Scaling laws for the catastrophic collisions of asteroids

Collisions of asteroids have traditionally been studied through laboratory experiments involving targets with masses some 15 to 20 orders of magnitude less than the bodies they are intended to simulate. Here the problem of extrapolation of experimental results up to the size regimes of interest is considered. Scaling relations are developed for the shattering threshold and the size and velocity distributions of collisional fragments. A methodology which has often been used assumes that collisional outcomes (e.g., the size of the largest remaining fragment) are completely characterized by Q, the kinetic energy of the impactor normalized by the mass of the target body. This scaling is shown to be an unlikely special case of a more general scaling theory which indicates that collisional outcomes should depend on target size and encounter velocity, even when Q is held constant. In particular, as target size increases, the critical value of Q required to shatter a body, and the characteristic fragment velocities should initially decrease (in qualitative agreement with the recent model of Farinella et al., 1982) up to an asteroid size of perhaps 40 to 50 km; then, as the gravitational forces start to dominate, the value of Q will again increase (in qualitative agreement with the recent model of Davis et al., 1983 and 1985).