Joshua E. Colwell

Stellar occultation measurements of Saturn’s rings from the Cassini Ultraviolet Imaging Spectrograph (UVIS) High Speed Photometer (HSP) reveal dozens of regions in the heart of the B ring that are opaque at the sensitivity level of the measurements (Colwell et al. 2021, 2024). We have identified 24 regions with an average transparency less than 0.25%, including 9 with an average transparency less than 0.1% and radial extents greater than 10 km. There are broad expanses within these opaque regions where the transparency is indistinguishable from zero. Occultations of the star Beta Centauri with photon count rates of up to 600 per 1 ms integration period provide the strongest constraints on transparency. The elevation angle B (sin(B)=0.918) and brightness of Beta Centauri make it best suited among the UVIS stellar occultations to probe high optical depth regions. The radial resolution of these occultations is about 8 meters, while the azimuthal resolution in the frame co-moving with the ring particles is ~20-30 meters. Each measurement samples an area in the frame of the particles of ~400 m2. The average transparency of each designated opaque region is > 0 because of the presence of narrow, transient features we dub “phantoms” that increase the average transparency of the region. Phantoms are analogous to the ghosts observed by Baillie et al. (2013) and Green et al. (2024) in the C ring and Cassini Division. While ghosts were defined by being indistinguishable from 100% transparent in an otherwise moderate optical depth background, phantoms are defined as narrow, non-axisymmetric features with non-zero transparency in an otherwise opaque background (Figure 1).Figure 1: Stacked and offset (by 5%) transparency profiles of the O1 opaque region in Saturn’s B2 ring region as measured by the Cassini UVIS HSP. The different curves are for different observations of the same star system: Beta Centauri. Narrow spikes in transparency are identified as phantoms. The vertical dashed lines indicate the boundaries of the opaque region. We observed phantoms that are only a single HSP measurement, indicating a radial extent of < 10 m, but most span several measurements. Phantom transparencies range from a few percent up, but in some cases the transparency exceeds 10% and can approach 50%. There are 15 measurements of the B ring with Beta Centauri occultations, and the phantoms do not repeat from one occultation to the next, highlighting their non-axisymmetric and/or temporally variable nature. The Beta Centauri system has two stars that contribute to the HSP signal, and the projected separation of these two components in the ring plane enables us to place limits on the azimuthal extent of some phantoms, which in some cases is < 100 m. At some locations there are clusters of phantom-like features that appear in all occultations at the same location. We identify these axisymmetric features “grassy regions” based on their appearance in plots of transparency (Figure 2). The average transparency of grassy regions is still low, typically less than 2%. The opaque regions are bordered by regions of higher optical depth (Figure 2).Figure 2: Stacked and offset (by 5%) transparency profiles of an opaque region in Saturn’s B2 ring region as measured by the Cassini UVIS HSP. The different curves are for different observations of the same star system: Beta Centauri. This opaque region is home to phantoms (isolated spikes in transparency) as well as several grassy regions, identifiable by their presence at the same location in all occultation profiles and marked by blue dashed lines. The measured optical depth of the opaque stretches between phantoms can exceed 6 in some cases (see for example the bottom curve in Figure 2 between 100,250-100,300 km). N-body simulations have struggled to reproduce such high optical depths. Individual ring particles in the simulations clump together due to their mutual gravity, leaving gaps which elevate the overal transparency of the region. We will present results of simulations with modified boundary conditions and particle properties to try to explain the existence of such high-optical-depth regions and the sub-structure of phantoms and grassy regions within them. Baillié, K., J. E. Colwell, L. W. Esposito, and M. C. Lewis 2013. Meter-sized Moonlet Population in Saturn’s C Ring and Cassini Division. Astron J. 145, 171, doi:10.1088/0004-6256/145/6/171. Colwell, J. E., M. Brooks, R. Jerousek, C. Coleman, M. S. Tiscareno, K.-M. Aye, M. Lewis, L. W. Esposito 2021. Irregular Structure in the Core of Saturn’s B Ring. 2021 AGU Fall Meeting P34A-01. Colwell, J. E., A. Goforth, C. Coleman, R. Jerousek, L. W. Esposito 2024. Saturn’s B Ring Phantoms: Frequency, Properties, and Distribution. 2024 Meeting of the American Geophysical Union. P33F-2931. Green, M. R., J. E. Colwell, L. W. Esposito, R. G. Jerousek 2024. Particle sizes in Saturn’s rings from UVIS stellar occultations 2. Outlier populatinos in the C ring and Cassini Division. Icarus 416, 116081.

