Tidal Friction and Tidal Lagging. Applicability Limitations of a Popular Formula for the Tidal Torque

TL;DR

This paper critically examines a widely used formula for tidal torque, highlighting its limitations due to assumptions of frequency-independent lag and rheology, and clarifies the correct theoretical framework for modeling tidal friction.

Contribution

It provides a clear, self-consistent summary of tidal friction theory and demonstrates the conditions under which the popular torque formula is valid or invalid.

Findings

The common formula assumes a frequency-independent time lag.

The formula is only valid for bodies with hypothetical rheology.

Using the formula with realistic rheologies leads to inaccuracies.

Abstract

Tidal torques play a key role in rotational dynamics of celestial bodies. They govern these bodies' tidal despinning, and also participate in the subtle process of entrapment of these bodies into spin-orbit resonances. This makes tidal torques directly relevant to the studies of habitability of planets and their moons. Our work begins with an explanation of how friction and lagging should be built into the theory of bodily tides. Although much of this material can be found in various publications, a short but self-consistent summary on the topic has been lacking in the hitherto literature, and we are filling the gap. After these preparations, we address a popular concise formula for the tidal torque, which is often used in the literature, for planets or stars.We explain why the derivation of this expression, offered in the paper by Goldreich (1966; AJ 71, 1 - 7) and in the books by Kaula (1968, eqn. 4.5.29), and Murray & Dermott (1999, eqn. 4.159), implicitly sets the time lag to be frequency independent. Accordingly, the ensuing expression for the torque can be applied only to bodies having a very special (and very hypothetical) rheology which makes the time lag frequency independent, i.e, the same for all Fourier modes in the spectrum of tide. This expression for the torque should not be used for bodies of other rheologies. Specifically, the expression cannot be combined with an extra assertion of the geometric lag (or the phase lag) being constant, because at finite eccentricities the said assumption is incompatible with the constant-time-lag condition.