Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 4

Let the stellar radius and temperature be $R_{\ast }$ and $T_{\ast }$. Assume Bond albedo $A_B$, isotropic stellar emission, blackbody planetary emission, and complete redistribution over the tidally locked planet. The absorbed stellar power isAdding the isolated internal luminosity $4\pi R_p^2\sigma T_0^4$ and balancing the total against $4\pi R_p^2\sigma T_{\mathrm{eq}}^4$ gives the planetary equilibrium temperatureIf heat is reradiated uniformly only over the dayside, replace the denominator $4a^2$ by $2a^2$. The latter is often more plausible for inefficient redistribution on a tidally locked bare planet.

Use a single-layer greenhouse model whose atmosphere is transparent to incident stellar light and has infrared emissivity $ϵ$. Let $T_s$ and $T_a$ be the surface and atmospheric temperatures. Atmospheric balance givesAt the top of the atmosphere, the escaping flux is the directly transmitted surface radiation plus upward atmospheric emission:ThereforeFor a perfectly infrared-opaque one-layer atmosphere, $T_s=2^{1/4}{T}_{\mathrm{eq}}$; for $ϵ=0$, $T_s=T_{\mathrm{eq}}$.

In vacuum, a narrow ray bundle conserves both power and geometrical etendue $dA\cos \theta d\Omega$. Their ratio, the specific intensity, is therefore independent of source-receiver distance. Equivalently, geometric dilution reduces received power and apparent solid angle by the same inverse-square factor.

For an isotropically emitting isothermal blackbody atmosphere, the outward surface flux iswhere the last equality is bolometric. Multiplication by the emitting area gives the luminosity of a spherical blackbody

At a molecular line, the large opacity moves the optical-depth-one surface to lower pressure and higher altitude than the neighboring continuum. The Eddington-Barbier relation makes the emergent intensity approximately the Planck function at that layer. In an ordinary outward-cooling atmosphere, the line-forming layer is cooler and the feature is in absorption. In an atmospheric thermal inversion, it is hotter, so the spectral-line emission from an atmospheric thermal inversion exceeds the continuum brightness and the feature appears in emission.

Factors that can create or suppress inversions include the abundance of high-altitude optical absorbers such as TiO, VO, or atomic metals; the host star's irradiation level and spectral energy distribution; and clouds, hazes, composition, and day-night circulation, all of which alter where stellar and thermal radiation are absorbed.

At fixed temperature and pressure, a closed reacting system is in thermochemical equilibrium when its Gibbs free energy is minimal subject to elemental-abundance constraints. For every independent reaction,with a positive second variation in every allowed direction.

An atmosphere can be driven into disequilibrium chemistry in an exoplanet atmosphere by vertical mixing faster than chemical conversion, which causes chemical quenching; ultraviolet atmospheric photochemistry; and atmospheric escape. Lightning, energetic particles, horizontal transport, condensation, and rainout provide further mechanisms.