Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 315 / 1

For the plane-parallel radiative transfer equationangular integration gives the zeroth radiation-field moment equationThe net radiative heating per unit volume is thereforeRadiative equilibrium requires it to vanish, equivalently that the frequency-integrated radiative flux be independent of depth:In local thermodynamic equilibrium with coherent isotropic scattering,where ${\omega }_{\nu }$ is the single-scattering albedo. Since ${\alpha }_{\nu }(1-{\omega }_{\nu })={\alpha }_{\nu ,\mathrm{abs}}$, the condition becomesConservative scattering redistributes directions but contributes no net material heating.

Put $\mu =\cos \theta$. For $I_{\nu }=A_{\nu }+B_{\nu }{\mu }^3$, the mean intensity isbecause the cubic term is odd. The K-integral, or second angular moment, isbecause the ${\mu }^5$ contribution is also odd. HenceThis intensity obeys the Eddington closure approximation even though it is not isotropic.

Deep in an optically thick grey atmosphere, write $I_{\nu }=B_{\nu }+\delta I_{\nu }$ and retain the first spatial-gradient correction in the transfer equation:Angular and frequency integration then gives the radiative diffusion fluxFor a thin plane-parallel atmosphere, constant Rosseland mean opacity $\kappa$, negligible external irradiation, and radius nearly equal to $R$, radiative equilibrium gives $F=L/(4\pi R^2)$. ThereforeWith hydrostatic balance $dP/dz=-\rho g$, the equivalent pressure form iswhere $g=GM/R^2$.