Stellar Structure and Evolution

Stellar structure and evolution

Warrick Ball

November 3 & 4, 2014 Contents

1 Stellar modelling 4 1.1 Basic assumptions ...... 4 1.1.1 Mass conservation ...... 6 1.1.2 Hydrostatic equilibrium ...... 6 1.1.3 Energy generation (and conservation) ...... 7 1.1.4 Energy transport ...... 7 1.2 Composition equations ...... 10 1.3 Matter equations ...... 11 1.3.1 Opacity ...... 12 1.3.2 Nuclear reactions ...... 13 1.3.3 Neutrino loss rates ...... 15 1.3.4 Equation of state ...... 16 1.4 Boundary conditions ...... 18 1.4.1 Centre ...... 18 1.4.2 Surface ...... 18 1.4.3 Composition ...... 18 1.4.4 Initial models ...... 19

2 Stellar evolution 20 2.1 Characterizing stellar evolution ...... 20 2.1.1 The Hertzsprung–Russell diagram ...... 20 2.1.2 Colour–magnitude diagrams ...... 22 2.1.3 The ρ-T diagram ...... 23 2.1.4 Kippenhahn diagrams ...... 23 2.2 Evolution of a Sun-like star ...... 25 2.2.1 The main sequence ...... 25 2.2.2 The red giant branch ...... 25 2.2.3 The helium ﬂash and core helium burning ...... 27 2.2.4 Thermal pulses and the asymptotic giant branch ...... 28 2.2.5 Envelope expulsion and the white dwarf cooling track ...... 29 2.3 More massive stars ...... 30 2.3.1 Convective cores on the main sequence ...... 30 2.3.2 Non-degenerate helium ignition ...... 30

1 2.3.3 Mass loss on the main sequence ...... 31 2.3.4 Carbon burning and beyond ...... 34 2.4 Parting thoughts ...... 34

2 Prologue

About the course

These lecture notes are from the two lectures I gave at the DWIH Winter School in 2014. They were forked from the notes I had so far prepared for the course on stellar modelling taught at Göttingenin August 2014. There are two lectures, distinct in content and character. The first lecture aims to introduce the equations of stellar structure and evolution, with a keen focus on the principles and concepts behind the models, rather than detailed derivations to study the theory in depth. The second lecture aims to simply outline how we study stars and what the models say about the evolutionary states of stars in different phases. I tend to use the Sun as a fiducial case, both since it’s a special target and because it happens to undergo many distinct phases of evolution. I then identify things that change in stars of increasing mass.

3 Chapter 1

Stellar modelling

Over the course of the 20th century, a basic formulation of stellar structure and evolution emerged that 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 is well-studied, and serves as the starting point for studying the stars. If you read a paper and the writer doesn’t specify precisely how their models are calculated, they probably use the standard picture. Conversely, authors should be specific about non-standard properties (or properties for which there is no standard!). We will now consider how this standard model is constructed, and only allude to where “non-standard” modelling can enter. There is a long line of fine textbooks on stellar structure and evolution, arguably starting with the theoretical works of Eddington (1926) and Chandrasekhar (1939, 1957). The post-war era saw the rise of computers, and textbooks shifted in flavour towards formulating the equations, solving them 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 of Kippenhahn & 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 Jørgen Christensen-Dalsgaard1 and Onno Pols 2. I’m a particular fan of Onno Pols’ notes, which are near textbook quality and have excellent figures, many of which I have borrowed. If you do use these notes 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 several assumptions. We assume that a star is

1. a ﬂuid;

2. spherically symmetric;

1http://astro.phys.au.dk/~jcd/evolnotes/LN_stellar_structure.pdf 2http://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/

4 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 done explicitly in the textbook by Collins (1989)3.

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

3. Specifically, this assumption is intended to mean that gravity is the only external body force that 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 of the star where large scale flows exist, but there are always equal upward and downward flows, 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 infinitesimal neighbourhood of that shell. In addition, we assume that particle velocities are distributed like Maxwell distributions and that the radiation field is given by a blackbody at the same local 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 species decouple, above the photosphere.

