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 $F_*=\sigma T_*^4(R_*/a)^2$, the globally averaged absorbed stellar flux is
$$
S=\frac{1-\beta}{4}F_*.
$$
The atmosphere absorbs $\alpha\sigma T_{\rm surf}^4$ from the surface and emits from both faces, so
$$
\alpha\sigma T_{\rm surf}^4=2\alpha\sigma T_a^4,
\qquad T_a^4=\frac12T_{\rm surf}^4.
$$
Surface balance is
$$
S+\alpha\sigma T_a^4=\sigma T_{\rm surf}^4.
$$
Therefore the <single-layer greenhouse model> gives
$$
\boxed{
T_{\rm surf}=T_*\sqrt{\frac{R_*}{2a}}
\left(\frac{1-\beta}{1-\alpha/2}\right)^{1/4}}.
$$
Visible scattering cools the surface, whereas infrared absorption and downward re-emission warm it.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-315/4/c/solution