Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 (Extrasolar Planets: Atmospheres and Interiors)

Let ${\tau }_{\nu }$ increase downward and let $\mu >0$ be the outward direction cosine. Assume a plane-parallel, static, nonscattering atmosphere in local thermodynamic equilibrium, so the source function is $B_{\nu }[T(P)]$. Hydrostatic balance givesThe formal solution of the radiative transfer equation between $P_1>P_2$, corresponding to ${\tau }_1>{\tau }_2$, isFor an isothermal atmosphere, $B_{\nu }(T)$ is constant andIf the lower boundary is itself thermalized, $I_{\nu ,1}=B_{\nu }(T)$ and the upward intensity remains the blackbody value.

At secondary eclipse the removed monochromatic planetary flux is $\pi B_{\lambda }(T_p)R_p^2/d^2$, while the stellar flux is $\pi B_{\lambda }(T_s)R_s^2/d^2$. The thermal exoplanet secondary eclipse depth is thereforeA hot Jupiter has tiny visible thermal contrast, rapidly improving contrast toward the infrared, and molecular absorption or emission features superposed on its continuum. Reflected light may add a visible component. In the Rayleigh-Jeans law limit, $B_{\lambda }(T)\propto T/{\lambda }^4$, soa wavelength-independent asymptote for ideal blackbodies.

The peak near $0.5\mu \mathrm{m}$ can be reflected starlight carrying the stellar spectral shape, while the $1\mu \mathrm{m}$ peak can be thermal emission from a very hot, young, or strongly irradiated planet. Wien displacement law then suggests $T_s\simeq 5800\mathrm{K}$ and $T_p\simeq 2900\mathrm{K}$, so $T_p/T_s\simeq 1/2$.

Assume both objects are in the far-infrared Rayleigh-Jeans law regime. Fromone obtains $R_p/R_s\simeq \sqrt{2\times 10^{-3}}\simeq 0.045$. With $R_s=0.5R_{\odot }$ and $R_J\simeq 0.10R_{\odot }$,of order a Neptune radius. This estimate is sensitive to the peak interpretation and blackbody assumptions.

If the system transits, secondary-eclipse emission spectroscopy is especially effective because it separates planetary light from starlight and directly measures the favorable $10^{-3}$ infrared contrast. For a sufficiently wide orbit, direct-imaging spectroscopy is preferable because the hot planet is self-luminous and can be spatially separated from its star.

The measured mass and radius illustrate the exoplanet interior-composition degeneracy. Three plausible structures are:

An exoplanet transmission spectrum can distinguish a large-scale-height hydrogen atmosphere from compact steam or carbon-dioxide atmospheres. Molecular abundances, clouds, atmospheric mean molecular mass, and atmospheric escape constrain the envelope mass and evolution. An exoplanet emission spectrum constrains temperature, albedo, and internal cooling. A detected hydrogen-rich atmosphere favors the first scenario, a water-dominated high-molecular-weight spectrum supports the second, and a thin secondary atmosphere supports the third, although clouds can preserve degeneracies.

Potential surface habitability requires a stable liquid-water interface, a temperature and pressure outside runaway-greenhouse and high-pressure-ice regimes, and an ${\mathrm{H}}_2$ inventory large enough for collision-induced greenhouse warming but not so deep that pressure and temperature sterilize the ocean. Long-term atmospheric retention, tolerable stellar ultraviolet and particle flux, nutrient cycling, and chemical disequilibrium compatible with metabolism are also needed. A location in the circumstellar habitable zone alone is insufficient.

For $M_p=5M_{\oplus }$ and $R_p=2R_{\oplus }$,Taking $T=300\mathrm{K}$ and hydrogen-rich mean molecular weight $\mu =2.3$ gives the atmospheric scale heightIf a strong feature spans $N_H=5$ scale heights and $R_s=0.5R_{\odot }$, the atmospheric spectral-feature amplitude isClouds, hazes, refraction, and smaller molecular abundance reduce this estimate.

Hydrostatic equilibrium and mass conservation areFor $P=K{\rho }^{1+1/n}$, setEliminating $m$ gives the Lane-Emden equationIf ${\xi }_1$ is its first zero, thenEliminating ${\rho }_c$ at fixed $K$ givesso one convenient choice is $\boxed{\alpha =n-1,\beta =3-n}$. Equivalently, one may take $\beta =1$ and $\alpha =(n-1)/(3-n)$.

The index $n=0$ describes an incompressible constant-density body, approximating a weakly compressed small rocky planet. The index $n=3/2$ describes a fully convective monatomic ideal gas or nonrelativistic electron degeneracy, as in a low-mass star, brown dwarf, or nonrelativistic white dwarf.

