The dark-side illumination of Saturn's rings

Using Focas and Dollfus' (1969) measurements, the effective optical thickness of Saturn's rings along the cross-section studied is evaluated from intensity of radiation transmitted through the rings. The most probable value, including also the contribution of gaps, is 0.2. A large fraction of dark-side illumination is produced by single scattering in gaps having an optical thickness 10^–4 to 10^–3.     

The photometric function for Saturn's rings

An inhomogeneous vertical distribution of matter in Saturn's rings is capable of producing the tilt effect which is observed in ring B. In this model the effective thickness of the rings is 8 to 9 times the radius of the particles.     

On the formation of Saturn's rings

It is shown that the formation of Saturn's ring C can be explained by the action of solar radiation pressure on the small ring particles. If the age of the rings is 1.6×10⁸ yr, the predicted optical thickness of ring C, as a function of the distance from the planet, can then be shown to be in agreement with the measured one. The disruption of a solid satellite as the origin of the rings is shown to be quite plausible. If, in Roche's limit, the molecular cohesion is taken into account, the disruption distance of a satellite having the mass of the rings seems to be in agreement with the average distance of the ring system.     

Radar imaging of Saturn's rings

We present delay-Doppler images of Saturn's rings based on radar observations made at Arecibo Observatory between 1999 and 2003, at a wavelength of 12.6 cm and at ring opening angles of 20.1°?|B|?26.7°. The average radar cross-section of the A ring is ∼77% relative to that of the B ring, while a stringent upper limit of 3% is placed on the cross-section of the C ring and 9% on that of the Cassini Division. These results are consistent with those obtained by Ostro et al. [1982, Icarus 49, 367-381] from radar observations at |B|=21.4°, but provide higher resolution maps of the rings' reflectivity profile. The average cross-section of the A and B rings, normalized by their projected unblocked area, is found to have decreased from 1.25±0.31 to 0.74±0.19 as the rings have opened up, while the circular polarization ratio has increased from 0.64±0.06 to 0.77±0.06. The steep decrease in cross-section is at variance with previous radar measurements [Ostro et al., 1980, Icarus 41, 381-388], and neither this nor the polarization variations are easily understood within the framework of either classical, many-particle-thick or monolayer ring models. One possible explanation involves vertical size segregation in the rings, whereby observations at larger elevation angles which see deeper into the rings preferentially see the larger particles concentrated near the rings' mid-plane. These larger particles may be less reflective and/or rougher and thus more depolarizing than the smaller ones. Images from all four years show a strong m=2 azimuthal asymmetry in the reflectivity of the A ring, with an amplitude of ±20% and minima at longitudes of 67±4° and 247±4° from the sub-Earth point. We attribute the asymmetry to the presence of gravitational wakes in the A ring as invoked by Colombo et al. [1976, Nature 264, 344-345] to explain the similar asymmetry long seen at optical wavelengths. A simple radiative transfer model suggests that the enhancement of the azimuthal asymmetry in the radar images compared with that seen at optical wavelengths is due to the forward-scattering behavior of icy ring particles at decimeter wavelengths. A much weaker azimuthal asymmetry with a similar orientation may be present in the B ring.     

Saturn's rings in the thermal infrared

This paper reviews our current knowledge of Saturn's rings’ physical properties as derived from thermal infrared observations. Ring particle composition, surface structure and spin as well as the vertical structure of the main rings can be determined. These properties are the key to understand the origin and evolution of Saturn's rings. Ring composition is mainly constrained by observations in the near-infrared but the signature of some probable contaminants present in water ice may also be found at mid-infrared wavelengths. The absence of the silicate signature limits nowadays their mass fraction to 10^−7±1. Recent measurements on the thermal inertia of the ring particle surface show it is very low, of the order of 5±2 Jm⁻² K⁻¹ s^−1/2. New models and observations of the complete crossing of the planetary shadow are needed to attribute this low value either to compact regoliths covered by cracks due to collisions and thermal stresses or to large fluffy and irregular surfaces. Studies of the energy balance of ring particles show a preference for slowly spinning particles in the main rings. Supplementary observations at different phase angles, showing the temperature contrast between night and day sides of particles, and new models including finite spin and thermal inertia, are needed to constrain the actual spin distribution of ring particles. These results can then be compared to numerical simulations of ring dynamics. Many thermal models have been proposed to reproduce observations of the main rings, including alternative mono- or many-particles-thick layers or vertical heterogeneity, with no definitive answer. Observations on the lit and dark faces of rings as a function of longitude, at many incidence and emission angles, would provide prime information on the vertical thermal gradient due to interparticle shadowing from which constraints on the local vertical structure and dynamics can be produced. Future missions such as Cassini will provide new information to further constrain the ring thermal models.     

