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

Let the stellar radius and temperature be $R_{\ast }$ and $T_{\ast }$. Assume Bond albedo $A_B$, isotropic stellar emission, blackbody planetary emission, and complete redistribution over the tidally locked planet. The absorbed stellar power isAdding the isolated internal luminosity $4\pi R_p^2\sigma T_0^4$ and balancing the total against $4\pi R_p^2\sigma T_{\mathrm{eq}}^4$ gives the planetary equilibrium temperatureIf heat is reradiated uniformly only over the dayside, replace the denominator $4a^2$ by $2a^2$. The latter is often more plausible for inefficient redistribution on a tidally locked bare planet.