Evolution of Planetesimal Velocities Based on Three-Body Orbital Integrations and Growth of Protoplanets

We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10-30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies.     

Constraints to Uranus' Great Collision III: The Origin of the Outer Satellites

The theory of satellite loss resulting from a giant impact on Uranus (Parisi and Brunini 1997, Planet. Space Sci.45, 181-187) is revisited, in the light of the discovery of its five outer moons (Gladman et al. 1998, Nature392, 897-899; Gladman et al. 2000, Icarus147, 320-324; erratum 148, 320). Physical conditions and dynamical constraints in the great collision scenario and restrictions in the possible mechanisms for the origin of the outer uranian satellites are obtained from the knowledge of their actual orbital properties. We conclude that the existence of these moons implies that their origin must be connected to a breakup process. Other scenarios for their origin cast serious doubts on the occurrence of a giant collision at the end of Uranus' formation process to account for its large spin axis obliquity.     

The Existence of a Planet beyond 50 AU and the Orbital Distribution of the Classical Edgeworth-Kuiper-Belt Objects

We study the effects of a Mars-like planetoid with a semimajor axis at about ∼60 AU orbiting embedded in the primordial Edgeworth-Kuiper belt (EKB). The origin of such an object can be explained in the framework of our current understanding of the origin of the outer Solar System, and a scenario for the orbital transport mechanism to its present location is given. The existence of such an object would produce a gap in the EKB distribution with an edge at about 50 AU, which seems to be in agreement with the most recent observations. No object at low eccentricity with semimajor axis beyond 50 AU has been detected so far, even though the present observing capabilities would allow an eventual detection (B. Gladman et al. 1998, Astron. J.116, 2042-2054; D. Jewitt et al. 1998, Astron. J.115, 2125-2135; E. I. Chiang and M. E. Brown 1999, Astron. J.118, 1411-1422; R. L. Allen et al. 2000, Astrophys. J.549, 241-244; C. A. Trujillo et al. 2001, Astron. J.122, 457-473; B. Gladman et al. 2001, Astron. J.122, 1051-1066; C. A. Trujillo and M. E. Brown 2001, Astrophys. J.554, 95-98). Finally, ranges for the magnitude and proper motion of the proposed object are given.     

Iron(VI) Seems an Unlikely Explanation for Viking Labeled Release Results

Tsapin et al. (2000, Icarus147, 68-78) propose the strong oxidant ferrate(VI) to explain the Viking Labeled Release Mars life detection results. However, their data do not support that theory. Further, sensitive IR searches for oxidants on Mars found none, and Viking produced physical evidence against an oxidizing surface. Finally, Tsapin et al. (2000, Icarus147, 68-78) report no precautions to prevent microbial contamination from confounding their results.     

Origin and sustainability of the population of asteroids captured in the exterior resonance 1:2 with Mars

At present, approximately 1500 asteroids are known to evolve inside or sticked to the exterior 1:2 resonance with Mars at a ? 2.418 AU, being (142) Polana the largest member of this group. The effect of the forced secular modes superposed to the resonance gives rise to a complex dynamical evolution. Chaotic diffusion, collisions, close encounters with massive asteroids and mainly orbital migration due to the Yarkovsky effect generate continuous captures to and losses from the resonance, with a fraction of asteroids remaining captured over long time scales and generating a concentration in the semimajor axis distribution that exceeds by 20% the population of background asteroids. The Yarkovsky effect induces different dynamics according to the asteroid size, producing an excess of small asteroids inside the resonance. The evolution in the resonance generates a signature on the orbits, mainly in eccentricity, that depends on the time the asteroid remains captured inside the resonance and on the magnitude of the Yarkovsky effect. The greater the asteroids, the larger the time they remain captured in the resonance, allowing greater diffusion in eccentricity and inclination. The resonance generates a discontinuity and mixing in the space of proper elements producing misidentification of dynamical family members, mainly for Vesta and Nysa-Polana families. The half-life of resonant asteroids large enough for not being affected by the Yarkovsky effect is about 1 Gyr. From the point of view of taxonomic classes, the resonant population does not differ from the background population and the excess of small asteroids is confirmed.     

