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

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 τ , during which the orbital angular momentum precesses about the total angular momentum by an angle α . This defines two distinct frequencies that vary on the radiation-reaction time scale: the nutation frequency ω ≡2 π /τ and the precession frequency Ω ≡α /τ . 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 Ω -n ω for integer n . As black holes inspiral, they can pass through nutational resonances (Ω =n ω ) 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^-3 radians for nutational resonances at binary separations r >10 M . 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.