We present a comparative occultation analysis of narrow, eccentric ringlets in the Saturnian and Uranian ring systems, focusing on how mesoscale structure and particle size distributions vary with orbital phase (true anomaly). Our study combines the full suite of Cassini stellar and radio occultation profiles of Saturn’s narrow ringlets, obtained by Cassini’s UVIS (λ = 150 nm), VIMS (λ = 2.84 μm), and RSS (Ka-, X-, S-band experiments). We compare our results with a reanalysis of Voyager 2 PPS and RSS occultations of Uranus’s ε, α, β, η, and δ rings.We group Saturn ringlet profiles into four orbital quadrants—periapsis, apoapsis, and the streamline divergence/convergence sectors—to isolate phase-dependent structure (Figure 1). In each quadrant, we model viewing-geometry dependent normal optical depth using the irregularly spaced, three component, granola bar self-gravity wake model of Esposito et al. (2025) (Figure 2). The model fits the dimensional ratios and cant angles of aligned structures such as self-gravity wakes, particle-depleted lanes, or both. Following Green et al. (2024), we compute the normalized central moments of UVIS HSP occultation transparencies to constrain the size of the largest ring particles (amax) and to characterize the clumpiness of the ringlets with orbital phase. We compare these moments to predictions from Esposito et al. (2025) for wake-dominated rings. Simultaneously, we perform multi-wavelength particle size distribution fits by selecting UVIS, VIMS, and RSS profiles at similar orbital phase and with similar viewing geometries to constrain the power-law index (q) and minimum particle radius (amin) (Figure 3).For the Uranian rings, we use the σ-Sagittarii and β-Persei Voyager 2 PPS occultations to extract radial optical depth profiles of the dense narrow rings, supplemented by RSS S/X-band differential attenuation. While the geometry does not allow for true-anomaly-resolved comparisons, we compare Voyager 2 PPS occultation statistics and RSS S and X band optical depths to enable structural comparison.Our approach incorporates a self-consistent model of wavelength and viewing-geometry dependent optical depths with an analysis of photon counting statistics (Cassini UVIS HSP and Voyager 2 PPS). We find that Saturn’s ringlets exhibit significant variation in both self-gravity wake dimensions and inferred size distributions between quadrants (Figure 4).By applying a common framework across two planetary ring systems, we aim to identify techniques to constrain the phase-dependence of the particle size distribution, surface densities, and structure of dense narrow rings from in-situ occultations. This approach is intended not only to advance current understanding but also to inform the design of future occultation experiments at Uranus.Figure 1. Ringlet edge radii from UVIS, VIMS, and RSS occulations colored by sine of the ring opening angle for the Maxwell Ringlet. Occultations are separated into quadrants (Groups 1 – 4) in true anomaly for orbital phase-dependent analysis.Figure 2. The irregularly spaced granola bar self-gravity wake model of Esposito et al. (2025). 𝐸[𝜆T] is the expected value of the Toomre critical wavelength or the largest unstable wavelength toward gravitational collapse (Julian and Toomre 1966): λΤ = 4π2GΣ/κ2 where Σ is the local surface mass density and κ is the local epicyclic frequency. The black dashed arrow is the line of sight from Cassini to the occulted star (or to Earth in the case of radio occultations. The red square is the area over which light is collected during a single integration of the UVIS instrument. (see Jerousek et al. (2024) for details on the integration area).Figure 3. The Maxwell ringlet contains a prominent spiral density wave driven by an outer Lindblad resonance (OLR). We fit optical depths in the first trough of the wave. Structure nearly disappears at apoapsis when particle streamlines are most widely separated. At other orbital phases, the model best matches a structure with broad radial spacing and variable vertical extent.Figure 4. Example of particle size distribution fits for the Maxwell ringlet at two orbital phases.. Best-fitting parameters are consistent with Jerousek et al. (2020) for the background C ring near periapsis where q is much larger. Ring particle radii generally range from several