Measuring the undetectable: Proper motions and parallaxes of very faint sources

The near future of astrophysics involves many large solid-angle, multi-epoch, multiband imaging surveys. These surveys will, at their faint limits, have data on a large number of sources that are too faint to be detected at any individual epoch. Here, we show that it is possible to measure in multi-epoch data not only the fluxes and positions, but also the parallaxes and proper motions of sources that are too faint to be detected at any individual epoch. The method involves fitting a model of a moving point source simultaneously to all imaging, taking account of the noise and point-spread function (PSF) in each image. By this method it is possible to measure the proper motion of a point source with an uncertainty close to the minimum possible uncertainty given the information in the data, which is limited by the PSF, the distribution of observation times (epochs), and the total signal-to-noise in the combined data. We demonstrate our technique on multi-epoch Sloan Digital Sky Survey (SDSS) imaging of the SDSS Southern Stripe (SDSSSS). We show that with our new technique we can use proper motions to distinguish very red brown dwarfs from very high-redshift quasars in these SDSS data, for objects that are inaccessible to traditional techniques, and with better fidelity than by multiband imaging alone. We rediscover all 10 known brown dwarfs in our sample and present nine new candidate brown dwarfs, identified on the basis of significant proper motion.