  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.
