Steady-state size distribution for the self-similar collision cascade

Dohnanyi (1969, J. Geophys. Res. 74, 2531-2554) analytically obtained the steady-state mass distribution of the collisional fragmentation cascade as n(m) = Am--alpha, where the power law exponent alpha is very nearly 11/6. In the present study, we investigated the generality of Dohnanyi's result of alpha = 11/6 and clarified what essentially determines the value of the exponent alpha. We first derived new basic equations describing the evolution of the mass distribution in the collision cascade. The new basic equations are independent of the model of collisional outcomes and, hence, enable us to investigate the general properties of the collision cascade. As the steady-state solution to the derived basic equations, we obtained a power law mass distribution under the single assumption that the collisional outcome is selfsimilar. The results are summarized as follows: the power law exponent alpha of the mass distribution is exactly independent of the collisional outcome model as long as the model is selfsimilar and the value of alpha is directly determined only by the mass-dependence of the collision rate. (C) 1996 Academic Press, Inc.