Cassini states for black-hole binaries

Cassini states correspond to equilibria of the spin axis of a celestial body when its orbit is perturbed. They were initially described for planetary satellites, but the spin axes of black-hole binaries also present this kind of equilibria. In previous works, Cassini states were reported as spin-orbit resonances, but actually the spin of black-hole binaries is in circulation and there is no resonant motion. Here we provide a general description of the spin dynamics of black-hole binary systems based on a Hamiltonian formalism. In absence of dissipation the problem is integrable and it is easy to identify all possible trajectories for the spin for a given value of the total angular momentum. As the system collapses due to radiation reaction, the Cassini states are shifted to different positions, which modifies the dynamics around them. This is why the final spin distribution may differ from the initial one. Our method provides a simple way of predicting the distribution of the spin of black-hole binaries at the end of the inspiral phase.