Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 4 g Solution Created 2026-10-03 Updated 2026-10-06
- Muted molecular-band amplitudes in the near-infrared, for example weak water structure around : an optically thick exoplanet cloud deck raises the continuum and hides part of the gas column.
- A nearly grey transit-radius continuum across optical/near-infrared wavelengths, roughly --, when particles are large relative to the wavelengths. Suppressed optical atomic-line wings provide a related indication that deep high-pressure layers are hidden.
- A radius increasing toward blue/ultraviolet wavelengths, roughly --, for small-particle scattering of light. In a simple Rayleigh scattering limit, , and the scattering slope of a transmission spectrum is .
A second approach uses the dayside spectrum and brightness obtained with exoplanet secondary eclipses, optionally supported by orbital phase variations. Its three useful aerosol signatures are:
- Enhanced reflected-light geometric albedo or characteristic colour in the visible, roughly --, for a scattering exoplanet cloud. Absorbing atmospheric haze can instead lower the reflected flux, so high albedo is not a universal aerosol rule.
- Weak near-/mid-infrared gas features and a cloud-set continuum brightness temperature, roughly --, when an opaque cloud moves the thermal photosphere to a different pressure. Changes in the exoplanet thermal phase curve can support a spatially patchy cloud interpretation, though circulation also affects it.
- Broad condensate-dependent infrared spectral structure, for example a silicate resonance around , if particles are sufficiently small and the optical-depth/temperature geometry permits the feature. Its presence and sign depend on particle composition, size and vertical thermal structure.
Transmission measures aerosol extinction along the limb; eclipse/reflection measurements test dayside scattering and emission. No single flattened feature proves clouds: small atmospheric scale height, low molecular abundances, stellar heterogeneity and thermal-gradient degeneracies must be considered. Joint wavelength coverage and geometries make the inference stronger.
Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 4 i Solution Created 2026-10-03 Updated 2026-10-06
- Bulk composition and material compressibility. A larger iron-core fraction usually makes an exoplanet interior denser and smaller, whereas silicate or water-rich interiors are larger. The planetary mass-radius relation therefore differs among compositions; mass and radius alone retain an exoplanet interior-composition degeneracy.
- A gaseous envelope and its retention. Even a modest hydrogen-helium mass fraction can substantially increase radius through the transit-radius contribution of a gaseous envelope. Its mean molecular weight, total atmospheric mass and opacity determine how far the slant-optical-depth surface lies above the condensed interior. Atmospheric escape can strip that envelope, producing a much smaller object without a comparable loss of core mass.
- Thermal and irradiation history. Higher interior entropy, youth and stellar heating tend to expand an envelope; Kelvin-Helmholtz contraction reduces its size over time. Stellar high-energy exposure affects the envelope through atmospheric escape. Exoplanet clouds and wavelength-dependent opacity shift the measured transit radius even when the deep interior is unchanged.
Bulk composition, envelope fraction/composition, and thermal/irradiation history are three independent controls. If mass is not fixed, the mass itself is an additional major variable. The label super-Earth does not ensure a rocky composition or an Earth-like atmosphere, and a radius alone does not determine which of these effects dominates.
Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 4 j Solution Created 2026-10-03 Updated 2026-10-06
Three research directions natural in the 2017 setting are:
- Atmospheric characterization of small planets, extending beyond bright hot Jupiters to super-Earth-sized and terrestrial bodies. The aim is to determine whether they retain light envelopes or heavier secondary atmospheres and to understand exoplanet habitability. Potential exoplanet biosignatures require a chemical and stellar context because abiotic processes can mimic individual molecules; future infrared observatories such as the then-planned James Webb Space Telescope offered a route to better spectra, without guaranteeing a biological interpretation.
- Connecting measured composition to planet formation and planetary migration. Atmospheric metallicity of a giant planet, atmospheric carbon-to-oxygen ratio, orbital architecture and bulk mass-radius measurements can constrain accretion of gas and solids. Atmospheric condensate rainout, disequilibrium chemistry in an exoplanet atmosphere and exoplanet interior-composition degeneracy make a one-to-one mapping from a measured ratio to a birth location unreliable; comparative samples and coupled models are needed.