Near-Infrared Spectrophotometry of Phobos and Deimos

We have observed the leading and trailing hemispheres of Phobos from 1.65 to 3.5 μm and Deimos from 1.65 to 3.12 μm near opposition. We find the trailing hemisphere of Phobos to be brighter than its leading hemisphere by 0.24±0.06 magnitude at 1.65 μm and brighter than Deimos by 0.98±0.07 magnitude at 1.65 μm. We see no difference larger than observational uncertainties in spectral slope between the leading and trailing hemispheres when the spectra are normalized to 1.65 μm. We find no 3-μm absorption feature due to hydrated minerals on either hemisphere to a level of ∼5-10% on Phobos and ∼20% on Deimos. When the infrared data are joined to visible and near-IR data obtained by previous workers, our data suggest the leading (Stickney-dominated) side of Phobos is best matched by T-class asteroids. The spectral slope of the trailing side of Phobos and leading side of Deimos are bracketed by the D-class asteroids. The best laboratory spectral matches to these parts of Phobos are mature lunar soils and heated carbonaceous chondrites. The lack of 3-μm absorption features on either side of Phobos argues against the presence of a large interior reservoir of water ice according to current models of Phobos' interior (F. P. Fanale and J. R. Salvail 1989, Geophys. Res. Lett.16, 287-290; Icarus88, 380-395).     

The angular momenta of solar system bodies: Implications for asteroid strengths

The angular momentum H is plotted versus mass M for the planets and for all asteroids with known rotation rates and shapes, primarily taken from D. C. McAdoo and J. A. Burns [Icarus18, 285–293 (1973)]. An asteroid's angular momentum is derived from its rotation rate as determined by the period of its lightcurve, its shape as indicated by the lightcurve amplitude, and where possible its size as given by polarimetry or radiometry. The asteroid is assumed to be rotating about its axis of maximum moment of inertia. As previously found by F. F. Fish [Icarus7, 251–256 (1967]) and W. K. Hartmann and S. M. Larson [Icarus7, 257–260 (1967)], H is approximately proportional to $\text{M}{}^{\mathrm{<mtext>5</mtext><mtext>3</mtext>}}$, which shows that the asteroids and most planets spin with nearly the same rate. The very smallest asteroids on the plot deviate from the above reaction, usually containing excess angular momentum. This suggests that collisions have transferred substantial angular momentum to the smallest asteroids, perhaps causing their internal stress states to be substantially modified by centrifugal effects.The forces produced by gravitation are then compared to centrifugal effects for a rotating, triaxial ellipsoid of density 3 g cm^?3. For all asteroids with known properties the gravitational attraction is shown to be larger than the centrifugal acceleration of a particle on the surface: thus the observed asteroid regoliths are gravitationally bound. Poisson's equation for the gravitational potential is investigated and it is shown by mathematical and physical arguments that any arbitrarily shaped ellipsoid with the attractive surface force boundary condition found above will have only attractive internal forces. Thus the internal stress states in asteroids are always compressive so that asteroids could be internally fractured without losing their integrity.     

Thermal inertia of near-Earth asteroids and implications for the magnitude of the Yarkovsky effect

Thermal inertia determines the temperature distribution over the surface of an asteroid and therefore governs the magnitude the Yarkovsky effect. The latter causes gradual drifting of the orbits of km-sized asteroids and plays an important role in the delivery of near-Earth asteroids (NEAs) from the main belt and in the dynamical spreading of asteroid families. At present, very little is known about the thermal inertia of asteroids in the km size range. Here we show that the average thermal inertia of a sample of NEAs in the km-size range is . Furthermore, we identify a trend of increasing thermal inertia with decreasing asteroid diameter, D. This indicates that the dependence of the drift rate of the orbital semimajor axis on the size of asteroids due to the Yarkovsky effect is a more complex function than the generally adopted D⁻¹ dependence, and that the size distribution of objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less skewed to smaller sizes than generally assumed. We discuss how this fact may help to explain the small difference in the slope of the size distribution of km-sized NEAs and main-belt asteroids.     

