Collisional model of asteroids and their debris

A model for colliding objects in the asteroidal belt is formulated. An integro-differential equation describing the evolution of a system of particles undergoing inelastic collisions and fragmentation is derived and solved for steady-state conditions. It is found that the number density of particles per unit volume in the mass range m to m + dm is Am−a dm, where A and α are constants (provided that certain conditions are satisfied). The population index α can then be derived theoretically; for asteroids and their debris, α = 1.837, in agreement with an empirical fit to the observed distribution. Various statistical properties of the distribution can be derived from the model. It is found that, for asteroidal objects, catastrophic collisions constitute the most important physical process determining particle lifetimes and the form of the particle distribution for particles sufficiently large that radiation effects are unimportant. The lifetime of the largest asteroids is found to be of the same order of magnitude as the probable lifetime of the solar system; therefore, some of the largest asteroids may have survived since the time of creation, whereas most smaller ones have not and are collisional fragments, according to the present model.