Stellar structure and evolution

Warrick Ball

November 3 & 4, 2014

Contents

1 Stellar modelling 41.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2 Hydrostatic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 Energy generation (and conservation) . . . . . . . . . . . . . . . . . . . . . 71.1.4 Energy transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Composition equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Matter equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Opacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.3 Neutrino loss rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.4 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.1 Centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.2 Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.4 Initial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Stellar evolution 202.1 Characterizing stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 The Hertzsprung–Russell diagram . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 Colour–magnitude diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.3 The ρ-T diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.4 Kippenhahn diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Evolution of a Sun-like star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.1 The main sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 The red giant branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.3 The helium flash and core helium burning . . . . . . . . . . . . . . . . . . 272.2.4 Thermal pulses and the asymptotic giant branch . . . . . . . . . . . . . . . 282.2.5 Envelope expulsion and the white dwarf cooling track . . . . . . . . . . . . 29

2.3 More massive stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Convective cores on the main sequence . . . . . . . . . . . . . . . . . . . . 302.3.2 Non-degenerate helium ignition . . . . . . . . . . . . . . . . . . . . . . . . 30

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2.3.3 Mass loss on the main sequence . . . . . . . . . . . . . . . . . . . . . . . . 312.3.4 Carbon burning and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Parting thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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Prologue

About the course

These lecture notes are from the two lectures I gave at the DWIH Winter School in 2014. Theywere forked from the notes I had so far prepared for the course on stellar modelling taught atGottingen in August 2014.

There are two lectures, distinct in content and character. The first lecture aims to introducethe equations of stellar structure and evolution, with a keen focus on the principles and conceptsbehind the models, rather than detailed derivations to study the theory in depth. The secondlecture aims to simply outline how we study stars and what the models say about the evolutionarystates of stars in different phases. I tend to use the Sun as a fiducial case, both since it’s a specialtarget and because it happens to undergo many distinct phases of evolution. I then identify thingsthat change in stars of increasing mass.

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Chapter 1

Stellar modelling

Over the course of the 20th century, a basic formulation of stellar structure and evolution emergedthat is able to explain most observational results (or, rather, the observed properties of most stars).This is sometimes called classical, standard or even vanilla stellar structure and evolution. It iswell-studied, and serves as the starting point for studying the stars. If you read a paper and thewriter doesn’t specify precisely how their models are calculated, they probably use the standardpicture. Conversely, authors should be specific about non-standard properties (or properties forwhich there is no standard!). We will now consider how this standard model is constructed, andonly allude to where “non-standard” modelling can enter.

There is a long line of fine textbooks on stellar structure and evolution, arguably starting withthe theoretical works of Eddington (1926) and Chandrasekhar (1939, 1957). The post-war era sawthe rise of computers, and textbooks shifted in flavour towards formulating the equations, solvingthem numerically, and interpreting the results. Notable subsequent works include Schwarzschild(1958) and Cox & Giuli (1968), among many others. But since it’s publication, the work ofKippenhahn & Weigert (1990) has been definitive, and a second edition was recently released(Kippenhahn et al. 2012).

There are also many lecture notes available online, but two notable entries are those of JørgenChristensen-Dalsgaard1 and Onno Pols 2. I’m a particular fan of Onno Pols’ notes, which are neartextbook quality and have excellent figures, many of which I have borrowed. If you do use thesenotes at any point, also glance at some of the exercises to test yourself.

1.1 Basic assumptions

To formulate equations that represent the structure and evolution of a star, we make severalassumptions. We assume that a star is

1. a fluid;

2. spherically symmetric;

1http://astro.phys.au.dk/~jcd/evolnotes/LN_stellar_structure.pdf2http://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/

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3. self-gravitating;

4. stationary on dynamical timescales; and

5. in local thermal equilibrium

We can correspondingly write down the consequences of these assumptions.

1. Our formulation will start essentially from the equations of fluid dynamics. This is doneexplicitly in the textbook by Collins (1989)3.

2. The structure will depend only on the radial co-ordinate in the star, or any function thatvaries monotonically with radius (e.g. the mass or pressure). This assumption also requiresthat the star is not rotating.

3. Specifically, this assumption is intended to mean that gravity is the only external body forcethat we consider in the fluid equations. We exclude electric or magnetic fields.

4. We neglect net velocities and accelerations in the star. As we will see, there are regions ofthe star where large scale flows exist, but there are always equal upward and downwardflows, so the net flow (averaged over a shell) is zero.

5. For each shell of material, we can define a local temperature that is constant in an infinitesimalneighbourhood of that shell. In addition, we assume that particle velocities are distributedlike Maxwell distributions and that the radiation field is given by a blackbody at the samelocal temperature.

The assumptions may have seemed quite mild at first but keep in mind precisely what they imply.Let us also give a moment’s thought to when these assumptions will fail.

1. Generally speaking, the fluid approximation is excellent until different particle speciesdecouple, above the photosphere.

2. Stars generally rotate, but the amount of rotation is often small enough that the structureof the star is not significantly altered. However, rotation induces fluid instabilities that leadto additional mixing of material, and this can be significant even when the structural effectof rotation is small.

3. Again, in truth we know that the Sun has a magnetic field and we have strong evidencethat many stars with convective envelopes do too. In addition, many stellar remnants (e.g.white dwarfs, pulsars) have strong magnetic fields. So they are present but, as with rotation,probably not at a structurally-important level.

3Collins’ book is out of print and available freely from http://ads.harvard.edu/books/1989fsa..book/

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4. There are two places where this breaks down. The more benign occurrence is at the end ofa massive star’s life, when iron has been produced in the core, which ultimately begins tocollapse. The other widespread case is the occurrence of convection. Material simultaneouslyflows upwards and downwards, with zero net mass flux but effective heat (and composition)transport. Stellar models circumvent this problem by employing simplified steady-stateapproximations of the structure in convective zones, but in principle there can be structurallysignificant velocities.

5. Local thermodynamic equilibrium applies well throughout most of most stellar modelsbut breaks down near the surface, where the surface boundary conditions of the stellarmodel must be applied. The radiation field diverges from the Planck function, owing to thedevelopment of spectral lines in the lower density atmospheres.

