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Incoming links: Ohmic heating of a giant planet
Past exam of the mathematics course of the University of Cambridge 2015 iii Paper 59 4 j Solution Created 2026-10-03 Updated 2026-10-06
The hot-Jupiter radius inflation problem is that some strongly irradiated giant planets have radii much larger than standard age-, mass- and composition-dependent cooling models predict. Greater internal entropy generally means a larger radius at fixed mass. The two broad classes of explanation are retaining existing heat by delaying cooling and depositing additional energy into the deep planet.
- Enhanced atmospheric opacity slows radiative leakage and keeps the deep interior hot. Required enrichment or persistent cloud opacity must be compatible with composition and spectra; adding heavy material also tends to increase density. Insulation can preserve initial heat but cannot necessarily reinflate an already cooled planet.
- Layered convection in a giant planet uses a stabilizing composition gradient and double-diffusive layers to reduce heat transport. The needed gradient and layer structure must survive mixing, and their efficiency is model-dependent; very inefficient transport cannot simply be assumed for every planet.
- Tidal heating dissipates orbital or spin energy, often requiring maintained eccentricity or obliquity. Nearly circular, synchronized planets have little of the simplest eccentricity-tide power, so a pumping mechanism or different tidal configuration is needed to explain them.
- Ohmic heating of a giant planet dissipates wind-induced electric currents in a conducting, magnetized atmosphere and interior. It requires appropriate ionization, conductivity, magnetic field, wind speeds and sufficiently deep deposition; magnetic drag can limit the available mechanical power.
Past exam of the mathematics course of the University of Cambridge 2016 iii Paper 315 3 a Solution Created 2026-10-03 Updated 2026-10-06
The two relevant categories are strongly irradiated, often transiting hot Jupiters, and young directly imaged exoplanets with large residual-entropy radii. The young objects are inflated relative to an old cold mass-radius sequence; a large young radius is not automatically anomalous relative to an age-appropriate formation model.
For a transit, the depth gives after geometric and limb darkening corrections; an independently estimated stellar radius sets . A measured planet mass then tests its location on the planetary mass-radius relation. For a directly imaged object, distance and integrated spectral flux give luminosity; a fitted effective temperature suppliesThe imaged radius is model-dependent because temperature, gravity and exoplanet clouds are inferred from its spectrum; its mass may also depend on age and evolutionary models unless dynamically measured.
Two broad explanatory classes are slower loss of existing heat and addition of heat to the deep interior. Specific delayed-cooling proposals are enhanced atmospheric opacity, which restricts radiative loss, and composition gradients producing layered convection in a giant planet. Specific heating proposals are tidal heating and Ohmic heating from currents induced by magnetized atmospheric winds, the proposed Ohmic heating of a giant planet. Heating must reach or influence sufficiently deep layers to maintain the interior specific entropy; merely heating the optically thin upper atmosphere is not equivalent. For young distant objects, retention of formation heat, described by hot and cold starts of a giant planet, itself explains much of the large radius without requiring the hot-Jupiter heating mechanisms.