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1 Microphysics ASTR 501 Stellar Physics 1 1 Microphysics • equation of state, density: ρ(P, T, X) • radiative absorption coefficient, opacity: κ(P, T, X) • rate of energy production: (P, T, X) (nuclear physics) Nuclear physics also responsible for composition changes dXj = Fˆj · Xj (1) dt burn while the total composition change also includes a mixing term: dX ∂ dX i = 4πr2ρ D i (2) dt mix ∂m dm 1.1 Equation of state For the fluid to have an EOS it must be collisional: lregion λ (3) where λ is the mean free path. P from equation of state, consider three cases: ASTR 501 Stellar Physics 2 (a) ideal gas equation of state: p = R ρT (b) barotropic equation of state: pressure p depends only on ρ, as for example • isothermal: p ∼ ρ • adiabatic: p = Kργ Plus non-ideal effects, e.g. electron degeneracy. 1.2 Ideal gas equation of state Assumption: internal energy entirely in kinetic energy according to kinetic theory: = CVT , where CV is the specific heat at constant volume. This can be seen from the first law of thermodynamics dQ = d + pdV , where V is the specific volume, d if we express d = dT dT . d Similarly, the specific heat at constant pressure Cp = dT + R where R is the gas constant. This follows from the ideal gas equation of state. [†13] It also follows that Cp − CV = R [†14], and the ratio of specific heats is defined as C γ = p . (4) CV ASTR 501 Stellar Physics 3 α CV can be related to R as CV = 2 R , where α is the number of degrees of freedom 1 per particle, since the energy per particle per degree of freedom is 2kT where k is 2+α the Boltzmann constant, and R = nk. Thus, γ = α , which means that for a 5 monoatomic gas γ = 3. Using these quantities the ideal gas equation of state can be expressed in terms of γ and the internal energy : p = (5) (γ − 1)ρ 1.3 Barotropic EOS: adiabatic case For an adiabatic change: dQ = 0 which allows to derive the barotropic EOS for an ideal gas p = Kργ (6) where K is a constant. A fluid element behaves adiabatically if K does not change as the element evolves. A reversible, adiabatic process is isentropic (dQ = T dS = 0) and therefore an adiabatic stratification has S = const. Note, that a relation of the form Eq. (6) is called a polytropic relation, and in 5 general γ is then the polytropic index. It does not have to be the adiabatic γad = 3 ASTR 501 Stellar Physics 4 as it may also describes other stratifications or non-ideal gases, as for example fully degenerate electron gases (for which in the non-relativistic case γ = 5/3 as well, see later). 1.4 Mean molecular weight [e.g. Kippenhahn & Weigart, Sect. 13.1] Ideal gas equation of state (stellar evolution version): R? P = nkT = ρT (7) µ where ρ = nµMu, k Boltzmann constant, R? = k/Mu universal mass constant and Mu atomic mass unit. Mean molecular weight: dimensionless, average number of atomic mass units per particle !−1 X Xi(1 + Zi) µ = (8) µ i i e.g. ionised H µ = 0.5. ASTR 501 Stellar Physics 5 1.5 Radiation pressure Photons will contribute considerably to the pressure in the stellar interior. The radiation pressure of what is practically a black body is 1 a P = U = T 4 rad 3 3 and the total pressure is R? a P = P + P = ρT + T 4 gas rad µ 3 Pgas with the often used gas pressure fraction β = P . 1.6 Electron degeneracy [See Prialnik, Sect. 3.1-3.5 and Appendix B, De Boer & Seggewiss, Sect. 4.4.2.] At high ρ the electron-degenerate gas dominates the pressure. For a fully non- relativistic degenerate gas[†41] 5/3 P = K1ρ (9) while for a relativistic degenerate gas 4/3 P = K2ρ . (10) ASTR 501 Stellar Physics 6 1.7 The ρ − T diagram You can delineate areas in the ρ − T diagram by equating the pressure for neigh- boring regimes. Of interest are the evolution of the central ρc and Tc for a single star [cf. opacity] as well as the profiles of stars in the ρ − T diagram. 1.8 Opacities [de Boer & Seggewiss, Sect. 2.8 & 4.4.1, Kippenhahn & Weigart, Sect. 5.1.3 & 17.] Various atomic physics processes cause absorption and scattering of EM waves (radiation) in stellar interiors and atmospheres. ASTR 501 Stellar Physics 7 1.8.1 Rosseland mean These processes are frequency dependent (κν) and for use in stellar evolution an appropriate averaging over frequency is required. The Rossland mean 1 π Z ∞ 1 ∂B = 3 dν (11) κ acT 0 κν ∂T where B(ν, T ) is the Planck function or the intensity of black-body radiation[†38]. Rosseland mean favours frequency ranges of maximum energy flux, and thus is appropriate to descibe transport. 