Collisions in the asteroid belt

Abstract

Öpik's theory of collision of planetary bodies has been generalized to include the case where the orbits of both colliding bodies are ellipses. The resulting expression can be written in a form similar to Öpik's result for the crossing of an elliptical orbit with a circular orbit. However, in this generalized case the quantities appearing in the expression refer to the collision probability for a given orientation of the perihelion of one of the bodies (field body); to the x component of velocity of the other body alone (test body) in the reference frame of the orbit of the field body, and to the properly modified expression for the relative velocity of the two bodies. Furthermore, even for a given orientation of the perihelion of the field body, the expression now has four values; the total probablity is the sum of these values. This result must finally be integrated over all possible orientations of the perihelion of the field body and over all possible relative orientations of the nodes of the two bodies. This expression has been used to calculate the collision probability of meteorites (test bodies) with given orbital elements moving in a field of bodies having orbital elements corresponding to those of the visible asteroids. It is found that this collision probability is nearly independent of the orbital elements of the test body. This is in general agreement with previous approaches to this problem by Piotrowski and by Arnold; however, there are differences in details. The results of these calculations have been applied to the problem of the suppression of old cosmic-ray ages by mutual destruction of stone meteorites in the asteroid belt, under a range of assumptions regarding the size-frequency distribution of matter in the asteroid belt. It is found that the necessary very short (∼30 million year) destruction lifetimes are obtained only if the increase in the number of small bodies with decreasing radius is nearly that allowed by an upper limit to the area of the asteroid belt determined by the intensity of the gegenschein, and then only for a very small value of the minimum mass needed to destroy a body of a given mass by collision.