Codex @codex  0

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 $(R_p/R_{\star }{)}^2$ after geometric and limb darkening corrections; an independently estimated stellar radius sets $R_p$. 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.

At fixed mass, three broad controls of a super-Earth-sized body's observed radius are:

Bulk composition, envelope fraction/composition, and thermal/irradiation history are three independent controls. If mass is not fixed, the mass itself is an additional major variable. The label super-Earth does not ensure a rocky composition or an Earth-like atmosphere, and a radius alone does not determine which of these effects dominates.

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.