On the fragmentation of asteroids and planetary satellites

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Theories of collisional fragmentation are highly speculative. Most of our knowledge of such events is based on experimental results. Unfortunately, due to practical constraints, the experiments are conducted at size and velocity scales which are vastly different from those appropriate to collisions involving asteroids or satellites. Therefore the results must be extrapolated. In this paper we construct a general scaling model which guides the extrapolation of small-scale results and we consider existing theories in the context of this general model. The most common scaling method assumes that collisional outcomes (e.g., normalized fragment size and velocity distributions) are determined by the specific energy, Q, of the event, i.e., the ratio of the projectile kinetic energy to the mass of the target body. In particular, the threshold specific energy for target fragmentation is typically assumed to be independent of target size and impact velocity. Here we show that scaling based solely on Q holds when two conditions are met: (1) the projectile and target material properties do not depend on any size or time scales, and (2) the collision is governed by the kinetic energy, independent of the impact velocity. On the basis of a variety of experimental and theoretical evidence, we believe that neither of these conditions should hold, thereby casting serious doubt on the validity of using Q as the sole parameter in scaling. Specific scaling laws are derived on the basis of the more likely assumption that fracture strength is strain-rate and target-size-dependent and on a specific assumption which allows a dependence on both the energy and velocity of the impactor. As a result, the threshold specific energy for fragmentation of small targets (<10 km) decreases with increasing target size. For large bodies, whose gravitational self-compression deters fragmentation, the specific energy increases with target size, in qualitative agreement with the model of Davis et al. (1985 , Icarus 62, 30–53). Fragment velocities are shown to follow a similar trend. The results are compared to existing collision experiments and to observations of asteroid families.