Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 4

Specific intensity is defined by $dE=I_{\nu }\cos \theta dAd\Omega d\nu dt$. A ray bundle in free space expands in area while its solid angle contracts by the same factor, soThus specific intensity does not obey an inverse-square law; the flux of an unresolved source does because its apparent solid angle scales as distance${}^{-2}$.

For a cold medium with coherent, isotropic, conservative scattering, the source function is the mean intensity $J_{\nu }$. With scattering optical depth increasing along the ray,andThe unscattered pencil beam is attenuated by $e^{-\tau }$ and reappears as a diffuse halo in other directions. Coherent scattering preserves frequency, and conservative scattering preserves total luminosity when all outgoing directions are collected.

For $A+B\rightleftharpoons C+D$, the forward and reverse rates areAt thermochemical equilibrium, $r_f=r_r$ by detailed balance, whileafter the appropriate standard-concentration factors are included. The standard reaction Gibbs free energy isConsequentlyFor other stoichiometries, the same argument uses the corresponding activity product and its standard-state factors.

Assume uniform global temperatures, all visible light not scattered back to space is absorbed by the surface, and use Kirchhoff's law of thermal radiation so that the atmospheric infrared emissivity is $\alpha$. If $F_{\ast }=\sigma T_{\ast }^4(R_{\ast }/a{)}^2$, the globally averaged absorbed stellar flux isThe atmosphere absorbs $\alpha \sigma T_{\mathrm{surf}}^4$ from the surface and emits from both faces, soSurface balance isTherefore the single-layer greenhouse model givesVisible scattering cools the surface, whereas infrared absorption and downward re-emission warm it.

On a planetary mass-radius relation, compressed rocky planets grow sublinearly, approximately $R\propto M^{0.25-0.3}$. Adding a hydrogen-helium envelope produces a rapid radius increase toward sub-Neptunes and gas giants. Around a few Jupiter masses the radius is nearly constant and then decreases as electron degeneracy pressure becomes important, approximately approaching the nonrelativistic degenerate scaling $R\propto M^{-1/3}$.

Brown dwarfs occupy roughly $13$ to $75$--$80$ Jupiter masses, with deuterium burning near the lower conventional boundary and sustained hydrogen burning beginning at the hydrogen-burning minimum mass. Low-mass main-sequence stars then have radii that increase with mass. Thus an isolated-body sketch has a rising rocky branch, a broad giant-planet/brown-dwarf radius maximum and decline, followed by a rising stellar branch.

Hot-Jupiter radius inflation places strongly irradiated hot Jupiters above the isolated giant-planet sequence. Irradiation retards cooling and contraction; additional proposed contributions include tidal heating, Ohmic dissipation, atmospheric circulation depositing energy at depth, enhanced opacity, and residual youth.

Young giant planets retain high formation entropy and radiate gravitational and thermal energy as they contract. Their infrared self-luminosity, especially at wide angular separation from a young nearby star, made the first directly imaged exoplanets much easier to detect than mature reflected-light planets. The inferred brightness depends on whether formation followed a high-entropy hot start or a low-entropy cold start.

By $10^{10}$ years, deuterium burning and most rapid Kelvin-Helmholtz contraction have ended. The intrinsic luminosity is governed mainly by the remaining interior entropy and ionic heat capacity, slow contraction supported by partially degenerate electrons, and the atmospheric opacity that controls escape of heat. Composition-dependent processes such as helium rain can add energy. For an irradiated planet, absorbed and reradiated starlight may dominate the observed luminosity, but it does not equal the planet's intrinsic cooling luminosity.