millimeters up to 5 – 10 m.References:Green, M. R., Colwell, J. E., Esposito, L. W., & Jerousek, R. G. (2024). Particle sizes in Saturn’s rings from UVIS stellar occultations 2. Outlier Populations in the C ring and Cassini Division. Icarus, 416, 116081.Esposito, L. W., Colwell, J. Ε., Eckert, S., Green, M. R., Jerousek, R. G., & Madhusudhanan, S. (2025). Statistics of Saturn's ring occultations: Implications for structure, dynamics, and origins. Icarus, 429, 116386.Jerousek, R. G., Colwell, J. E., Esposito, L. W., Tiscareno, M. S., Lewis, M. C., Pohl, L., & Benavides, D. A. (2024). The smallest structures in Saturn’s rings from UVIS stellar occultations. Icarus, 415, 116069.

The global energy budget is pivotal to understanding planetary evolution and climate behaviors. Assessing the energy budget of giant planets, particularly those with large seasonal cycles, however, remains a challenge without long-term observations. Evolution models of Saturn cannot explain its estimated Bond albedo and internal heat flux, mainly because previous estimates were based on limited observations. Here, we analyze the long-term observations recorded by the Cassini spacecraft and find notably higher Bond albedo (0.41 ± 0.02) and internal heat flux (2.84 ± 0.20 Wm⁻²) values than previous estimates. Furthermore, Saturn’s global energy budget is not in a steady state and exhibits significant dynamical imbalances. The global radiant energy deficit at the top of the atmosphere, indicative of the planetary cooling of Saturn, reveals remarkable seasonal fluctuations with a magnitude of 16.0 ± 4.2%. Further analysis of the energy budget of the upper atmosphere including the internal heat suggests seasonal energy imbalances at both global and hemispheric scales, contributing to the development of giant convective storms on Saturn. Similar seasonal variabilities of planetary cooling and energy imbalance exist in other giant planets within and beyond the Solar System, a prospect currently overlooked in existing evolutional and atmospheric models.

The varying geometry of Cassini star occultations by Saturn’s rings constrains both the size and shape of structures that block starlight. Statistics of UVIS star 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 of gaps, ghosts and clumps. The widths W and separation S of rectangular clumps play an analogous role to the relative size of the particle shadows, δ. In the first model considered, our calculations are based on the moments of the transparency T in that part of the ring Asampled 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.These new approaches compare well to the formula for excess variance from Showalter and Nicholson in the region where all are accurate, δτ≪1. Skewness for small τ has a different sign for transparent and opaque structures, distinguishing 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 measured optical depth is correlated with particle size. For a linear dependence Reff = 12 * (τ – 0.08) + 1.8m from Jerousek’s results, we match the curvature of normalized excess variance, the skewness and the kurtosis in the region between 78,000 and 84,600 km from Saturn.Statistics calculated from the granola bar model give different predictions from individual particles. The different τ dependence suggests that the wave crests compress the gaps more than the wakes, and produce more regularity among the clumps; and larger and more opaque self-gravity wakes in the wave crests, with transparent ghosts. The UVIS observations fall between the most regular and the most irregular granola bar models.We compare selected occultations (Eckert etal 2020) at different values of the elevation B to estimate the flattening and axial ratio of ring particles and clumps. In Ring C, we find spheres: The statistical measures from multiple occultations follow the expected dependence on sin B, e.g. Showalter & Nicholson (1990). However, in the Janus 2:1 and Mimas 5:3 density waves, the excess variance for stars β Cen, λ Sco and σ Sgr shows no B dependence. This is exactly the expectation for completely flat (H/W =0) self-gravity wakes that we have derived from the autocovariance of the wake shadows. A closer analysis of this particular case gives H/W < 0.04, different from Colwell etal (2007), suggesting wakes are more like linguine than granola bars.

