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.     

On the azimuthal brightness variations of Saturn's rings

We present a simple, semiquantitative explanation that accounts both for the presence of the azimuthal brightness variations in Saturn's ring A and for their absence in ring B. Our explanation avoids any ad hoc reliance on albedo variations and/or synchronous rotation of ring particles. Instead, it requires only some degree of self-gravitation between nearby orbiting bodies. A bias in the particle distribution and corresponding photometric effects are thereby produced—the latter corresponding very closely to the variations observed in ring A. Their absence in ring B is primarily a consequence of the higher optical thickness and decreasing importance of self-gravitation in that ring.     

Infrared brightness temperature of Saturn's rings based on the inhomogeneous multilayer assumption

Models of Saturn's rings based on the classical multilayer assumption have been studied in the infrared. Thermal energy balance of Saturn's rings is treated rigorously by solving the infrared radiative transfer equations. It was found that a homogeneous multilayer model is incompatible with the observed infrared brightness variation of the A and B rings, although it can fit that of the C ring. The alternative inhomogeneous multilayer model with dark particles within a bright haze of small icy particles is presented in order to satisfy the available infrared data of the A, B, and C rings. The results based on the inhomogeneous multilayer model may be summarized as follows: The observed infrared brightness data of the three rings are explained in terms of the different optical thickness without having significant differences in the ring-particle properties, such as albedo, spin rate, and sizes. But each ring contains a different amount of bright haze particles and their concentration within the rings depends on whether or not dark particles emit radiation mostly from one hemisphere (slow rotator and/or low thermal inertia). If a dark particle is an isothermal radiator, the possible ranges of A₁ and A₂ for all three rings are given by 0.9 ? A₁ ? 0.95 and 0.0 ? A₂ ? 0.15, where A₁ and A₂ are the bolometric bond albedos of a bright haze and a dark particle, respectively. The possible ranges of the optical thickness ratio X of the dark particle layer to the total ring layer for the rings A, B, and C are given by 0.65 ? X ? 0.75, 0.8 ? X ? 0.9, and 0.8 ? X ? 1.0, respectively. If a dark particle is a slow rotator, we obtain 0.9 ? A₁ ? 0.95 and 0.0 ? A₂ ? 0.4 for all three rings. The ranges of X for the rings A, B, and C are given by 0.35 ? X ? 0.7, 0.65 ? X ? 0.9, and 0.35 ? X ? 1.0, respectively. In this paper, for the first time, a consistent model is presented which is applicable to all three rings from the multilayer point of view.     

Saturn's rings: Infrared brightness variation with solar elevation

The variation in infrared equilibrium brightness temperature of Saturn's A, B, and C rings is modeled as a function of solar elevation B′ with respect to the ring plane. The basic model includes estimates of minimum and maximum interparticle shadowing in a monolayer approximation. Simple laboratory observations of random particle distributions at various illumination angles provide more realistic shadowing functions. Radiation balance calculations yield the physical (kinetic) temperature of particles in equilibrium with radiation from the Sun, Saturn, and neighboring particles. Infrared brightness temperatures as a function of B′ are then computed and compared to the available 20-μm data (Pioneer results are also briefly discussed). The A and B rings are well modeled by an optically thick monolayer, or equivalently, a flat sheet, radiating from one side only. This points to a temperature contrast between the two sides, possibly due to particles with low thermal inertia. Other existing models for the B ring are discussed. The good fit for the monolayer model does not rule out the possibility that the A and B rings are many particles thick. It could well be that a multilayer ring produces an infrared behavior (as a function of tilt angle) similar to that of a monolayer. The C ring brightness increases as B′ decreases. This contrast in behavior can be understood simply in terms of the low C ring optical depth and small amount of interparticle shadowing. High-albedo particles (A?0.5) can fit the C ring infrared data if they radiate mostly from one hemisphere due to slow rotation or low thermal inertia (or both). Alternatively, particles isothermal over their surface (owing to a rapid spin, high inertia, or small size), and significantly darker (A?0.3) than the A and B ring particles, can produce a similar brightness variation with ring inclination. In any case, the C ring particles have significantly hotter physical temperatures than the particles in the A and B rings, whether or not the rings form a monolayer.     

