# AST 1410 – Stars (2018)

## 1 Syllabus

Lectures
M12 and W11-1 (i.e., 2 hr, with coffee break), AB 113
Lecturer
Marten van Kerkwijk, MP 1203B, 416-946-7288, mhvk@astro (.utoronto.ca)
Office hours
Drop by my office, or by appointment
Web page
http://www.astro.utoronto.ca/~mhvk/AST1410/
notes
pdf

Stellar astrophyiscs – the success story of 20th century astronomy – requires a synthesis of most of basic physics (thermodynamics, quantum mechanics, and nuclear physics). It underlies nearly all of astronomy, from reionisation to galaxy evolution, from interstellar matter to planets, and from supernovae and planetary nebulae to white dwarfs, neutron stars, and black holes.

In this course, we will review these successes (roughly first four weeks) and then discuss current topics and remaining puzzles (in four two-week series, detailed content depending on interest).

### Underlying Physics

Master equations
equilibria; timescales; mass-radius and mass-luminosity relations; Hertzbrung-Russell diagram, common threads in stellar evolution, features in stellar evolution.
Equation of state
fermions and bosons; pressure and energy density; ideal gas; (completely and partially) degenerate gas; radiation pressure; Boltzman distribution; Saha equation.
Heat loss
radiative diffusion; conduction; opacity sources; Schwarzschild and Ledoux criteria; mixing length theory; convective flux; stellar context for convection; semi-convection.
Energy production
nuclear binding energy; Coulomb barrier; reaction channels (PP, CNO, He and beyond, D/Li burning, s-/r-/p-processes); rates and neutrinos.

### Stellar Evolution Themes

Low-mass stars
Hayashi track; Li burning; (former) solar neutrino problem; pressure ionization and thermal ionization; convection zone advance; rotational evolution; shell burning; core-mass radius, luminosity relations; helium core flash; thermal flashes; RGB/AGB winds; production of intermediate-mass elements; white dwarfs.
High-mass stars
CNO burning and core convection; Eddington luminosity and formation/mass loss of high mass stars; nucleosynthetic yield of high mass stars; rotational evolution; feedback to the galaxy; electron-capture, core collapse and pair instability SNe; Pop III stars; neutron stars, black holes.
Binary evolution
frequency of binarity; tidal synchronization and circularization; Roche lobe overflow; conservative and non-conservative mass transfer; common-envelope evolution; mergers; blue and red stragglers.
Nucleosynthesis
production of the elements; explosive nucleosynthesis; r and s process; core-collapse supernovae; thermonuclear supernovae; various dredge-ups; thermal pulses.

### Course texts

The main book we will use is Stellar Structure and Evolution (KW; Kippenhahn & Weigert for first, plus Weiss for second edition; Springer-Verlag, 1990, 2012). Especially for those who did not take undergraduate astrophysics, I strongly recommend An Introduction to Modern Astrophysics, by Carroll & Ostlie (2nd edition; Cambridge University Press, 2017). This book introduces more empirical knowledge (and jargon) assumed known in KW, and is used for the UofT undergraduate courses AST 221 and AST 320 (I'll try to refer to relevant notes from those classes; have a look at the parts on stars from the AST 320 notes and mini problem sets).

### Evaluation

• Two problem sets (20% total), due two weeks after posting.
• Presentations (20% total), one short ones explaning a specific concept (8 min., plus 7 min. discussion; see list of topics) and one long one on a more advanced topic (list of topics; 20 min., plus 10 min. discussion).
• Computer project (20%), using mesa to investigate a specific part of stellar evolution.
• Final exam (40%; likely oral, depending on number of students).

## 2 Master equations

Equilibria; timescales; mass-radius and mass-luminosity relations; Hertzbrung-Russell diagram, common threads in stellar evolution, features in stellar evolution.

### Mon 7 Jan

Textbook
KW 1–2.3, 3.1 [and AST320 notes 1]
Exercises
• What mass-radius relation would you expect for a set of stars with the same central temperature? Would more massive stars have higher or lower central density and pressure?
• AST320 mini problem sets: I, II
• General qualifier questions: Galactic:13, Stars:10,11, Physics:8 (first part)
• Derive the Virial Theorem (assuming ideal gas), following hint below.
• What is a star?
• me: ball of gas
• wiki:Star: A star is a massive, luminous sphere of plasma held together by gravity.
• Implication: P high → T high → loose heat

Unless high P without high T → degeneracy (brown dwarf, degenerate helium core, white dwarf).

• Basics of evolution: contraction, heat up, fusion

When does it stop?

• Where do we see stars

Check wiki:HRD. Why are there limits?

• left: Tc very high → fusion (think of contraction gone too far)
• top: L too high, matter blown away (LEdd)
• right: T profile too steep → convection
• Virial theorem: Eint=-½ Egrav; Etot=Eint+Egrav=½ Egrav
• Roughly, one has Egrav ≈ GM2/R, Eint ≈ N⟨eint⟩ = (M/μmH)3/2 kT, hence kT ≈ GMμmH/R.
• To derive it formally, multiply HE by r on both sides and integrate over sphere; use that for ideal gas U = 3/2 nkT = 3/2 P.