Theoretical profile of Saturn's rings

Steady-state solutions for the optical thickness of Saturn's rings are studied in terms of Hämeen-Anttila's (1983) theory of bimodal gravitating systems. The elastic properties of particles determine the behaviour of the rarefied mode (gaps), while the dense mode (ringlets) depends on the size and the internal density of the particles. In the outer parts of the rings the dense mode is unstable against the growth of gravitational perturbations. Inside the Roche distance this produces only very narrow ring-shaped configurations with helical orbits around them, and the system is not destroyed. The outer boundary of the rings corresponds to the distance beyond which the gravitational instability transforms the dense mode into strictly local condensations (moons). The inner boundary of the ring system is caused by the absence of dense mode near Saturn. The rarefied mode is stable in a larger region.     

East-west asymmetry of Saturn's rings

The energy distribution curves of eleven Ap stars, three Am stars, four normal A stars and one F0 V magnetic star have been obtained between 478 nm and 680 nm. For four of the Ap stars, two Am stars and all the four normal A stars, the effective temperatures are believed to have been estimated for the first time. For the rest, these estimates are expected to be an improvement over previously available values.It is concluded that the Ap and Am stars are not much different from the normal A stars in so far as their temperatures are concerned.     

The influence of radiation pressure and drag on Saturn's rings

The influence of radiation pressure and drag on the optical thickness of ring C is calculated as a function of the saturnocentric distance. The results are compared with Franklin and Cook's 1958 data. It seems probable that drag exerts more important effects than radiation pressure, at least in inner parts of ring C. The drag effect might also explain the existence of the Cassini division.     

The east-west asymmetry of Saturn's rings: New measurements

Measurements made during 1976–1979 at the Aarne Karjalainen Observatory show slight east-west asymmetry of Saturn's ring B.     

On photometric properties of Saturn's rings

The photometric observations of Saturn's rings made by Camichel (1958a), and Focas and Dollfus (1969) are studied. It has been found that, at large elevation angles multiple scattering between ring particles, and at small elevation angles deviation from Seeliger's principal photometric theory can explain the observations. The geometric albedo 0.82 and the Bond albedo 0.90 have been suggested for the ring particles. The optical thickness of ring B is found to be 1.25 and that of ring A 0.30.     

The influence of particle adhesion on the stability of agglomerates in Saturn's rings

In planetary rings, binary collisions and mutual gravity are the predominant particle interactions. Based on a viscoelastic contact model we implement the concept of static adhesion. We discuss the collision dynamics and obtain a threshold velocity for restitution or agglomeration to occur. The latter takes place within a range of a few cm s⁻¹ for icy grains at low temperatures. The stability of such two-body agglomerates bound by adhesion and gravity in a tidal environment is discussed and applied to the saturnian system. A maximal agglomerate size for a given orbit location is obtained. In this way we are able to resolve the borderline of the zone where agglomerates can exist as a function of the agglomerate size and thus gain an alternative to the classical Roche limit. An increasing ring grain size with distance to Saturn as observed by the VIMS-experiment on board the Cassini spacecraft can be found by our estimates and implications for the saturnian system will be addressed.     

