Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 4 a Solution Created 2026-10-03 Updated 2026-10-06
The usual major carbon/oxygen reservoirs in a hydrogen-rich hot Jupiter are water, carbon monoxide and methane, with their relative importance set by thermochemical equilibrium. This is not a universal ranking for every temperature and composition: nitrogen molecules or carbon dioxide can exceed a strongly depleted member of this trio.
At about one bar, the useful net reaction isThe rightward reaction is exothermic. Cooler gas favors methane and water; warming favors carbon monoxide and suppresses methane. The CO/CH4 crossover is of order and shifts with pressure, elemental inventory and metallicity; it is not a universal temperature. Water remains an important oxygen reservoir for oxygen-rich compositions but can dissociate at sufficiently high temperature.
Increasing the atmospheric metallicity of a giant planet raises the available carbon and oxygen. In a dilute, H2-dominated regime, major CO and H2O abundances roughly increase with the enrichment factor, and cool-regime CH4 does likewise. CO2 can rise faster, approximately quadratically in suitable warm regimes. At very high enrichment, the H2 fraction and mean molecular weight also change, invalidating simple linear scalings.
At high temperature and atmospheric carbon-to-oxygen ratio below unity, CO binds much of the carbon, leaving excess oxygen for H2O. As C/O approaches or exceeds one, CO consumes nearly all available oxygen and H2O is strongly depleted; excess carbon enhances CH4, hydrogen cyanide and acetylene. Cooler CH4-dominated chemistry uses less oxygen in CO and can leave more water even at the same elemental ratio. These trends are carbon partition and atmospheric water abundance, and assume equilibrium rather than vertical or horizontal quenching.
Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 4 b Solution Created 2026-10-03 Updated 2026-10-06
- Vertical chemical quenching: when the eddy mixing time becomes shorter than the chemical relaxation time, transported gas retains a deeper abundance rather than reaching local thermochemical equilibrium. For example, carbon monoxide–methane quenching can maintain excess carbon monoxide in cool upper giant-planet gas where equilibrium favors methane.
- Atmospheric photochemistry: ultraviolet photons dissociate molecules and drive reaction networks away from thermal equilibrium. For example, irradiation of CH4-containing gas can produce acetylene and hydrocarbon precursors of an atmospheric haze.
- Horizontal chemical quenching: winds cross a day-night temperature gradient faster than reactions can reset the composition. For example, CO-rich dayside gas can reach a cooler hot Jupiter nightside without converting most of its carbon to CH4.
These are three distinct ways to maintain disequilibrium chemistry in an exoplanet atmosphere; each compares chemistry with a different transport or irradiation process.
Past exam of the mathematics course of the University of Cambridge 2016 iii Paper 315 4 j Solution Created 2026-10-03 Updated 2026-10-06
Three solar-system effects illustrate chemical disequilibrium. First, vertical mixing carries carbon monoxide from hot deep regions into cooler observable layers of giant planets, where its local-equilibrium abundance would be small. Second, phosphine survives in the observable atmospheres of Jupiter and Saturn because deep supply competes with its destruction, making it a tracer of transport. Third, methane photolysis creates hydrocarbons such as ethane and acetylene, and can help form upper-atmospheric atmospheric hazes in giant planets and Titan.
Three exoplanet effects are complementary. First, carbon monoxide–methane quenching can keep carbon monoxide abundant and methane depleted in cool observable gas, including hot-Neptune/giant atmospheres that would locally favor methane. Second, atmospheric photochemistry dissociates methane/ammonia and produces radicals, hydrogen cyanide, hydrocarbons and possibly atmospheric hazes, changing spectral bands from their equilibrium strengths. Third, horizontal chemical quenching can preserve hot-dayside carbon chemistry on the cooler nightside/terminator and reduce the abundance contrast expected from separate local-equilibrium columns.
Transport, photon-driven reactions and finite chemical adjustment times change the observable composition. The effects are conditional on the species' chemical timescale relative to mixing, advection and irradiation; they do not imply that every molecule is out of equilibrium everywhere.
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.
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.