Heating of an inflated giant planet

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.