Orbital-plane precessional resonances for binary black-hole systems

We derive a new class of post-Newtonian precessional resonances for binary black holes (BBHs) with misaligned spins. According to the orbit-averaged spin-precession equations, the angle between the orbital angular momentum $\mathbf{L}$ and the total angular momentum $\mathbf{J}$ oscillates with a period $\tau$ during which time $\mathbf{L}$ precesses about $\mathbf{J}$ by an angle $\alpha$. If $\alpha$ is a rational multiple of 2$\pi$, the precession of $\mathbf{L}$ will be closed indicating a resonance between the polar and azimuthal evolution of $\mathbf{L}$. If $\alpha$ is an integer multiple of 2$\pi$, the misalignment between the angular momentum $\Delta\mathbf{L}$ radiated over the period $\tau$ and $\mathbf{J}$ will be minimized, as will the opening angle of the cone about which $\mathbf{J}$ precesses in an inertial frame. However, the direction of $\Delta\mathbf{L}$ will remain nearly fixed in an inertial frame over many precessional periods, causing the direction of $\mathbf{J}$ to tilt as inspiraling BBHs pass through such a resonance. Generic BBHs encounter many such resonances during an inspiral from large separations. We derive the evolution of $\mathbf{J}$ near a resonance and assess their detectability by gravitational-wave detectors and astrophysical implications.