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≡2π/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-nω for integer n. As black holes inspiral, they can pass through nutational resonances (Ω=nω) 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>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.