The power requirement function simply becomes: PU = R\ Thus, to determine the expected value of this function over a constrained field of view: where lj was previously defined [9]. The above function may then be averaged over the whole orbit in a manner similar to equations III.l and III.2, yielding the average power to illuminate the area in the field of view. We note that if what was required was the average power to illuminate a randomly located target within the field of view, the procedure would have been to determine (I) as in equation III.3. Then letting Pr=P/4, the average evaluated power over the whole orbit would be IV. Analysis and Optimization of Search Patterns for Lost Satellites (Eccentric orbits where e>0.1) Since this paper is primarily concerned with radial orbital statistics, this section will not address circular orbits (i.e., where the orbit eccentricity is zero). The rationale for not addressing lost satellites in circular orbits is that the satellite is equally likely (in time) to be found anywhere within the torroidal generated volume; thus, an optimal volume search strategy may not exist. Consider that the orbital plane of a satellite is well known, and that the orbital parameters semimajor axis (a) and eccentricity (e) are also known (or can be accurately guessed). Assume further that no other orbital parameters about the satellite are known. We wish to develop a schema to search for the satellite in space. What we would like to do is search the smallest feasible volume of space which has the greatest probability of containing the satellite. It seems natural that we define an efficiency factor, such that: We now look for a search pattern which optimizes £ [9]. This can be performed numerically rather easily. One starts with r0 = u(l—e) and incrementing 3r0 throughout the whole orbit so that the total volume of the orbit is covered. Then one increments r0 to the next inner value (r0 = u(l — e) +J) and continues the process until
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