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This report is a review of Darwin's classical theory of bodily tides in which we present the analytical expressions for the orbital and rotational evolution of the bodies and for the energy dissipation rates due to their tidal interaction. General formulas are given which do not depend on any assumption linking the tidal lags to the frequencies of the corresponding tidal waves (except that equal frequency harmonics are assumed to span equal lags). Emphasis is given to the cases of companions having reached one of the two possible final states: (1) the super-synchronous stationary rotation resulting from the vanishing of the average tidal torque; (2) capture into the 1:1 spin-orbit resonance (true synchronization). In these cases, the energy dissipation is controlled by the tidal harmonic with period equal to the orbital period (instead of the semi-diurnal tide) and the singularity due to the vanishing of the geometric phase lag does not exist. It is also shown that the true synchronization with non-zero eccentricity is only possible if an extra torque exists opposite to the tidal torque. The theory is developed assuming that this additional torque is produced by an equatorial permanent asymmetry in the companion. The results are model-dependent and the theory is developed only to the second degree in eccentricity and inclination (obliquity). It can easily be extended to higher orders, but formal accuracy will not be a real improvement as long as the physics of the processes leading to tidal lags is not better known.

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We examine means for driving hydrothermal activity in extraterrestrial oceans on planets and satellites of less than one Earth mass, with implications for sustaining a low level of biological activity over geological timescales. Assuming ocean planets have olivine-dominated lithospheres, a model for cooling-induced thermal cracking shows how variation in planet size and internal thermal energy may drive variation in the dominant type of hydrothermal system-for example, high or low temperature system or chemically driven system. As radiogenic heating diminishes over time, progressive exposure of new rock continues to the current epoch. Where fluid-rock interactions propagate slowly into a deep brittle layer, thermal energy from serpentinization may be the primary cause of hydrothermal activity in small ocean planets. We show that the time-varying hydrostatic head of a tidally forced ice shell may drive hydrothermal fluid flow through the seafloor, which can generate moderate but potentially important heat through viscous interaction with the matrix of porous seafloor rock. Considering all presently known potential ocean planets-Mars, a number of icy satellites, Pluto, and other trans-neptunian objects-and applying Earth-like material properties and cooling rates, we find depths of circulation are more than an order of magnitude greater than in Earth. In Europa and Enceladus, tidal flexing may drive hydrothermal circulation and, in Europa, may generate heat on the same order as present-day radiogenic heat flux at Earth's surface. In all objects, progressive serpentinization generates heat on a globally averaged basis at a fraction of a percent of present-day radiogenic heating and hydrogen is produced at rates between 10(9) and 10(10) molecules cm(2) s(1).

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[1] Interior models of a differentiated Titan with an internal ammonia-water ocean and chondritic radiogenic heat production in an undifferentiated rock + iron core have been calculated. We assume thermal and mechanical equilibrium and calculate the structure of the interior as a function of the thickness of an ice I layer on top of the ocean as well as the moment of inertia factor and the tidal Love numbers for comparison with Cassini gravity data. The Love numbers are linearly dependent on the thickness of the ice I shell at constant rheology parameters but decrease by one order of magnitude in the absence of an internal ocean. Ice shell thicknesses are between 90 and 105 km for models with 5 wt.% ammonia and for core densities between 3500 and 4500 kg m À3 . For 15 wt.% ammonia, the shell is 65 to 70 km thick. We use a strongly temperature-dependent viscosity parameterization of convective heat transport and find that the stagnant lid comprises most of the ice I shell. Tidal heating in the warm convective sublayer is of minor importance. The internal ocean is several hundred kilometers thick, and its thickness decreases with increasing thickness of the ice shell. Core sizes vary from 1500 to 1800 km radius with associated moment of inertia factors of 0.30 ± 0.01.