Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 4 / c

In vacuum, a narrow ray bundle conserves both power and geometrical etendue $dA\cos \theta d\Omega$. Their ratio, the specific intensity, is therefore independent of source-receiver distance. Equivalently, geometric dilution reduces received power and apparent solid angle by the same inverse-square factor.

For an isotropically emitting isothermal blackbody atmosphere, the outward surface flux iswhere the last equality is bolometric. Multiplication by the emitting area gives the luminosity of a spherical blackbody