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.     

Magnetospheric environments of outer planet rings: Influence of Saturn's axially symmetric magnetic field

The Jovian and Uranian rings exist within severe energetic particle and plasma environments where magnetosphere-related losses of small ring particles and surface reflectance alteration by sputtering are likely to be important. In contrast, the main Saturnian rings exist within a zone where magnetospheric losses and surface alteration effects are negligible, primarily because of solid-body absorption of inwardly diffusing magnetospheric particles. It is shown here that solid-body absorption of radially diffusing ions is a much more efficient process in the inner Saturnian magnetosphere than in the inner Jovian and Uranian magnetospheres because of the near axial symmetry of the planetary magnetic field with respect to the rotational equatorial plane. This is especially true for continuous rings (as opposed to satellites) for which the approximate time scale against absorption is the particle bounce period in an axially symmetric field, whereas it is the particle drift period in an asymmetric field. Assuming comparable diffusion rates, inward transport of magnetospheric particles is much more strongly inhibited in the inner Saturnian magnetosphere than in the inner magnetospheres of Jupiter and Uranus. This remains true when only rings of comparable widths and optical depths are considered (e.g., the F ring at Saturn and the ϵ ring at Uranus). The most extreme possible consequence of this difference in solid-body absorption efficiency may have been the preferential development of a radially extensive, optically thick ring system at Saturn where magnetospheric losses are minimized in comparison to those at Jupiter and Uranus. A more definite consequence is that the Uranian rings are most probably directly exposed to nearly the same proton fluxes measured at Voyager 2's closest approach. Exposure of ring particle surfaces to radiation belt ion fluxes therefore remains as a viable explanation for the low albedos of the Uranian rings.     

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.     

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.     

Saturn's sweeper moons predicted

The recent observation of the absorption of radiation belts in the vicinity of Saturn's bright rings and historical observations of the ring system make the following related results apparent:

- The gaps in the rings are caused by the presence of at least 6 small, extremely dense and probably electrically charged ‘sweeper’ moons which effectively sweep the ring matter clean from the gaps. This is known due to the fading of the inner ring edges whereas the outer edges are well defined. Their orbital periods will differ from the expected Keplerian periods if the moons and Saturn do possess electric fields.

- Absorption of radiation belts near the rings (of Jupiter also) implies that the ring particles themselves are not absorbing the radiation but the small moons are. This is consistent with the observed radiation belt absorption near the outer Saturnian moons.

- If electric fields of the sweeper moons cause the ring edge fading as observed (and not simply gravitational), then Saturn itself must maintain an electric field in its vicinity by way of a sizeable proton wind to affect the uneven ring edge fading and will be surrounded by an H⁺ cloud at least to approximately the A-ring. this is consistent with the detection of an H⁺ cloud surrounding Saturn (Weiseret al., 1977, p. 755). The other possibility is that these moons are extremely dense and have very large internal magnetic fields.

- Because of their location, these moons must be captured and if very dense as believed, may be core remnants of a nova.

     

Far-infrared and submillimeter observations of the planets

New broadband observations in several passbands between 30 and 500 μm of Mercury, Venus, Mars, Jupiter, Saturn, and Uranus are presented. The best agreement between the data and various thermal models of Mars, Jupiter, and Uranus is obtained with a slightly cooler absolute temperature scale than that previously adopted by Armstrong et al. (1972). The effective temperature of Uranus is 58 ± 2°K, which is in agreement with its solar equilibrium temperature. The existence of an internal energy source of Saturn has been reconfirmed and must lie within the range of 0.9 to 3.2 times the absorbed solar flux. A depression exists in the spectra of Jupiter, Saturn, and Uranus between 80 and 300 μm, which may be a result of NH₃ opacity.     

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.     

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.     

