Dynamical resonance locking in tidally interacting binary systems

Abstract

We examine the dynamics of resonance locking in detached, tidally interacting binary systems. In a resonance lock, a given stellar or planetary mode is trapped in a highly resonant state for an extended period of time, during which the spin and orbital frequencies vary in concert to maintain the resonance. This phenomenon is qualitatively similar to resonance capture in planetary dynamics. We show that resonance locks can accelerate the course of tidal evolution in eccentric systems and also efficiently couple spin and orbital evolution in circular binaries. Previous analyses of resonance locking have not treated the mode amplitude as a fully dynamical variable, but rather assumed the adiabatic (i.e. Lorentzian) approximation valid only in the limit of relatively strong mode damping. We relax this approximation, analytically derive conditions under which the fixed point associated with resonance locking is stable, and further check these analytic results using numerical integrations of the coupled mode, spin, and orbital evolution equations. These show that resonance locking can sometimes take the form of complex limit cycles or even chaotic trajectories. We provide simple analytic formulae that define the binary and mode parameter regimes in which resonance locks of some kind occur (stable, limit cycle, or chaotic). We briefly discuss the astrophysical implications of our results for white dwarf and neutron star binaries as well as eccentric stellar binaries.