Saturn's rings and Pioneer 11

Quantitative predictions of the diffuse reflection and transmission properties of Saturn's rings, relevant to the September 1979 Pioneer 11 flyby, are presented. Predictions are based on an elementary anisotropic scattering model. Interparticle separations are considered to be sufficiently large that mutual shadowing is negligible. Likely ranges in both the single scattering albedo and perpendicular optical thickness of the ring are considered. Situations of pronounced back-scattering and of isotropic scattering are treated individually. Spacecraft measurement of the radiation suffering diffuse scattering by the ring can provide a useful test of the basic ring model.     

Plasma clouds in Saturn's rings

The expansion and ionization of vapor produced by impacts of meteorites on Saturn's rings is described. There is an “impact plasma” produced in the initial collision, and a “secondary plasma” produced by subsequent ionization of the neutral gas ejecta. The dynamics of these plasma clouds, their size, density, and life time are calculated. It is suggested that large clouds, produced by meter-sized meteorites (or a collection of such clouds produced, e.g., by the impact of a swarm of meteorites) may lead to the formation of spokes by the mechanism discussed in Goertz, C. K., and Morfill, G. E. (Icarus53, 219–229, 1983).     

The velocity dispersion in Saturn's rings

The surface of Mercury exhibits a global tectonic system consisting of an ancient set of NE and NW trending lineaments and a younger set of planimetrically arcuate escarpments interpreted as thrust or high-angle reverse faults. The trends, distribution, and age relations of these tectonic features can be explained by a combination of tidal despinning and global contraction of the planet. In our model, early tidal despinning resulted in conjugate shear fractures trending roughly N60°E and N60°W which were subsequently modified by a variety of surface processes to produce the presently visible set of lineaments. Continued despinning plus global contraction produced thrust faults with roughly north-south trends. Final contraction may have postdated despinning and produced randomly oriented thrust faults. All of these events predated the formation of Caloris basin, because basin-associated deposits blanket both lineaments and arcuate thrust faults.     

Delay-Doppler radar observations of Saturn's rings

The relative radar reflectiveness of Saturn's classical ring sections were estimated from delay-Doppler observations made at 12.6-cm wavelength. The A and B rings are responsible for most, if not all, of the radar echo. The average radar reflectivity per unit projected area of the A ring is nearly (～90%) as large as that of the B ring. The outer half of the B ring contains the most reflective part of the ring system. There is no firm evidence for detection of radar backscattering from particles interior to the B ring, exterior to the A ring, or from the planet itself. The radar reflectivity of the C ring is certainly no more than one-half that of the B ring, and probably is much less. Unexpectedly large amounts of power at Doppler shifts near the center of the echo spectrum, previously reported at both λ3.5 and λ12.6 cm for ring-plane tilt angles δ ≥ 24.4°, are not apparent in λ3.5- or 12.6-cm spectra obtained at δ ≤ 21.4°.     

Anisotropic optical scattering within Saturn's rings

Visual photometric function data for Saturn's rings are analyzed in terms of elementary anisotropic scattering radiative transfer models which involve the Henyey-Greenstein function. Limits are placed on the combinations of single scattering albedo and backscattering directivity which are permitted by observation. Particles with both microscopic and macrscopic lunar-like scattering properties are excluded by the analysis. Results are consistent with the ring particles being nearly pure spherical conglomerates of H₂O frost.     

A theoretical photometric function of Saturn's rings

A photometric theory of Saturn's rings is developed on the assumption that partially elastic collisions have brought all the ring particles into the same plane. The resulting photometric function explains the tilt effect of the rings, but the opposition peak must originate in the particles themselves.     

