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 3 b ii Solution Created 2026-10-03 Updated 2026-10-05
For solar elemental abundances, take a molecular hydrogen- and helium-dominated atmosphere with atmospheric carbon-to-oxygen ratio near . The principal oxygen-bearing molecule is generally water; the carbon and nitrogen carriers depend on both temperature and pressure.
At the cool 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 , 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 low-pressure region. The exact partition requires the equilibrium constant for .
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.
Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 315 3 b iv Solution Created 2026-10-03 Updated 2026-10-05
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 , 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 givesWhere carbon monoxide and water each scale roughly linearly with enrichment , carbon dioxide consequently scales approximately as . 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.
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 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.