-radiation interactions. At this point, the lami-nar structure of the stellar layers becomes disrupted by convective motions,which become the dominant energy transfer mechanism. Convection can beimportant in the external layers of for instance low-mass stars, introducingperturbations in the stellar atmosphere, or in the core of high-mass stars,homogenizing the chemical composition at their centres.

1.4 Stellar timescales

The timescale related to the importance of gravity is the free-fall timescaleff , which gives a measure of the readability of gravity to introduce changesin the stellar structure. One can derive this timescale, to zeroth order, forinstance by means of dimensional arguments, making use of the stellar massM and radius R, and the gravitational constant G:

ff (R3/GM)1/2 1.6 103(R/R)3/2(M/M)1/2 s (2)

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where means the solar value. This timescale is also known as the dynamicaltimescale. Under pressure perturbations of a magnitud exceeding that ofgravity, the characteristic timescale will be similar.

Another important timescale corresponds to how long a star may shinethanks to its gravitational energy reservoir. This reservoir can be derived bydimensional arguments as

M2G/R 4 1048(M/M)2(R/R)1 erg (3)

For the Sun, given the solar luminosity L 4 1033 erg s1, one obtainsKH /L 3 107 yr. This timescale is similar to the so-calledthermal timescale th, the time a star could sustain a certain luminosityradiating its internal energy (i.e. cooling down).

For most of the star lifetime, its light is fueled through nuclear processes.The nuclear timescale nuc turns out to be much longer than both ff andKH, and can be estimated if one assumes that 0.1% of the SunMc2 turnsinto energy. This results in nuc 1010 yr, KH. Therefore, the stellarmaterial can quickly adapt through contractions to the slow exhaustion ofnuclear fuel, in a time KH.

1.5 Hydrostatic equilibrium

If stars are to be stable for long periods of time, like for instance the Sunduring the time period Life has existed on Earth, they have to be approxi-mately in equilibrium. This equilibrium is reached by the force exerted bypressure P , generated by a negative pressure gradient with radius, againstgravity within the star.

To know the mechanical structure of the star one needs an equation ofstate linking P with the stellar density and temperature T . The lastquantity is linked also with the production and the transport of energywithin the star. In most cases, the stellar gas can be considered ideal, soP = k T/mH , with a minor contribution coming from radiation pressure,although in high-mass stars radiation may be a dominant source of pressure.Sometimes the gas may have no temperature dependence, as in the interiorof degenerate stars under the Pauli exclusion principle 1 like white dwarfsand neutron stars.

In hydrostatic equilibrium, in stars dominated by thermal pressure, theinternal energy is minus one half of the gravitational energy (gravitational

1No two fermions of a quantum system can occupy the same quantum state, generatingpressure even under zero temperature (see Sect. 18).

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energy is taken negative). This is the result of the so-called virial theorem.Interestingly, under gravitational contraction, the star radiates only one halfof the gravitational energy that is lost, and the other half goes to heat thestar, so the specific heat of a star in hydrostatic equilibrium is negative.

1.6 Stellar evolution

When the nuclear burning has consumed about 10% of the stellar hydrogen,the star moves away from the main sequence, cooling down, contracting andsubsequently heating up, expanding then its envelope and burning heavierelements. The nuclear reactions stop at a certain point depending on thestellar mass, and then can only be stopped by gas degeneracy.

For stars withM < 8M, a mass MMCh will be ejected in the formof a planetary nebula, whereMCh = 1.4M is the Chandrasekhar mass, andthe remaining object will be a (degenerate) white dwarf.

For M > 8M, the core will burn all the way up to forming an ironcore. Beyond this point, photo-disintegration of iron and light elements, astrongly endothermic process, makes the core become unstable and collapse,usually leading to a supernova explosion.

Collapse leads to the formation of a neutron star if the remaining object isnot too massive (M 1.43M). In this case, the collapse is stopped by thePauli principle applied to neutrons that form from protons through electroncapture and with an abundant production of neutrinos. If the remainingobject is too massive to be sustained by neutron gas degeneration, it will keepcontracting until forming a black hole. The formation of neutron stars andblack holes are accompanied, although not always, by a supernova explosion.

The compact remnants of stars can play an important role in the Uni-verse from a physical but also a purely phenomenological point of view (seePart IV).

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Figure 1: Structural components of the Milky Way.

Part I

The Interstellar Medium and Star

Formation

2 The Interstellar Medium

2.1 The galactic structure

The Galaxy is composed of normal matter that emits and absorbs light. Alarge fraction of the Galaxy mass is however in the form of dark matter, ofunclear nature, but required to explain the dynamics of Galaxy rotation. TheGalaxy also contains a supermassive black hole at its centre of 4106M.

Normal matter is mostly in the form of neutral and ionized atomic gas,molecular gas, dust, stars, and the end products of stellar evolution. TheGalaxy structural components are a central bulge populated mainly by old

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Figure 2: Components of the interstellar medium in the Milky Way.

stars (intermediate Population II), a bar of few kpc of half-length, and adisc that presents several spiral arms and hosts younger stars (Population I)than in the bulge, with the youngest ones (extreme Population I) closer tothe galactic plane and centre than the older ones (intermediate PopulationI). Finally, the oldest stars are located in the globular clusters that ploughthrough the galactic halo, a large region of very hot and diluted gas thatsurrounds the inner galactic structures. Figure 1 provides with an overviewof the main structural components of the Galaxy.

2.2 The content of the interstellar medium

The galactic disc contains neutral and ionized hydrogen gas, closer to andfurther up from the galactic plane, respectively. These gas components oc-cupy most of the disc volume. The disc, in particular the spiral arms, hostcompact and dense regions of molecular clouds, dense ionized hydrogen, anddust. The gas and the dust of the galactic disc obscure the light coming fromthe central regions of the Galaxy, reddening it because of the larger opacity

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at shorter wavelengths.

Despite the hostile environmental conditions of the interstellar mediumto the presence of molecules, these can form because of the protection toUV radiation provided by dust. Dust also enhances the molecule formationrate providing a site for atom binding. Given their large densities and lowtemperatures, the clouds of molecular hydrogen are suitable sites for stellarformation. Therefore, star formation activity takes place in the galacticdisc, mainly in the spiral arms, thus providing to these structures a brighteraspect than the neighboring disc regions. The origin of the spiral armsis still unclear, although it might be the manifestation of a density wavepropagating outwards.

For illustrative purposes, in Fig. 2 the components of the interstellarmedium of the Milky Way are presented together with their main properties.

In the following, before discussing the basics of stellar formation, weprovide with a generic explanation of the virial theorem, an important lawof energy balance in objects in dynamical quasi-equilibrium. This will be auseful tool to follow the collapse of molecular clouds into stars, and also theenergy evolution of stars themselves.

3 The Virial Theorem

The virial theorem of classical mechanics is very useful to understand theenergy balance of a gas spherical cloud, because it allows to link the totalenergy E of the cloud and its internal U and gravitational energy . Thistheorem will be useful when studying stars, but we introduce it here becauseit is also relevant to the energetic balance of gas when collapsing into a star.

Let us start from the following equation

4 r3P

m= Gm

r(4)

where m is the mass within a radius r in spherical symmetry, and rewrite itas

m(4 r3 P ) 4 r2 3P r

m= Gm

r(5)

Equation (4) is derived from momentum conservation in the Lagrangianformulation in hydrostatic equilibrium, to be seen in Sect. 5.3, by multiplyingits both sides by 4 r3. This equation is of primary importance regardingthe stellar structure and its validity implies zero acceleration, i.e. hydrostaticequilibrium, at any point within the cloud.

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Integrating overM0 dm, using the equation for mass conservation(

m

r

)t= 4 r2 (6)

discussed in Sect. 5.1, and assuming that the surface pressure is zero, onegets from Eq. (5)

3

M0

P

dm+ = 0 (7)

which is the virial theorem in astrophysics. This equation can be rewrittenintroducing the gas internal energy through the equation of state P/ =( 1)u:

3

M0

P

dm+ = 3

M0

( 1)u dm+ = (8)

3 ( 1) M0

u dm+ = 3 ( 1)U + = 0or

U = 13 ( 1) (9)

where u is the specific internal energy. For the total energy, which is E =U +, we have the relation

E = U + =3 43 ( 1) = (4 3)U (10)

For an ideal mono-atomic gas, = cP /cv = 5/3, U = 12 and E =U , and for a photon gas, = 4/3, U = and E = 0. The virialtheorem has important consequences regarding stability and the effects ofthe gravitational contraction of the stellar material.

