  Two forms of ohmic heating of astrophysical secondaries have received
particular attention: unipolar-generator heating with currents running between
the primary and secondary; and magnetic induction heating due to the primary's
time-varying field. Neither appears to cause significant dissipation in the
contemporary solar system. But these discussions have overlooked heating
derived from the spatial variation of the primary's field across the interior
of the secondary. This leads to Lorentz force-driven currents around paths
entirely internal to the secondary, with resulting ohmic heating. We examine
three ways to drive such currents, by the cross product of: (1) the secondary's
azimuthal orbital velocity with the non-axially symmetric field of the primary;
(2) the radial velocity (due to non-zero eccentricity) of the secondary with
the primary's field; or (3) the out-of-plane velocity (due to non-zero
inclination) with the primary's field. The first of these operates even for a
spin-locked secondary whose orbit has zero eccentricity, in strong contrast to
tidal dissipation. We show that Jupiter's moon Io today could dissipate about
600 GW (more than likely current radiogenic heating) in the outer hundred
meters of its metallic core by this mechanism. Had Io ever been at 3 jovian
radii instead of its current 5.9, it could have been dissipating 15,000 GW.
Ohmic dissipation provides a mechanism that could operate in any solar system
to drive inward migration of secondaries that then necessarily comes to a halt
upon reaching a sufficiently close distance to the primary.