For constant density ${\rho }_c$,Imposing $P(R_p)=0$ and using $g=G{M}_p/R_p^2=4\pi G{\rho }_c{R}_p/3$ gives

Assume a spherical blackbody planet, uniform reradiation, constant Bond albedo $A_B$, and negligible internal luminosity. Radiative equilibrium givesWith $L_s=K{M}_s^{\beta }$,The inner and outer circumstellar habitable zone boundaries are obtained by setting $T=T_{\max }$ and $T=T_{\min }$ respectively:Real boundaries require wavelength-dependent albedo, clouds, greenhouse effects, and the stellar spectrum, which this equilibrium-temperature model neglects.

For an Earth-mass planet with a nitrogen-carbon-dioxide-water atmosphere, commonly quoted conservative Solar boundaries are approximatelyThe inner edge is governed by moist-greenhouse water loss and ultimately the runaway greenhouse. The outer edge is the maximum greenhouse outer habitable-zone limit, where additional carbon dioxide condenses or increases reflected light faster than it increases infrared trapping. Optimistic empirical boundaries, based on recent Venus and early Mars, are broader, roughly $0.75$ to $1.77\mathrm{AU}$. The values depend on clouds, planetary mass, atmospheric inventory, and climate assumptions.

First, an Earth-like atmosphere produces signals of only order parts per million in transit and planet-star contrasts near $10^{-10}$ in reflected light, demanding extreme photon collection, calibration, and starlight suppression. Second, stellar spots, faculae, flares, and spectral variability can imitate or overwhelm atmospheric features, especially around active M dwarfs. Third, long orbital periods provide few transits, while clouds, hazes, refraction, and overlapping molecular bands make the resulting sparse spectra difficult to interpret uniquely.

Essential requirements include a persistent liquid solvent, a usable free-energy source, accessible biogenic elements, and environmental stability over evolutionary times. Earth supplies liquid water, sunlight and chemical redox gradients, carbon-nitrogen-phosphorus chemistry, and billion-year climate stability. Subsurface oceans may instead use tidal or radiogenic energy, while a Hycean planet is a proposed hydrogen-covered ocean environment.

Four major modifiers are atmospheric mass and greenhouse composition, which set surface pressure and temperature; host-star ultraviolet, flares, and winds, which affect photochemistry and escape; interior evolution, volcanism, and carbon cycling, which replenish gases and stabilize climate; and orbital or rotational architecture, including eccentricity, obliquity, synchronous rotation, giant impacts, and stabilizing or destabilizing companions. A magnetic field may reduce some charged-particle erosion but is neither sufficient nor universally necessary.

No single molecule establishes life. A persuasive exoplanet biosignature requires atmospheric and planetary context and exclusion of abiotic production.

An Earth analogue has an iron-silicate interior and a thin secondary nitrogen-dominated atmosphere with water and carbon dioxide. Contextual coexistence of oxygen or ozone with methane and surface-water indicators would be a target biosignature. Its small scale height and extreme reflected-light contrast put a true Sun-Earth analogue beyond routine current atmospheric work, but future large direct-imaging missions and the Extremely Large Telescope target nearby terrestrial planets.

A temperate M-dwarf rocky planet has a similar solid interior but may retain carbon-dioxide, nitrogen, or water atmospheres under synchronous rotation. Water, carbon dioxide, methane, and contextual oxygen chemistry are observable targets. Its small host gives deeper transits, so the James Webb Space Telescope and large ground telescopes can test nearby systems, although stellar activity and atmospheric erosion complicate interpretation.

A Hycean planet or water-rich sub-Neptune has a water-rich interior or ocean below a hydrogen-rich atmosphere. Methane, carbon dioxide, ammonia, water, and proposed sulfur-bearing biosignatures must be interpreted against abiotic photochemistry. Its large radius and hydrogen scale height make transmission spectroscopy comparatively favorable for JWST and future Ariel space telescope surveys, but whether a clement ocean exists beneath the atmosphere remains model-dependent.

The formal solution of the radiative transfer equation and the Eddington-Barbier relation imply that each wavelength samples the temperature near its own photosphere. High opacity moves the emitting level upward. If temperature decreases upward, a molecular band samples cooler gas than the continuum and appears in absorption. If an atmospheric thermal inversion makes temperature increase upward, the same band can appear in emission.

In hot Jupiters, strong optical and ultraviolet absorption by TiO, VO, atomic metals, metal hydrides, or hydrogen-minus opacity can heat low-pressure gas; low infrared cooling efficiency and stellar activity also matter. Solar-System inversions arise from selective absorption, including ozone ultraviolet heating on Earth, methane and hydrocarbons on giant planets, and aerosols or dust. Around M dwarfs, water, methane, carbon dioxide, and hazes absorb the star's near-infrared radiation high in a sub-Neptune atmosphere.

