## 5. Hyades: Results from Hipparcos dataIn this section we briefly discuss the application of the procedures described in the previous sections to the actual Hipparcos observations of the Hyades cluster. As the purpose is not to make an in-depth study of the cluster kinematics, we restrict the application to the basic cluster model. ## 5.1. The formal maximum-likelihood solutionStarting from the sample Hy0 defined in Sect. 4, we made a succession of solutions for decreasing rejection threshold . Results for the common cluster parameters and , with formal error estimates, are given in the upper part of our Table 3. The corresponding estimates of the centroid radial velocity (Eq. A.16) are also given. As expected, the estimated dispersion decreases with , while the solution for is relatively stable after the first few (worst) outliers have been removed. The formal errors also decrease in the sequence of solutions, but as discussed in Sect. 4 this does not necessarily reflect the true uncertainties.
The rapid decrease in from
to 20 (with
Part of the large dispersion in the full Hy0 sample may be caused by double and multiple stars, in particular astrometric binaries with deviating proper motions for the centre of light. Many of the stars in the sample Hy0 are actually known visual or spectroscopic binaries, and several more were indicated in the Hipparcos Catalogue as possible astrometric binaries or suspected resolved systems. The most doubtful cases are those flagged in columns s or u in Table 3 of Perryman et al. (1998), and those visual binaries having both a separation less than 20 arcsec and a magnitude difference less than 4 mag. Removing these stars results in a list of 120 a priori `clean' Hyades members (sample `Hy1'). Solutions starting from this sample are shown in the lower part of Table 3. The results for the Hy1 sample are not significantly different from those of Hy0, except for a smaller velocity dispersion for in the range 15 to 30 (where the selection in Hy1 remains practically the same). This reduction in is probably real and caused by a smaller proportion of astrometric binaries in sample Hy1. The Monte Carlo experiments described in Sect. 4.2 suggest that a cut-off limit around might be optimal. Since Hy0 and Hy1 give rather consistent results at this limit, we adopt as the preferred solution the one retaining the larger number of stars, i.e. the one (Hy0, ) marked with a box in Table 3. In that solution the formal covariance matrix for the three parameters , , and (the upper-left part of in Eq. A.17) is which however should be multiplied by
to give the actual uncertainties
according to Sect. 5.2. We note that the principal axes of the
error ellipsoid are nearly aligned (within
) with the triad
defined in Appendix A.4
(footnote): the longest axis (corresponding to a velocity uncertainty
of 0.37 km s ## 5.2. Velocity dispersion and error calibrationIf the proper-motion residuals from the adopted solution in
Table 3 are analysed as described in Sect. 4.3 we find an
rms velocity dispersion of
km s As previously mentioned, the distribution of the goodness-of-fit
statistics in Fig. 5 indicates
a non-Gaussian velocity dispersion with a spread in
of the order of
. This is however for the original
sample of 197 stars, i.e. before the rejection procedure, and the
situation may be very different for the final sample of 168 stars. We
have investigated the distribution of proper-motion residuals in that
sample in order to see if there is evidence for a deviation from a
Gaussian velocity dispersion. To this end we generated synthetic
samples with Gaussian and other distributions, added random
observational errors, and compared the resulting distributions with
the observed distribution. Based on the Kolmogorov-Smirnov test (Press
et al. 1992), a purely Gaussian velocity dispersion cannot be ruled
out. However, a visibly much better agreement was obtained by assuming
a log-normal dispersion with median value
km s ## 5.3. Cluster expansionThe adopted solution does not take into account effects of a
possible systematic velocity pattern within the Hyades cluster. In
particular, it assumes that there is no net expansion or contraction
of the cluster (). In Paper I it
was shown that cluster expansion will bias the astrometric
radial-velocity estimate by , where
is the distance to the star and
the expansion rate. If the expansion
rate of the Hyades cluster equals the inverse cluster age, then
km s ## 5.4. Spatial correlationsOne further approximation in the present method needs to be
discussed. It concerns our neglecting the possible correlations among
the astrometric data for different stars [Eq. (9)]. It is well
known that the observational technique used by Hipparcos tends to give
positive correlations among the data for stars within an area of the
sky comparable with the instrument's field of view,
(Lindegren 1988; ESA 1997,
Vol. 3, Ch. 16-17). The degree of correlation in the
Hipparcos data and the extent to which it affects e.g. the
determination of cluster distances is however controversial (Narayanan
& Gould 1999b; van Leeuwen 1999a). In the case of the Hyades
cluster, the parallax residuals from the adopted ML solution make it
possible to obtain a rough estimate of the spatial correlations, since
these residuals are dominated by the parallax errors in the Hipparcos
Catalogue. We made a correlation analysis of the normalised residuals
as function of the angular
separation between stars. The normalisation was made in order to get
roughly equal weight to the residuals. For pairs with separation in
the intervals 0-0.5 Unless these correlations are included in the observation model, or
in the simulations, it is difficult to assess how they would affect
the solution. Probably, the main effect of their omission is that the
errors of the solution are underestimated, similar to the case for the
mean parallax of a cluster (Lindegren 1988). However, the
moving-cluster method depends on the determination of proper-motion
A more complete treatment of the correlations will in practice require the present ML solution to be re-formulated along the principles described by van Leeuwen & Evans (1998), i.e. by treating the Hipparcos Intermediate Astrometric Data (ESA 1997, Vol. 17, Disk 5) as (correlated) observations, instead of the catalogued parallax and proper-motion values. Such an exercise is however beyond the scope of this paper. © European Southern Observatory (ESO) 2000 Online publication: April 17, 2000 |