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Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 315 3 a Solution Created 2026-10-03 Updated 2026-10-05
Four mechanisms that can produce disequilibrium chemistry in an exoplanet atmosphere are:
- Atmospheric photochemistry. Stellar ultraviolet photons initiate reactions whose products need not follow local thermochemical equilibrium. The ozone layer on Earth is a solar-system example. Calculations for HD 189733 b predict enhanced hydrogen cyanide and acetylene from the processing of methane and ammonia; these are model examples rather than assertions of an unambiguous detection.
- Vertical transport and chemical quenching. When the eddy mixing time is shorter than the chemical relaxation time, gas retains a deeper abundance above its chemical quench level. The excess carbon monoxide in Jupiter's cool atmosphere exemplifies carbon monoxide–methane quenching. Models of HD 189733 b predict quenched methane and ammonia abundances differing from their local chemical equilibrium values. The enhancement or depletion depends on the underlying atmospheric pressure-temperature profile.
- Horizontal chemical quenching. If the atmospheric advection time is short, winds move chemically processed gas into regions with different irradiation or temperature faster than it can re-equilibrate. Transport of gas within Earth's ozone layer moves material away from its local photochemical production regions. Models of HD 209458 b show that dayside carbon monoxide-rich composition can persist into the cooler nightside instead of forming the local chemical equilibrium amount of methane.
- Condensation with sedimentation or rainout. Finite-rate cloud formation can depart from phase equilibrium, while atmospheric condensate rainout removes elements from a layer and changes its gas composition. Earth's water atmospheric cold trap limits the supply of water to the stratosphere. In HD 209458 b models, titanium-bearing condensates can settle and suppress upper-atmospheric titanium monoxide. The remaining gas can still be in local chemical equilibrium with its depleted inventory: rainout is an open-column effect, not necessarily a failure of equilibrium among all gas reactions.
The exoplanet transport and photochemical examples follow kinetic atmosphere calculations and models including horizontal transport; the condensate example is examined in cold-trap calculations.
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.