The extended Gaia-PS1-SDSS (GPS1+) proper motion catalog

The GPS1 catalog was released in 2017. It delivered precise proper motions for around 350 million sources across three-fourths of the sky down to a magnitude of $r\sim20$\,mag. In this study, we present GPS1+ the extension GPS1 catalog down to $r\sim22.5$\,mag, based on {\it Gaia} DR2, PS1, SDSS and 2MASS astrometry. The GPS1+ totally provides proper motions for $\sim$400 million sources with a characteristic systematic error of less than 0.1\masyr. This catalog is divided into two sub-samples, i.e., the primary and secondary parts. The primary $\sim$264 million sources have either or both of the {\it Gaia} and SDSS astrometry, with a typical precision of 2.0-5.0 \masyr. In this part, $\sim$160 million sources have {\it Gaia} proper motions, we provide another new proper motion for each of them by building a Bayesian model. Relative to {\it Gaia}'s values, the precision is improved by $\sim$0.1\,dex on average at the faint end; $\sim$50 million sources are the objects whose proper motions are missing in {\it Gaia} DR2, we provide their proper motion with a precision of $\sim$4.5\masyr; the remaining $\sim$54 million faint sources are beyond {\it Gaia} detecting capability, we provide their proper motions for the first time with a precision of 7.0 \masyr. However, the secondary $\sim$136 million sources only have PS1 astrometry, the average precision is worse than 15.0 \masyr. All the proper motions have been validated using QSOs and the existing {\it Gaia} proper motions. The catalog will be released on-line and available via the VO-TAP Service, or via the National Astronomical Data Center serviced by China-VO: https://nadc.china-vo.org/data/data/gps1p/f.