3 Exploitation of the moving-cluster method

   
3.1 Basic cluster model

The mathematical procedure described in Paper II yields maximum-likelihood estimates for the space velocity of the cluster centroid $\widehat{\vec{v}}_0$, for the internal velocity dispersion of stars within the cluster $\widehat{\sigma}_v$, and for the kinematically improved parallax of each star $\widehat{\pi}_i$, $i=1,\,2,\,\dots\,n$. (We reserve index 0 for the centroid. The caret $\widehat{\phantom{x}}$ signifies an estimated value.) Although additional model parameters could be included, e.g. to describe a possible rotation or non-isotropic dilation of the cluster, the present studies are restricted to the ``basic cluster model'' in which no such systematic velocity patterns are assumed. In Paper II it was shown, through Monte Carlo simulations of the Hipparcos observations of the Hyades cluster, that the presence of any reasonable amount of rotation and shear in the actual cluster will not significantly bias the solution for the centroid velocity, even though the analysis is restricted to the basic model. One important exception concerns the possible isotropic expansions of gravitationally unbound associations. These could indeed introduce significant biases, which are discussed separately in Sect. 5.7.

   
3.2 Observational data

The Hipparcos Catalogue (ESA 1997) provided input data for each star i in the form of positions in barycentric right ascension ${\alpha}_i$, declination ${\delta}_i$, trigonometric parallax $\widetilde{\pi}_i$, the proper-motion components $\widetilde{\mu}_{\alpha i}$ and $\widetilde{\mu}_{\delta i}$, and standard deviations and correlation coefficients for the latter three. (The tilde $\widetilde{\phantom{x}}$ signifies an observed value, the uncertainty of which needs to be taken into account in the estimation procedure.) The positions and proper motions in the Hipparcos Catalogue are referred to the barycentric ICRS reference system, and consequently all resulting velocities are also in that system.

For each cluster or association, an initial sample of probable member stars was identified from the literature, mainly from studies based on Hipparcos data. However, since the mathematical formalism for obtaining radial velocities is strictly applicable only to stars sharing the same average velocity vector (with a random spread about that value), cluster non-members and binary stars in non-modelled orbits must be removed from the sample as far as possible. This was done using an iterative rejection procedure described in Paper II. Monte Carlo simulations (Sect. 4.2 in that paper) showed that this procedure works best when adopting the goodness-of-fit rejection limit $g_{\rm lim}=15$. As illustrated there for the Hyades, this gave the lowest scatter in the centroid radial velocity, as well as in other quality indicators. Therefore, unless otherwise stated, all samples discussed here were cleaned according to this criterion.

Solutions were primarily obtained using data from the main Hipparcos catalogue but, for some clusters, we used also data from the Tycho-2 catalogue (Høg et al. 2000). The accuracies in the latter are generally somewhat worse, but since its proper motions incorporate about a century of ground-based observations, the segregation of long-period binaries may be improved.

   
3.3 Clusters and associations studied

Table 4 in Paper I lists some 15 clusters and associations, for which Hipparcos-type accuracies could potentially yield viable astrometric radial velocities, i.e. with standard errors less than a few km s^-1. In practice the resulting accuracies depend on several additional factors, not considered in that survey, such as the number of member stars actually observed by Hipparcos, the position of the cluster on the sky, the statistical correlations among the astrometric data, and the procedures used to obtain a clean sample. The simplest way to find out whether our method ``works'' on a particular group of stars is to make a trial solution. We have done that for all the potentially interesting clusters and associations, based on the list in Paper I supplemented with data from the compilations in the Hipparcos Input Catalogue (Turon et al. 1992), by Robichon et al. (1999), and by de Zeeuw et al. (1999).

The original criterion for including a cluster or association in the present study was that it yielded a valid solution with the basic cluster model (Sect. 3.1), including a non-zero estimate of the velocity dispersion. This turned out to be the case for four clusters (Ursa Major, Hyades, Coma Berenices, and Praesepe) and four associations (Lower Centaurus Crux, Upper Centaurus Lupus, Upper Scorpius, and ``HIP 98321''). For two more, the Pleiades cluster and the $\alpha$ Persei association, reasonable solutions could be obtained by assuming zero velocity dispersion in the maximum-likelihood procedure (note that the dispersion could still be estimated from the proper-motion residuals, as explained in Sect. 3.4). Considering the astrophysical importance of these clusters, they were therefore included in the study. A separate solution was also made for the Sco OB2 complex (Sect. 5.4).

   
3.4 Mathematical bias and noise in the solutions

Resulting space velocities and internal velocity dispersions for the 11 clusters and associations are in Table 1. $\widehat{v}_{0x}$, $\widehat{v}_{0y}$, and $\widehat{v}_{0z}$ are the equatorial components (ICRS coordinates) of the estimated space velocity of the cluster centroid, while $\widehat{\sigma}_{\rm v}$ is the velocity dispersion in each coordinate, i.e. the standard deviation of peculiar velocities along a single axis.

