MISS DISTANCE – GENERALIZED VARIANCE NON - CENTRAL CHI DISTRIBUTION Ken

In many current practical applications, the probability of collision is often considered as the most meaningful criterion for determining the risk to a spacecraft. However, when the miss distance is very small (say 100 m or less), the collision probability may not comfortably serve well as the only metric for measuring the risk. In these cases, it is also important to know the statistics of the miss distance distribution, particularly the confidence limits that bound the miss distance. One approach to determine the distribution of the miss distance is to perform rather time-consuming Monte Carlo simulations on the two conjuncting objects by choosing random initial conditions at the epoch commensurate with the covariance. The orbit propagations must be of very high precision because accuracies down to the meter range are required. The simulated results are then fitted with a Weibull curve which, unfortunately, does not bring out the salient features of the distribution. The analysis here formulates the problem analytically. For this, we consider the miss distance as given by a non-central chi distribution with unequal variances. This method eliminates the need to perform an inordinate amount of computation, thus reducing the desired results in a timely manner by several orders of magnitude. Moreover, this generalized variance non-central chi distribution possesses the requisite features of the miss distance statistics. INTRODUCTION In a short-term encounter between two orbiting objects, the nominal miss distance is determined by propagating their orbits from initial conditions at some epoch. When they are in the vicinity of the point of closest approach, the integration steps are smaller and the minimum separation is determined using interpolation and possibly some iteration. However, this result provides only a nominal minimum separation at the nominal time of closest approach and not a distribution of what the miss distance would be. If we desire this information, then we would have to perform Monte Carlo simulations by choosing random initial conditions at the epoch through the generation of Gaussian random numbers using the covariances associated with their orbit determinations. For each such choice of initial conditions, we would have to repeat the orbit propagations and subsequent minimum separation computations. This is obviously a very timeconsuming process because we need approximately one million runs to obtain a meaningful result. An alternative approach is to propagate the two nominal orbits to some instant of time when they are both in the encounter region. In this region, the rectilinear approximation is valid just as assumed in all the collision probability computations. Thus, with this knowledge of the ingress initial conditions, we may use the analytical methods of projective geometry to determine the minimum separation and also the nominal time when this occurs. With the ingress covariances, we can then choose seeds to perform the Monte Carlo simulations. Note that we have eliminated the orbit propagations which form the bulk of the previous effort. For each choice of ingress initial conditions, we use the analytical formulation to determine the minimum separation. Note