1.1.1 Mass conservation

The mass dm contained in a shell at radius r, with width dr and local density ρ(r) is simply

dm = 4πρ(r)r2dr (1.1)

Written differentially,

dm

dr= 4πρ(r)r2 (1.2)

Since the density is always positive, the function m(r) is monotonic. We can make m the dependentvariable (instead of r) and instead write the equation as

dr

dm=

1

4πρ(r)r2(1.3)

The formulation with r or m as the dependent variables are known as using Eulerian or Lagrangianco-ordinates. Early codes tended to use m as the dependent co-ordinate because the surfaceboundary is then fixed at the total mass of the star, M∗. Were we to use r as the dependentvariable, then the surface boundary is the total stellar radius R∗, which moves over the star’s life.

In reality, modern codes use neither of these formulations, usually instead using some functionof monotonic variables (pressure P , temperature T , mass m and radius r).

1.1.2 Hydrostatic equilibrium

The general equation of motion for material in the star is

ρ

(∂~v

∂t+(~v · ~∇

)~v

)= −~∇p+ ~f. (1.4)

Normally, the pressure p is a tensor but the pressure is taken to be isotropic and reduces to ascalar. First, we suppose that the only body force is the gravity, so that ~f = −ρ~g. Second, we

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suppose that the system is dynamically stationary and not rotating, so that ~v = 0. In this case,we write

~∇p = −ρ~g (1.5)

which is known as the equation of hydrostatic equilibrium. Finally, we suppose that the system isspherically symmetric, so that we can write

dp

dr= −ρg = −Gmρ

r2(1.6)

This equation—hydrostatic equilibrium under spherical symmetry—is just a statement of thebalance of forces, which is itself a statement of the conservation of linear momentum.

Note that these two equations—hydrostatic equilibrium and mass conservation—are said todescribe the mechanical (rather than thermal) structure of the star. If the pressure P is purely afunction of ρ (as is the case for e.g. polytropes), then the structure can be solved at this point.

In addition, these equations alone are enough for us to derive the Virial Theorem: a usefulresult that relates total energies in the star. We shall not derive it here, but I recommend lookingit up in any standard reference. In fact, Collins has written a book on just this result (Collins1989?).

1.1.3 Energy generation (and conservation)

The energy content of the star at a given layer is modified only by creation or destruction ofenergy, so we write

∂L

∂m= ε = εnuc − εν − T

∂s

∂t(1.7)

which simply represents the conservation of energy. I have expanded the total energy generationrate ε into three components:

• the nuclear energy generation rate, which describes energy created by the fusion of elementsinto more strongly bound nuclei;

• the neutrino loss rate, which describes energy lost to the creation of neutrinos (and antineu-trinos); and

• the gravitational energy generation rate, which is equal to dQ = TdS/dt by definition anddescribes energy that is released or absorbed by the contraction or expansion of the star.

1.1.4 Energy transport

There are three main ways to transport heat through a star, and we will now work through themin sequence.

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Radiation

The most natural mechanism is radiation: the core of the star is hot, dense and producing energyin nuclear reactions. This energy can basically shine out. We treat this process in the diffusionlimit, because the mean free path of a photon is small and is scattered, absorbed, and re-radiatedmany times as it literally diffuses out of the star.

We start with Fick’s law, with the radiative flux diffusing along the energy density gradient.

~Fν = −Dν~∇uν (1.8)

The diffusion coefficient is given by

Dν =1

3

c

κνρ(1.9)

where

a =4σBc

(1.10)

is the radiation. The energy density of the radiation field is

uν(T ) =4π

cBν(T ) =

4π

c

2hν3

c2

1

ehνkT − 1

(1.11)

We will also use the facts that∫Bν(T )dν =

2π4

15

k4T 4

h3c2=σ

πT 4 (1.12)

and therefore∫dB

dTdν =

8π4

15

k4T 3

h3c2=

4σ

πT 3 (1.13)

because ν and T are independent variables.Now, we can integrate Fick’s law over frequency and apply the assumption of spherical

symmetry.

L

4πr2=

∫Fνdν = −

∫Dd

druνdν (1.14)

= −∫

c

3κνρ

4π

c

dBν

dT

dT

drdν (1.15)

= −4π

3ρ

dT

dr

∫1

κν

dBν

dTdν (1.16)

= −4π

3ρ

dT

dr

1

κR

4σ

πT 3 (1.17)

= −16σT 3

3ρκR

dT

dr(1.18)

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where we have defined the Rosseland mean opacity by

1

κR=

∫1κν

dBdTdν∫

dBdTdν

(1.19)

In stellar modelling, we traditionally define a dimensionless temperature gradient

∇ =d log T

d logP(1.20)

for a given process, and write the temperature gradient as

∂T

∂r= −∇GmρP

Tr2(1.21)

For the case of radiation, we thus have

∇rad =d log T

d logP=P

T

dT

dr

dr

dP(1.22)

=P

T

L

4πr23ρκR16σT 4 r2

Gmρ(1.23)

=3κLP

64πσGmT 4(1.24)

.

Convection

Consider a blob4 of material somewhere in a chemically homogeneous region of our star. If weperturb this blob upward adiabatically (so no heat is exchanged with the surroundings), thenthe blob will expand as it rises, because it’s hotter than its surroundings. However, because ofthe density gradient, the surrounding material may or may not be more dense than the blob.If the surroundings are less dense than the blob (i.e. the blob is denser), then it sinks after itsperturbation and returns to its starting point. If, however, the surroundings are more dense (i.e.the blob is less dense), then the blob is positively buoyant and will continue to rise. Equivalently,if we perturb the blob inwards, it first contracts, and again may be positively or negativelybuoyant, depending on the density gradient. This situation is unstable, and we call this convectiveinstability.

I omit here a detailed derivation of the stability criterion, but it can be found in any standardtextbook or set of lecture notes. The criterion for instability turns out to be

∇rad > ∇ad (1.25)

or, equivalently,

∂s

∂r< 0 (1.26)

4The professionals might call this a parcel of material, but it’s really just a blob.

