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

Assume an ideal-gas atmosphere of constant mean molecular mass and approximately constant gravity. If temperature decreases linearly with altitude, write $dT/dz=-\Gamma$. Combining the ideal-gas law with hydrostatic equilibrium givesThe endpoint conditions determine the power law directly:

Differentiate the result of part i and use hydrostatic equilibrium:Thus the constant atmospheric lapse rate is

Apply the formal solution of the radiative transfer equation successively to the two isothermal layers at normal incidence. In local thermodynamic equilibrium,For optically thin layers this becomesThis assumes no scattering and negligible reflection between layers.

With $I_{\nu ,0}=B_{\nu }(T_0)$, the sign of each first-order term is set by whether that layer is hotter or colder than the incident blackbody.

The corresponding temperature profiles, from bottom to top, are respectively monotone decreasing, decreasing then increasing, and increasing throughout.

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.