Numerical exploration of planetary arc dynamics

Abstract

Ground-based observations have revealed in 1984 and 1985 the existence of interrupted ring-like structures, or “arcs” around Neptune. Recent observations by the Voyager 2 spacecraft in August 1989 have confirmed the presence of stable, dense arcs, embedded in a faint continuos ring. These structures would be destroyed in a few years if they followed unperturbed Keplerian motions. We study here the stabilizing effect of corotation and Lindblad resonances on planetary arcs, along the lines of previous analytical work by Lissauer ( Nature (London) 318, 1985, 544–545) and Goldreich, Tremaine, and Borderies ( Astron. J. 92, 1986, 490–494). We first describe analytically the responce of a test particle to the combined effects of corotation and Lindblad resonances, caused by Neptunian satellites. The evolution of the particle is shown to be described by two coupled dynamical systems, the coupling depending on collective effects in the arc. This yields a formula for the energy provided by the Lindblad resonance, which is actually used to confine the arc material around the corotation resonance radius. We show in particular that the gradient of the torque density across the arc must be negative for the latter to be stable. This confirms in a general framework the constraints, previously given by Lin, Papaloizou, and Ruden ( Mon. Not. R. Astron. Soc. 227, 1987, 75–95), on the relative positions of the corotation and the Lindblad resonances for the arc to be stable. We test our results with a direct numerical simulation, which takes into account inelastic collisions between identical spherical particles. Two configurations are studied: (1) an arc at a L 4 Lagrange point of a satellite, further perturbed by an isolated Lindblad resonance with a second satellite, and (2) an arc at an isolated corotation resonance with a single satellite on an eccentric orbit. Our main results are: (a) the corotation points alone are unstable against dissipative collisions, (b) a stable arc must be submitted to a negative gradient of torque density, and (c) such a stable arc reaches a limit cycle where the energy provided by the resonant satellite is balanced by the energy dissipated by collisions.