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

3. Again, in truth we know that the Sun has a magnetic ﬁeld and we have strong evidence that 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/

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

5. Local thermodynamic equilibrium applies well throughout most of most stellar models but breaks down near the surface, where the surface boundary conditions of the stellar model must be applied. The radiation ﬁeld diverges from the Planck function, owing to the development 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 = 4πρ(r)r2 (1.2) dr Since the density is always positive, the function m(r) is monotonic. We can make m the dependent variable (instead of r) and instead write the equation as

dr 1 = (1.3) dm 4πρ(r)r2

The formulation with r or m as the dependent variables are known as using Eulerian or Lagrangian co-ordinates. Early codes tended to use m as the dependent co-ordinate because the surface boundary is then ﬁxed at the total mass of the star, M∗. Were we to use r as the dependent variable, 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 function of 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

∂~v ρ + ~v · ∇~ ~v = −∇~ p + f.~ (1.4) ∂t

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

6 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 is spherically symmetric, so that we can write dp Gmρ = −ρg = − (1.6) dr r2 This equation—hydrostatic equilibrium under spherical symmetry—is just a statement of the balance 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 to describe the mechanical (rather than thermal) structure of the star. If the pressure P is purely a function 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 useful result that relates total energies in the star. We shall not derive it here, but I recommend looking it up in any standard reference. In fact, Collins has written a book on just this result (Collins 1989?).

1.1.3 Energy generation (and conservation) The energy content of the star at a given layer is modiﬁed only by creation or destruction of energy, so we write ∂L ∂s = = − − T (1.7) ∂m nuc ν ∂t which simply represents the conservation of energy. I have expanded the total energy generation rate into three components:

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

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

• the gravitational energy generation rate, which is equal to dQ = T dS/dt by deﬁnition and describes 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 them in sequence.

7 Radiation The most natural mechanism is radiation: the core of the star is hot, dense and producing energy in nuclear reactions. This energy can basically shine out. We treat this process in the diffusion limit, because the mean free path of a photon is small and is scattered, absorbed, and re-radiated many 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 1 c Dν = (1.9) 3 κνρ where 4σ a = B (1.10) c is the radiation. The energy density of the radiation field is 4π 4π 2hν3 1 uν(T ) = Bν(T ) = 2 hν (1.11) c c c e kT − 1 We will also use the facts that Z 2π4 k4T 4 σ B (T )dν = = T 4 (1.12) ν 15 h3c2 π and therefore Z dB 8π4 k4T 3 4σ dν = = T 3 (1.13) dT 15 h3c2 π because ν and T are independent variables. Now, we can integrate Fick’s law over frequency and apply the assumption of spherical symmetry. L Z Z d = F dν = − D u dν (1.14) 4πr2 ν dr ν Z c 4π dB dT = − ν dν (1.15) 3κνρ c dT dr 4π dT Z 1 dB = − ν dν (1.16) 3ρ dr κν dT 4π dT 1 4σ = − T 3 (1.17) 3ρ dr κR π 16σT 3 dT = − (1.18) 3ρκR dr

8 where we have deﬁned the Rosseland mean opacity by

R 1 dB 1 dν = κν dT (1.19) κ R dB R dT dν In stellar modelling, we traditionally deﬁne a dimensionless temperature gradient d log T ∇ = (1.20) d log P for a given process, and write the temperature gradient as ∂T GmρP = −∇ (1.21) ∂r T r2 For the case of radiation, we thus have d log T P dT dr ∇ = = (1.22) rad d log P T dr dP P L r2 = 3ρκ 16σT 4 (1.23) T 4πr2 R Gmρ 3κLP = (1.24) 64πσGmT 4 .

Convection Consider a blob4 of material somewhere in a chemically homogeneous region of our star. If we perturb this blob upward adiabatically (so no heat is exchanged with the surroundings), then the blob will expand as it rises, because it’s hotter than its surroundings. However, because of the 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 its perturbation 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 negatively buoyant, depending on the density gradient. This situation is unstable, and we call this convective instability. I omit here a detailed derivation of the stability criterion, but it can be found in any standard textbook or set of lecture notes. The criterion for instability turns out to be

∇rad > ∇ad (1.25) or, equivalently, ∂s < 0 (1.26) ∂r 4The professionals might call this a parcel of material, but it’s really just a blob.

