AST 1410 – Stars (2022)

Underlying Physics

Stellar Evolution Themes

Course texts

Evaluation

Mon 10 Jan

• 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?

• Virial theorem: E_int=-½ E_grav; E_tot=E_int+E_grav=½ E_grav
  
  • Roughly, one has E_grav ≈ GM²/R, E_int ≈ N⟨e_int⟩ = (M/μm_H)3/2 kT, hence kT ≈ GMμm_H/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 12 Jan

• Where do we see stars
  

  Check wiki:HRD. Why are there limits?

  • left: T_c very high → fusion (think of contraction gone too far)
  • top: L too high, matter blown away (L_Edd)
  • right: T profile too steep → convection
• 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, l_mfp = 1/σn = 1/κρ, n = aT⁴; hence, F_rad = 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 …)

Mon 17 Jan

• 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/μm_H)kT:
    • MC+HE: kT ≈ GMμm_H/R
    • MC+HE+TE: L ≈ acG⁴m_H⁴ μ⁴M³/κ

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

• 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

  One can also derive other properties; see KWW 20.

• Real M-R and M-L relations
  
  • ε steep function of T → M/R nearly constant. Reality R ∝ M^0.8.
  • 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 denser lump with fusion (say He core): ``star inside a star''

Wed 19 Jan

• Stability
  

  Upon compression, ρ∝R^-3. Thus, for an adiabatic perturbation, P ∝ ρ^γ ∝ 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.

• Timescales
  
  • Dynamical: τ_dyn ≈ 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? τ_ff ≈ (R/g)^1/2 ≈ 1/(Gρ)^1/2
    • Gravity drops away? τ_expl ≈ R(P/ρ)^-1/2 ≈ R/c_s ≈ 1/(Gρ)^1/2
  • Thermal: t_th ≈ E_th/L (≈ GM²/RL for whole star)
    

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

    Hence, timescale τ_adj ≈ Ud³/Fd² ≈ 3Ud³/vl_mfpUd ≈ d²/vl_mfp ≈ (l_mfp/v)(d/l_mfp)² (random walk: t_step N_steps)

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

  • Nuclear: t_nuc ≈ E_nuc/L
    

Mon Jan 24

• Polytropes: P = Kρ^γ ≡ Kρ^1+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), E_g, etc.; see AST 320 notes 4, esp. Table 4.1.

• EOS: Pressure integral: P = (1/3) ∫_p n_pv_ppdp
  
  • NR: v_pp→p²/m=2e_p → P=(2/3)U → Virial Theorem: E_g=-2E_i, E_tot=(1/2)E_g
  • ER: v_pp→cp=e_p → P=(1/3)U → Virial Theorem: E_g=-E_i, E_tot=0
  • generally, n_p=n(e_p)g(4πp²/h³)dp, n(e_p)=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/μm_H)kT (μ here is mean molecular weight)
  • photons: -1, μ=0 → get BB → P=(1/3)aT⁴
  • electrons: +1: completely degenerate → fill up to p_F = h(3n/4πg)^1/3
    • NRCD: P=K₁(ρ/μ_em_H)^5/3, K₁=(3/4πg)^2/3(h²/5m_e)≈2.34×10^-38 N m³
    • ERCD: P=K₂(ρ/μ_e)^4/3 K₂=(3/4πg)^1/3(hc/4)≈2.45×10^-26 N m²
  • Complications: molecular/atomic/nuclear dissociation, pair formation
  • Combinations
    • Simplest: whichever dominates, or at least add radiation
    • Rough estimate everywhere (Paczynski 1983ApJ...267..315P:
      • P = P_e+P_i+P_rad = P_e+(ρ/μ_im_H)kT +(1/3)aT⁴
      • P_e = (P²_e,ideal+P²_e,cd)^1/2, P_e,ideal=(ρ/μ_em_H)kT
      • P_e,cd = (P^-2_e,nrcd+P^-2_e,ercd)^-1/2
    • EOS from look-up table
      • for completely ionised gas: Helmholtz (Timmes+Swesty 2000ApJS..126..501T; includes pair formation, explanation in Timmes+Arnett 1999ApJS..125..277T)
      • MESA

Wed Jan 26

• Another way to think about ionisation, etc.
  

  (Different from how I discussed it in class, which was based on 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 \(N\), \(\log N!=N\log N - N\). 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!

Mon Jan 31

• Radiative flux: F_rad=-(1/3) (c/κρ) dU_rad/dr
  

  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/κ_ν)(dB_ν/dT)dν
  

Wed Feb 2

• Criterion for convection: -(1/γ)dlnP/dr > dlnρ/dr
  

  Schwarzschild criterion

  Ignoring composition gradients → ∇_ad<∇_rad,
  where ∇_ad=(dlnT/dlnP_ad=1-1/γ and ∇_rad=(dlnT/dr)_rad/(dlnP/dr)

  Ledoux criterion

  With composition gradients → ∇_ad<∇_rad-f∇_μ,
  where ∇_μ=dlnμ/dlnP and f=(∂lnρ/∂lnμ)/(-∂lnρ/∂lnT); f=1 for fully-ionised ideal gas.

