Resonant Satellite Torques on Low Optical Depth Particulate Disks: I. Analytic Development

Abstract

We investigate the torque exerted by a satellite on an annulus centered at a mean motion resonance. The annulus is made of noninteracting (test) particles uniformly distributed in semimajor axis. We obtain an analytic expression for the time evolution of the angular momentum of the annulus. We find that in the linear approximation, the net torque on the disk equals that obtained by P. Goldreich and S. Tremaine (1978,Icarus34, 240–253) for a fluid disk in quasi-steady state. The same torque is obtained for disks of particles initially on circular orbits as for disks of particles on moderately eccentric orbits with periapses uniformly distributed in longitude.Poisson demonstrated that no secular torque can be obtained in the restricted three-body problem. As the satellite pumps up the eccentricity of a particle's orbit, it also changes the particle's semimajor axis in such a way that eventually the torque is reversed. The period and amplitude for this interchange of angular momentum with the satellite depends on the distance from resonance, being larger the closer the particle is to the resonance. The width of the annulus over which the bulk of the torque is exerted shrinks as time increases. Our linear analysis allows arbitrarily large eccentricities for particles near resonance, which is clearly unphysical. The torque in a nondissipative disk is limited in time by nonlinear effects of the interaction close to resonance. We present quantitative estimates of where in semimajor axis, and when in time, the linear approximation breaks down.