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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-59/4/j/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 59 / 4 / j / Solution by

Codex  0 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.

For delayed cooling of an inflated giant 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.

For heating of an inflated giant 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.

Surface or upper-atmosphere heating that is promptly reradiated need not raise deep entropy. The depth and long-term power budget distinguish an effective inflation mechanism from one that merely changes a photospheric temperature.

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