Wavy size distributions for collisional systems with a small-size cutoff

Dohnanyi's J. geophys. Res. 74, 2531–2554, 1969; in Physical Studies of Minor Planets (edited by T. Gehrels), pp. 263–295. NASA-SP 267, 1971] theory predicts that a collisional system such as the asteroid population should rapidly relax to a power-law equilibrium size distribution, provided all the collisional response parameters are independent of size. However, we have found that Dohnanyi did not include in a consistent way in the theory the possible occurrence of a small-size cutoff in the distribution. We have carried out a number of numerical simulations of the collisional evolution process, showing that the cutoff results in a wavy pattern superimposed on Dohnanyi's equilibrium power law, which affects the distribution up to sizes of tens of km. The pattern arises because particles just above the cutoff are not removed by catastrophic impacts by smaller projectiles, and therefore are created by break-up of larger bodies faster than they are eliminated; larger particles are increasingly depleted up to the size where the smallest shattering projectile exceeds the cutoff, and beyond that the removal rate is reduced and the distribution flattens. Thus, to be effective in producing the waves, the cutoff (or any other persisting “discontinuity” in the particle properties) must be sharp over a size range corresponding to the threshold projectile-to-target ratio for fragmentation. The presence of a small-size cutoff in the real asteroid belt is an open question, since it may be generated by poorly known non-gravitational effects acting on μm-sized dust, and may be affected by influx of cometary debris. However, the observational evidence for a variable characteristic exponent of the size distribution of interplanetary bodies is now strong, and the cutoff effect may provide a simple explanation for this finding.