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Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 4 e Solution Created 2026-10-03 Updated 2026-10-06
The circumstellar habitable zone is the range of orbital distances where a terrestrial planet with a specified atmospheric inventory can retain liquid water at its surface. It is a conditional climate criterion, not a guarantee of life or a requirement for every possible subsurface habitat.
Four influential factors are:
- The incident stellar energy and spectrum, including orbital distance and long-term stellar evolution.
- Atmospheric pressure and composition, greenhouse effect, Bond albedo and clouds, which set the relation between absorbed light and surface temperature.
- The water and volatile inventory, together with planetary mass and the ability to retain or replenish an atmosphere.
- Internal and surface evolution, including plate tectonics, outgassing and the carbonate-silicate cycle, which can regulate climate over geological time.
For a Sun-like present-day star and an Earth-like planet, a useful conservative range is approximately –. The inner limit depends on the adopted moist or runaway greenhouse condition, and the outer limit on the maximum greenhouse outer habitable-zone limit. More restrictive water-loss choices put the inner edge near . Empirical optimistic limits based on past Venus and Mars are about –. These are model conventions rather than exact universal boundaries.
Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 4 f Solution Created 2026-10-03 Updated 2026-10-06
Three routes away from local thermochemical equilibrium are:
- Vertical transport and quenching. When the eddy mixing time becomes shorter than the chemical relaxation time, gas retains a deeper composition. In cool, directly imaged giant exoplanets, carbon monoxide–methane quenching can preserve CO and suppress the methane expected in cool equilibrium layers. In Jupiter, CO carried from deeper hot layers, and phosphine maintained against upper-atmosphere chemical loss, are examples of transported disequilibrium species.
- Horizontal transport and quenching. Atmospheric advection time shorter than the local chemical relaxation time carries abundances between regions of different temperature. A hot Jupiter can carry CO-rich dayside gas into cooler nightside gas that would otherwise favour methane. In the Solar System, upper-atmosphere CO produced on the illuminated side of Venus can be redistributed to its dark side by circulation; this is an example of nonlocal production and transport, rather than local dark-side thermochemical equilibrium.
- Atmospheric photochemistry. Ultraviolet photons dissociate or ionize molecules and initiate reaction networks. Models of irradiated hydrogen-rich exoplanet atmospheres can produce enhanced hydrogen cyanide and hydrocarbon precursors from methane/nitrogen chemistry; whether these products accumulate depends on ultraviolet flux and transport. The terrestrial atmosphere gives a clear Solar-System example: oxygen photodissociation and subsequent reactions maintain the ozone layer, which is not a purely thermochemical-equilibrium abundance.
Vertical mixing, horizontal advection, and photochemistry supply three mechanisms and examples in both exoplanets and Solar-System atmospheres. A photochemical steady state balances production and loss; it is different from a Gibbs free energy minimum. These examples describe mechanisms and model expectations rather than asserting unique observational attribution for every planet.