Probability of Collision of satellites and space debris for short-term
  encounters: Rederivation and fast-to-compute upper and lower bounds
Ricardo Ferreira Cláudia Soares Marta Guimarães
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.