Computing the Instantaneous Collision Probability between Satellites using Characteristic Function Inversion

TL;DR

This paper presents a numerical method based on characteristic function inversion to accurately compute the instantaneous collision probability between satellites with Gaussian uncertainties, applicable in short-term encounters and implemented in Python and R.

Contribution

It demonstrates how to apply characteristic function inversion to calculate collision probabilities without relying on analytical approximations, enhancing accuracy for satellite collision risk assessment.

Findings

Method accurately computes collision probabilities in short-term scenarios.

Applicable to both 2D and 3D collision probability evaluations.

Implementation available in Python and R for practical use.

Abstract

The probability that two satellites overlap in space at a specified instant of time is called their instantaneous collision probability. Assuming Gaussian uncertainties and spherical satellites, this probability is the integral of a Gaussian distribution over a sphere. This paper shows how to compute the probability using an established numerical procedure called characteristic function inversion. The collision probability in the short-term encounter scenario is also evaluated with this approach, where the instant at which the probability is computed is the time of closest approach between the objects. Python and R code is provided to evaluate the probability in practice. Overall, the approach has been established for over fifty years, is implemented in existing software, does not rely on analytical approximations, and can be used to evaluate two and three dimensional collision probabilities.