Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 (Extrasolar Planets: Atmospheres and Interiors)

For the two-zone cloudy annulus transmission model, assume a uniform stellar disc, an opaque planet below $R_p$, $0\le H_c\le H$, and constant total slant optical depth within each radial zone. Neglect scattered light entering the beam and the planet's own emission. A ray transmits the fraction $e^{-\tau }$, so each zone blocks its projected area multiplied by $1-e^{-\tau }$.

The opaque disc, cloudy annulus and clear annulus have areas $\pi R_p^2$, $\pi [(R_p+H_c{)}^2-R_p^2]$ and $\pi [(R_p+H{)}^2-(R_p+H_c{)}^2]$, respectively. Thus the exoplanet transmission spectrum, expressed as total transit depth, isHere ${\tau }_{\lambda ,c}$ means the total optical depth on rays assigned to the cloudy zone. If it denotes cloud extinction alone, its exponent must instead contain the sum of cloud and gas optical depths. For a geometrically thin exoplanet atmosphere, the two atmospheric prefactors become $2R_p{H}_c/R_s^2$ and $2R_p(H-H_c)/R_s^2$.

An opaque exoplanet cloud deck gives the simpler expressionThe exoplanet cloud deck raises the wavelength-independent occulting radius and suppresses the clear atmospheric contribution. If $H_c=H$, the spectrum is flat at $(R_p+H{)}^2/R_s^2$; if $H_c=0$, the ordinary annulus model for transmission spectroscopy is recovered. Stellar limb darkening and varying tangent-ray optical depth would require an intensity-weighted radial integral rather than this constant-depth model.

Assume an isothermal atmosphere in hydrostatic equilibrium, with constant gravity $g$, mean molecular weight $\mu$, and an ideal gas equation of state. Use $h$ for the atmospheric scale height, to distinguish it from the total observable height $H$. Thenso the cloud-top pressure in an isothermal atmosphere isAn opaque, approximately grey exoplanet cloud deck can flatten the exoplanet transmission spectrum if the remaining gas above it has little wavelength-dependent extinction. If its top nearly reaches the observable atmospheric extent, $H_c\simeq H$, this gives $P_c\simeq P_s{e}^{-H/h}$. More generally, a top at $N$ atmospheric scale heights has $P_c/P_s=e^{-N}$; illustrative values $N=5$ and $10$ give $6.7\times 10^{-3}$ and $4.5\times 10^{-5}$.

A flat exoplanet transmission spectrum and $P_s$ alone do not determine a unique cloud-top pressure. The temperature, gravity, mean molecular weight and cloud altitude, or equivalent opacity information, are also needed. A large mean molecular weight can itself suppress spectral features, so the exoplanet cloud deck interpretation is conditional.

Assume that the optical continuum is dominated by Rayleigh scattering by particles much smaller than the wavelength, and that their column abundance is fixed. Their cross-section, hence the slant optical depth, scales as ${\tau }_{\lambda }\propto {\lambda }^{-4}$. Differentiating the annulus model for transmission spectroscopy exactly gives the logarithmic slope of total transit depthFor $A\ge 0$ and $B>0$, the slope lies between $-4$ and zero, but generally depends on wavelength. In the optically thin limit, ${\delta }_{\lambda }\simeq A+B{\tau }_{\lambda }$, so its slope is $-4B{\tau }_{\lambda }/(A+B{\tau }_{\lambda })$. It vanishes as the atmospheric signal becomes negligible compared with the opaque disc; it also approaches zero in the saturated optically thick limit.

The familiar value $m\simeq -4$ applies to the optically thin atmospheric excess,or formally to a model in which $A$ is negligible. For an ordinary thin exoplanet atmosphere, however, $B/A\simeq 2H/R_p\ll 1$, so neglecting the opaque planetary baseline in the total transit depth is not usually justified. A universal $m=-4$ for the logarithm of the total transit depth does not follow from the stated model.