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To determine how much heat is transported by convection, we need a theory of convection.The standard theory used in stellar models is the (frighteningly?) simple mixing length theory(MLT Bohm-Vitense 1958). In MLT, it’s presumed that the rising and falling blobs move somecharacteristic distance (the mixing length) before dispersing into their surroundings, mixing heatand material. The mixing length `MLT is usually taken to be proportional to the pressure scaleheight HP ≡ P/(dP/dr). The constant of proportionality is typically denoted α and is determinedby calibrating stellar models to the Sun.

MLT is quite basic, and has many flaws. For a start, there are several slightly different versionsof the theory, with corresponding slightly different predictions of the convective flux, dependingon things like the shapes of the blobs, their level of opacity, and so on. What’s more, the theoryis completely local, and therefore allows situations where the convection acceleration goes tozero at the convective boundary, but the convective velocity might be non-zero. Thus, the blobspotentially penetrate into convectively stable regions. We call this process overshooting, and treatit with somewhat ad hoc parametrizations.

This begs the question: why has MLT persisted so long, and how does it work at all? MLTis relatively easy to implement, and it turns out that MLT is usually efficient, and drives thetemperature gradient very close to its adiabatic value. Thus, one could compute fairly accuratestellar models by simpling assuming that convection zones are perfectly mixed and adiabaticallystratified. MLT does fail, however, when convection is not efficient, as is the case near the surfacesof Sun-like stars. This has important consequences for oscillation frequencies.

What of other theories of convection? Some exist: see for example the work of Xiong et al.and Canuto et al. (both the older full spectrum turbulence and the more recent Reynolds stressmodel).

Conduction

At very high densities, heat is transferred by the jostling of particles against one another. Thisis the process we know as thermal conduction. It is also regarded as a diffusion process and,consequently, one can derive an effective opacity, just as we did for radiative heat transfer. Thisconductive opacity is also provided as a component of tables that are interpolated.

1.2 Composition equations

In the absence of any mixing, change in number density ni of species i is

∂ni∂t

=∑j

rji −∑k

rik (1.27)

where rij is the rate at which species i is transformed into species j. The number density is definedby

ni =ρXi

mi

(1.28)

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where mi is the mass of a particle of species i. so

∂Xi

∂t=mi

ρ

(∑j

rji −∑k

rik

)(1.29)

Convective mixing is also modelled as a diffusion process. If it isn’t clear why this makes anysense, imagine a convective zone of pure hydrogen in which we insert a layer of pure helium. Thehelium will be transported upwards and downwards by the convective flows. Similarly, at thelocation of the helium layer, hydrogen is introduced from above and below. Hence, this is basicallya diffusion process, and contributes a mixing term

∂Xi

∂t=

∂

∂m

(Dconv

∂Xi

∂m

)(1.30)

where Dconv is a convective diffusion coefficient, usually something like vc`MLT/3, where vc and`MLT are the convective velocity and mixing length. In practice, the coefficient is basically just bigenough to instantaneously mix convective zones but not so big as to cause numerical problems.

There are other mixing processes that take place in stars, notably gravitational settling: lightermaterial diffuses towards the surface of the star; heavier material towards the centre. These arealso modelled as diffusion processes. But remember that the diffusion coefficient can be differentfor each species, although for convection it is the same.

1.3 Matter equations

Although we have technically now defined all the differential equations of stellar structure, thereare several functions that we have not yet described. They are

• the opacity κ,

• the nuclear energy generation rate εnuc,

• the neutrino energy loss rate εν ,

• the nuclear reaction rates Ri,j, and

• the equation of state, which relates the pressure, density and temperature.

None of these is computed a priori in stellar evolution codes. Instead, we rely on precompileddata, usually in the form of tables. For each, there are many sources of these tables from variousresearch groups around the world who make it their business to perform the detailed calculationsof these quantities. Stellar evolution codes then interpolate in tables of data to determine thevalues (and often their derivatives) at the relevant points in the star.

The first four are generally regarded as functions of density, temperature and composition. Theequation of state is a slightly special case. At its simplest, it can be thought of as a function thatgives the pressure as a function of density, temperature and composition, too, but in truth the

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Figure 1.1: Left : Plots of Rosseland mean opacity versus temperature for selected values of log ρ,indicated on each curve. Right : Surface of Rosseland mean opacity as a function of density andtemperature. The thick lines indicate profiles of stellar models of the indicated masses.

equation of state package involves a large number of thermodynamic variables. e.g. specific heatcapacities cP , and so on. Some analytic approximations to the equation of state exist, but most relyon precomputed functions to approximate some terms. In the past, the analytic approximationswere popular because they are much faster than interpolating in tables, but this has become lessimportant as computers have become faster and able to manage much larger tables. The mostpopular analytic approximations (e.g. EFF and CEFF) are known to be less accurate than thelatest tabulated data.

1.3.1 Opacity

Recall that the opacity was defined in the diffusion approximation for radiative heat transport. Itis computed by taking detailed account of several processes that absorb (and therefore re-emit) orscatter light.

• Electron scattering. At low energies, this is simply Thomson scattering, and the opacityis given by κ = σTne/ρ, where σT is the Thomson cross-section and ne is the free electronnumber density. In fully ionized material, the electron density is proportional to the density,so the electron scattering opacity is constant. At higher energies, electron scattering becomesCompton scattering, and decreases with temperature.

• Bound-bound scattering. These are quantum transitions from one bound state to another.e.g. hydrogen Lyman lines. These require detailed calculations to know what the relevantenergy levels are, for material in which the ionization state also depends on temperature.

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• Bound-free scattering or photo-ionization. This is the complete liberation of an electronfrom a nucleus. This is possible for a range of photon energies above some minimum value,and generally contributes more to the opacity than bound-bound transitions.

• Free-free scattering. In essence, inverse bremsstrahlung: instead of an electron deceleratingin the presence of a nucleus, and emitting radiation, it accelerates and absorbs radiation.

• H−. At low temperatures, an equilibrium abundance of H− ions form. This decreases thefree electron population enough to affect the bound- and free-free opacities. This effectdominates the atmospheres of F, G and K-type stars (3000 K . Teff . 6000 K).