### Wed 9 Jan

Textbook
KW 2.4, 3.3, 5.1.1, 5.1.2, 20.2.2 [AST 320 notes 2]
Exercises
• AST320 mini problem sets: III, VI
• Calculate dynamical time for the Sun (⟨ρ⟩≈1 g/cm3), a neutron star (∼1014 g/cm3), and the Universe as a whole (∼1 m-3≈10-30 g/cm3).
• Structure equations: MC, HE, TE, EB
• mass conservation: ∂m/∂r = 4πr2ρ
• Hydrostatic equilibrium: ∂P/∂r = -gρ = -Gmρ/r2
• Thermal equilibrium: ∂T/∂r = -(3/4ac)(κρ/T3)(ℓ/4πr2); easiest to derive from general diffusion equation: $$j = -\frac13 vl_{mfp} \nabla{}n$$; for radiation, v=c, lmfp = 1/σn = 1/κρ, n = aT4; hence, Frad = ℓ/4πr2 = -c/3κρ ∂(aT4)/∂r, from which one can solve for ∂T/∂r.
• Energy balance: ∂ℓ/∂r = 4πr2ρε
• To solve, needs
• equation of state P(ρ,T,abundances)
• opacity κ(ρ,T,abundances)
• energy generation rate ε(ρ,T,abundances)
• If T (dependence) known, can solve MC, HE separately.
• Independent variable r or m (or P, or …)
• Homology

If two stars have the same structure, i.e., m'(r')/M'=m(r)/M for all r'/R' = r/R, then:

• MC: ρ'(r')/ρ(r) = (M'/M)(R'/R)-3
• HE: P'(r')/P(r) = (M'/M)2(R'/R)-4
• Luminosity of a star
• Simple estimates/scalings:
• MC: ρ ≈ M/R3
• HE: P ≈ GM2/R4
• TE: L ≈ acRT4/κρ
• Combining with ideal gas law P=(ρ/μmH)kT:
• MC+HE: kT ≈ GMμmH/R
• MC+HE+TE: L ≈ acG4mH4 μ4M3

Note: what is radiated does not depend on how energy is generated; star has to provide the energy, whether by contraction or fusion

• Real M-R and M-L relations
• ε steep function of T → M/R nearly constant. Reality R ∝ M0.8.
• κ not constant (higher at low T, but convection moderates the effect) → L ∝ M4
• contribution of degeneracy → steeper at low M
• contribution of radiation pressure → flatter at high M.
• Substantial difference: inert lump inside → shell source

Or with fusion: star inside a star''

• Timescales
• Dynamical: τdyn ≈ 1/(Gρ)1/2

What if not in HE?

Equation of motion: ρ∂v/∂t = ρ∂2r/∂t2 = -∇P + ρ∇Φ = -∂P/∂r -ρg

• Pressure drops away? τff ≈ (R/g)1/2 ≈ 1/(Gρ)1/2
• Gravity drops away? τexpl ≈ R(P/ρ)-1/2 ≈ R/cs ≈ 1/(Gρ)1/2
• Thermal: tth ≈ Eth/L (≈ GM2/RL for whole star)

If not in TE over some distance d: F = -(vlmfp/3)∇U ≈ (vlmfp/3)U/d (where vlmfp=c/κρ for diffusion by radiation).

Hence, timescale τadj ≈ Ud3/Fd2 ≈ 3Ud3/vlmfpUd ≈ d2/vlmfp ≈ (lmfp/v)(d/lmfp)2 (random walk: tstep Nsteps)

Timescale for radiative damping of pulsations? Higher order → smaller d → faster damping.

• Nuclear: tnuc ≈ Enuc/L

## 3 Equation of state

Fermions and bosons; pressure and energy density; ideal gas; (completely and partially) degenerate gas; radiation pressure; Boltzman distribution; Saha equation.

### Mon Jan 14

Textbook
25.3 (esp. 25.3.2), 19.1–19.4, 19.9 [AST 320 notes 4]
Exercises
• Write your own polytrope integrator (you'll need it for the first problem set; I suggest using python; if you are clueless, have a look at my simple integrator for an isothermal atmosphere)
• Use it to calculate the radius of a star with a solar mass and with central density and pressure like the Sun, for n=1.5 and 3.
• AST320 mini problem sets: V
• General qualifier questions: Physics:6
• Stability

Upon compression, ρ∝R-3. Thus, for an adiabatic perturbation, P ∝ ργ ∝ R-3γ (with γ a suitable average over the star). To keep in HE, P should increase as R-4 or faster, i.e., 3γ>4 or γ>4/3.

• Polytropes: P = Kρ^γ ≡ Kρ1+1/n

For K, γ, n ≡ 1/(γ-1) constant, can integrate HE+MC.

Examples:

• Constant density (incompressible fluid)
• Isothermal (part)
• Completely convective
• Degenerate (K fixed)

For given K, n, know ρ(r), P(r), Eg, etc.; see AST 320 notes \#4, esp. Table 4.1.

