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Astron. Astrophys. 356, 1119-1135 (2000)

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2. Overview of the method

The aim of the present method is to estimate the radial velocities of the stars in a moving cluster, based exclusively on astrometric data (mainly proper motions and trigonometric parallaxes). In effect, it reverses the well-known `moving-cluster method' (Binney & Merrifield 1998) in which the distances to the stars in a moving cluster are derived from the proper motions and (spectroscopic) radial velocities: if instead the distances are known from the trigonometric parallaxes, the radial velocities follow. The classical application of the moving-cluster method, e.g. by van Bueren (1952), proceeds in three steps: firstly, the convergent point is located from the directions of the proper motions; secondly, the common space velocity (in km s-1) of the stars is derived from the spectroscopic radial velocities using the adopted convergent point; and finally the distances to the individual stars are computed from their observed proper motions and the adopted space velocity. The resulting distance estimates are referred to as `kinematic distances' or `kinematic parallaxes', as they are based on the kinematic parameters of the cluster. Murray & Harvey (1976) gave a rigorous formulation of the technique, in which the conventional division into the three steps mentioned above was replaced by a simultaneous least-squares solution for the kinematic parameters and the (kinematic) parallax of each individual star.

The overall approach of our method is similar to that of Murray & Harvey (1976), in that a simultaneous solution is made for the kinematic parameters of the cluster and the individual parallaxes. The main differences with respect to the classical formulation are: (1) spectroscopic radial velocities are not used as input to the solution; (2) trigonometric parallaxes are on the other hand used as input; (3) the kinematic model of the cluster is refined by including an internal velocity dispersion and a possible first-order velocity field. In addition, a special procedure for rejection of outliers (e.g. non-member stars) is incorporated and extensive Monte Carlo simulations are used to validate the results.

Both the classical moving-cluster method and the present method can be regarded as special cases of the more general `statistical parallax' method (Hawley et al. 1986; Strugnell et al. 1986; Popowski & Gould 1998; Narayanan & Gould 1999a), in which distances and kinematic parameters may be derived from a combination of astrometric, spectroscopic, and photometric data.

A detailed description of our method is given in the following sections and in Appendix A. The flow diagram for the computations is shown in Fig. 1, including references to the sections of this paper where the different steps are described. The most important parts of the procedure are summarised below.

[FIGURE] Fig. 1. Flow diagram for the computation of astrometric radial velocities and kinematically improved parallaxes of a moving cluster. The section where each computational step is described is indicated on the chart.

Input data and initial stellar sample: The present method assumes that a provisional sample of stars likely to be members of the cluster has been identified and that accurate astrometric data (positions, proper motions and trigonometric parallaxes) are available for this sample. The initial sample may be defined by means of classic kinematic and photometric criteria, e.g. as described by Platais et al. (1998).

Maximum-likelihood estimation: At the core of the computation is the maximum-likelihood (ML) estimation algorithm, which fits the parameters of the cluster model to the observed astrometric data. The model parameters are, usually, the common space velocity vector of the cluster, the internal velocity dispersion, and the distances to the individual stars. The output consists of the fitted (estimated) model parameters with their estimated uncertainties (standard errors).

Star selection and model validation: The ML algorithm assumes that the observational data conform to the statistical descriptions that are part of the model. Because the model is restricted to the cluster itself, this assumption is not valid for non-member stars. Moreover, some member stars (in particular binaries) may have deviating motions that are not included in the statistical modelling of the cluster. Residuals of the fit are used to assess the validity of the model as a whole and its applicability to the individual stars. Stars with large residuals do not conform to the model and are therefore rejected.

Simulation: The procedures above can be applied not only to observed astrometric values (e.g. from the Hipparcos Catalogue), but also to simulated data. The simulation starts with the cluster model and assumed model parameters, from which the `true' (error-free) astrometric data are computed. Adding Gaussian measurement noise according to the assumed standard errors of the observations finally gives the simulated astrometric data. Contaminating field stars or anomalous velocity distributions can be included.

Parameter validation: The simulated observations are used to validate the ML estimation procedure: the assumed parameters should, ideally, be recovered by the estimation. Because of the added measurement noise this will at best happen as the mean result of many simulation/estimation experiments. The general technique to study properties of an estimation procedure by repeated simulation experiments is known as Monte Carlo simulation (Press et al. 1992). It is particularly important to find out whether the estimated parameters are systematically different from the assumed ones, i.e. if they are biased. In our case a significant bias was found in the ML estimation of the internal velocity dispersion, and a special procedure was devised to circumvent this effect. Monte Carlo simulations are also used to determine the uncertainties of the estimated parameters.

End products: The final model parameters, corrected for bias if necessary, provide a kinematic characterisation of the cluster as well as kinematically improved individual distances to the member stars. Projecting the common cluster velocity vector onto the line of sight of each star provides an estimate of the star's radial velocity. The error of the resulting astrometric radial velocity has two parts: a (systematic) part which derives from the estimation error of the cluster velocity vector, and a (random) part due to the peculiar velocity of the star.

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© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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