A numerical study of vector resonant relaxation

Stars bound to a supermassive black hole interact gravitationally. Persistent torques acting between stellar orbits lead to a rapid resonant relaxation of the orbital orientation vectors ('vector' resonant relaxation) and slower relaxation of the eccentricities ('scalar' resonant relaxation), both at rates much faster than two-body or non-resonant relaxation. We describe a new parallel symplectic integrator, N-RING, which follows the dynamical evolution of a cluster of N stars through vector resonant relaxation, by averaging the pairwise interactions over the orbital period and periapsis precession time-scale. We use N-RING to follow the evolution of clusters containing over 104 stars for tens of relaxation times. Among other results, we find that the evolution is dominated by torques among stars with radially overlapping orbits, and that resonant relaxation can be modelled as a random walk of the orbit normals on the sphere, with angular step size ranging from similar to 0.5-1 rad. The relaxation rate in a cluster with a fixed number of stars is proportional to the root mean square (rms) mass of the stars. The rms torque generated by the cluster stars is reduced below the torque between Kepler orbits due to apsidal precession and declines weakly with the eccentricity of the perturbed orbit. However, since the angular momentum of an orbit also decreases with eccentricity, the relaxation rate is approximately eccentricity-independent for e less than or similar to 0.7 and grows rapidly with eccentricity for e greater than or similar to 0.8. We quantify the relaxation using the autocorrelation function of the spherical multipole moments; this decays exponentially and the e-folding time may be identified with the vector resonant relaxation time-scale.