• Damped and driven oscillation
  

  Can be driven when the gradient is in between the Schwarzschild and Ledoux criteria; see KWW 6.2 and 6.3.

Mon Feb 7

• Opacities
  

  Discussed KWW 17, AST 320 notes 5, including why electron-scattering opacity (in area/mass) is independent of density, while most other sources scale with density.

• 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 \(\ell_{\rm mix}/2\); 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.

• Scalings for conduction
  

  Yansong 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 mean-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.

Wed Feb 9

Mon Feb 14 (first half)

Mon Feb 14 (second half)

Mon Feb 28

• More on tunneling
  

  See wiki:Quantum_tunneling; hyperphysics:quantum/barr; similar to wiki:Evanescent_field#Evanescent-wave_coupling relevant for the two-prism example discussed in class.

  Note the link between fusion and radioactivity, which was solved by Gamov 1928ZPhy...51..204G (download PDF via UofT library; in German, but perhaps do-able even if you don't know it).

• EB revisted: ∂l/∂m = ε-ε_ν+ε_g
  

  Specify contributions from neutrino losses and contraction/expansion

Wed Mar 2

• 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., \[ \langle\sigma{}v\rangle=\sqrt{\frac{8}{\pi\mu}}\left(\frac{1}{kT}\right)^{3/2} \int_E S(E)\exp(-E/kT-b/\sqrt{E}) dE \] Normally, \(S(E)\) can be taken out of the integral and one finds the Gamov peak, with height \(\exp(-3E_0/kT)\), with \(3E_0/kT=-19.721(\mu/m_u)^{1/3}(Z_{a}Z_{b})^{2/3}T_{7}^{-1/3}\).

  But resonances can be 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 \(\exp(-C/T^{1/3})\).

Mon Mar 7 - Main Sequence

• 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 KWW and AST 320 notes); beyond that the core has to contract and either ignite He fusion or become degenerate.

Wed Mar 9 - Giant Branch and Helium Flash

• 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≈GM_cμm_H/R_c(H_P/R_c), where the ratio of the scale height to the core radius, \(H_{P}/R_{c}\), is constant for homologous stars.

  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 M_o, somewhat off centre. Since the core is degenerate, a thermonuclear runaway ensues, though it does not become dynamically unstable.

Mon Mar 14 - Intermediate Mass Giants, Blue Loops

Wed Mar 16 - Asymptotic Giant Branch: Thermal Pulses and M-L relation

• 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 \[ L_{\rm Edd} = \frac{4\pi{}cGM}{\kappa}. \]

• 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.)

Mon Mar 21 - Nucleosynthesis on the AGB; symptotic Giant Branch

Wed Mar 23 - Overall Evolution, Supernovae

Mon Mar 28 - Mass transfer: stability and effects on orbit

• Angular momentum loss
  

  Two stars can be driven closer by angular-momentum loss. For gravitational radiation (in a circular orbit), \[ -\frac{\dot J}{J}=\frac{32 G^3}{5c^5}\frac{M_1M_2(M_1+M_2)}{a^4}, \] implying a merger time of \(1.05\times10^7{\rm\,yr}(M/M_\odot)^{-2/3}(\mu/M_\odot)^{-1}(P/{\rm1\,hr})^{8/3}\), where \(\mu=M_1M_2/(M_1+M_2)\) is the reduced mass, and \(P\) the orbital period. Thus, to merge within a Hubble time requires periods less than \(\sim\!0.5{\rm\,d}\).

  For binaries with low-mass stars, angular momentum can also be lost by ``magnetic braking'' – a solar-like wind coupled to a magnetic field. This mechanism is usually described by semi-empirical relations, which are calibrated using the rotational evolution of single stars and using population synthesis models for binaries.

• 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

  • Effect on stellar radius
    

    If the mass is lost from the outside of a star, the star becomes initially smaller, but on a hydrodynamic timescale it will partially re-expand in responds to the decreased pressure. Which effect dominates depends on the internal structure of the star. Generally, for thermal envelopes, the stars shrinks inside its Roche lobe, re-expanding only on the thermal timescale, typically to nearly its original size (especially for giants). However, a complication for thermal-timescale mass transfer is that, if the secondary is substantially less massive, it cannot accrete sufficiently fast and will bloat itself. For massive stars, this leads to contact, and almost certainly further mass loss and/or a merger. If this can be avoided, then eventually the two stars have equal mass, after which further mass transfer leads to expansion of the orbit, and eventually the donor will regain thermal equilibrium. After that, any further mass transfer is on its evolutionary timescale.

    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 \[ 2\left(q-\frac{5}{6}\right)<-\frac{1}{3} \Leftrightarrow q<\frac{2}{3}\qquad\hbox{for}\quad n=1.5. \]

    

    Figure 3: \(\zeta_{\rm L}\equiv\partial\ln R_{\rm L}/\partial\ln M_2\) as a function of mass ratio, with all mass transfer through a single channel: conservative (cons); isotropic wind from donor star (wind); isotropic re-emission of matter, from vicinity of `accreting' star (iso-r). (Also shown is a ring formation, indicative of mass loss from an outer Lagrange point). From Soberman et al., 1997A&A...327..620S, their Fig. 4.

Wed Mar 30 - Common envelope evolution