The maximum-likelihood estimation tends to underestimate the velocity dispersion, as examined through Monte Carlo simulations in Paper II. In Appendix A.4 of that paper we gave an alternative procedure to estimate the velocity dispersion from the proper-motion residuals perpendicular to the centroid velocity projected on the sky. This was shown to give nearly unbiased results. The velocity dispersions $\widehat{\sigma}_{\rm v}$ in Table 1 and elsewhere in this paper have therefore been estimated through this alternative procedure.

As was also described in Paper II, the radial-velocity errors among individual stars in the same cluster are not statistically independent, but may carry a significant positive correlation. For each star, the error contains a [nearly constant] component being the uncertainty in the cluster velocity as a whole, plus a random component corresponding to the physical velocity dispersion among the individual stars. Averaging over many stars in a given cluster averages away the influence of the velocity dispersion, but has only little effect on the error in the radial velocity of the cluster centroid. This quantity, discussed already in Paper I (e.g. its Table 4), is therefore a limiting accuracy in the average astrometric radial velocity of stars in any one cluster. For certain applications, effects of this noise can be lessened by averaging over different clusters (whose errors are not correlated), e.g., when searching for systematic differences between astrometric and spectroscopic radial-velocity values.

It should be noted that the selection process used to arrive at the final sample may have a significant impact on the estimated internal velocity dispersion. The cleaning process successively removes those stars that deviate most from the mean cluster velocity, thus successively reducing  $\widehat{\sigma}_v$ for the ``cleaner'' samples. While designed to remove non-members and other outliers, this procedure naturally affects also the mean characteristics of the remaining stars. For example, in the case of the Hyades, stars in the outskirts of the cluster are preferentially removed during the rejection procedure, meaning that the resulting clean samples more or less correspond to the stars within the tidal radius. In the case of the OB associations, we obtain velocity dispersions of about 1 km s^-1, a factor of two lower than the estimates for the Orion Nebula Cluster (Jones & Walker 1988; Tian et al. 1996) and other nearby associations (Mathieu 1986). Thus, although we believe that the velocity dispersions reported here correctly characterize the retained samples, they are not necessarily representative for the cluster or association as a whole.

   
3.5 Calculation of astrometric radial velocities

   
3.5.1 The stringent definition of `radial velocity'

Recognizing the potential of astrometric radial velocities determined without spectroscopy, a resolution for their stringent definition was adopted at the General Assembly of the International Astronomical Union held in 2000. This resolution (Rickman 2001) defines the geometric concept of radial velocity as $v_{\rm r} = {\rm d}b/{\rm d}t_{\rm B}$, where b is the barycentric coordinate distance to the object and $t_{\rm B}$ the barycentric coordinate time (TCB) for light arrival at the solar system barycentre. This definition is analogous to the conventional understanding of proper motion as the rate of change in barycentric direction with respect to the time of light reception at the solar-system barycentre.

In this work, we follow this IAU definition of ``astrometric radial velocity''. The difference with respect to alternative possible definitions is on the order of $v_{\rm r}^2/c$, with c = speed of light (Lindegren et al. 1999; Lindegren & Dravins, in preparation). Most population I objects (including all clusters and associations considered in this paper) have low velocities, $\vert v_{\rm r}\vert<50$ km s^-1, resulting in only very small differences, <10 m s^-1, between possible alternative definitions.

   
3.5.2 Radial velocities for individual stars

In the basic cluster model, the estimated radial velocity of an individual star is given by

$$\widehat{v}_{{\rm r}i} = \vec{r}_i^\prime\widehat{\vec{v}}_0 \, ,$$ (1)

where $\vec{r}_i$ is the unit vector towards star i and $\widehat{\vec{v}}_0$ is the estimated space velocity of the cluster as a whole (actually of its centroid). The prime ($^\prime$) denotes the transpose of the vector. In terms of the equatorial coordinates $(\alpha_i,\delta_i)$ of the star we have

$$\widehat{v}_{{\rm r}i} = \widehat{v}_{0x}\cos\delta_i\cos\a... ...} \cos\delta_i\sin\alpha_i + \widehat{v}_{0z}\sin\delta_i\, ,$$ (2)

where ( $\widehat{v}_{0x}, \widehat{v}_{0y}, \widehat{v}_{0z}$) are the equatorial velocity components as listed in Table 1 (or as determined by other means). We emphasize that Eq. (2) applies to any star that shares the cluster motion, irrespective of whether that star was present in the database used to determine the cluster motion in the first place.

The standard error $\epsilon(\widehat{v}_{{\rm r}i})$ of the individual radial velocity is computed from

$$\epsilon(\widehat{v}_{{\rm r}i})^2 = \vec{r}_i^\prime {\rm Cov}(\widehat{\vec{v}}_0)\vec{r}_i + \widehat{\sigma}_v^2 \, ,$$ (3)

where ${\rm Cov}(\widehat{\vec{v}}_0)$ is the $3\times 3$ submatrix in ${\rm Cov}(\widehat{\vec{\theta}})$ of all model parameters, referring to the centroid velocity (cf. Eq. (A18) in Paper II). The first term in Eq. (3) represents the uncertainty in the radial component of the common cluster motion, while the second represents the contribution due to the star's peculiar motion.