Comparison between the Shkuratov and Hapke Scattering Theories for Solid Planetary Surfaces: Application to the Surface Composition of Two Centaurs

This project examines the different approaches which deal with the theory of radiative transfer on atmosphereless bodies. We present the relative merits of two scattering theories based on the equivalent slab model: the extensively used Hapke theory (Hapke 1981, J. Geophys. Res.86, 3039-3054) and the Shkuratov theory (Shkuratov et al. 1999, Icarus141, 132-155). We found that their main difference is the role of the phase function of individual particles of regolith, which is predicted (and generally forward directed) in the case of the Shkuratov model instead of being a free parameter as formulated in the Hapke model. We also emphasize that different assumptions as to the manner in which different constituents are physically mixed in either model have a substantial effect on the synthetic spectra inferred. This leads to a significant extension of the validity of Hapke's or similar practical approaches to areas where these approaches are valid.We used two objects (the Centaurs 5145 Pholus and 8405 Asbolus) as examples. Previous modeling of the spectra of these two bodies with the Hapke approach gave suspect results in terms of the derived grain sizes, which were smaller than the wavelength, violating key assumptions of the model (Cruikshank et al. 1998, Icarus135, 389-407 for Pholus; Barucci et al. 2000, Astron. Astrophys.357, L53-56 for Asbolus). We considered several different types of powdered surfaces to interpret the surface composition of these two Centaurs. The effect of fine-scale contamination of water ice grains by small amounts of carbon and/or tholins is also explored. We can explain the strong red color and the rich near-infrared spectral signatures of Pholus using a five-component surface (contaminated water ice, amorphous carbon, Titan tholin, olivine, and methanol ice) where the grain sizes are consistent with the model assumptions. These components are similar to those inferred by Cruikshank et al. (1998), but we obtain very different grain sizes and relative abundances. For example, we obtain a relative abundance of water ice on the surface of Pholus of about 40% instead of 6% found with the Hapke model. Organic and carbonaceous components change by similar amounts. In the case of Asbolus, a tholin and amorphous carbon areal mixture can reproduce the spectrum, with water remaining at 9% or less. Using the albedo published by Fernandez et al. (2002, Astron. J.123, 1050-1055) which is higher than most workers assume for Centaurs and Kuiper belt objects, a surface composition similar to that of Pholus is found. It appears that model-based uncertainties in relative compositions must be regarded with more attention.     

The Diagnosis of Complex Rotation in the Lightcurve of 4179 Toutatis and Potential Applications to Other Asteroids and Bare Cometary Nuclei

The analysis of radar observations of the asteroid 4179 Toutatis by Hudson and Ostro (1995, Science270, 84-86) yielded a complex spin state. We revisit the visible lightcurve data on Toutatis (Spencer et al. 1995, Icarus117, 71-89) to explore the feasibility of using a rotational lightcurve to recover the signature of an excited spin state. For this, we apply Fourier transform and CLEAN algorithm (WindowCLEAN). WindowCLEAN yields clear and precise frequency signatures associated with the precession of the long axis about the total angular momentum vector and a combination of this precession and rotation about the long axis. For a long-axis mode state, our periodicities for Toutatis yield a mean long-axis precession period, P_φ, of 7.38 days and a rotation period around the long axis, P_ψ, of 5.38 days, which compare well with the respective periods of 7.42 and 5.37 days derived by Ostro et al. (1999, Icarus137, 122-139) and represent an independent confirmation of these values. We explain why the dramatic change in the Earth-Toutatis-Sun geometry during the time that the lightcurve was obtained has little effect on the final results obtained. Using the Toutatis example as a guide, we discuss the capabilities as well as the limitations on deriving information about complex spin states from asteroidal lightcurves.     

Search for the 3.4-μm C-H Spectral Bands on Low-Albedo Asteroids

A report of the detection of the C-H hydrocarbon band complex at 3.4 μm in an asteroid spectrum, by D. P. Cruikshank and R. H. Brown (1987, Science238, 183-184) is not confirmed by recent data of higher quality. Spectra of the same asteroid and six other low-albedo asteroids do not show this feature, which if present would indicate the presence of hydrocarbons and might link these asteroids with certain classes of carbonaceous meteorites.     