On a temperate planet, a stratospheric inversion can provide a cold trap that limits water loss and can alter photochemistry and circulation. Excess upper-atmosphere heating can instead weaken the cold trap, promote escape, or create hostile surface conditions. Earth's ozone-heated stratosphere is the standard example of a habitable planet with an inversion.

Assume the lower atmosphere is optically thick, hydrogen-helium dominated, convective, and approximately ideal with constant $\gamma =7/5$. The convective temperature gradient is the dry adiabatIf $T_0$ is the temperature at $P_0=0.1\mathrm{bar}$ at the base of the inverted stratosphere, thenIn a radiative lower layer with constant upward flux $F$ and Rosseland opacity ${\kappa }_R$, diffusion would instead giveThe actual profile follows the shallower stable radiative gradient until it exceeds the adiabatic value, where convection begins.

Strongly irradiated giant exoplanets commonly develop broad eastward equatorial atmospheric superrotation, producing an eastward thermal hot-spot offset. Their large permanent day-night forcing also drives global overturning, waves, and imperfect heat redistribution measured by phase curves. Jets, vortices, clouds, and storms can vary with time and produce changing phase curves or eclipse maps.

Solar-System giants receive much weaker day-night forcing, rotate rapidly, and often emit substantial internal heat. They consequently show many narrow alternating zonal jets, long-lived vortices such as Jupiter's Great Red Spot, moist convection, and banded clouds. Both groups exhibit waves, jets, storms, and equatorial dynamics, but hot Jupiters are dominated by stellar irradiation and often have fewer, broader jets, whereas Solar-System giants are shaped more strongly by rapid rotation and internal convection.

Three observable signatures of disequilibrium chemistry in an exoplanet atmosphere are vertically quenched carbon monoxide in methane-favored cool layers or methane in carbon-monoxide-favored hot layers; quenched ammonia-nitrogen chemistry; and photochemical products such as hydrogen cyanide, sulfur dioxide, hydrocarbons, or hazes. Their abundances constrain mixing, ultraviolet irradiation, reaction kinetics, and atmospheric escape.

Solar-System methane and gravity-field measurements imply that Jupiter is enriched in heavy elements by a few times solar, Saturn by roughly an order of magnitude, and Uranus and Neptune by tens of times solar. This motivates an inverse planetary mass-metallicity trend: lower-mass planets acquire a larger heavy-element fraction from solids relative to hydrogen-helium gas. Carbon-to-oxygen and nitrogen abundances also constrain ice lines, planetesimal accretion, core erosion, and migration.

Uniform exoplanet transmission and emission surveys can test whether this trend is universal across mass, equilibrium temperature, age, and host metallicity. Measurements of water, methane, carbon monoxide, carbon dioxide, ammonia, and sulfur species can separate bulk enrichment from carbon-to-oxygen ratio, clouds, and disequilibrium, thereby constraining where and how planets formed.

The Hubble Space Telescope uses ultraviolet through near-infrared transit and eclipse spectroscopy. It established detections of alkali metals, water, aerosols, and escaping hydrogen but has a small aperture and restricted continuous infrared coverage. High-resolution ground facilities such as the Very Large Telescope separate Doppler-shifted planetary lines from telluric and stellar spectra and directly image young giants; the atmosphere and thermal background limit broad-band precision. The James Webb Space Telescope uses NIRISS, NIRSpec, NIRCam, and MIRI for roughly visible-to-mid-infrared transit, eclipse, phase-curve, and direct-imaging spectroscopy. It has measured broad molecular inventories, including the clear carbon-dioxide feature and photochemical sulfur dioxide in WASP-39 b.

JWST improves on earlier facilities through its 6.5-metre collecting area, cold space environment, stable time-series spectroscopy, and broad infrared wavelength coverage spanning several bands of water, carbon dioxide, methane, carbon monoxide, ammonia, and sulfur compounds. The Ariel space telescope is designed for a uniform atmospheric census of about one thousand planets, while the Extremely Large Telescope will add spatial resolution and high-dispersion spectroscopy for nearby giant and terrestrial planets.

Three directions enabled or sharply advanced by JWST are chemically complete retrievals across multiple molecular bands rather than isolated detections; atmospheric tests of smaller and cooler sub-Neptunes and rocky planets; and measurements of photochemistry, clouds, heat redistribution, and vertical thermal structure through repeated transit, eclipse, and phase-resolved spectra.