Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 315 / 4

Transmission spectroscopy, eclipse/emission spectroscopy and direct-imaging spectroscopy probe atmospheres in complementary geometries.

A exoplanet transmission spectrum uses the wavelength dependence of primary-transit depth to measure slant-path absorption at the limb. Its long path length gives sensitivity to trace absorbers, but only transiting systems qualify; exoplanet clouds, stellar heterogeneity and the reference-radius/abundance degeneracy complicate interpretation, and the result samples the terminator rather than the dayside.

An exoplanet secondary eclipse isolates the dayside thermal/reflected spectrum by removing the planet behind the star. It constrains brightness temperatures, molecules and thermal inversions, but the planet/star contrast is small, a suitable orbital geometry is needed, and temperature-abundance degeneracies remain. A exoplanet thermal phase curve extends this method to longitudinal information.

Exoplanet direct imaging spatially separates a planet from its star and permits repeated spectroscopy without requiring a transit. It is effective for young, luminous, widely separated giant planets, but demands high contrast and angular separation; old small planets close to their stars are difficult. Inferred gravities, radii and masses can depend strongly on atmosphere and evolutionary models.

Four mechanisms are Jeans escape, hydrodynamic escape, Roche-lobe overflow and nonthermal atmospheric escape.

Jeans escape is loss from the high-speed tail of a nearly Maxwell-Boltzmann velocity distribution near the exobase. It is sensitive to the escape parameter ${\lambda }_J=G{M}_p\mu m_H/(k_B{T}_{\mathrm{exo}}{R}_{\mathrm{exo}})$ and favors light species in a hot, weakly bound atmosphere.

Hydrodynamic escape occurs when strong XUV heating drives a bulk outflow, which can entrain heavier species. An illustrative energy-limited scale is $\dot{M}\sim \eta \pi R_p^3{F}_{\mathrm{XUV}}/(G{M}_p{K}_{\mathrm{tide}})$ when the absorption and planet radii are comparable; efficiency, radiative losses and recombination can invalidate that simple limit.

Roche-lobe overflow removes gas through the low effective-potential barrier toward the inner Lagrange point when the extended atmosphere approaches the Roche lobe. Tides can also assist a wind before actual overflow.

Nonthermal atmospheric escape includes ion pickup, sputtering, charge exchange and energetic photochemical products. Stellar-wind interactions and photoionization supply particles with escape energies not represented by the local thermal tail. These mechanisms can act together rather than constituting four mutually exclusive evolutionary states.

Three useful approaches are self-consistent one-dimensional models, atmospheric retrievals and three-dimensional circulation models.

A one-dimensional radiative-convective equilibrium model solves energy balance and radiative transfer, typically with a prescribed gravity, irradiation and composition or a chemical-equilibrium/kinetic calculation. It predicts physically coupled temperature profiles and spectra economically, but averages away longitude/latitude structure and inherits opacity, exoplanet cloud and chemistry assumptions.

An atmospheric retrieval parametrizes temperature, gas abundances, exoplanet clouds and other nuisance quantities and fits them to observations through a forward radiative-transfer model. It quantifies parameter constraints and degeneracies without enforcing every theoretical relation, but limited data can leave nonunique answers and priors/model structure can dominate.

A general circulation model solves rotating fluid dynamics with radiative heating/cooling and, when feasible, chemistry and exoplanet cloud transport. It predicts winds, phase curves and horizontal contrasts, but is computationally costly and still needs approximations for unresolved turbulence, exoplanet cloud microphysics and chemical networks. The approaches answer different questions and can be used jointly.

In the 2016 observational context, appropriate choices are as follows.

For atomic species, use optical transit spectra of Na I near $0.589\mu \mathrm{m}$ and K I near $0.770\mu \mathrm{m}$, with Hubble Space Telescope/STIS or ground-based high-resolution instruments on Very Large Telescope/Keck telescope-class telescopes. Neutral/ionized ultraviolet metal lines offer additional diagnostics with Hubble Space Telescope. Resolving a line's shape and distinguishing stellar contamination matters as much as locating its wavelength.