1.8.2 Processes The main processes contributing to opacity are[†39] • electron scattering • free-free transitions • bound-free transitions • bound-bound transitions • negative H ion H− ASTR 501 Stellar Physics 8 1.9 Ionization [This material is covered, e.g., in Kippenhahn & Weigert, Sect. 14.1 and 14.2. More realistic cases (ionisation of H only). Mixtures of H and He are treated in the following chapters.] Ionization will change the number of particles per unit mass that contribute to the pressure, and is therefore important for the EOS. Also, the degree of ionisation x enters the calculation of the bound-free opacity κbf (Sect. 1.8.2). Most important are the ionisation of H and He because they dominate the abun- dance distribution of the stellar atmosphere and envelope of most stars. Calculating the degree of ionisation requires to determine the number of atoms in an ionized state through writing down the Boltzmann formula n g s = s e−Ψs/kT (12) n0 g0 1.10 Nuclear physics and nucleosynthesis ASTR 501 Stellar Physics 9 1.10.1 Nuclear reactions, networks and numerical solution In terms of the number density Nj of species j we collect all production and destruction terms of reactions of the type k + l → j + l dN j = N N < σv > −N N < σv > +... + N λ − N λ (13) dt k l kl,j j l jl,n i i,j j j,m where < σv > is the product of the cross section and the relative velocity in the center-of-mass system averaged over the appropriate distribution function and λ is the rate for β decays. Reaction rates can be obtained, for example, at http://pntpm.ulb.ac.be/ Nacre/nacre d.htm with the corresponding publication Angulo etal. (1999), see http://adsabs.harvard.edu/abs/1999NuPhA.656....3A. Chart of isotopes and types of reactions. Solution of nuclear networks: Newton-Raphson is an easy and stable possibility, al- though only 1st order accurate. See Numerical Recipies (Press etal) for details. ASTR 501 Stellar Physics 10 1.10.2 Nucleosynthetic processes [e.g. Wallerstein etal 1997, Rev Mod Phys, vol 69, 995 or textbook Rolfs & Rodney: Cauldrons in the Cosmos] • Hydrogen burning (conversion of H to He) 12 16 • Helium burning (He C, O, ...) • Carbon, oxygen and neon burning (production of 16 ≤ A ≤ 28) • Silicon burning (production of 28 ≤ A ≤ 60) • The s-, r- and p-processes (production of A ≥ 60) • The l-processes (production of reactive light elements D, Li, Be, B) 2 Macrophysics: convection 2.1 Conservation laws ASTR 501 Stellar Physics 11 2.1.1 Mass conservation Eulerian: ∂ρ + ∇ · (ρu) = 0 (14) ∂t 2.1.2 Momentum equation Lagrangian: Du ρ = −∇p + ρg (15) Dt Gravitational acceleration given by gravitational potential Ψ: g = −∇Ψ (16) GM Spherically symmetric potential Ψ = − r Otherwise solve Poisson equation: ∇2Ψ = 4πGρ (17) where G is the gravitational constant and ρ is the mass density. ASTR 501 Stellar Physics 12 2.1.3 Conservation of energy Total energy per unit volume: 1 E = p u2 + Ψ + (18) 2 (for see Eq. (5)). D Change of internal energy Dt from the first law of thermodynamics, and momentum and continuity equation the energy equation: ∂E ∂Ψ + ∇ · [(E + p)u] = −ρQ˙ + ρ (19) ∂t cool ∂t 2.2 Convection 2.2.1 Stability Temperature gradients: • radiative: ∂T 3 κρL = − (20) ∂r rad 16πac r2T 3 ASTR 501 Stellar Physics 13 • convective: ∂T 1 T dP = 1 − (21) ∂r ad γ P dr With this the Schwarzschild condition for convective instability is: ∂T ∂T > (22) ∂r rad ∂r ad ∂lnT [The gradient are often written in terms of ∇ ≡ ∂lnP .] 2.2.2 Hydrodynamics Solving the equations from Sect. 2.1 for convectively unstable regions in stars demonstrates the hydrodynamic and turbulent nature of convection. [watch movie]. 2.2.3 Mixing-length theory [see ASTR 404 script Sec. 2.6 and references there] • Ftot = Frad + Fconv ASTR 501 Stellar Physics 14 • consider radiation as diffusion and convective heat transport as advection in coherent blobs that rise for a certain distance and then disperse • Radiative flux in the presence of a temperature gradient: 4acG T 4m F = ∇ (23) 3 κP r2 • if ∇ < ∇rad need to find Fconv by making some local assumptions on properties and velocities of convective blobs 3 Stellar evolution Now, putting it all together: 3.1 The equations Assume 1D spherical symmetry and hydrostatic equillibrium: u = 0 (static) and ∂ ∂t = 0 (equilibrium). Then the conservation laws become: dP GM = − r ρ (24) dr r2 ASTR 501 Stellar Physics 15 dM r = 4πr2ρ (25) dr dL r = 4πr2ρ (26) dr plus the temperature gradients (Sect.
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