3.1 Consequences of the virial theorem

3.1.1 Stability

For the formation of a gas cloud in hydrostatic equilibrium (e.g. a proto-star or a star), the gas is to be gravitationally bound, which implies E < 0.Otherwise, the extra kinetic energy2 will drive the gas to infinity, escapingfrom the gravitational pull. The previous analysis using the virial theoremshows that for the gas to be stable, > 4/3, since otherwise E 0 after thegas should have relaxed and it is clearly not a stable configuration.

2In this case, this kinetic energy is associated to thermal microscopic motion or heat,i.e. internal energy.

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3.1.2 Gravitational contraction and energy release

If the gas cloud (e.g. a proto-star or a star) suffers a contraction witha timescale much longer than the dynamical timescale (e.g. the free-falltimescale -see Sect. 6.1-), it will smoothly move through different states eachfulfilling the virial theorem. A contraction is to change the relation of E,U and because of the modification in and thereby in the other energycomponents:

GM2

R GM

2

R2(R) (11)

for a contraction from R+R R. Here, we have made use of dimensionalanalysis to derive the functional relation between , M and R. This, plusthe relations established by the virial theorem, imply that for > 4/3

E =3 43 ( 1) < 0 (12)

and

U = 13 ( 1) > 0 (13)

The impact of a contraction is a reduction of the total energy even whenthe star heats up (U > 0). The energy must therefore be lost in some wayto allow the contraction to occur, and in general it will be radiated awaywith a luminosity

L = dEdt

= 3 43 ( 1)

d

dt> 0 (14)

Interestingly, the loss of energy implies nevertheless an increase in T .

During the star formation, the main energy source is gravitational energy,which under a low radiation opacity (see Sect. 4.1.3) is quickly radiatedaway allowing for fast gravitational collapse. When stars are in the mainsequence, the energy carried away by black body radiation is balanced bynuclear reactions. It is when the nuclear fuel starts to be scarce that starsmake use again of their gravitational energy reservoir through contraction,at least in their central regions.

3.1.3 Exercise: the stellar mean temperature

The virial theorem can be used to derive a minimum stellar mean tempera-ture. From Eq. (7) and assuming that the star is formed by an ideal gas of

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particles of mass mH,

= M0

Gm

rdm = 3

M0

P

dm = 3

M0

k T

mHdm (15)

one can derive the minimum mean temperature of the star as follows. First,

M0

Gm

rdm = 3

M0

k T

mHdm >

M0

Gm

Rdm =

GM2

2R(16)

Rearranging, one can get that the mean temperature must fulfill

< T >=1

M

M0

T dm >GM mH

6 k R 4 106K (17)

or 2 106 K for a fully ionized hydrogen plasma in eH+ energy equipar-tition ( = 0.5).

4 Star Formation

4.1 Protostar formation

Stars form through the gravitational collapse of interstellar molecular clouds.The collapse of these clouds is affected by other factors beside gravity, likerotation, the gas cooling efficiency, level of ionization and turbulent motion,the magnetic field, etc. In what follows, we will characterize the processof star formation in its early stages, although these factors will be mostlyneglected. For simplicity, we will also assume spherical symmetry and gashomogeneity, and negligible gas velocities and pressure gradients.

4.1.1 The Jeans criterion

Sir James Jeans investigated in 1902 the study of cloud collapse through ananalysis of the stability of the gas. This analysis was based on the virialtheorem, which for a mono-atomic, non-relativistic, adiabatic ( = 5/3) gascan be written as

2U + = 0 (18)

where U is the (internal) kinetic energy and the potential energy of thegas elements within a sphere of gas in equilibrium of a certain radius Rc.If the quantity 2U + > or < 0, then the gas within the sphere is not inequilibrium, and it will tend to change. In particular, for 2U + < 0, the

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gas will contract until reaching a new equilibrium state, if possible, for which2U + = 0.

For a spherical cloud of constant density c and total massMc = (4/3) R3cc

the gravitational potential energy is

= 35

GM2cRc

(19)

and the cloud internal kinetic energy

U =3

2Nc k Tc =

3

2

McmH

k Tc (20)

where Nc is the number of cloud particles. From these equations, the condi-tion for cloud collapse can be derived as

MJ (

5 k TcGmH

)3/2 ( 34c

)1/2(21)

for the minimum mass, and for the minimum radius

RJ (

15 k Tc4GmHc

)1/2(22)

For the case when the external pressure P0 is relevant one can rewrite thedensity as = P0/v

2T (Bonnor-Ebert mass) and obtain the following relation:

MBE cBE v4T

P1/20 G

3/2(23)

where vT =k Tc/mH is the isothermal sound speed. With cBE = 1.18,

one gets the maximum mass an isothermal cloud in pressure equilibrium withthe medium can have.

4.1.2 Homologous collapse

Under the assumptions stated above, the collapse takes place in free fall aslong as heat (the internal kinetic energy) is released in the form of radiationimmediately after it is produced, because of a pressure gradient being keptwell below gravity. Free fall means

d2r

dt2= Gm(r)

r2(24)

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clearly out of hydrostatic equilibrium.For a sphere of initial radius Rc and density c, Eq. (24) can be rewritten

asdr

dt

d2r

dt2=

(4

3GcR

3c

)1

r2dr

dt(25)

which solution for the velocity is

dr

dt=

[8

3GcR

2c

(Rcr 1

)]1/2(26)

and after some development one can derive an implicit formula for r(t) andthereby the free fall time

tff = (3

32

1

Gc

)1/2(27)

This formula already shows few important things. In principle the collapsetakes place everywhere at the same pace in an homogeneous cloud fulfillingthe Jeans condition, because the contraction rate does not depend on the

initial radius. It is called homologous collapse. However, having 1/2c dividing

already shows that the collapse will take place faster in regions of largerdensity (e.g. the cloud core(s)).

4.1.3 Fragmentation of collapsing clouds

As the cloud collapses gravitationally under isothermal collapse, in whichheat is radiated away and T is kept to some equilibrium constant value, MJand tff decrease with increasing c. This implies that, once a region of gashas started to contract, smaller regions inside can contract at faster ratesunder some perturbation. This process may seem unstoppable and veryefficient producing stars, but this is not the case. At some point, however,the collapse becomes less radiative because the heat produced starts to gettrapped within the collapsing gas through absorption processes.

Under the cloud conditions, collisions are rare and magnetic pressure isnot yet relevant, so the collapse gets blocked through radiation absorption:i.e. the cloud becomes optically thick to its own radiation. At this point,the radiation heats the medium and the pressure supports the cloud, whichbecomes almost adiabatic (little heat is lost). For a purely adiabatic gas inwhich no heat is released, one can write T 1 and MJ (34)/2. TheJeans mass then grows for = 5/3, so the fast contraction should eventuallystop. Note that the transition between a radiative and an adiabatic gas isexpected to be smooth.

18

One can find the minimum Jeans mass making the cloud luminosity (theescaping radiation) taken as a black body equal to the gravitational energyto be evacuated per second. If smaller, the contraction gets almost blocked.If the cloud is surrounded by more radiating material, the escaping radia-tion efficiency is diminished with respect to the black-body one, so one canintroduce the parameter e: the fraction of the luminosity that is allowed toescape. This yields a radiated luminosity:

Lrad = 4 R2Je T

4 (28)

The free-fall luminosity can be derived from the available internal energywithin the jeans radius: Lff U/tff , where U 310GM2J/RJ = /2,i.e.:

Lff G3/2(MJ/RJ)5/2 (29)Making Lrad = Lff , one can see that collapse is blocked by inefficient coolingat

MJ min 0.03(

T 1/4

e1/29/4

)M (30)

This simple estimate already gives values of 0.1 1M, although morecomplex modeling predicts a minimum mass of 0.01M. This shows thatfragmentation stops at masses of the order of those of stellar or sub-stellarobjects.

