Nutational resonances, transitional precession, and precession-averaged evolution in binary black-hole systems

Abstract

In the post-Newtonian (PN) regime, the time scale on which the spins of binary black holes precess is much shorter than the radiation-reaction time scale on which the black holes inspiral to smaller separations. On the precession time scale, the angle between the total and orbital angular momenta oscillates with nutation period $\ensuremath{\tau}$, during which the orbital angular momentum precesses about the total angular momentum by an angle $\ensuremath{\alpha}$. This defines two distinct frequencies that vary on the radiation-reaction time scale: the nutation frequency $\ensuremath{\omega}\ensuremath{\equiv}2\ensuremath{\pi}/\ensuremath{\tau}$ and the precession frequency $\mathrm{\ensuremath{\Omega}}\ensuremath{\equiv}\ensuremath{\alpha}/\ensuremath{\tau}$. We use analytic solutions for generic spin precession at 2PN order to derive Fourier series for the total and orbital angular momenta in which each term is a sinusoid with frequency $\mathrm{\ensuremath{\Omega}}\ensuremath{-}n\ensuremath{\omega}$ for integer $n$. As black holes inspiral, they can pass through nutational resonances ($\mathrm{\ensuremath{\Omega}}=n\ensuremath{\omega}$) at which the total angular momentum tilts. We derive an approximate expression for this tilt angle and show that it is usually less than ${10}^{\ensuremath{-}3}$ radians for nutational resonances at binary separations $r>10M$. The large tilts occurring during transitional precession (near zero total angular momentum) are a consequence of such states being approximate $n=0$ nutational resonances. Our new Fourier series for the total and orbital angular momenta converge rapidly with $n$ providing an intuitive and computationally efficient approach to understanding generic precession that may facilitate future calculations of gravitational waveforms in the PN regime.