# AST 1410 – Stars (2013)

## 1 Syllabus

Lectures
M10, W10–12 (with coffee break), AB 113 (except: M11 on 14 Jan., AB 101H on 23 Jan. and 6 Mar.)
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

### 1.1 Synopsis

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; Addison-Wesley, 2006). 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

• Three problem sets (30% total), due two weeks after posting.
• Presentations (20% total), two short ones explaning a specific concept (8 min., plus 7 min. discussion) and one long one on a more advanced topic (list to come; 20 min., plus 10 min. discussion).
• Computer project (20%), using mesa to investigate a specific part of stellar evolution. (If you are running Ubuntu or Debian, you may find my installation instructions handy.)
• Final exam (20%; 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.

### 2.1 Mon 14 Jan

Textbook
KW 1–3 (can skip 2.5–2.7, 3.2) [and/or AST320 notes 1]
Exercises
• Derive the Virial Theorem (assuming ideal gas)
• 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 set I, II

#### 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

#### Basics of evolution: contraction, heat up, fusion

When does it stop?

#### Where do we see stars

Check wiki:HRD. Why are there limits?

• top: L too high, matter blown away (LEdd)
• right: T profile too steep → convection
• left: Tc very high → fusion (think of contraction gone too far)

#### Structure equations: MC, HE, TE, EB

• mass conservation: ∂m/∂r = 4πr2ρ
• Hydrostatic equilibrium: ∂P/∂r = -gρ = -Gmρ/r2 (multiply with 4πr2dr on both sides)
• Thermal equilibrium: diffusion; Frad = -c/3κρ ∂(aT4)/∂r (can solve for ∂T/∂r); have Frad = l/4πr2
• Energy balance: ∂l/∂r = 4πr2ρε
• To solve, needs
• equation of state P(ρ,T)
• opacity κ(ρ,T)
• energy generation rate ε(ρ,T)
• If T (dependence) known, can solve MC, HE separately
• Independent variable r or m (or P, or …)
• Simple estimates/scalings:
• ρ ≈ ⟨ρ⟩ = M/(4πR3/3)
• P ≈ GM2/R4
• kT ≈ GMmH/R

#### Virial theorem: Eint=-½ Egrav; Etot=Eint+Egrav=½ Egrav

To derive: multiply HE by r and integrate over sphere; use that for ideal gas U = 3/2 nkT = 3/2 P

### 2.2 Wed 16 Jan

Textbook
KW 6.4, 19.1–19.4, 19.9 [AST 320 notes 2, 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)
• AST320 mini problem set VI

#### Luminosity of a star

TE: l/4πr2 = -c/3κρ ∂(aT4)/∂r. Hence L ≈ (4πac/3) (RT4/κρ). Using that ρ ∝ M/R3 and T ∝ M/R, one finds L ∝ M3/⟨κ⟩

#### 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) → L~M4
• contribution of radiation pressure → flatter at high M.
• contribution of degeneracy → steeper at low M

#### Substantial difference: inert lump inside → shell source

star inside a star''; sketch potential

#### Timescales

• Dynamical: τdyn ≈ 1/(Gρ)1/2
What if not in HE? 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

Dynamical time for Sun (⟨ρ⟩≈1 g/cm3), NS (∼1014 g/cm3), Universe? (∼1 m-3≈10-30 g/cm3)

• 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

#### Polytropes: P = Kργ ≡ 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 AST320 \#4, esp. Table 4.1.

#### 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

#### 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.

## 3 Equation of state

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

### 3.1 Mon Jan 21

Textbook
KW 4, 25.3, 33.2

#### First law of thermodynamics: du = Tds -Pdv (u,s,v per unit mass)

• For small changes: Tds = δq
• If particle numbers change, add: ∑iμidNi

#### Gravithermal specific heat: c* = cP(1-∇ad/∇gr)

If no EB from fusion, 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 (ττth), negligible effect; for fast ones (τ τth), changes will be nearly adiabatic.

• Homologous change for sphere (KW 25.3)
For small size change, homology gives dρ/ρ=-3dr/r dP/P=-4dr/r. For a general EOS ρ ∝ PαT, dρ/ρ=αdP/P-δdT/T=-3dr/r=(3/4)dP/P, and thus dP/P=δ/(α-3/4)dT/T.

• Ideal gas: α=δ=1, ∇ad=2/5, hence c*=cP(1-8/5)<0
• Degenerate gas: α\to3/5, δ\to0, c*>0
• 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). So, ∇gr = (αD/r-1/4)/(δD/r) = (α-r/4D)/δ and c*=cP(1-∇adδ/(α-r/4D)).

