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

The near future of astrophysics involves many large solid-angle, multi-epoch, multi-band imaging surveys. These surveys will, at their faint limits, have data on large numbers 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 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 point-spread function, 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 imaging of the SDSS Southern Stripe. We show that we can distinguish very red brown dwarfs by their proper motions from very high-redshift quasars more than $1.6\mag$ fainter than with traditional technique on these SDSS data, and with better better fidelity than by multi-band imaging alone. We re-discover all 10 known brown dwarfs in our sample and present 9 new candidate brown dwarfs, identified on the basis of high proper motion.