4 Accuracy of astrometric radial velocities

From the cluster simulations and subsequent application of the maximum-likelihood method (Sect. 2) the astrometric radial velocities are estimated for the individual stars (or systems), $\widehat{v}_{{\rm r}i}$. Of course, the true radial velocities $\overline{v}_{{\rm r}i}$ are also known directly from the simulation. Thus the estimation errors $\Delta_{ij}=\widehat{v}_{{\rm r}i}-\overline{v}_{{\rm r}i}$are known. Here, index j is used to distinguish the different realisations of the cluster model. With  $\langle~\rangle_k$ denoting an average over index k, the following statistics are computed:

$$\Delta_j = \big\langle \Delta_{ij} \big\rangle_i$$ (4)

is the "cluster bias" in realisation j (i.e., the common error for all stars in the cluster);

$$\epsilon_{\rm int} = \Big\langle (\Delta_{ij}-\Delta_j)^2 \Big\rangle_{ij}^{1/2}$$ (5)

is the "internal standard error" of the astrometric radial velocities (i.e., the dispersion of the individual values around the cluster bias); and

$$\epsilon_{\rm tot} = \Big\langle \Delta_{ij}^2 \Big\rangle_{ij}^{1/2}$$ (6)

is the "total standard error" of the astrometric radial velocities (i.e., including the cluster bias). Clearly $\epsilon _{\rm int}$ is the relevant statistic for the precision of relative astrometric radial velocities within a given cluster, while  $\epsilon _{\rm tot}$ is relevant for the accuracy of absolute astrometric radial velocities. Both $\epsilon _{\rm int}$ and  $\epsilon _{\rm tot}$can be computed for various subsets depending on observable quantities such as the goodness-of-fit measure g_i, radial distance r, and mass or absolute magnitude. An interesting question is whether it is possible to observationally define subsets with reduced  $\epsilon _{\rm int}$ or  $\epsilon _{\rm tot}$.

The results presented below are based on solutions using the rejection limit $g_{\rm lim}=15$, although the results for $g_{\rm lim}=10$ are very similar. Any conclusions from these simulations are also applicable to the astrometric radial velocities published in Madsen et al. (2002).

4.1 Standard errors versus goodness-of-fit

In Fig. 2 (top) the internal and total standard errors of the astrometric radial velocities are shown versus the goodness-of-fit g_i. The absence of any significant trend shows that g_i is not a useful criterion for selecting "good" astrometric radial velocities. Even stars with g_i>10 are not worse than the rest in terms of radial-velocity precision. This somewhat counter-intuitive result can be understood if the line-of-sight component of the peculiar velocities is statistically independent of the tangential component. This is obviously the case for truly random motions, but one might expect that large proper-motion errors caused by photocentric motion in binaries should be correlated with large errors in the radial component.

Figure 2: Standard errors of the astrometric radial velocities as function of the goodness-of-fit measure gi (top) and distance from the cluster centre r (bottom). Open circles show the internal standard errors  $\epsilon _{\rm int}$ (i.e., for the relative velocities within the cluster); filled circles show the total standard errors  $\epsilon _{\rm tot}$ (i.e., for the absolute velocities). The dashed line is the expected relation from the Plummer model.

4.2 Standard errors versus radius

The bottom part of Fig. 2 shows the internal and total standard errors of the astrometric radial velocities versus the distance r from the cluster centre. In this case the standard errors clearly decrease from the centre out to 7-8 pc radius, after which they seem to increase again.

The initial decrease (for r<8 pc) is roughly in agreement with the Plummer model in Eq. (1) for $M\simeq 460~M_\odot$ and  $r_{\rm c}\simeq 2.7$ pc.

4.3 Standard errors versus mass and absolute magnitude

In Fig. 3, the internal standard errors of the astrometric radial velocities are plotted versus the true masses of the stars or systems (top) and versus the absolute magnitudes (bottom). The sample is divided at 3 pc (see Sect. 3.6.2). Inside 3 pc there is a clear difference in the velocity dispersion between the highest masses and 1 $M_\odot$, although not as much as for a full equipartition of kinetic energy ( $\sigma _{\rm v} \propto M^{-1/2}$). The effect is much smaller outside of 3 pc. The velocity dispersion also seems to decline again for stars with masses less than 1 $M_\odot$.

The effect can still be seen when the dispersion is plotted versus absolute magnitude instead of mass (Fig. 3, bottom), although the trend is less clear because of the many binary systems, for which there is no unique correspondence between system mass and total luminosity.

Together with the results of the previous section we can conclude that the practical minimum for the internal error of the astrometric radial velocities in the Hyades is around 0.20 km s^-1, which is achieved for stars at an intermediate distance ($\simeq$2-3 core radii) from the cluster centre. At that distance there is little equipartition of kinetic energy, so it does not matter much if more or less massive stars are selected.