- Resolving atmospheric structure in space and time. Exoplanet thermal phase curves, repeated spectra, high-resolution planetary spectroscopy and exoplanet direct imaging can test day-night heat redistribution, winds, patchy exoplanet clouds, atmospheric escape and chemical heterogeneity. The challenge is to distinguish actual variability or circulation from changes in stellar light and instrumental calibration.
Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 315 3 b iii Solution Created 2026-10-03 Updated 2026-10-05
Four possible molecular signatures of disequilibrium chemistry in an exoplanet atmosphere, measured through an exoplanet transmission spectrum or exoplanet emission spectrum, are:
- Excess carbon monoxide in the cool upper layers, where local chemical equilibrium would place most carbon in methane. This can indicate carbon monoxide–methane quenching from deeper hot gas.
- Suppressed methane bands relative to the same cool equilibrium model, consistent with transport preventing complete carbon monoxide conversion or with photochemical loss.
- An ammonia abundance inconsistent with the local equilibrium nitrogen partition, potentially recording nitrogen–ammonia quenching. Its direction must be evaluated for the actual atmospheric pressure-temperature profile; a depleted value is possible when hotter, molecular nitrogen-rich gas is transported upward.
- Enhanced hydrogen cyanide or acetylene bands, consistent with atmospheric photochemistry acting on the transported carbon and nitrogen reservoirs.
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.
Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 315 4 a Solution Created 2026-10-03 Updated 2026-10-05
Three observational signatures constrain exoplanet atmospheric dynamics:
- A displaced maximum of the exoplanet thermal phase curve. If the thermal maximum precedes exoplanet secondary eclipse, the hottest region is displaced east of the substellar longitude in the usual synchronously rotating geometry. The observed offset for HD 189733 b is consistent with eastward heat advection and atmospheric superrotation; the early infrared phase-curve measurement established this effect.
- Warm nightside emission and a reduced day-night contrast. The amplitude of an exoplanet thermal phase curve constrains day-night heat redistribution. For HD 189733 b, the same observations found hemisphere brightness temperatures roughly and , supporting substantial transport to the nightside instead of purely local reradiation.
- Atmospheric line shifts after orbital motion is removed. A residual Doppler shift measures winds projected along the line of sight. A carbon monoxide signal blueshifted by roughly in HD 209458 b was interpreted as day-to-night winds in high-resolution transit spectroscopy. Rotational broadening and different limb velocities can supply additional dynamical information.
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.
Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 315 4 e Solution Created 2026-10-03 Updated 2026-10-05
Three major research directions, appropriate to the scientific questions posed in 2018, are:
- Characterizing small-planet atmospheres and habitability. Measure an exoplanet transmission spectrum and an exoplanet emission spectrum to establish whether rocky planets retain atmospheres, determine their composition and climate, and test exoplanet habitability. Interpreting an exoplanet biosignature requires a planetary and stellar context: photochemical false positives, exoplanet cloud obscuration and atmospheric escape can all affect apparently promising molecules.
- Connecting atmospheres and interiors to planet formation. Combine planetary mass-radius relations with ages, host-star abundances, atmospheric metallicity of a giant planet and atmospheric carbon-to-oxygen ratio to test growth and migration histories. Exoplanet interior-composition degeneracy, chemical processing and atmospheric condensate rainout prevent a molecular abundance or bulk radius from being a unique record of birth conditions.
- Understanding atmospheric circulation and thermal evolution across populations. Combine exoplanet thermal phase curves, Doppler spectroscopy and pressure-dependent spectra to test day-night heat redistribution, exoplanet cloud formation and disequilibrium chemistry in an exoplanet atmosphere. Relate these measurements to hot-Jupiter radius inflation, internal cooling and atmospheric escape, using comparative observations to distinguish mechanisms instead of fitting each planet in isolation.
These are research goals rather than claims that the relevant mechanisms or habitable atmospheres have already been established.