### Wed Jan 16

Textbook
Discussed in class: KW 13–16 (AST 320 notes 3); not discussed but to be read: KW 4, 25.3, 33.2
Exercises
• For classical particles, show that n(p)dp is a Maxwellian, and that one recovers the ideal gas law.
• For photons, μ=0. Show that Uνdν equals the Planck functions, and that its integral equals aT4 (note: ∫0^∞ dx x3/(exp(x)-1)=π4/15)
• AST320 mini problem sets: IV, VII
• EOS: Pressure integral: P = (1/3) ∫p npvppdp
• NR: vpp→p2/m=2ep → P=(2/3)U → Virial Theorem: Eg=-2Ei, Etot=(1/2)Eg
• ER: vpp→cp=ep → P=(1/3)U → Virial Theorem: Eg=-Ei, Etot=0
• generally, np=n(ep)g(4πp2/h3)dp, n(ep)=1/[exp((e-μ)/kT) ± 1]
+1:Fermions; -1: bosons; μ: chemical potential; g: number of internal states (such as spin)
(For a nice description of the meaning of The elusive chemical potential μ, see Baierlein 2001AmJPh..69..423B.)
• ignore ±1: classical particles → ideal gas law: P=(ρ/μmH)kT (μ here is mean molecular weight)
• photons: -1, μ=0 → get BB → P=(1/3)aT4
• electrons: +1: completely degenerate → fill up to pF = h(3n/4πg)1/3
• NRCD: P=K1(ρ/μemH)5/3, K1=(3/4πg)2/3(h2/5me)nil≈2.34×10-38 N m3
• ERCD: P=K2(ρ/μe)4/3 K2=(3/4πg)1/3(hc/4)≈2.45×10-26 N m2
• Complications: molecular/atomic/nuclear dissociation, pair formation
• Combinations
• Rough estimate everywhere (Paczynski 1983ApJ...267..315P:
• P = Pe+Pi+Prad = Pe+(ρ/μimH)kT +(1/3)aT4
• Pe = (P2e,ideal+P2e,cd)1/2, Pe,ideal=(ρ/μemH)kT
• Pe,cd = (P-2e,nrcd+P-2e,ercd)-1/2
• EOS from look-up table
• Another way to think about ionisation, etc. (not discussed in class)

(A bit different from KW 14)

Consider a fixed volume V at a fixed temperature T (or, equivalently, constant ρ and T). In thermal equilibrium, systems go to their most probable state, i.e., one maximizes entropy, $$S=k \log Z$$, where $$Z$$ is the partition function, a sum over all possible states i, weighted by $$\exp(-E_i/kT)$$. Usually, one can split contributions, e.g., for non-interacting photons, ions, and electrons, one has $$Z=Z_{\gamma}\times{}Z_{e}\times{}Z_{i}$$ (and thus $$S=k\sum\log Z$$).

In the volume, for one particle at some momentum $$p$$, the number of phase space elements available is $$(V/h^3)\times4\pi{}p^{2}dp$$, with a probability $$\exp(-\epsilon_p/kT)$$. The total number of phase space elements is thus $$\sim{}(V/h^3)p_{th}^3$$, where $$p_{th}$$ is some typical momentum associated with the temperature. Doing the integral gives the Maxwellian and $$p_{th}=\sqrt{2\pi{}mkT}$$. Maybe more insightful is follow Baierlein 2001AmJPh..69..423B and define a typical size, $$\lambda_{th}\equiv{}h/p_{th}$$, the thermal De Broglie'' wavelength. Then, the number of possible states is simply $$V/\lambda_{th}^3$$. For a set of N identical particles, the contribution to the partition function is thus $Z_N=\frac{[g(V/\lambda_{th}^3)\exp(-\epsilon/kT)]^N}{N!},$ where $$g$$ is the number of internal states, the factorial $$N!$$ ensures we do not overcount states where two particles are swapped, and $$\epsilon$$ is an energy cost beyond thermal kinetic energy there may be for having this particle.

