Saturn raises a time-dependent tide on its small moon Enceladus, due to the eccentricity of the orbit. As shown in a companion paper (Goldreich et al.), the resulting tidal heating drives Enceladus into a limit cycle, in which its eccentricity and shell thickness vary in tandem, on a timescale of ~ 10 Myr. The limit cycle explains a variety of observed phenomena on Enceladus, including its large luminosity and cracked ice shell. Here we derive the tidal heating rate needed for that study, starting from a simple first-principles derivation of Enceladus's tidal response. Enceladus is comprised of three layers: a rocky core, an outer ice shell, and an ocean sandwiched in between. Tides force the shell to librate and distort, which generates heat. We calculate the libration amplitude and tidal heating rate by minimizing the sum of elastic and gravitational energies. The final expressions are analytic, and account for the finite hardness of the shell, and for resonant libration. Although we specialize to Enceladus, our approach may be extended to other bodies that have a similar three layer structure, such as Europa and Titan.
Yoram Lithwick (Mon,) studied this question.