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

Hydrostatic balance and the ideal-gas adiabatic temperature gradient implyThe radiative region is stable while $K\rho /T^3<g/c_p$. Equality at the radiative-convective boundary, together with $\rho =\mu P/(\mathcal{R}T)$, givesFor radiative diffusion carrying intrinsic flux $F_{\mathrm{int}}=\sigma T_{\mathrm{int}}^4$, $K=3\kappa F_{\mathrm{int}}/(16\sigma )$, soFor an irradiated hot Jupiter with $g\simeq 10\mathrm{m}{\mathrm{s}}^{-2}$, ${\nabla }_{\mathrm{ad}}\simeq 2/7$, $\kappa \simeq 10^{-2}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$, $T_{\mathrm{rc}}\simeq 1500\mathrm{K}$, and $T_{\mathrm{int}}\simeq 150$--$200\mathrm{K}$, this gives roughly $P_{\mathrm{rc}}\sim 50$--$200\mathrm{bar}$.