## Appendix A: Implementation of the maximum-likelihood solution## A.1. Observational dataThe 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 ( Units are the milliarcsec (mas) for the parallax,
mas yr The normal triad introduced in Sect. 3.4 is computed from the right ascension and declination in Fields H8-9 of the Hipparcos Catalogue. The equatorial Cartesian components of these vectors are ## A.2. Solution of the maximum-likelihood equationsThe ML estimate is obtained by
finding the maximum of , or,
equivalently, the minimum of 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 stars takes a
fraction of a second on a standard PC, and the time grows only
linearly with Recall that the dimension of is
for the basic cluster model, and
or
for the models including systematic
velocity patterns (Sect. 3.3). Define the With , and we have from Eq. (17) for . Differentiating a second time, now with respect to , , and using that and , we find The main complication in solving the ML equations arises because depends on the model parameters and 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 allows considerable simplification of the calculations. We note that the dependence on and is only through the variable . Writing Eq. (16) as , where , we find by direct expansion of the determinant from which the second term in Eq. (A.3) can be calculated. Using the Woodbury formula (Press et al. 1992) we also find 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: The matrix is symmetric and, if all the parameters are estimable, positive definite. Thus a wide range of numerical methods are available to solve the system and to compute the inverse matrix for Eq. (1). Efficient implementations should however take advantage of the sparse structure of : it contains non-zero elements only along the diagonal and in a border of width along the right and bottom edges. The system of equations may be partitioned where the dimensions of the submatrices are indicated above and to the left of the system. is diagonal and therefore its inversion is trivial. With this in mind we can eliminate to obtain a system of dimension , from which is solved. Back-substitution then gives For fixed the number of
operations required to solve this system grows linearly with the
number of stars ( ## A.3. Standard errors of the estimated quantitiesAfter convergence to the ML estimate , elements of the covariance matrix may be computed after partitioning in analogy with Eq. (A.8). The covariance of the cluster parameters is thus found to be from which the covariance of is obtained as The standard errors of the estimated parameters are computed from the diagonal elements of : More generally, the standard error of any quantity which is a function of the estimated model parameters, say , is computed from For instance, the estimated radial velocity of the cluster centroid is given by the projection of along the direction to the centroid, , and its standard error (in the basic cluster model) is therefore given by 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 may be used. For 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 residualsAs discussed in Sect. 4.3, the velocity dispersion of the
cluster () can be estimated from an
analysis of the proper-motion residuals perpendicular to the centroid
velocity projected on the sky. For each star ( are unit vectors in the plane of the sky, oriented perpendicular
and parallel to the plane containing the observer, the star, and
.
we compute the peculiar velocities in the direction as 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 regarded as independent observations of a Gaussian population with mean value zero and standard deviation . Each observation has the standard error . The required estimate is obtained by solving the non-linear equation [This equation has no solution if , which corresponds to an rms proper-motion residual smaller than that expected from the observational uncertainties. In that case we set .] The uncertainty of the estimate, obtained by means of Eq. (1), is given by ## Appendix B: NotationsThis 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 © European Southern Observatory (ESO) 2000 Online publication: April 17, 2000 |