Statistics of Saturn's Ring Occultations: Implications for Structure, Dynamics and Origins

The Cassini star occultations by Saturn's rings constrain both the size and shape of structures that block starlight. Statistics of UVIS occultations measure structures as small as meters, on times scales of minutes to decades. We calculate the excess variance, skewness and kurtosis including the effects of irregular particle shadows, along with a granola bar model (GBM) for gaps, ghosts and clumps. We then use the statistics of ring occultations observed by the Cassini UVIS to characterize structures in Saturn's rings. Skewness for small τ has a different sign for transparent and opaque structures, and can distinguish gaps from clumps. The higher order central moments are more sensitive to the extremes of the size distribution and opacity.

We explain the upward curvature of the dependence of normalized excess variance for Saturn's background C ring by the observation of Jerousek etal(2018) that the increased optical depth is directly correlated with effective particle size. For a linear dependence Reff = 12 * (τ – 0.08) + 1.8m from Jerousek's results, we match both the curvature of normalized excess variance and the skewness in the region. This explanation has no free parameters and requires no gaps or ghosts (Baillie etal 2013). Another check of our method is provided by the C ring ramp (just inside Saturn's B ring): it is smooth, with no undulations, and provides a simple case to examine the higher moments of the occultation statistics in order to infer the particle size and shape. We assume that the particle size distribution and ring structure do not depend on τ, only on the number density of particles. To calculate the expected variance, skewness and kurtosis, we use the moments approach of Showalter and Nicholson (1990), extended to higher moments and removing their restrictions on fractional particle area δ <<1 and line-of-sight optical depth τ <<1. We include Poisson contributions, but ignore Sheppard's corrections for data compression; and use the exact formulas, not Taylor expansions. The measured occultation statistics show the expected extrema and zero crossings. Excess variance (NEV) gives δ=0.05, Reff=2.3m; ΓX gives δΓ=0.07, from amax=5m & q=3.1; 𝛫X gives δ𝛫=0.06, but we do not achieve a good fit for 𝛫X (τ). These size distribution results are similar to those derived for the background C-ring, indicating a similar origin for the ramp. For a power law size distribution, we expect δ𝛫> δΓ > δ. The more platykurtic distribution of star count statistics may be due to a size distribution, including gaps, ghosts and clumps. For a granola bar model of self-gravity wakes in the A ring, we find W = 18-29m; S+W ~ 60m; H/W< 0.12, thus H < 4m. These results are consistent with a simple dynamical model of the rings, analogous to an ecological Predator-Prey interaction. Perturbed by passing density waves, self-gravity wakes grow and erode on orbital timescales with a full amplitude of 60%, and a phase lag 𝚫φ ~45°.