• Molecular lines. At very low temperatures (T . 3000 K), molecules begin to form. Theyhave additional quantum states given by the rotational or vibrational state of the molecule,and all of these lines contribute to the opacity. Typical contributing molecules include H2,TiO2, H2O and CN.

1.3.2 Nuclear reactions

Introductory notes

Before diving directly into the nuclear reaction chains relevant for stellar evolution, a few pointsare in order.

First, the nuclear reactions are presented in a characteristic notation. A reaction

A + B→ C + D (1.31)

is denoted

A(B,C)D (1.32)

This is useful because, if D participates in a subsequent reaction, we can write it immediatelyafter.

Note also that when a chemical species is written, e.g. 12C, we mean only its nucleus. For thisreason, 1H and p are interchangeable, as are α and 4He. I have tried to consistently use one, butthere’s nothing fundamentally wrong with using both.

Hydrogen burning

Stars spend most of their lives burning hydrogen into helium in their cores. The net reaction isalways of the form 41H→ 4He. This net reaction occurs through two primary channels: the ppchains, in which protons and other light elements are combined; and the CNO cycle, in whichC, N, O and possibly F act as catalysts onto which protons are added until an alpha particle isemitted. (See Figures 1.3 & 1.4.)

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Figure 1.2: Binding energy per nucleon, as a function of atomic mass number. Note the peak atiron, indicating that no further energy can be liberated. Also, note that 8Be is less strongly bound(per nucleon) than an alpha particle, hence its rapid decay.

Figure 1.3: Diagram of reactions that constitute the proton-proton chains.

Figure 1.4: Diagram showing the reactions that constitute the CNO cycles. The lines are alldirected upward, indicating alpha emission.

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Helium burning

There is no stable state of 8Be, so we cannot straightforwardly fuse two helium nuclei togetheron the way to heavier products. But recall that by “unstable”, what we really mean is that thefusion product (some excited state of 8Be) has a very short lifetime, but at sufficient density andtemperature, it is time enough for a third helium nucleus (alpha particle) to join the excitedberyllium nucleus, forming 12C. Because of the need for three helium nuclei to combine more orless instantaneously, this process is called the triple alpha process.

I like to make an historical remark about this process. In the early days of detailed stellarmodelling, the laboratory data on 12C did not indicate that there was an appropriate excited statevia which the triple alpha process could occur. As a result, the reaction rates were very low, andthe stellar models wouldn’t form carbon. Fred Hoyle, who contributed a phenomenal amountof work in the area of stellar modelling and stellar nucleosynthesis5 argued that such a statemust exist, else carbon would simply never have been created. On closer inspection, calculationsindicated that indeed, the relevant state exists, and it sometimes therefore known as the Hoylestate.

Because the triple alpha process already requires high temperatures and densities to overcomethe beryllium barrier, subsequent alpha-captures occur nearly immediately. First, one sees

12C(α, γ)16O (1.33)

followed by

16O(α, γ)20Ne (1.34)

Thus, stellar cores, after helium burning, are typically a mixture of carbon, oxygen and, at highermasses (and thus core temperatures and densities), neon. So, for example, one never encounters apure carbon white dwarf, only carbon-oxygen (or sometimes oxygen-neon) white dwarfs.

1.3.3 Neutrino loss rates

Whenever neutrinos are formed, it is assumed that they escape from the star without interactingwith any other material. This energy is thus lost, which is made explicit in the energy equation.Most of these neutrinos are associated with nuclear reactions, but there are also independentprocesses that can take place at high temperatures (& 3× 108 K).

The first process is known as pair production. At sufficiently high temperatures (T > 109 K),photons are sufficiently energetic to spontaneously form electron-positron pairs, which mostlyannihilate nearly immediately. i.e.

γ + γ → e− + e+ → γ + γ (1.35)

However, once in long while (about one reaction in every 1022), the annihilation instead releases aneutrino-antineutrino pair. This pair is lost, taking the energy of the photons with them.

5Some people believe that Fred Hoyle was shunned from sharing the 1984 Nobel Prize in Physics because of hisother, more controversial, opinions, and his abrasive nature. The 1983 Nobel Prize went to two other scientistswho had worked on stellar structure and evolution: Subramanyan Chandrasekhar and Willie Fowler.

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The second process is the Urca process, which produces neutrinos and antineutrinos from theequilibrium of forward and inverse beta decay in some material. The general form of the forwardbackward beta decays are

A + e− → B + ν (1.36)

B→ A + e− + ν (1.37)

or vice versa. The net result is the continuous release of neutrinos and antineutrinos, which arelost.

1.3.4 Equation of state

Computing the equation of state accurately, quickly and in such a way that preserves fundamentalthermodynamic identities is a complicated problem, but several basic components can be identified.

• Ideal gas pressure. Recall the definition of the partial pressure contributed by a chemicalspecies i that behaves like an ideal gas:

Pi = nikBT =ρXikBT

µimp

(1.38)

where ni is particle number density, kB is Boltzmann’s constant and mp is the mass of aproton. This defines

1

µi=nimp

ρXi

(1.39)

i.e. number of particles per proton mass. In other words, µi is the number of proton massesper particle, or the molecular weight of species i. Because protons are about as heavy asneutrons, and both are much heavier than electrons, we basically count the number ofnucleons per free particle. Consider these few cases.

Chemical species Proton masses Free particles µ

Neutral hydrogen 1 1 1Ionized hydrogen 1 2 2Neutral helium 4 1 4

Singly-ionized helium 4 2 2Doubly-ionized helium 4 3 4/3

Metal, atomic number Z about 2Z Z + 1 about 2

The total pressure is just the sum of them partial pressures, so we write

Pg =∑i

Pi =ρkT

mp

∑i

Xi

µi≡ ρkT

µmp

(1.40)

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Figure 1.5: Plots of the occupation number of momentum states for electron gases in differentstates. The dashed lines show the distribution predicted by the Maxwell distribution, and the solidlines show the distribution including degeneracy. Electrons are forced to occupy higher momentumstates, which exerts a pressure that we identify as degeneracy pressure.

which defines the (harmonic) mean molecular weight: 1/µ =∑

iXi/µi.

