Use a <single-layer greenhouse model> whose atmosphere is transparent to incident stellar light and has infrared emissivity $\epsilon$. Let $T_s$ and $T_a$ be the surface and atmospheric temperatures. Atmospheric balance gives
$$
\epsilon\sigma T_s^4=2\epsilon\sigma T_a^4,
\qquad
T_a^4=\frac12T_s^4.
$$
At the top of the atmosphere, the escaping flux is the directly transmitted surface radiation plus upward atmospheric emission:
$$
\sigma T_{\rm eq}^4=(1-\epsilon)\sigma T_s^4
+\epsilon\sigma T_a^4.
$$
Therefore
$$
\boxed{T_s=\frac{T_{\rm eq}}{(1-\epsilon/2)^{1/4}}}.
$$
For a perfectly infrared-opaque one-layer atmosphere, $T_s=2^{1/4}T_{\rm eq}$; for $\epsilon=0$, $T_s=T_{\rm eq}$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-315/4/b/solution