6 Conclusions

A dynamical model of the Hyades cluster, based on N-body simulations using the NBODY6 code, has been fitted to the astrometric information available in the Hipparcos and Tycho-2 catalogues in order to study the accuracy of astrometric radial velocities. The number of stars as function of magnitude, their three-dimensional distribution, and the distribution of proper motions have been adequately reproduced by the model, as well as basic binary statistics. No spectroscopic radial velocities have been used in the present study (except for the initial membership determination by Perryman et al. 1998) meaning that the results should be directly comparable with the astrometrically determined radial velocities of Hyades stars by Lindegren et al. (2000) and Madsen et al. (2002).

From the simulations it is concluded that the velocity dispersion of the Hyades decreases from $\sigma_{\rm v}\simeq 0.35$ km s^-1 at the centre of the cluster to nearly 0.2 km s^-1 at 7-8 pc from the centre. Outside the tidal radius of 10-11 pc, the dispersion slightly increases again. Compared with previous studies of the velocity dispersion in the centre of the Hyades, the results here indicate a somewhat higher value.

The internal velocity dispersion contributes to the random errors of the astrometric radial velocities with the same magnitude. This is significantly less than the $\sigma_{\rm v}=0.49$ km s^-1 estimated in Madsen et al. (2002) directly from the Hipparcos observations. This discrepancy can be understood with reference to Table 3 as an overestimation from the observed data when the less strict rejection limit $g_{\rm lim}=15$ was used. Thus the previous estimate of the internal standard error (due to the dispersion) can now be almost halved.

In fact, stars with an expected velocity dispersion as low as 0.20 km s^-1 can be selected for studies that compare astrometric and spectroscopic radial velocities in order to disclose astrophysical phenomena causing spectroscopic line shifts. However, it should be remembered that the total standard error, including the uncertainty of the motion of the cluster centroid, is still of order 0.55-0.65 km s^-1 (Fig. 2, bottom), in agreement with the previous estimate.

Attempts to see a radial dependence of the velocity dispersion with Hipparcos and Tycho-2 astrometry have been inconclusive. The observed relation is essentially flat for the most optimal sample. Given the uncertainty of the estimated velocity dispersions when the stars are divided into radial shells, this result is not surprising. Similar examples can be found in the simulations. Only when the mean relation is computed from several realisations of the cluster model do the variations become clear. In particular, it appears that the structure of dispersion/radius relation reported by Madsen et al. (2001) does not reflect typical dynamical properties of the cluster, but could result by chance or from some (unknown) mechanism related to the photocentric motions of undetected binaries.

The fit has yielded an estimate of the initial cluster mass of 1100-1200 $M_\odot$ and of the initial multiplicity, which appears to be very high (possibly near 100%, if brown-dwarf companions are included). The current cluster mass is estimated to be $\simeq$460 $M_\odot$with a tidal radius of $\simeq$11 pc and a mean velocity dispersion within r<3 pc of 0.32 km s^-1.

Some of the differences between observations and simulations could be due to some of the non-modelled features discussed in Sect. 5, which would lead to a higher initial particle number in the model and which might also solve some of the discrepancies noted in the binary statistics. The development of numerical tools such as NBODY6 to include e.g. a time-dependent tidal field would allow an improved realism of the Hyades model, and to study the effect on the accuracy of astrometric radial velocities from assumed negligible contributions to the velocity field with respect to the Hipparcos precisions.

The method used to estimate astrometric radial velocities discussed in Sect. 2 cannot eliminate of the error contribution from the internal dynamics of the cluster, no matter how precise the astrometry might be. The velocity dispersion therefore sets a fundamental limit on the accuracy of astrometric radial velocities, and as a consequence the results from the simulations presented here also apply to planned astrometric space missions such as GAIA (Perryman et al. 2001), even though it has been Hipparcos observations of the Hyades that have been simulated.

The Hipparcos and Tycho-2 catalogues contain the best available astrometry to study the internal velocity structure of the nearest open cluster, the Hyades. To study it in greater detail, even better astrometry is needed. The GAIA mission, in combination with improved N-body simulations, will make it possible to observe directly the internal velocity field of the Hyades, and give us insight in the kinematics of the Hyades in particular and open clusters in general.

Acknowledgements

I thank Sverre Aarseth for making NBODY6 freely available, Tim Adams for helping me with the code, and Lennart Lindegren, Melvyn B. Davies, and Dainis Dravins for useful comments and valuable suggestions.