Physical processes in Jupiter's ring: Clues to its origin by Jove!

The particles making up the Jovian ring may be debris which has been excavated by micrometeoroids from the surfaces of many unseen (R ? 1 km) parent bodies (or “mooms” as we will occasionally call them) residing in the ring. A distribution of particle sizes exists: large objects are sources for the small visible ring particles and also account for the absorption of charged particles noted by Pioneer; the small grains are generated by micrometeoroid impacts, by jostling collisions among different-sized particles, and by self-fracturing due to electrostatic stresses. The latter are most effective in removing surface asperities to thereby produce smooth and crudely equidimensional grains. The presence of intermediate-sized (radius of several to several hundred microns) objects is also expected; these particles will have a total area comparable to the area of the visible ring particles. The nominal size (?2 μm) of the visible particles derived from their forward-scattering characteristics is caused, at least in part, by a selection effect but may also reflect a fundamental grain size or the preferential generation of certain sizes along with the destruction of others. The tiny ring particles have short lifetimes (?10²?10³ years) limited by erosion due to sputtering and meteoroid impacts. Plasma drag significantly modifies orbits in ～10² years but Poynting-Robertson drag is not effective (T_PR ～ 10⁵ years) in removing debris. The ring width is influenced by the distribution of source satellites, by the initial ejection velocity off them, by electromagnetic scattering, and by solar radiation forces. In the absence of electromagnetic forces, debris will reimpact a mother satellite or collide with another particle in about 10 years. A relative drift between different-sized particles, caused by a lessened effective gravity due to the Lorentz force, will substantially shorten these times to less than a month. The ring thickness is determined by a balance between initial conditions (abetted perhaps by electromagnetic scattering) and collisional damping; existence of the “halo” over the diffuse disk compared to its relative absence over the bright ring indicates the presence of mooms in the bright ring but not in the faint disk. Small satellites (R ? 1 km) will not reaccumulate colliding dust grains whereas satellites having the size of J14 or J16 may be able to do so, depending upon their precise shape, size, density, and location. Visible ring structure could indicate separate source satellites. The particles in the faint inner disk are delivered from the bright ring by orbital evolution principally under plasma drag. The halo is comprised of small particles (～0.1 μm) partially drawn out of the faint disk by interactions with the tilted Jovian magnetic field.     

Density waves in Cassini UVIS stellar occultations: 1. The Cassini Division

We analyze density waves in the Cassini Division of Saturn's rings revealed by multiple stellar occultations by Saturn's rings observed with the Cassini Ultraviolet Imaging Spectrograph. The dispersion and damping of density waves provide information on the local ring surface mass density and viscosity. Several waves in the Cassini Division are on gradients in the background optical depth, and we find that the dispersion of the wave reflects a change in the underlying surface mass density. We find that over most of the Cassini Division the ring opacity (the ratio of optical depth to surface mass density) is nearly constant and is ∼5 times higher than the opacity in the A ring where most density waves are found. However, the Cassini Division ramp, a 1100-km-wide, nearly featureless region of low optical depth that connects the Cassini Division to the inner edge of the A ring, has an opacity like that of the A ring and significantly less than that in the rest of the Cassini Division. This is consistent with particles in the ramp originating in the A ring and being transported into the Cassini Division through ballistic transport processes. Damping of the waves in the Cassini Division suggests a vertical thickness of 3–6 m. Using a mean opacity of 0.1 cm²/g we find the mass of the Cassini Division, excluding the ramp, is 3.1×¹⁰¹⁶ kg while the mass of the Cassini Division ramp, with an opacity of 0.015 cm²/g, is 1.1×¹⁰¹⁷ kg. Assuming a power-law size distribution for the ring particles, the larger opacity of the main Cassini Division is consistent with the largest ring particles there being ∼5 times smaller than the largest particles in the ramp and A ring.     

On the structure of Saturn's rings and the ‘real’ rotational period for the planet

An analysis of the Lowell Observatory photographic plates of Saturn gave the following results: (1) ring A and B show peculiar brightness distributions around the planet, from which we conclude that both are composed of particles in synchronous rotation. (2) The leading side of the particles in ring A is brighter than the trailing side by about 4%, which may indicate an interaction between such particles and the interplanetary medium. (3) Scans of the rings across the major axis show a small (～0.3″) region of enhanced brightness, from which we derive a value ofT _s =10^h13 _. ^m 8±5 _. ^m 4 for the actual planetary rotational period of Saturn. (4) In order to explain the synchronous rotation, the particles in ring A have to be at least 42 m in diameter.     

