3 The intermediate astrometric data or abscissa residuals

 

3.1 The general solution

 The details of the intermediate astrometric data file are described in Volume 1, Sect. 2.8, the file itself is available on disc 5 of the ASCII CD-ROM set. The file contains the data from which the astrometric parameters for single stars, as presented in the main catalogue, were derived through the fitting of one of the models described in Sect. 2.4. The model used is indicated for each star, both in the main catalogue and the intermediate astrometric data file.

A model fitted to the abscissae is described in the form of the expected changes in the abscissa positions, v_i, as a result of changes in each of the model parameters a_j. Collecting all abscissa residuals in a vector ${\vec{v}}$, the following relation is obtained:  

$$\Delta {\vec{v}} = \sum_{j=1}^n{\partial {\vec{v}}\over \partial a_j}\Delta a_j + \epsilon,$$ (2)

which is solved through minimizing the $\chi^2$, the sum of the squares of the residuals:  

$$\chi^2\! =\!\! \left(\Delta {\vec{v}} - \sum_{j=1}^n{\partia... ...{j=1}^n{\partial {\vec{v}}\over \partial a_j}\Delta a_j\right),$$ (3)

as a function of the parameter corrections $\Delta a_j$.${\bf V}$ is the covariance matrix of the abscissa residuals. In the Hipparcos Catalogue the standard errors on the parameters have been calculated assuming the predicted $\chi^2$ value, i.e. using an assumed standard error for the solution of 1.0 rather than the observed value, which may be larger or smaller than 1.0. In the presence of the variance matrix ${\bf U} = {\bf V}^{-1}$, the minimization of $\chi^2$ results in k=1, n equations of the type:  

$$\sum_{j=1}^n\left({\partial{\vec{v}}\over\partial a_k}\right... ...ec{v}}\over\partial a_k}\right)^\prime {\bf U} \Delta{\vec{v}},$$ (4)

which constitute the so-called normal equations. The matrix ${\bf U}$ is symmetric and can be factorized as:  

$${\bf U} = {\bf T}^\prime {\bf T},$$ (5)

where ${\bf T}$ is a lower triangular matrix (see e.g. Bierman 1977). The matrix ${\bf T}$ is called the square root of ${\bf U}$, and can be obtained using e.g. the Cholesky decomposition algorithm, as described by Bierman. After replacing ${\bf U}$ by ${\bf T}^\prime {\bf T}$ and some reorganization the following equations are obtained (k=1,n):  

$$\sum_{j=1}^n\left({\bf T}{\partial {\vec{v}}\over\partial a_... ...rtial a_k}\right)^\prime \left({\bf T} \Delta {\vec{v}}\right).$$ (6)

Thus, by multiplying the left- and right-hand sides of the observation equations, as defined by Eq. (2), by the matrix ${\bf T}$, which is the Cholesky square root of the inverse of the covariance matrix ${\bf V}$, a set of de-correlated and properly weighted observation equations is obtained, ready to be incorporated in the traditional least-squares solution. Note that the square root of a matrix is not unique, and therefore different ways exist to de-correlate the same set of observation equations. Also, when the covariance matrix is diagonal, the square root is too, and the multiplication of the observation equations reduces to a straight forward weighting by the inverse of its standard error for each observation.

3.2 The covariance matrix

  The observation equations for a single star consist in fact of two sets of correlated observation equations, one set from each consortium. Data obtained for observations on reference great circle k by FAST will be indicated by F_k, and for NDAC by N_k. The counter i (or j) refers to an observation equation independent of its origin. The covariance matrix ${\bf V}$ for a single star solution is built up as follows. The diagonal elements are given by:  

$$V_{ii} = \sigma_{N_k}^2,\qquad V_{jj} = \sigma_{F_k}^2$$ (7)

for observation i obtained by NDAC and observation j obtained by FAST. If two observations i and j originate from NDAC and FAST reductions on the same reference great circle k, additional off-diagonal elements have to be added to the covariance matrix, representing the correlation between these observations:  

$$V_{ij} = V_{ji} = q_{NF_k}\sigma_{N_k}\sigma_{F_k},$$ (8)

where q_NFk is the correlation coefficient between the FAST and NDAC reduction results for orbit number k. The values of q_NFk have been determined empirically from the FAST and NDAC abscissa residuals as function of magnitude, time and the estimated standard errors on the abscissa by FAST and NDAC and are provided in the abscissa records.

When only these consortia correlations are considered, the correlations between observations can be incorporated in the observation equations by multiplying pairs of correlated observations equations with the Cholesky square root of the inverse of their covariance matrix:  

$${\bf T} = \left(\begin{array} {cc} {1\over\sigma_{F_k}} & 0 ... ...} & {1\over \sigma_{N_k}\sqrt{1-q_{NF_k}^2}}\end{array}\right),$$ (9)

where the first equation is assumed to originate from FAST, and the second from NDAC. The resulting pair of observations are uncorrelated and properly weighted, and can be treated as any other observation equations in a least squares solution. It is clear that, if the correlation coefficient q_NFk is equal to zero, Eq. (9) reduces to a simple scaling of each equation by the inverse of the square root of the variance of the standard error on the observation it represents.

3.3 Astrometric parameter models

  The abscissa file provides $\partial v\over \partial a_i$ for the solution of the five astrometric parameters: corrections to the mean position (${\rm d}\alpha\cos\delta$, ${\rm d}\delta$), the assumed parallax (${\rm d}\pi$) and the assumed linear proper motions (${\rm d}\mu_{\alpha}\cos\delta$, ${\rm d}\mu_{\delta}$). Provisions have also been made for variable proper motions, the so-called 7- and 9-parameter solutions (see Sect. 2.4 and Vol. 1, Sect. 2.8). The information for the construction of the covariance matrix ${\bf V}$ (standard errors and correlation coefficients) is included in the abscissa records (fields IA9 and IA10 in Table 2.8.3 of Vol. 1).

The abscissa residuals presented in the intermediate astrometric data file are given relative to an implementation of the five astrometric parameters given in the header records. These five parameters may be part of a more complicated solution, such as a 7- or 9-parameter solution or an orbital motion solution. In these cases the abscissa residuals were obtained through implementing only the first five parameters of the solution, as given in each header record. In the 7- and 9-parameter solutions the first- and second-order time dependence of the proper motion were referred to a reference time such that the effect of these parameters on the basic 5-parameter model would be very small. This leads to the following results:

• In case the original solution was a 5-parameter solution, the proper application of a 5-parameter solution to the residuals must give negligible corrections to the astrometric parameters (only rounding-off errors), and reproduce the standard errors on those parameters and their covariances as given in the main catalogue;
• In the case of 7- or 9-parameter or orbital motion solutions, the application of the original solution must produce negligible corrections for the first 5-parameters, and reproduce the remaining parameters, together with all standard errors and the full covariance matrix. When applying a 5-parameter solution to the residuals of the partially implemented 7- or 9-parameter solution, the expected results are small corrections to the 5-parameters and an increase in the standard error of the solution.
• In the case of a stochastic solution (``X" in field H59 of the main catalogue, and in field IH8 in the header records of the intermediate astrometric data), the original solution can be recovered by adding in quadrature a ``cosmic noise'' to the standard errors of the abscissae, and applying the standard 5-parameter solution to the result. The level of the ``cosmic noise'' was adjusted such as to give the expected $\chi^2$ for the solution.

In the case of double star solutions (types ``C'' and ``V'') the abscissa residuals (often only given for the FAST data) are mostly of little meaning, and should not be used. For these stars the Transit Data (Vol. 1, Sect. 2.9) should be used instead.