"RADIAL-VELOCITY  MEASURE"

A stringent definition for spectroscopic measurements


Observations of wavelength or frequency shifts in astronomical spectra permit to analyze various processes causing these shifts.  For high-accuracy studies, the fundamental concept of "radial velocity" must be precisely defined, in particular with respect to relativistic velocity effects and measurements made inside gravitational fields.

   Why the need for a stringent definition?

  Spectroscopic measurement precisions are now reaching (and exceeding) levels of meters per second.  However, irrespective of measurement precision, such values cannot easily be intercompared or transferred to "absolute" numbers, simply because there does not exist any sufficiently stringent definition of "radial velocity" as such.

  Naively, "radial velocity" equals the line-of-sight component of the stellar velocity vector, measured by the apparent Doppler shifts of, e.g., stellar spectral lines.  Although many physical effects in stellar atmospheres contribute to the line shifts, those could in principle be corrected for, leaving the "true" (center-of-mass) velocity.  However, also this concept becomes ambiguous at accuracy levels around 10 or 100 m/s.

  For example, radial velocity may be defined as the rate of change in "distance" with respect to "time".  But is this the time of light emission (at the object) or light reception (at the observer)?  The former seems natural if radial velocity is considered a "property" of the star, while the latter is more natural for the observer.  The finite speed of light, c, causes a difference of second order in velocity (V2/c), exceeding 100 m/s for V > 173 km/s.  The naive notion of "radial velocity" as the line-of-sight component of the stellar velocity vector is thus ambiguous already in a classical (non-relativistic) formulation.

  Similar differences exist between the classical and the relativistic Doppler formulae, and depend on how the second-order (including the transverse) Doppler effect is treated.  Thus, the determination of the radial velocity component cannot be separated from the determination of the transverse one, requiring knowledge also of the object's proper motion, and distance. The inclusion (or not) of terms representing the transverse Doppler shift can lead to differences of 100 m/s for stars moving at 250 km/s relative to the Sun.

  In a framework of special or general relativity, the observed shift depends on additional factors, such as the gravitational potential of the source and, ultimately, the cosmological redshift.  The Hubble parameter is around 70 km/s/Mpc = 70 m/s/kpc.  To what extent does also the local space to nearby stars take part in the general expansion of the Universe?  What is an actual "velocity", and what is a change of spatial coordinates?  Since such factors are generally not known to the spectroscopic observer, it is impossible to convert the observed shift into a precise kinematic quantity.

  The gravitational redshifts in the Hertzsprung-Russell diagram change by three orders of magnitude between white dwarfs (some 30 km/s) and supergiants (some 30 m/s).  For the Sun, the shift is 636.1 m/s for light escaping from the solar photosphere to infinity, and 633 m/s for light intercepted at the Earth: the Earth's location inside the solar gravitational potential blueshifts stellar photons by 3 m/s (and this quantity changes slightly during the year, due to the eccentricity of the Earth's orbit).

  The prospect of measuring spectroscopic radial velocities at gravitational potentials other than that at the Earth, is becoming very real.  Locations that have been discussed for future space telescopes include orbits outside the asteroid belt, in order to decrease the zodiacal-light contamination.  There, at perhaps 600 solar radii (3 AU) from the Sun, the solar redshift has decreased to 1 m/s.  All objects, when observed from such a space telescope, will thus show a redshift of some 2 m/s, relative to measurements made near the Earth.

  Also concepts such as "line of sight" need specification: from exactly which vantage point should radial velocities be defined?   As seen from the Earth, the radial component of any stellar velocity vector changes during the year, as the exact angle under which this vector is observed, changes due to the parallax, when viewed from different positions along the Earth's orbit.


   The stringent definition of "barycentric radial-velocity measure"

  To enable high-accuracy studies of radial velocities; to permit accurate comparisons between different observers, and to allow high-accuracy studies of various physical processes which affect astronomical wavelengths and frequencies, a resolution was adopted by a number of Divisions and Committees of the International Astronomical Union, at a special session during its XXIVth General Assembly in Manchester (August 2000).  The adopted resolution defines a stringent "barycentric radial-velocity measure" which, in principle, permits all types of spectroscopic observations to be placed on a common scale with an accuracy around 1 m/s, or perhaps better.  (An improvement by some two orders of magnitude relative to the past, where the ambiguities could typically reach 100 m/s.)

  The examples above show that spectroscopic measurements alone cannot very accurately determine the radial motion of an object.  What can be derived from radial-velocity measurements is the wavelength shift zB reduced to the solar system barycenter.  (Different observations should be standardized through transformation to a fictitious observer located there, but unaffected by the gravitational field of the solar system.)  For convenience, the shift can be expressed in velocity units as czB, analogous to the case in cosmology.  The epoch of any observation is naturally defined as the corresponding barycentric time of light arrival.

  This quantity czB is conceptually well-defined, in contrast to the usual Vrad.  Although it approximately corresponds to "radial velocity", its precise interpretation is model-dependent and one should therefore avoid calling it just "radial velocity".  The term "barycentric radial-velocity measure" was adopted for czB, emphasizing both its connection with the traditional spectroscopic method and the fact that it is not quite the radial velocity in the usual sense.

