2 Cluster kinematics derived from astrometry

Since an ultimate aim of the present programme is to confront spectroscopic measurements of line shifts in stellar spectra with independent measurements of the stellar motions, it is essential that the kinematic data, including the radial velocities, are derived without using the spectroscopic Doppler effect. Dravins et al. (1999b) describe several methods to derive the radial motion of stars by purely geometric means, i.e. using astrometric data. Of these, the moving-cluster method has been successfully applied to several open clusters and OB associations, in particular the Hyades (Lindegren et al. 2000; Madsen et al. 2002). The principle of the moving-cluster method is very simple: let $\theta$ be the angular size of the cluster and R its distance. Assuming its linear size $R\theta$ to be constant, we have $\dot{R}\theta+R\dot{\theta}=0$, where the dot signifies time derivative. Since R is known from trigonometric parallaxes, the astrometric radial velocity of the cluster follows as $\dot{R}=-R\dot{\theta}/\theta$.

In practice, several kinematic parameters are simultaneously estimated from the astrometric data of the cluster member stars, using the method of maximum likelihood (Lindegren et al. 2000). Some features of the method, relevant for the subsequent discussion, are recalled hereafter.

The estimated parameters include the common space velocity of the cluster ($\vec{v}_0$), the individual stellar parallaxes ($\pi_i$ for star i), and the internal velocity dispersion ( $\sigma_{\rm v}$). The astrometric radial velocity of an individual star i is then calculated as $\widehat{v}_{{\rm r}i}=\vec{r}_i'\widehat{\vec{v}}_0$, where $\vec{r}_i$is the unit vector towards the star and the caret  $\widehat{\phantom{x}}$  signifies estimated quantities. As part of the procedure, improved parallaxes  $\widehat{\pi}_i$ are obtained for the individual stars. In the Hyades, these are 2-5 times more precise than the original Hipparcos parallaxes which have errors around 1-1.5 mas. The improvement results from a combination of trigonometric and kinematic parallaxes, where the latter follow from the proper-motion components along the cluster motion, which are inversely proportional to distance. The kinematically improved parallaxes allow a very precise mapping of the spatial structure of the cluster. The maximum likelihood estimate of  $\sigma_{\rm v}$ is unfortunately biased. Instead the proper motions perpendicular to the cluster motion are used to estimate the velocity dispersion according to the method described in Lindegren et al. (2000), Appendix A.4. For each star, a goodness-of-fit statistic g_i is also obtained from the maximum-likelihood estimation (see Lindegren et al. 2000 for a thorough discussion of g_i). The statistic is primarily used to reject stars whose astrometric data do not fit the cluster model well enough; a rejection limit of $g_{\rm lim}=15$ was normally used, although a stricter limit (10) or no limit at all ($\infty$) were also tried. For the retained stars, the g_i values (which are then $\le g_{\rm lim}$) could be regarded as a quality index, with a lower value indicating a better fit to the cluster model.

The error in the estimated astrometric radial velocity, $\widehat{v}_{{\rm r}i}$, has two parts. The first part is due to the error in the common space motion of the cluster, $\widehat{\vec{v}}_0$. Its uncertainty depends on global properties of the cluster such as its distance, angular extent, and richness, as well as on the accuracy of the astrometric data. The second part is due to the star's peculiar motion relative to the cluster centroid. This part depends only on the dispersion of the peculiar motions along the line of sight, which for a uniform, isotropic velocity dispersion equals  $\sigma_{\rm v}$. In most of the clusters for which the method has been applied, the main uncertainty comes from the first part, i.e. the error in the cluster's space motion. In the Hyades, however, the uncertainty in  $\widehat{\vec{v}}_0$ is small enough (0.36 km s^-1along the line-of-sight; Madsen et al. 2002) that the total uncertainty in the astrometric radial velocities is dominated by the contribution from the internal velocity dispersion (0.49 km s^-1according to the estimate in the same source).

On the other hand, the assumption of a constant and isotropic velocity dispersion throughout the cluster may be rather simplistic. Theoretically, one expects at least a variation with distance r from the centre of the cluster, and possibly also a variation with stellar mass due to the equipartition of kinetic energy. For instance, in a simple Plummer (1915) potential we have

$$\sigma_{\rm v}^2(r) = \frac{GM}{6\sqrt{r_{\rm c}^2+r^2}}$$ (1)

(Gunn et al. 1988; Spitzer 1987), where M is the cluster mass and $r_{\rm c}$ the core radius ($\simeq$3 pc for the Hyades). According to Eq. (1), $\sigma_{\rm v}$ should decrease by one third as one moves two core radii away from the centre, and become even smaller further out in the cluster; but this trend is obviously broken at some distance by tidal forces. Clearly, these effects must be also reflected in the accuracy of the astrometric radial velocities. Attempts to measure the radial variation of dispersion in the Hyades from astrometry were inconclusive (Madsen et al. 2001), but N-body simulations could help to establish to what extent such variations exist in real clusters.