Ideal gas: α=δ=1, unstable if D/r<1/4. (think limit D\to0: upon expansion, P does not change, ρ decreases, hence T increases).

#### EB revisted: ∂l/∂m = ε-εν+εg

Specify contributions from neutrino losses and contraction/expansion

### 3.2 Wed Jan 23

Textbook
KW 13–16 (AST 320 notes 3)
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-PS III, 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)(4πp2/h3)dp, n(ep)=1/[exp((e-μ)/kT) ± 1]
+1:Fermions; -1: bosons; μ: chemical potential
(For a nice description of the meaning of chemical potential μ, see Baierlein 2001AmJPh..69..423B.)
• ignore \pm1: 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/μe)5/3, K1=(3/π)2/3(h2/20me)(1/mH)5/3\approx9.91\times1012 (cgs)
• ERCD: P=K2(ρ/μe)4/3 K2=(3/π)1/3(hc/8)(1/mH)4/3\approx1.231\times1015 (cgs)
• 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)

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, $$\sum_{left}\mu=\sum_{right}\mu$$. (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!

#### Short presentations

• Nathan: ionisation (KW 14, include why it happens at kT<<χH; pressure ionisation – also AST 320 mini-PS VII; extension to nuclei, KW 34.2.2).
• Stephen: crystallisation (KW 16.4, both effect on EOS and latent heat; classics: Salpeter 1961ApJ…134..669S and Van Horn 1968ApJ…151..227V; also Shapiro+Teukolsky 1983bhwd.book…..S)
• Adam: neutron-star interior (KW 36, Lattimer & Prakash 2004Sci…304..536L, references therein; Shapiro+Teukolsky 1983bhwd.book…..S; effects of ever-increasing Fermi energy from outside to in; neutron drip; reason why neutron gas not like degenerate gas, symmetry energy; some history (but incorrect for WDs, see Salpeter classic above): Cameron 1959ApJ…130..884C).

## 4 Heat loss

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

### 4.1 Mon Jan 28

Textbook
Discussed in class: KW 5, 6.1; not discussed but to be read: KW 17 (AST320 notes 5, 6)
Exercises
• AST320 mini-PS VI

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

#### 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 a fully-ionised ideal gas.

### 4.2 Wed Jan 30

Textbook
KW 6.2–6.5, 7, 30.4, 24 (AST 320 notes 6, 7)
Exercises
• AST320 mini-PS V

#### Scalings for conduction

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, $$\kappa_{e}\propto{}T^2/\rho^{4/3}$$ (the former two are as mentioned by Elliot). 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.

#### Short presentations

• Lisa: low-temperature opacities (KW 17.5, 7, H-; molecules, dust; ask RayJay/Ernst for reference on BD atmospheres?)
• Elliot: conduction (KW 17.6; estimates of mfp via Cox+Giuli reference?)
• Bob: real convection (nice overview, Spruit 1997MmSAI..68..397S (for outer zones); Arnett+Meakin 2011IAUS..271..205A (inner zones are different))

## 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.

### 5.1 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.

### 5.3 Mon Feb 11

Textbook
KW 18 (except 18.5.3, beyond helium)
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 Charles 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 Theme 1: low-mass stars

### 6.1 Wed Feb 13 – low-mass stars

Textbook
KW1 32.1–32.2/KW2 33.1–33.3. Smooth evolution off main sequence; dominance of core mass; determines luminosity and radius

#### Short presentations

• Heidi: r, s processes

### 6.2 Mon Feb 25

Red giant branch, more detail: dredge-up and bump.'' he flash for Wednesday, but does it encounter H? Maybe for zero-met or with very thin H env (hot flashers)

## 7 Theme 2: high-mass stars

### 7.1 Mon Mar 4

End of the main sequence for intermediate-mass stars

### 7.4 Wed Mar 13

Very useful review of the difficulties in modelling massive stars: Langer 2012ARA&A..50..107L

#### Short presentations

• Nathan: formation of massive stars (main source: Zinnecker+Yorke 2007ARA&A..45..481Z)
• Heidi: Zero-metallicity stars (see AST 320 mini-PS XI, XII).
• Adam: core collapse (proper, not the shock)

## 8 Theme 3: binary evolution

### 8.1 Mon Mar 25

Text
notes for my transient class (pdf), in particular Sec. 10.

### 8.2 Wed Mar 27

Text
above notes. For a recent review on common-envelope evolution, see Ivanova+ 2013A&ARv..21…59I.

Date: 2013-04-16T10:26-0400

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