Tidal interactions of a Maclaurin spheroid. II: Resonant excitation of modes by a close, misaligned orbit

TL;DR

This paper models tidal interactions in a Maclaurin spheroid, revealing that higher-degree modes and inertial mode resonances can significantly enhance tidal dissipation in close, misaligned orbits, especially relevant for giant planets.

Contribution

It provides a comprehensive formalism for decomposing tidal forces into normal modes and demonstrates the importance of higher-degree and inertial mode resonances in tidal dissipation.

Findings

Resonant excitation of modes can increase tidal power by up to three orders of magnitude.

Higher-degree modes, especially l=4, significantly contribute to tidal dissipation in close orbits.

Inertial modes can be strongly resonant, affecting orbital evolution.

Abstract

We model a tidally forced star or giant planet as a Maclaurin spheroid, decomposing the motion into the normal modes found by Bryan (1889). We first describe the general prescription for this decomposition and the computation of the tidal power. Although this formalism is very general, forcing due to a companion on a misaligned, circular orbit is used to illustrate the theory. The tidal power is plotted for a variety of orbital radii, misalignment angles, and spheroid rotation rates. Our calculations are carried out including all modes of degree $l \le 4$, and the same degree of gravitational forcing. Remarkably, we find that for close orbits ($a/R_* \approx 3$) and rotational deformations that are typical of giant planets ($e\approx 0.4$) the $l=4$ component of the gravitational potential may significantly enhance the dissipation through resonance with surface gravity modes. There are also a large number of resonances with inertial modes, with the tidal power being locally enhanced by up to three orders of magnitude. For very close orbits ($a/R_* \approx 3$), the contribution to the power from the $l=4$ modes is roughly the same magnitude as that due to the $l = 3$ modes.