A numerical study of vector resonant relaxation
  Stars bound to a supermassive black hole interact gravitationally. Persistent
torques acting between stellar orbits lead to the 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
timescale. We use N-ring to follow the evolution of clusters containing over
10^4 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 0.5-1 radian. The
relaxation rate in a cluster with a fixed number of stars is proportional to
the 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<0.7 and grows rapidly with
eccentricity for e>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 timescale.