9 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 Böhm-Vitense1958). In MLT, it’s presumed that the rising and falling blobs move some characteristic distance (the mixing length) before dispersing into their surroundings, mixing heat and material. The mixing length `MLT is usually taken to be proportional to the pressure scale height HP ≡ P/(dP/dr). The constant of proportionality is typically denoted α and is determined by calibrating stellar models to the Sun. MLT is quite basic, and has many flaws. For a start, there are several slightly different versions of the theory, with corresponding slightly different predictions of the convective flux, depending on things like the shapes of the blobs, their level of opacity, and so on. What’s more, the theory is completely local, and therefore allows situations where the convection acceleration goes to zero at the convective boundary, but the convective velocity might be non-zero. Thus, the blobs potentially penetrate into convectively stable regions. We call this process overshooting, and treat it with somewhat ad hoc parametrizations. This begs the question: why has MLT persisted so long, and how does it work at all? MLT is relatively easy to implement, and it turns out that MLT is usually efficient, and drives the temperature gradient very close to its adiabatic value. Thus, one could compute fairly accurate stellar models by simpling assuming that convection zones are perfectly mixed and adiabatically stratified. MLT does fail, however, when convection is not efficient, as is the case near the surfaces of 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 stress model).

Conduction At very high densities, heat is transferred by the jostling of particles against one another. This is 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. This conductive 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 X X = r − r (1.27) ∂t ji ik j k where rij is the rate at which species i is transformed into species j. The number density is deﬁned by

ρXi ni = (1.28) mi

10 where mi is the mass of a particle of species i. so ! ∂Xi mi X X = r − r (1.29) ∂t ρ ji ik j k

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

∂X ∂ ∂X i = D i (1.30) ∂t ∂m conv ∂m 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 big enough 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: lighter material diffuses towards the surface of the star; heavier material towards the centre. These are also modelled as diffusion processes. But remember that the diffusion coefficient can be different for 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, there are several functions that we have not yet described. They are

• the opacity κ,

• the nuclear energy generation rate nuc,

• the neutrino energy loss rate ν,

• 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 precompiled data, usually in the form of tables. For each, there are many sources of these tables from various research groups around the world who make it their business to perform the detailed calculations of these quantities. Stellar evolution codes then interpolate in tables of data to determine the values (and often their derivatives) at the relevant points in the star. The ﬁrst four are generally regarded as functions of density, temperature and composition. The equation of state is a slightly special case. At its simplest, it can be thought of as a function that gives the pressure as a function of density, temperature and composition, too, but in truth the

11 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 and temperature. 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 heat capacities cP , and so on. Some analytic approximations to the equation of state exist, but most rely on precomputed functions to approximate some terms. In the past, the analytic approximations were popular because they are much faster than interpolating in tables, but this has become less important as computers have become faster and able to manage much larger tables. The most popular analytic approximations (e.g. EFF and CEFF) are known to be less accurate than the latest tabulated data.

1.3.1 Opacity Recall that the opacity was deﬁned in the diﬀusion approximation for radiative heat transport. It is computed by taking detailed account of several processes that absorb (and therefore re-emit) or scatter light.

• Electron scattering. At low energies, this is simply Thomson scattering, and the opacity is given by κ = σT ne/ρ, where σT is the Thomson cross-section and ne is the free electron number 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 becomes Compton 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 relevant energy levels are, for material in which the ionization state also depends on temperature.

12 • Bound-free scattering or photo-ionization. This is the complete liberation of an electron from 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 decelerating in 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 the free electron population enough to affect the bound- and free-free opacities. This effect dominates 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. They have 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 points are 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 immediately after. Note also that when a chemical species is written, e.g. 12C, we mean only its nucleus. For this reason, 1H and p are interchangeable, as are α and 4He. I have tried to consistently use one, but there’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 is always of the form 41H → 4He. This net reaction occurs through two primary channels: the pp chains, in which protons and other light elements are combined; and the CNO cycle, in which C, N, O and possibly F act as catalysts onto which protons are added until an alpha particle is emitted. (See Figures 1.3 & 1.4.)

