Tidal friction in close-in satellites and exoplanets. The Darwin theory re-visited

TL;DR

This paper reviews Darwin's classical tidal theory, providing analytical expressions for orbital and rotational evolution, emphasizing final states like super-synchronous rotation and spin-orbit resonance, and discussing energy dissipation without assuming specific lag-frequency relations.

Contribution

It offers a generalized, assumption-light analytical framework for tidal evolution, focusing on final states and energy dissipation, with explicit formulas applicable to close-in satellites and exoplanets.

Findings

Analytical expressions for orbital and rotational evolution.

Conditions for super-synchronous rotation and spin-orbit resonance.

Energy dissipation controlled by specific tidal harmonics.

Abstract

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) the capture into a 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.