The Cassini Ultraviolet Imaging Spectrograph (UVIS) High Speed Photometer (HSP) observed 15 occultation traces of the bright star Hadar (Beta Centauri) in 2008-2009 at an elevation above the ring plane of 66.7 degrees. The combination of the high signal from this star (up to 600 counts per msec integration period) and its high elevation above the rings means these occultations provide the strongest constraints on the transparency of high optical depth regions as observed by UVIS. We use the excess variance that is introduced into the UVIS stellar occultation counting rates by the presence of ring particles to determine when the observed signal is purely background with no transmitted starlight. When the observed variance is equal to the mean, we identify the region as opaque. We have identified several regions that are opaque in the UVIS occultations in the B2 and B3 regions (Marouf et al. 2006). When comparing these regions between occultations of the same star, we discovered non-repeating, narrow regions where the transparency jumps from less than 1% up to as much as 20%. These regions are typically less than 100 m in radial extent and frequently less than 50 m. These regions do not repeat at the same location between occultations. In addition, we are able to take advantage of the fact that Hadar is a binary star and both components produce measurable signals in the HSP data. These partially transparent features produce two distinct signals in the occultation data if the azimuthal extent of the feature is at least as large as the projected separation of the two stars onto Saturn’s rings. In many cases we find that there is only one feature observed, indicating that the features are not only narrow in the radial direction but also of limited (< 100 m) extent in the azimuthal direction. We refer to these features as “phantoms” due to their similarity to the completely transparent regions identified in the C ring plateaus dubbed “ghosts” (Baillie et al. 2013). Figure 1 shows an example of an opaque region in the B2 region of the B ring in all 15 occultations of Hadar. In all but two of the 15 profiles there is a narrow spike with a transparency of 5% up to nearly 20% in the region between 100,090 km and 100,130 km. If a transparent region is observed in all occultations at the same location, we identify that as a boundary between two opaque regions rather than as a phantom. The region between ~100,140 – 100,160 km in Figure 1 has complicated structure that is not exactly repeated between the occultations but marks the boundary between two opaque regions. The phantoms, observed between 100,090 km and 100,130 km, do not repeat and are generally very narrow. The first 15 km of the next opaque region, seen from 100,165-100,180 km in Figure 1 are virtually free of phantom features. The projected separation of the two components of Hadar is less than 50 m in the azimuthal direction, and frequently phantoms are only seen to pass behind one of the two stars. The actual transparency of narrow phantoms is thus approximately twice as high as that shown in Figure 1, because the transparency is based on the combined brightness of the two components. Baillie et al. (2013) suggested that ghosts are the nearly empty openings in the background ring carved out by small moonlets akin to the propeller features imaged in the A ring. The situation may be similar for the phantoms, but the dynamics of neighboring ring particles resulting from embedded moonlets in the dense B ring may not produce the same orbital perturbations that give rise to propeller features. We will describe the distribution and characteristics of opaque regions and phantoms in the B2 and B3 regions.Figure 1: Transparency of 100 km of the B2 region of Saturn’s B ring showing an opaque region extending from approximately 100084 km to 100140 km, and part of another beginning at about 100162 km. Each curve is offset by 5% for clarity.Bibliography:Baillié, K., J. E. Colwell, L. W. Esposito, and M. C. Lewis 2013. Meter-sized Moonlet Population in Saturn’s C Ring and Cassini Division. Astron J.145, 171, doi:10.1088/0004-6256/145/6/171. Marouf, E. A., French, R. G., Rappaport, N. J., McGhee, C. A., Wong, K., Thomson, F. S., Anabtawi, A. 2006. Structure and properties of Saturn’s Ring B from Cassini radio occultations. Bull. Am. Astron. Soc. 38, 552.