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.     

Thermal emission from a multiple scattering model of Saturn's rings

Models of Saturn's B ring have been investigated which include the shadowing mechanism, realistic phase functions for the ring particles, and the effects of multiple scattering and a particle size dispersion. These models are based on the assumption that the rings form a layer many particles thick. A power law relation dn ∝ ?^?s is used for the size dispersion law of the ring particles, where dn is the number of particles with radii between ? and ? + d?. In the calculation of the infrared brightness temperature of the rings, the effect of mutual heating among the ring particles is considered quantitatively for the first time. The parameters of the polydisperse s = 2 model can be chosen to satisfy both optical (λ ? 1.1 μ) and infrared data, but the situation could be much clarified if a good phase curve for the rings were available in the red, if the ring brightness were known accurately for λ > 1 μ, and if it could be established whether the ring particles are rotating synchronously.     

Ring and plasma: The enigmae of Enceladus

The E ring associated with the Kronian moon Enceladus has a lifetime of only a few thousand years against sputteringly by slow corotating O ions. The existence of the ring implies the necessity for a continuous supply of matter. Possible particle source mechanisms on Enceladus include meteoroidal impact ejection and geysering. Estimates of ejection rates of particulate debris following small meteoroid impact are on the order of 3 × 10^?18 g cm^?2 sec^?1, more than an order of magnitude too small to sustain the ring. A geyser source would need to generate a droplet supply at a rate of approximately 10^?16 g cm^?2 sec^? in order to account for a stable ring. Enceladus and the ring particles also directly supply both plasma and vapor to space via sputtering. The absence of a 60 eV plasma at the Voyager 2 Enceladus L-shell crossing, such as might have been expected from sputtering, cannot be explained by absorption and moderation of plasma ions by ring particles, because the ring is too diffuse. Evidently, the effective sputtering yield in the vicinity of Enceladus is on the order of, or smaller than, 0.4, about an order of magnitude less than the calculated value. Small scale surface roughness may account for some of this discrepancy.     

Rotation rate and velocity dispersion of planetary ring particles with size distribution II. Numerical simulation for gravitating particles

We examine rotation rates of gravitating particles in low optical depth rings, on the basis of the evolution equation of particle rotational energy derived by Ohtsuki [Ohtsuki, K., 2006. Rotation rate and velocity dispersion of planetary ring particles with size distribution. I. Formulation and analytic calculation. Icarus 183, 373-383]. We obtain the rates of evolution of particle rotation rate and velocity dispersion, using three-body orbital integration that takes into account distribution of random velocities and rotation rates. The obtained stirring and friction rates are used to calculate the evolution of velocity dispersion and rotation rate for particles in one- and two-size component rings as well as those with a narrow size distribution, and agreement with N-body simulation is confirmed. Then, we perform calculations to examine equilibrium rotation rates and velocity dispersion of gravitating ring particles with a broad size distribution, from 1 cm up to 10 m. We find that small particles spin rapidly with 〈ω²〉^1/2/Ω?¹⁰²-¹⁰³, where ω and Ω are the particle rotation rate and its orbital angular frequency, respectively, while the largest particles spin slowly, with 〈ω²〉^1/2/Ω?1. The vertical scale height of rapidly rotating small particles is much larger than that of slowly rotating large particles. Thus, rotational states of ring particles have vertical heterogeneity, which should be taken into account in modeling thermal infrared emission from Saturn's rings.     

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.     

Saturn's rings: II. Condensations of light and optical thickness of Cassini's division

“Condensations” of light have been observed when Saturn's rings are seen almost edge on, and the Sun and the Earth are on opposite sides of the ring plane. These condensations are associated with ring C and Cassini's division. If the relative brightness between the two condensations and the optical thickness of ring C are known, we can calculate the optical thickness of Cassini's division, τ_CASS. Using Barnard's and Sekiguchi's measurements, we have obtained 0.01 ? τ_CASS ? 0.05. A brightness profile of the condensations which agrees well with visual observations is also presented.We are able to set an upper limit of about 0.01 for the optical thickness of any hypothetical outer ring. This rules out a ring observed by C. Cragg in 1954, but does not eliminate the D′ ring observed by Feibelman in 1967.It is known that the outer edge of ring B is almost at the position of the 1/2 resonance with Mimas. Franklin, Colombo, and Cook explained this fact in 1971, postulating a total mass of ring B of 10^?6M_SATURN. We have derived a formula for the mass of the rings, which is a linear function of the mean particle size. We find that 10^?6M_SATURN implies large particles (～70m). If the particles are small (～10cm), as currently believed, the total mass of ring B is not enough to shift the outer edge. We conclude that the above explanation and current size estimates are inconsistent.     

