Rotation and figure evolution in the creep tide theory. A new approach and application to Mercury

TL;DR

This paper introduces a new differential equation formulation of the creep tide theory to study planetary rotation and figure evolution, applying it to Mercury to estimate its relaxation factor and compare with observational data.

Contribution

It presents a coupled system of equations for rotation and figure evolution, and a method to determine the relaxation factor gamma for non-rigid bodies in spin-orbit resonance.

Findings

Derived gamma range for Mercury's rotational evolution

Found Mercury's figure coefficients match Darwin-Kaula model

Observed current Mercury flattenings exceed tidal theory predictions

Abstract

This paper deals with the rotation and figure evolution of a planet near the 3/2 spin-orbit resonance and the exploration of a new formulation of the creep tide theory (Folonier et al. 2018). This new formulation is composed by a system of differential equations for the figure and the rotation of the body simultaneously (which is the same system of equations used in Folonier et al. 2018), different from the original one (Ferraz-Mello, 2013, 2015a) in which rotation and figure were considered separately. The time evolution of the figure of the body is studied for both the 3/2 and 2/1 spin-orbit resonances. Moreover, we provide a method to determine the relaxation factor gamma of non-rigid homogeneous bodies whose endpoint of rotational evolution from tidal interactions is the 3/2 spin-orbit resonance, provided that (i) an initially faster rotation is assumed and (ii) no permanent components of the flattenings of the body existed at the time of the capture in the 3/2 spin-orbit resonance. The method is applied to Mercury, since it is currently trapped in a 3/2 spin-orbit resonance with its orbital motion and we obtain 4.8 times 10 -8 s -1 lower than gamma lower than 4.8 times 10 -9 s -1 . The equatorial prolateness and polar oblateness coefficients obtained for Mercury's figure with such range of values of gamma are the same as the ones given by the Darwin-Kaula model (Matsuyama and Nimmo 2009). However, comparing the values of the flattenings obtained for such range of gamma with those obtained from MESSENGER's measurements (Perry et al. 2015), we see that the current values for Mercury's equatorial prolateness and polar oblateness are 2-3 orders of magnitude larger than the values given by the tidal theories.