It is also worth noting, despite the simplifying assumptions above, thatthe inclusion of the magnetic field may be needed to slow down the initialstages of the collapse of clouds. Otherwise, dense cores would become stars insuch a short timescales that no forming stars would be visible in the Galaxy,which is not the case. Including the magnetic field, the minimum mass fora cloud to start to collapse is

MB 70M(

B

10G

)(R

1 pc

)2(31)

The magnetic field dynamical effect is mediated through its interactionwith a relatively small fraction of ionized gas. Ions get coupled then to theneutrals through collisions. Given the particular geometry of the magneticfield due to B = 0 (B consists of close loops of field lines), strong asym-metries in the ions and neutrals arise (e.g. ambipolar diffusion) and affectthe B-field geometry.

Other processes also influence the dynamical importance of B, like mag-netic reconnection, which occurs when different polarity lines get too close

19

and there are not enough charges to sustain the associated currents in idealplasma conditions. Cloud turbulent motions and rotation are also influencedby, and influence, B.

4.1.4 Exercise: the Eddington luminosity

The Eddington Luminosity is an important concept in astrophysics. It entersin different fields, from star formation to stellar winds and accretion physics.Here a brief description is provided.

Let us start assuming that a cloud of gas is sustained by radiation pres-sure, which in spherical symmetry implies that the latter balances gravity:

cLEdd =

c

SdS =

V2dV = 4G

VdV = 4GM (32)

where F is the radiation luminosity (erg s1), the gravitational potential, the opacity coefficient of matter to radiation, and c the speed of light.Equation (33) leads to

LEdd =4GMc

1.3 1038

(M

M

)erg/s (33)

for the case of ionized hydrogen and Thomson (Compton) scattering, forwhich = Th/mp, and Th = 6.65 1025 cm2.

The Eddington luminosity (or Eddington limit) changes depending theopacity coefficient, which can be much larger for instance if bound-freeatomic transitions are included. In particular, massive star winds are drivenby the excitation of certain ressonant atomic transitions in elements such asC and N. These are processes with a cross-section much larger than that ofThomson scattering of electrons.

Part II

Stellar Structure

5 Hydrostatic Equilibrium

5.1 Mass conservation equation

Stars can be approximated as spheres formed by spherical shells. Underspherical symmetry, the properties of the stellar matter are homogeneous

20

within spherical shells of constant radius r and thickness dr. Since the starcan change with time, the matter conditions depend also on t. Anotherapproach to characterize the stellar structure is to take a spherical shell ofconstant mass dm containing inside it matter with constant mass m. Sucha shell may change its radius and width with time because of the changingconditions of the star, e.g. through compression or expansion of the gas, butm and dm would be constant. The first approach is known as Eulerian, inwhich the independent coordinate is r, and Lagrangian the second one, inwhich m is the independent coordinate.

Under the Eulerian description, we write the total mass within a certainradius r at a time t as m(r, t). An infinitesimal mass variation dmr can becomputed varying r by dr at constant t, and then letting matter flow at aconstant r during dt. These yields two components for the mass variation(i.e. at constant t and at constant r), to be added to obtain dmr:

dmr =

(m

r

)tdr +

(m

t

)rdt = 4 r2dr 4 r2vrdt (34)

where the in front of the second RHS term comes from the fact that forpositive vr values matter is leaving the region within r. The first term in theRHS (

m

r

)t= 4 r2 (35)

is the mass conservation equation for a static system. In Figs. 3 and 4, weshow examples of Eulerian and Lagrangian coordinates, respectively.

Given the equality3(

t

(m

r

)t

)r

=

(

r

(m

t

)r

)t

(36)

and Eq. (35), one can write

t= 1

r2(r2 vr)

r(37)

which is the radial component (there is spherical symmetry) of the masscontinuity equation of hydrodynamics:

t= (v) (38)

with v as the velocity vector.

3The symbols r and t in the partial derivatives have been dropped for simplicity.

21

r+ r

r

R(t)

Figure 3: Stellar shell of mass dmr(r, t) and constant dr. The locationcoordinate r does not change with t although R(t) does.

m + m

m

M = R/

Figure 4: Stellar shell of mass of constant dm (so dr(m, t)). The locationcoordinate m nor M do not change with t.

22

5.2 Momentum conservation equation and hydrostatic equi-

librium

Typically stars shine for long periods of time, so they must be stable andtherefore their internal forces are in balance for long. In such a case, themomentum transfer between neighboring shells is to be negligible. Thisbalance is reached thanks to the two relevant forces at work:

a) a gradient-pressure force related to the energy flux within the star(otherwise it wont shine), generated through a shell of thickness dr:

dFP = SPr

dr (39)

with S = 4 r2, andb) the gravitational force dFG operating on that shell, presented in what

follows.

5.2.1 The gravitational field

The gravitational field can be described by the gravitational potential func-tion (r, t), solution of the Poisson equation:

2 = 4G (40)

where 2 is the Laplacian operator and G the gravitational constant. Inspherical symmetry, one can write

1

r2

r

(r2

r

)= 4G (41)

from which one can easily derive the acceleration

g = g r = = r

r = G (m+ C(0))r2

r = Gmr2

r (42)

where m = r0 4r

2(r)dr, and C(0) = 0 since otherwise, g forr 0. The solution to Eq. (41) is

=

r0

Gm

r2dr + constant, with () = 0. (43)

The gravitational force can thus be written as

dFG = g dm = g S dr = Gmr2

S dr (44)

23

5.2.2 Hydrostatic equilibrium

One can relate now the acceleration to the pressure gradient and gravity ina shell at r and thickness dr

d2r

dt2dr = (dFG + dFP )/S = Gm

r2dr P

rdr (45)

i.e. the expression of the momentum conservation in spherical symmetry inhydrodynamics under a gravitational potential :

dv

dt v

t+ v v = P (46)

Equation (45) can be rewritten as

P

r= Gm

r2 d

2r

dt2(47)

In hydrostatic equilibrium, the acceleration term in Eq. (47) must benegligible, and also the pressure-gradient and gravity terms must be roughlyequal, i.e.:

d2r/dt2 Pr Gm

r2(48)

Without strong accelerations and tensions, the stellar interior conditionschange smoothly in a succession of quasi-equilibrium states. This allows fora simple treatment of stellar evolution in which different regions of the starchange quasi-simultaneously with time.

A quasi-equilibrium treatment to stellar evolution holds as long as no dy-namically relevant process becomes dominant over the others. For instance,a sudden increase in the energy injection in the stellar core could, besideother important effects, lead to a quick expansion of the stellar core itself, orits external layers. In addition, too little energy injection, due for instance toefficient cooling, can lead to the contraction of the stellar core or the wholestar.

5.3 The Lagrangian formulation

Instead of r, one can use m as the spatial coordinate. This ensures thatthe coordinate remains constant with time (in absence of matter sources orsinks). Now, the interval of the spatial coordinate is not 0 r R (with Ras the stellar radius, which may actually change with time) but 0 m M ,

24

with M being of course constant. The partial derivatives in Lagrangiancoordinates are (

m

)t=

(r

m

)t

(

r

)t

(49)

(

t

)m=

(

r

)m

(r

t

)m+

(

t

)r

(50)

Note that here (r/t)m, being this derivative attached to a fixed mass ele-ment located at m, is the velocity at which this mass element moves radiallyinside the star.

Applying Eq. (49) to m and using the first RHS term in Eq. (35) one canderive the mass conservation equation in the Lagrangian formulation

1 =

(m

m

)t=

(r

m

)t

(m

r

)t r

m=

1

4 r2(51)

and thus (from Eq. 49)

m=

1

4 r2

r(52)

The momentum conservation equation is therefore:

P

m=

1

4 r2

P

r= Gm

4 r4(53)

It is worth noting that computing partial time derivatives in the Lagrangianformulation, (/t)m, which are the rate of change with time of a quantityat rest with respect to the flow, is much simpler than in the Eulerian formu-lation, (/t)r, because of the convective terms present in the latter (i.e. ingeneral there is a mass flux at a given r, but not, by definition, at a givenm).