In a stratified isothermal atmosphere, the related scattering slope of a transmission spectrum is instead $d{R}_{\mathrm{tr}}/d\ln \lambda =-4h$, obtained by moving the effective tangent level to keep its slant optical depth of order unity. Consequently $d\ln \delta /d\ln \lambda =-8h/R_{\mathrm{tr}}$ there. The radius slope, atmospheric-excess slope and total-depth slope are different observables.

This is a semi-grey irradiated atmosphere in radiative equilibrium. The quantities have these meanings:

The model assumes a static plane-parallel atmosphere, separate grey stellar and thermal bands with constant opacities, local thermodynamic equilibrium for thermal emission, negligible scattering, and transport by radiative transfer rather than convection. The Eddington closure approximation sets the thermal angular-moment ratio to $1/3$, and the upper boundary supplies the usual $2/3$ term. Treating the incoming radiation with representative direction cosine $1/\sqrt{3}$ gives its attenuation $e^{-\sqrt{3}\gamma \tau }$. Intrinsic flux enters from below, while the imposed external stellar flux is absorbed from above.

With constant gravity and hydrostatic equilibrium, $\tau \simeq {\kappa }_{\mathrm{IR}}P/g$, allowing conversion to an atmospheric pressure-temperature profile. This last relation additionally assumes constant thermal opacity. The grey treatment describes the energy balance approximately; it does not resolve individual molecular absorption lines.

The thermal optical depth increases inward, so an atmospheric thermal inversion requires $dT/d\tau <0$. Differentiating the semi-grey irradiated atmosphere profile givesTherefore the inversion criterion with intrinsic planetary flux isAn inverted layer exists at the top precisely whenFor $T_{\mathrm{int}}>0$, it extends overprovided the logarithm is positive. Otherwise the profile has no inverted interval. For a strongly irradiated hot Jupiter, $T_{\mathrm{int}}\ll T_{\mathrm{irr}}$ and the threshold is approximately $\gamma >1$; the intrinsic flux eventually restores an inward-increasing temperature at greater depth.

Large visible absorption opacity deposits stellar heat high in the exoplanet atmosphere. Gas-phase titanium monoxide and vanadium monoxide are candidate absorbers in sufficiently hot layers. An atmospheric cold trap or other condensate loss can remove them, while sufficiently high temperature and vigorous mixing can help keep them in the gas. The condition concerns absorbing opacity: highly reflective scattering alone does not deposit the required heat.

For uniform composition, the Schwarzschild criterion gives stability against convection whenA parcel displaced upward expands approximately adiabatically. If the ambient logarithmic temperature gradient is shallower than the adiabatic temperature gradient, the parcel cools more than its surroundings, becomes denser, and sinks back. Equality is neutral stability; a steeper gradient drives buoyant growth.

For a diatomic ideal gas with active translational and rotational degrees of freedom, $c_p/c_v=7/5$ and ${\nabla }_{\mathrm{ad}}=2/7$. If the infrared opacity is constant, $P\propto \tau$, so the profile's radiative temperature gradient isAn atmospheric thermal inversion has negative ${\nabla }_{\mathrm{rad}}$ and is stable by this criterion. Composition gradients require the Ledoux criterion; molecular dissociation and changing heat capacities can also change the numerical adiabatic temperature gradient.

Take $z$ to increase inward, so the positive coefficient $A$ describes an inward-increasing temperature. Constant gravity and hydrostatic equilibrium give $dP/dz=\rho g$, henceMatching the radiative temperature gradient to the adiabatic temperature gradient gives the formal local boundary relationThe same result follows from radiative diffusion: for constant upward thermal flux $F$ and Rosseland mean opacity $\kappa$, $A=3\kappa F/(16{\sigma }_{\mathrm{SB}})$.

