FORMULATION OF COLLISION PROBABILITY WITH TIME-DEPENDENT PROBABILITY DENSITY FUNCTIONS

This paper is primarily concerned with the formulation of the probability of collision between two space orbiting objects when their probability density functions (pdfs) are time-dependent. Some recent papers addressing this problem have dealt with the concept of integrating the flux of time-dependent pdfs through a hemispherical surface over a period of time. This latter approach is permissible only for the relatively simple case of time-independent pdfs and relative motion which does not lead to self-intersection of the cylindrical tube. However, it is not even valid for time-independent pdfs and self-intersecting integration volumes. It is definitely not consistent with the basic tenets of probability theory for the case of time-dependent pdfs. Several concrete examples are provided to illustrate this point.