The microwave opacity of Saturn's rings at wavelengths of 3.6 and 13 cm from Voyager 1 radio occultation

Radio occultation observations of Saturn's rings with Voyager 1 provided independent measurements of complex (amplitude and phase) microwave extinction and near-forward scattering cross section of the rings at wavelengths (λ) of 3.6 and 13 cm. The ring opening was 5.9°. The normal microwave opacities, τ[3.6] and τ[13], provide a measure of the total cross-sectional area of particles larger than about 1 and 4 cm radius, respectively. Ring C exhibits gently undulating (～ 1000 km) structure of normal opacity τ[3.6] ? 0.25 except for several narrow imbedded ringlets of less than about 100 km width and τ[3.6] ～ 0.5 to 1.0. The normalized differential opacity Δτ/τ[3.6], where Δτ = τ[3.6] ? τ[13], is about 0.3 over most of ring C, indicating a substantial fraction of centimeter-size particles. Some narrow imbedded ringlets show marked increases in Δτ/τ[3.6] near their edges, implying an enhancement in the relative population of centimeter-size and smaller particles at those locations. In the Cassini division, several sharply defined gaps separate regions of opacity τ ～ 0.08 and τ ～ 0.25; the opacity in the Cassini Division appears to be nearly independent of λ. The boundary features at the outer edges of ring C and the Cassini Division are remarkably similar in width and opacity profile, suggesting a similar dynamical control. Ring A appears to be nearly homogeneous over much of its width with 0.6 < τ[3.6] < 0.8 but with considerable thickening, to τ[3.6] ～ 1.0, near its inner boundary with the Cassini division. Normalized differential opacity decreases from ～0.3 at the inner and outer edges of ring A to Δτ/τ[3.6] ～ 0 at a point about one-third of the distance from the inner edge to the outer. The inner one-fourth of ring B has τ[3.6] ～ 1.0, except very near the boundary with ring C, where it is greater. The outer three-fourths of ring B has $\text{τ[3.6] ? 1.2}$. The differential opacity for the inner one-fourth of ring B is Δτ/τ[3.6] ～ 0.15. There are no gaps in ring B exceeding about 2 km in width. Ring F was observed at 3.6 cm as a single ringlet of radial width $\text{? 2}\text{km}$, but was not detected in 13 cm data.     

W-shaped occultation signatures: Inference of entwined particle orbits in charged planetary ringlets

Observed W-shaped occultation signatures of certain narrow ringlets in the ring systems of Saturn and Uranus imply a concentration of material near their inner and outer radial edges. A model is proposed where edge bunching is a natural consequence of particles in entwined elliptical orbits, with the same particles alternately defining both edges. While such orbits cross over in radius, collisions would not occur if they have small inclinations, the same fixed argument of periapse ω, and other parameters whereby the particles would “fly in formation” along compressed helical paths relative to the core of the ringlet, which is taken to be a circle in the equatorial plane. For this model to match the observed ring thickness and ringlet widths, orbit inclinations i must be much smaller than their eccentricities e, which themselves would be very small compared to unity. Thus, the meridional cross section of the resultant torus would be a very thin ellipse of thickness proportional to i∣cos ω∣, tilted slightly from the equatorial plane by (i/e)∣sin ω∣ radians. However, gravitational perturbations due to the oblateness of the planet would cause a secular change in ω so that this cross section would collapse periodically to a tilted line, and collisions would then occur. If this collapse could be prevented, the torus could remain in a continuous state of nearly zero viscosity. Stabilization against collapse appears possible due to several remarkable characteristics that are added to the model when the particles are electrically charged. First, because of inherent features of the torus structure, a weak electric force could counter the key effect of the vastly larger oblateness force. Second, because the electric perturbation also affects i, there is a large region in ω,i space where stability against cross-sectional collapse is automatic. For this region, the thickness of the elliptical cross section would expand and contract in concert with the way that the major axis of the ellipse rocks back and forth relative to the equatorial plane. The period of these “rocking and breathing” changes would be from 1 to 3 weeks for a torus in the C ring of Saturn, for example. The electric effects could change considerably without driving the parameters of the torus from the stable domain where cross-sectional collapse does not occur. While specialized and in several important ways still incomplete, the proposed model could account for the W-shaped patterns and explain how very dense ringlets might endure without energy loss due to collisions. It also appears to be capable of explaining the observed sorting of particles by size within a ringlet. Several characteristics of the model suggest definitive tests of its applicability, including its prediction that a nonsymmetrical W-shaped occultation signature could be reversed a half orbit away, and that grazing solar illumination of tilted ringlets might cast shadows that change with time in a prescribed way.     