6 Dynamical, Thermal and Nuclear timescales

6.1 The dynamical timescale

The equilibrium of a star, which as mentioned should last for a very long timeif we account for its observed stable appearance, is possible because of thebalance between different forces: gravity and the pressure gradient generatedby the energy flow outwards (recall that we have neglected already someelements, like rotation, the magnetic field, etc.). If the star did not presentsuch an energy flow, gravity would take over and the stellar material would

25

fall in at the free-fall speed in an accelerated fashion. The associated time,or free-fall timescale, can be calculated from a modified Eq. (45)

2r

t2= Gm

r2(54)

which can be simplified through r R, m M and 2/t2 = 1/2ff toobtain

ff =

(R2

GM

)1/2= 1.6 103

(M

M

)1/2 ( RR

)3/2s (55)

One can, on the other hand, assume that gravity is negligible. Then, thepressure gradient will lead to accelerated expansion

2r

t2= P

r(56)

which can be approximated through the modifications 2/t2 = 1/2exp and1/P/r = P/R yielding

exp (R

cs

)(57)

where cs = (P/)1/2 is the (averaged) sound speed in the stellar interior.

Given the approximations adopted, this expansion timescale corresponds ofcourse to the one of a homogeneous gas cloud with radius R and sound speedcs.

In hydrostatic equilibrium, tff texp, and both therefore determine thedynamical timescale (d). Recall that a smooth stellar evolution will requirethis evolution, characterized by the energy source timescale (either the ther-mal, tth, or the nuclear timescale, nuc), to be much longer than d. Onlyduring explosive events the stellar evolution has d as its natural timescale.

6.2 The thermal timescale

The thermal or Kelvin-Helmholtz (KH) timescale is the timescale of a radi-ating gas sphere when emission is fed by contraction (gravitational energy).It can be estimated using the virial theorem. Let us start characterizing thegravitational energy

=

M0

Gm

rdm q GM

2

R= 3.8 1048 q

(M

M

)2 ( RR

)1erg (58)

26

where q = 3/5 for a star homogeneous in density, and in general q > 1/2.As shown above, L = dE/dt (although it may be also defined as L =dU/dt), and from the virial theorem dE = d/2 (for = 5/3), where d =(q GM2/R2) dR, so

L = 12qGM2

R

1

R

dR

dt(59)

Defining th/KH = R/(dR/dt), one obtains

th/KH

L 1

2qGM2

RL(60)

For the Sun, q = 3/2, which yields

th/KH 2 107(M

M

)2 ( RR

)1 ( LL

)1yr (61)

Therefore, gravity may feed the Sun activity for twenty million years, whichis too short to sustain the Sun during its estimated lifetime.

Around the 1900s, gravitational contraction was postulated as a candi-date for the solar energy supply, but the geological and fossil evidence onEarth showed that a much longer and energetic process had to be feeding theSun light. Gravity however can indeed play an important role in producinglight or providing structural stability for certain objects, like for instancein early-stage star formation or in brown dwarfs (none or too little nuclearactivity), or after the main sequence (when the nuclear fuel exhausts).

6.3 The nuclear timescale

The nuclear timescale can be estimated simply dividing the energy availablefor nuclear reactions (Enuc) by the star luminosity: nuc = Enuc/L. Thestar goes of course through different stages of evolution and both Enuc andL change with time, but still one can focus on the hydrogen burning phase,being the longest, as the most characteristic one.

To estimate Enuc though, it is necessary to account for the fact that onlythe innermost stellar hydrogen is converted into helium, typically about 10%:

Enuc,H 0.1XM QH (62)where X is the initial hydrogen mass fraction, and QH = 6.3 1018 erg g1is the energy released by 1 g of H when turning into He. Therefore

nuc,H 9 109 XX

M

M

(L

L

)1yr (63)

27

and adopting the main sequence star mass-luminosity relation (L 4 1033 (M/M)

3.5 erg/s) and fixing X = X

nuc,H = 9 109(M

M

)2.5yr (64)

Equation (64) clearly shows that the most massive stars, say of M > 10M,will last only for 106 yr, whereas stars with M < 0.8M live for periodslonger than the age of the Universe.

Equation (64) also shows that in general nuc,H th ff . This impliesthat the evolution of the star, strongly related to its energy supply, will besmooth, and the overall stellar structure will quickly adapt to changes on thelatter. These timescale-relations will allow simplifications on the structureequations and their solution.

7 Mechanical Structure

The gravitational field,

=

20

Gm

r2dr + constant, with () = 0 (65)

the mass conservation equation

m

r= 4 r2 (66)

and the momentum conservation equation under hydrostatic equilibrium

P

r= Gm

r2(67)

provide with a mechanical description of the stellar interior. Note that Eq. 65is contained in Eq. 67, so they are actually two equations with three un-knowns: m, and P . Therefore, to close the system, an additional equationis needed, the equation of state of the gas, which will leave two partial dif-ferential equations with two unknowns. The central stellar point, r = 0, issingular in this representation, but it can be solved using Taylor series forthe quantities when r 0.

28

7.1 The equation of state

A family of equations of state which are common in stellar astrophysics isthat of the barytropic models, in which = (P ), a special case of which isthe polytropic model

P = K (68)

suitable for certain stars, like white dwarfs. Other special types of starscan be described through Eq. (68), solving the Lane-Emden equations (seeSect. 8).

Unfortunately, in most cases the temperature cannot be eliminated throughapproximations. Then (P, T ) has T as an explicit variable, and additionalequations are required, because the thermal and the mechanical structure,and the generation and transport of energy, are intimately coupled. A simpleequation of state with explicit T-dependence is that of an ideal gas:

P =k

mH T (69)

with being the mean molecular weight and mH the hydrogen mass. Notethat the ideal gas approximation is justified as long as particles can be con-sidered as point-like and their interactions are elastic.

7.2 Exercise: an estimate of the central pressure and tem-

perature

Before deriving the equation of the thermal structure and characterizing thestellar composition (both required to fully describe the star), lets estimateas an exercise the central pressure (Pc) and temperature (Tc) in the star.

For the pressure, one can simplify Eq. (67) through mM/2, r R/2,and = 3M/4 R3:

P

r Pc

R 3

2

GM2

R5(70)

and thus

Pc 32

GM2

R4= 5.4 1015

(M

M

)2 ( RR

)4dyn cm2 (71)

and for the temperature

Tc mHk

Pc

=2mHk

GM

R 2.3 107

(M

M

)(R

R

)1K (72)

Note that more accurate calculations predict for the Sun Pc, = 2.7 1017 dyn cm2 and Tc, = 1.5 107 K.

29

7.3 Exercise: a lower limit on the central pressure

A lower limit on the central pressure of the star can be also derived, usingnow the conservation momentum equation in Lagrangian coordinates:

P

m= Gm

4 r4(73)

and integrating it overM0 dm M

0

P

mdm = Ps Pc =

M0 Gm4 r4

dm (74)

where Ps 0 dyn cm2 is the surface pressure. Then one can use that1/r4 > 1/R4, and thus Eq. (74) yields

Pc >GM2

8 R4= 4.4 1014

(M

M

)2 ( RR

)4dyn cm2 (75)

about 10 times smaller than the value derived in Sect. 7.2.

8 The Polytropic Star

8.1 The Polytropic gas equation

The assumption of a relation between density and pressure as given inEq. (68) allows the construction of a simple model for a gas cloud in hydro-static equilibrium with few analytical solutions: polytropic indexes n = 0, 1and 5, where Pn = K

n and being = (n + 1)/n the adiabatic index, and

a simple numerical solution in the remaining cases. As we shall see, despiteits simplicity the polytropic gas may be a fair approximation in a number ofinteresting cases.

To obtain the equation describing the structure of a polytropic star, onecan start by rewriting the equation for hydrostatic equilibrium (Eq. 67),deriving both sides in radius, and using the mass conservation equation(Eq. 35), one gets

d

dr

(r2

dP

dr

)= Gdm

dr= G(4r2) (76)

Note that, since in hydrostatic equilibrium dP/dr = d/dr, one canrewrite Eq. (76) as

1

r2d

dr

(r2d

dr

)= 4G (77)

30

Figure 5: Solutions for Dn() (n()) for the cases n = 0 (pink), 1 (blue), 2(green), 3 (yellow), 4 (red), and 5 (brown).

which is the Poissons equation in the spherically symmetric case (recallEq. 41).