Excitation of Orbital Inclinations of Asteroids during Depletion of a Protoplanetary Disk: Dependence on the Disk Configuration

The pumping up of orbital inclinations of asteroids caused by sweeping secular resonances associated with depletion of a protoplanetary disk is discussed, focusing on the dependence on the disk inclinations and surface density distribution. The asteroids have large mean inclinations that cannot be explained by present planetary perturbations alone. It has been suggested that the sweeping secular resonances caused by disk depletion are responsible for these high inclinations. Nagasawa et al. (2000, Astron. J.119, 1480-1497) showed that the inclinations of asteroids are pumped up if the disk is depleted in an inside-out manner on a time scale longer than 3×10⁵ years. Their assumed disk midplane is not on the invariant plane. However, it should be affected by the inclination of the disk plane. Here we investigate the dependence on the disk inclinations. We assume a disk depletion model in which the disk inside the jovian orbit has been removed and the residual outer disk is uniformly depleted. We calculate the locations of the secular resonances and the excitation magnitude of the inclinations with analytical methods. We found that the inclinations are pumped up to the observational level for a depletion time scale longer than 10⁶ years in the case of the disk plane that coincides with the invariant plane. The required time scale is longest (3×10⁶ years) if the disk plane coincides with the jovian orbital plane. However, it is still within the observationally inferred depletion time scale. We also studied dependence on a disk surface density gradient and found that the results do not change significantly as long as the inner disk depletion is faster than the outer disk one.     

Radiation pressure and dust particle dynamics

The dynamics of small dust grains orbiting a planet are investigated when solar radiation pressure forces are added to the planet's gravitational central field. In the first part a set of differential equations is derived in a reference frame linked to the solar motion. The complete solution of these equations is given for particles lying in the planet's orbital plane, and we show that the orbital eccentricity may undergo considerable variation. At the same time the pericenter longitude librates or circulates according to initial conditions. With this result we establish a criterion for any orbiting particle (because of its highly eccentric orbit) to collide with its planet's atmosphere. The case of inclined orbit is studied through a numerical integration and allows us to draw conclusions related to the stability of the orbital plane. All solutions are periodic, with the period being independent of the initial conditions. This last point permits us to investigate the different time scales involved in that problem. Finally, the Poynting-Robertson drag is included, along with the radial radiation pressure forces, and the secular trend is considered. A coupling effect between the two components is ascertained, yielding a systematic behavior in the eccentricity and thus in the pericenter distance. Our solutions generalize the results of S. J. Peale (1966, J. Geophys. Res.71, 911–933) and J. A. Burns, P. Lamy, and S. Soter (1979, Icarus40, 1–48) by allowing eccentricities to be large (of order 1) and inclinations to be nonzero and by considering Poynting-Robertson drag.     

Carbon Monoxide on Jupiter: Evidence for Both Internal and External Sources

Thirteen lines of the CO band near 4.7 μm have been observed on a jovian hot spot at a resolution of 0.045 cm⁻¹. The measured line profiles indicate that the CO mole fraction is 1.0±0.2 ppb around the 6-bar level and is larger in the upper troposphere and/or stratosphere. An external source of CO providing an abundance of 4⁺³₋₂×10¹⁶ molecules cm⁻² is implied by the observations in addition to the amount deposited at high altitude by the Shoemaker-Levy 9 collision. From a simple diffusion model, we estimate that the CO production rate is (1.5-10)×10⁶ molecules cm⁻² s⁻¹ assuming an eddy diffusion coefficient around the tropopause between 300 and 1500 cm² s⁻¹. Precipitation of oxygen atoms from the jovian magnetosphere or photochemistry of water vapor from meteoroidal material can only provide a negligible contribution to this amount. A significant fraction of the CO in Jupiter's upper atmosphere may be formed by shock chemistry due to the infall of kilometer- to subkilometer-size Jupiter family comets. Using the impact rate from Levison et al. (2000, Icarus143, 415-420) rescaled by Bottke et al. (2002, Icarus156, 399-433), this source can provide the observed stratospheric CO only if the eddy diffusion coefficient around the tropopause is 100-300 cm² s⁻¹. Higher values, ∼700 cm² s⁻¹, would require an impact rate larger by a factor of 5-10, which cannot be excluded considering uncertainties in the distribution of Jupiter family comets. Such a large rate is indeed consistent with the observed cratering record of the Galilean satellites (Zahnle et al. 1998, Icarus136, 202-222). On the other hand, the ∼1 ppb concentration in the lower troposphere requires an internal source. Revisiting the disequilibrium chemistry of CO in Jupiter, we conclude that rapid vertical mixing can provide the required amount of CO at ∼6 bar for a global oxygen abundance of 0.2-9 times the solar value considering the uncertainties in the convective mixing rate and in the chemical constants.     

