JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. E4, PAGES 7387-7401, APRIL 25, 1993 Origins of the Rings of Uranus and Neptune 2. Initial Conditions and Ring Moon Populations JOSHUA E. COLWELL and LARRY W. ESPOSITO Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Colorado The smallest moons of the Jovian planets are unlikely to survive intact the flux of cometary impactors in the outer solarsystem for billions of years. In paper I (Colwell and Esposito, 1992) we showed that the small moons of Uranus and Neptune are fragments or rubble pile agglomerations left over from some older, larger population of satellites. Catastrophic fragmentation occurs in ~ 108 years for these ring moons. The fate of the debrisfollowing a fragmenting impact is central to understanding the evolution of these satellites and the hypothesized origin of rings from their debris. In this extension of our earlier work we examine the possible effects of the velocity distribution of fragments following a catastrophic fragmentation on satellite dinxinution via a collisional cascade. We compare our results with those presented in paper 1. Fragment velocities are critical in the evolution of the collisional cascade because of the possibility of reaccretion following disruption. Using our simulations of the collisional cascade including the effects of the fragment velocity distribution we estimate an unseen population of moons in the I to 10 km size range of ~ 1000 at Uranus and at Neptune. Using our model fragment velocity distribution we calculate the initial phase space distribution of the new ring particles. This provides a physically realistic initial condition for simulations of the collisional evolution of planetary rings. We find that a narrow ring with a characteristic width of ~ 50 km is a natural outcome of the catastrophic disruption of satellites. 1. INTRODUCTION The idea of a collisional cascade of planetary satellites waspresented in Colwelland Esposito [1992](paper 1 here- after). Fragmentationrates of the small moonsof Uranus and Neptune were calculated based on an assumed impactor flux, physicalmodels of the fragmentation process, and lab- oratory experiments of catastrophic fragmentation. These rates were considerablyshorter than the age of the solar sys- tem for the smallest moons. This suggests their existence today requires that they be (1) fragments from some larger moon(s) disrupted sometime in the relatively recent past,or (2) rubble pilesthat haveundergone oneor morefragmen- tation events only to teaccrete into a gravitationally bound agglomeration of debris. In paper I we consideredthe sec- ond possibility by using an ad hoc increase of a factor of 100 in Q*, the specificenergy required for disruption. In this extension of that work we examine the effects of different fragment velocity distributions on the evolution of the colli- sionalcascade. Rather than using a larger Q* for disruptive impacts, we model the fragment velocity distribution explic- itly and allow only those fragments with velocities greater than some critical velocity (such as,for example, the escape velocity of the original moon) to escape the moon. Becauseour knowledge of the fragment velocities is lim- ited, we have chosen three different velocity distribution models to check the dependence of our results on the par- ticular velocity distribution used. The collisional cascade proceeds more slowly when fragments are required to exceed Copyright 1993 by the American Geophysical Union Paper number 93JE00329. 0148-0227/93/ 93J E- 00329505.00 escapevelocity, but derived ages of the smallest moons are still less than the age of the solar system. We find a break in the size distribution of the fragments produced by catas- trophic fragmentation in this collisional cascade at the size where gravity induces reaccretion. This size is a function of the velocity distribution and. the unmodeled accretion pro- cess of disrupted planetary satellites. In the next section we describe our velocity distribution models, and in section 3 we outline how they are incorporated into the Markov chain and Monte Carlo simulations of the collisional cascade (pa- per 1), and present the results of the new collisional cascade simulations. With a particular model of the fragment mass and veloc- ity distribution we are in a position to explore the details of planetary ring creation from satellite disruption. In sec- tion 4 we compute the three-dimensional initial distribution of debris from a satellite disruption in the absence of inter- particle collisions. These distributions show how initial ring width varies with the fragment velocity distribution. We explore the variations in the new ring phase space densities due to different impact conditions. In section 5 we discuss our results and present a particular model for the origin of the rings of Uranus and Neptune that is consistent with our detailed simulations of the collisional cascade. 2. FRAGMENT MASS AND VELOCITY DISTRIBUTIONS Velocity distributions for the ejecta from hypervelocity cratering impacts have been extensively studied and consis- tently show a•distribution of the form V --• f(> v)= Vr (1) for v > v*, where f(> v) is the mass fraction of ejecta with velocities greater than v, and v* is the minimum ejecta veloc- ity [e.g. , Gault et al., 1963; StSj•ter et al., 1975; Hartmann, 1985] The wlue of 7 is often t•ken to be 2.25 from the impact 7387