About halfway out from the Sun's centre

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Energy transport and the complete stellar model

Problem

About halfway out from the Sun's centre, solar Model S gives ρ=1.3×103 kg m−3\rho = 1.3\times10^{3}\,\mathrm{kg\,m^{-3}}, T=4.0×106 KT = 4.0\times10^{6}\,\mathrm{K}, P=7.2×1013 PaP = 7.2\times10^{13}\,\mathrm{Pa}, a mass coordinate m=0.89 M⊙m = 0.89\,M_\odot and a local luminosity L=1.0 L⊙L = 1.0\,L_\odot. Take X=0.70X = 0.70, Z=0.02Z = 0.02 and a Gaunt factor gˉff=1\bar g_{\rm ff} = 1. ((a)) Use Chapter 4's Kramers law, κˉff≈3.68×1018 gˉff(1−Z)(1+X) ρ T−7/2\bar\kappa_{\rm ff} \approx 3.68\times10^{18}\,\bar g_{\rm ff}(1-Z)(1+X)\,\rho\,T^{-7/2} in SI units, to compute the free-free opacity at this point. ((b)) Taking κ≈κˉff\kappa \approx \bar\kappa_{\rm ff}, use the radiative-gradient formula in mass coordinate to compute ∇rad\nabla_{\rm rad}, and state whether the point is radiative or convective. ((c)) Say which region of the opacity map this point occupies, and which mass-luminosity homology exponent (electron-scattering or Kramers) is the relevant one to use near it.

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