13 Figure 1.2: Binding energy per nucleon, as a function of atomic mass number. Note the peak at iron, 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 all directed upward, indicating alpha emission.

14 Helium burning There is no stable state of 8Be, so we cannot straightforwardly fuse two helium nuclei together on the way to heavier products. But recall that by “unstable”, what we really mean is that the fusion product (some excited state of 8Be) has a very short lifetime, but at suﬃcient density and temperature, it is time enough for a third helium nucleus (alpha particle) to join the excited beryllium nucleus, forming 12C. Because of the need for three helium nuclei to combine more or less 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 stellar modelling, the laboratory data on 12C did not indicate that there was an appropriate excited state via which the triple alpha process could occur. As a result, the reaction rates were very low, and the stellar models wouldn’t form carbon. Fred Hoyle, who contributed a phenomenal amount of work in the area of stellar modelling and stellar nucleosynthesis5 argued that such a state must exist, else carbon would simply never have been created. On closer inspection, calculations indicated that indeed, the relevant state exists, and it sometimes therefore known as the Hoyle state. Because the triple alpha process already requires high temperatures and densities to overcome the 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 higher masses (and thus core temperatures and densities), neon. So, for example, one never encounters a pure 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 interacting with 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 independent 8 processes that can take place at high temperatures (& 3 × 10 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 mostly annihilate 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 a neutrino-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 his other, more controversial, opinions, and his abrasive nature. The 1983 Nobel Prize went to two other scientists who had worked on stellar structure and evolution: Subramanyan Chandrasekhar and Willie Fowler.

15 The second process is the Urca process, which produces neutrinos and antineutrinos from the equilibrium of forward and inverse beta decay in some material. The general form of the forward backward 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 are lost.

1.3.4 Equation of state Computing the equation of state accurately, quickly and in such a way that preserves fundamental thermodynamic 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 chemical species i that behaves like an ideal gas:

ρXikBT Pi = nikBT = (1.38) µimp

where ni is particle number density, kB is Boltzmann’s constant and mp is the mass of a proton. This deﬁnes 1 n m = i p (1.39) µi ρXi

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

Chemical species Proton masses Free particles µ Neutral hydrogen 1 1 1 Ionized hydrogen 1 2 2 Neutral helium 4 1 4 Singly-ionized helium 4 2 2 Doubly-ionized helium 4 3 4/3 Metal, atomic number Z about 2ZZ + 1 about 2

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

X ρkT X Xi ρkT P = P = ≡ (1.40) g i m µ µm i p i i p

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

P which deﬁnes the (harmonic) mean molecular weight: 1/µ = i Xi/µi. Note that the changing mean molecular weight of the core (owing to nuclear reactions) is the main driver of stellar evolution on the main sequence. In the Sun, the mean molecular weight goes roughly from 0.6 to 1.34. So at constant pressure, the central density would nearly double over the main sequence!

• Radiation pressure. Remember that we have assumed an isotropic radiation ﬁeld everywhere in the star, in equilibrium with the gas. The radiation ﬁeld contributes to the pressure it’s own radiation pressure equal to aT 4/3, where a is the radiation constant. 3 Inside a star, it turns out that Pg/Pr ≈ ρ/T is (very!) roughly constant, which one can use to construct a basic stellar model. In fact, opacity tables are actually tabulated in the variable R ∝ ρ/T 3 to reduce some of the interpolation error and decrease the size of the tables necessary to cover stellar interiors.

• Degeneracy. When electrons are packed very close together, their allowed quantum states begin to overlap. That is, instead of each atom having its own quantum levels, which must respect the Pauli exclusion principle, the atoms begin to share states, decreasing the number of quantum state available overall. This forces the electrons to occupy higher energy (or higher 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 ﬁeld is 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

17 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 slightly reduces the radiation pressure.

