Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 1 / c / Solution

Assume that both objects emit as blackbodies, take $T_{\ast }=5778\mathrm{K}$ and $R_p/R_{\ast }\simeq R_J/R_{\odot }=0.100$, and use $T_p=1250\mathrm{K}$ because the line-free window sees the $1$-bar continuum photosphere. Across a narrow bin, the thermal eclipse depth isAt $\lambda =17\mu \mathrm{m}$ this givesIntegrating the Planck law over the full $1\mu \mathrm{m}$ bin changes this narrow-bin estimate only slightly.