Post‐Newtonian Theory for Precision Doppler Measurements of Binary Star Orbits

Abstract

The determination of velocities of stars from precise Doppler measurements is described here using relativistic theory of astronomical reference frames so as to determine the Keplerian and post-Keplerian parameters of binary systems. Seven reference frames are introduced: (i) proper frame of a particle emitting light, (ii) the star-centered reference frame, (iii) barycentric frame of the binary, (iv) barycentric frame of the Galaxy, (v) barycentric frame of the Solar system, (vi) geocentric frame, and (vii) topocentric frame of observer at the Earth. We apply successive Lorentz transformations and the relativistic equation of light propagation to establish the exact treatment of Doppler effect in binary systems both in special and general relativity theories. As a result, the Doppler shift is a sum of (1) linear in c−1 terms, which include the ordinary Doppler effect and its variation due to the secular radial acceleration of the binary with respect to observer; (2) terms proportional to c−2, which include the contributions from the quadratic Doppler effect caused by the relative motion of binary star with respect