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^-α, where the power law exponent α is very nearly 11/6. In the present study, we investigated the generality of Dohnanyi's result of α = 11/6 and clarified what essentially determines the value of the exponent α. 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 self-similar. The results are summarized as follows: the power law exponent α of the mass distribution is exactly independent of the collisional outcome model as long as the model is self-similar and the value of α is directly determined only by the mass-dependence of the collision rate.