Simplified Derivation of the Collision Probability of Two Objects in Independent Keplerian Orbits

Youngmin Jeongahn, Renu Malhotra

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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 and Wetherill, the collision probability was derived by linearizing the motion near the collision points, and there is now a vast amount of 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.

Original languageEnglish (US)
Article number235
JournalAstronomical Journal
Issue number5
StatePublished - May 2017


  • celestial mechanics
  • meteorites meteors meteoroids
  • minor planets asteroids: general
  • planetary systems

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science


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