Post-Newtonian theory for precision Doppler measurements of binary star orbits

The determination of velocities of stars from precise Doppler measurements is described here using a relativistic theory of astronomical reference fra mes to determine the Keplerian and post-Keplerian parameters of binary syst ems. Seven reference frames are introduced: (1) the proper frame of a parti cle emitting light, (2) the star-centered reference frame, (3) the barycent ric frame of the binary, (4) the barycentric frame of the Galaxy, (5) the b arycentric frame of the solar system, (6) the geocentric frame, and (7) the topocentric frame of observer at the Earth. We apply successive Lorentz tr ansformations and the relativistic equation of light propagation to establi sh 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-l terms, which include the ordinary Doppler effect and its variation due to the secular radial acceleration of the binary with respec t to observer; (2) terms proportional to c(-2), which include the contribut ions from the quadratic Doppler effect caused by the relative motion of bin ary star with respect to the solar system, the motion of the particle emitt ing light and diurnal rotational motion of observer, orbital motion of the star around the binary's barycenter, and the orbital motion of the Earth; a nd (3) terms proportional to c-2, which include the contributions from reds hifts due to gravitational fields of the star, the star's companion, the Ga laxy, the solar system, and the Earth. After parameterization of the binary 's orbit, we find that the presence of periodically changing terms in the D oppler shift enables us to disentangle different terms and measure, along w ith the well-known Keplerian parameters of the binary, four additional post -Keplerian parameters, which characterize (1) the relativistic advance of t he periastron; (2) a combination of the quadratic Doppler and gravitational shifts associated with the orbital motion of the primary relative to the b inary's barycenter and the companion's gravitational field, respectively; ( 3) the amplitude of the "gravitational lensing" contribution to the Doppler shift; and (4) the usual inclination angle of the binary's orbit, i. We br iefly discuss the feasibility of practical implementation of these theoreti cal results, which crucially depends on further progress in the technique o f precision Doppler measurements.