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 Colombos 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.
Many proposed scenarios for black hole (BH) mergers involve a tertiary companion that induces von Zeipel-Lidov-Kozai (ZLK) eccentricity cycles in the inner binary. An attractive feature of such mechanisms is the enhanced merger probability when the o
ctupole-order effects, also known as the eccentric Kozai mechanism, are important. This can be the case when the tertiary is of comparable mass to the binary components. Since the octupole strength [$propto (1-q)/(1+q)$] increases with decreasing binary mass ratio $q$, such ZLK-induced mergers favor binaries with smaller mass ratios. We use a combination of numerical and analytical approaches to fully characterize the octupole-enhanced binary BH mergers and provide analytical criteria for efficiently calculating the strength of this enhancement. We show that for hierarchical triples with semi-major axis ratio $a/a_{rm out}gtrsim 0.01$-$0.02$, the binary merger fraction can increase by a large factor (up to $sim 20$) as $q$ decreases from unity to $0.2$. The resulting mass ratio distribution for merging binary BHs produced in this scenario is in tension with the observed distribution obtained by the LIGO/VIRGO collaboration, although significant uncertainties remain about the initial distribution of binary BH masses and mass ratios.
