Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 4 / a / Solution

Specific intensity is defined by $dE=I_{\nu }\cos \theta dAd\Omega d\nu dt$. A ray bundle in free space expands in area while its solid angle contracts by the same factor, soThus specific intensity does not obey an inverse-square law; the flux of an unresolved source does because its apparent solid angle scales as distance${}^{-2}$.

For a cold medium with coherent, isotropic, conservative scattering, the source function is the mean intensity $J_{\nu }$. With scattering optical depth increasing along the ray,andThe unscattered pencil beam is attenuated by $e^{-\tau }$ and reappears as a diffuse halo in other directions. Coherent scattering preserves frequency, and conservative scattering preserves total luminosity when all outgoing directions are collected.