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

Assume uniform global temperatures, all visible light not scattered back to space is absorbed by the surface, and use Kirchhoff's law of thermal radiation so that the atmospheric infrared emissivity is $\alpha$. If $F_{\ast }=\sigma T_{\ast }^4(R_{\ast }/a{)}^2$, the globally averaged absorbed stellar flux isThe atmosphere absorbs $\alpha \sigma T_{\mathrm{surf}}^4$ from the surface and emits from both faces, soSurface balance isTherefore the single-layer greenhouse model givesVisible scattering cools the surface, whereas infrared absorption and downward re-emission warm it.