Note that the changing mean molecular weight of the core (owing to nuclear reactions) isthe main driver of stellar evolution on the main sequence. In the Sun, the mean molecularweight goes roughly from 0.6 to 1.34. So at constant pressure, the central density wouldnearly double over the main sequence!

• Radiation pressure. Remember that we have assumed an isotropic radiation field everywherein the star, in equilibrium with the gas. The radiation field contributes to the pressure it’sown radiation pressure equal to aT 4/3, where a is the radiation constant.

Inside a star, it turns out that Pg/Pr ≈ ρ/T 3 is (very!) roughly constant, which one canuse to construct a basic stellar model. In fact, opacity tables are actually tabulated in thevariable R ∝ ρ/T 3 to reduce some of the interpolation error and decrease the size of thetables necessary to cover stellar interiors.

• Degeneracy. When electrons are packed very close together, their allowed quantum statesbegin to overlap. That is, instead of each atom having its own quantum levels, which mustrespect the Pauli exclusion principle, the atoms begin to share states, decreasing the numberof quantum state available overall. This forces the electrons to occupy higher energy (orhigher momentum) states, which exerts a kind of pressure, known as degeneracy pressure.When the degeneracy pressure is large, the material is described as degenerate.

• Pair-production. At very high temperatures, the energy density of the radiation fieldis sufficiently high that pairs of photons can spontaneously form electron-positron pairs,that quickly annihilate. Almost all of these annihilations produce photons again, but a

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small number (that increases with temperature) instead annihilate and create a neutrino-antineutrino pair, both of which are lost from the star. The loss of the photons slightlyreduces the radiation pressure.

1.4 Boundary conditions

The solution of a set of differential equations is only possible given the appropriate number ofboundary conditions, which we describe here. In boundary-valued problems, these can be specifiedat either end of the solution domain. i.e. at either the surface or the centre of the star.

1.4.1 Centre

At the centre of the star, the formal boundary conditions are simply the co-ordinate requirementsthat m, r and L are all zero. (This gives us two boundary conditions.) In practice, the use ofr = 0 cause problems because of terms that go like 1/r. This can be overcome either by carefullychoosing different co-ordinates or by setting the interior boundary condition to be the averageover the space inside the innermost meshpoint.

If you were inclined to model a star outside the core, then one could also specify finite valuesm, r and L at the centre. This could be done to model, for example, the behaviour of materialbeing accreted onto a neutron star.

1.4.2 Surface

The derivation of the surface boundary conditions requires a more detailed analysis of radiativetransport, since the photons begin to escape from the star and into the vacuum of space. Astandard set of boundary conditions is provided by the Eddington grey atmosphere, which is grey(opacity is constant in frequency), plane-parallel and at constant surface gravity (has negligiblemass and radial extent). These assumptions lead to the conditions

m = M∗ (1.41)

L = 4πR2T 4 (1.42)

pg = 2/3(g/κ− F/c) (1.43)

at r = R∗.

1.4.3 Composition

Each composition equation is of second-order in space, so we need two boundary conditions.The conditions are simply that there is no composition flux across the centre or the surface. i.e.composition is lost out of the star. Mathematically, this is imposed by

∂Xi

∂r= 0 at r = 0, R∗ (1.44)

for each species.

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1.4.4 Initial models

We also need an initial condition because of the existence of time derivatives. The necessaryinitial conditions can be inferred from the time derivatives in the stellar model. First, each of thecomposition equations involves a time derivative, so we need to known the initial distribution ofcomposition throughout the star (usually taken as homogeneous). Second, there is the gravitationalenergy generation term, which basically means that we must specify the energetic state of themodel. Once this is done, however, the model and its subsequent evolution are defined.

Typical initial models for evolutionary calculations are:

• Approximate pre-main-sequence (PMS) models. PMS stars, if sufficiently cool, are expectedto be fully convective and well-approximated by n = 3/2 polytropes, constant entropysolutions, or other simple models. Thus, an initial model can be estimated by rescaling sucha model, and allowing it to first relax to a true solution of the equations before allowingevolution to proceed.

• Zero-age main-sequence models. If we suppose that the composition in a star is homogeneousand that the thermal state is such that there is no gravitational contraction, then the solutionis unique, and known as a zero-age main-sequence (ZAMS) model. Some authors specifythat the composition of catalytic abundances (i.e. CNO) take their equilibrium values,which avoids the ZAMS model first having to establish these abundances at the start of theevolution.

Note that a real star never actually reaches such a ZAMS phase because nuclear reactionsbegin gently at the end of the PMS contraction. Even so, this phase should be relativelyshort and inconsequential, so it is often ignored. If, however, you are concerned with veryaccurate ages (e.g. solar models), it is worth noting exactly where the zero age is defined.

• Any previous model. Obviously, any previous stellar model is a valid starting point fora calculation, even if one is manipulating the equations to suppress or enhance variousprocesses. The manipulation of the equations to rapidly create a suitable model (realistic ornot!) is sometimes known as stellar engineering.

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Chapter 2

Stellar evolution

This lecture is crudely divided into two parts. First, we review some diagnostics of stellarevolution. These are various plots (some observationally motivated) that we use to determinewhat’s happening to a star as it evolves. Some are familiar, some not, but I’ll go through them allfor completeness, even though you’ve probably already seen a lot colour-magnitude diagrams...

In the second part, we discuss the details of a 1 M� model, which happens to experience manydistinct phases of evolution that are worth discussing. It’s a useful reference point that we thenextend to higher mass (and other conditions) to explore what happens differently in the lives ofthose stars.

2.1 Characterizing stellar evolution

2.1.1 The Hertzsprung–Russell diagram

The original Hertzsprung–Russell diagram (Russell 1914) was a plot of absolute visual magnitudeagainst spectral type. The modern version tends to show absolute magnitude against colour, orluminosity against effective temperature. Observers tend to use the former; modellers the latter.In either case, and given enough numbers, several clear trends appear. These are all widely known,but are included here for completeness.

In the modern plot, made using Hipparcos targets for which the distances are accurate tobetter than 5%, we identify the following notable features.

• The main sequence is obvious, going roughly from faint red to bright blue. Most of the starsin the sample are found along this strip.

• Connected near the Sun’s position is the red giant branch, along which stars have similarspectral types (G, K, sometimes M) but become very bright.

