Collision Probabilities for Keplerian Orbits

We present a new derivation of the probability of collisions between spherical satellites occupying Keplerian orbits. The equations follow from the central concept of the instantaneous collision rate, an expression that describes the occurrence of collisions by using a Dirac [delta]-function. The derivation proceeds by showing how this instantaneous collision rate can be averaged over orbital mean anomaly angles and, additionally, over orbital precession angles to generate expressions appropriate for intermediate and long time scales. Collision rates averaged over mean anomalies tend to be non-zero during relatively brief collision seasons, when the peak collision probability may exceed the long-term average by several orders of magnitude. Derived precession-angle averages have a functional form similar but not identical to the collision probability expression derived using the spatial density approach of Kessler (Icarus, 48: 39-48, 1981), and the two methods have been found to yield numerical results to within 1% for all cases examined. [PUBLICATION ABSTRACT]