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5 Non-modelled effects

The validity of the conclusions above depends critically on the realism of the N-body simulations. A number of non-modelled effects, and their possible impact on the results, are briefly considered below.

Time-dependent tidal field: When star clusters move through the galactic disk, they are subject to tidal shocks, and shock heating from the bulge. These effects are important to consider here since they increase the random motion of the stars. For globular clusters it has been found that tidal shocks accelerate significantly both core collapse and evaporation (Gnedin et al. 1999).

In the case of open clusters, Bergond et al. (2001) estimated that those with high-z oscillations lose some 10-20% of the mass integrated over the lifetime of the cluster, mainly in low-mass stars, through disk-shocking. The Hyades have a low vertical velocity (W=6 km s-1 relative to the LSR), and therefore only oscillates with an amplitude of about 50 pc in z. Since this is small compared with the thickness of the disk, the disk-crossings should not cause much additional heating. The radial oscillations in the galactic plane, having an amplitude of 2 kpc, may be more important. The present N-body model assumes that the cluster moves in a circular galactic orbit. Thus it cannot be excluded that it underestimates the mass loss by perhaps some 5-10% of the initial mass. Preferentially, the lowest-mass stars leave the cluster, forming tidal tails (Combes et al. 1999). Although this would slightly affect the estimation of the velocity dispersion, it would have only a very small effect on the number of observed stars of spectral type earlier than M0.

  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{H4071F3a.eps}\par\vspace*{2mm}
\includegraphics[width=8.8cm,clip]{H4071F3b.eps}
\end{figure} Figure 3: The internal standard error of the astrometric radial velocities, $\epsilon _{\rm int}$, as function of stellar mass (top) and absolute magnitude (bottom). Circles refer to stars inside 3 pc of the cluster centre, crosses to those outside 3 pc. For binaries, m is the total mass of the system and MV the total absolute magnitude. The dashed line is the curve  $\sigma _{\rm v} \propto M^{-1/2}$.

Molecular clouds: Terlevich (1987) studied open cluster N-body models with initially 1000 particles and moving in a circular orbit at 10 kpc from the galactic centre (i.e., assumptions comparable with this work). She concluded that the timescale for encounters with giant molecular clouds is of the same order of magnitude as the present age of the Hyades. Since such an encounter would probably be catastrophic, it can be assumed that the Hyades have not been exposed to such a meeting. More abundant are encounters with smaller interstellar clouds. They will not shorten the lifetime of open clusters significantly but may contribute to the tidal heating of the outer regions in a given cluster. Wielen (1975) stated that gravitational shocks due to interstellar clouds will produce a significant flattening (up to 1:2) of the halo of the cluster perpendicular to the galactic plane. For the Hyades the flattening is 1:1.5 (Perryman et al. 1998). Since the galactic tidal field is also contributing to the flattening, it is doubtful if the Hyades have had any but minor interactions with interstellar clouds.

Perryman et al. (1998) examined the possibility that the Hyades recently experienced an encounter with a massive object causing a tidal shear in the outer regions of the cluster, but excluded it based on the impulsive approximation (Spitzer 1958; Binney & Tremaine 1987). Lindegren et al. (2000) included more velocity components in their model to test for non-isotropic dilation, and concluded that if such an effect existed it had to be higher than 0.01 km s-1 pc-1 to be detected with Hipparcos data. Effects from a tidal heating are thus not detectable in the Hyades with current astrometric precision.

Brown dwarfs: Despite extensive searches, no single-star brown dwarf (BD) candidate has been found in the Hyades (Reid & Hawley 1999; Gizis et al. 1999; Dobbie et al. 2002). Reid & Hawley (1999) found that the lowest-mass Hyades candidate star (LH 0418+13) has a mass of 0.083 $M_\odot$, placing it very close to the hydrogen-burning limit. The only promising candidate brown dwarf in the Hyades is the unresolved companion in the short-period system RHy403 (Reid & Mahoney 2000). Of course, the faintness of these substellar objects make them hard to observe, but still, the conclusion seems to be that the number today is quite small.

Adams et al. (2002) performed extensive simulations with a modified version of NBODY6 to model the brown dwarf population in open clusters, and concluded that the effects of different brown-dwarf populations were minimal, leaving the dynamics of the cluster largely unchanged.

The IMF in the version of NBODY6 used here cannot produce brown dwarfs, so this must be considered when defining the initial binary fraction. The IMF for brown dwarfs, or substellar masses, is very uncertain. Kroupa (2001) argues that a power-law value of  $\alpha=0.3\pm 0.7$ is the most reasonable. Since stellar masses with $M<0.08~M_{\odot}$ are not produced in the code, one must represent the star-BD binary systems either as single stars or by overproducing binaries with secondary components slightly above the BD limit. Thus an initial binary fraction of 86% was assumed, which corresponds approximately to unity if brown dwarfs had been included. Based on the investigations of Adams et al. (2002), and considering that Hipparcos did not observe stars less massive than M0 stars in the Hyades, the above approximation should be sufficient for the present purpose.

Cluster rotation: Gunn et al. (1988) did a comprehensive study of the rotation of the Hyades, but had to conclude that it was at most of the same size as their statistical error. Nonetheless they stated that their results suggested a cluster rotation, but not higher than 0.015 km s-1 pc-1.

Perryman et al. (1998) did a thorough study of the velocity residuals and concluded that they were consistent with a non-rotating system and the given observational errors. Lindegren et al. (2000) tested the Hyades for rotation by assuming solid-body rotation parameters, but found that it was too small to be detected, setting an upper limit of 0.01-0.02 km s-1 pc-1. If this upper limit should equal the true rotation of the Hyades, then the effect is non-negligible at 10 pc compared to the internal error. But there seems to be nothing in the present study suggesting such a rotation.

But probably the solid-body assumption is too simple. In the globular cluster $\omega$ Centauri, Merritt et al. (1997) found that only at small radii could the rotation be approximated by a solid-body. Beyond that the rotation falls off. Einsel & Spurzem (1999) did theoretical investigations on the influence of rotation on the dynamical evolution of collisional stellar systems, that could explain the findings by Merritt et al. (1997). In fact, it seems that only inside the half-mass radius is it reasonable to talk about a solid-body rotation (cf. Kim et al. 2002).

Although it is unlikely that the cloud in which the Hyades formed had zero angular momentum, there currently exists no certain measure of the rotation. In the model, it is instead assumed that the effects are sufficiently small and can be ignored.

Expansion: During the evolution of a cluster parts of it expand and parts of it contract. Under the assumption that the relative expansion rate equals the inverse age of the cluster, Dravins et al. (1999b) estimated that an isotropic expansion of the Hyades would lead to a bias in the astrometric radial velocity of 0.07 km s-1 of the centroid velocity. This is completely negligible and any expansion effects have been ignored.

To summarise, it appears that none of these non-modelled effects would affect the results very significantly. While the modelling of tidal fields and brown dwarfs could be improved, the possible effect of cloud encounters remains an uncertainty which cannot easily be included in the modelling of a specific cluster such as the Hyades. Although NBODY6 allows encounters with interstellar clouds, the option has not been used in the present study.


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