Eccentric features in Saturn's outer C ring

A systematic search has been made for as yet unrecognized eccentric and inclined features in Saturn's outer C ring. The radii of all sharp-edged features in the outer C ring were measured in Voyager data consisting of six high-resolution images, the Photopolarimeter occultation data, and the Radio Science λ3.6-cm occultation data corrected for the effects of diffraction. Besides the well-known Maxwell ringlet at 87,491 km (1.450R_s), whose eccentric shape and kinematics have already been studied, two other narrow ringlets at 88,716 km (1.470R_S), and 90,171 km (1.495R_S) have been found to be demonstrably eccentric. The former has a mean width of ∼16 km and is located within a gap ∼30 km wide. The latter has a mean width of ∼62 km and is only partially isolated: its outer edge is defined by a gap ∼15 km wide. Though a coincidence of these two gaps with the Mimas 3:1 inner vertical and inner Lindblad resonances has been noted by previous workers, we find that neither ringlet shows conclusive evidence for the anticipated resonantly forced distortions. The 1.495R_S ringlet is best fitted by a model describing a freely precessing Keplerian ellipse with a radial amplitude of 2.8 ± 0.5 km. Neither a resonant forcing nor a free precession model fitted to the 1.470R_S ringlet provides conclusive results, though the latter is marginally better, yielding an amplitude no larger than ∼2.2 km. These two newly identified eccentric ringlets are compared with the previously studied Titan and Maxwell ringlets (C. Porco, P. D. Nicholson, N. Borderies, G. E. Danielson, P. Goldreich, J. B. Holberg, and A. L. Lane, Icarus 60 (1984), 1–16) and with the Uranian α, β, and ϵ ring.     

Morphology and variability of the Titan ringlet and Huygens ringlet edges

We present a forward modeling approach for determining, in part, the ring particle spatial distribution in the vicinity of sharp ring or ringlet edges. Synthetic edge occultation profiles are computed based on a two-parameter particle spatial distribution model. One parameter, h, characterizes the vertical extent of the ring and the other, δ, characterizes the radial scale over which the ring optical depth transitions from the background ring value to zero. We compare our synthetic occultation profiles to high resolution stellar occultation light curves observed by the Cassini Ultraviolet Imaging Spectrograph (UVIS) High Speed Photometer (HSP) for occultations by the Titan ringlet and Huygens ringlet edges.More than 100 stellar occultations of the Huygens ringlet and Titan ringlet edges were studied, comprising 343 independent occultation cuts of the edges of these two ringlets. In 237 of these profiles the measured light-curve was fit well with our two-parameter edge model. Of the remaining edge occultations, 69 contained structure that could only be fit with extremely large values of the ring-plane vertical thickness (h > 1 km) or by adopting a different model for the radial profile of the ring optical depth. An additional 37 could not be fit by our two-parameter model.Certain occultations at low ring-plane incidence angles as well as occultations nearly tangent to the ring edge allow the direct measurement of the radial scale over which the particle packing varies at the edge of the ringlet. In 24 occultations with these particular viewing geometries, we find a wide variation in the radial scale of the edge. We are able to constrain the vertical extent of the rings at the edge to less than ∼300 m in the 70% of the occultations with appropriate viewing geometry, however tighter constraints could not be placed on h due to the weaker sensitivity of the occultation profile to vertical thickness compared to its sensitivity to δ.Many occultations of a single edge could not be fit to a single value of δ, indicating large temporal or azimuthal variability, although the azimuthal variation in δ with respect to the longitudes of various moons in the system did not show any discernible pattern.     

