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Abstract

We address the occurrence of narrow planetary rings under the interaction with shepherds. Our approach is based on a Hamiltonian framework of non-interacting particles where open motion (escape) takes place, and includes the quasi-periodic perturbations of the shepherd's Kepler motion with small and zero eccentricity. We concentrate in the phase-space structure and establish connections with properties like the eccentricity, sharp edges and narrowness of the ring. Within our scattering approach, the organizing centers necessary for the occurrence of the rings are stable periodic orbits, or more generally, stable tori. In the case of eccentric motion of the shepherd, the rings are narrower and display a gap which defines different components of the ring.

Multiple Components in Narrow Planetary Rings

“…1(a) we show the histogram of hti for the n 0 trapped region; Fig. 1(b) shows a detail of the corresponding ring [9]. We note that Fig.…”

“…The scattering approach to narrow rings [9] considers the scattering dynamics of a (ring) particle in a planar system with some intrinsic rotation (external force). In the planetary case, this rotation comes from the orbital motion of one or several satellites.…”

“…The ring is typically narrow, sharp-edged, and noncircular; this scenario is generic [14]. For more than two DOF, e.g., a nonuniform rotation on a Kepler elliptic orbit, the ring displays further structure [9].…”

“…For the disk on a circular orbit, the simple periodic orbits are the radial collision orbits. They and their linear stability are given by [9] J 2! 2 d R ÿ d 2 1 tancos 2 ÿ2 ; (1)…”

“…Each one is related to a different region in phase space; this is a consequence of the large gap opened by the 1:3 SR. These components are entangled and form a braided ring [9].…”