• Displaced slightly towards the blue near the middle of the red giant branch is a clearoverdensity of stars, known as the red clump.

• For mid-temperature stars, there is a region between that main sequence and red giantbranch that has few stars. This is known as the Hertzsprung gap. It appears because stars

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Figure 2.1: Two Herzsprung–Russell diagrams. Left : Russell’s original HR diagram, from 1914.The main sequence is reasonably clear. Less clear, but arguably present, is the red giant branchor red clump. Right : A modern HR diagram, compiled using stars from the Hipparcos sample forwhich distances are precise to better than 5%. Several regions of interest are indicated.

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Figure 2.2: A colour-magnitude diagram for the globular cluster M3. Several interesting featuresare indicated.

evolve quickly through this phase, so we are simply unlikely to catch them at this point intheir evolution.

• At the faint blue end, we see a small number of white dwarfs. These are the remains of thecores of stars smaller than about 8 M�. Their radiation is produced purely from gravitationalcontraction, and they slowly cool and dim.

2.1.2 Colour–magnitude diagrams

The Hertzsprung–Russell diagram requires absolute magnitudes (or luminosities), which themselvesrequire knowledge of the distances to the stars. However, if we look at a single cluster (or othermutually proximate group of stars), then the distances will be roughly the same, so similar trendsshould emerge. In the case of a star cluster, we also expect that the stars have the same age andcomposition. Such plots for individual clusters are colour-magnitude diagrams, and show somefeatures that are distinct from (but related to) the HR diagram.

• The main sequence ends abruptly at the most massive stars that have not yet evolved ontothe red giant branch. This is known as the main sequence turnoff. More massive (brighter)stars (should!) have already evolved off the main sequence; less massive (fainter) stars haveyet to do so.

• We again see a clear red giant branch.

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• We also find a number of stars that have crossed back into the blue, forming a nearlyhorizontal line in colour-magnitude space. This is known as the horizontal branch.

• Curiously, there are clearly some stars that appear to be on the main sequence, but brighterthan the main sequence turnoff. These are the blue stragglers. Although there is reasonableconcensus that these stars have interacted with another star and mixed hydrogen back intothe core, their existence and behaviour is still a hot topic of research.

2.1.3 The ρ-T diagram

Plots of density ρ against the temperature T inside the star give us two useful diagnostics. Thevalue of both comes from our knowing in roughly what density and temperature regimes certainprocesses (mostly from nuclear reactions and the equation of state) take hold.

First, we can plot profiles of a star at particular points in time. Then by examining the profile,we can see where parts of the star are evolving in different ways. For example, it is straightforwardto identify a degenerate core, or multiple burning shells.

Second, we can plot the central density and temperature over time, which roughly tells usthrough what phases of evolution the star goes. We will see immediate what material was burnedat the core, and if it experienced degenerate burning. Note that these plots won’t tell us directlyabout reactions outside the core, which may also be important.

2.1.4 Kippenhahn diagrams

Kippenhahn diagrams are two-dimensional plots that show how the boundaries between qual-itatively distinct parts of the star evolve. The x-axis is usually the age of the star (but couldbe anything that defines a sequence of models: Fig. 2.3 uses mass) and the y-axis the massco-ordinate. Then one plots the mass co-ordinates of certain boundaries, usually the boundariesof the convective zones and contours of the nuclear energy generation rate. Thus, a Kippenhahndiagram shows quite neatly where a star is convective, where it is burning nuclear fuel, and howthis changes over time.

This is a bit confusing at first, so let us walk through how one might construct a Kippenhahndiagram, just for the convective boundary at first. Let us suppose we have a sequence of stellarmodels from an evolutionary track. Each model in the sequence has some age, and we can look inthat model to find the convective boundaries. (We could find where ∇−∇ad = 0.) We wouldthen go to our diagram and, for each convective boundary we find, plot a point with the age ofthe star and the mass co-ordinate of the boundary. We then move on to the next model in thetrack and do the same thing, and proceed over the whole sequence. Joining the points will showhow the boundaries evolve.

Kippenhahn diagrams can be a bit tricky to plot and they can also get a bit complicated. Butthey are similarly very rich, and one can quickly identify several events in a star’s life based onthe history as shown in the Kippenhahn diagram.

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Figure 2.3: Kippenhahn diagram showing the zero-age main-sequence structure of stars at roughlysolar metallicity. The grey shading indicates convectively unstable regions. The lines show contoursof constant fractional radius (dashed blue) and fractional luminosity (solid red). Very low massstars are completely convective. As the mass increases, the core first becomes radiative and theconvective envelope becomes shallower. Once the CNO cycles dominate energy production, thecore is convective.

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Figure 2.4: Kippenhahn diagram for the evolution of a 1 M� star. Note that the x-axis (age) isbroken at 11 Gyr.

2.2 Evolution of a Sun-like star

A solar-mass star goes through many distinct phases of evolution, so we use it as an anchor andthen look at differences in other stars.

2.2.1 The main sequence

A star like the Sun burns hydrogen into helium through the proton-proton chains. The ratiosbetween the energy from the chains ppI:ppII:ppIII is about 85:15:0.02. The temperature gradientenforced by the proton-proton chains is steeper than the radiative gradient, so the core is radiative(convectively stable). At the same time, the Sun has a shrinking convective envelope. In it’scurrent state, the convective envelope has a depth of about 30% by radius but only 5% by mass.

2.2.2 The red giant branch

The Sun will take roughly 10 Gyr to deplete hydrogen at the core, after which hydrogen continuesto burn in a shell around the helium core. The shell gradually becomes thinner but the whole starbrighter, and it ascends the red giant branch, whose temperature is roughly set by convection andH− opacity in the atmosphere. At the same time, the helium core grows and become increasinglydegenerate.

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Figure 2.5: Hydrogen abundance as a fraction of mass for a 1 M� model, at various stages of corehydrogen burning. The core is radiative, so material is unmixed. The centre is the hottest, densestregion, so burns hydrogen faster.

The convective envelope also penetrates inwards, to some maximum depth. Along the way, itreaches into material that was affected by nuclear reactions on the main sequence, and is thereforedepleted in H, enriched in He, and has slightly modified metal abundances. The convection zonemixes these to the surface where they can be seen. The event of the surface abundances changinglike this is known as dredge up.

