Take a single point source moving with constant velocity through the Milky Way. From L2 the source will produce a wiggly track, shown by the dashed back line, which is a linear combination of the source position, parallax ellipse and proper motion vector.
To demonstrate how the astrometry is fit in practice, we show the expected observations and astrometric uncertainty for a hypothetical source. The source is given proper motion produces a trajectory from south-east to north-west. Adding the parallax ellipse generates a spiralling apparent position observed by Gaia throughout DR2 given by the black dashed line. The source position is a linear combination of position, parallax and proper motion!
Gaia scans this region of the sky 15 times in DR2 shown by the blue and red arrows for scans from FoV1 and FoV2, respectively. Each scan improves the constraint on each of the five astrometry parameters the uncertainties for which are given in the bottom panel.
The astrometry is fit with linear regression over the five parameters. This has two nice properties: the uncertainties are Gaussian and parameter independent! So the measurement uncertainty can be predicted from the scanning law...provided we know the position uncertainty of each observation...
Measuring a 1D position
In order to predict the astrometry uncertainty, we need to know the measurement uncertainty of every observation. Gaia has 9 columns of CCDs used to measure astrometry enabling up to 9 measurements per scan. We use the published covariance to estimate how the along-scan measurement uncertainty of the CCDs changes with source apparent magnitude.
The red line in the top panel of the figure gives the running median of the measurement error. The blue line is the "robust scatter estimate" from Figure 9 of Lindegren 2018. The difference between our fits is due to the slightly different statistic used, ours has bad observations implicitly downweighted. We now have everything we need to evaluate the expected uncertainty.
The Astrometry Spread Function is the expected astrometric uncertainty for a point source moving without acceleration. This is the 5D equivalent of a Line/Point Spread Function.
The plot on the left shows the ASF correlation coefficients with our predictions in the lower left and the medians of observations in the upper right for G=18-18.1. The agreement down to sub-degree scales is fantastic!
If you take an image of a star and find the spread is broader than the PSF, you might infer that the object is an extended source such as a Galaxy. Similarly, if the astrometric error is larger than the ASF then the object might be extended or have some extra non-linear motion such as a component of a binary orbiting it's system barycenter.
We use the ASF and published Gaia astrometry to estimate the Unit Weight Error which shows the extent of excess error in the source. This statistic is also published with internal Gaia data but doing it with the ASF is a useful proof of concept. The spread of UWE is greatest at G ∼ 13 and narrows to fainter magnitude, a clear signature of excess error that is resolvable at brighter magnitude but becomes increasingly dominated by photon count noise for fainter sources.
About 79% of sources in Gaia DR2 have 5D astrometry (positions, parallaxes and proper motions). One of the limiting factors behind missing sources is a cut on astrometric_sigma5d_max. This is the maximum eigenvalue of the scaled astrometry covariance matrix. With our ASF, we can predict this quantity as a function of position and apparent magnitude and therefore work out the probability that a source will survive the cut. However, this requires modelling the along-scan error distribution as a function of magnitude which we only do approximately. Therefore these results are more a proof of concept for the astrometry selection function.