The Christiansen Effect in Saturn’s narrow dusty rings and the spectral identification of clumps in the F ring

Stellar occultations by Saturn’s rings observed with the Visual and Infrared Mapping Spectrometer (VIMS) onboard the Cassini spacecraft reveal that dusty features such as the F ring and the ringlets in the Encke and the Laplace Gaps have distinctive infrared transmission spectra. These spectra show a narrow optical depth minimum at wavelengths around 2.87 μm. This minimum is likely due to the Christiansen Effect, a reduction in the extinction of small particles when their (complex) refractive index is close to that of the surrounding medium. Simple Mie-scattering models demonstrate that the strength of this opacity dip is sensitive to the size distribution of particles between 1 and 100 μm across. Furthermore, the spatial resolution of the occultation data is sufficient to reveal variations in the transmission spectra within and among these rings. In both the Encke Gap ringlets and F ring, the opacity dip weakens with increasing local optical depth, which is consistent with the larger particles being concentrated near the cores of these rings. The Encke Gap ringlets also show systematically weaker opacity dips than the F ring and Laplace Gap ringlet, implying that the former has a smaller fraction of grains less than ∼30 μm across. However, the strength of the opacity dip varies most dramatically within the F ring; certain compact regions of enhanced optical depth lack an opacity dip and therefore appear to have a greatly reduced fraction of grains in the few-micron size range. Such spectrally-identifiable structures probably represent a subset of the compact optically-thick clumps observed by other Cassini instruments. These variations in the ring’s particle size distribution can provide new insights into the processes of grain aggregation, disruption and transport within dusty rings. For example, the unusual spectral properties of the F-ring clumps could perhaps be ascribed to small grains adhering onto the surface of larger particles in regions of anomalously low velocity dispersion.     

Ringlets of the major planets which may consist of coagulated particles

A ringlet of Saturn, Uranus, Neptune or Jupiter may be composed of particles held in contact by their mutual gravitation, without relative motion. Lacking tensile strength, each part of the ringlet orbits as if it were a separate particle, but all parts are constrained to the same orbit by their contacts. Slight shear strength prevents flow. This configuration is stable inside Roche's limit, and outside an inner limit within which it would scatter. These limits depend on the density of the ringlet. Conversely, for an observed radius in a ring, a range of possible density is calculated. For Saturn's ring system, the density of a ringlet at the inner edge of the C ring must be at least 2.0 g cm^-3 and in the outer F ring not more than 0.73. For Uranus, the inner ring must be at least 2.3, and the outer between 1.0 and 2.3. Jupiter's ring must be in the range 1.4 to 3.9, and Neptune's, in the range 0.6 to 1.5. In extended crowded regions of a ring system, the gaps between ringlets must be at least 38% as wide as the ringlets, in the outer portions of the system, and wider than that at smaller radii. Certain observations can be explained by this model, including the sharp edges of the rings, a long life of the system, the possible existence of a partial ring, asymmetry of brightness of Saturn ring A, and forward scattering of radio waves.     

Profiling Saturn's rings by radio occultation

Diffraction of radio waves is a prominent phenomenon in Voyager 1 radio occultation measurements of Saturn's rings. It limits the effective radial resolution of observed signal intensity and phase to the characteristic Fresnel scale F, which is set by the geometry and wavelength, Λ. For the two Voyager wavelengths at Saturn, F ≅ 9–15 km at 3.6 cm Λ, and F ≅ 17–29 km at 13 cm Λ. This limitation can be largely removed by inverse-Fresnel filtering of the complex (i.e., amplitude and phase) observed signals. An Huygens-Fresnel formulation of the diffracted signal in terms of a circularly symmetric, complex gray-screen model of the rings, valid to second order in phase, leads to an exact Fresnel transform solution for the complex transmittance of the screen, which is useful for analysis. Extension of the formulation to fourth order in the phase of the transform kernel provides a practical implementation where the final resolution is limited by uncertainties in system parameters and noise. Consideration of the effects of uncertainties in the geometry, finite width of the data window employed, analytical approximations used, profile reconstruction fidelity required, system thermal noise, and system phase stability shows the phase stability and thermal noise to be the most critical factors for realistic systems. For Voyager at Saturn, phase instability limits radial resolution to values of the order of F/90, or about 200 m for optically thin rings. For more opaque rings, useful signal-to-noise ratios are the limiting factor: the resolution achieved at 3.6 cm Λ is typically 200–400 m over Ring C and the Cassini Division, 1–4 km over Ring A, and is greater than about 4 km over Ring B. For Voyager 2 at Uranus, the achievable resolution at 3.6 cm Λ is set by system phase stability and should approach 30 m as long as the normal opacity does not exceed ∼2. Reconstructed profiles of limited regions of Saturn's rings illustrate the technique. These reveal a remarkable array of small-scale (∼1 km) ring structures, including very sharp edges, narrow ringlets, gaps with distinctive edge profiles, wakes of embedded satellites, bending waves, density waves, and many unidentified wave-like phenomena. Profiles reconstructed over the full extent of the rings are available currently at 4.2 km and 900 m resolutions, and will be available presently at 400 m resolution.     

