Dynamics of Colombo’s top: tidal dissipation and resonance capture, with applications to oblique super-Earths, ultra-short-period planets and inspiraling hot Jupiters

Abstract

ABSTRACT We present a comprehensive theoretical study on the spin evolution of a planet under the combined effects of tidal dissipation and gravitational perturbation from an external companion. Such a ‘spin + companion’ system (called Colombo’s top) appears in many [exo]planetary contexts. The competition between the tidal torque (which drives spin-orbit alignment and synchronization) and the gravitational torque from the companion (which drives orbital precession of the planet) gives rise to two possible spin equilibria (‘tidal Cassini Equilibria’, tCE) that are stable and attracting: the ‘simple’ tCE1, which typically has a low spin obliquity, and the ‘resonant’ tCE2, which can have a significant obliquity. The latter arises from a spin-orbit resonance and can be broken when the tidal alignment torque is stronger than the precessional torque from the companion. We characterize the long-term evolution of the planetary spin (both magnitude and obliquity) for an arbitrary initial spin orientation, and develop a new theoretical method to analytically obtain the probability of resonance capture driven by tidal dissipation. Applying our general theoretical results to exoplanetary systems, we find that a super-Earth (SE) with an exterior companion can have a substantial probability of being trapped in the high-obliquity tCE2, assuming that SEs have a wide range of primordial obliquities. We also evaluate the recently proposed ‘obliquity tide’ scenarios for the formation of ultra-short-period Earth-mass planets and for the orbital decay of hot Jupiter WASP-12b. We find in both cases that the probability of resonant capture into tCE2 is generally low and that such a high-obliquity state can be easily broken by the required orbital decay.