Exoplanet interior

An exoplanet interior is the structure and composition beneath the layers directly sampled by observations. Mass and radius constrain the mean mass density, but a physical model also needs hydrostatic equilibrium, an equation of state, and assumptions about composition and thermal history. Dense material can be concentrated in a core or mixed into surrounding layers.

Solid-planet calculations show how iron-, silicate- and water-rich bodies occupy different broad mass-radius relations, with compression preventing one constant-density scaling from applying to every mass.^[ref] A light gas envelope can substantially alter the observed radius. Exoplanet interior-composition degeneracy therefore remains even when both mass and radius are measured precisely.

An idealized self-gravitating interior with constant mass density. Negligible surface pressure and spherical hydrostatic equilibrium give $M=4\pi \rho R^3/3$ and the uniform-density planetary pressure profile. It is the $n=0$ polytrope interpretation, rather than a literal substitution into $P=K{\rho }^{1+1/n}$. It approximates weak compression and misses real compositional stratification.

For a spherical incompressible planetary interior, $M(r)=4\pi \rho r^3/3$ gives $dP/dr=-4\pi G{\rho }^{2}r/3$. With $P(R)=0$, $P(r)=2\pi G{\rho }^2(R^2-r^2)/3$. The surface gravity $g_s=4\pi G\rho R/3$ then yields $P_c=3g_s^2/(8\pi G)$. Add an imposed surface pressure to the entire profile if needed.

The hydrostatic pressure support equation in enclosed-mass coordinate is $dP/dm=-Gm/(4\pi r^4)$. Since $r(m)\le R$, integration from surface pressure zero to the centre givesIf mass density decreases outward, the mean density within $r$ is at least the global mean, so $r(m)\le R(m/M{)}^{1/3}$. This improves the bound towith equality for uniform mass density. Centrally concentrated planets need larger central pressure to support their inner mass.

A two-layer constant-density planet has a uniform core density ${\rho }_1$ and uniform mantle density ${\rho }_2$. Hydrostatic pressure follows by integrating $dP/dr=-G\rho (r)M(r)/r^2$ separately in each layer and matching pressure at their boundary.

The interior-composition degeneracy is the failure of mass and radius alone to determine unique layer mass fractions. Atmospheric mean molecular mass and composition, stellar abundances, tidal response, and thermal evolution add independent constraints.

A mass-radius relation connects the sizes of a class of self-gravitating objects with their masses. It encodes both material properties and the mechanism supporting the object against gravity. For an illustrative sequence of uniform, fixed mass density $\rho$, the identity $M=4\pi \rho R^3/3$ gives $R\propto M^{1/3}$. Real self-compression changes that scaling.

A planetary mass-radius relation depends strongly on composition and any gas envelope. In compact objects, electron degeneracy pressure can make a larger mass correspond to a smaller radius. A relation therefore applies to a stated family under stated assumptions, rather than every object obeying one universal size law. Deviations can test an equation of state, but can also reflect differences in composition or age.

For a specified age and irradiation, a hydrogen–helium sequence rises at low mass, has radii near Jupiter's across giant planets and brown dwarfs, and declines as electron degeneracy pressure becomes important. Above the hydrogen-burning minimum mass, sustained fusion supports a rising low-mass stellar branch.

A planetary mass-radius relation compares the sizes of bodies at a given mass. Rock compression, volatile layers, gaseous envelopes, thermal entropy, irradiation, and electron degeneracy produce distinct branches and substantial composition degeneracy.

A low-mean molecular weight hydrogen-helium envelope can add a large radius while adding a small fraction of the mass. The exoplanet transit photometry radius is the altitude where tangential optical depth becomes order unity, so temperature, opacity, and exoplanet clouds can move the measured radius above the deep material radius. Atmospheric loss and cooling change this contribution over time.

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.

Delayed cooling preserves a young giant planet's large radius by reducing the escape of internal energy. Enhanced atmospheric opacity, an irradiation-maintained outer radiative blanket, and composition gradients that inhibit convection can slow Kelvin-Helmholtz contraction. These mechanisms retain existing heat rather than supplying a new deep heat source.

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.

A giant planet can also remain inflated when power is deposited sufficiently deep to alter its internal energy budget. Candidate sources include tidal heating, Joule heating by wind-driven currents, and dissipation of circulation-driven mechanical energy. The depth and efficiency of deposition matter, as discussed in models of hot-Jupiter heating.

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.