Turbulent viscosity and lifetime of Saturn's rings

The viscosity (the angular momentum flux) in the disk of mutually gravitating particles of Saturn's rings is investigated. The hydrodynamic theory of the gravitational Jeans-type instability of small gravity perturbations (e.g., those produced by spontaneous disturbances) of the disk is developed. It is suggested that in such a system the hydrodynamic turbulence may arise as a result of the instability. The turbulence is related to stochastic motions of “fluid” elements. The objective of this paper is to show that in the Jeans-unstable Saturnian ring disk the turbulent viscosity may exceed the ordinary microscopic viscosity substantially. The main result of local N-body simulations of planetary rings by Daisaka et al. (2001. Viscosity in a dense planetary ring with self-gravitating particles. Icarus 154, 296-312) is explained: in the presence of the gravitationally unstable density waves, the effective turbulent viscosity ν_eff is given as ν_eff=CG²Σ²/Ω³, where G, Σ, and Ω are the gravitational constant, the surface mass density of a ring, and the angular velocity, respectively, and the nondimensional correction factor C≈10. We argue that both Saturn's main rings and their irregular of the order of 100 m or even less fine-scale structure (being recurrently created and destroyed on the time scale of an order of Keplerian period ) are not likely much younger than the solar system.     

Gravitational accretion of particles in Saturn's rings

Gravitational accretion in the rings of Saturn is studied with local N-body simulations, taking into account the dissipative impacts and gravitational forces between particles. Common estimates of accretion assume that gravitational sticking takes place beyond a certain distance (Roche distance) where the self-gravity between a pair of ring particles exceeds the disrupting tidal force of the central object, the exact value of this distance depending on the ring particles' internal density. However, the actual physical situation in the rings is more complicated, the growth and stability of the particle groups being affected also by the elasticity and friction in particle impacts, both directly via sticking probabilities and indirectly via velocity dispersion, as well as by the shape, rotational state and the internal packing density of the forming particle groups. These factors are most conveniently taken into account via N-body simulations. In our standard simulation case of identical 1 m particles with internal density of solid ice, ρ=900 kg m⁻³, following the Bridges et al., 1984 elasticity law, we find accretion beyond a=137,000-146,000 km, the smaller value referring to a distance where transient aggregates are first obtained, and the larger value to the distance where stable aggregates eventually form in every experiment lasting 50 orbital periods. Practically the same result is obtained for a constant coefficient of restitution εn=0.5. In terms of rp parameter, the sum of particle radii normalized by their mutual Hill radius, the above limit for perfect accretion corresponds to rp<0.84. Increased dissipation (εn=0.1), or inclusion of friction (tangential force 10% of normal force) shifts the accretion region inward by about 5000 km. Accretion is also more efficient in the case of size distribution: with a q=3 power law extending over a mass range of 1000, accretion shifts inward by almost 10,000 km. The aggregates forming in simulations via gradual accumulation of particles are synchronously rotating.     

Brightness of Saturn's rings with decreasing solar elevation

Early ground-based and spacecraft observations suggested that the temperature of Saturn's main rings (A, B and C) varied with the solar elevation angle, B^′. Data from the composite infrared spectrometer (CIRS) on board Cassini, which has been in orbit around Saturn for more than five years, confirm this variation and have been used to derive the temperature of the main rings from a wide variety of geometries while B^′ varied from near −24^° to 0^° (Saturn's equinox).Still, an unresolved issue in fully explaining this variation relates to how the ring particles are organized and whether even a simple mono-layer or multi-layer approximation describes this best. We present a set of temperature data of the main rings of Saturn that cover the ∼23^°—range of B^′ angles obtained with CIRS at low (α∼30^°) and high (α≥120^°) phase angles. We focus on particular regions of each ring with a radial extent on their lit and unlit sides. In this broad range of B^′, the data show that the A, B and C rings’ temperatures vary as much as 29-38, 22-34 and 18-23 K, respectively. Interestingly the unlit sides of the rings show important temperature variations with the decrease of B^′ as well. We introduce a simple analytical model based on the well known Froidevaux monolayer approximation and use the ring particles’ albedo as the only free parameter in order to fit and analyze this data and estimate the ring particle's albedo. The model considers that every particle of the ring behaves as a black body and warms up due to the direct energy coming from the Sun as well as the solar energy reflected from the atmosphere of Saturn and on its neighboring particles. Two types of shadowing functions are used. One analytical that is used in the latter model in the case of the three rings and another, numerical, that is applied in the case of the C ring alone. The model lit side albedo values at low phase are 0.59, 0.50 and 0.35-0.38 for the A, B and C rings, respectively.     