Let's apply this to pair creation, assuming some mix of photons, ions, electrons and electron-positron pairs. Assuming a dilute plasma, their contributions to $$Z$$ can be split, i.e., $$Z=Z_{\gamma}\times{}Z_{e}\times{}Z_{i}\times{}Z_{\pm}$$ (of course, the physical picture is that there is a formation rate from the interactions of two photons, balanced by an annihilation rate; for the statistics, we are only concerned about the final equilibrium ). Since the electrons and positrons are independent, $$Z_{\pm}=Z_{+}\times{}Z_{-}$$, with both given by the above equation with $$\epsilon=m_{e}c^2$$, but with $$N_{+}=N_{-}=N_{\pm}$$. Hence, $$Z_{+}=Z_{-}$$, and to find the number of particles, we can just find the maximum of $$S_{+}=k\log{}Z_{+}$$, i.e., $\frac{\partial{}S_{+}}{\partial{}N_{+}} = \frac{\partial{}k\log Z_{+}}{\partial{}N_{+}} = \frac{\partial}{\partial{}N_{+}}kN_{+}\left[\log\left(g\frac{V}{\lambda_{th}^3}\right)-\frac{m_{e}c^2}{kT} -\log N_{+}-1\right]=0,$ where we used that for large $$N$$, $$N!=N\log N - N$$. Solving this for $$N_{+}$$, one finds $N=g\frac{V}{\lambda_{th}^3}\exp(-m_{e}c^2/kT).$ Equivalently, one has $$n\equiv{}N/V=g\exp(-m_{e}c^2/kT)/\lambda_{th}^3$$, which has the nice implication that for classical particles, the probability for one with given internal state to exist in a given volume element $$\lambda_{th}^3$$ is simple $$\exp(-\epsilon/kT)$$. Thus, for this very small volume, the probability becomes significant for $$kT\approx{}m_{e}c^2$$. But when does the number of pairs become significant on larger scales? One measure to use is when $$n_{\pm}=n_e$$, i.e., when $$\exp(-m_{e}c^2/kT)=_{}\lambda_{th}^3 n_e/g$$. For electrons ($$m=m_e$$), one has $$\lambda_{th}=2.4\times10^{-10}T_9^{-1/2}$$ cm, and $$n_e=\rho/\mu_e{}m_H=6\times10^{23}(\rho_2/\mu_e)$$ cm-3, so it requires $$T_{9}\approx{}m_{e}c^{2}/k(11.7+\log{}gT_9^{1/2}/\rho_2)\approx0.6$$, quite consistent with KW, Fig. 34.1.

One can treat ionisation similarly, writing $$Z_H=Z_{0}\times{}Z_{p}\times{}Z_{e}$$. We need to use that $$N_{p}=N_{e}=N_{H}-N_0$$. Doing a similar derivations as above, one derives the Saha equation. Again, ionisation is well before $$kT\approx{}\chi$$. One consequence of this, is that if one, e.g., wants to know the population in excited states in hydrogen, it is easier to do this relative to the ionised state (since by the time you can excite even to the first excited state with $$\epsilon_2=\chi_H(1-1/4)=10.2$$ eV, hydrogen is mostly ionised). For given state $$s$$, one thus writes $$n(H_0,s)/n_{p} = (g_s/g_{p}g_{e}n_{e}\lambda_{th}^3)\times\exp((\chi-\epsilon_2)/kT)$$.

Finally, back to the chemical potential $$\mu$$ (and Baierlein 2001AmJPh..69..423B). In terms of above quantities, one finds $$\mu=\epsilon+kT\log(g\lambda_{th}^3/n)$$, but $$\mu$$ also enters all thermodynamic potentials (internal energy U, enthalpy H, Helmholtz free energy F, Gibbs free energy G), as an additional term $$\dots+\mu{}dN$$, i.e., the energy required to add one particle. In particular, for constant T, V, Helmholtz is handiest: $$F(T,V,N)=PV+\sum_{i}\mu_{i}N_{i}$$ (and $$dF=PdV+\sum_{i}\mu_{i}dN_i$$). For pair plasma, minimizing $$F$$ for $$N_{+}=N_{-}$$ (holding $$T$$, $$V$$, other $$N$$ constant), one requires $$\mu_{+}+\mu_{-}=0$$. With the above microscopic definition of $$\mu$$, one recovers the solution. Similarly, for ionisation, $$\mu_0=\mu_p+\mu_e$$. In general, for any reaction left↔right, one expects that in equilibrium, $$\sum_{\rm left}\mu=\sum_{\rm right}\mu$$. (In that sense, the above are missing photons – but these have $$\mu_\gamma=0$$.)

All the above was for classical particles, but the same holds for non-classical ones (except of course that one cannot assume a Maxwellian once particles start to overlap, $$\lambda_{th}\approx{}d=n^{-1/3}$$). For completely degenerate neutron gas, where $$\mu=\epsilon_F$$, one now trivially finds that there will be a contribution of protons and electrons such that $$\mu_n=\mu_p+\mu_e$$. (Here, there is no $$\mu_\nu$$, since the neutrinos escape; for a hot proto-neutron star, where the neutrino opacity is still high, one does need to include it.) Remember, however, that above we derive a final, equilibrium state. The process to get there can be slow – not all baryons are in the form of iron yet!

• Gravothermal specific heat: c* = cP(1-∇ad/∇gr) (KW 4)

If fusion does not balance luminosity, heat/cool at a rate ∂q/∂t = ε-∂l/∂m. Define: εg = -∂q/∂t = -cPT [(1/T)∂T/∂t - (∇ad/P)∂P/∂t]

Here, following KW, S4.1, we used: dq = du+P dv = … = cP(dT-(∂T/∂P)sdP) = cPT(dT/T-∇addP/P), with ∇ad=(∂lnT/∂lnP)s

Important for contraction, on timescale τ≈τth. For slower changes (τ≫τth), negligible effect; for fast ones (τ≪τth), changes will be nearly adiabatic.

