In the post-Newtonian (PN) regime, the timescale on which the spins of binary
black holes precess is much shorter than the radiation-reaction timescale on
which the black holes inspiral to smaller separations. On the precession
timescale, 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 timescale: 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.Comment: 18 pages, 9 figures, version published in PR