This is an exploration of dynamic tides on elastic bodies. The body is thought of as a dynamical system described by its modes of oscillation. The dynamics of these modes are governed by differential equations that depend on the rheology. The modes are damped by dissipation. Tidal friction occurs as exterior bodies excite the modes and the modes act back on the tide raising body. The whole process is governed by a closed set of differential equations. Standard results from tidal theory are recovered in a two-timescale approximation to the solution of these differential equations.

Goldreich (Goldreich, P. [1967]. J. Geophys. Res. 72, 3135) showed that a lunar core of low viscosity would not precess with the mantle. We show that this is also the case for much of lunar history. But when the Moon was close to the Earth, the Moon’s core was forced to follow closely the precessing mantle, in that the rotation axis of the core remained nearly aligned with the symmetry axis of the mantle. The transition from locked to unlocked core precession occurred between 26.0 and 29.0 Earth radii, thus it is likely that the lunar core did not follow the mantle during the Cassini transition. Dwyer and Stevenson (Dwyer, C.A., Stevenson, D.J. [2005]. An Early Nutation-Driven Lunar Dynamo. AGU Fall Meeting Abstracts GP42A-06) suggested that the lunar dynamo needs mechanical stirring to power it. The stirring is caused by the lack of locked precession of the lunar core. So, we do not expect a lunar dynamo powered by mechanical stirring when the Moon was closer to the Earth than 26.0-29.0 Earth radii. A lunar dynamo powered by mechanical stirring might have been strongest near the Cassini transition.

Coupled thermal-orbital histories of early lunar evolution are considered in a simple model. We consider a plagioclase lid, overlying a magma ocean, overlying a solid mantle. Tidal dissipation occurs in the plagioclase lid and heat transport is by conduction and melt migration. We find that large orbital eccentricities can be obtained in this model. We discuss possible consequences of this phase of large eccentricities for the shape of the Moon and geochronology of lunar samples. We find that the orbit can pass through the shape solution of Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M. [2006]. Science 313, 652), but we argue that the shape cannot be maintained against elastic deformation as the orbit continues to evolve.

The main equations in the paper “Episodic volcanism of tidally heated satellites with application to Io” by Ojakangas and Stevenson [Icarus 66, 341 358] are presented; numerical integration of these equations confirms the results of Ojakangas and Stevenson [Icarus 66, 341 358] for Io. Application to Enceladus is considered. It is shown that Enceladus does not oscillate about the tidal equilibrium in this model by both new nonlinear stability analysis and numerical integration of the model equations.

The tidal evolution through several resonances involving Mimas, Enceladus, and/or Dione is studied numerically with an averaged resonance model. We find that, in the Enceladus Dione 2:1 e-Enceladus type resonance, Enceladus evolves chaotically in the future for some values of k/Q. Past evolution of the system is marked by temporary capture into the Enceladus Dione 4:2 ee-mixed resonance. We find that the free libration of the Enceladus Dione 2:1 e-Enceladus resonance angle of 1.5° can be explained by a recent passage of the system through a secondary resonance. In simulations with passage through the secondary resonance, the system enters the current Enceladus Dione resonance close to tidal equilibrium and thus the equilibrium value of tidal heating of 1.1(18,000/Q) GW applies. We find that the current anomalously large eccentricity of Mimas can be explained by passage through several past resonances. In all cases, escape from the resonance occurs by unstable growth of the libration angle, sometimes with the help of a secondary resonance. Explanation of the current eccentricity of Mimas by evolution through these resonances implies that the Q of Saturn is below 100,000. Though the eccentricity of Enceladus can be excited to moderate values by capture in the Mimas Enceladus 3:2 e-Enceladus resonance, the libration amplitude damps and the system does not escape. Thus past occupancy of this resonance and consequent tidal heating of Enceladus is excluded. The construction of a coherent history places constraints on the allowed values of k/Q for the satellites.

The heating in Enceladus in an equilibrium resonant configuration with other saturnian satellites can be estimated independently of the physical properties of Enceladus. We find that equilibrium tidal heating cannot account for the heat that is observed to be coming from Enceladus. Equilibrium heating in possible past resonances likewise cannot explain prior resurfacing events.