Jupiter's ring system: New results on structure and particle properties

A reexamination of the Voyager images has yielded a refined understanding of Jupiter's diffuse ring system. The system is composed of a relatively bright narrow ring and inner toroidal halo, in addition to the exterior “gossamer” ring discussed elsewhere (Showalter et al., 1985, Nature 316, 526–528). The previously suspected inner disk is absent. The main ring is ∼7000 km wide and has an abrupt outer boundary at a radius of 129,130 ± 100 km. Visible in the ring are several narrow bright features, which may bear some relationship to Adrastea and Metis; these features appear to be narrower and relatively brighter in backscatter. The smallest ring particles obey a power law size distribution, and have an optical depth of 1–6 × 10⁻⁶ for grains up to 100 μm in radius. The largest bodies are dark, rough, and red, and of comparable total optical depth. The halo arises at the bright ring's inner boundary and rapidly expands inward to a ∼20,000-km full thickness, but remains symmetric about the ring plane. It disappears from sight at a radius of 90,000 km, roughly halfway between the main ring and the planet's cloudtops. The halo particles are not predominantly Rayleigh scatterers; they appear to obey a size distribution similar to that of the micron-sized population in the main ring, and comprise a similar optical depth.     

Saturn's rings, the Yarkovsky effects, and the Ring of Fire

Saturn's icy ring particles, with their low thermal conductivity, are almost ideal for the operation of the Yarkovsky effects (photon thrust due to temperature gradients across the ring particles). An extremely simple case of the Yarkovsky effects is examined here, in which orbital evolution is computed as though each particle travels around Saturn alone in a circular orbit, so that there are no collisions, shadowing, or irradiance from other particles; nor are resonances, tumbling, or micrometeoroid erosion considered. The orbital evolution for random spin orientations appears to be a competition between two effects: the seasonal Yarkovsky effect, which makes orbits contract, and the Yarkovsky-Schach effect, which makes orbits expand. There are values of the far infrared and visible particle albedos for which (working radially out from the planet) the along-track particle acceleration S is negative, then positive, and then negative again; the region for which S>0 is interpreted as a region where stable rings are possible. Typical timescales for centimeter-sized particles to travel half a Saturn radius are ¹⁰⁷-¹⁰⁸ yr. Collisions, shadowing, and resonances may lengthen the timescales, perhaps considerably. It is speculated here that the C ring may be depleted of particles because of the seasonal Yarkovsky effect, and small particles that are present in the C ring ultimately fall on Saturn, possibly creating a “Ring of Fire” as they enter the planet's atmosphere.     

Total particulate mass in Enceladus plumes and mass of Saturn’s E ring inferred from Cassini ISS images

The eclipse mosaic (PIA08329) of the Saturn system, taken on September 15, 2006 when Cassini was in Saturn’s shadow, contains numerous color images of the Enceladus plume and the E ring at phase angles ranging from 173° to 179°. These forward-scattering observations sample the diffraction peak for particle radii in the 1–5 μm range. The phase angle dependence and total brightness are sensitive indicators of the total mass of solid material in the plume. We fit the data with a variety of particle shapes and size distributions, and find that the median radius of the equivalent-volume sphere is 3.1 μm, with an uncertainty of ±0.5 μm. The total mass of particles in the plume is (1.45 ± 0.5) × 10⁵ kg. We have not considered variations with altitude in the particle size and shape distribution, and we leave that for another paper. We find that the brightness of the E ring varies with position in the orbit, not only because of the viewing geometry, e.g., variations in phase angle, but also because of some unknown intrinsic variability. The total mass of solid material in the E ring is (12 ± 5.5) × 10⁸ kg. For the plume, the production rate of particles – the mass per unit time leaving the vents is 51 ± 18 kg s⁻¹. We estimate that 9% of these particles are escaping from Enceladus, implying lifetimes of ∼8 years for the E ring particles. Based on three comparisons with vapor amounts from ultraviolet spectroscopy, the ice/vapor ratio is in the range 0.35–0.70. This high ratio poses a problem for theories in which particles form by condensation from the gas phase, and could indicate that particles are formed as spray from a liquid reservoir.     

Venera 9, 10: Is there a dust ring around Venus?

