How terrestrial planets traverse spin-orbit resonances: A camel goes through a needle's eye

TL;DR

This study models the spin-orbit resonance evolution of terrestrial planets like Mercury, revealing that such planets are almost always captured in the 3:2 resonance at current eccentricities, with a significant probability of passing through higher resonances.

Contribution

The paper introduces a comprehensive harmonic expansion method to analyze tidal torques, providing new insights into the probability and conditions of resonance capture for terrestrial planets.

Findings

Mercury-like planets are always captured in the 3:2 resonance at current eccentricity.

The probability of capture in the 2:1 resonance is approximately 23%.

Resonance capture is inevitable for eccentricities between 0.2 and 0.41.

Abstract

The dynamical evolution of terrestrial planets resembling Mercury in the vicinity of spin-orbit resonances is investigated using comprehensive harmonic expansions of the tidal torque taking into account the frequency-dependent quality factors and Love numbers. The torque equations are integrated numerically with a small step in time, includng the oscillating triaxial torque components but neglecting the layered structure of the planet and assuming a zero obliquity. We find that a Mercury-like planet with its current value of orbital eccentricity (0.2056) is always captured in the 3:2 resonance. The probability of capture in the higher 2:1 resonance is approximately 0.23. These results are confirmed by a semi-analytical estimation of capture probabilities as functions of eccentricity for both prograde and retrograde evolution of spin rate. As follows from analysis of equilibrium torques, entrapment in the 3:2 resonance is inevitable at eccentricities between 0.2 and 0.41. Considering the phase space parameters at the times of periastron, the range of spin rates and phase angles, for which an immediate resonance passage is triggered, is very narrow, and yet, a planet like Mercury rarely fails to align itself into this state of unstable equilibrium before it traverses the 2:1 resonance.