1.4 Boundary conditions

The solution of a set of diﬀerential equations is only possible given the appropriate number of boundary conditions, which we describe here. In boundary-valued problems, these can be speciﬁed at 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 requirements that m, r and L are all zero. (This gives us two boundary conditions.) In practice, the use of r = 0 cause problems because of terms that go like 1/r. This can be overcome either by carefully choosing diﬀerent co-ordinates or by setting the interior boundary condition to be the average over the space inside the innermost meshpoint. If you were inclined to model a star outside the core, then one could also specify ﬁnite values m, r and L at the centre. This could be done to model, for example, the behaviour of material being accreted onto a neutron star.

1.4.2 Surface The derivation of the surface boundary conditions requires a more detailed analysis of radiative transport, since the photons begin to escape from the star and into the vacuum of space. A standard 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 negligible mass 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 ∂X i = 0 at r = 0,R (1.44) ∂r ∗ for each species.

18 1.4.4 Initial models We also need an initial condition because of the existence of time derivatives. The necessary initial conditions can be inferred from the time derivatives in the stellar model. First, each of the composition equations involves a time derivative, so we need to known the initial distribution of composition throughout the star (usually taken as homogeneous). Second, there is the gravitational energy generation term, which basically means that we must specify the energetic state of the model. 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 expected to be fully convective and well-approximated by n = 3/2 polytropes, constant entropy solutions, or other simple models. Thus, an initial model can be estimated by rescaling such a model, and allowing it to ﬁrst relax to a true solution of the equations before allowing evolution to proceed.

• Zero-age main-sequence models. If we suppose that the composition in a star is homogeneous and that the thermal state is such that there is no gravitational contraction, then the solution is unique, and known as a zero-age main-sequence (ZAMS) model. Some authors specify that 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 the evolution. Note that a real star never actually reaches such a ZAMS phase because nuclear reactions begin gently at the end of the PMS contraction. Even so, this phase should be relatively short and inconsequential, so it is often ignored. If, however, you are concerned with very accurate 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 for a calculation, even if one is manipulating the equations to suppress or enhance various processes. The manipulation of the equations to rapidly create a suitable model (realistic or not!) is sometimes known as stellar engineering.

19 Chapter 2

Stellar evolution

This lecture is crudely divided into two parts. First, we review some diagnostics of stellar evolution. These are various plots (some observationally motivated) that we use to determine what’s happening to a star as it evolves. Some are familiar, some not, but I’ll go through them all for 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 many distinct phases of evolution that are worth discussing. It’s a useful reference point that we then extend to higher mass (and other conditions) to explore what happens diﬀerently in the lives of those 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 magnitude against spectral type. The modern version tends to show absolute magnitude against colour, or luminosity 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 to better than 5%, we identify the following notable features. • The main sequence is obvious, going roughly from faint red to bright blue. Most of the stars in the sample are found along this strip. • Connected near the Sun’s position is the red giant branch, along which stars have similar spectral types (G, K, sometimes M) but become very bright. • Displaced slightly towards the blue near the middle of the red giant branch is a clear overdensity of stars, known as the red clump. • For mid-temperature stars, there is a region between that main sequence and red giant branch that has few stars. This is known as the Hertzsprung gap. It appears because stars

20 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 branch or red clump. Right: A modern HR diagram, compiled using stars from the Hipparcos sample for which distances are precise to better than 5%. Several regions of interest are indicated.

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

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

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

2.1.2 Colour–magnitude diagrams The Hertzsprung–Russell diagram requires absolute magnitudes (or luminosities), which themselves require knowledge of the distances to the stars. However, if we look at a single cluster (or other mutually proximate group of stars), then the distances will be roughly the same, so similar trends should emerge. In the case of a star cluster, we also expect that the stars have the same age and composition. Such plots for individual clusters are colour-magnitude diagrams, and show some features 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 onto the 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 have yet to do so.

• We again see a clear red giant branch.

