The probability of beam on beam intercept in the vehicle recapture problem

An attempt is made to correctly state the analysis of a probability problem which has been classically formulated in error. The general problem is the probability distribution associated with the time-to-intersection of two different periodic trains of window functions. The specific problems chosen here to illustrate the more general problems deal with the idealized telemetry recapture of a tumbling satellite with a directed gain reception antenna and a tracking station with a high gain, narrow beam, scanning acquisition antenna. The present analysis gives the intuitively obvious results for certain special limiting cases and specifies outcomes clearly different from the classical treatment which always predicts success after a long enough search time. In particular, for the case where one window train is a harmonic of the other there is clearly a nonzero probability that there will never be an intersection due to randomness in the relative phasing (the classical treatment specifies a probability that approaches zero in time for the intersection).