Cassini’s Ultraviolet Imaging Spectrograph (UVIS) observed 276 stellar occultations of Saturn’s rings over the course of its mission. During these occultations, UVIS’ High-Speed Photometer (HSP) collected photon count measurements through the rings at a typical radial resolution of 10 meters from a wide range of viewing angles. Because photon counts are Poisson distributed, the variance of the starlight is approximately equal to its mean in the absence of occulting ring material. When the star passes behind the rings from the point-of view of the spacecraft, the finite sizes of the ring particles result in an excess variance above the mean caused by correlation in the blocking of photons. Showalter and Nicholson (1990, Icarus, 87, 285) and Colwell et al. (2018, Icarus, 300, 150) interpreted this excess variance in terms of an effective particle or clump size, RE, which depends on the length scale of shadows cast by particles and clumps. In the A ring however, where ring particles aggregate into trailing spiral structures called self-gravity wakes (Colwell et al., 2006, Colwell et al., 2007, Hedman et al., 2007, Nicholson and Hedman (2010), the assumptions of Showalter and Nicholson (1990) and Colwell et al. (2018) of uncorrelated spherical particles are invalid.Here we expand their analyses to account for the presence of self-gravity wakes, ephemeral agglomerations of ring particles under the competing influence of their mutual self-gravity and Keplerian shear, by introducing free parameters S(wake separation), and W (wake width) into a direct calculation of higher order statistical moments of excess variance, skewness, and kurtosis from the expectation value of the random variate, ring transparency (T). We use the granola bar model of self-gravity wakes (Colwell et al. 2006) to calculate the autocovariance of the measured signal. In this work we compare the values of these higher order moments in the peaks and troughs of the Janus 2:1, Pandora 5:4, Janus 5:4, and Janus 6:5 density waves, which exist in regions in which self-gravity wakes are prominent. We compare the best-fit combinations of S and W to the excess variance for many combinations of S and W from different ring regions to reveal trends in the variation of wake properties by ring region. The sum S+W is constrained by the measured dispersion of the waves which gives the local surface mass density. We use this value to determine the Toomre most-unstable wavelength which we assume to be equal to the self-gravity wake wavelength (S+W) (Julian and Toomre, 1966). The second order moment, skewness (S), measures the asymmetry of the distribution and is indicative of either large clumps or small gaps in the rings nicknamed "ghosts" (Baillie et al., 2012). The presence of a few small gaps (high transparency outliers in the distribution) may lead to a positive skewness. The presence of too many gaps, however, increases the symmetry of the distribution as the gaps are no longer outliers. We take the ratio of S to τ to account for the correlation between the two variables (S is proportional to τ when τ ≳ 1). We occasionally observe that the ratio in the troughs differs by more than one standard deviation from the ratio in the peaks of the density waves. Our results indicate that either there are more gaps (positive outliers) present in the troughs than in the peaks or that there are so many gaps in the peaks that they no longer stand out as outliers of the distribution.We apply this analysis technique to a multitude of occultations across a variety of ring elevation (B) and azimuthal (f) angles to survey wake parameter variation across these regions. ReferencesBaillié, K., Colwell, J.E., Lissauer, J.J., Esposito, L.W., Sremčević, M., 2011. Waves in Cassini UVIS stellar occultations 2: The C ring. Icarus 216, 292-308.Colwell, J.E., et al., 2006. Gravitational wakes in Saturn’s A ring measured by stellar occultations from Cassini. Geophys. Res. Lett. 33.Colwell, J.E., Esposito, L.W., Sremčević, M., Stewart, G.R., McClintock, W.E., 2007. Self-gravity wakes and radial structure of Saturn’s B ring. Icarus 190, 127-144. Colwell, J. E., Esposito, L. W., Cooney, J. Particle sizes in Saturn’s rings from UVIS stellar occultations 1. Variations with ring region. Icarus 300, 150. Hedman, M. M., et al., 2007. Self-gravity wake structures in Saturn’s A ring revealed by Cassini-VIMS. The Astronomical Journal 133(6), 2624-2629.Jerousek, R. G., Colwell, J. E., Nicholson, P. D., Hedman, M. M., Esposito, L. W., 2016. Small Particles and Self-Gravity Wakes in Saturn’s Rings from UVIS and VIMS Stellar Occultations. Icarus 279, 36-50.Julian, W.H. and Toomre, A. (1966). Non-axisymmetric responses of differentially rotating disks of stars. Astrophys. J., 146. 