On obtaining the forward phase functions of Saturn ring features from radio occultation observations

The near-forward scattering functions of particles in Saturn ring features are related to 3.6 cm radio occultation power spectra by a Fredholm integral equation of the first kind. The equation reduces to an algebraic system of equations whose solution by usual inversion techniques (i.e., least mean squares) is precluded by the near singularity of the forward transformation matrix. The instabilities are reduced by applying a combination of constrained linear inversion and a filtering algorithm based on eigenvector decomposition of the matrix, which yields derived phase functions valid over the range of zero to about 12 mrad. These functions represent the collective forward diffraction lobe of particles greater than about 1 m in radius. Multiple scattering of the signal is a significant effect, and the measured phase functions must be adjusted to obtain the singly scattered component. This single-scattering correction is examined for two physical ring models, (a) the monolayer and (b) the classical discrete random slab, and the fraction of opacity in submeter particles for each model for specific ring features is estimated. Four representative regions of the rings approximately between 1.3 and 1.4R_s, 1.5 and 1.52R_s, 2.0 and 2.02R_s, and 2.08 and 2.16R_s have been studied in detail and single-scatter phase functions produced. Each of these features exhibits effective particle sizes in the range of 3–6 m radius. The approximate fractions of optical thickness due to the submeter particles in each of these regions are 0.58, 0.54, 0.23, and 0.0, respectively, for the many-particle-thick model, and 0.67, 0.67, 0.50, and 0.50, for the monolayer model.     

The rings of Saturn: Two-frequency radar observations

Observations of 3.5- and 12.6-cm radar echoes from the rings of Saturn suggest that no significant difference in scattering properties exists in this wavelength interval. The echoes are largely unpolarized at both wavelengths, and yield a radar cross section at 3.5 cm of 7.32 ± 0.84 × 10⁹ km² for each polarization. The combined radar cross sections for both polarizations correspond to 1.37 ± 0.16 times the optically observed projected A- and B-ring areas (excluding that part of the rings shadowed by the planet). The shape of the echo spectrum is compatible with a homogeneous ring scattering model, except in having excess power at frequencies near the center of the spectrum. A number of possible explanations for the observed scattering properties are explored.     

Saturn's rings: Particle composition and size distribution as constrained by microwave observations: I. Radar observations

We have calculated the radar backscattering characteristics of a variety of compositional and structural models of Saturn's rings and compared them with observations of the absolute value, wavelength dependence, and degree of depolarization of the rings' radar cross section (reflectivity). In the treatment of particles of size comparable to the wavelength of observation, allowance is made for the nonspherical shape of the particles by use of a new semiempirical theory based on laboratory experiments and simple physical principles to describe the particles' single scattering behavior. The doubling method is used to calculate reflectivities for systems that are many particles thick using optical depths derived from observations at visible wavelengths. If the rings are many particles thick, irregular centimeter- to meter-sized particles composed primarily of water ice attain sufficiently high albedos and scattering efficiencies to explain the radar observations. In that case, the wavelength independence of radar reflectivity implies the existence of a broad particle size distribution that is well characterized over the range 1 cm ? r ? m by n(r)dr = n₀r^?3dr. A narrower size distribution with $\text{a}\text{～ 6}\text{cm}$ is also a possibility. Particles of primarily silicate composition are ruled out by the radar observations. Purely metallic particles, either in the above size range and distributed within a many-particle-thick layer or very much larger in size and restricted to a monolayer, may not be ruled out on the basis of existing radar observations. A monolayer of very large ice “particle” that exhibit multiple internal scattering may not yet be ruled out. Observations of the variation of radar reflectivity with the opening angle of the rings will permit further discrimination between ring models that are many particles thick and ring models that are one “particle” thick.     

