Aggregation and fragmentation kinetics: Application to planetary rings
(Joint work with Anna Bodrova, Paul Krapivsky and Juergen Schmidt)
A simple model of ballistic aggregation and fragmentation is proposed. The model is characterized by two energy thresholds, Eagg and Efrag, which demarcate different types of impacts: if the kinetic energy of the relative motion of a colliding pair is smaller than Eagg or larger than Efrag, particles respectively merge or break; otherwise they rebound. We assume that particles are formed from monomers which cannot split any further and that in a collision-induced fragmentation the larger particle splits into fragments. We start from the Boltzmann equation for the mass-velocity distribution function and derive Smoluchowski-like equations for concentrations of particles of different mass. We analyze these equations analytically, solve them numerically and perform Monte Carlo simulations. We consider a wide class of fragmentation models, including the complete disaggregation into monomers at collisions, fragmentation by half and fragmentation with a power-law distribution of the debris sizes. For several models we obtain an analytical solution for the time evolution of the size distribution of the aggregates and their steady-state distribution. In particular we show, that for any fragmentation model with a predominance of small debris at a collision, the resulting size distribution n(R) obeys a power-law n(R) ~ R-α with an exponential cut-off. Moreover, the exponent of the power law α does not depend on a particular fragmentation model. These predictions for the particle size distribution, and especially for the exponent α = 2.75 are in a very good agreement withthe observational data for the size distribution of particles in planetary rings.