Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 315 / 2

Let brackets denote number densities and impose a local photochemical steady state. The atomic-oxygen and ozone balances areSubtracting gives $J_1[{\mathrm{O}}_2]=K_4[\mathrm{O}][{\mathrm{O}}_3]$. If photodissociation is the dominant direct ozone loss, $J_3[{\mathrm{O}}_3]\gg K_4[\mathrm{O}][{\mathrm{O}}_3]$, the second balance becomes $K_2[\mathrm{O}][{\mathrm{O}}_2][\mathrm{M}]\simeq J_3[{\mathrm{O}}_3]$. Eliminating atomic oxygen yields the Chapman ozone equilibriumHigh in the atmosphere ultraviolet photons make $J_1$ large but the third-body density is small; low down, $[\mathrm{M}]$ is large but O2-dissociating ultraviolet radiation has been absorbed. Their product peaks at intermediate altitude, producing an ozone layer.

Comparable hydrostatic thermal escape requires comparable Jeans escape parameter $\lambda =G{M}_{p}m/(k_{B}T{R}_{\mathrm{exo}})$. For the same escaping species and $R_{\mathrm{exo}}\simeq R_p$,Thus Jupiter at 5 au needs an exobase temperature at least about thirty times Earth's at the same irradiation to have comparable Jeans escape flux.

For the inner planet, the usable EUV power is $\eta \pi R_{\mathrm{exo}}^2{F}_{\mathrm{EUV}}$. If the binding energy per unit escaping mass is $G{M}_p/R_p$, energy-limited atmospheric escape givesThe time to lose a fraction $x$ of the planetary mass is thereforeThis neglects Roche-lobe reduction, radiative cooling, changes in radius and flux, and the planet's orbital evolution. If the gas is lifted only from $R_{\mathrm{exo}}$, replace $R_p$ in the denominator by $R_{\mathrm{exo}}$.

At exoplanet secondary eclipse, the full-phase planet-star flux ratio is the sum of reflected and thermal light. Approximating both bodies as unresolved blackbodies and taking wavelength-independent geometric albedo,The first term is a flat reflected-light level under the stated constant-albedo assumption. At short wavelength the cool planet lies in the Wien limit, so thermal emission is exponentially suppressed and reflection dominates. At long wavelength both spectra enter the Rayleigh-Jeans law, givingThe sketch therefore starts on the reflected plateau, rises where planetary thermal emission becomes important, and asymptotically approaches the long-wavelength plateau. This neglects spectral albedo features, phase dependence, stellar lines, and a nonisothermal planetary photosphere.

At fixed temperature and pressure, thermochemical equilibrium minimizes the Gibbs free energy subject to elemental conservation. For every independent reaction with stoichiometric coefficients ${\nu }_i$,Equivalently, forward and reverse rates satisfy detailed balance. For ideal gases,