Conditions of Passage and Entrapment of Terrestrial Planets in Spin-orbit Resonances

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, including 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 a current value of orbital eccentricity (0.2056) is always captured in 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 evolutions of spin rate. As follows from analysis of equilibrium torques, entrapment in 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 2:1 resonance.