Statistical Analysis of Elliptical Keplerian Orbits, With Applications to Search and Surveillance Algorithms KITT C. CARLTON-WIPPERN Summary This paper addresses the motion of satellites in elliptical Keplerian orbits as a problem of random variable analysis. Concentration is placed on time as the random variate. Essential probability functions and significant statistics on radial distance are formally derived or calculated. The analysis is then applied to specific problems of interest. For surveillance, average area in view, constrained field of view, average distance to a randomly located target on a spherical surface (below the satellite) and average power requirements for target and area illumination are discussed. For search algorithms, search volumes, general and specific search patterns and efficiencies are discussed. Tables of comparisons between two distinct volume search patterns are presented to illustrate decision making on optimal search strategies. I. Introduction This paper is primarily concerned with finding radial statistics for elliptical Keplerian orbits. Since for unperturbed orbits all motion is confined to a plane, we will be concerned with finding radial averages as a function of semi-major axis (a) and eccentricity (e) only. The other Keplerian orbital elements define only the initial position of the satellite in the orbital plane (Mo) and the orientation of the orbital plane in space (i, O), Q). We start by assuming there exists a relation between two arbitrary variables x and y, such that: where it is understood that f~' exists, is single valued, integratable [1] and the limits of integration (a<y<b) set bounds on the permissible values^ may have [2], Thus,j is said to be probabilizable; and the set of real numbers, a<y<b, are the Borel set associated with y [3]. Kitt C. Carlton-Wippern, Assistant Professor (Adjoint), Department of Space Science, College of Engineering and Applied Science, University of Colorado at Colorado Springs, USA.
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