Magnetic induction heating of planetary satellites: Analytical formulae and applications

Calculations of the magnetic induction heating of planetary satellites to date have been done using approximate or numerical models. Here we present analytical formulae that may be used for either conducting metallic cores (spheres) or conducting water oceans or rock mantles (spherical shells). The formula for the magnetic induction heating of conducting spheres has been in the classic electromagnetic literature for nearly a century, but appears to have been overlooked by astrophysicists. Analogous formulae for conducting spherical shells are derived in this paper. We apply these formulae to calculate induction heating by alternating magnetic fields as seen in the frame of an orbiting satellite. These alternating fields may arise from inclined or horizontally displaced planetary magnetic dipoles, as well as eccentric or inclined satellite orbits. We derive analytical formulae for these, making it easy to investigate quickly the importance of magnetic induction heating for many different system configurations. Magnetic induction heating of planetary satellites in our current Solar System appears unimportant for each of these effects, but may have been greater in the past, in exoplanetary systems, or for other celestial binaries.