There is an important limitation to treating $A$ as constant over the entire radiative layer. Integration from an irradiated outer boundary givesFor a diatomic ideal gas, ${\nabla }_{\mathrm{ad}}=2/7>1/4$, so this profile cannot actually reach a radiative-convective boundary. Formally, imposing a constant ${\nabla }_{\mathrm{ad}}$ would givewhich is positive only for $0<{\nabla }_{\mathrm{ad}}<1/4$. A finite boundary for a normal molecular atmosphere requires additional opacity, flux, or thermodynamic variation. The local matching formula is usable near a real boundary, but constant $A$ is not a complete global model of it. This is the convective stability of a constant-opacity irradiated atmosphere.

For the intended order-of-magnitude scaling, suppose the local values of $g/A$ and ${\nabla }_{\mathrm{ad}}$ are comparable for Jupiter and a hot Jupiter, and assume $T_b$ scales with planetary equilibrium temperature. Equal absorbed-flux factors around the same stellar luminosity give $T_b\propto a^{-1/2}$. Taking $a_J=5.2\mathrm{AU}$ and a representative close-in orbit $a_h=0.05\mathrm{AU}$ yieldsThis illustrates how irradiation can push a boundary much deeper. It is a conditional estimate calibrated from the supplied $0.1\mathrm{bar}$ reference, not a self-consistent prediction of the globally constant-$A$ model. Different intrinsic cooling flux, opacity, gravity, or atmospheric metallicity of a giant planet can substantially alter it.

Four mechanisms that can produce disequilibrium chemistry in an exoplanet atmosphere are:

The exoplanet transport and photochemical examples follow kinetic atmosphere calculations and models including horizontal transport; the condensate example is examined in cold-trap calculations.

Take the logarithm to base ten and write pressure in units of $1\mathrm{bar}$, so its argument is dimensionless. Continuity of the atmospheric pressure-temperature profile givesThusIf the logarithm means $\ln$, the equivalent constant is $C=900/\ln 100\simeq 195.4\mathrm{K}$. A plot of temperature against logarithmic pressure is vertical in each isothermal region and straight between the two endpoints.

The processes can be organized by the supplied pressure ranges, although the exact boundaries require reaction rates, irradiation and mixing information:

The profile identifies plausible chemical regimes, but does not fix their transition pressures by itself. In particular, cloud formation depends on the species-specific condensation curve, and ultraviolet processing depends on shielding.

For solar elemental abundances, take a molecular hydrogen- and helium-dominated atmosphere with atmospheric carbon-to-oxygen ratio near $0.55$. The principal oxygen-bearing molecule is generally water; the carbon and nitrogen carriers depend on both temperature and pressure.

At the cool $500\mathrm{K}$ observable upper layers, methane is the expected main carbon reservoir under chemical equilibrium, with abundant water and much less carbon monoxide. At the hotter layers approaching $1400\mathrm{K}$, carbon monoxide becomes the main carbon carrier and water contains much of the oxygen not bound in it. The relevant law of mass action follows fromLower temperature favors the exothermic methane-forming direction, while increasing pressure favors the side with fewer molecules. This explains why the change of dominant carrier cannot be specified by temperature alone.

Nitrogen is distributed between molecular nitrogen and ammonia. Cooling favors ammonia, but low pressure favors molecular nitrogen; it is therefore unsafe to call ammonia dominant throughout the $500\mathrm{K}$ low-pressure region. The exact partition requires the equilibrium constant for ${\mathrm{N}}_2+3{\mathrm{H}}_2\rightleftharpoons 2{\mathrm{NH}}_3$.

Carbon dioxide is usually a minor constituent at solar composition; hydrogen cyanide and acetylene are much less abundant than the principal carbon carriers in this oxygen-rich equilibrium case. Condensation can remove refractory species where a condensation curve is crossed. The robust cool-atmosphere expectation is a molecular hydrogen–helium background with methane and water, changing toward carbon monoxide in hotter layers. Precise mixing ratios require thermodynamic data and an explicitly specified elemental inventory after rainout.

