Exoplanet interior

An exoplanet interior is inferred from mass, radius, age, irradiation, and atmospheric composition. Distinct mixtures of metal, rock, water, and hydrogen-helium can share the same bulk mass and radius.

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 compares the characteristic radius of a class of self-gravitating objects with mass. Its form records the equation of state and composition.

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 anomalously large radius of many strongly irradiated gas giants relative to isolated cooling models. Delayed cooling and energy deposition by tides, atmospheric circulation, or electrical currents are leading contributions.

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.

Atmospheric winds across a magnetic field can induce electric currents. Their Ohmic heating converts mechanical energy into heat, at local rate $q=\mathbf{J}\cdot \mathbf{E}=J^2/\sigma$ in the simplest resistive description. To maintain hot-Jupiter radius inflation, current paths and deposited heat must influence the deep interior; dissipation restricted to optically thin upper layers need not maintain the planet's entropy.