The fundamental definition of "radial velocity''

Accuracy levels of metres per second require the fundamental concept of``radial velocity'' for stars and other distant objects to be examined,both as a physical velocity, and as measured by spectroscopic andastrometric techniques. Already in a classical (non-relativistic)framework the line-of-sight velocity component is an ambiguous concept,depending on whether, e.g., the time of light emission (at the object)or that of light detection (by the observer) is used for recording thetime coordinate. Relativistic velocity effects and spectroscopicmeasurements made inside gravitational fields add further complications,causing wavelength shifts to depend, e.g., on the transverse velocity ofthe object and the gravitational potential at the source. Aiming atdefinitions that are unambiguous at accuracy levels of 1 ms-1, we analyse different concepts of radial velocity andtheir interrelations. At this accuracy level, a strict separation mustbe made between the purely geometric concepts on one hand, and thespectroscopic measurement on the other. Among the geometric concepts wedefine kinematic radial velocity, which corresponds most closely to the``textbook definition'' of radial velocity as the line-of-sightcomponent of space velocity; and astrometric radial velocity, which canbe derived from astrometric observations. Consistent with thesedefinitions, we propose strict definitions also of the complementarykinematic and astrometric quantities, namely transverse velocity andproper motion. The kinematic and astrometric radial velocities depend onthe chosen spacetime metric, and are accurately related by simplecoordinate transformations. On the other hand, the observationalquantity that should result from accurate spectroscopic measurements isthe barycentric radial-velocity measure. This is independent of themetric, and to first order equals the line-of-sight velocity. However,it is not a physical velocity, and cannot be accurately transformed to akinematic or astrometric radial velocity without additional assumptionsand data in modelling the process of light emission from the source, thetransmission of the signal through space, and its recording by theobserver. For historic and practical reasons, the spectroscopicradial-velocity measure is expressed in velocity units asczB, where c is the speed of light and zB is theobserved relative wavelength shift reduced to the solar-systembarycentre, at an epoch equal to the barycentric time of light arrival.The barycentric radial-velocity measure and the astrometric radialvelocity are defined by recent resolutions adopted by the InternationalAstronomical Union (IAU), the motives and consequences of which areexplained in this paper