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David J. Stevenson modeled a magnetic mechanism for hot-Jupiter radius inflation with Konstantin Batygin. Their calculation coupled winds, a planetary magnetic field and electrical conductivity to estimate how induced electric currents deposit Ohmic heating in a giant planet's interior.^[ref]

The location of the dissipated energy is crucial. Heating a thin upper atmospheric layer need not replace the deep energy lost as the planet contracts; a useful inflation model must connect atmospheric forcing to the interior heat budget.

Hot-Jupiter radius inflation is the unexpectedly large radius of some strongly irradiated giant exoplanets compared with ordinary cooling and contraction models at the inferred age. A giant planet loses internal energy through its exoplanet atmosphere; slowing that loss or depositing additional energy sufficiently deep can keep its interior hotter and more extended.

Atmospheric opacity and day-night heat redistribution affect the boundary condition seen by the interior. Analytic irradiated-atmosphere work shows why deep temperatures depend on global transport as well as incident power, while also illustrating the difficulty of explaining observed radii by simple opacity changes alone.^[ref] An expanded radius is thus a constraint on the thermal history, rather than direct evidence for one unique heating mechanism.

A stabilizing composition gradient can suppress large overturning motions while permitting double-diffusive convection in small layers separated by interfaces. If heat loss is less efficient than in a single well-mixed adiabat, the planet retains more entropy and contracts more slowly. This is a proposed delayed-cooling contribution to hot-Jupiter radius inflation, with an effectiveness that depends on the gradient and layer structure.

Winds in a weakly ionized exoplanet atmosphere can drive electric currents through a planetary magnetic field. Their Ohmic heating is a proposed power source for hot-Jupiter radius inflation. A model must determine the electrical conductivity, the complete current paths and the depth at which the power is deposited.^[ref]

In a nonrelativistic, isotropically conducting fluid, let $\mathbf{v}$ be its velocity, $\mathbf{E}$ the laboratory electric field, $\mathbf{B}$ the magnetic field, $\mathbf{J}$ the current density and $\sigma >0$ the electrical conductivity. The simple resistive Ohm's law isThe irreversible heat production per unit volume and time is thereforeThe laboratory electrical power $\mathbf{J}\cdot \mathbf{E}$ alone also contains mechanical work by the Lorentz force:Thus identifying $\mathbf{J}\cdot \mathbf{E}$ directly with resistive heat requires an appropriate rest frame or a vanishing mechanical-work term. The total deposited power is $L_{\mathrm{Ohm}}=\int q_{\mathrm{Ohm}}dV$. A large value confined to optically thin upper layers need not maintain the deep interior's entropy; magnetic braking and current closure also limit what a prescribed wind can supply.

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:

For heating of an inflated giant planet:

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.

Hot-Jupiter radius inflation is the excess radius of many irradiated gas giants relative to standard cooling models of the same mass, composition and age. A young gas giant starts with high entropy and a large radius, but ordinarily contracts through Kelvin-Helmholtz contraction. An old inflated object must either retain that thermal reservoir unusually well or receive power that affects the deep interior.

The first class is delayed cooling of an inflated giant planet. Two specific mechanisms are increased atmospheric opacity, which makes internal radiation escape less readily, and inhibited interior convection caused by composition gradients, potentially with layered transport. Both slow entropy loss. Stellar irradiation can also maintain an outer radiative blanket, but an appropriate irradiated boundary is already included in many baseline models and does not alone explain every extreme radius.

The second class is heating of an inflated giant planet. Two mechanisms are tidal heating from eccentricity, obliquity or other time-dependent tidal forcing, and Joule heating from wind-driven electric currents in an ionized atmosphere coupled to the magnetic field. Mechanical energy carried inward from atmospheric circulation is another candidate. Heating must be deposited sufficiently deeply and with sufficient power; superficial absorption that is promptly reradiated is not equivalent to deep interior heating. A circular synchronized isolated orbit does not automatically provide a persistent tidal source.

The classes can coexist and are tested through radius-age-irradiation trends, orbital properties and the required energy budget. Merely changing an observed transit altitude by a few atmospheric scale heights is generally distinct from inflating the bulk giant-planet interior.

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:

The conventional deuterium-burning mass near $13$ 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:

Three major research directions, appropriate to the scientific questions posed in 2018, are:

These are research goals rather than claims that the relevant mechanisms or habitable atmospheres have already been established.

Exoplanet transit photometry discovers a giant planet through a periodic flux decrement and measures $R_p/R_{\star }\simeq \sqrt{\delta }$. Stellar-radius estimates turn this into a planetary radius. The radial-velocity method discovers the stellar orbital reflex motion and constrains $M_p\sin i$. Follow-up exoplanet transit photometry is required to measure a geometric radius when such a planet also transits; a radial-velocity detection alone gives no direct size. Alternatively, exoplanet direct imaging finds young luminous giants, whose sizes are inferred less directly from luminosity, temperature, distance, and a planetary mass-radius relation or atmosphere model.

The two broad explanations for hot-Jupiter radius inflation are retention of primordial heat and addition of new interior power.

Delayed cooling of an inflated giant planet can arise from enhanced atmospheric opacity, which slows radiative escape; an irradiation-maintained radiative blanket, which insulates the convective interior; or compositional stratification and inefficient layered convection, which inhibit the outward transport of heat. These alter the rate of Kelvin-Helmholtz contraction.

Heating of an inflated giant planet can arise from tidal heating maintained by eccentricity or obliquity; Joule heating of currents driven by atmospheric winds through a magnetic field; or downward transport and dissipation of atmospheric mechanical energy generated by irradiation. To affect radius, the energy must be deposited at a depth and rate that changes the interior cooling balance. Simply absorbing starlight high in the atmosphere does not automatically supply deep heating. These are proposed mechanisms with different efficiencies, not six universally established contributions in every inflated planet.

On a planetary mass-radius relation, compressed rocky planets grow sublinearly, approximately $R\propto M^{0.25-0.3}$. Adding a hydrogen-helium envelope produces a rapid radius increase toward sub-Neptunes and gas giants. Around a few Jupiter masses the radius is nearly constant and then decreases as electron degeneracy pressure becomes important, approximately approaching the nonrelativistic degenerate scaling $R\propto M^{-1/3}$.

Brown dwarfs occupy roughly $13$ to $75$--$80$ Jupiter masses, with deuterium burning near the lower conventional boundary and sustained hydrogen burning beginning at the hydrogen-burning minimum mass. Low-mass main-sequence stars then have radii that increase with mass. Thus an isolated-body sketch has a rising rocky branch, a broad giant-planet/brown-dwarf radius maximum and decline, followed by a rising stellar branch.

Hot-Jupiter radius inflation places strongly irradiated hot Jupiters above the isolated giant-planet sequence. Irradiation retards cooling and contraction; additional proposed contributions include tidal heating, Ohmic dissipation, atmospheric circulation depositing energy at depth, enhanced opacity, and residual youth.