Now, using Pn = K(n+1)/nn in Eq. (76), one can write:(

n+ 1

n

)K

r2d

dr

[r2(1n)/nc

dcdr

]= 4Gc (78)

which can be simplified through n(r) = cDn(r)n:[

(n+ 1)

(K

(1n)/nc

4G

)]1

r2d

dr

[r2dDndr

]= Dnn (79)

where Dn is a dimensionless function of values between 0 and 1 with c beingthus the maximum density.

31

Figure 6: Comparison of the polytropic solution (n = 3) with an accuratemodel for the density, inner mass, pressure, and temperature in the Suninterior.

Since [(n+ 1)(K(1n)/nc /4G)] = 2n has distance-squared units, one

can rewrite Eq. (79) using a dimensionless space coordinate = r/n as

1

2d

d

[2dDnd

]= Dnn (80)

Note that n is a sort of characteristic distance for the functionDn(r) beyondwhich Dn decreases strongly.

The boundary conditions of the dimensionless Eq. (80) are the follow-ing. First, one can see that Dn(1) = 0, where 1 = R/n, i.e. at thestellar surface density goes to zero. Second, it can be shown from dP/dr =

32

Gm/r2 < 0 and P (n+1)/n that d/dr must be negative, which addedto the fact that (r < R) > 0 but finite, implies that (0) is maximal (andequal to c). Thus, the second boundary condition is dDn/d|=0 = 0.

8.2 Solutions of the polytropic gas equation

For illustrative purposes, we present in Fig. 5 the graphical representationsof the solutions Dn() of the Lane-Emden dimensionless equation for thecases n = 0, 1, 2, 3, 4, and 5. These n-values correspond to = , 2, 3/2,4/3, 5/4, and 6/5, respectively. Note that the physical solutions are confinedto the first region with 0, Dn > 0 and dDn/d < 0.

An ideal mono-atomic adiabatic gas would correspond to the case n = 1.5( = 5/3; not shown in the figure). This solution provides a good descriptionof the interior of a white dwarf, which behaves as a non-relativistic polytropicgas (see Sect. 18).

For n = 3 ( = 4/3), the solution would correspond to a star in radiativeequilibrium, i.e. in which the gas pressure is proportional to the radiationpressure (see below). This solution also applies to relativistic degeneratestars (not discussed in these notes). In fact, in Fig. 6 we show a comparisonof a polytropic model (n = 3) with an accurate calculation of the Sun interior,and they are surprisingly similar.

For n > 5, the solution yields a diverging stellar mass so n = 5 is anupper-limit, which actually corresponds to the case with 1 = . Theparticular case n = 0 has = c, P0 = , and 0 = . Seeminglyunphysical, taking n = 0 in fact gives the description of the structure of aconstant (thereby 0 = ) incompressible (so P0 = ) sphere of infiniteextension.

8.2.1 Exercise: the Eddingtons radiative equilibrium star

The Eddington standard model associated to a star in radiation equilibriumhas the same relation between pressure and density as that shown by apolytropic gas with n = 3. This can be shown making the gas and theradiation pressure proportional:

Pgas Prad (81)

as the star is in radiation equilibrium, and using the ideal gas approximation

Pgas = kT (82)

33

So, from Eq. (82) and

Prad =4

3cT 4 (83)

one getsP 4/3 (84)

i.e. a polytropic index n = 3.

9 The Thermal Structure

The hydrostatic equilibrium assumption gives the possibility to characterizethe density and pressure within the star (only if P (, )), but the tempera-ture cannot be determined from the mechanical equations alone if one goesbeyond the polytropic approximation; their determination requires to modelthe energy generation, linked to nuclear processes and sometimes contrac-tion, and transfer, linked to the temperature gradient and the specific energytransport mode. In short, one needs to model the thermal structure.

9.1 The energy conservation equation

As we know, stars produce large amounts of energy for very long time inter-vals while keeping their main properties more or less constant. The energyconservation law thus implies that the star must balance the energy loss withenergy generation.

For a star with spherical symmetry (i.e. radial energy transport) andnegligible time variations, one considers for the energy injection rate permass unit (erg g1 s1 in cgs)

= [(r), T (r), Xi(r)] [(m), T (m), Xi(m)] (85)

which, through the energy conservation law, globally implies a total energy-generation luminosity

L =

R0

4r2(r)dr =

M0

(m) dm (86)

The energetic balance must however operate locally as well, so at eachradius the in- and out-going luminosities must equal each other in absenceof sources/sinks of energy, or must be balanced by the generated luminositydLr at r:

dLr = (Lr + dLr) Lr = 4 r2 dr (87)

34

or put in different terms

Lrr

= 4 r2 Lrm

= (88)

with the latter expressed in Lagrangian coordinates. This is the energeticbalance equation. Partial derivatives are used to take into account that inreality there might be a time dependence. In Fig. 7 an illustration of theincoming and outgoing energy rates in a shell at radius r and width dr isshown.

r+ r

Lr

r

Lr+ r

Figure 7: Stellar shell of width dr at radius r through which energy, withincoming and outgoing rates Lr and Lr+dr, respectively, propagates.

In fact, should also account for losses, because nuclear reactions (andother processes; see below) produce neutrinos in addition to photons andmassive particles. Since the interior of "normal" stars is transparent to neu-trinos, they escape carrying energy away. In the context of nuclear reactions,the energy loss rate must be included as :

Lrr

= 4 r2( ) Lrm

= (89) has the same dependencies as .

When the star, either locally or globally, is not able to sustain the energybalance through nuclear reactions, gravity becomes dominant and contrac-tion begins. According to the virial theorem, half of the released gravitational

35

energy goes away as radiation, and the rest goes to the internal energy ofthe star.

We note that in some phases of stellar evolution, contraction can heat thestellar core enough to trigger nuclear reactions in the layers just surroundingthe core. In this way the hydrogen-burning period is extended at the expenseof the contracting stellar core.

If one drops the assumption of constancy of the mechanical structure,dLr may be positive even if no nuclear reactions took place. This quantityalso depends on stellar cooling, heating, and expansion or contraction.

Assuming a slow modification of the shell thermodynamical conditions,one can use the first law of thermodynamics

q

t T s

t=

u

t+ P

v

t=

u

t P2

t(90)

where magnitudes are per mass unit. The energy conservation law showsthat heat will increase or decrease as energy is generated or lost in theshell through cooling or heating and expansion or contraction. This canbe rewritten as

q

t= ( ) Lr+dr Lr

4 r2 dr= ( ) 1

4 r2

L

r(91)

or equivalently

L

r= 4 r2

[( ) T s

t

]= 4 r2

[( ) u

t+

P

2

t

](92)

L

m= ( ) u

t+

P

2

t(93)

where the last two equations describe the energetic balance in Eulerian andLagrangian coordinates in a more general case. It is worth noting that Lrdoes not necessarily grow with r, or dLr > 0. If a star suffered strongexpansion, Lr may decrease with r, i.e. Lr/r < 0 because energy isinvested in exerting work. In some cases, for instance under very strongneutrino losses, one may find that Lr < 0.

The energetic balance equation (Eq. 89) does not determine the luminos-ity of a star, but reflects the luminosity gradient dependency on the differentenergy gains and losses. To characterize the energy flux and thereby Lr, oneneeds to specify the energy transport mechanism, which to its turn dependson the temperature gradient within the star.

36

10 Energy Transport

The temperature gradient depends on how energy is transported within thestar. In the conditions of stellar interiors, the relevant processes are radia-tion in an optically thick regime, conduction via particle collisions (electronsmostly), and convection when the temperature gradient is too steep. Theefficiency of all three processes is proportional to T .