22 • We also ﬁnd a number of stars that have crossed back into the blue, forming a nearly horizontal 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 brighter than the main sequence turnoﬀ. These are the blue stragglers. Although there is reasonable concensus that these stars have interacted with another star and mixed hydrogen back into the 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. The value of both comes from our knowing in roughly what density and temperature regimes certain processes (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 straightforward to identify a degenerate core, or multiple burning shells. Second, we can plot the central density and temperature over time, which roughly tells us through what phases of evolution the star goes. We will see immediate what material was burned at the core, and if it experienced degenerate burning. Note that these plots won’t tell us directly about 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 qualitatively distinct parts of the star evolve. The x-axis is usually the age of the star (but could be anything that defines a sequence of models: Fig. 2.3 uses mass) and the y-axis the mass co-ordinate. Then one plots the mass co-ordinates of certain boundaries, usually the boundaries of the convective zones and contours of the nuclear energy generation rate. Thus, a Kippenhahn diagram shows quite neatly where a star is convective, where it is burning nuclear fuel, and how this changes over time. This is a bit confusing at first, so let us walk through how one might construct a Kippenhahn diagram, just for the convective boundary at first. Let us suppose we have a sequence of stellar models from an evolutionary track. Each model in the sequence has some age, and we can look in that model to find the convective boundaries. (We could find where ∇ − ∇ad = 0.) We would then go to our diagram and, for each convective boundary we find, plot a point with the age of the star and the mass co-ordinate of the boundary. We then move on to the next model in the track and do the same thing, and proceed over the whole sequence. Joining the points will show how the boundaries evolve. Kippenhahn diagrams can be a bit tricky to plot and they can also get a bit complicated. But they are similarly very rich, and one can quickly identify several events in a star’s life based on the history as shown in the Kippenhahn diagram.

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

24 Figure 2.4: Kippenhahn diagram for the evolution of a 1 M star. Note that the x-axis (age) is broken 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 and then 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 ratios between the energy from the chains ppI:ppII:ppIII is about 85:15:0.02. The temperature gradient enforced 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’s current 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 continues to burn in a shell around the helium core. The shell gradually becomes thinner but the whole star brighter, and it ascends the red giant branch, whose temperature is roughly set by convection and H− opacity in the atmosphere. At the same time, the helium core grows and become increasingly degenerate.

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

The convective envelope also penetrates inwards, to some maximum depth. Along the way, it reaches into material that was affected by nuclear reactions on the main sequence, and is therefore depleted in H, enriched in He, and has slightly modified metal abundances. The convection zone mixes these to the surface where they can be seen. The event of the surface abundances changing like this is known as dredge up. After the convective envelope retreats, there is a small composition jump at the depth of its maximum penetration. At some point, the hydrogen-burning shell cross this layer, and the stellar structure abruptly changes, though only slightly. Even so, this is enough to cause a small downward 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 number of 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. As the star becomes larger and brighter, the outermost layers experience an ever weaker gravitational field, and an ever stronger outward radiative force. Thus, material is driven off, enough to potentially 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 one can 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 a strong burning shell with a composition gradient, but never becomes a giant. This is known as the red giant problem (or very occasionally erythrogigantism) and, though unanswered, few in the scientific community nowadays consider it a worthwhile problem...

26 Figure 2.6: Detailed plot of stellar evolution during the helium flash. The shaded regions show where 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 against the axis on the right. Finally, the remaining lines show the radii of the surface (dot-dashed) and outermost helium-burning layer (dashed), to be read against the extra axis to the left. Note how the 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 slightly cooler toward the centre, where the cooling is most efficient. Eventually, some off-centre layer of the star is hot enough to ignite helium. At first, the commencement of nuclear reactions heats the neighbouring layers, so they too start helium burning. As a result, a small region of the 9 core suddenly starts burning enough helium to produce about 10 L : the brightness of a small galaxy! 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 continues until the centre itself ignites, and the core begins to burn helium stably. The change in structure weakens 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 quickly and with such extreme properties that most codes balk at the problem and crash. So this picture is simply our best model at the moment. While the broad trend of ignition starting suddenly and off centre is probably right, precisely how the burning shifts to the core may yet change as our models and computers improve. After the flash, the star burns helium stably in the core. The triple-alpha process causes

