Title: Simplified derivation of the collision probability of two objects in independent Keplerian orbits,
Abstract: Many topics in planetary studies demand an estimate of the collision
probability of two objects moving on nearly Keplerian orbits. In the classic
works of Öpik (1951) and Wetherill (1967), the collision probability was
derived by linearizing the motion near the collision points, and there is now a
vast literature using their method. We present here a simpler and more
physically motivated derivation for non-tangential collisions in Keplerian
orbits, as well as for tangential collisions that were not previously
considered. Our formulas have the added advantage of being manifestly symmetric
in the parameters of the two colliding bodies. In common with the
Öpik-Wetherill treatments, we linearize the motion of the bodies in the
vicinity of the point of orbit intersection (or near the points of minimum
distance between the two orbits) and assume a uniform distribution of impact
parameter within the collision radius. We point out that the linear
approximation leads to singular results for the case of tangential encounters.
We regularize this singularity by use of a parabolic approximation of the
motion in the vicinity of a tangential encounter.