10.1 Energy transfer: photon diffusion

The opacity coefficient to photons allows the derivation of the characteristicmean free path = 1/ (within the stellar interior), where [(r), T (r), Xi(r)](neglecting for the moment the frequency dependence) with units cm2 g1.For the Sun, 0.4 cm2 g1, and thus one obtains for the average Sundensity 2 cm. Since R and the medium in which photons arescattered is basically isotropic, the photons propagate through the stellarmaterial in a diffusive fashion, and the straight distance covered in this waycan be written as l N1/2 , where N is the number of collisions. For theSun radius and the simplified parameters above, one derives an N 1021and a Sun-crossing time of thousands of years, although the actual time is infact of about a million years, because the diffusion time in the inner regionsis much longer.

As mentioned, the diffusive process is almost isotropic, but some level ofanisotropy/inhomogeneity is to be present, in particular in the temperaturespatial distribution, if energy is to be transported outwards. The radialvariation of the temperature can be estimated as T/r (Tc Ts)/R.The resulting (radial) flux can be estimated assuming black body radiationand taking the temperature at a radius r and at r +

4. In this way, oneobtains

Frad F+rad(r r) Frad(r + r) (94)

[(T + T )4 T 4] T 4 TT

where T/T (T/Tr) 1010.For the average temperature of the Sun, the flux that can be extracted

by radiation due to the temperature gradient is 1011 erg cm2 s1, whichis of the order of the flux at the Sun surface, taken as an approximate

4Photons cannot propagate further than , so there will be no contribution to theflux from further distances.

37

representative value. Therefore, the radiation transfer seems to be efficientto transfer the required energy from the Sun core to the surface.

Let us elaborate a bit further on the description of the diffusion approx-imation for the radiation transfer in stellar interiors. Under this approxima-tion, one can find an expression for the temperature gradient when energytransport is radiative as follows. The diffusion energy flux is

Frad = Du (95)where D = c/3 is the diffusion coefficient and u will be taken as the blackbody energy density. This yields for the flux outwards at a given radius

Frad = kradT with krad = 4 a c3

T 3

a = 4/c (96)

and adopting Frad = Lr/4 r2 one can finally derive the temperature gradient

under radiation transport in spherical symmetry:

T

r= 3

4 a c

T 3Lr

4 r2(97)

orT

m= 3

642 a c

r4LrT 3

(98)

This completes the thermal structure description for stellar material inapproximate radiative equilibrium. Note that the diffusion approximationfails when approaching the surface (and the stellar atmosphere; see Sect. 12),because becomes of the order of the typical region size and then matterand radiation cannot be in equilibrium.

10.1.1 Exercise: the thermal adjustment timescale

Rearranging Eq. 98, and using Eq. 93 without the energy injection/loss andadiabatic terms (valid for most of a static star excepting the nucleus), one canderive a relation between the energy diffusion through radiation transportand the internal energy evolution:

m

(

T

m

)= cv

T

twith =

642a c

3

r4T 3

(99)

Such an equation allows one to estimate the time required by the star tochange its temperature significantly through radiation diffusion, or adjust-ment timescale (adj). This can be done adopting averaged values for thequantities:

m

(

T

m

)=

T

M2(100)

38

cvT

t cv T

adj(101)

and

Lr = Tm

TM M L

T(102)

so

adj cvT ML

uML

UL

(103)

which is of the order of th as derived above. This means that dimensionalanalysis works, because a substantial change in T implies an even strongerchange in L, and thus adj th. Note that adj is also the time needed bya sudden fluctuation in T to propagate as a fluctuation in radiation throughthe star.

10.1.2 The Rosseland mean opacity

At this point, it is worth noting that the opacity coefficient is actually afunction of frequency, , but often the relevant quantities are integratedover photons of any frequency. In particular, Frad allows the derivation ofa frequency-weighted and averaged opacity coefficient, the Rosseland meanopacity. This is done when comparing the specific flux with its bolometricform:

Frad =

0Frad, d =

0Du/r d =

0 c3

4

c

dBdT

T

rd

(104)

= 43

0

1

dBdT

d Tr

= kradTr

with

krad =4

3

0

1

dBdT

d =4 a c

3

T 3

(105)

where1

=

0 (1/)(dB/dT ) d

0 (dB/dT ) d(106)

In this way, the mean opacity coefficient is only a function of temperature,which gives a quite good approximation when matter and radiation are inthermodynamical equilibrium, in which most of the photons have frequenciesassociated to the black body temperature. For non-zero metallicity, the meanopacity coefficient is to be calculated for the different elements and summedup weighting with the relative mass fraction.

39

10.2 Energy transfer: conduction

Conduction is a mean of energy transport like radiation diffusion but with gasparticles as carrying the energy. The longer the mean free path, the larger isthe difference in energy between the medium particles and those coming fromhotter regions, making the energy transport more effective. The transport ofenergy through conduction is strongly suppressed in most of cases becauseof the short mean free path of the involved particles, part 106 cm for anopacity coefficient 6 103 cm2 g1 cond 6 105 cm2 g1.

In some exceptional situations the particle mean free path may be longerthan that of photons, as happens in the degenerate electron gas of a whitedwarf, in which electrons suffer collisions very seldom to keep their momen-tum constant in a very populated momentum space.

Being a diffusive process in a plasma in thermodynamical equilibrium,conduction can be treated together with radiation transport using an opacitycoefficient of the form

1

=

1

rad+

1

cond(107)

in which the opacity will be dominated by min(rad, cond).

10.3 Energy transfer: convection

10.3.1 The stability Schwarzschild criterion

A very large temperature gradient generate convective instabilities that formconvective cells or bubbles. These bubbles carry the energy instead of radia-tion from hotter to cooler regions. One can estimate the critical temperaturegradient at which convection starts. Let us begin by assuming that a cellof material starts its way up to larger r. If the bubble rises up fast enough,it wont exchange heat and the process will be adiabatic. In addition, thespeed of the bubble will be much slower than the sound speed (which recallis of the order of the free-fall velocity), so pressure in the bubble will keepbalanced by the external pressure.

If the bubble density becomes progressively smaller than the density inthe surroundings, an Archimedes force will push the former further up. Oth-erwise, the bubble will eventually sink back. Therefore, the condition forstability can be written as

r

(T

r

)b

(109)

where the RHS term can expressed as

(T

r

)b=

(T

r

)ad

=

(1 1

)T

P

P

r(110)

with the stability condition becoming

T

r>

(1 1

)T

P

P

r(111)

It is important to keep in mind that these derivatives are negative, becauseT < 0 with growing r. Therefore, in absolute values one has

Tr S) (126)

anddPraddz

=Fradc

(127)

where < I > is the solid-angle -averaged intensity, Frad the radiation fluxtowards the observer, and Prad the radiation pressure.

In equilibrium, assuming that the stellar radiation is Planckian with tem-perature Teff , Frad = T

4eff and constant with z, which implies from Eq. (126):

< I >= S, and thus from Eq. (127)

Prad =Frad z

c+ C (128)

The integration constant C from Eq. (128) can be derived using the Edding-ton approximation, which is based on assuming that I can be taken as twohalf-hemisphere isotropic components, Iout in the z

+-direction, and Iin inthe z-direction. This approximation renders

< I >=1

2(Iout + Iin) (129)

Frad = (Iout Iin) (130)

50

Prad =2

3c(Iout + Iin) (131)

Since at z = 0 Iin = 0, one can derive Prad(0) =23cFrad, so finally Prad =

T 4eff

c (z +23). The derivation can be completed assuming local thermody-

namical equilibrium, i.e. Prad =43c T

4, which yields

T 4 =3

4T 4eff(z +

2

3) (132)

12.3 Mechanical structure

The distribution of density and pressure can be found adopting an equation ofstate for the atmosphere gas (P (, T )) and assuming hydrostatic equilibrium:

dP

dr=Gm(r)

r2 (133)

where r is the stellar radius and m(r) is the stellar mass within r, althoughthis equation can be simplified very accurately for the stellar atmosphere as(r z)

dP

dz=GMR2

(134)

where gs =GMR2

is the surface gravity. This equation is difficult to solveunless one makes a number of approximations, e.g. a relatively simple equa-tion of state, adopting a constant Rosseland mean opacity, an isothermalatmosphere, etc.