Saturn's rings: Particle size distributions for thin layer models

The small physical thickness of Saturn's rings requires that radio occultation observations be interpreted using scattering models with limited amounts of multiple scatter. A new model in which the possible order of near-forward scatter is strictly limited allows for the small physical thickness, and can be used to relate Voyager 1 observations of 3.6-and 13-cm wavelength microwave scatter from Saturn's rings to the ring particle size distribution function n(a), for particles with radius 0.001 ≤ a ≤ 20 m. This limited-scatter model yields solutions for particle size distribution functions for eight regions in Saturn's rings, which exhibit approximately inverse-cubic power-law behavior, with large-size cutoffs in particle radius ranging from about 5 m in ring C to about 10 m in parts of ring A. The power-law index is about 3.1 in ring C, about 2.8 in the Cassini division, and increases systematically with radial location in ring A from 2.7 at 2.10R_s to slightly more than 3.0 at 2.24R_s. Corresponding mass densities are 32–43 kg/m² in ring C, 188 kg/m² in the Cassini division, and 244–344 kg/m² in ring A, under the assumption that the material density of the particles is 0.9 g/cm³. These values are a factor of 1 to 2 lower than first-order mass loading estimates derived from resonance phenomena. In view of the uncertainties in the measurements and in the linear density wave model, and the strong arguments for icy particles with specific gravity not greater than about 1, we interpret this discrepancy as being indicative of possible differences in the regions studied, or systematic errors in the interpretation of the scattering results, the density wave phenomena, or some combination of the above.     

Saturn's dynamic D ring

The Cassini spacecraft has provided the first clear images of the D ring since the Voyager missions. These observations show that the structure of the D ring has undergone significant changes over the last 25 years. The brightest of the three ringlets seen in the Voyager images (named D72), has transformed from a narrow, <40-km wide ringlet to a much broader and more diffuse 250-km wide feature. In addition, its center of light has shifted inwards by over 200 km relative to other features in the D ring. Cassini also finds that the locations of other narrow features in the D ring and the structure of the diffuse material in the D ring differ from those measured by Voyager. Furthermore, Cassini has detected additional ringlets and structures in the D ring that were not observed by Voyager. These include a sheet of material just interior to the inner edge of the C ring that is only observable at phase angles below about 60°. New photometric and spectroscopic data from the ISS (Imaging Science Subsystem) and VIMS (Visual and Infrared Mapping Spectrometer) instruments onboard Cassini show the D ring contains a variety of different particle populations with typical particle sizes ranging from 1 to 100 microns. High-resolution images reveal fine-scale structures in the D ring that appear to be variable in time and/or longitude. Particularly interesting is a remarkably regular, periodic structure with a wavelength of ∼30 km extending between orbital radii of 73,200 and 74,000 km. A similar structure was previously observed in 1995 during the occultation of the star GSC5249-01240, at which time it had a wavelength of ∼60 km. We interpret this structure as a periodic vertical corrugation in the D ring produced by differential nodal regression of an initially inclined ring. We speculate that this structure may have formed in response to an impact with a comet or meteoroid in early 1984.     

Microwave brightness of Saturn's rings

It is shown that a lower limit exists on the microwave brightness of the rings of Saturn, if they are assumed to be composed of Mie scatterers of geological composition. The lower limit (about 15°K) is due to scattering of planetary microwave emission. Significant variation of brightness with azimuth along the rings is expected if the particles are typically of 2–3cm radius. Implications for the multiple-scattering hypothesis of the radar cross section of the rings are noted.     

Energy-dependent variability from accretion flows

We develop a formalism to calculate energy-dependent fractional variability (rms) in accretion flows. We consider rms spectra resulting from radial dependences of the level of local variability (as expected from the propagation of disturbances in accretion flows) assuming the constant shape of the spectrum emitted at a given radius. We consider the cases when the variability of the flow is either coherent or incoherent between different radial zones. As an example of local emission, we consider blackbody, Wien and thermal Comptonization spectra. In addition to numerical results, we present a number of analytical formulae for the resulting rms. We also find an analytical formula for the disc Wien spectrum, which we find to be a very good approximation to the disc blackbody. We compare our results to the rms spectrum observed in an ultrasoft state of GRS 1915+105.     

Photogrammetric measurements of Olympus Mons on Mars

Mariner 9 took many pictures of the giant Olympus Mons during its year in orbit around Mars. Control points have been identified on the top of Olympus Mons, on the volcanic shield, and on the surrounding plains, and their locations have been measured on the television pictures. These measurements were used to compute the aerographic coordinates and the planetary radii of the points. The radii at some of the points were derived from radar elevation measurements and from radio occultation measurements. The mountain rises about 21 km above its base.     