Theory of radio occultation by Saturn's rings

The radio occultation technique is developed here as a new method for the study of the physical properties of planetary ring systems. Particular reference is made to geometrical and system characteristics of the Voyager dual-wavelength (13 and 3.6 cm) experiment at Saturn. The rings are studied based on the perturbations they introduce in the spectrum of coherent sinusoidal radio signals transmitted through the rings from a spacecraft in the vicinity of the planet to Earth. Two separate signal components are identified in a perturbed spectrum: a sinusoidal component that remains coherent with the incident signal but is reduced in intensity and possibly changed in phase, and a Doppler-broadened incoherent component whose spectral shape and strength are determined by the occultation geometry and the radial variation of the near-forward radar cross section of illuminated ringlets. Both components are derived in terms of the physical ring properties starting from a conventional radar formulation of the problem of single scattering on ensembles of discrete scatterers, which is then generalized to include near-forward multiple scattering. The latter is accomplished through special solutions of the equation of transfer for particles that are larger than the wavelength. When the occultation geometry is optimized, contributions of an individual ringlet to a perturbed spectrum can be identified with radial resolution on the order of a few kilometers for the coherent component and a few hundred kilometers for the incoherent one, thus permitting high-resolution reconstruction of the radial profile of the optical depth, as well as reconstruction of the radar cross section of resolved ringlets. Simultaneous estimates of the optical depth and radar cross section of a ringlet at 3.6 cm-gl allow separation of its aerial density and particle size, if the particles are of known material and form a narrow size distibution with radii greater than several tens of centimeters. This separation is also achieved for radii ?10 cm from differential effects on the coherent signal parameters at 3.6- and 13-cm wavelengths. For the more general case of a broad size distribution modeled by a power law, the absence of differential effects on the coherent signal binds the minimum size to be ?10 cm. In this case, the radius inferred from an estimate of the radar cross section represents an equivalent radius, which is strongly controlled by the maximum size of the distribution provided that the power index is in the range 3 to 4. On the other hand, detection of differential coherent signal extinction determines an upper bound on the maximum size and a lower bound on the power index, assuming water-ice particles. These bounds, together with an inferred equivalent size, constrain the size distribution at both its small and large ends.     

Bending waves and the structure of Saturn's rings

The surface mass density profiles at four locations within Saturn's rings are calculated using Voyager spacecraft images of spiral bending waves. Bending waves are vertical corrugations in Saturn's rings which are excited at vertical resonances of a moon, e.g., Mimas, whose orbit is inclined with respect to the mean plane of the rings. Bending waves propagate toward Saturn by virtue of the rings' self-gravity; their wavelength depends on the local surface mass density of the rings. Observations of bending waves can thus be used to determine the surface density in regions of Saturn's rings near vertical resonances. The average surface density of the outer B ring near Mimas' 4:2 inner vertical resonance is 54 ± 10 g cm^?2. Surface density in this region probably varies by ～ 30% over radial length scales of tens of kilometers; and irregular radial structure is present on similar length scales in this region. Surface densities ranging from 24 g cm^?2 to 45 g cm^?2 are found in the A ring. Small scale variations in surface density are not seen in the A ring, consistent with its more uniform optical appearance.     

The vertical structure and thickness of Saturn's rings

We have considered the steady state vertical structure of Saturn's rings with regard to whether collapse to a monolayer due to collisions between particles, the end state predicted by Jeffreys (1947a), may be prevented by any of a variety of mechanisms. Given a broad distribution of particle sizes such as a typical power law n(R) = n₀R^?3, it is found that gravitational scattering of small particles by large particles maintains a true ring thickness of several times the radius of the largest particles, or many times the radius of the smallest particles. Thus the “many-particle-thick” condition which best satisfies optical observations, such as the opposition effect, may be reconciled with ongoing particle collisions. If we consider the obvious sources of energy available for such a process, we find that a ring thickness of only tens of meters may be sustained over the lifetime of the solar system. This implies a maximum particle size on the order of a few meters.     

Comment on radar scattering from Saturn's rings

Transparent particle scattering is proposed to explain the unexpectedly large radar cross section recently observed for Saturn's rings. According to this theory, only 10% of the optically observed material in the A, B, and C rings need consist of smooth ice fragments larger than 8 cm in radius to yield the radar results.     

Photometry of the unlit face of Saturn's rings

From our telescopic observations of Saturn's rings in 1966, 1979, and 1980, the luminance of the unlit face at λ = 0.58 μm is derived as a function of the height B′ of the Sun above the lit face. A maximum is reached at B′ = 1.9° and a decrease is observed for larger values of B′. Ring B is 1.8 time less bright than ring A and Cassini division. The unlit/lit luminances ratios for the two rings merged together is 8% at B′ = 1.0° and 3% at B′ = 2.8°. The larger value at more grazing incidence is related to the photometric “opposition effect” which reflects more of the incident light backward into the ring plane when the height of the sun is small; the light so reflected is again reflected and scattered and a certain flux reaches the unlit face to escape toward the observer. The unlit face luminances for blue and for yellow light indicate a contribution by micron size particles. The Saturn globe produces a ring illumination which, observed from the Earth, amounts to 1.8 × 10^?3 of the disk center reflectance. The rings observed exactly edge-on do not disappear but a faint lineament remains, which produces a flux of (0.30 ± 0.15) 10^?3 times the brightness of a segment of 1 arcsec width at Saturn disk center; illuminations of rings' borders or particles outside the exact ring plane are indicated.     