The rotation of small asteroids

The addition of the unbiased sample of R. P. Binzel and J. D. Mulholland (Icarus56, 519–533) nearly triples the sample size of photoelectrically determined rotational parameters for main belt asteroids with estimated diameters (D) ≤30 km. Nonparametric stattistical tests which require no assumptions about the distributions or variances of the samples are used to examine rotational parameters for all D ≤ 30 km asteroids. A comparison of photoelectric and photographic results shows that the techniques have a highly significant difference in the range of detected frequencies. This difference does not allow photographic and photoelectric observations to be combined for meaningful statistical tests since a photographic bias toward smaller sample variances can induce statistical results that appear overly significant. Photographic observations also show a highly significant bias toward detecting asteroids with larger lightcurve amplitudes. The fit of a Maxwellian to the observed rotational frequency distribution can be rejected at a highly significant confidence level but the observed distribution can be acceptably fit by two Maxwellian distributions, which is consistent with the hypothesis that there are separate populations of slow and fast rotating asteroids. The frequency distributions of <15 km main belt asteroids and Earth and Mars crossers are not found to differ significantly. However, the larger mean lightcurve amplitude of the Earth and Mars crossing asteroids is found to be statistically significant. This latter result is interesting in view of the lack of any strong inverse amplitude versus diameter relation for small asteroids. No significant diameter dependence on rotational frequency is seen among only D ≤ 30 km asteroids. However, the inverse frequency versus diameter relation for D ≤ 120 km asteroids found by S. F. Dermott, A. W. Harris, and C. D. Murray (Icarus, in press) is found to be statistically significant using a linear least-squares analysis of photoelectric data only. No significant diameter dependence on rotational lightcurve amplitude is seen using linear least-squares analysis of photoelectric data for D≤30 and D≤90 km asteroids. However, a significant inverse amplitude versus diameter relation is found when this analysis is extended to D≤120 km asteroids. This finding may be consistent with the hypothesis of Dermott et al. that near 120 km there is a transition between primordial asteroids and their collisional fragments.     

Shear Instabilities as a Probe of Jupiter's Atmosphere

Linear wave patterns in Jupiter's clouds with wavelengths strongly clustered around 300 km are commonly observed in the planet's equatorial atmosphere (F. M. Flasar and P. J. Gierasch, 1986, J. Atmos. Sci.43, 2683-2707). We propose that the preferred wavelength is related to the thickness of an unstable shear layer within the clouds (A. P. Ingersoll and D. W. Koerner 1989, Bull. Am. Astron. Soc.21, 943). We numerically analyze the linear stability of wavelike disturbances that have nonzero horizontal phase speeds in Jupiter's atmosphere and find that, if the static stability in the shear layer is very low (but still nonnegative), a deep vertical shear layer like the one measured by the Galileo probe (D. H. Atkinson et al. 1998, J. Geophys. Res.103, 22911-22928) can generate the instabilities. The fastest growing waves grow exponentially within an hour, and their wavelengths match the observations. Close to zero values of static stability that permit the growth of instabilities are within the range of values measured by the Galileo probe in a hot spot (A. Seiff et al. 1998, J. Geophys. Res.103, 22857-22889). Our model probes Jupiter's equatorial atmosphere below the cloud deck and suggests that thick regions of wind shear and low static stability exist outside hot spots.     

