Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 315 / 4 / a / Solution

In an infrared-opaque single-layer greenhouse model, the atmospheric layer obeys $2\sigma T_a^4=\sigma T_s^4$. The outgoing planetary flux is $\sigma T_a^4$, while the globally averaged absorbed stellar flux is $(1-A_B)L/(16\pi a^2)$. Radiative equilibrium therefore givesHence the orbit at which the prescribed surface temperature can be maintained isThis assumes uniform redistribution, constant Bond albedo, unit longwave emissivity, a transparent atmosphere to starlight, and no internal heat. Without the greenhouse layer, replace $8\pi$ by $16\pi$.