For any star, its component of radial motion may become visible as a secular change in its parallax and proper motion. Although the theoretical possibility of thus deducing radial velocities was realized long ago, only Hipparcos space astrometry reached an accuracy level permitting a meaningful search for differences to spectroscopic velocities. There exist different ways to astrometrically determine stellar radial motion:
(A) Changing trigonometric parallax: This very direct method requires extremely accurate measurements, e.g., even for Barnard's star the expected parallax rate is only 34 microarcsec/yr.
(B) Proper motion varies as the angle to the space-velocity vector changes. Combining Hipparcos data with old stellar positions reaches accuracies around 10 km/s.
(C) Observations of the changing angular separation of stars sharing the same space velocity (e.g., in a moving cluster), yields high accuracies (0.4 km/s) already with Hipparcos data. (D.Dravins, L.Lindegren & S.Madsen: Astrometric Radial Velocities. I. Non-Spectroscopic Methods for Measuring Stellar Radial Velocity; A&A 348, 1040, 1999 )
Moving clusters, whose stars are grouped around the same average velocity vector, offer a special possibility to determine astrometric radial velocities. As the cluster moves in the radial direction, it appears to contract or expand at a rate equal to the relative rate of change in distance. This can be converted to a linear velocity (in km/s) if the stellar distances are known, e.g., from trigonometric parallaxes.
Figure: The moving-cluster method for determining astrometric radial velocities. Stars in open moving clusters share the same velocity vector. Parallaxes give the distance, while proper-motion vectors show the fractional change with time of the cluster's angular size. The latter equals the time derivative of distance, yielding the radial velocity.
In essence, this inverts the classical problem of determining the cluster distance from proper motions and radial velocities, which (due to geometric projection effects) change across the angle subtended by the cluster. With the distance accurately known, one solves the problem for the radial velocity instead. A stringent application of the method, however, demands a careful consideration of the stellar dynamics in the cluster, possible cluster expansion, and other effects. (L.Lindegren, S.Madsen & D.Dravins: Astrometric Radial Velocities. II. Maximum-Likelihood Estimation of Radial Velocities in Moving Clusters, A&A 356, 1119, 2000)
The application of Hipparcos data to the Ursa Major, Hyades, Coma Berenices, Pleiades, and Praesepe clusters, and for the Scorpius-Centaurus, Alpha Persei, and 'HIP 98321' associations has given astrometric radial velocities for more than 1,000 stars. The radial motion of the Hyades cluster is determined to within 0.4 km s-1 (standard error), and that of its individual stars to within 0.6 km s-1. For other clusters, typical accuracies are a few km s-1. A comparison of these astrometric values with spectroscopic values shows a good general agreement and, in the case of the best-determined Hyades cluster, also permits searches for subtle astrophysical differences, such as evidence for enhanced convective blueshifts of F-dwarf spectra, and decreased gravitational redshifts in giants.
Figure: Map of the full sky, showing those stars in clusters and associations, whose radial velocities were astrometrically determined from Hipparcos data. Symbol shape identifies different clusters; symbol size denotes apparent magnitude Hp , while symbol shading denotes B-V (note how some clusters are dominated by very blue stars). The Aitoff projection in equatorial coordinates is used.
Figure: Proper motions of the program stars over 200,000 years. Best radial-velocity accuracy is obtained in rich nearby clusters with large angular extent, and large proper motions. However, the accuracy in the largest associations (Ursa Major, Scorpius-Centaurus) is limited by the partly unknown expansion of these systems. Stellar paths in the Ursa Major group (green) cover large areas of the sky. The thickness of the proper-motion vectors is inversely proportional to stellar distance: the closest star is Sirius and the two next ones are faint red dwarfs. Proper motions vary greatly among different clusters. (S.Madsen, D.Dravins & L.Lindegren: Astrometric Radial Velocities. III. Hipparcos Measurements of Nearby Star Clusters and Associations, A&A 381, 446, 2002 )
Differences between astrometric and spectroscopic velocities permit to better understand physical processes on stellar surfaces (convective lineshifts); stellar interior structure (gravitational redshifts); and to understand which spectral features are the best sensors for the true stellar center-of-mass motion and/or its variations, e.g. for studying small internal motions in star clusters, or detecting the slight variations induced by exoplanets.
Figure: Systematic differences between spectroscopic radial velocities, and true stellar motions. The plot shows the differences between spectroscopic velocities in the Hyades (mainly from Gunn et al., AJ 96, 198, 1988), and astrometric determinations. An increased blueshift of spectral lines in stars somewhat hotter than the Sun (B-V approx. 0.3-0.5) is theoretically predicted from hydrodynamic models (Dravins & Nordlund, A&A 228, 203, 1990) due to their more vigorous surface convection, causing greater convective blueshifts. Gravitational redshifts of white-dwarf spectra place them far off main-sequence stars. (S.Madsen, D.Dravins & L.Lindegren: Astrometric Radial Velocities. III. Hipparcos Measurements of Nearby Star Clusters and Associations, A&A 381, 446, 2002)
A stringent definition of 'astrometric radial velocity'
The radial velocity determined through purely geometric methods does, in principle, depend on the coordinate frames and timescales chosen, why there is a need for a stringent definition for the fundamental concept of "astrometric radial velocity". A resolution for its stringent definition was adopted at the IAU XXIV:th General Assembly in Manchester.
Effects due to relativity and gravity influence the wavelength displacements of stellar spectral lines. The formula is for the weak-field post-Newtonian approximation, neglecting higher-order terms of order 1/c3.
(Lennart Lindegren & Dainis Dravins: The fundamental definition of 'radial velocity', Astron.Astrophys. 401, 1185-1201, 2003 )
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