• Homologous change for sphere in HE (KW 25.3.4)

For small size change, homology gives dρ/ρ = -3dr/r, dP/P = -4dr/r and thus dρ/ρ = ¾dP/P. For a general EOS ρ ∝ PαT and thus dρ/ρ = αdP/P - δdT/T. Combining the two, we find that to keep HE, the temperature change has to be related to the pressure change by dT/T = (α-¾)/δ dP/P ≡ ∇gr dP/P.

Where we defined a temperature gradient ∇gr=(∂lnT/∂lnP)gr that is required for the star to remain in hydrostatic equilibrium.

Hence, the size change causes the change in heat per unit mass to be related to the temperature change via dq = cPdT(1-∇ad/∇gr) ≡ c* dT, with c* = cP(1-∇ad/∇gr) Compare with cP at constant P, or cV at constant volume; here it is at "constant hydrostatic equilibrium".

• Ideal gas: α=δ=1, ∇ad=2/5, ∇gr=1/4, hence c*=cP(1-8/5)<0: removing heat causes the temperature to increase.
• Degenerate gas: α→3/5, δ→0, c*>0: removing heat causes the temperature to decrease.
• Homologous change for shell (KW 33.2)

Shell of width D at r, with D≪r, mass m~ρr2D. Suppose expands with dr=dD, then dρ/ρ=-dD/D. If outside expands homologously, still have dP/P=-4dr/r=-4(dD/D)(D/r). Hence, ∇gr = (αD/r-¼)/(δD/r) = (α-r/4D)/δ and c*=cP(1-∇adδ/(α-r/4D)).

Ideal gas: α=δ=1, unstable if D/r<¼. (think limit D→0: upon expansion, P does not change, ρ decreases, hence T must increase to keep HE).

## 4 Heat loss

Radiative diffusion; conduction; opacity sources; Schwarzschild and Ledoux criteria; mixing length theory; convective flux; stellar context for convection; semi-convection.

### Mon Jan 21

Textbook
Discussed in class: KW 5.1; to be discussed as short presentations (and to be read): KW 5.2, 17 (AST320 notes 5)
Exercises
• Redo AST320 mini-PS VI, and think through what changes if you assume Kramers-like opacities (see KW 17.2–3).
• General qualifier questions: Stars:4

Like general diffusion equation: j = -(1/3) vl ∇n

• Eddington equation: dT/dr = -(3/4ac)(κρ/T3)(l/4πr2)
• Rosseland mean: 1/⟨κ⟩ = (π/acT3)∫ν(1/κν)(dBν/dT)dν

### Wed Jan 23

Textbook
KW 6.1–6.5, 7, 30.4 (AST 320 notes 6)
Exercises
• Criterion for convection: -(1/γ)dlnP/dr > dlnρ/dr
Schwarzschild criterion
Ledoux criterion
where ∇μ=dlnμ/dlnP and f=(∂lnρ/∂lnμ)/(-∂lnρ/∂lnT); f=1 for fully-ionised ideal gas.
• Convective flux

Generally, one can write the flux as, $F_{\rm conv} = \rho \overline{v}_{\rm conv} \Delta q = \rho \overline{v}_{\rm conv} c_P \Delta T,$ where $$\overline{v}_{\rm conv}$$ is a suitable average'' of the convective velocity.

In terms of the gradients, one finds $F_{\rm conv} = \rho \overline{v}_{\rm conv} c_P T \frac{\ell_{\rm mix}}{2H_P}\left(\nabla-\nabla_{\rm ad}\right),$ where $$\ell_{\rm mix}$$ is the mixing length, usually parametrized as a fraction of the scale height, i.e., $$\ell_{\rm mix}\equiv\alpha_{\rm mix}H_P$$, with $$\alpha_{\rm mix}$$ the mixing length parameter.

The estimate of $$\overline{v}_{\rm conv}$$ is the tricky part. We follow the AST 320 notes and balance buoyancy ($$Vg\Delta\rho=\rho Vg\Delta T/T$$) and friction ($$-A\rho v^2$$); evaluate velocity at $$l_{\rm mix}/2$$; define $$V/A=\beta\ell_{\rm mix}$$, where $$\beta$$ is a shape factor; and find $v_{\rm conv}^2 = \frac{\beta g}{H_P}\frac{\ell_{\rm mix}^2}{2} \left(\nabla-\nabla_{\rm ad}\right).$ This leads to a convective flux given by $F_{\rm conv} = \rho c_P T \alpha_{\rm mix}^2 \sqrt{\frac{\beta g H_P}{8}} \left(\nabla-\nabla_{\rm ad}\right)^{3/2}.$ Fortunately, the difficulty does not matter much: in the interiors of stars, convection is so efficient that the final temperature gradient ends up being essentially the adiabatic one. This is why we can treat completely convection stars as constant-entropy polytropes. But near the atmosphere, this is no longer true.

### Mon Jan 28

Textbook
KW 10, 24 (AST 320 notes 9, 7): Hayashi line.
Exercises
• Check you understand the qualitative shapes of proto-stellar tracks (AST 320 notes, Fig. 7.3)

For fun, you could also have a look at the paper where the Hayashi line turned out to be important: 2000ApJ...529..428V (and the acknowledgement of the referee).

