A numerical study of vector resonant relaxation

Kocsis, Bence and Tremaine, S. (2015) A numerical study of vector resonant relaxation. MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, 448 (4). pp. 3265-3296. ISSN 0035-8711

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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 ori- entation vectors (“vector” resonant relaxation) and slower relaxation of the eccen- tricities (“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 reso- nant relaxation, by averaging the pairwise interactions over the orbital period and periapsis-precession timescale. We use n-ring to follow the evolution of clusters con- taining 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 ∼ 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.

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