Four surveys in which the geometrical parameters were suitable for observations on weak scattering objects were carried out by the Venera 9, 10 orbiters using 3000–8000 Å spectrometers. The results of one survey can be explained by a dust layer at the height of sighting h = 100–700 km. Its absence in other sessions suggests a ring structure. The spectrum of dust scattering is a power function of the wavelength with the index varying from ?2.1 at 100km to ?1.3 at 500km. A method is proposed for obtaining the optical thickness, density and size distribution of dust particles from the scattering spectra. For m > 10^?14 g the number of dust particles with a mass higher than m is proportional to m^?1.3. The radial optical thickness τ is 0.7 × 10^?5 at 5000 Å assuming the geometric thickness δ to be 100 km. The maximum optical thickness along the normal to the plane of the ring is τ_n = 4 × 10^?6. The mass of the ring is 20 tons or 5 × 10^?3 g cm^?1 per unit circumference length; the maximum mass in a column normal to the ring plane is 10^?10g cm^?2; the maximum density (for δ = 100 km) is 10^?17 g cm^?3. A satellite of Venus gradually destroyed by temperature effects and by meteorite streams and plasma fluxes is suggested as the source of dust in the ring. One of 1 km radius could sustain such a ring for a billion years. The zodiacal light intensity near Venus is estimated.     

How the Enceladus dust plume feeds Saturn’s E ring

Pre-Cassini models of Saturn’s E ring [Horányi, M., Burns, J., Hamilton, D., 1992. Icarus 97, 248-259; Juhász, A., Horányi, M., 2002. J. Geophys. Res. 107, 1-10] failed to reproduce its peculiar vertical structure inferred from Earth-bound observations [de Pater, I., Martin, S.C., Showalter, M.R., 2004. Icarus 172, 446-454]. After the discovery of an active ice-volcanism of Saturn’s icy moon Enceladus the relevance of the directed injection of particles for the vertical ring structure of the E ring was swiftly recognised [Juhász, A., Horányi, M., Morfill, G.E., 2007. Geophys. Res. Lett. 34, L09104; Kempf, S., Beckmann, U., Moragas-Klostermeyer, G., Postberg, F., Srama, R., Economou, T., Schmidt, J., Spahn, F., Grün, E., 2008. Icarus 193, 420-437]. However, simple models for the delivery of particles from the plume to the ring predict a too small vertical ring thickness and overestimate the amount of the injected dust.Here we report on numerical simulations of grains leaving the plume and populating the dust torus of Enceladus. We run a large number of dynamical simulations including gravity and Lorentz force to investigate the earliest phase of the ring particle life span. The evolution of the electrostatic charge carried by the initially uncharged grains is treated selfconsistently. Freshly ejected plume particles are moving in almost circular orbits because the Enceladus orbital speed exceeds the particles’ ejection speeds by far. Only a small fraction of grains that leave the Hill sphere of Enceladus survive the next encounter with the moon. Thus, the flux and size distribution of the surviving grains, replenishing the ring particle reservoir, differs significantly from the flux and size distribution of the particles freshly ejected from the plume. Our numerical simulations reproduce the vertical ring profile measured by the Cassini Cosmic Dust Analyzer (CDA) [Kempf, S., Beckmann, U., Moragas-Klostermeyer, G., Postberg, F., Srama, R., EconoDmou, T., Smchmidt, J., Spahn, F., Grün, E., 2008. Icarus 193, 420-437]. From our simulations we calculate the deposition rates of plume particles hitting Enceladus’ surface. We find that at a distance of 100 m from a jet a 10 m sized ice boulder should be covered by plume particles in 10⁵-10⁶ years.     

A Study of Saturn's Ring Phase Curves from HST Observations

Solar phase curves between 0.3° and 6.0° and color ratios at wavelengths λ=0.336 μm and λ=0.555 μm for Saturn's rings are presented using recent Hubble Space Telescope observations. We test the hypothesis that the phase reddening of the rings is less due to collective properties of the ring particles than to the individual properties of the ring particles. We use a modified Drossart model, the Hapke model, and the Shkuratov model to model reddening by either intraparticle shadow-hiding on fractal and normal surfaces, multiple scattering, or some combination. The modified Drossart model (including only shadowing) failed to reproduce the data. The Hapke model gives fair fits, except for the color ratios. A detailed study of the opposition effect suggests that coherent backscattering is the principal cause of the opposition surge at very small phase angles. The shape of the phase curve and color ratios of each main ring regions are accurately represented by the Shkuratov model, which includes both a shadow-hiding effect and coherent backscatter enhancement. Our analysis demonstrates that in terms of particle roughness, the C ring particles are comparable to the Moon, but the Cassini division and especially the A and B ring particles are significantly rougher, suggesting lumpy particles such as often seen in models. Another conspicuous difference between ring regions is in the effective size d of regolith grains (d∼λ for the C ring particles, d∼1-10 μm for the other rings).     