After the convective envelope retreats, there is a small composition jump at the depth ofits maximum penetration. At some point, the hydrogen-burning shell cross this layer, and thestellar structure abruptly changes, though only slightly. Even so, this is enough to cause a smalldownward jump in luminosity, and this is actually statistically significant in star clusters. That is,if one counts stars along various parts of the HR diagram, one sees a small increase in the numberof stars around the luminosity bump, basically confirming this prediction of the models.

In addition, even low mass stars can experience significant mass loss on the red giant branch. Asthe star becomes larger and brighter, the outermost layers experience an ever weaker gravitationalfield, and an ever stronger outward radiative force. Thus, material is driven off, enough topotentially remove something like 25% of the stellar mass from the surface.

As a final note on the red giant branch, it is still unclear why exactly stars become red giants.This might seem like an odd question at first, because that’s simply what the models do, but onecan engineer stellar models with similar properties to red giants that do not become red giants.For example, if one makes a stellar model of about 0.5 M� containing pure helium, it also has astrong burning shell with a composition gradient, but never becomes a giant. This is known asthe red giant problem (or very occasionally erythrogigantism) and, though unanswered, few in thescientific community nowadays consider it a worthwhile problem...

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Figure 2.6: Detailed plot of stellar evolution during the helium flash. The shaded regions showwhere the star is convective (light blue) or undergoing energetically significant nuclear reactions(red), in terms of mass co-ordinate, read against the left axis. The solid lines show the total(orange), hydrogen-burning (green) and helium-burning (blue) luminosities, to be read againstthe axis on the right. Finally, the remaining lines show the radii of the surface (dot-dashed) andoutermost helium-burning layer (dashed), to be read against the extra axis to the left. Note howthe helium flash progresses as a series of subflashes that gradually approach the stellar core.

2.2.3 The helium flash and core helium burning

As the star ascends the red giant branch, the core becomes hotter and denser. It is degenerate,and heat is transported most efficiently through conduction, rendering the core nearly isothermal.At these temperatures, neutrino cooling becomes non-negligible, so the the core is actually slightlycooler toward the centre, where the cooling is most efficient. Eventually, some off-centre layer ofthe star is hot enough to ignite helium. At first, the commencement of nuclear reactions heatsthe neighbouring layers, so they too start helium burning. As a result, a small region of thecore suddenly starts burning enough helium to produce about 109 L�: the brightness of a smallgalaxy! But this probably lasts only seconds, and the luminosity never manifests at the surface.In addition, the ignition causes the core to expand, cool, and halt the progress of the reactions.

Of course, over time, the same thing just happens again: the core once again contracts, heats,and ignites helium, but this time slightly closer to the core. This series of subflashes continuesuntil the centre itself ignites, and the core begins to burn helium stably. The change in structureweakens the hydrogen-burning shell, and the total luminosity actually decreases.

It’s worth noting that the helium flash is notoriously difficult to model. It happens so quicklyand with such extreme properties that most codes balk at the problem and crash. So this pictureis simply our best model at the moment. While the broad trend of ignition starting suddenly andoff centre is probably right, precisely how the burning shifts to the core may yet change as ourmodels and computers improve.

After the flash, the star burns helium stably in the core. The triple-alpha process causes

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Figure 2.7: Schematic Kippenhahn diagram of two thermal pulses..

as super-radiative temperature gradient, so the core is convective during helium burning. Thehydrogen-burning shell is not extinguished, but proceeds gently at the same time as core heliumburning. In a star like the Sun, this phases takes about 1 Gyr. The star will appear as part of thered clump.

2.2.4 Thermal pulses and the asymptotic giant branch

After helium is depleted in the core, it continues to burn in a shell. This means there are twoburning shells: helium and hydrogen. It turns out that this situation becomes unstable, andthe star undergoes thermal pulses as it ascends something like the red giant branch (called theasymptotic giant branch).

The sequence of events in a thermal pulse is as follows. (See Fig. 2.7.) Suppose we start thecycle with a degenerate CO core, a He shell, a hydrogen-burning shell and a convective envelope.Slowly, the hydrogen-burning shell advances outwards, and the CO core (and the surroundinghelium) contracts and heats up, until helium stars to burn at the bottom of the helium shell. Thisdrives an intershell convective zone and suppresses the hydrogen-burning shell. The helium-burningshell then advances outwards, and the convective envelope penetrates inwards. Eventually, thehelium burning shell peters out, and the hydrogen-burning shell regains it strength. But now thehydrogen-burning is slightly higher up in the star, and the CO core has grown.

Thermal pulses are important because they potentially mix processed material to the surface.This happens because the intershell convective zone mixes the helium region to a point from whichthe inward penetration of the envelope convection zone can bring it to the surface.

Thermal pulses are also difficult to model, so codes might not agree on how many pulses

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Figure 2.8: Plot of streamlines (i.e. the location of a given shell of mass) from a simulation of asimple stellar envelope driven by sinusoidal oscillations at the bottom. Although not realistic, thisshows how material might gradually flow from the surface of a thermally-pulsing star in denseshells.

happen, or exactly how long they last. But, as in the helium flash, it seems relatively clear thatthermal pulses happen, much as described in the models.

2.2.5 Envelope expulsion and the white dwarf cooling track

The thermal pulses are thought to drive strong mass loss. Owing to the pulsations, these areexpected to appear as a series of shells of material, rather than a continuous loss like on the redgiant branch. These shells of material manifest themselves as the planetary nebula. Thus, at thecentre of each planetary nebula, there is presumably some “dying” star or its white dwarf remnant,gradually exposing itself to the void.

This core, once exposed, is a CO white dwarf. As its initially hot surface is exposed, the starrockets across to the hot part of the HR diagram, reaching surface temperatures around 105 K.However, the white dwarf gradually cools, roughly following a power law in age. Limited by theage of the Universe, most observed white dwarfs have temperatures like 104 K, but one (BPM37093) seems to have cooled enough that the core has partially crystallized.

