Stellar Aberration and Parallax: A Tutorial
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Stellar Aberration and Parallax: A Tutorial Malcolm D. Shuster! Abstract Formulas for the observed distortions of star directions due to the motion of the spacecraft are developed within a framework suited to attitude determination activities. In particular, the expressions for these distortions are given in terms of a direction-cosine matrix and a rotation vector. Different mechanizations of the stellar aberration and parallax effects and their correction are discussed, as are details and trade-offs in the implementation of the algorithms in attitude determination.
Introduction The use of star cameras' in spacecraft attitude determination requires that account be taken of the effect of the motion of the spacecraft on measured star directions. To ignore such effects for Earth-orbiting spacecraft is to introduce errors as large as 26 arc seconds into the attitude estimate. Unfortunately, while most threeaxis attitude-determination methods nowadays require vector inputs, the effect on these vectors from the aberration of starlight and of parallax are usually presented in books on Observational Astronomy [1] in terms of scalar quantities, for which the correct interpretation of signs is not always transparent. Also, the correct derivation of stellar aberration is possible only within the framework of Special Relativity, which requires a familiarity with Physics usually foreign to aerospace engineers. A purely Newtonian computation of the aberration of starlight will, in fact, yield the same result (to order vic) as Special Relativity. The classical Newtonian result has been known since 1727 [2]. A Newtonian derivation appears also in the Explanatory Supplement to the Astronautical Almanac' [3]. Such an approach,
'Director of Research, Acme Spacecraft Company, 13017 Wisteria Drive, Box 328, Germantown, Maryland 20874. email: [email protected]. 2The term "star camera" denotes here any sensor which measures the direction of a star. 3The Explanatory Supplement to the Astronautical Almanac gives both Newtonian and relativistic derivations, both of which are too short to be really useful explanatorily.
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unfortunately, rests on intermediate steps which contain physical errors as large as the stellar aberration effect itself. For example, while Newtonian Mechanics gives the correct direction of the velocity for the aberrant photon, it also gives a value for the magnitude of that velocity which can be wrong (fractionally) by the same amount as the real effect on the direction. This writer does not wish to present incorrect intermediate results simply because one final result happens to agree with the correctly derived expression. Such an approach in 1727 can be excused but not the application of that approach a century after the development of Special Relativity. Some physicists may even argue that the correct calculation requires General Relativity. However, the additional improvements due to General Relativity are of order V 2/C2 , which can be ignored in attitude if not in orbit work. The present w
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