## The fundamental definition of “radial velocity”

Lund Observatory, Box 43, 22100 Lund, Sweden e-mail: lennart@astro.lu.se; dainis@astro.lu.se

Corresponding author: L. Lindegren, lennart@astro.lu.se

Received:
4
September
2002

Accepted:
31
January
2003

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 and
astrometric 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 the
time coordinate. Relativistic velocity effects and spectroscopic
measurements made inside gravitational fields add further complications,
causing wavelength shifts to depend, e.g., on the transverse velocity
of the object and the gravitational potential at the source.
Aiming at definitions that are unambiguous at accuracy levels of 1 m s^{-1}, we analyse different concepts of radial velocity
and their interrelations. At this accuracy level, a strict separation
must be made between the purely geometric concepts on one hand, and
the spectroscopic measurement on the other. Among the geometric
concepts we define *kinematic radial velocity*, which corresponds
most closely to the “textbook definition” of radial velocity as the
line-of-sight component of space velocity; and *astrometric
radial velocity*, which can be derived from astrometric observations.
Consistent with these definitions, we propose strict definitions also of
the complementary kinematic and astrometric quantities, namely transverse
velocity and proper motion. The kinematic and astrometric radial
velocities depend on the chosen spacetime metric, and are accurately
related by simple coordinate transformations. On the other hand, the
observational quantity that should result from accurate spectroscopic
measurements is the *barycentric radial-velocity measure*.
This is independent of the metric, and to first order
equals the line-of-sight velocity. However, it is not a physical
velocity, and cannot be accurately transformed to a kinematic or
astrometric radial velocity without additional assumptions and
data in modelling the process of light emission from the source,
the transmission of the signal through space, and its recording by
the observer. For historic and practical reasons, the spectroscopic
radial-velocity measure is expressed in velocity units as *cz*_{B}, where *c* is the speed of light and *z*_{B} is
the observed relative wavelength shift reduced to the solar-system
barycentre, at an epoch equal to the barycentric time of light
arrival. The barycentric radial-velocity measure and the astrometric
radial velocity are defined by recent resolutions adopted by the
International Astronomical Union (IAU), the motives and
consequences of which are explained in this paper.

Key words: techniques: radial velocities / techniques: spectroscopic / astrometry / reference systems / stars: kinematics / methods: data analysis

*© ESO, 2003*