Stochastic simulations of the collisional fragmentation of the small moons of Neptune and Uranus confirm the conclusions of Smith et al. <1986, 1989> that many of these moons cannot have survived intact since the end of planetary formation. We perform two types of stochastic simulations of the collisonal history of small moons. Monte Carlo simulations in which only the largest surviving fragment from each disruption is followed show bimodal probability distributions for the size of the largest fragment. Once the moon is destroyed the first time, the collisional cascade to smaller sizes proceeds relatively quickly. A Markov chain approach allows us to follow the size distribution from each disruption to arbitrarily small sizes. These simulations show that the more numerous smaller fragments can outlive the largest fragment followed in the Monte Carlo formalism. We find that the evolution of moon populations from catastrophic fragmentation is more complex than can be described by a simple break-up ''time scale.'' We rederive cratering rates using the method of Smith et al. <1986, 1989> for all the satellites of Uranus and Neptune with an improved crater scaling and modified impactor distributions.

These changes produced a higher number of predicted craters larger than the radii of the small moons of Uranus and Neptune than derived by Smith et al. <1986, 1989>. Our more detailed simulations of the catastrophic fragmentation process also show a higher rate of disruptions than estimated by Smith et al. <1986, 1989>. The smallest observed moons at Uranus and Neptune have calculated lifetimes against catastrophic fragmentation of less than 5¿108 years. If we require that half the mass of a satellite is given escape velocity we find lifetimes for Cordelia (1986U7), Naiad (1989N6), and Thalassa (1989N5) of 9¿108, 2¿109, and 4¿109 years, respectively. We conclude that we are observing a collisionally evolved small satellite population around Neptune and Uranus and that some observed moons are gravitationally bound rubble piles that have undergone multiple disruptions. In order for the satellites to be primordial the population of planet family comets in the outer solar system would have to be smaller than current estimates by at least a factor of 5.