Past exam of the mathematics course of the University of Cambridge 2019 iii Paper 315 1 c Solution Created 2026-10-03 Updated 2026-10-05
In the annulus model for transmission spectroscopy, an added aerosol optical depth changes the signal to . Large particles can supply nearly wavelength-independent extinction. An opaque high exoplanet cloud deck then masks deeper gas, flattens the optical exoplanet transmission spectrum, and weakens atomic or molecular features.
Small particles can instead produce a rising transit radius toward short wavelengths. In an isothermal atmosphere, the slant optical depth of an isothermal atmosphere isTaking the effective radius near gives . For Rayleigh scattering, , so the scattering slope of a transmission spectrum isA steeper-than-expected optical slope or suppressed gas features can therefore indicate atmospheric haze or exoplanet clouds. These signatures are not unique: high mean molecular mass reduces , gas itself can produce Rayleigh scattering, and stellar surface heterogeneity can mimic slopes. Consistent optical and infrared features help distinguish these explanations.
Past exam of the mathematics course of the University of Cambridge 2019 iii Paper 315 4 e Solution Created 2026-10-03 Updated 2026-10-05
First refine the ephemeris, planetary mass, stellar radius, and stellar variability using exoplanet transit photometry and the radial-velocity method. Then combine observations that probe different regions rather than relying on one spectrum. A present-day programme could use the following complementary measurements; in the 2019 setting of the paper, James Webb Space Telescope observations would have been a future capability.
- Exoplanet transmission spectrum at roughly – with the Hubble Space Telescope or optical ground-based spectroscopy: constrain exoplanet clouds, atmospheric haze, the scattering slope of a transmission spectrum, and sodium or potassium absorption.
- Near-infrared exoplanet transmission spectrum with NIRISS at – and NIRSpec modes covering roughly –: measure water, carbon monoxide, carbon dioxide, and methane bands, then constrain atmospheric metallicity of a giant planet and atmospheric carbon-to-oxygen ratio through a joint atmosphere model.
- Exoplanet secondary eclipses at near- and mid-infrared wavelengths with NIRSpec and MIRI, especially about – for the latter's time-series low-resolution mode: infer brightness temperatures, the vertical pressure-temperature structure, and whether an atmospheric thermal inversion turns bands into emission.
- A full-orbit exoplanet thermal phase curve in one or more infrared bands with the James Webb Space Telescope: constrain day-night heat redistribution, nightside emission, and the offset of the hottest region, with different bands probing different pressures.
- High-resolution near-infrared spectroscopy around molecular bands such as carbon monoxide near using CRIRES on the Very Large Telescope: resolve the planetary Doppler shift and seek wind velocities or rotation broadening after accounting for the orbital velocity.
- Ultraviolet transit spectroscopy with the Hubble Space Telescope, or ground-based near-infrared helium spectroscopy at : search for atmospheric escape and an extended upper atmosphere.
Together these address aerosols, molecular composition, elemental enrichment, vertical thermal structure, horizontal heat transport, winds, and escape. Repeat key events and monitor stellar activity, since stellar contamination and instrumental trends can imitate atmospheric signals. Use actual brightness, saturation limits, and predicted feature amplitudes to choose observing modes and exposure times.
The wavelength ranges and time-series capabilities are documented in the NIRISS SOSS guide, NIRSpec overview, MIRI spectroscopy guide, and ESO's CRIRES description.
Reflected-light planetary phase curve 2026-10-06
The orbital variation of a planet's reflected stellar light as the visible illuminated area changes. In a simple spherical model, , where and is the star-planet-observer angle. Exoplanet clouds can alter the geometric albedo and the angular scattering law; this observable differs from an exoplanet thermal phase curve.
A low-mean molecular weight hydrogen-helium envelope can add a large radius while adding a small fraction of the mass. The exoplanet transit photometry radius is the altitude where tangential optical depth becomes order unity, so temperature, opacity, and exoplanet clouds can move the measured radius above the deep material radius. Atmospheric loss and cooling change this contribution over time.