An Evolutionary Turbulent Model of Saturn's Subnebula: Implications for the Origin of the Atmosphere of Titan

We have elaborated an evolutionary turbulent model of the subnebula of Saturn derived from that of Dubrulle (1993, Icarus106, 59-76) for the solar nebula, which is valid for a geometrically thin disk. We demonstrate that if carbon and nitrogen were in the form of CO and N₂, respectively, in the early subnebula, these molecules were not subsequently converted into CH₄ and NH₃ during the evolution of the disk, contrary to the current scenario initially proposed by Prinn and Fegley (1981, Astrophys. J., 249, 308-317). However, if the early subnebula contained some CH₄ and NH₃, these gases were not subsequently converted into CO and N₂. We argue that Titan must have been formed from planetesimals migrating from the outer part of the subnebula to the present orbit of the satellite. These planetesimals were relics of those embedded in the feeding zone of Saturn prior to the completion of the planet and contained hydrates of NH₃ and clathrate hydrates of CH₄. It is shown that, for plausible abundances of CH₄ and NH₃ in the solar nebula at 10 AU, the masses of methane and nitrogen trapped in Titan were higher than the estimate of masses of these components in the primitive atmosphere of the satellite. If our scenario is valid and if our turbulent model properly describes the structure and the evolution of the actual subnebula of Saturn, the Xe/C ratio should be six times higher in Titan's atmosphere today than in the Sun, while the current scenario would probably result in a quasi solar Xe/C ratio. The mass spectrometer and gas chromatograph instrument aboard the Huygens Titan probe of the Cassini mission has the capability of measuring this ratio in 2004, thus permitting us to discriminate between the current scenario and the one proposed in this report.     

The accuracy of proper orbital elements and the properties of asteroid families: Comparison with the linear theory

The accuracy and reliability of the proper orbital elements used to define asteroid families are investigated by simulating numerically the dynamical evolution of families assumed to arise from the “explosion” of a parent object. The orbits of the simulated family asteroids have then been integrated in the frame of the elliptic restricted three-body problem Sun-Jupiter-asteroid, for times of the order of the circulation periods of perihelia and nodes. By filtering out short-periodic perturbations, we have monitored the behavior of the proper eccentricities and inclinations, computed according to the linear secular perturbation theory. Significant long-period variations have been found especially for families having nonnegligible eccentricities and/or inclinations (like the Eos family), and strong disturbances due to the proximity of mean motion commensurabilities with Jupiter have been evidenced (for instance, in the case of the Themis family). These phenomena can cause a significant “noise” on the proper eccentricities and inclinations, probably affecting in some cases the derived family memberships. They can also give rise to a spurious anisotropy in the fragment ejection velocity fields computed from the dispersion in proper elements observed in each family, and this could explain the puzzling anisotropies of this kind actually found in real families by D. Brouwer (1951, Astron. J. 56, 9–32) and by V. Zappalà, P. Farinella, Z. Knežević, and P. Paolicchi (1984), Icarus 59, 261–285).     

Moonlet Collisions and the Effects of Tidally Modified Accretion in Saturn's F Ring

We both test and offer an alternative to a meteoroid bombardment model (M. R. Showalter 1998, Science282, 1099-1102) and suggest that anomalous localized brightenings in the F ring observed by Voyager result from disruptive collisions involving poorly consolidated moonlets, or “rubble piles.” This model can also explain the transient events observed during ring plane crossing. We have developed an evolutionary model that considers both the competing effects of accretion and disruption at the location of the F ring. Our numerical model is a Markov process where probabilities of mass transfer between the states of the system form a “transition matrix.” Successive multiplications of this matrix by the state vector generate expectation values of the distribution after each time step as the system approaches quasi-equilibrium. Competing effects of accretion and disruption in the F ring are found to lead to a bimodal distribution of ring particle sizes. In fact, our simulation predicts the presence of a belt of kilometer-sized moonlets in the F ring. These moonlets may continually disrupt one another and re-accrete on short time scales. We also agree with J. N. Cuzzi and J. A. Burns (1988, Icarus74, 284-324), who suggest that the classical F ring itself may be the consequence of a relatively recent collision between two of the largest of these yet unseen objects. Cassini observations can confirm the existence of the moonlet belt by directly observing these objects or the waves they create in the rings.