Four possible molecular signatures of disequilibrium chemistry in an exoplanet atmosphere, measured through an exoplanet transmission spectrum or exoplanet emission spectrum, are:

The first two signatures can be consequences of the same process and are not independent evidence for two mechanisms. Changes in atmospheric carbon-to-oxygen ratio, atmospheric metallicity of a giant planet, exoplanet cloud coverage, and the atmospheric pressure-temperature profile can mimic abundance changes. A convincing inference compares several molecular bands with a chemically consistent equilibrium model rather than identifying one unusual band alone.

At fixed temperature and pressure, the atmospheric carbon-to-oxygen ratio controls the division of the elemental inventory among carbon monoxide, methane, water and other molecules. For an oxygen-rich mixture, water can remain after carbon monoxide consumes its share of oxygen. Increasing the atmospheric carbon-to-oxygen ratio reduces the oxygen available for water, particularly in the hotter layers where carbon monoxide is stable. Near or above unity there, water can become strongly depleted while excess carbon feeds methane, hydrogen cyanide and acetylene. This familiar hot carbon-rich behavior should not be applied unchanged to every cool layer: at $500\mathrm{K}$, methane formation can leave substantial water even at high atmospheric carbon-to-oxygen ratio.

Increasing the atmospheric metallicity of a giant planet at fixed elemental ratios increases the heavy-element inventory relative to molecular hydrogen and helium. Abundances of water and the major carbon-bearing molecules generally rise while molecular hydrogen remains dominant. In that regime, the law of mass action for $\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{CO}}_2+{\mathrm{H}}_2$ givesWhere carbon monoxide and water each scale roughly linearly with enrichment $Z$, carbon dioxide consequently scales approximately as $Z^2$. This scaling changes when molecular hydrogen ceases to dominate or chemical partitioning changes. Higher enrichment can also favor carbon monoxide over methane at fixed conditions, increase mean molecular weight, and reduce the atmospheric scale height and transmission-feature amplitudes. Cloud formation and atmospheric condensate rainout alter the observable elemental ratios relative to the bulk ones.

Use the standard local chemical quench level approximation: carbon monoxide-rich gas from the hot deep region is transported upward, and conversion becomes slower as the gas enters the cooler layers. Assume that neither a faster loss process nor strong compositional fractionation removes carbon monoxide above the quench level. For an effective mixing length $L$, the eddy mixing time is ${\tau }_{\mathrm{mix}}\simeq L^2/K_{zz}$, so outrunning conversion near $1\mathrm{bar}$ requiresThe usual order-of-magnitude choice is $L\simeq h_b$, the local atmospheric scale height at $T_b=1400\mathrm{K}$. Equal planetary mass and radius give the same gravity as Jupiter; with the same mean molecular weight, the atmospheric scale height scales linearly with temperature. To use the supplied $30\mathrm{km}$ reference, additionally adopt a representative Jovian reference temperature $T_J=150\mathrm{K}$. ThenUsing $k_{B}T/(\mu m_{H}g)$ directly with $\mu \simeq 2.3$ and Jovian $g\simeq 25\mathrm{m}{\mathrm{s}}^{-2}$ gives $h_b\simeq 200\mathrm{km}$ and $K_{zz}\gtrsim 4\times 10^5{\mathrm{m}}^2{\mathrm{s}}^{-1}$, the same order of magnitude. The supplied Jovian height is approximate and does not specify its reference temperature. Inserting $30\mathrm{km}$ unchanged for the hot gas would instead give $9\times 10^3{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and neglect this temperature scaling.

The pressure separation is relevant to a stronger, whole-column transport estimate. With constant gravity and mean molecular weight, the profile givesIf one additionally requires diffusion through the entire column within $10^5\mathrm{s}$, a sufficient conservative condition is $K_{zz}\gtrsim \Delta z^2/{\tau }_{\mathrm{chem}}\sim 10^7{\mathrm{m}}^2{\mathrm{s}}^{-1}$. It is not a necessary local quench condition: the chemical relaxation time is expected to become much longer in the cooler gas, so a longer total transit time can still preserve carbon monoxide.

