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Astron. Astrophys. 356, 1119-1135 (2000)

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6. Conclusions

We have developed and implemented a rigorous maximum-likelihood algorithm for estimating the kinematic parameters of an open cluster or association from astrometric data, i.e. without using spectroscopic information. The cluster parameters comprise, in the basic model, the individual distances to the stars, the centroid space velocity vector and the internal velocity dispersion. Extended models include cluster rotation and other components of a first-order systematic velocity field, except the isotropic dilation (expansion or contraction) which cannot be determined purely by astrometry. From the estimated cluster parameters, the radial velocities of the individual stars are obtained independent of spectroscopic data. The solution also provides kinematically improved parallaxes for the individual stars.

Extensive Monte Carlo simulations have been used to test the ML algorithm and its statistical properties. The main conclusions are (1) that the basic kinematic parameters are estimated without significant bias, except for the internal velocity dispersion, which is normally underestimated; (2) that a sequential rejection procedure based on the goodness-of-fit statistics [FORMULA] is effective in eliminating otherwise contaminating outliers such as caused by astrometric binaries or non-member stars; (3) that a posteriori analysis of the proper motion residuals yields an improved (nearly unbiased) estimate of the internal velocity dispersion; and (4) that Monte Carlo simulations are essential for validating the model assumptions and for providing realistic confidence limits for the estimated parameters.

In this paper the method has been applied to the Hyades. On the assumption that the cluster has no net expansion or contraction, we find the centroid space velocity [FORMULA] km s-1, where the uncertainties have been corrected by the factor 1.28 derived in Sect. 5.2. The covariance of this estimate is given by Eq. (23) multiplied by [FORMULA]. The internal velocity dispersion for the adopted sample of 168 stars is estimated at [FORMULA] km s-1, although the distribution of proper-motion residuals suggests a slightly non-Gaussian velocity distribution. This could be characterised as a log-normal spread of dispersions having a median value [FORMULA] km s-1 and [FORMULA]. The resulting astrometric radial velocities have a (position-dependent) uncertainty of about 0.68 km s-1 for the individual values and 0.47 km s-1 for the centroid radial velocity. The greater errors for the individual radial velocities arise from quadratically adding the radial uncertainty of the stellar peculiar velocities (0.49 km s-1) to that of the projected centroid velocity. If known binaries are removed, a smaller dispersion of 0.37 km s-1 is found, corresponding to a standard error of 0.60 km s-1 for the individual radial velocities.

The distributions of [FORMULA] values (Fig. 5) contain evidence for a much more strongly non-Gaussian velocity distribution in the original sample of 197 stars. Possible effects of internal velocity fields are further addressed in the subsequent Paper III, in which this method will be applied also to other nearby clusters and associations.

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© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000