# Simplified derivation of the collision probability of two objects in independent Keplerian orbits

## TL;DR

This paper introduces a simplified, symmetric derivation of collision probabilities for two objects in Keplerian orbits, including tangential collisions, improving upon classic linearized methods.

## Contribution

A new, more physically motivated derivation that handles tangential collisions and ensures symmetry between colliding bodies, extending previous linearized approaches.

## Key findings

- Provides formulas for collision probabilities in non-tangential and tangential cases.
- Regularizes singularities in tangential encounter calculations.
- Offers a more straightforward derivation compared to classic methods.

## 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 \"Opik (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 \"Opik-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.