Brownian motion of main-belt asteroids on human timescales

The explosion of high-precision astrometric data on main-belt asteroids (MBAs) enables new inferences on gravitational and non-gravitational forces present in this region. We estimate the size of MBA motions caused by mutual gravitational encounters with other MBAs that are either omitted from ephemeris models or have uncertain mass estimates. In other words, what is the typical Brownian motion among MBAs that cannot be predicted from the ephemeris, and therefore serves as noise on inferences from the MBAs? We estimate the RMS azimuthal shift $\sigma_\phi$ of this ``Brownian noise'' by numerical estimation of the distribution of impulse sizes and directions among known MBAs, combined with analytical propagation into future positional uncertainties. At current levels of asteroid-mass knowledge, $\sigma_\phi$ rises to $\approx2$ km or $\approx1$ mas over $T=10$ yr, increasing as $T^{3/2},$ large enough to degrade many inferences from Gaia and LSST MBA data. LSST data will, however, improve MBA mass knowledge enough to lower this Brownian uncertainty by $\ge4\times.$ Radial and vertical Brownian noise at $T=10$ yr are factors $\approx7$ and $\approx45,$ respectively, lower than the azimuthal noise, and grow as $T^{3/2}$ and $T^{1/2}.$ For full exploitation of Gaia and LSST MBA data, ephemeris models should include the $\approx1000$ largest asteroids as active bodies with free masses, even if not all are well constrained. This will correctly propagate the uncertainties from these 1000 sources' deflections into desired inferences. The RMS value of deflections from less-massive MBAs is then just $\sigma_\phi\approx60$ m or 30 $\mu$as, small enough to ignore until occultation-based position data become ubiquitously available for MBAs.