5. Hyades: Results from Hipparcos data
In 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 solution
Starting 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.
Table 3. Results obtained with real Hipparcos data for the Hyades cluster. ML estimates were calculated, starting both from the full sample (Hy0, 197 stars) and from the reduced sample with only bona fide single stars (Hy1, 120 stars). The basic cluster model with parameters was used, and results are given as function of the rejection limit . n is the number of stars in the sample after rejection of outliers. is the estimated internal velocity dispersion, and , , , are the estimated components of the centroid space velocity along the equatorial (ICRS) axes and in the radial direction (, ). Uncertainties () are the formal standard errors in respective solution. The adopted solution, shown in a box, is for the larger sample (Hy0) using the cut-off value indicated as optimal by Monte Carlo simulations. This solution does not include any correction for possible internal velocity field, e.g. representing cluster expansion (see Sect. 5.3). As discussed in Sect. 5.2 our final estimate for the velocity dispersion is km s-1, while the formal errors on the velocity components need to be multiplied by 1.28.
The rapid decrease in from to 20 (with n decreasing from 191 to 171) suggests that most of the dispersion can be attributed to a relatively small fraction of the stars. To examine this further, we show in Fig. 5 the distribution of values in the successive solutions. The distributions are rather different from the truncated exponentials expected for a Gaussian velocity dispersion (Fig. 3), but qualitatively similar to the ones in Fig. 4, for a continuous (log-normal) mix of dispersions.
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-1) is along the direction toward the cluster centroid; the shortest axis (corresponding to an uncertainty of 0.03 km s-1) is along , perpendicular to both and .
5.2. Velocity dispersion and error calibration
If 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-1 perpendicular to the projected space velocity of the cluster. On the assumption of an isotropic dispersion we take this to be our best estimate of for the adopted sample of 168 stars. A Monte Carlo simulation with this dispersion accurately reproduces both and the formal errors in Table 3, but the scatter among the solutions also show that the formal errors should be increased by a factor 1.28 to reflect the true uncertainties of the estimates.
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-1 and . Roughly speaking, the observed velocity dispersion can thus be interpreted as a mixture of Gaussian distributions, with one third of the stars having a dispersion less than 0.31 km s-1, one third between 0.31 and 0.47 km s-1, and one third greater than 0.47 km s-1. It is tempting to relate this mixture to the range of stellar masses () in the sample, where equipartition of energy would require . By dividing the sample according to absolute magnitude we do find some positive correlation between and , but only at a hardly significant standard deviation. For the solution with starting from sample Hy1, i.e. excluding known binaries, an overall dispersion of km s-1 is found, but without visible correlation with . Thus we find no convincing support for the hypothesis that depends on the stellar mass.
5.3. Cluster expansion
The 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-1. However, at least the inner part of the Hyades cluster is gravitationally bound, and so is not expected to expand. The actual state of the cluster in terms of expansion or contraction is essentially unknown and may depend on the selection of stars. We have therefore chosen not to apply any corresponding correction to the data in Table 3.
5.4. Spatial correlations
One 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 0-0.5o, 0.5-1.0o and 1-2o, we found the sample correlation coefficients , , and , respectively. The positive values for separations depend on a small number of pairs involving the two stars HIP 19834 and 19862, both of which have strong positive residuals ( and mas), but which are otherwise unremarkable. If a more robust estimator is used, such as the median product of the residuals, then no correlation at all is found at these separations. We conclude that the spatial correlations, if they exist, are probably less than , with a range of about . Similar correlations were found by van Leeuwen (1999b).
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 gradients across a cluster, and these first moments are generally much less sensitive to local correlations than are the zeroth moments (e.g. the mean parallax). For instance, if we assume that the correlation in the Hyades cluster is for separations up to , and zero otherwise, then the standard error of the mean parallax may be underestimated by as much as 19 per cent, while the standard errors of the gradients are only underestimated by 1.5 to 4 per cent (along and , respectively, assuming the spatial distribution of the final sample of 168 stars). We therefore conclude that the effects of the correlations on the ML solution are relatively small for very extended objects such as the Hyades and nearby OB associations, the only objects for which the method is expected to yield significant results with Hipparcos data. Neglecting the correlations should therefore be a reasonable first approximation in our applications of the method.
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