27 Figure 2.7: Schematic Kippenhahn diagram of two thermal pulses.. as super-radiative temperature gradient, so the core is convective during helium burning. The hydrogen-burning shell is not extinguished, but proceeds gently at the same time as core helium burning. In a star like the Sun, this phases takes about 1 Gyr. The star will appear as part of the red 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 two burning shells: helium and hydrogen. It turns out that this situation becomes unstable, and the star undergoes thermal pulses as it ascends something like the red giant branch (called the asymptotic giant branch). The sequence of events in a thermal pulse is as follows. (See Fig. 2.7.) Suppose we start the cycle 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 surrounding helium) contracts and heats up, until helium stars to burn at the bottom of the helium shell. This drives an intershell convective zone and suppresses the hydrogen-burning shell. The helium-burning shell then advances outwards, and the convective envelope penetrates inwards. Eventually, the helium burning shell peters out, and the hydrogen-burning shell regains it strength. But now the hydrogen-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 which the 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

28 Figure 2.8: Plot of streamlines (i.e. the location of a given shell of mass) from a simulation of a simple stellar envelope driven by sinusoidal oscillations at the bottom. Although not realistic, this shows how material might gradually ﬂow from the surface of a thermally-pulsing star in dense shells. happen, or exactly how long they last. But, as in the helium ﬂash, it seems relatively clear that thermal 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 are expected to appear as a series of shells of material, rather than a continuous loss like on the red giant branch. These shells of material manifest themselves as the planetary nebula. Thus, at the centre 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 star rockets 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 the age of the Universe, most observed white dwarfs have temperatures like 104 K, but one (BPM 37093) seems to have cooled enough that the core has partially crystallized.

29 Figure 2.9: Hydrogen abundance as a fraction of mass for a 3 M model, at various stages of core hydrogen burning. The core is convective, so material is mixed. The convective boundary slowly retreats, 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 on metallicity) tend to burn hydrogen through the CNO cycle, which has a steep temperature gradient and drives a convective core. Over the stars life, the core boundary tends to move inward, leaving a graded 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 for hydrogen burning, the whole star contracts slightly before the core boundary layer is hot enough to burn hydrogen. This contraction appears as a small movement towards higher temperature in the 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 convective envelope, this means that stars about to 2 M generally ignite helium at a particular mass: about 0.48 M .

30 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 Myr and 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 in the core. In fact, as the mass becomes larger and helium ignition happens earlier, we eventually ﬁnd stars that start burning helium in the Hertzsprung gap, before they even reach the red giant branch!

2.3.3 Mass loss on the main sequence As stars become very massive, mass loss becomes important even on the main sequence. In a 15 M star, the effect is still modest, with the star losing a few per cent of its total mass. At 60 M , however, mass loss is sufficient to strip about a quarter of the mass, allowing one to see layers that were previously mixed with the hydrogen-burning core. As the evolution proceeds through 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 basis of strong, broad emission lines including He, C, N, O and heavier elements, but conspicuously lacking in hydrogen lines. The emission lines are formed in the thick, fast wind being driven off the surface. They are very hot, with surface temperatures of order 105 K. There are several subclasses, depending on which lines are present with what strength, and this is thought to be a consequence of mass loss removing ever deeper layers of the stars.

31 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 Myr and 4.32 Myr. Note also that the total mass decreases signiﬁcantly on the main sequence, and so much so during helium burning that the entire hydrogen envelope is removed by 4 Myr.

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

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

33 2.3.4 Carbon burning and beyond

CO cores never get hot enough to burn carbon unless the stellar mass exceeds about 8 M . As with helium burning, the lightest carbon-burners ignite carbon in a degenerate core. However, it’s currently unclear whether this flash proceeds as a series of subflashes (as in the helium flash), or if the burning stars off centre and gradually proceeds to the centre, without halting between. This latter phenomenon is known as a carbon flame. Any star that starts burning carbon will ultimately go through all the nuclear reactions up to iron. Per particle, iron is the most tightly bound nucleus, so one cannot extract energy by fusing or dividing it. The nuclear reactions proceed in a layer of shells, each proceeding reaction more quickly than the last. Ultimately, the iron core becomes too massive for even electron degeneracy to support it against gravity, and it collapses, marking the onset of a supernova, and ultimately leaving 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 assumptions we initially made. Similarly, the different phases of evolution span many qualitatively distinct forms, but many are 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.

34 Bibliography

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