Bodily tides near the 1:1 spin-orbit resonance. Correction to Goldreich's dynamical model

TL;DR

This paper corrects and extends Goldreich's model of bodily tides near the 1:1 spin-orbit resonance, addressing the unphysical assumptions in the MacDonald torque approach and analyzing the effects of triaxiality and dissipation on spin evolution.

Contribution

It introduces necessary modifications to the MacDonald torque model for accurate spin-orbit coupling near resonance, incorporating effects of triaxiality and dissipation.

Findings

Corrected the MacDonald torque model for physical consistency.

Derived expressions for libration frequency, damping, and orientation.

Showed stability of pseudosynchronous spin depends on dissipation model.

Abstract

Spin-orbit coupling is often described in the "MacDonald torque" approach which has become the textbook standard. Within this method, a concise expression for the additional tidal potential, derived by MacDonald (1964; Rev. Geophys. 2, 467), is combined with an assumption that the Q factor is frequency-independent (i.e., that the geometric lag angle is constant in time). This makes the approach unphysical because MacDonald's derivation of the said formula was implicitly based on keeping the time lag frequency-independent, which is equivalent to setting Q to scale as the inverse tidal frequency. The contradiction requires the MacDonald treatment of both non-resonant and resonant rotation to be rewritten. The non-resonant case was reconsidered by Efroimsky & Williams (2009; CMDA 104, 257), in application to spin modes distant from the major commensurabilities. We continue this work by introducing the necessary alterations into the MacDonald-torque-based model of falling into a 1:1 resonance. (For the original version of the model, see Goldreich 1966; AJ 71, 1.) We also study the effect of the triaxiality on both circulating and librating rotation near the synchronous state. Circulating rotation may evolve toward the libration region or toward a spin rate larger than synchronous (pseudosynchronous spin). Which behaviour depends on the eccentricity, the triaxiality of the primary, and the mass ratio of the secondary and primary bodies. The spin evolution will always stall for the oblate case. For small-amplitude librations, expressions are derived for the libration frequency, damping rate, and average orientation. However, the stability of pseudosynchronous spin hinges upon the dissipation model. Makarov and Efroimsky (2012; arXiv:1209.1616) have found that a more realistic dissipation model than the corrected MacDonald torque makes pseudosynchronous spin unstable.