For molecules, use near/mid-IR spectroscopy: water bands near $1.4$, $1.9$ and $2.7\mu \mathrm{m}$, CO near $2.3$ and $4.6\mu \mathrm{m}$, methane near $3.3\mu \mathrm{m}$, and ${\mathrm{CO}}_2$ near $4.3$ or $15\mu \mathrm{m}$. Hubble Space Telescope/WFC3 samples water around $1.1$--$1.7\mu \mathrm{m}$; ground-based Very Large Telescope/CRIRES or Keck telescope/NIRSPEC spectra target accessible near-IR windows such as CO at $2.3\mu \mathrm{m}$. Warm Spitzer Space Telescope provides $3.6$ and $4.5\mu \mathrm{m}$ photometry, while cryogenic Spitzer Space Telescope spectra/longer-wavelength data are archival by 2016. A broadband eclipse point alone does not uniquely identify a molecule.

For atmospheric escape, use ultraviolet transit spectroscopy of H I Ly$\alpha$ at $121.6\mathrm{nm}$, and suitable C/O/metal lines, with Hubble Space Telescope/STIS or COS. Extended, velocity-shifted absorption beyond the optical planet suggests an escaping envelope, although stellar variability, geocoronal emission and interstellar absorption require treatment. Ground-based H$\alpha$ at $656.3\mathrm{nm}$ probes excited hydrogen and is a complementary rather than interchangeable measure of neutral-hydrogen escape.

Three signatures are a displaced exoplanet thermal phase curve maximum, the day-night temperature contrast, and wind-induced Doppler shifts.

The maximum of an exoplanet thermal phase curve need not coincide with secondary eclipse. An eastward hot spot and a maximum before eclipse indicate eastward advection/atmospheric superrotation, provided reflected light and exoplanet clouds are accounted for.

The amplitude and shape of an exoplanet thermal phase curve measure day-night heat redistribution: a warm nightside and reduced day-night contrast imply effective transport relative to radiative relaxation time in a planetary atmosphere. Wavelength-dependent contrasts can reveal different circulation efficiencies at different contribution pressures.

High-resolution transmission or emission spectra can show line-centroid shifts/broadening relative to the orbital velocity. A limb blueshift is consistent with a dayside-to-nightside flow toward the observer; rotation and limb asymmetry affect the pattern. Such Doppler spectroscopy probes velocity directly, while phase curves probe the transported temperature pattern.

Three observable signatures are a blue optical transmission slope, muted molecular transmission features, and enhanced reflected optical light.

Small-particle atmospheric haze can increase the apparent transit radius toward short wavelengths, roughly $0.3$--$1\mu \mathrm{m}$. For Rayleigh scattering-like extinction, $\sigma \propto {\lambda }^{-4}$ and an isothermal atmospheric scale height model gives $d{R}_p/d\ln \lambda \simeq -4H$. Larger particles instead produce a flatter, approximately grey continuum.

An optically thick exoplanet cloud deck can truncate the slant path through gas and flatten molecular bands in a exoplanet transmission spectrum, for example over $0.4$--$2.5\mu \mathrm{m}$ and the water band around $1.4\mu \mathrm{m}$. The observed quantity is reduced band-to-continuum radius contrast, not necessarily a low molecular abundance.

exoplanet cloud scattering can produce a high geometric albedo and distinctive optical secondary eclipses/phase curves around $0.4$--$0.9\mu \mathrm{m}$. Longitudinal exoplanet cloud variation can shift a reflected-light phase maximum. A spectral slope or flat spectrum alone is not a unique exoplanet cloud identification: particle size, gas scattering, metallicity and stellar contamination must be considered together.

A super-Earth is a planet more massive than Earth but below the usual ice-giant scale, conventionally about $1$--$10M_{\oplus }$. The term does not guarantee an Earth-like or rocky composition; water-rich planets and objects with small H/He envelopes can lie in this range.