HST observations of azimuthal asymmetry in Saturn's rings

From 378 Hubble Space Telescope WFPC2 images obtained between 1996-2004, we have measured the detailed nature of azimuthal brightness variations in Saturn's rings. The extensive geometric coverage, high spatial resolution (), and photometric precision of the UBVRI images have enabled us to determine the dependence of the asymmetry amplitude and longitude of minimum brightness on orbital radius, ring elevation, wavelength, solar phase angle, and solar longitude. We explore a suite of dynamical models of self-gravity wakes for two particle size distributions: a single size and a power law distribution spanning a decade in particle radius. From these N-body simulations, we calculate the resultant wake-driven brightness asymmetry for any given illumination and viewing geometry. The models reproduce many of the observed properties of the asymmetry, including the shape and location of the brightness minimum and the trends with ring elevation and solar longitude. They also account for the “tilt effect” in the A and B rings: the change in mean ring brightness with effective ring opening angle, |Beff|. The predicted asymmetry depends sensitively on dynamical ring particle properties such as the coefficient of restitution and internal mass density, and relatively weakly on photometric parameters such as albedo and scattering phase function. The asymmetry is strongest in the A ring, reaching a maximum amplitude A∼25% near a=128,000 km. Here, the observations are well-matched by an internal particle density near 450 kg m⁻³ and a narrow particle size distribution. The B ring shows significant asymmetry (∼5%) in regions of relatively low optical depth (τ∼0.7). In the middle and outer B ring, where τ?1, the asymmetry is much weaker (∼1%), and in the C ring, A<0.5%. The asymmetry diminishes near opposition and at shorter wavelengths, where the albedo of the ring particles is lower and multiple-scattering effects are diminished. The asymmetry amplitude varies strongly with ring elevation angle, reaching a peak near |Beff|=10° in the A ring and at |Beff|=15-20° in the B ring. These trends provide an estimate of the thickness of the self-gravity wakes responsible for the asymmetry. Local radial variations in the amplitude of the asymmetry within both the A and B rings are probably caused by regional differences in the particle size distribution.     

Voyager observations of the azimuthal brightness variations in Saturn's rings

This paper presents some Voyager observations of the azimuthal brightness variations in Saturn's ring A. Measurements in reflected light are in general agreement with Earth-based studies. The unique contribution of Voyager—images of the rings in light transmitted through them—shows the brightness variations also to be present, but they have a decidedly greater amplitude and differ in phase by ∼65° from those seen in reflexion. The photometric behavior on both sides can probably be qualitatively explained by the extensive presence of particle wakes in ring A.     

A micrometeorite erosion model and the age of Saturn's rings

The sharp, about 100-km-wide, transition between Saturn's C and B rings is at the inner stability limit of small (micrometer or less) highly charged debris from micrometeorite bombardment of the main ring bodies. The latter vary from about 1 cm to 5 m in radius. In the C ring this charged debris escapes from the ring plane to Saturn along magnetic field lines because of gravitational pull, thus producing a net mass loss. But in the B ring the debris oscillates stably back and forth through the ring plane until reabsorbed by a large ring body. In this model we assume that what is now the B and C rings was initially formed as one ring with the optical thickness of the present B ring. We estimate the C ring net erosion rate and determine the ring age, assuming that the mass influx is small compared with the erosion flux. The erosion rate has been calculated with the use of presently observed micrometeorite fluxes. We also use the best present estimates of the size distribution and total mass eroded by a micrometeorite of a given size and energy. We find that the ring age is between 4.4 and 67 myr. In either case the age is much younger than the 4.5 byr of the solar system. The sharpness of the transition between the B and C rings indicates that the principal mass loss is carried by particles moving at a few meters per second with respect to the parent bodies from which they were eroded.     

Exploring local N-body simulations of Saturn's rings

The dynamical behavior of low and moderately high optical depth regions of Saturn's ring system of discrete, mutually gravitating, and inelastically colliding particles is studied by simplified local N-body simulations in Hill's linearized equations context. The focus is on a statistical analysis of time-evolution of fine-scale structures seen in the simulations and the comparison between theoretical predictions and computer experiments. Prospects for the Cassini spacecraft mission are briefly summarized.