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Figure 2.9: Hydrogen abundance as a fraction of mass for a 3 M� model, at various stages of corehydrogen burning. The core is convective, so material is mixed. The convective boundary slowlyretreats, leaving a sloped hydrogen profile at the end of central hydrogen burning.

2.3 More massive stars

Finally, let us consider a number of key differences that exist for stars more massive than the Sun.We will start with stars only slightly more evolved and proceed to ever higher masses.

2.3.1 Convective cores on the main sequence

In the Sun, we previously noted that the core is radiative. More massive stars (depending onmetallicity) tend to burn hydrogen through the CNO cycle, which has a steep temperature gradientand drives a convective core. Over the stars life, the core boundary tends to move inward, leaving agraded composition profile. Still, at the end of the core burning, some of the core is still convective,and is therefore depleted of hydrogen all at once. With no location immediately suitable forhydrogen burning, the whole star contracts slightly before the core boundary layer is hot enoughto burn hydrogen. This contraction appears as a small movement towards higher temperature inthe HR diagram, known as a blue hook.

2.3.2 Non-degenerate helium ignition

For stars up to about 2 M�, helium core degeneracy always sets in before helium burning begins.Since the properties of the degenerate core are largely insensitive to the burning shell and convectiveenvelope, this means that stars about to 2 M� generally ignite helium at a particular mass: about0.48 M�.

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Figure 2.10: Kippenhahn diagram for a 15 M� star. Bracket like shapes indicate convective zones;diagonally-striped regions show burning regions. Note that the x-axis (age) is broken at 11 Myrand 13.7 Myr.

In more massive stars, helium core degeneracy doesn’t set it before helium burning begins.Thus, these stars do not experience helium flashes, but instead star burning helium steadily inthe core. In fact, as the mass becomes larger and helium ignition happens earlier, we eventuallyfind stars that start burning helium in the Hertzsprung gap, before they even reach the red giantbranch!

2.3.3 Mass loss on the main sequence

As stars become very massive, mass loss becomes important even on the main sequence. In a15 M� star, the effect is still modest, with the star losing a few per cent of its total mass. At60 M�, however, mass loss is sufficient to strip about a quarter of the mass, allowing one to seelayers that were previously mixed with the hydrogen-burning core. As the evolution proceedsthrough helium burning, so much mass is lost that the hydrogen-burning shell is itself removed,exposing the helium-rich envelope below.

Such objects are what we see as Wolf-Rayet stars. These were first classified on the basisof strong, broad emission lines including He, C, N, O and heavier elements, but conspicuouslylacking in hydrogen lines. The emission lines are formed in the thick, fast wind being drivenoff the surface. They are very hot, with surface temperatures of order 105 K. There are severalsubclasses, depending on which lines are present with what strength, and this is thought to be aconsequence of mass loss removing ever deeper layers of the stars.

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Figure 2.11: Kippenhahn diagram for a 60 M� star. Bracket like shapes indicate convective zones;diagonally-striped regions show burning regions. Note that the x-axis (age) is broken at 3.5 Myrand 4.32 Myr. Note also that the total mass decreases significantly on the main sequence, and somuch so during helium burning that the entire hydrogen envelope is removed by 4 Myr.

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Figure 2.12: Schematic representation of the layered structure of an evolved, massive star, not farfrom core collapse.

Figure 2.13: Plot of initial mass versus final mass for stars at solar metallicity, with colouredregions indicating the expected composition of each layer following core collapse. (Figure from apresentation by Marco Limongi.)

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2.3.4 Carbon burning and beyond

CO cores never get hot enough to burn carbon unless the stellar mass exceeds about 8 M�. Aswith helium burning, the lightest carbon-burners ignite carbon in a degenerate core. However, it’scurrently unclear whether this flash proceeds as a series of subflashes (as in the helium flash), or ifthe burning stars off centre and gradually proceeds to the centre, without halting between. Thislatter phenomenon is known as a carbon flame.

Any star that starts burning carbon will ultimately go through all the nuclear reactions up toiron. Per particle, iron is the most tightly bound nucleus, so one cannot extract energy by fusingor dividing it. The nuclear reactions proceed in a layer of shells, each proceeding reaction morequickly than the last. Ultimately, the iron core becomes too massive for even electron degeneracyto support it against gravity, and it collapses, marking the onset of a supernova, and ultimatelyleaving a neutron star or black hole remnant.

2.4 Parting thoughts

It is difficult to neatly summarize all of stellar evolution, but an important overall principle is:A star’s structure is determined by how the macrophysics responds to the microphysics.That is, a star takes whatever form because the opacities, nuclear reactions and equation of

state (among other things) are dictating how the fluid arranges itself, given the simple assumptionswe initially made.

Similarly, the different phases of evolution span many qualitatively distinct forms, but manyare neatly summarized by considering what is burning, and where? Thus, in the life of the Sun,we see a progression of

• core hydrogen burning on the main sequence,

• shell hydrogen burning on the red giant branch,

• core helium burning (with hydrogen shell burning) in the red clump,

• helium and hydrogen shell burning on the asymptotic giant branch, and finally

• a quiescent (non-burning) core remnant on a white dwarf cooling track.

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Bibliography

Bohm-Vitense, E. 1958, Zeit. Ast., 46, 108

Chandrasekhar, S. 1939, An Introduction to the Study of Stellar Structure, ed. Chandrasekhar, S.(Chicago Univ. Press, Chicago)

Chandrasekhar, S. 1957, An Introduction to the Study of Stellar Structure, ed. Chandrasekhar, S.(Dover Publications, New York)

Collins, G. W. 1989, The fundamentals of stellar astrophysics (W. H. Freeman and Co., New York)

Cox, J. P. & Giuli, R. T. 1968, Principles of Stellar Structure (Gordon & Breach, New York)

Eddington, A. S. 1926, The Internal Constitution of the Stars (Cambridge University Press,Cambridge)

Kippenhahn, R. & Weigert, A. 1990, Stellar Structure and Evolution (Springer-Verlag, Berlin)

Kippenhahn, R., Weigert, A., & Weiss, A. 2012, Stellar Structure and Evolution, Astronomy andAstrophysics Library (Springer-Verlag, Berlin)

Russell, H. D. 1914, Nature, 93, 252

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