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Overcoming Separation Between Counterparts Due to Unknown Proper Motions in Catalogue Cross-Matching

To perform precise and accurate photometric catalogue cross-matches -- assigning counterparts between two separate datasets -- we need to describe all possible sources of uncertainty in object position. With ever-increasing time baselines between observations, like 2MASS in 2001 and the next generation of surveys, such as the Vera C. Rubin Observatory's LSST, Euclid, and the Nancy Grace Roman telescope, it is crucial that we can robustly describe and model the effects of stellar motions on source positions in photometric catalogues. While Gaia has revolutionised astronomy with its high-precision astrometry, it will only provide motions for ~10% of LSST sources; additionally, LSST itself will not be able to provide high-quality motion information for sources below its single-visit depth, and other surveys may measure no motions at all. This leaves large numbers of objects with potentially significant positional drifts that may incorrectly lead matching algorithms to deem two detections too far separated on the sky to be counterparts. To overcome this, in this paper we describe a model for the statistical distribution of on-sky motions of sources of given sky coordinates and brightness, allowing for the cross-match process to take into account this extra potential separation between Galactic sources. We further detail how to fold these probabilistic proper motions into Bayesian cross-matching frameworks, such as those of Wilson & Naylor. This will vastly improve the recovery of e.g. very red objects across optical-infrared matches, and decrease the false match rate of photometric catalogue counterpart assignment.