  The spectroscopic "barycentric radial-velocity measure" is thus defined as czB, where c is the speed of light and zB is the observed relative wavelength shift reduced to the solar system barycenter.  To first order, czB equals the line-of-sight velocity, but its precise interpretation is model dependent.  The index B for barycentric signifies that - in contrast to the case in cosmology - the velocity is referred to the solar-system barycenter, not the cosmological microwave background.

  The text of the formal definition follows, as adopted at the IAU XXIVth General Assembly in Manchester:


   IAU definition of "barycentric radial-velocity measure"

Resolution No. C1  on the Definition of a Spectroscopic "Barycentric Radial-Velocity Measure"

Divisions I, IV, V, VI, VII, IX and X, and Commissions 8, 27, 29, 30, 31, 33, 34 and 40 of the International Astronomical Union

Recognizing

1. That recently improved techniques for determining radial velocities in stars and other objects, reaching and exceeding precision levels of meters per second, require the definition of "radial velocity" to be examined;

2. That, due to relativistic effects, measurements being made inside gravitational fields, and alternative choices of coordinate frames, the naive concept of radial velocity being equal to the time derivative of distance, becomes ambiguous at accuracy levels around 100 m/s;

Considering

1. That, although many effects may influence the precise shifts of spectroscopic wavelengths and frequencies, only local ones (i.e. arising within the solar system, and depending on the gravitational potential of the observer, and the observer's position and motion relative to the solar-system barycenter) can in general be reliably calculated;

2. That, although the wavelength displacement (or frequency shift) corrected for such local effects can thus be derived from spectroscopic measurements, the resulting quantity cannot unambiguously be interpreted as a radial motion of the object;

Therefore recommend

That, whenever radial velocities are considered to a high accuracy, the spectroscopic result from a measurement of shifts in wavelength or frequency be given as the "barycentric radial-velocity measure" czB, after correcting for gravitational effects caused by solar-system objects, and effects by the observer's displacement and motion relative to the solar-system barycenter.

Here, c equals the conventional speed of light = 299,792,458 m/s, and zB = (lambda - lambda0)/lambda0, where lambda0 is the rest-frame wavelength and lambda the wavelength observed by a hypothetical observer at zero gravitational potential, located at, and being at rest with respect to, the solar-system barycenter.  The epoch of the observation equals the barycentric time of light arrival.

The radial-velocity measure czB is expressed in velocity units: to first order in zB it coincides with the classical concept of "radial velocity", while avoiding the implicit interpretation as physical motion.  The solar-system barycenter is defined by Resolution A4 adopted at the IAU XXI:th General Assembly in 1991, and supplemented by Resolution B6 at the IAU XXIII:th General Assembly in 1997.

(end of resolution text)


   Background comments

  The definition implies that high-accuracy radial-velocity observations should be reduced to the solar-system barycenter according to procedures based on general relativity and using constants and ephemerides consistent with the required accuracy.

  The definition avoids any discussion on what the "true" radial velocity of the object would be.  The transformation between the spectroscopically determined barycentric radial-velocity measure czB and the physical velocity of the object depends, e.g., on the metric used to express the velocity, and on factors which may be (partly) unknown, such as the gravitational conditions outside the solar system and the object's transverse velocity.  The interpretation of czB in terms of radial motion is thus model-dependent and cannot be treated in isolation from, e.g., the tangential motion.

  The radial-velocity measure may be different for different spectral lines in the spectrum of the same object.  For example, lines that are formed in different convective layers of a stellar atmosphere usually show different amounts of convective lineshifts; lines formed at different radial distances show different gravitational redshifts, etc.  To accurately obtain the actual radial motion of the object will, in principle, require a modeling of the object's motion, together with the emission of the radiation, and its subsequent propagation to the observer.

  The speed of light (c = exactly 299,792,458 m/s) is introduced in the expression for czB, only for the purpose of converting the dimensionless measure zB into to a convenient quantity that conforms with the "ordinary" radial velocity.  The radial-velocity measure therefore obtains physical dimensions of SI meters per SI second.  (However, when appropriate, one could specify only the measure zB, as now often the case when considering large cosmological velocities.)

  The definition leaves "uncorrected" all the (largely unknown) effects originating from outside the solar system.  These include shifts related to the gravitational potential of the Milky Way Galaxy (these shifts may be comparable to the solar gravitational redshift, but depend on the [unknown] distribution of masses within and outside our Galaxy), gravitational lensing (causing time-varying relativistic time delay) and cosmological effects (the extent to which the "local" space partakes in the general expansion of the Universe is a non-trivial question).

  For work at modest accuracies, the proposed definition implies no change of existing procedures, nor of any published radial-velocity values.  The use of the "barycentric radial-velocity measure" will only be required when absolute accuracies on the sub-km/s are needed.


   The definition of "astrometric radial velocity"

  The above discussion only concerns spectroscopically determined quantities.  It is also possible to determine radial velocity through purely geometric methods, i.e. using astrometry.  Since those deduced values depend on the conventions for the coordinate frames or timescales chosen, another stringent definition of astrometric radial velocity has been adopted.


   Main publication

  Lennart Lindegren &  Dainis Dravins: The fundamental definition of  'radial velocity'
Astron.Astrophys. 401,  1185-1201 (2003), [PDF, 280 kb]


Comments are welcome to Dainis Dravins,  dainis@astro.lu.se, and to Lennart Lindegren,  lennart@astro.lu.se

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