Resonant Relaxation in Protoplanetary Disks

Abstract

Resonant relaxation is a novel form of gravitational relaxation that arises in nearly Keplerian disks such as protoplanetary disks. Resonant relaxation does not affect the semimajor axes of the particles but enhances relaxation of particle eccentricities and inclinations. The equilibrium state after resonant relaxation is a Rayleigh distribution, with the mean-square eccentricity and inclination inversely proportional to mass. The rate of resonant relaxation depends strongly on the precession rate of the disk. If the precession due to the disk's self-gravity is small compared to the total precession, then the relaxation is concentrated near the secular resonance between each pair of interacting bodies; on the other hand, if the precession rate is dominated by the disk's self-gravity, then relaxation occurs through coupling to the large-scale low-frequency m = 1 normal modes of the disk. Depending on the disk properties-especially the mean-square eccentricity-resonant relaxation may be either stronger or weaker than the usual nonresonant relaxation: for the terrestrial planets, nonresonant relaxation dominates in the early stages of planetesimal growth, but resonant relaxation may dominate in the late stages when the mean-square eccentricity is higher.