Some consequences of meteoroid impacts on Saturn's rings

High-velocity impacts of interplanetary meteoroids on Saturn's rings are discussed. It is shown that the neutral gas emitted by impact vaporization may be responsible, to a large part, for the observed neutral ring atmosphere. Both the predicted neutral gas injection rate and the gas temperature (or kinetic energy) are compatible with the measurements (see Broadfoot, A. L., B. R. Sandel, D. E. Shemansky, J. B. Holberg, G. R. Smith, D. F. Strobel, J. C. McConnell, S. Kumar, D. M. Hunten, S. K. Atreya, T. M. Dohnahne, H. W. Moos, J. L. Bertaux, J. E. Blamont, R. B. Pomphrey, and S. Linik, Science212, 206–211, 1981). Heavy ejecta particles produce a particulate ring “halo”. The physical properties of this halo are calculated, and it appears to be identical with the tenous particle population discussed by Baum and Kreidl (1982). Erosion of Saturn's ring particles, the resulting mass balance, and regolith formation are estimated. This provides some constraints on surface properties and optical albedo.     

Saturn's rings: Resonance about an oblate planet

Classical resonance theory is extended to include corrections due to Saturn's oblateness. A single classical resonance splits into a band structure, with individual resonances almost evenly spaced in radius from the planet. When applied to Saturn's rings this theory predicts, in detail, the structure of Cassini's division.     

Spokes in Saturn's rings: Dynamical and reflectance properties

Orbital velocities and relative reflectances of a preliminary sample of 15 Saturnian spokes recorded in the Voyager 2 low-resolution ring movie have been examined. While 13 spokes exhibit the expected Keplerian velocities, 2 anomalous spokes deviate from this motion. For approximately 2 hr after their formation these spokes exhibit corotational motion and, only then, accelerate to Keplerian speeds. Only 1 of the 2 accelerating spokes is within view of the Voyager cameras throughout its lifetime. When first seen this spoke appears on the morning ansa of the B ring with a 0.02 contrast; it gains in contrast throughout its corotational phase, reaching a maximum of 0.09 during its velocity transition. The spoke then loses contrast during its Keplerian phase, dropping to 0.02 in the last visible image. Thus a correlation between the contrast and the anomalous dynamical phases of this feature is observed. The radial reflectance profile, measured when the spoke is approximately 1 hr old, suggests discrete sources for spoke material in regions of maximum contrast within the B ring; a lower limit of 3 × 10¹¹ g can be established for the mass at this point. The behavior of this spoke seems to be explained by the plasma cloud model of Goertz and Morfill (1983, Icarus 55, 111–123). The atypical dynamics of these 2 spokes suggest that they are generated by plasma clouds of unusually high charge density, while the contrast of these features appears to depend more on ring particulate concentration than on plasma cloud density.     

Saturn's rings: 3-mm observations and derived properties

We have used 3-mm Saturn observations, obtained from 1965 through 1977 and with Jupiter as a reference, to derive a ring brightness temperature of 18 ± 8°K. Thebrightness temperature of the disk of Saturn is 156 ± 9° K. Part of the ring brightness (≈62K) may be accounted for as disk emission which is scattered from the rings; the remainder (12 ± 8° K we attributed to ring particle thermal emission. Because this thermal component brightness temperatures is so much less than the particle physical temperature, limits are placed on the mean size and composition of the ring particles. In particular, as found by others, the particles cannot be rocky, but must be either metallic or composed of extremely low-loss dielectric material such as water ice. If the particles are pure water ice, for example, then a simple slab model and a multiple-scattering model both give upper limits to the particle sizes of ≈ 1 m, a value three times smaller than previously available. The multiple-scattering model gives a particle single-scattering albedo at 3 mm of 0.83±0.13.     

Ballistic transport in Saturn's rings: An analytic theory

Ejecta from impacts of micrometeoroids on Saturn's ring particles will, in most cases, remain in orbit about Saturn and eventually be reaccreted by the rings, possibly at a different radial location. The resulting mass transport has been suggested as the cause of some of the features observed in Saturn's rings. Previous attempts to model this transport have used numerical simulations which have not included the effects of the angular momentum transport coincident with mass transport. An analytical model for ballistic mass transport in Saturn's rings is developed. The model includes the effects of angular momentum advection and shows that the net material movement due to angular momentum advection is comparable to that caused by direct ballistic mass transport.