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

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Appendix A: Implementation of the maximum-likelihood solution

A.1. Observational data

The most promising star clusters and associations for the determination of astrometric radial velocities are identified in Table 3 of Paper I. Using astrometric data from the Hipparcos Catalogue (ESA 1997), solutions for some of these clusters will be discussed in Paper III. Here we identify the data items in the Hipparcos Catalogue that are required as input to the computations.

For each star (i) the main observational data required are the trigonometric parallax [FORMULA] and the proper motion components [FORMULA] and [FORMULA]. These may be taken directly from Fields H11-13 of the Hipparcos Catalogue and define the arrays [FORMULA] [cf. Eq. (6)]. The standard errors of the observables [FORMULA], [FORMULA] and [FORMULA] are given in Fields H16-18 of the catalogue. These are used together with the correlation coefficients [FORMULA] (Field H24), [FORMULA] (H27) and [FORMULA] (H28) to compute the [FORMULA] covariance matrix [FORMULA]. The elements are

[EQUATION]

Units are the milliarcsec (mas) for the parallax, mas yr-1 for the proper motion components (where 1 yr is exactly 365.25 days), and km s-1 for linear velocities. In this system the astronomical unit is [FORMULA] km yr s-1.

The normal triad [FORMULA] introduced in Sect. 3.4 is computed from the right ascension [FORMULA] and declination [FORMULA] in Fields H8-9 of the Hipparcos Catalogue. The equatorial Cartesian components of these vectors are

[EQUATION]

A.2. Solution of the maximum-likelihood equations

The ML estimate [FORMULA] is obtained by finding the maximum of [FORMULA], or, equivalently, the minimum of [FORMULA] in Eq. (18). Standard numerical packages for non-linear minimisation can be used to achieve this more or less efficiently. Since many thousand minimisations are required in the Monte Carlo simulations, it is not without interest to have a computationally efficient implementation. We have found that a standard Newton-Raphson iterative method performs very well if advantage is taken of the sparse structure of the linearised equations, as described below. A complete minimisation for [FORMULA] stars takes a fraction of a second on a standard PC, and the time grows only linearly with n.

Recall that the dimension of [FORMULA] is [FORMULA] for the basic cluster model, and [FORMULA] or [FORMULA] for the models including systematic velocity patterns (Sect. 3.3). Define the m-dimensional function [FORMULA] so that [FORMULA] is a system of non-linear equations to be solved. In the Newton-Raphson iterative method (Press et al. 1992, Ch. 9.6) an improvement [FORMULA] to the current approximation [FORMULA] is obtained by solving the linearised equations [FORMULA]. Writing [FORMULA] for the Hessian matrix the improvement is therefore [FORMULA]. This equation is iterated until [FORMULA] is negligible. While [FORMULA] must of course be calculated rigorously in order that the final [FORMULA] should represent the true ML solution, it is possible to use various approximations in the calculation of [FORMULA], as long as the procedure converges. In fact, it is usually advantageous to neglect the terms in [FORMULA] that depend linearly on the residuals (Press et al. 1992, Ch. 15.5). Thus, we introduce the [FORMULA] matrix [FORMULA], also required for estimating the covariance of the ML estimate according to Eq. (1). The system to be solved in each iteration is then [FORMULA].

With [FORMULA], [FORMULA] and [FORMULA] we have from Eq. (17)

[EQUATION]

for [FORMULA]. Differentiating a second time, now with respect to [FORMULA], [FORMULA], and using that [FORMULA] and [FORMULA], we find

[EQUATION]

The main complication in solving the ML equations arises because [FORMULA] depends on the model parameters [FORMULA] and [FORMULA] through Eq. (16); hence the second and third terms on the right-hand sides of Eqs. (A.3) and (A.4). Fortunately the structure of [FORMULA] allows considerable simplification of the calculations. We note that the dependence on [FORMULA] and [FORMULA] is only through the variable [FORMULA]. Writing Eq. (16) as [FORMULA], where [FORMULA], we find by direct expansion of the determinant

[EQUATION]

from which the second term in Eq. (A.3) can be calculated. Using the Woodbury formula (Press et al. 1992) we also find

[EQUATION]

from which the third term in Eq. (A.3) is obtained. Application of the last two formulae to the second and third terms in Eq. (A.4) gives:

[EQUATION]