The usual quench estimate is $K_{zz}$ of order $10^6{\mathrm{m}}^2{\mathrm{s}}^{-1}$ under the stated scale-height assumptions. A unique bound for survival to $10^{-3}\mathrm{bar}$ cannot be inferred from one reaction time without assumptions about its variation, the mixing length and upper-atmospheric losses.

Three observational signatures constrain exoplanet atmospheric dynamics:

Opacity, exoplanet cloud distributions, and pressure-dependent brightness temperature affect the phase-curve interpretation. These measurements constrain circulation through atmospheric models; a single offset or line shift does not uniquely determine a three-dimensional wind field.

The geometric albedo $A_g$ compares the planet's reflected brightness at full phase with that of a perfectly reflecting Lambertian surface in the form of a flat disc of the same projected area, illuminated by the same incident flux. The Bond albedo $A_B$ is the fraction of total incident stellar power reflected in all directions and over all wavelengths. They need not be equal: the planetary phase integral and the incident spectral weighting connect them.

For a blackbody star, the incident flux at orbital distance $a$ isAssume negligible intrinsic luminosity, thermal emissivity one, and efficient day-night heat redistribution giving uniform emission over the whole planet. The absorbed and emitted powers areUsing the Stefan–Boltzmann law gives the planetary equilibrium temperatureFor a representative emitting area encoded by a stellar flux redistribution factor $f$, the equivalent expression iswith $f=1/4$ for global uniform emission and $1/2$ for uniform dayside-only emission. The Bond albedo, rather than the geometric albedo, belongs in the energy budget. Internal heat, wavelength-dependent emissivity and imperfect redistribution require corresponding modifications.

Use the single-layer greenhouse model with a blackbody surface and one isothermal atmospheric layer. The layer is transparent to incoming stellar radiation, absorbs a fraction $\alpha$ of the surface's thermal radiation, and has the same thermal emissivity $\alpha$ by Kirchhoff's law of thermal radiation. Here $0\le \alpha \le 1$, and $T_e$ is the planetary equilibrium temperature defined by the absorbed global stellar flux ${\sigma }_{\mathrm{SB}}{T}_e^4$. Neglect convection, latent heat transport and intrinsic heat.

The layer absorbs $\alpha {\sigma }_{\mathrm{SB}}{T}_s^4$ and emits $\alpha {\sigma }_{\mathrm{SB}}{T}_a^4$ upward and downward. For $\alpha >0$, its energy balance givesThe surface receives the absorbed stellar flux plus downward atmospheric emission, soEliminating $T_a$ yieldsThe outgoing top-of-atmosphere flux is $(1-\alpha ){\sigma }_{\mathrm{SB}}{T}_s^4+\alpha {\sigma }_{\mathrm{SB}}{T}_a^4={\sigma }_{\mathrm{SB}}{T}_e^4$, confirming overall energy conservation. At $\alpha =0$, $T_s=T_e$ and the decoupled layer's temperature is not determined; at $\alpha =1$, $T_s=2^{1/4}{T}_e$. This warming is the restriction of thermal escape by the absorbing and emitting layer.

A mass-radius curve of solar-composition substellar objects must specify age, irradiation and composition. For a mature, weakly irradiated hydrogen–helium sequence, the qualitative mass-radius relation has three main trends:

The conventional deuterium-burning mass near $13$ Jupiter masses marks a commonly used giant-planet/brown dwarf classification scale, not a sharp discontinuity in radius or a universal formation boundary. Young brown dwarfs are larger and cool by contraction; irradiation, a heavy-element core and hot-Jupiter radius inflation shift planets away from a single solar-composition curve.

The drawing is a qualitative mature sequence, not an interpolation of a specified evolutionary model grid. Three relevant observations, using representative historical measurements, are:

Three major research directions, appropriate to the scientific questions posed in 2018, are:

These are research goals rather than claims that the relevant mechanisms or habitable atmospheres have already been established.