4.5.7 Correlations in the Gaia data
Author(s): Lennart Lindegren
Correlations among the astrometric data exist on several different levels, e.g.:

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withinsource correlations (between the different astrometric parameters of the same source);

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betweensource correlations (between the same astrometric parameter, e.g. parallax, for different sources);

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general correlations (between arbitrary astrometric parameters of different sources).
Withinsource correlations are always provided, when relevant, in the Gaia Archive. They are estimated from the $5\times 5$ normal matrix of the individual sources (or the $6\times 6$ normal matrix for sixparameter solutions). Because they are computed by neglecting the betweensource correlations, they are only approximations of the actual withinsource correlations, but probably sufficiently good for all astrophysical applications. They are principally needed when transforming the five astrometric parameters to other representations (e.g. calculation of galactic coordinates, tangential velocity, or epoch transformation).
Betweensource correlations are important when calculating quantities that depend on several sources, such as the mean parallax or internal kinematics of a stellar cluster. In the final Gaia data such correlations are expected to be important mainly on small angular scales (less than a few degrees), but in Gaia DRe they could exist on much larger angular scales (tens of degrees).
Betweensource correlations and general correlations are much harder to estimate than the withinsource correlations, principally because their rigorous calculation would involve the inversion of extremely large matrices. Approximate methods to estimate general correlations exist (e.g. Holl and Lindegren 2012; Holl et al. 2012a) but have not been implemented in the Gaia data processing. Empirically, the betweensource correlations can be estimated by analysing the spatial correlations of the astrometric residuals, or from statistical analysis of the parallaxes and proper motions for distinct groups of sources, such as in stellar clusters and quasars. For Gaia DR3 the betweensource correlations have only been investigated by means of the quasars, as reported in Sections 5.6 and 5.7 of Lindegren et al. (2021b).