The matrix [FORMULA] is symmetric and, if all the parameters are estimable, positive definite. Thus a wide range of numerical methods are available to solve the system [FORMULA] and to compute the inverse matrix [FORMULA] for Eq. (1). Efficient implementations should however take advantage of the sparse structure of [FORMULA]: it contains non-zero elements only along the diagonal and in a border of width [FORMULA] along the right and bottom edges. The system of equations may be partitioned

[EQUATION]

where the dimensions of the submatrices are indicated above and to the left of the system. [FORMULA] is diagonal and therefore its inversion is trivial. With this in mind we can eliminate [FORMULA] to obtain a system of dimension [FORMULA],

[EQUATION]

from which [FORMULA] is solved. Back-substitution then gives

[EQUATION]

For fixed [FORMULA] the number of operations required to solve this system grows linearly with the number of stars (n).

A.3. Standard errors of the estimated quantities

After convergence to the ML estimate [FORMULA], elements of the covariance matrix [FORMULA] may be computed after partitioning in analogy with Eq. (A.8). The covariance of the cluster parameters [FORMULA] is thus found to be

[EQUATION]

from which the covariance of [FORMULA] is obtained as

[EQUATION]

The remaining parts of [FORMULA] are

[EQUATION]

The standard errors of the estimated parameters are computed from the diagonal elements of [FORMULA]:

[EQUATION]

More generally, the standard error of any quantity which is a function of the estimated model parameters, say [FORMULA], is computed from

[EQUATION]

For instance, the estimated radial velocity of the cluster centroid is given by the projection of [FORMULA] along the direction to the centroid, [FORMULA],

[EQUATION]

and its standard error (in the basic cluster model) is therefore given by

[EQUATION]

For the radial-velocity estimates of the individual stars, computed according to Eq. (20), the contribution from the radial component of the star's peculiar velocity must however also be taken into account. To sufficient approximation the formula

[EQUATION]

may be used. For [FORMULA] the estimated value corrected for bias should be used, or (better) the value estimated from proper motion residuals according to Appendix A.4. It should be noted that the individual radial-velocity errors in a given cluster are strongly correlated due to the common error resulting from the estimated centroid velocity.

A.4. Velocity dispersion estimated from residuals

As discussed in Sect. 4.3, the velocity dispersion of the cluster ([FORMULA]) can be estimated from an analysis of the proper-motion residuals perpendicular to the centroid velocity projected on the sky. For each star (i) we defined the normal triad [FORMULA] through Eq. (A.2). Having determined the centroid velocity [FORMULA] by means of the ML solution, we can now define the supplementary triad [FORMULA], where

[EQUATION]

are unit vectors in the plane of the sky, oriented perpendicular and parallel to the plane containing the observer, the star, and [FORMULA].  4 The proper motion residuals from the ML solution are contained in the vectors [FORMULA]. Introducing the auxiliary vector

[EQUATION]

we compute the peculiar velocities in the [FORMULA] direction as

[EQUATION]

and their uncertainties as

[EQUATION]

In the latter expression, only the proper-motion errors are taken into account.

To estimate the true velocity dispersion, we use the ML method with [FORMULA] regarded as independent observations of a Gaussian population with mean value zero and standard deviation [FORMULA]. Each observation has the standard error [FORMULA]. The required estimate [FORMULA] is obtained by solving the non-linear equation

[EQUATION]

[This equation has no solution if [FORMULA], which corresponds to an rms proper-motion residual smaller than that expected from the observational uncertainties. In that case we set [FORMULA].] The uncertainty of the estimate, obtained by means of Eq. (1), is given by

[EQUATION]

Appendix B: Notations

This Appendix summarises notations used in this paper. Equation numbers refer to the place where the notation is defined or first used.

Lower-case bold-face letters designate vectors and arrays, upper-case bold-face letters tensors and matrices. Ordinary-face italics are used for scalar variables. The prime (´) denotes the transpose of a matrix as well as the scalar product of vectors. For the generic variable x the true value is designated [FORMULA], the observed value [FORMULA] and the estimated value [FORMULA]. [FORMULA] is the uncertainty (standard error) of x, [FORMULA] the sample mean value and [FORMULA] the expectation. [FORMULA] is the covariance operator. [FORMULA] is the statistical correlation (from -1 to [FORMULA]) between x and y. Pr denotes probability.


[TABLE]

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© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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