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

Optical depth increases inward, so an atmospheric thermal inversion occurs where temperature decreases with $\tau$. Differentiation givesTherefore the inversion condition in the observable atmosphere isIn particular, an inversion reaches the top when $\beta C>B$. A hot Jupiter can satisfy this when visible absorbers such as atomic metals, metal oxides, or negative hydrogen absorb incident starlight above the infrared photosphere. This makes the shortwave opacity large relative to the thermal opacity and deposits heat at low pressure.

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.