Low-energy charged particle ring around equator in the altitude range 400–1100 km

Measurements of charged particle fluxes at energies >-13 MeV (if protons), by means of a detector system of high geometrical factor (950 cm² sr), flown on OGO-6 satellite, reveals a ring of low energy charged particles around equator with fluxes of the order of 50–70 particles (m^–2 s^–1 sr^–1), in the altitude range of 400–1100 km. The ring of charged particles exists below the inner radiation belt and is restricted to ±4° of the geomagnetic equator. Distribution of the maximum flux with geomagnetic latitude andL is presented. Comparison of the observed fluxes with earlier measurements of low energy particles, reveals a differential energy spectrum of the type KE^– with the exponent nearly equal to 2.4 to 3.     

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.     

Analysing the New Saturnian Rings, R/2004 S1 and R/2004 S2

The Cassini-Huygens arrival into the Saturnian system brought a large amount of data about the satellites and rings. Two diffuse rings were found in the region between the A ring and Prometheus. R/2004 S1 is coorbital to Atlas and R/2004 S2 is close to Prometheus. In this work we analysed the closest approach between Prometheus and both rings. As a result we found that the satellite removes particles from R/2004 S2 ring. Long-term numerical simulations showed that some particles can cross the F ring region . The well known region of the F ring, where small satellites are present and particles are being taking from the ring, gains a new insight with the presence of particles from R/2004 S2 ring. The computation of the Lyapunov Characteristic Exponent reveled that the R/2004 S2 ring lies in a chaotic region while R/2004 S1 ring and Atlas are in a stable region. Atlas is responsible for the formation of three regimes in the R/2004 S1 ring, as expected for a satellite embedded in a ring.     

CASSINI CIRS OBSERVATIONS OF A ROLL-OFF IN SATURN RING SPECTRA AT SUBMILLIMETER WAVELENGTHS

The Cassini Composite Infrared Spectrometer (CIRS) spatially resolved Saturn’s main rings in the far-infrared, measuring the spectrum from 20 to 400 wavenumbers (cm⁻¹) (tens of microns to submillimeter wavelengths). We find a spectral roll-off below 50 cm⁻¹ (200 μm) for each of the A, B and C rings. From these data we derive temperatures and emissivities for each ring. Mie calculations of individual water ice particles show a natural variation in the optical properties of the rings similar to the roll-off we observe in our data. A simple radiative transfer model placing a distribution of water ice particles randomly in a layer provides a good fit to the data and illustrates one possible interpretation of the results. This is most likely only part of the explanation for the roll-off effect as the impact of shape, surface, and composition variations have been left for future analysis.     

A multilayer model for thermal infrared emission of Saturn’s rings. III: Thermal inertia inferred from Cassini CIRS

The thermal inertia values of Saturn’s main rings (the A, B, and C rings and the Cassini division) are derived by applying our thermal model to azimuthally scanned spectra taken by the Cassini Composite Infrared Spectrometer (CIRS). Model fits show the thermal inertia of ring particles to be 16, 13, 20, and 11 J m⁻² K⁻¹ s^−1/2 for the A, B, and C rings, and the Cassini division, respectively. However, there are systematic deviations between modeled and observed temperatures in Saturn’s shadow depending on solar phase angle, and these deviations indicate that the apparent thermal inertia increases with solar phase angle. This dependence is likely to be explained if large slowly spinning particles have lower thermal inertia values than those for small fast spinning particles because the thermal emission of slow rotators is relatively stronger than that of fast rotators at low phase and vise versa. Additional parameter fits, which assume that slow and fast rotators have different thermal inertia values, show the derived thermal inertia values of slow (fast) rotators to be 8 (77), 8 (27), 9 (34), 5 (55) J m⁻² K⁻¹ s^−1/2 for the A, B, and C rings, and the Cassini division, respectively. The values for fast rotators are still much smaller than those for solid ice with no porosity. Thus, fast rotators are likely to have surface regolith layers, but these may not be as fluffy as those for slow rotators, probably because the capability of holding regolith particles is limited for fast rotators due to the strong centrifugal force on surfaces of fast rotators. Other additional parameter fits, in which radii of fast rotators are varied, indicate that particles less than ∼1 cm should not occupy more than roughly a half of the cross section for the A, B, and C rings.