### 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 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 language | English (US) |
---|---|

Article number | 235 |

Journal | Astronomical Journal |

Volume | 153 |

Issue number | 5 |

DOIs | |

State | Published - May 1 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Astronomy and Astrophysics
- Space and Planetary Science

### Cite this

**Simplified Derivation of the Collision Probability of Two Objects in Independent Keplerian Orbits.** / Jeongahn, Youngmin; Malhotra, Renu.

Research output: Contribution to journal › Article

*Astronomical Journal*, vol. 153, no. 5, 235. https://doi.org/10.3847/1538-3881/aa6aa7

}

TY - JOUR

T1 - Simplified Derivation of the Collision Probability of Two Objects in Independent Keplerian Orbits

AU - Jeongahn, Youngmin

AU - Malhotra, Renu

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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.

AB - 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.

KW - celestial mechanics

KW - meteorites meteors meteoroids

KW - minor planets asteroids: general

KW - planetary systems

UR - http://www.scopus.com/inward/record.url?scp=85019039976&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019039976&partnerID=8YFLogxK

U2 - 10.3847/1538-3881/aa6aa7

DO - 10.3847/1538-3881/aa6aa7

M3 - Article

AN - SCOPUS:85019039976

VL - 153

JO - Astronomical Journal

JF - Astronomical Journal

SN - 0004-6256

IS - 5

M1 - 235

ER -