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 gives
$$
\frac{\sigma T_s^4}{2}
=\frac{(1-A_B)At^{-\beta}}{16\pi a^2}.
$$
Hence the orbit at which the prescribed surface temperature can be maintained is
$$
\boxed{
a(t)=\left[\frac{(1-A_B)A}
{8\pi\sigma T_s^4}\right]^{1/2}t^{-\beta/2}}.
$$
This 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 $16\pi$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-315/4/a/solution