### Wed Jan 30

Textbook
KW 17 (again; AST 320 notes 5)
Exercises
• Low temperature opacities: see Bethany's notes.
• Scalings for conduction

Aarya gave a nice qualitative introduction (see also KW 17.6). Here, a somewhat more mathematical one.

Generally, the flux is $$F=-\frac13vl\nabla{}U$$. It can be separated in different components. For photons, we saw $$U=aT^4$$, $$v=c$$ and $$l=1/\sigma{}n$$ and hence one has $$F=-(4ac/3)(T^3/\sigma{}n)\nabla{}T$$ (where usually we write $$\sigma{}n=\kappa\rho$$, but it is easier not to do so here). Given the definition of conductivity through $$F=-k\nabla{}T$$, one infers an equivalent conductivity $$k_{\gamma}=(4ac/3)(T^3/\sigma{}n)$$.

For particles, $$U=\frac32nk_{B}T$$ and thus $$F=-\frac13vln\frac32k_{B}\nabla{}T$$. Again writing $$l=1/n\sigma$$, one finds $$k=\frac13\frac23k_{B}(v/\sigma)$$. For an ideal, completely ionised gas, $$v\propto{}T^{1/2}$$ and $$\sigma\sim{}Z^{2}e^4/(kT)^{2}\propto1/T^2$$. Hence, $$k\propto{}T^{5/2}$$.

For degenerate material, we should consider ions and electrons separately. The ions still have very short mean-free path, so do not contribute much. For the electrons, only a small fraction $$kT/E_{F}$$ near the Fermi surface carries any heat, i.e., $$U_{e}\sim{}n_{e}(kT/E_F)kT$$, and thus $$\nabla{}U\sim{}n_{e}(k_{B}T/E_{F})\nabla{}T$$. Furthermore, those electrons have velocity depending on density, not temperature. Their free path still is $$l=1/n_{i}\sigma$$ ($$n_i$$ the ion density), but now $$\sigma\sim{}Z^{2}e^4/E_{F}^{2}\propto{}1/E_{F}^2$$, and thus $$k_{e}\propto{}(v/\sigma{}n_{i})n_{e}(k_{B}T/E_{F})\propto{}vE_{F}T$$. For non-relativistic electrons, $$v\propto{}\rho^{1/3}$$ and $$E_{F}\propto\rho^{2/3}$$, so $$k_{e}\propto\rho{}T$$. For relativistic particles, $$v\to{}c$$ and $$E_{F}\propto\rho^{1/3}$$, so $$k_{e}\propto\rho^{1/3}T$$.

Writing in terms of an equivalent opacity, $$\kappa=(4ac/3)(T^3/k\rho)$$, one finds for the ionised ideal gas, the opacity for electrons scales as $$\kappa_{e}\propto{}T^{1/2}/\rho$$, for non-relativistic degenerate electrons, $$\kappa_{e}\propto{}T^2/\rho^2$$, and for relativistic degenerate electrons, $$\kappa_{e}\propto{}T^2/\rho^{4/3}$$. Note that the photon opacity should also be affected, since photons can only interact with electrons near the Fermi surface, so $$l_{\gamma}\sim1/\sigma{}n_{e}(kT/E_{F})$$. Equivalently, one can write that the effective opacity scales as $$T/E_{F}\propto{}T/\rho^{2/3}$$ (non-relativistic) or $$T/\rho^{1/3}$$ (relativistic). At high densities, however, electron conduction will still win because of its steeper dependence on ρ.

## 5 Energy production

Nuclear binding energy; Coulomb barrier; reaction channels (PP, CNO, He and beyond, D/Li burning, s-/r-/p-processes); rates and neutrinos.

### Mon Feb 4

Textbook
KW 18.5.1 (p-p and CNO cycles). Slowness of p-p compared to

Li+p and D+p due to weak reaction.

• EB revisted: ∂l/∂m = ε-ενg

Specify contributions from neutrino losses and contraction/expansion

### Wed Feb 6

Textbook
KW 18
Exercises
• AST320 mini-PS XI, XII
• Temperature dependence

Generally, we write the cross section $$\sigma(E)=(S(E)/E)\exp(-b/\sqrt{E})$$, and integrate over $$E$$ to get $$\langle\sigma{}v\rangle$$, i.e., $\langle\sigma{}v\rangle=\sqrt{\frac{8}{\pi\mu}}\left(\frac{1}{kT}\right)^{3/2} \int_E S(E)\exp(-E/kT-b/\sqrt{E}) dE$ We discussed how normally $$S(E)$$ can be taken out of the integral and one finds the Gamov peak, with height $$\exp(-3E_0/kT)$$, with $$3E_0/kT=-19.721(\mu/m_u)^{1/3}(Z_{a}Z_{b})^{2/3}T_{7}^{-1/3}$$.