The complete covariance matrix ${\rm Cov}(\widehat{\vec{v}}_0)$ is only given in the electronic (extended) version of Table 1. The printed Table 1 gives (following the $\pm$ symbol) the standard errors $\epsilon(\widehat{v}_{0x})$ etc. of the vector components; these equal the square roots of the diagonal elements in ${\rm Cov}(\widehat{\vec{v}}_0)$. Also given for each cluster is the standard error of the radial component $\widehat{v}_{0{\rm r}}=\vec{r}_0'\widehat{\vec{v}}_0$ of the centroid motion. This was computed from

$$\epsilon(\widehat{v}_{0{\rm r}})^2 = \vec{r}_0^\prime {\rm Cov}(\widehat{\vec{v}}_0)\vec{r}_0 \, ,$$ (4)

where $\vec{r}_0$ is the unit vector towards the adopted centroid position $(\alpha_0,\delta_0)$ specified in the table. $\widehat{v}_{0{\rm r}}$ can be regarded as an average radial velocity for the cluster as a whole, and its standard error (squared) can be regarded as a typical value for the first term in Eq. (3). Thus, for any star not too far from the cluster centroid, the total standard error of its astrometric radial velocity can be approximately computed as $[\epsilon(\widehat{v}_{0{\rm r}})^2+\widehat{\sigma}_{\rm v}^2]^{1/2}$, using only quantities from the printed Table 1.

   
3.6 Kinematically improved parallaxes

Our maximum-likelihood estimation of the cluster space motions also produces estimates of the distances to all individual member stars. A by-product of this moving-cluster method is therefore that individual stellar distances are improved, sometimes considerably, compared with the original trigonometric determinations. This improvement results from a combination of the trigonometric parallax $\pi_{\rm trig}$ with the kinematic (secular) parallax $\pi_{\rm kin}=A\mu/v_{\rm t}$ derived from the star's proper motion $\mu$ (scaled with the astronomical unit A) and tangential velocity $v_{\rm t}$, the latter obtained from the estimated space velocity vector of the cluster. The calculation of secular parallaxes and kinematic distances to stars in moving clusters is of course a classical procedure; what makes our ``kinematically improved parallaxes'' different from previous methods is that the values are derived without any recourse to spectroscopic data (for details, see Papers I and II).

De Bruijne (1999b) applied the present method to the Scorpius OB2 complex in order to study its HR diagram by means of the improved distances. His ``secular parallaxes'' are essentially the same as our ``kinematically improved parallaxes'', being based on the same original formulation by Dravins et al. (1997). The main differences are in the choice of rejection criteria (de Bruijne uses $g_{\rm lim}=9$ versus our 15) and in the practical implementation of the solution (downhill simplex versus our use of analytic derivatives). De Bruijne also made extensive Monte Carlo simulations which demonstrated that the distance estimates are robust against all systematic effects considered, including cluster expansion. For the accuracy of the secular parallaxes, de Bruijne (1999b) used a first-order formula (his Eqs. (16) and (17)) which explicitly includes a contribution from the (assumed) internal velocity dispersion, but neglects the contribution from the trigonometric parallax error (cf. Eq. (11) in Paper I). By contrast, our error estimates are derived directly from the maximum-likelihood solution (Paper II, Appendix A.3), which in principle takes into account all modelled error sources but in practice underestimates the total error as discussed below. As a result, our error estimates are somewhat smaller than those given by de Bruijne (1999b).

The standard errors for the estimated parallaxes given in this paper are the nominal ones obtained from the maximum-likelihood estimation, which could be an underestimation of the actual errors. Determination of realistic error estimates would require extensive Monte-Carlo experiments based on a detailed knowledge of the actual configuration of stars, their kinematic distributions, etc. This information is in practice unavailable except in idealised simulations, and we therefore choose not to introduce any ad hoc corrections for this. As an example of the possible magnitude of the effect, the Hyades simulations in Paper II could be mentioned: in that particular case, the nominal standard errors required a correction by a factor 1.25 to 1.28 in order to agree with the standard deviations in the actual sample.

Figure 3: Astrometrically determined radial velocities compared with spectroscopic values from the literature, for stars in four open clusters. The diagonal lines follow the expected relation $v_{\rm r}({\rm astrom})\simeq v_{\rm r}({\rm spectr})$. Black symbols denote single stars while certain or suspected binaries are in grey. The top three frames are kinematic solutions obtained from Hipparcos data only. In some cases, including the Pleiades (bottom), somewhat better accuracies are reached using data from the Tycho-2 catalogue, which incorporates almost a century of proper-motion data.