A grey atmosphere is in radiative equilibrium

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres · Radiation, radiative transfer and stellar atmospheres

Stars from the Inside, Ch Radiative transfer solved: the grey atmosphere, the diffusion limit and the curve of growth

Problem

A grey atmosphere is in radiative equilibrium, so the frequency-integrated source function equals the mean intensity, S=JS = J (all the energy absorbed is re-emitted), and the total flux F=4πHF=4\pi H is constant with depth. The Eddington closure sets K=J/3K=J/3, and the surface has no incoming radiation, so H(0)=J(0)/2H(0) = J(0)/2. ((a)) Using the moment equations dH/dτ=J−S\mathrm{d}H/\mathrm{d}\tau = J-S and dK/dτ=H\mathrm{d}K/\mathrm{d}\tau = H together with these closures, derive T4(τ)=34Teff4(τ+23)T^4(\tau) = \tfrac34 T_{\rm eff}^4\left(\tau+\tfrac23\right), and hence the surface temperature T(0)=2−1/4TeffT(0) = 2^{-1/4}T_{\rm eff}. Then use the resulting linear source function in the Eddington-Barbier relation to derive I(0,μ)/I(0,1)=(2+3μ)/5I(0,\mu)/I(0,1) = (2+3\mu)/5. ((b)) A B-type star has Teff=15 000 KT_{\rm eff} = 15\,000\,\mathrm{K}. In the Eddington approximation, find its surface temperature T(0)T(0), and find the bolometric ratio I(0,0.4)/I(0,1)I(0,0.4)/I(0,1) that this model predicts at μ=0.4\mu=0.4.

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