Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 2 / c / ii / Solution

Convection begins when the radiative temperature gradient equals the adiabatic temperature gradient ${\nabla }_{\mathrm{ad}}$. To convert optical depth into pressure, assume a pressure-law opacity $\kappa ={\kappa }_0{P}^n$ and constant gravity. Hydrostatic balance givesHenceThe exact radiative-convective boundary is the positive solution ofDeep enough that the exponential term is negligible,which requires $n+1>4{\nabla }_{\mathrm{ad}}$. This exposes why constant opacity is inadequate for a molecular atmosphere: its limiting radiative gradient is $1/4$, below ${\nabla }_{\mathrm{ad}}\simeq 2/7$.

In a grey scaling, $A\sim T_{\mathrm{eq}}^4$ measures irradiation and $B\sim T_{\mathrm{int}}^4$ measures intrinsic flux. ThusFor a hot Jupiter with $T_{\mathrm{eq}}\sim 1500\mathrm{K}$ and $T_{\mathrm{int}}\sim 100\mathrm{K}$, irradiation pushes the boundary to hundreds of bars for typical increasing opacity. Jupiter has $T_{\mathrm{eq}}$ and $T_{\mathrm{int}}$ both of order $10^2\mathrm{K}$ and becomes convective near the bar scale. The estimate is order-of-magnitude because real opacities depend on both pressure and temperature.