The ballistic transport instability in Saturn's rings - I. Formalism and linear theory

Abstract

Planetary rings sustain a continual bombardment of hypervelocity meteoroids that erode the surfaces of ring particles on time-scales of 105-107years. The debris ejected from such impacts re-accretes on to the ring, though often at a slightly different orbital radius from the point of emission. This 'ballistic transport' leads to a rearrangement of the disc's mass and angular momentum, and gives rise to a linear instability that generates structure on relatively large scales. It is likely that the 100-km wavetrains in Saturn's inner B-ring and the plateaus and 1000-km undulations in Saturn's C-ring are connected to the non-linear saturation of the instability. In this paper the physical problem is reformulated so as to apply to a local patch of disc (the shearing sheet). This new streamlined model helps facilitate our physical understanding of the instability, and also makes more tractable the analysis of its non-linear dynamics. We concentrate on the linear theory in this paper, showing that the instability is restricted to a preferred range of intermediate wavenumbers and optical depths. We subsequently apply these general results to the inner B-ring and the C-ring and find that in both regions the ballistic transport instability should be near marginality, a fact that may have important consequences for its prevalence and non-linear development. Owing to damping via self-gravity wakes, the instability should not be present in the A-ring. A following paper will explore the instability's non-linear saturation and how it connects to the observed large-scale structure. © 2012 The Authors Monthly Notices of the Royal Astronomical Society © 2012 RAS.