A numerical study of vector resonant relaxation

TL;DR

This paper introduces a new parallel symplectic integrator, N-ring, to simulate vector resonant relaxation in star clusters around supermassive black holes, revealing the dominant role of overlapping orbits and the dependence of relaxation rates on eccentricity.

Contribution

The paper presents a novel N-ring integrator for efficiently modeling vector resonant relaxation in large star clusters, with detailed analysis of relaxation dynamics and dependencies.

Findings

Relaxation dominated by stars with overlapping orbits.

Resonant relaxation modeled as a random walk of orbit normals.

Relaxation rate nearly independent of eccentricity for e<0.7, increases for e>0.8.

Abstract

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.