Statistics of Saturn Ring Occultations

We calculate the excess variance, excess skewness and excess kurtosis including the effects of cylindrical shadows, along with gaps, ghosts and clumps (all calculated for the granola bar model for rectangular clumps and gaps). The widths and separation of the clumps play an analogous role to the relative size of the particle shadows, δ. Wherever the rings have significant gaps or clumps, those will dominate the statistics over the individual ring particles shadow contribution. In the first model considered, our calculations are based on the moments of the transparency T in that part of the ring sampled by the occultation, thus extending the work of  Showalter and Nicholson (1990) to larger τ and δ, and to higher central moments, without their simplifying assumptions. We also calculate these statistics using an approach based on the autocovariance, autocoskewness and autocokurtosis. This  may be more intuitive, and can be extended to other transparency distributions, e.g., those provided by gaps, ghosts, clumps and granola bars. In a third method, we have refined an overlap correction for multiple shadows, which is important for larger optical depth. This correction is calculated by summing a geometric series, and is similar to the empirical formula, eq. (22) in Colwell et al (2018). These 3 new approaches compare well to the formula for excess variance from Showalter and Nicholson in the region where all are accurate, that is δτ≪1. 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.

As a check, we can 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 R_eff = 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 between 78,000 and 84,600km from Saturn. This explanation has no free parameters and requires no gaps or ghosts (Baillie etal 2013) in this region of Saturn’s C ring.