APPENDIX B ABERRATION ERROR The formula, for the pointing error of an ARA due to the transverse component, v^, of its velocity relative to the pilot source is obtained from the velocity addition formulas of special relativity as follows. Let the pilot source be at rest at the origin of the X, Y, Z coordinate system (CS) as shown in Figure 11, and let the spacecraft carrying the ARA be at rest at the origin of the X’, Y’, Z’ CS. We assume that the spacecraft, along with its CS, moves at constant velocity v with respect to the pilot source and its CS, and we orient the spacecraft CS so that the X’ axis coincides with the velocity vector []. We also orient the X, Y, Z axes of the pilot source CS parallel to the [] axes, respectively, of the spacecraft CS. Finally, for the sake of brevity, let [] (the derivation for the general case, [], is a straightforward but repetitious extension of the less general one that follows). The angle between the pilot source to spacecraft vector ([]) and the Y axis is denoted by ip. The velocity addition equations relating the velocity, [], of a point in the X, Y, Z (pilot source) CS to its velocity, [] in the [] (spacecraft) CS are [14]:
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