Past exam of the mathematics course of the University of Cambridge 2016 iii Paper 315 2 c Solution Created 2026-10-03 Updated 2026-10-06
The inferred dayside water mixing ratio is about times below the solar-equilibrium reference. Several explanations are possible. A high atmospheric carbon-to-oxygen ratio, especially near or above unity in hot carbon monoxide-dominated gas, locks much of the oxygen in CO and leaves little water. A genuinely oxygen-poor or low-metallicity envelope also reduces water, although bulk elemental abundances must apply to both hemispheres. Atmospheric photochemistry can destroy water at low pressures. In portions of the dayside that are substantially hotter than the stated minimum, thermal dissociation can also reduce water, especially at low pressure; alone does not establish strong dissociation throughout the emitting region. Temperature/opacity degeneracies or incomplete treatment of exoplanet clouds and horizontal structure in an atmospheric retrieval can bias the inferred abundance.
At the cooler terminator, is only a factor of five below the reference. Moderately reduced oxygen abundance, an enhanced C/O ratio, and exoplanet cloud/atmospheric haze dilution of spectral features are plausible explanations. Water is comparatively stable at , so strong thermal dissociation is not the natural explanation there. A exoplanet transmission spectrum samples a slant path through the limb, and the degeneracy among exoplanet cloud height, reference pressure and gas abundance can mimic a lower mixing ratio.
The two measurements need not describe the same pressure range or longitude. Local dissociation/photochemistry on the hotter dayside and reformation on the cooler limb can produce a real spatial difference. Even at one elemental C/O ratio, hot carbon monoxide-rich chemistry can leave less oxygen for water than cooler methane-rich chemistry, an example of carbon partition and atmospheric water abundance. Transport can modify or homogenize those tendencies, depending on the reaction and advection timescales. Alternatively, inconsistent assumptions in the emission and transmission retrievals can create an apparent discrepancy. A reconciliation should use one bulk elemental inventory, separate dayside/limb temperature profiles and contribution pressures, and consistent exoplanet cloud and transport physics; it should not assign independent planetary metallicities to the two hemispheres.
Past exam of the mathematics course of the University of Cambridge 2016 iii Paper 315 4 c Solution Created 2026-10-03 Updated 2026-10-06
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.