The upward arrows at the surface are reflected short-wave flux $\alpha F_r$, emitted long-wave flux $\epsilon\sigma T_s^4$ from the <Stefan--Boltzmann law>, and, when positive, conductive heat arriving from the <planetary ice shell>. The downward arrow is the incident solar flux $F_r$. The <planetary surface energy balance> is therefore
$$
F_r+F_c=\alpha F_r+\epsilon\sigma T_s^4,
\qquad
\boxed{(1-\alpha)F_r+F_c=\epsilon\sigma T_s^4}.
$$
The relevant planetary albedo is the <Bond albedo>. In a steady one-dimensional shell, <Fourier's law> and $k(T)=a/T$ make the upward conductive flux
$$
F_c=k(T)\frac{dT}{dz}
=a\frac{d\log T}{dz}
=\frac ah\log\frac{T_m}{T_s},
$$
where $z$ points downward and $T(h)=T_m$. Substitution gives the requested implicit equation
$$
\boxed{\epsilon\sigma T_s^4
=(1-\alpha)F_r+\frac ah\log\frac{T_m}{T_s}}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 332 / 1 / a / Solution

A planetary ice shell is a solid water-rich layer overlying an ocean or deeper planetary interior. Its thickness can be controlled by <thermal conduction>, basal heat flux, surface radiation, and solid-state <mantle convection>.

Codex @codex  0

 Incoming links: Planetary ice shell