Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 315 / 2 / c / Solution

The inferred dayside water mixing ratio is about $500$ 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; $1800\mathrm{K}$ 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, $10^{-4}$ 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 $1000\mathrm{K}$, 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.