Formation of the regular satellite systems and rings of the major planets

In this paper, we apply the ideas presented by one of us (Prentice, 1978a, b) for the development of the proto-solar cloud into a system of Laplacian rings to the development of the protoplanetary clouds which ultimately led to Jupiter, Saturn and Uranus. We show that if one accepts this scenario — especially the idea of supersonic turbulence in the proto-planetary clouds — one can satisfactorily explain, on the basis of fixing a single adjustable parameter, both the geometric precession of the orbital radii of the regular satellite systems of these three planets and the chemical composition and mass distribution of these satellites. We suggest that thermal stirring in the proto-planetary cloud in the vicinity of the surface of the planet may be responsible for the smaller masses of some of the inner satellites as well as for the formation of the rocky rings of Uranus. The icy rings of Saturn are suggested to be the product of condensation processes in a continuous gaseous disc within the Roche limit of the planet.     

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 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.     

Satellite formation,II

The satellite formation model of Harris and Kaula (Icarus24, 516–524, 1975) is extended to include evolution of planetary ring material and elliptic orbital motion. This model is more satisfactory than the previous one in that the formation of the moon begins at a later time in the growth of the earth, and that a significant fraction of the lunar material is processed through a circumterrestrial debris cloud where volatiles might have been lost. Thus the chemical differences between the earth and moon are more plausibly accounted for. Satellites of the outer planets probably formed in large numbers throughout the growth of those planets. Because of rapid inward evolution of the orbits of small satellites, the present satellite systems represent only satellites formed in the last few percent of the growths of their primaries. The rings of Saturn and Uranus are most plausibly explained as the debris of satellites disrupted within the Roche limit. Because such a ring would collapse onto the planet in the course of any significant further accretion by the planet, the rings must have formed very near or even after the conclusion of accretion.     

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.     

Rings Around Planets, Atmospheric Superrotation, and Their Great Spots

A common explanation is offered of the facts that the (outer) planets Jupiter, Saturn, Uranus, and Neptune (i) have ring systems inside their magnetospheres, (ii) show alternating atmospheric super- and sub-rotation in latitude belts, and (iii) have great (coloured) whirling spots in their atmospheres, at latitudes of maximal shear flow. The common reason, so we argue, is the action of magnetic torques between the various ring systems in non-synchronous rotation which drive electric currents, help ionize the orbiting gas, and redistribute angular momentum. Very similar reasoning has been used earlier – though incorrect in detail – to explain the complicated system of torsional oscillations in the solar convection zone.     

On the actual existence of the periodic and long-period comet families of Uranus

Aphelion distances of the known periodic comets in the range 12–26 AU are analyzed. The aphelia of 12 of the 38 known comets are found to be concentrated at 19.23–20.91 AU, i.e., near the heliocentric distance of Uranus, which seems unlikely to be a coincidence. It is shown by testing that there is also a significant redundancy of distant nodes of the periodic comets’ orbits in the region of motion of Uranus. This is confirmed by the analysis of the MOIDs in the comet-Uranus system. The values of the Tisserand constant for some of the comets exhibit less dispersion relative to Uranus than to Saturn, Jupiter, and the Earth. We selected 20 long-period comets with distant nodes near the region of motion of Uranus. It is shown that, given a uniform spatial distribution, there must be 12 such nodes. Considering the distant nodes and the MOIDs, the planet is likely to have a dynamical connection with the selected group of comets. The distant nodes and perihelia of both periodic and long-period comets are found to be redundant in the directions 76° and 256°, which is qualitatively consistent with the hypothesis of eruptive origin of comets.     