12.4 Processes shaping spectroscopic lines

Lines have a natural broadening due to the Heisenberg principle, but ther-mal or larger scale motion (e.g. turbulence, rotation) can have also impactthrough Doppler boosting. Collisional or pressure broadening can also takeplace, in which nearby ions affect the orbitals of atoms through their electricfields, either by collision or near influence.

51

Part III

Stellar Evolution

13 Pre-main Sequence Evolution

When the collapsed gas has reached a pressure gradient strong enough tobalance gravity, the protostar evolution becomes characterized not by thefree-fall timescale but by the Kelvin-Helmholtz one, i.e. the time neededby the gas in hydrostatic equilibrium to re-adjust itself as radiation coolingproceeds. For a 1 M star, this phase lasts about 20 million years.

13.1 The Hayashi track

As gas contracts, and due to inefficient cooling (see above), the temperaturerises until the nuclear reactions can take place in the center of the protostar.From this point on, the protostar follows a specific path in the H-R diagram,the Hayashi track (see Fig. 10).

Figure 10: The Hayashi track in the Hertzsprung-Russell diagram.

In the late stages of gravitational collapse, the H opacity in the outer

52

stellar layers increases so much that convection sets in. The convection zonecan reach very deep into the star, even down to the center, and its impact onthe stellar structure is manifested as an almost vertically downwards pathin the H-R diagram, known as the Hayashi track.

To the right of the Hayashi track, neither radiation nor convection in thestellar interior are adequate to sustain hydrostatic equilibrium, and thereforestars are not viable there.

13.2 Evolution of a Sun-like star

At the beginning of the Hayashi track and for about one million years, con-vection is the main energy transport mechanism in stars of 1 M becauseof the high H opacity. During this period, deuterium burning occurs, butgiven the small 21H abundances these reactions only slightly slow down thecollapse.

With the increase of temperature along the Hayashi track, the stellarcore becomes radiative. The growth of such a radiative layer increases theenergy transport efficiency, and the stellar luminosity increases at the end ofthe track together with the effective temperature. The star is still shrinking,and gravitational collapse is still a source of energy, but at this stage the coretemperature is high enough for the first two steps of the PP I chain to operate.The CNO mechanism can also work, which causes an increase in luminosityand temperature. Given the strong temperature dependence of CNO, someconvection is triggered again in the core. At this point, the nuclear energyproduction is so large that the core is forced to expand, reducing slightlythe total luminosity and effective temperature when the star approaches themain sequence because energy goes to make work. When 126 C gets too scarceand CNO stops, the stellar core has already reached a stable nuclear burningstate and the PP I chain works in full.

Interestingly, the Kelvin-Helmholtz timescale, being a quite crude esti-mate, is surprisingly quite similar to the time needed for stars to go throughthis pre-main sequence step.

13.3 Lower and higher mass stars

For lower/higher mass stars, the pre-main sequence phase is much longer/shorter.

For very low-mass stars, the CNO mechanism cannot operate, whichmeans that no increase of luminosity and temperature occurs at the end ofthe Hayashi track. It also implies that these stars remain fully convectiveuntil they enter into the main sequence. For objects between 0.013 and

53

0.072 M, nuclear reactions are not enough to reach the main-sequence,although deuterium (> 0.013M) and Lithium (> 0.06M) burning cantake place. This type of objects is known as brown dwarf, and were observedfor the first time in 1995.

Massive stars have the CNO cycle as the dominant nuclear process, andthe core remains convective even after the main sequence starts.

14 Main Sequence

14.1 The zero age main sequence

Stars of homogeneous chemical composition that have just started hydrogenburning in a stable form are in the zero age main sequence, or ZAMS. Centraltemperatures 2 107 K determine the border between the dominant PP-chains and the CNO cycle. The latter mechanism, as shown above, is sosensitive to T that energy generation is very steep in the stellar core, leadingto a very strong T -gradient and thereby convection. Under the PP-chains,the much smoother T -dependence allows otherwise for radiative cores. Thisis a characteristic that low- (cool) and high-mass (hot) stars keep duringthe main sequence. In addition, large ionization zones in the envelope oflow-mass stars lead to a small adiabatic coefficient (large cP) and thus toconvection, whereas the shallow ionization shell in high-mass stars allows forradiative energy transport in most of the envelope. In the lowest-mass stars,convection in the envelope gets so deep into the star that its whole interiorbecomes convective.

The star final approach to the zero age main sequence is marked by adrop in luminosity, which happens when the present 12C starts to burn bythe CNO mechanism (as described in Sect. 13.2). The strong T -dependenceof the process leads to a convective core. Since convection requires energyto take place, the energy available for radiation is reduced. Since the stellarcontraction effectively stops when convection sets in, there is the mentioneddrop in luminosity. The scarce amount of 12C available implies a rathershort-lived phase, and afterwards H-burning proceeds as the primary energysource in low-mass stars, where the core becomes radiative again, Otherwise,in high-mass stars the core keeps convective, as the CNO cycle can proceedfurther.

After ZAMS, stars become basically static, compensating their energylosses through nuclear burning and evolving only in composition.

54

Figure 11: Hertzsprung-Russell diagram formed by stars of different masseslocated at the main sequence as well as at further stages of their evolution.

Figure 12: Path of a Sun-like star through the Hertzsprung-Russell diagramfor different stages of its evolution (pre-, post-, and main sequence).

14.2 Low-mass main sequence

As hydrogen turns into helium increases, and according to the ideal gaslaw, unless and/or T in the core also increase simultaneously, there will

55

Figure 13: Different mass star paths through the Hertzsprung-Russell dia-gram (from Zeilek & Smith 1987).

be a reduction in gas pressure. This leads to a compression of the core, anda subsequent enhancement of and T by the virial theorem. The growthof T means an increase in the nuclear reaction rate, and the luminosity ofthe star increases despite the reduction in the hydrogen mass fraction in thecore.

The core hydrogen depletion does not imply the end of the nuclear ac-tivity, because the core temperature has grown to the point that a hydrogenshell surrounding a predominantly (inert) helium core begins to burn. At thispoint the luminosity generated exceeds the one produced before in the core,but part of the energy goes to exert work on the external layers. Thus, thelatter will expand, and the effective temperature will decrease. A Sun-likestar in the H-R (Hertzsprung-Russell; see Fig. 11) diagram moves red-/right-wards with increasing luminosity at this stage, which is called the sub-giantbranch. This can be seen in Figs. 12 and 17.

56

The non-burning helium core is isothermal and in hydrostatic equilib-rium, which means an inwards increase of density. As the hydrogen burningshell continues burning, the isothermal helium core grows while the star red-dens because of the lower Teff . There is a limiting mass at which the isother-mal core cannot sustain any more a quasi-static equilibrium, the Schnberg-Chandrasekhar mass, from which on the core collapses on a Kelvin-Helmholtztimescale. For stars below 1.2M, this defines the end of the main sequence,moment at which a 1M star is 10 Gyr old, has spent 12% of its hydro-gen, doubled its radius, and tripled its luminosity, with a surface temperaturereduced by 16%, from the ZAMS.

Remarkably, the Schnberg-Chandrasekhar limit can be avoided (soft-ened) if the core is so compact that the gas of free electrons is (partially)degenerate, because the degeneracy pressure can help to stop the collapse.This effect is already evident in Sun-like stars, and in lighter objects the coremay not exceed the Schnberg-Chandrasekhar mass at all before the nextstage of nuclear burning commences.

14.3 High-mass main sequence

The main sequence evolution of stars with masses significantly higher than1M is similar to that of lower-mass stars, but with the difference thatconvection homogenizes the core chemical composition and prevents the for-mation of a hydrogen burning shell as early. Instead of that, when thehydrogen fraction reaches about a few per cent, the entire star begins tocontract. It releases gravitational energy, enhances the luminosity, and alsothe temperature (due to the R-reduction). This point of overall contractiondefines the end of the main sequence for high-mass stars. Since the ZAMS,the stellar radius of a 5M ( 7 107 yr) has been reduced by half, thecentral temperature grown by a 30%, and the luminosity doubled.

