An Introductory Analysis of Satellite Collision Probabilities KITT C. CARLTON-WIPPERN* Summary: This paper addresses a probabilistic approach in assessing the probabilities of a satellite collision occurring due to relative trajectory analyses and probability density functions representing the satellites’ position/momentum vectors. The paper is divided into 2 parts: Static and Dynamic Collision Probabilities. In the Static Collision Probability section, the basic phenomenon under study is: given the mean positions and associated position probability density functions for the two objects, calculate the probability that the two objects collide (defined as being within some distance or volume of each other). The paper presents the classic Laplace problem of the probability of arrival, using standard uniform distribution functions. This problem is then extrapolated to show how "arrival" can be classified as "collision", how the arrival space geometries map to collision space geometries and how arbitrary position density functions can then be included and integrated into the analysis. In the Dynamic Collision Probability section, the nature of collisions based upon both trajectory and energy considerations is discussed, and that energy states alone cannot be used to completely describe whether or not a collision occurs. This fact invalidates some earlier work on the subject and demonstrates why Liouville's theorem cannot be used in general to describe the constant density of the position/momentum space in which a collision may occur. Future position probability density functions are then shown to be the convolution of the current position and momentum density functions (linear analysis), and the paper further demonstrates the dependency of the future position density functions on time. Strategies for assessing the collision probabilities for two point masses with uncertainties in position and momentum at some given time, and these integrated with some arbitrary impact volume schema, are then discussed. This presentation concludes with the formulation of a high level design for a Collision Requirements and Satellite Trajectory Analysis (or CRYSTAL) computer program, prototype software package. Static Collision Probability For the static case, the basic phenomenon under study is given the mean position and associated position probability function for two objects, calculate the probability that the two objects collide (or are within some distance or volume of each other). t*U of Colorado at Colorado Springs
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