The second data release from the Gaia mission (DR2) provides a comprehensive and unprecedented picture of the motions of astronomical sources in the plane of the sky, extending from the solar neighborhood to the outer reaches of the Milky Way. I present proper motion measurements based on Gaia DR2 for 17 ultra-faint dwarf galaxies within 100 kpc of the Milky Way. I compile the spectroscopically-confirmed member stars in each dwarf bright enough for Gaia astrometry from the literature, producing member samples ranging from 2 stars in Triangulum II to 68 stars in Bootes I. From the spectroscopic member catalogs I estimate the proper motion of each system. I find good agreement with the proper motions derived by the Gaia collaboration for Bootes I and Leo I. The tangential velocities for 14 of the 17 dwarfs are determined to better than 50 km/s, more than doubling the sample of such measurements for Milky Way satellite galaxies. The orbital pericenters are well-constrained, with a median value of 38 kpc. Only one satellite, Tucana III, is on an orbit passing within 15 kpc of the Galactic center, suggesting that the remaining ultra-faint dwarfs are unlikely to have experienced severe tidal stripping. As a group, the ultra-faint dwarfs are on high-velocity, eccentric, retrograde trajectories, with nearly all of them having space motions exceeding 370 km/s. In a low-mass (M_vir = 0.8 x 10^12 M_sun) Milky Way potential, eight out of the 17 galaxies lack well-defined apocenters and appear likely to be on their first infall, indicating that the Milky Way mass may be larger than previously estimated or that many of the ultra-faint dwarfs are associated with the Magellanic Clouds. The median eccentricity of the ultra-faint dwarf orbits is 0.79, similar to the values seen in numerical simulations, but distinct from the rounder orbits of the more luminous dwarf spheroidals.
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