Error characterization of the

Context. Accurate characterization of the astrometric errors in the forthcoming Gaia Catalogue will be essential for making optimal use of the data. This includes the correlations among the estimated astrometric parameters of the stars as well as their standard uncertainties, i.e., the complete (variance-)covariance matrix of the relevant astrometric parameters. Aims. Because a direct computation of the covariance matrix is infeasible due to the large number of parameters, approximate methods must be used. The aim of this paper is to provide a mathematical basis for estimating the variance-covariance of any pair of astrometric parameters, and more generally the covariance matrix for multidimensional functions of the astrometric parameters. The validation of this model by means of numerical simulations will be considered in a forthcoming paper. Methods. Based on simplifying assumptions (in particular that calibration errors can be neglected), we derive and analyse a series expansion of the covariance matrix of the least-squares solution. A recursive relation for successive terms is derived and interpreted in terms of the propagation of errors from the stars to the attitude and back. We argue that the expansion should converge rapidly to useful precision. The recursion is vastly simplified by using a kinematographic (step-wise) approximation of the attitude model. Results. Low-order approximations of arbitrary elements from the covariance matrix can be computed efficiently in terms of a limited amount of pre-computed data representing compressed observations and the structural relationships among them. It is proposed that the user interface to the Gaia Catalogue should provide the tools necessary for such computations