Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 (Extrasolar Planets: Atmospheres and Interiors)

Treat the morning and evening halves of the day-night terminator as independent, isothermal, hydrostatic atmospheres with the same reference radius $R_0$, reference pressure $P_0$, composition, gravity $g$, and extinction coefficient ${\kappa }_{\lambda }$. Their atmospheric scale heights areFor $H_i\ll R_0$, the slant optical depth of half $i$ at tangent altitude $z$ is approximatelyThe standard isothermal effective altitude is thereforeup to a wavelength-independent choice of reference radius. The two semicircular limbs add in projected area, so the exoplanet transmission spectrum isEquivalently, $R_{\mathrm{tr}}^2=[(R_0+z_m{)}^2+(R_0+z_e{)}^2]/2$. Differentiating with respect to $\log {\kappa }_{\lambda }$ givesThus a homogeneous retrieval measures, to leading order,provided the opacity and composition do not themselves differ between the two limbs.

Along a ray, the radiative transfer equation has the formal solution of the radiative transfer equationfor each isothermal region in local thermodynamic equilibrium, neglecting scattering. For a self-luminous atmosphere with no incident intensity from below at the relevant photosphere, and in the optically thick limit, $I_{\nu ,i}=B_{\nu }(T_i)$.

The observed radiative flux is the projected-disc integralThe inner region occupies projected fraction $f={\sin }^2{\theta }_T$, henceFor finite optical depths, each Planck law function in this formula is replaced by its corresponding emergent intensity above.

A homogeneous interpretation assigns the planet a brightness temperature $T_b$ satisfyingAt a fixed wavelength let $x=hc/(\lambda k_B)$ andInverting the Planck law givesAt $\lambda =20\mu \mathrm{m}$, $x=719.4\mathrm{K}$. With ${\theta }_T=\pi /4$, $T_1=1500\mathrm{K}$, and $T_2=1000\mathrm{K}$, one obtainsThe Rayleigh-Jeans law would give the nearly identical area-weighted estimate $1250\mathrm{K}$.

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.

LetThe enclosed mass isThis is continuous at the core boundary. Hydrostatic equilibrium gives $dP/dr=-Gm(r)\rho (r)/r^2$, with $P(R_p)=0$. DefineSince $F^{\prime }(r)=Gm(r)(A-br)/r^2$, the mantle pressure isInside the uniform core, the shell theorem gives $m(r)=4\pi {\rho }_c{r}^3/3$, so

The mantle density decreases outward when ${\rho }_m\ge 0$. Requiring a nonnegative surface density gives the immediate physical boundThe measured planetary mass also fixesEquivalently,and a viable model requires this value to obey the boxed bound.

Assume a thin isothermal atmosphere, so $g\simeq G{M}_p/R_p^2$ and the surface area are constant through it. Integrating hydrostatic equilibrium from base pressure $P_0$ to negligible top pressure gives atmospheric column mass $P_0/g$. ThereforeEquivalently, $M_{\mathrm{atm}}\sim 4\pi R_p^2{\rho }_{0}H$, with ${\rho }_0=P_0\mu m_H/(k_B{T}_0)$ and $H=k_B{T}_0/(\mu m_{H}g)$; the explicit temperature dependence cancels.

The location between pure-silicate and pure-water planetary mass-radius curves does not determine a unique composition. Possibilities include a rocky iron-silicate interior with a deep water layer, a rock-water mixture, or a mostly rocky planet whose small hydrogen-helium envelope inflates its radius. The atmosphere could accordingly be hydrogen-helium, steam-rich, carbon-dioxide-rich, or secondary gas released from the interior. Residence in the circumstellar habitable zone constrains stellar flux but does not by itself distinguish these models or guarantee surface liquid water.

For an atmosphere spanning fixed base and photospheric pressures,Thus two compositions haveAt equal temperature, a hydrogen-helium atmosphere with $\mu \simeq 2.3$ is abouttimes thicker than a steam atmosphere with $\mu \simeq 18$. This large difference is one reason atmospheric composition contributes strongly to the exoplanet interior-composition degeneracy.

The stellar flux at the orbit isLet $ϵ$ be the fraction of the planet's total absorbed power transported to the night side, and let $A_B$ be its Bond albedo. Equating transported power to nightside blackbody emission givesHence the nightside equilibrium temperature isPerfect global day-night heat redistribution has $ϵ=1/2$ and gives $T_n=T_s\sqrt{R_s/a}[(1-A_B)/4{]}^{1/4}$; without redistribution, stellar heating alone gives $T_n=0$ in this idealization. Efficient atmospheric or oceanic transport prevents volatile cold trapping and broadens the habitable region of a tidally locked planet, whereas a thin atmosphere can leave a frozen night side even when the day side is temperate.

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.