Tidal dissipation in a homogeneous spherical body. II. Three examples: Mercury, Io, and Kepler-10 b

TL;DR

This paper presents three case studies on tidal dissipation in homogeneous spheres, highlighting the sharp frequency dependence near resonances, the role of libration and triaxiality, and provides simplified formulas for rocky exoplanets, with implications for planetary evolution.

Contribution

It derives new formulas for tidal dissipation in homogeneous bodies, emphasizing the impact of libration and triaxiality, and applies these to Mercury, Io, and Kepler-10b.

Findings

Tidal dissipation sharply depends on frequency near resonances.

Stronger triaxiality increases tidal heating.

Kepler-10b likely experienced intense tidal heating leading to reshaping.

Abstract

In Efroimsky & Makarov (2014), we derived from the first principles a formula for the tidal heating rate in a tidally perturbed homogeneous sphere. We compared it with the formulae used in the literature, and pointed out the differences. Using this result, we now present three case studies - Mercury, Kepler-10b, and a triaxial Io. A very sharp frequency-dependence of k2/Q near spin-orbit resonances yields a similarly sharp dependence of k2/Q on the spin rate. This indicates that physical libration may play a major role in tidal heating of synchronously rotating bodies. The magnitude of libration in the spin rate being defined by the planet's triaxiality, the latter should be a factor determining the dissipation rate. Other parameters equal, a synchronously rotating body with a stronger triaxiality should generate more heat than a similar body of a more symmetrical shape. Further in the paper, we discuss scenarios where initially triaxial objects melt and lose their triaxiality. Thereafter, dissipation in them becomes less intensive; so the bodies freeze. The tidal bulge becomes a new permanent figure, with a new triaxiality lower than the original. In the paper, we also derive simplified, approximate expressions for dissipation rate in a rocky planet of the Maxwell rheology, with a not too small Maxwell time. The three expressions derived pertain to the cases of a synchronous spin, a 3:2 resonance, and a nonresonant rotation; so they can be applied to most close-in super-Earth exoplanets detected thus far. In such bodies, the rate of tidal heating outside of synchronous rotation is weakly dependent on the eccentricity and obliquity, provided both these parameters are small or moderate. According to our calculation, Kepler-10b could hardly survive the great amount of tidal heating without being synchronised, circularised and also reshaped through a complete or partial melt-down.