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Incoming links: HD 209458 b
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 4 a Solution Created 2026-10-03 Updated 2026-10-05
Three observational signatures constrain exoplanet atmospheric dynamics:
- A displaced maximum of the exoplanet thermal phase curve. If the thermal maximum precedes exoplanet secondary eclipse, the hottest region is displaced east of the substellar longitude in the usual synchronously rotating geometry. The observed offset for HD 189733 b is consistent with eastward heat advection and atmospheric superrotation; the early infrared phase-curve measurement established this effect.
- Warm nightside emission and a reduced day-night contrast. The amplitude of an exoplanet thermal phase curve constrains day-night heat redistribution. For HD 189733 b, the same observations found hemisphere brightness temperatures roughly and , supporting substantial transport to the nightside instead of purely local reradiation.
- Atmospheric line shifts after orbital motion is removed. A residual Doppler shift measures winds projected along the line of sight. A carbon monoxide signal blueshifted by roughly in HD 209458 b was interpreted as day-to-night winds in high-resolution transit spectroscopy. Rotational broadening and different limb velocities can supply additional dynamical information.
Opacity, exoplanet cloud distributions, and pressure-dependent brightness temperature affect the phase-curve interpretation. These measurements constrain circulation through atmospheric models; a single offset or line shift does not uniquely determine a three-dimensional wind field.
Past exam of the mathematics course of the University of Cambridge 2018 iii Paper 315 4 d Solution Created 2026-10-03 Updated 2026-10-05
A mass-radius curve of solar-composition substellar objects must specify age, irradiation and composition. For a mature, weakly irradiated hydrogen–helium sequence, the qualitative mass-radius relation has three main trends:
- At low giant-planet masses, weak compression allows a rough scaling at nearly fixed mean mass density. Radius rises toward a broad maximum around one to a few Jupiter masses.
- Through massive gas giants and brown dwarfs, radii remain near Jupiter's and eventually decline as compression and electron degeneracy pressure become important. A nonrelativistic degenerate stellar polytrope gives the approximate polytropic mass-radius relation ; finite thermal pressure and real equations of state flatten and modify this idealized scaling.
- Above the hydrogen-burning minimum mass, sustained hydrogen burning supplies the energy that supports a rising low-mass main-sequence branch. For near-solar composition, the threshold is around Jupiter masses (about solar masses), with composition-dependent variation. Red dwarfs then have radii increasing with mass, approximately linearly over part of the low-mass sequence.
The conventional deuterium-burning mass near Jupiter masses marks a commonly used giant-planet/brown dwarf classification scale, not a sharp discontinuity in radius or a universal formation boundary. Young brown dwarfs are larger and cool by contraction; irradiation, a heavy-element core and hot-Jupiter radius inflation shift planets away from a single solar-composition curve.
The drawing is a qualitative mature sequence, not an interpolation of a specified evolutionary model grid. Three relevant observations, using representative historical measurements, are:
- Inflated radii: HD 209458 b has a radius around at a mass around , illustrating hot-Jupiter radius inflation relative to a simple mature unirradiated sequence. Early transit photometry measured its unusually large radius.
- Compact metal-enriched planets: HD 149026 b was measured near and . Its small radius for its mass and irradiation implies substantial heavy-element enrichment in interior models, showing the importance of composition in the planetary mass-radius relation; see the discovery analysis.
- Jupiter-like sizes at substellar masses: CoRoT-3b was measured near but only , supporting the broad near-constant-radius region extending into brown dwarf masses. Its measured mass and radius also illustrate why mass alone does not establish a formation history.