As Ariel mentioned, resonances are important. The above holds if one's energy is in the far wing of a resonance, so that $$S(E)$$ indeed varies slowly. But if the resonance is inside the Gamov peak, it can dominate the energy dependence. In that case, one can consider it as a delta function, and the reaction rate will scale just with $$\exp(-E_{res}/kT-b/\sqrt{E_{res}})$$, i.e., the only temperature-dependent part comes from how many particles have the right energy. For this reason, the $$3\alpha$$ reaction rate has a term with $$\exp(-C/T)$$ instead of $$\exp(-C/T^{1/3})$$.

## 6 Evolution of single stars

### Mon Feb 25 - Main Sequence

Textbook
KW 28.2, 30, AST 320 notes 11.
Exercises
• Read also KW 28.1 and AST 320 notes 11; former helpful for the MESA problem set as well.
• Check that you understand the different ends of the main sequence for different masses listed in AST 320 notes 11, and how these relate to what one sees in the HRD (AST 320 notes Fig 10.2 is well worth studying in detail).
• Approach to the main sequence

Generally, contract until some fusion process can provide the luminosity radiated. On the way to the main sequence, D and Li are fused, but for most stars, the first fusion stage that can hold up the contraction for a little while is the first part of the CNO cycle, where C is turned into N (see Fig. 7.3 in the AST 320 notes). Only when the C is exhausted does the star contract further until either the p-p chain or the full CNO cycle takes over.

• On the main sequence

Hydrogen converted to Helium. In low-mass stars, radiative core so centre exhausts first. In more massive stars, convective core exhausts in one go, though the convection zone slowly becomes smaller during the main sequence. In detail, this depends on how convection actually happens, i.e., on overshooting and semi-convection.

For both, the luminosity increases slightly. Qualitatively, one can understand this from the increase in mean molecular weight μ. Naively, one would expect a decrease in radius, but changes in stellar structure counteract this (i.e., the star does not change homologously). Only in the final stages does the radius descrease a little.

• End of the main sequence

The core contracts and a shell around it ignites. In general, if a stable core can be formed, it will become isothermal. But there is a maximum (see KW and AST 320 notes); beyond that the core has to contract and either ignite He fusion or become degenerate.

### Wed Feb 27 - Giant Branch and Helium Flash

Textbook
KWW 33.1–6 (KW 32), AST 320 notes 12, low-mass giants.
Exercises
• Check that you understand the basic differences between fusion in main-sequence and giant stars, and in degenerate cores.
• Ensure you understand why for low-mass stars, the helium flash happens at a fixed core mass and luminosity, (nearly) independent of the stellar mass.
• Study the figures with evolutionary tracks and perhaps especially Fig. 12.4 in AST 320 notes.
• Giant stars

For a sufficiently dilute envelope (M small and/or R large), the properties of shell determined by the core only, as the envelope is all far away.'' In particular, kT≈GMcμmH/Rc.

As a consequence, if, e.g., the core contracts, T will go up and so will the luminosity, causing the envelope to expand: mirror principle.

• Helium flash

For low-mass stars, the degenerate helium core is at about the same temperature as the shell. Eventually, helium ignites, at a core mass of about 0.45 Mo, somewhat off centre. Since the core is degenerate, a thermonuclear runaway ensues, though it does not become dynamically unstable.

### Mon Mar 4 - Intermediate Mass Giants, Blue Loops

Textbook
KWW 31, AST 320 notes 12, intermediate-mass giants.
Exercises
• Check you understand what causes the first and second dredge up.
• To better understand the loops, read Lauterborn et al., 1971A&A....10...97L (for recent discussion on blue loops, see Walmswell et al. 2015MNRAS.447.2951W; for more general insights, Gautchy 2018arXiv181211864G).
• Study the figures with evolutionary tracks and perhaps especially Fig. 31.2 in KW.

### Wed Mar 6 - Thermal Pulses on the AGB

Textbook
KWW 33.3 (RGB bump), 34.1-34.4 (AGB thermal pulses), AST 320 notes 12, thermal pulses
Exercises

### Mon Mar 11 - Asymptotic Giant Branch

Textbook
KWW 34.4-5 (core-mass luminosity relation; nucleosynthesis)
Exercises
• Read also KWW 34.7 and ensure you understand the general evolution on the AGB.
• General qualifier questions: Stars:1, 9; part of Stars:20.
• Eddington luminosity

I find it easiest to derive from force balance (which makes sense only for optically thin material above a star's photosphere): $F_{\rm grav} = -\frac{GM}{R^2}m = F_{\rm rad} = \frac{L}{4\pi{}R^2}\sigma{}N$ where with $$m=\rho{}V$$ and $$\sigma{}N=\sigma{}nV=\kappa\rho{}V$$, one finds $L_{\rm Edd} = \frac{\pi{}cGM}{\kappa}.$

• End of the AGB

Near the end of the AGB, the luminosity from the shell approaches the Eddington lumnisity relevant for electron-scattering opacity. At the cool photosphere, the opacity generally is smaller except when it gets cold enough for dust grains to form. This becomes particularly easy after C has been dredged up. (And pulsations help too.)

