Strands and braids in narrow planetary rings: a scattering system approach

Abstract

We address the occurrence of narrow planetary rings and some of their structural properties, in particular when the rings are shepherded. We consider the problem as Hamiltonian scattering of a large number of non-interacting massless point particles in an effective potential. Using the existence of stable motion in scattering regions in this set up, we describe a mechanism in phase space for the occurrence of narrow rings and some consequences in their structure. We illustrate our approach with three examples. We find eccentric narrow rings displaying sharp edges, variable width and the appearance of distinct ring components (strands) which are spatially organized and entangled (braids). We discuss the relevance of our approach for narrow planetary rings.

Phase-space volume of regions of trapped motion: multiple ring components and arcs

“…The ring is typically narrow, sharp-edged and non-circular. These properties follow from the phase-space regions of trapped motion which are rather localized in phase space, the scattering dynamics, and the shape of the organizing centers (periodic orbits) of the regions of bounded motion, respectively (see Merlo and Benet 2007).…”

“…For systems with two degrees of freedom, the dynamics can be analyzed using standard methods, e.g., with a Poincaré section. In particular, for ε = 0 we consider the symplectic map of the surface of section onto itself, (α k+1 , p k+1 ) = P J (α k , p k ), defined when the particle collides with the disk, where p k = −d − R cos α k − v k sin(α k − θ k ) is kin to the angular momentum (Merlo and Benet 2007). The map is constructed by solving numerically the transcendental equation (using Newton's method to high accuracy) which determines the time for the next collision (see Benet et al 2005 for details); whenever there is a solution, the resulting outgoing variables are computed.…”

“…We checked this by considering a large number of consecutive bounces in the numerical experiments (ring particles must display at least 100,000 collisions with the disk to be considered trapped), and by confirming the appearance of a non-statistical peak in the decay-rate statistics (see Merlo 2004;Benet et al 2005 for details). Therefore, the theory can indeed be extended to systems with more degrees of freedom in terms of (a) the stable tori as the organizing centers of the dynamics and, (b) the regions of bounded motion that are found close to them which are defined now at least on an effective stability sense (Merlo and Benet 2007). Rings are then obtained by the construction described above.…”

“…More specifically, we consider the appearance of fine structure and its azimuthal dependence in the context of the narrow planetary rings, which are formed by the ensemble of non-escaping particles. Our study is based on the scattering approach (Merlo and Benet 2007), and we use as illustrative example an unrealistic system, namely, a planar circular hard disk whose center moves along a Keplerian closed trajectory. Despite of the non-realistic interactions considered in this example, we do obtain rings which display more than one component and arcs, features which are in qualitative agreement with observations of real narrow planetary rings (for an account of the observations see the recent book by Esposito 2006).…”

“…Robustness warranties that we can extend the model beyond the restricted three-body problem, and include the oblateness of the planet as well as other perturbations, e.g., the gravitational influence of other satellites. We refer the reader to Merlo and Benet (2007), where these and other physically relevant aspects of the scattering approach, in particular the predictions in connection to some structural properties of narrow rings, are discussed in detail.…”