A Gaia-PS1-SDSS (GPS1) Proper Motion Catalog Covering 3/4 of the Sky

We combine Gaia DR1, PS1, SDSS and 2MASS astrometry to measure proper motions for 350 million sources across three-fourths of the sky down to a magnitude of $m_r\sim20$\,. Using positions of galaxies from PS1, we build a common reference frame for the multi-epoch PS1, single-epoch SDSS and 2MASS data, and calibrate the data in small angular patches to this frame. As the Gaia DR1 excludes resolved galaxy images, we choose a different approach to calibrate its positions to this reference frame: we exploit the fact that the proper motions of stars in these patches are {\it linear}. By simultaneously fitting the positions of stars at different epochs of -- Gaia DR1, PS1, SDSS, and 2MASS -- we construct an extensive catalog of proper motions dubbed GPS1. GPS1 has a characteristic systematic error of less than 0.3 \masyr\, and a typical precision of $ 1.5-2.0$\masyr. The proper motions have been validated using galaxies, open clusters, distant giant stars and QSOs. In comparison with other published faint proper motion catalogs, GPS1's systematic error ($<0.3$ \masyr) should be nearly an order of magnitude better than that of PPMXL and UCAC4 ($>2.0$ \masyr). Similarly, its precision ($\sim 1.5$ \masyr) is a four-fold improvement relative to PPMXL and UCAC4 ($\sim 6.0$ \masyr). For QSOs, the precision of GPS1 is found to be worse ($\sim 2.0-3.0$\masyr), possibly due to their particular differential chromatic refraction (DCR). The GPS1 catalog will be released on-line and available via the VizieR Service and VO Service. (===GPS1 is available with VO TAP Query now, see http://www2.mpia-hd.mpg.de/~tian/GPS1/ for details=== )