15 Mass-loss in Low- and High-Mass Stars

15.1 Magnetic activity and mass-loss in low-mass stars

Typical mass-loss rates in Sun-like stars during the main sequence are 31014M yr1, too low to affect the stellar mass. The loss of material inlow-mass stars in the main sequence is linked with the magnetic activity ofthe stellar interior. The magnetic field has important consequences in boththe evolution of the star and also its external appearance.

57

Dynamo effects generate magnetic field in the convective regions of thestar by the motion of conducting plasma in a layer with differential rotation;induction and Coriolis forces play also a role in the magnetic field generation.The magnetic field has a complex structure as it is produced by currents ofdifferent orientations. The global current geometry is however dominatedby equatorial convection, which produces a global dipolar field. This dipo-lar field, however, gets bent in the azimuthal direction because of the stardifferential rotation, creating a significant toroidal field component.

The magnetic field produced in the convective layers of the star diffusesoutwards while is affected by turbulent convection. The magnetic tubeseventually cross the photo-sphere forming star spots, which are regions ofhigher magnetic line concentration and lower temperature (due to convectioninhibition) that appear as dark spots in the star surface.

For illustrative purposes, cartoons of the external magnetic field in theSun are shown in Fig. 14 for two different phases of its magnetic activity, atthe minimum and the maximum of solar magnetic activity. At its minimum,the field is rather ordered and star spots are rare. Around the maximum, themagnetic field is actually reversing its polarity and therefore becomes muchmore complex within the star and in its external manifestation, with anenhancement of star spots and other magnetic field-related phenomena likefaculae. In the Sun, the whole maximum-minimum-maximum cycle takes 22years.

Figure 14: For illustrative purposes, the external magnetic field in the Sun isshown in two different phases of its cycle, at the minimum and the maximumof solar magnetic activity.

58

The magnetic field lines that reach the stellar surface give rise to powerfuland violent phenomena, such as flares and coronal mass ejections, linked tothe dissipation of magnetic energy through magnetic reconnection. Somemagnetic lines are anchored in the star by both ends, but others open, oneend connecting with the external medium. The regions of open lines arelocations from which a fast wind, of velocity 750 km s1, is launched.In active regions, with a high concentration of closed magnetic loops, thestellar wind is slower, with velocities of 400 km s1, as the close magneticlines tend to trap the material in the stellar surroundings and the gas has topush against the field. The stellar wind is made of material of high enoughtemperature to escape the gravitational pull of the star, and carries withit the attached magnetic field. Angular momentum is transferred throughmagnetic lines to the expelled material, which slows down the rotation ofthe star. The magnetic field far from the star becomes dominantly toroidalas angular momentum is to be conserved in the stellar wind.

15.1.1 Exercise: a model for the stellar wind

The Parker model can be used to describe the corona of the star, where thewind is formed. Theoretical5 and observational considerations6 make the ap-proximation of an isothermal corona reasonable. This simplifies strongly thecorona characterization if one, in addition, assumes hydrostatic equilibrium:

P/r = GM/r2 (135)

Given that P = (n/)kT 2nkT , where n = /mH, one gets

(2nkT )/r = GMnmH/r2 (136)

with T constant, which yields as solutions:

n = n0 exp [(1R/r)] (137)

andP = P0 exp [(1R/r)] (138)

where = GMmH/2kTR.The obtained solutions for the corona density and pressure under the

assumption of hydrostatic equilibrium converge in the infinity to constant

5The corona is made of a highly conductive plasma with efficient cooling and heating.6Even at the Earth distance, the stellar wind temperature is only about 10 times lower

than in the corona.

59

values that are well above those in the interstellar medium. This inconsis-tency implies that hydrostatic equilibrium is no valid. Thus, the gas mustbe therefore out-flowing, and a hydrodynamical model is to be adopted todescribe the corona.

In reality, unlike T , density decreases very strongly in the corona. Con-vection leads to the formation of longitudinal waves that initially move out-wards at the sound speed, but given the strong density decay, their amplitudemotion accelerates to keep the mass, the momentum and the energy fluxesconstant. This implies that eventually a shock will form when the wave mo-tion becomes faster than the local sound speed. Such shocks of convectiveorigin heat and ionize the gas beyond the chromosphere.

The role of the magnetic field is important in the wind formation. In ad-dition to convection hydrodynamical waves, Alfven waves also form. Thesewaves are transversal perturbations of the magnetic lines that have the mag-netic tension as the propagation mediator. Although the corona is highlyconductive, some small resistivity remains, which renders Alfven waves dis-sipative, heating the corona as they go through it.

15.2 Stellar winds in high-mass stars

Main sequence high-mass stars are bright objects whose radiation field isstrong enough to overcome the Eddington limit in their atmospheres. TheEddington limit is imposed in this case by photon absorption through theexcitation of resonance lines in elements such as N, C, S and Si, which despiteof their relative low abundances are present in all massive stars. The crosssections of these resonant atomic transitions are orders of magnitude largerthan those of Thomson scattering of free electrons from ionized hydrogenand helium, by far dominant in the atmosphere of the massive star. Thismakes the driving force exciting these lines dominant over other light-matterinteractions, and allows the strong radiation field to carry away substantialamounts of matter forming a powerful and fast stellar wind. Note thatlower metalicities imply less resonance-line targets and hence a lower line-driven radiation pressure, so the metalicity is an important parameter thatdetermines the stellar wind. Rotation and the presence of magnetic fieldsalso affect the stellar wind production and properties, although they will notbe discussed here.

Unlike low-mass stars, high-mass stars can change their masses by afactor 2 during the main sequence. Typical mass-loss rates of OB starsgo from M 109 105M yr1, the earlier the type the higher the rate.For an O6, with 106M yr1 and Lifetime 106 yr, the total mass loss

60

is Mloss M, but for very early type stars the Mloss M .

15.3 Mass-loss and stellar evolution

A large fraction of the stellar mass in both low- and high-mass stars may belost through intense and discontinuous events after the main sequence. Boththe asymptotic giant branch (AGB) phase discussed in Sect. 16.1.2 for starsbelow several M, and the supergiant, the luminous blue variable (LBV)and the Wolf-Rayet (WR) phases for more massive stars, are characterizedby violent and strong mass loss, sometimes even explosive, as in LBVs.

In low-mass stars, the AGB phase is a necessary ingredient to link thestar mass with the mass of the white dwarf left at the end because of thevery small mass-loss rates during the main sequence. In high-mass stars,otherwise, the main sequence mass loss is already strong, being as importantas that in post-main sequence phases such as the supergiant or the WR one.It is still unclear however in massive stars whether the continuous, line-drivenwind is enough to explain the final masses before the star collapses into aneutron star or a black hole, or violent eruptions, like those of LBVs, areindeed needed to match the initial and the final masses of the object.

16 Post-Main Sequence Stellar Evolution

After the main sequence, the stellar core becomes structured in concentricshells in which hydrogen, helium, etc., burn at various times. This takesplace together with readjustments involving expansion or contraction of thecore or the envelope, and the development of extended convection zones. Atthe latest stages of evolution, extensive mass loss from the surface plays acritical role in determining the stars fate.

16.1 The fate of stars with M < 8M

16.1.1 Initial stages

Stars reach the hydrogen shell burning phase when the core hydrogen is al-most over. The helium core slowly grows, and when it reaches the Schnberg-Chandrasekhar limit, the core begins to contract quickly.

The core collapse leads to a nonzero T -gradient in the until-then isother-mal helium core, and the hydrogen burning shell T and grow, with itsenergy generation rapidly intensifying despite it also narrows. The extra

61

energy is mostly directed to stellar envelope expansion, and the stellar lumi-nosity decreases.

The envelope expansion (and H enhanced opacity) favors the occurrenceof convection in the external stellar layers, and the matter mixing reaches theregions chemically modified in the stellar interior. In the surface, abundanceratios are modified: for instance, lithium (still remaining some at the surface)is dragged inwards and quickly burned, and on the other hand the 32He-concentration grows. This process is known as the first dredge-up phase.

Envelope expansion leads to cooling of the outermost layers, and to a re-duction of the H ion opacity, so convection softens. On the other hand, thecore still on-going contraction and the hydrogen burning shell luminosity in-crease proceed further. Both effects combined im