The most direct constraints are transit radius and a mass from radial velocities or transit-timing variations, giving $\bar{\rho }=3M/(4\pi R^3)$. Comparing these with interior models constrains the possible iron core, silicate mantle, water layer and gas envelope, but many mixtures share one mass and radius.

Additional observables include atmospheric spectra/mean molecular weight and escape signatures, which test whether a low-density envelope is H/He or heavier volatiles; stellar Fe/Mg/Si abundances as priors on refractory composition; and age, irradiation and orbital history, which control thermal expansion and envelope survival. Where measurable, a Love number or related apsidal response constrains central concentration. Mass and radius constrain families of interiors, not a unique composition; atmospheric and dynamical information can reduce this exoplanet interior-composition degeneracy.

A primary planetary atmosphere is gas captured from the protoplanetary nebula, typically H/He-rich. A secondary planetary atmosphere is assembled or substantially replenished after formation by outgassing, volatile delivery and processing, rather than retaining the original nebular envelope.

Jupiter, Saturn, Uranus and Neptune retain primary H/He envelopes, although their heavy-element enrichment and deep interiors differ. Venus, Earth and Mars have predominantly secondary present-day atmospheres, shaped by volcanic outgassing, impacts, escape and, on Earth, biology. Mercury has a tenuous surface-derived exosphere rather than a substantial retained primary atmosphere. The classification describes origin and evolution, not simply whether a current molecule is heavy or light.

A thermal inversion is outward-increasing temperature over part of an atmosphere. In an irradiated radiative model, an upper absorber that captures incoming stellar energy efficiently compared with its infrared cooling can create an inversion. Early hot-Jupiter models proposed titanium monoxide/vanadium monoxide as strong visible absorbers, and an irradiation-based division between hotter titanium monoxide/vanadium monoxide-rich inverted atmospheres and cooler non-inverted atmospheres followed.

The simple temperature classification needed refinement. Condensation and vertical or day-night atmospheric cold traps can remove Ti/V even when the observed dayside is hot; settling competes with vertical mixing. Bulk C/O and metallicity determine whether oxygen-bearing absorbers are available, so irradiation alone does not specify their abundance. Photochemistry, stellar UV activity and alternative absorbers were also proposed to affect inversion formation or destroy an absorber; such correlations were hypotheses rather than a unique established cause.

Observationally, molecular bands in emission rather than absorption can diagnose an inverted contribution region, but sparse broadband eclipse measurements are degenerate with chemistry. By 2016, spectroscopic constraints had weakened some early inversion claims and supported the need for planet-specific opacity, abundance and transport models. The key development is from a single irradiation threshold to a composition- and transport-dependent visible/infrared opacity balance, tested by spectra rather than assuming every very hot planet must have the same inversion.

Three solar-system effects illustrate chemical disequilibrium. First, vertical mixing carries carbon monoxide from hot deep regions into cooler observable layers of giant planets, where its local-equilibrium abundance would be small. Second, phosphine survives in the observable atmospheres of Jupiter and Saturn because deep supply competes with its destruction, making it a tracer of transport. Third, methane photolysis creates hydrocarbons such as ethane and acetylene, and can help form upper-atmospheric atmospheric hazes in giant planets and Titan.

Three exoplanet effects are complementary. First, carbon monoxide–methane quenching can keep carbon monoxide abundant and methane depleted in cool observable gas, including hot-Neptune/giant atmospheres that would locally favor methane. Second, atmospheric photochemistry dissociates methane/ammonia and produces radicals, hydrogen cyanide, hydrocarbons and possibly atmospheric hazes, changing spectral bands from their equilibrium strengths. Third, horizontal chemical quenching can preserve hot-dayside carbon chemistry on the cooler nightside/terminator and reduce the abundance contrast expected from separate local-equilibrium columns.

Transport, photon-driven reactions and finite chemical adjustment times change the observable composition. The effects are conditional on the species' chemical timescale relative to mixing, advection and irradiation; they do not imply that every molecule is out of equilibrium everywhere.