Probability of Collision of satellites and space debris for short-term encounters: Rederivation and fast-to-compute upper and lower bounds

TL;DR

This work re-derives the short-term collision probability under the Akella–Alfriend assumptions from first principles, reformulating the problem in a 2D conjunction plane and reducing uncertainty to a Gaussian in that plane. By an eigen-decomposition of the projected covariance, the probability reduces to a 2D disk integral that can be bounded with simple products of 1D Gaussian integrals, yielding fast, accurate upper and lower bounds expressed via the error function. The approach preserves equivalence with the traditional AA formulation while offering substantial computational speedups—down to near real-time performance on large conjunction sets—without sacrificing accuracy, as validated on ESA CDM data. This enables scalable, real-time screening of space-surveillance conjunctions and more efficient collision-risk management for space debris.

Abstract

The proliferation of space debris in LEO has become a major concern for the space industry. With the growing interest in space exploration, the prediction of potential collisions between objects in orbit has become a crucial issue. It is estimated that, in orbit, there are millions of fragments a few millimeters in size and thousands of inoperative satellites and discarded rocket stages. Given the high speeds that these fragments can reach, even fragments a few millimeters in size can cause fractures in a satellite's hull or put a serious crack in the window of a space shuttle. The conventional method proposed by Akella and Alfriend in 2000 remains widely used to estimate the probability of collision in short-term encounters. Given the small period of time, it is assumed that, during the encounter: (1) trajectories are represented by straight lines with constant velocity; (2) there is no velocity uncertainty and the position exhibits a stationary distribution throughout the encounter; and (3) position uncertainties are independent and represented by Gaussian distributions. This study introduces a novel derivation based on first principles that naturally allows for tight and fast upper and lower bounds for the probability of collision. We tested implementations of both probability and bound computations with the original and our formulation on a real CDM dataset used in ESA's Collision Avoidance Challenge. Our approach reduces the calculation of the probability to two one-dimensional integrals and has the potential to significantly reduce the processing time compared to the traditional method, from 80% to nearly real-time.

Figures (6)

• Figure 1: Representation of the error vector between vector $z_0$ and vector $v$ times $t_{CPA}$.
• Figure 2: Graphical representation of the 2D ball centered at the origin with radius $R$, and the red and blue rectangle represent the upper-bound and lower-bound of the $P_C$, respectively. The red rectangle has side $2R$ and the blue rectangle has side $2R_I$ such that $R_I = R \cos(\frac{\pi}{4})$.
• Figure 3: Relative error of the probability of collision computed by the proposed method when compared to the base method, obtaining a maximum relative error in absolute value of $0.0008\%$.
• Figure 4: Difference between upper bounds computed by the proposed method and the naive one. It can be seen that the proposed method presents in most cases smaller values and therefore closer to the real probability value, serving as a tighter bound of the probability of collision.
• Figure 5: Cumulative distribution function of the processing times (in seconds) of the upper bound for the various implementations. It is possible to observe that the proposed method allows a faster processing time than the naive one, with the implementation using the error function being a little faster.

Theorems & Definitions (1)

• Remark 1