Stellar and planetary aberration.

A way of looking at the topics of stellar and planetary aberration is suggested that enables them to be viewed as closely related--both being conceived as dependent on source-detector relative velocity. We show that the long-range limiting case, stellar aberration, has certain uniquely subtle aspects, as well as some hypothesized characteristics that lend themselves to empirical testing.

Introduction

For some time the writer has been seeking improved understanding of the several kinds of aberration--these being rather superficially explained in most texts, particularly texts of relativity theory. Indeed, in some of the more highly regarded of these (e.g., Moller 1972) the phenomenon of stellar aberration is so slightingly treated that use of the formulas provided would literally yield wrong answers. During much of this investigation it seemed that the fault lay with special relativity theory or with the Lorentz transformation. This may still, or may not, be the case. The reader will have to judge. Interest particularly in stellar aberration has increased of late [cf., Ives 1950, Eisner 1967, Phipps 1989, Hayden 1993, etc.] With help from recent studies by Marmet (1994) and Sherwin (1993) it has become apparent to the writer that with suitable interpretation relativity theory can accommodate both principal types of optical aberration, planetary and stellar. The purpose of the present paper is to confirm this assertion, to exhibit stellar aberration as (in a sense) a limiting case of planetary aberration, to seek a physical model of the phenomenon, to describe observations that could be made to test the model, and to touch on residual aberration issues that remain unsettled in the context of relativity theory.

1. Aberration via Lorentz Transformation

First, let us deal with the mathematics--which is the easy part--so that we can get on to the real problem, which (as so often happens in physics) is interpretation. According to the special theory of relativity (Einstein 1905) both the Doppler effect and stellar aberration can be treated by a Lorentz transformation of the four-vector of light propagation,

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Inertial system K' moves with velocity v relatively to system K along the direction of their common x-axes. We consider both K and K' to be for the moment arbitrary inertial systems. Propagation vectors (wave-normal or ray-path vectors) describing the same given light beam are k' and k in K' and K, respectively. Applying a Lorentz transformation, we have

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From this it is verifiable that a directional turning of the k-vector, k'/k' [not equal to] k/k, is an unavoidable consequence of the Lorentz transformation, except in the special case of parallelism of the v- and k-vectors.

To specialize for simplicity to a specific model, let the x-direction (parallel to earth's orbital motion), in our "laboratory" (observatory) system K, lie in the horizontal plane of the ecliptic; let the z-axis lie in the vertical, defining the zenith direction; and (to eliminate diurnal effects) consider our telescope to be located at the north pole of a nonrotating earth, so that the pole of the ecliptic lies at the zenith. To treat stellar aberration, consider a star emitting light described by the k-vector to lie in the x-z plane at angle [alpha] measured from the horizontal. Then [k.sub.x] = kcos[alpha] = [??][omega]/c[??]cos[alpha]. In K' the corresponding angle [alpha]' obeys [k.sub.x] = [??][omega]'/c[??]cos[alpha]'. With these substitutions it follows from the first of the above Lorentz transformation equations that [??][omega]/[omega]'[??]cos[alpha]' = [gamma][??]cos[alpha] + [v/c][??] and from the last of them that [omega]'/[omega] = [gamma][??]1 + [v/c]cos[alpha][??]. On taking the quotient of these equations we obtain the well-known relation (Synge 1965)

(1a) cos[alpha]' = [cos[alpha] + [v/c]]/1 + [v/c]cos[alpha],

or its dual (symmetric with respect to interchange of K, K', with sign change of relative velocity),

(1b) cos[alpha] = cos[alpha]' - [v/c]]/1 - [v/c]cos[alpha]'.

Subtracting cos[alpha] from both sides of (1a), using a standard trigonometric identity (Peirce 1957), and introducing an aberration angle [epsilon] = [alpha] - [alpha], we obtain

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where [beta] = v/c. Observing that [epsilon] = O([beta]), we can expand the left-hand side of (2) as a power series in [epsilon],

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Substituting [epsilon] = a[beta] + b[[beta].sup.2] + c[[beta].sup.3] + O([[beta].sup.4]), where a,b,c are unknown coefficients, and equating the resulting [beta] - power series to the series expansion of the right-hand side of (2) in powers of [beta],

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we obtain, upon equating coefficients of successive powers of [beta], a…