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 a relativistic theory of astronomical reference frames to determine the Keplerian and post-Keplerian parameters of binary systems. Seven reference frames are introduced: (1) the proper frame of a particle emitting light, (2) the star-centered reference frame, (3) the barycentric frame of the binary, (4) the barycentric frame of the Galaxy, (5) the barycentric frame of the solar system, (6) the geocentric frame, and (7) the 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 to the solar system, the motion of the particle emitting light and diurnal rotational motion of observer, orbital motion of the star around the binary's barycenter, and the orbital motion of the Earth; and (3) terms proportional to c-2, which include the contributions from redshifts due to fields of the star, the star's companion, the Galaxy, the solar system, and the Earth. After parameterization of the binary's orbit, we find that the presence of periodically changing terms in the Doppler shift enables us to disentangle different terms and measure, along with the well-known Keplerian parameters of the binary, four additional post-Keplerian parameters, which characterize (1) the relativistic advance of the periastron; (2) a combination of the quadratic Doppler and shifts associated with the orbital motion of the primary relative to the binary's barycenter and the companion's 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 briefly discuss the feasibility of practical implementation of these theoretical results, which crucially depends on further progress in the technique of precision Doppler measurements.