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

In an infrared-opaque single-layer greenhouse model, the atmospheric layer obeys $2\sigma T_a^4=\sigma T_s^4$. The outgoing planetary flux is $\sigma T_a^4$, while the globally averaged absorbed stellar flux is $(1-A_B)L/(16\pi a^2)$. Radiative equilibrium therefore givesHence the orbit at which the prescribed surface temperature can be maintained isThis assumes uniform redistribution, constant Bond albedo, unit longwave emissivity, a transparent atmosphere to starlight, and no internal heat. Without the greenhouse layer, replace $8\pi$ by $16\pi$.

In an isothermal hydrostatic atmosphere, pressure falls as $P(z)=P_0{e}^{-z/H}$. If a water-band line core becomes optically thick at pressure $P_{\mathrm{line}}$ and an opaque cloud fixes the nearby continuum at $P_{\mathrm{cl}}$, the exoplanet transmission spectrum feature spansA two-scale-height feature therefore requiresTaking a representative near-infrared water-band pressure $P_{\mathrm{line}}\sim 1\mathrm{mbar}$ gives $P_{\mathrm{cl}}\sim 7\mathrm{mbar}$, so the appropriate estimate is an exoplanet cloud deck top near $10\mathrm{mbar}$. The numerical value scales directly with the assumed line-core pressure.

Other explanations include a high mean molecular weight, subsolar water abundance, a colder terminator, atmospheric haze, patchy two-limb clouds, stellar contamination, or instrumental systematics. Optical scattering slopes, broader James Webb Space Telescope molecular coverage, repeated transits, secondary-eclipse spectra, phase curves, and precise mass and radius measurements can distinguish these possibilities.

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 $P_{\mathrm{rc}}\sim 50$--$200\mathrm{bar}$.

Approximate the atmosphere above $R_p$ as isothermal with constant gravity and atmospheric scale height $H$, soVertical transport over one scale height has eddy mixing timeThe chemical quench level satisfies ${\tau }_{\mathrm{chem}}(z_q)={\tau }_{\mathrm{mix}}$. Since ${\tau }_{\mathrm{chem}}=\eta z$,Above this level, mixing is faster than reaction and freezes the deeper abundance of A. Larger $K_{zz}$ moves the quench level deeper and raises $P_q$. Important examples are carbon monoxide–methane quenching and nitrogen–ammonia quenching; phosphine destruction is another tracer of vertical quenching.

Exoplanet clouds and atmospheric haze add scattering and absorption opacity to an exoplanet transmission spectrum. A high opaque deck truncates the slant path and mutes molecular bands, while small aerosol particles can produce a blue scattering slope; patchiness creates mixtures of clear and cloudy limbs. In an exoplanet emission spectrum, aerosols move the photosphere to lower pressure, weaken or reshape molecular features, alter the geometric albedo, and can heat or cool layers depending on their shortwave and longwave absorption.

Observed aspects of exoplanet atmospheric dynamics include eastward equatorial atmospheric superrotation inferred from shifted thermal hot spots, day-night heat transport measured by phase-curve amplitude, and high-altitude winds measured from Doppler shifts of resolved spectral lines. Time-variable phase curves and eclipse maps also reveal changing cloud patterns and storms.