Saturn's ringlets

This paper suggests that Saturn's magnetic field is, in part, responsible for the very fine-scale radial features, or ringlets, seen in the ring-system. The planet's dipole field interacts with slight radial variations in plasma density, and the operation of an instability segregates the magnetic flux and plasma in the ring-plane into narrow alternating zones.We suggest that this mechanism may act by itself to give rise to the inner ringlets. At greater radial distances we believe it amplifies gravitational resonances.     

The fine structure of the Saturnian ring system

A dust disc within a planetary magnetosphere constitutes a novel type of dust-ring current. Such an azimuthal current carrying dust disc is subject to the dusty plasma analog of the well known finite-resistivity ‘tearing’ mode instability in regular plasma current sheets, at long wavelengths. It is proposed that the presently observed fine ringlet structure of the Saturnian ring system is a relic of this process operating at cosmogonic times and breaking up the initial proto-ring (which may be regarded as an admixture of fine dust and plasma) into an ensemble of thin ringlets. It is shown that this instability developes at a rate that is many orders of magnitude faster than any other known instability, when the disc thickness reaches a value that is comparable to its present observed value.     

The approximation of neutrino heat conduction with neutrino scattering

We have developed an algorithm for taking into account the neutrino scattering in the approximation of neutrino heat conduction. We show that in the case of incoherent neutrino scattering (e.g., by electrons), the coefficients of the temperature and chemical potential gradients are averaged over the neutrino energy using functions that can be found by numerically solving integral equations. The coherent scattering by free nucleons and atomic nuclei can be described by introducing a transport cross section. We suggest a new method for calculating the neutrino—electron scattering functions that is based on Fermi—Dirac functions of integer indices.     

Asteroseismology and Oblique Pulsator Model of beta Cephei

The evolution of halos consisting of weakly self-interacting dark matter particles is investigated using a new numerical Monte Carlo N-body method. The halos initially contain kinematically cold, dense r-1 power-law cores. For interaction cross sections sigma*=sigmawsi&solm0;mp>/=10-100 cm2 g-1, weak self-interaction leads to the formation of isothermal, constant-density cores within a Hubble time as a result of heat transfer into the cold inner regions. This core structure is in good agreement with the observations of dark matter rotation curves in dwarf galaxies. The isothermal core radii and core densities are a function of the halo scale radii and scale masses which depend on the cosmological model. Adopting the currently popular LambdaCDM model, the predicted core radii and core densities are in good agreement with the observations. For large interaction cross sections, massive dark halos with scale radii rs>/=1.4x104 cm2 g-1 (sigma*)-1 kpc could experience core collapse during their lifetime, leading to cores with singular isothermal density profiles.     

Saturn's rings: Particle composition and size distribution as constrained by observations at microwave wavelengths: II. Radio interferometric observations

The sizes, composition, and number of particles comprising the rings of Saturn may be meaningfully constrained by a combination of radar- and radio-astronomical observations. In a previous paper, we have discussed constraints obtained from radar observations. In this paper, we discuss the constraints imposed by complementary “passive” radio observations at similar wavelengths. First, we present theoretical models of the brightness of Saturn's rings at microwave wavelengths (0.34–21.0 cm), including both intrinsic ring emission and diffuse scattering by the rings of the planetary emission. The models are accurate simulations of the behavior of realistic ring particles and are parameterized only by particle composition and size distribution, and ring optical depth. Second, we have reanalyzed several previously existing sets of interferometric observations of the Saturn system at 0.83-, 3.71-, 6.0-, 11.1-, and 21.0-cm wavelengths. These observations all have spatial resolution sufficient to resolve the rings and planetary disk, and most have resolution sufficient to resolve the ring-occulted region of the disk as well. Using our ring models and a realistic model of the planetary brightness distribution, we are able to establish improved constraints on the properties of the rings. In particular, we find that: (a) the maximum optical depth in the rings is ～ 1.5 ± 0.3 referred to visible wavelengths; (b) a significant decrease in ring optical depth from λ3.7 to λ21.0 cm allows us to rule out the possibility that more than ～30% of the cross section of the rings is composed of particles larger than a meter or so; this assertion is essentially independent of uncertainties in particle adsorption coefficient; and (c) the ring particles cannot be primarily of silicate composition, independently of particle size, and the particles cannot be primarily smaller than ～0.1 cm, independently of composition.