Explicit expression of collision probability in terms of RSW and NTW components of relative position

The existing analytical and numerical computing methods for collision probability (Pc) provide sufficiently accurate results, but do not reveal the direct connection between Pc and conjunction geometry or error covariance. This paper derived an explicit expression of Pc under the circular orbit assumption, based on analysis of closest approach distance and Chan's analytical formulae for Pc. The Pc was expressed as an explicit function of the radial, in-track, cross-track components of relative position and error covariance. The explicit expression is simple and compact, relates Pc with components of relative position, which are two kinds of important conjunction risk criteria. By introducing non-zero flight-path angle and ratio of velocity's magnitude, a modified expression of Pc was presented for objects in eccentric orbits. For objects with eccentricity less than 0.01 (which account for 62% of all LEO objects), the relative error of the explicit expression is less than 0.107, or 0.252 for typical conjunction cases. What really matters in conjunction risk assessment is the order of magnitude rather than the specific value of Pc, the precision of the explicit expression is sufficient for conjunction risk assessment and decision-making for most LEO objects.