810-830.Nicholson, P. D., and Hedman, M. M., 2010. Self-Gravity Wake Parameters in Saturn’s A and B Rings. Icarus 206, 410-423.Showalter, M. R., and Nicholson, P. D., 1990. Saturn's rings through a microscope: Particle size constraints from the Voyager PPS scan. Icarus 87, 285-306.Spilker, L. J., et al., 2004. Saturn A ring surface mass densities from spiral density wave dispersion behavior. Icarus 171, 372–390.

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 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 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.

The Cassini cameras detected elongated structures in Saturn’s ring in highly perturbed regions near ring edges and within the strongest density waves. These ‘straw’ features are likely triggered by the periodic forcing arising from the nearby moons. We investigate the temporal response to this forcing by interpretation of the ring occultation counting statistics. The varying geometry of Cassini star occultations by Saturn’s rings constrains both the size and shape of structures that block starlight. Statistics of UVIS star 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 High Speed Photometer 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. 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 Cassini occultation statistics show the expected extrema and zero crossings. The observed excess variance shows aggregate growth following the passage of a density wave crest. For self-gravity wakes in the A ring, we find wake width W = 18-29m; typical wavelength S+W ~ 60m; H/W < 0.12, thus vertical height H < 4m. These results are consistent with a simple dynamical model of the rings, analogous to an ecological Predator-Prey interaction. Compression drives aggregation, which lags the forcing. Perturbed by passing density waves, self-gravity wakes grow and erode on orbital timescales with a full amplitude of 60%, and a phase lag

The varying geometry of Cassini star occultations by Saturn’s rings constrains both the size and shape of structures that block starlight. Statistics of UVIS star 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 of gaps, ghosts (local openings) and self-gravity wakes. In this model, the widths W and separation S of rectangular clumps play an analogous role to the  size of the particle shadows, R. In the first model considered, our calculations are based on the moments of the transparency T in the ring region sampled by the occultation, thus extending the work of  Showalter and Nicholson (1990) to larger τ  and fractional area 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.These new approaches compare well to the formula for excess variance from Showalter and Nicholson in the region where all are accurate, δτ≪1. Skewness for small τ has a different sign for transparent and opaque structures, distinguishing gaps from clumps. The higher order central moments are calculated from higher powers of the shadow size, thus more sensitive to the extremes of the size distribution. We explain the τ dependence of the excess variance for Saturn’s background C ring by the observation of Jerousek etal(2018) that the measured optical depth is correlated with particle size in the region between 78,000 and 84,600km from Saturn.Statistics calculated from the granola bar model give different predictions from those based on individual spherical particles. The density waves clearly show compression that triggers clump growth, as predicted by the Predator-Prey model (Esposito etal. 2012, Icarus 217, 103-114). The radial profiles and observed τ dependence suggest that the wave crests compress the gaps more than the wakes, along with broader self-gravity wakes in the wave crests, including transparent ghosts. The UVIS observations fall between the most regular and the most irregular granola bar models. Analysis of ring transparency favors irregularly-spaced elongated clumps. A closer analysis of this particular case gives H/W < 0.12, smaller than Colwell etal. (2007, Icarus 190, 127-144), suggesting wakes are more like linguine than granola bars.