Collisional evolution - an analytical study for the non steady-state mass distribution

To study the collisional evolution of asteroidal groups we can use an analy tical solution for the self-similar collision cascades. This solution is su itable to study the steady-state mass distribution of the collisional fragm entation. However, out of the steady-state conditions, this solution is not satisfactory for some values of the collisional parameters. In fact, for s ome values for the exponent of the mass distribution power law of an astero idal group and its relation to the exponent of the function which describes 'how rocks break' we arrive at singular points for the equation which desc ribes the collisional evolution. These singularities appear since some appr oximations are usually made in the laborious evaluation of many integrals t hat appear in the analytical calculations. They concern the cutoff for the smallest and the largest bodies. These singularities set some restrictions to the study of the analytical solution for the collisional equation. To overcome these singularities we performed an algebraic computation consi dering the smallest and the largest bodies and we obtained the analytical e xpressions for the integrals that describe the collisional evolution withou t restriction on the parameters. However, the new distribution is more sens itive to the values of the collisional parameters. In particular the steady -state solution for the differential mass distribution has exponents slight ly different from 11/6 for the usual parameters in the Asteroid Belt. The s ensitivity of this distribution with respect to the parameters is analyzed for the usual values in the asteroidal groups. With an expression for the mass distribution without singularities, we can evaluate also its time evolution. We arrive at an analytical expression giv en by a power series of terms constituted by a small parameter multiplied b y the mass to an exponent, which depends on the initial power law distribut ion. This expression is a formal solution for the equation which describes the collisional evolution. Furthermore, the first-order term for this solut ion is the time rate of the distribution at the initial time. In particular the solution shows the fundamental importance played by the exponent of th e power law initial condition in the evolution of the system. (C) 1999 Else vier Science Ltd. All rights reserved.