### Wed Mar 13 - Initial-Final Mass Relation and AGB Mass Loss

Textbook
KWW 34.6 (AGB mass loss, white dwarf initial-final mass relation)
Exercises

### Mon Mar 18 - Overall Evolution, Supernovae

Textbook
KWW 34.8, 35, 36.1, 36.3 (up to 36.3.4); AST 320 notes 13
Exercises
• Study both interior (ρ-T) and exterior (T-L) diagrammes in detail, ensuring you understand the basics. (Further nice ones in the first MESA paper: Paxton et al. 2011ApJS..192....3P.)
• AST320 mini problem sets: XIII

## 7 Binary evolution

Most stars increase in radius as they evolve, often drastically. If in a binary, they may at some point overflow their Roche lobes, leading to mass transfer to the companion. If this is stable, mass transfer will be on the evolutionary timescale. If unstable, it can be on the dynamical or thermal timescale. Masses transfer ceases when the star stops trying to expand; in giants, this is when most of the envelope has been transferred, and the remainder becomes so tenuous that it shrinks. Thus, one generally is left with just the core of the star. This process, and variations on it, is responsible for most of the more interesting stars we observe. For a general review, see Section 3 in Van den Heuvel, 2009ASSL..359..125V.

### Mass loss and tranfer

Consider a star that looses or transfers mass at some rate $$\dot M$$.

• Effect on orbit

The angular momentum of an orbit is given by $$J=(M_1M_2/M)\sqrt{GMa}$$, and thus, $\frac{\dot J}{J} = \frac{\dot M_1}{M_1}+\frac{\dot M_2}{M_2} -\frac{1}{2}\frac{\dot M}{M}+\frac{1}{2}\frac{\dot a}{a}$ With this, we can now consider several cases.

• Conservative mass transfer

Consider mass transfer from star 2 to star 1. If no mass and angular momentum is lost, then $$\dot M_1=-\dot M_2$$, $$\dot M=0$$, $$\dot J=0$$. Thus, $\frac{\dot a}{a} = 2\frac{M_{2}-M_{1}}{M_{1}M_2}\dot M_{2} = 2(q-1)\frac{\dot M_{2}}{M_{2}},$ where $$q=M_2/M_1$$ is the mass ratio between the donor (star 2) and the accretor (star 1). For donors less massive than the accretor, the orbit expands upon mass transfer (remember that $$\dot M_{2}<0$$).

Looking at the Roche lobe for a less massive donor, for which $$R_{L}\approx0.46a(M_{2}/M)^{1/3}$$, one finds $\frac{\dot R_{L}}{R_{L}} = \frac{\dot a}{a} +\frac{1}{3}\frac{\dot M_{2}}{M_{2}} = 2\left(q-\frac{5}{6}\right)\frac{\dot M_{2}}{M_{2}},$ showing that the Roche lobe, as expected, grows a little slower than the orbital separation. (An analysis valid for all $$q$$ would use the approximation of Eggleton 1983ApJ...268..368E, $$R_{L}/a\simeq 0.46q^{2/3}/[0.6q^{2/3}+\ln(1+q^{1/3})]$$.)

• Spherically symmetric wind

$$\dot M_{2}=\dot M$$, $$\dot M_{1}=0$$, $$\dot J=(\dot M_{2}/M_{2})(M_{1}/M)J$$. Hence, $\frac{\dot a}{a} = 2\left(\frac{M_{1} \dot M}{M_{2}M}-\frac{\dot M}{M_{2}}+\frac{\dot M}{2M}\right) =-\frac{\dot M}{M}.$ Thus, for mass loss ($$\dot M<0$$), the orbit expands.

• Spherically re-emitted wind

$$\dot M_{2}=\dot M$$, $$\dot M_{1}=0$$, $$\dot J=(\dot M_{2}/M_{1})(M_{2}/M)J$$ (idea is that accretor cannot handle mass transferred to it and re-emits it as a wind). Hence, $\frac{\dot a}{a} = 2\left(\frac{M_{2}\dot M}{M_{1}M}-\frac{\dot M}{M_{2}}+\frac{\dot M}{2M}\right) =\frac{2q^2-2-q}{1+q}\frac{\dot M}{M}.$ Hence, orbit expands for $$q<(1+\sqrt{17})/4=1.28$$ (with again a somewhat lower value for increasing Roche-lobe radius), i.e., it is less quickly unstable than for conservative mass transfer. For a more detailed analysis, see Soberman et al., 1997A&A...327..620S

Completely convective stars, or stars with deep convective layers, however, increase in size upon mass loss. For completely convective stars, which are described well by polytropes with $$P=K\rho^\gamma$$ with $$\gamma=\frac{5}{3}$$ (and thus $$n=1.5$$), this follows immediately from the mass radius relation: $$R\propto{}M^{-1/3}$$ (true for constant $$K$$, i.e., for constant entropy or completely degenerate, non-relativistic gas). Comparing this to the change in Roche lobe for conservative mass transfer, one sees that stability requires that $2\left(q-\frac{5}{6}\right)<-\frac{1}{3} \Leftrightarrow q<\frac{2}{3}\qquad\hbox{for}\quad n=1.5.$