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

The determination of velocities of stars from precise Doppler measurements isdescribed here using relativistic theory of astronomical reference frames so asto determine the Keplerian and post-Keplerian parameters of binary systems. Weapply successive Lorentz transformations and the relativistic equation of lightpropagation to establish the exact treatment of Doppler effect in binarysystems both in special and general relativity theories. As a result, theDoppler shift is a sum of (1) linear in $c^{-1}$ terms, which include theordinary Doppler effect and its variation due to the secular radialacceleration of the binary with respect to observer; (2) terms proportional to$c^{-2}$, which include the contributions from the quadratic Doppler effectcaused by the relative motion of binary star with respect to the Solar system,motion of the particle emitting light and diurnal rotational motion ofobserver, orbital motion of the star around the binary's barycenter, andorbital motion of the Earth; and (3) terms proportional to $c^{-2}$, whichinclude the contributions from redshifts due to gravitational fields of thestar, star's companion, Galaxy, Solar system, and the Earth. Afterparameterization of the binary's orbit we find that the presence ofperiodically changing terms in the Doppler schift enables us disentanglingdifferent terms and measuring, along with the well known Keplerian parametersof the binary, four additional post-Keplerian parameters, including theinclination angle of the binary's orbit, $i$. We briefly discuss feasibility ofpractical implementation of these theoretical results, which crucially dependson further progress in the technique of precision Doppler measurements.Comment: Minor changes, 1 Figure included, submitted to Astrophys.