The 1D Relativistic Doppler Formula is an Incorrect Approximation in Precise Radial Velocity Work

Abstract

Stellar Doppler velocimetry determines a star's radial velocity $v_r$ via measurement of a redshift, $z$. At precisions below 10 m s$^{-1}$ conversion between the two quantities is complex, and care must be taken to properly account for the effects of relativity. One particular aspect of the problem that bears repeating is that the one-dimensional version of the relativistic Doppler formula, which does not distinguish between the motion of the source and the observer, is incorrect in this context, and indeed does not even provide the correct coefficient for variations in the second-order terms involving $\beta$. Nonetheless, it is often useful to report a redshift in the more familiar units of velocity without a rigorous calculation, and much code already exists that does this. In these cases it is important to clearly document which formula is being used, and I recommend simply (and explicitly) using the approximation $v_r \approx cz$. This choice is trivially inverted, does not misrepresent the degree of relativistic rigor that has been applied in translating between redshift and radial velocity, and is, I believe, the most commonly followed convention in astronomy and cosmology. I also briefly discuss the differences between the Wright & Eastman barycentric correction procedure and the Lindegren & Dravins barycentric radial velocity measure.