Limitations to growth of microorganisms on Uranus,Neptune, and Titan

We have examined a wide range of physical, chemical, and thermal models of the atmosphere of Uranus. In that model, which we believe maximizes favorable conditions for the support of life [Weidenschilling and Lewis, Icarus20, 465–476 (1973)], we find the probability of growth of a contaminant terrestrial microorganisms to be nil. If, as is likely, conditions are even more extreme on Neptune, the probability of contamination of both of the outer planets Uranus and Neptune is nil. The Wiedenschilling and Lewis model guarantees the presence of water droplets through the temperature range 0 to 100°C; other published models add water liquid at higher temperatures or fail to provide liquid water at all within this temperature range. In this model the heavy elements (C, N, O, etc.) are enhanced in Uranus by a factor sufficient to form a deep massive cloud layer of aqueous ammonia solution droplets. We can estimate the probability of growth with respect to the following factors: the presence of stable liquid water, convection of parcels of atmosphere to lethally hot depths, solar energy sources reaching the water layer, organic molecular production by solar ultraviolet light, ammonia concentration at the water cloud level, ionic species distribution, and concentrations at the water cloud level. The evaluation of these factors suggests that most terrestrial life as we know it would be excluded on the basis of any one of them. We know of no organism that would be adapted to all the stringent Uranus conditions simultaneously. The discovery of even a single species of Earth organism that can survive or grow under allowable outer planetary conditions would establish new principles in biology.Titan, the methane-rich moon of Saturn, may be more hospitable for terrestrial organisms than any of the other objects of the outer solar system. Even there we see formidable barriers to the growth of an Earth organism in Titan's atmosphere. We recognize that revision of our views concerning Titan must occur as more is learned about this satellite.We advocate the abandonment, in principle, of the probabilistic approach to the estimation of growth of terrestrial organisms on spacecraft, planets, and satellites in the solar system. We do not support an approach which estimates probabilities of qualitatively unknown phenomena. We recommend a strategy which involves identification and intensive study of those organisms most likely to thrive under known conditions for each of the planets respectively. (Unknown environmental conditions may be allowed to vary optimally.) Some explicit areas for Earth-based experimentation are indicated.     

Some dynamical aspects of the accretion of Uranus and Neptune: The exchange of orbital angular momentum with planetesimals

The final stage of the accretion of Uranus and Neptune is numerically investigated. The four Jovian planets are considered with Jupiter and Saturn assumed to have reached their present sizes, whereas Uranus and Neptune are taken with initial masses 0.2 of their present ones. Allowance is made for the orbital variation of the Jovian planets due to the exchange of angular momentum with interacting bodies (“planetesimals”). Two possible effects that may have contributed to the accretion of Uranus and Neptune are incorporated in our model: (1) an enlarged cross section for accretion of incoming planetesimals due to the presence of extended gaseous envelopes and/or circumplanetary swarms of bodies; and (2) intermediate protoplanets in mid-range orbits between the Jovian planets. Significant radial displacements are found for Uranus and Neptune during their accretion and scattering of planetesimals. The orbital angular momentum budgets of Neptune, Uranus, and Saturn turn out to be positive; i.e., they on average gain orbital angular momentum in their interactions with planetesimals and hence they are displaced outwardly. Instead, Jupiter as the main ejector of bodies loses orbital angular momentum so it moves sunward. The gravitational stirring of planetesimals caused by the introduction of intermediate protoplanets has the effect that additional solid matter is injected into the accretion zones of Uranus and Neptune. For moderate enlargements of the radius of the accretion cross section (2–4 times), the accretion time scale of Uranus and Neptune are found to be a few 10⁸ years and the initial amount of solid material required to form them of a few times their present masses. Given the crucial role played by the size of the accretion cross section, questions as to when Uranus and Neptune acquired their gaseous envelopes, when the envelopes collapsed onto the solid cores, and how massive they were are essential in defining the efficiency and time scale of accretion of the two outer Jovian planets.     

On the formation of the rings of saturn

It is shown in this paper that a satellite which revolves round a primary in a circular orbit in tied revolution and which spirals within Roche's limit must form by its disintegration a ring of ellipse-like cross-section which is more than 11 times larger than the original spherical diameter of the disintegrating body. The individual particles have elliptic orbits with small eccentricities and revolve in different planes at small inclinations to each other. By inelastic collisions of particles in differently inclined orbits the original considerable thickness of the ring is very greatly reduced.

Applied to the system of Saturn it is assumed that the Rings A and B are formed by disintegration of two different satellites. It can be shown that the satellite A had an original diameter of 1721 km and a density of 0.95 g/cm³, the satellite B a diameter of 2463 km and a density of 1.95.

Thus the hypothesis of G. P. Kuiper, that the rings of Saturn are composed of H²O-snow contaminated by silicate dust (as Jupiter III), seems to fit very well.