  We present Hubble Space Telescope observations of active asteroid 313P/Gibbs
(formerly P/2014 S4) taken over the five month interval from 2014 October to
2015 March. This object has been recurrently active near perihelion (at 2.4 AU)
in two different orbits, a property that is naturally explained by the
sublimation of near surface ice but which is difficult to reconcile with other
activity mechanisms. We find that the mass loss peaks near 1 kg s$^{-1}$ in
October and then declines over the subsequent months by about a factor of five,
at nearly constant heliocentric distance. This decrease is too large to be
caused by the change in heliocentric distance during the period of observation.
However, it is consistent with sublimation from an ice patch shadowed by local
topography, for example in a pit like those observed on the nuclei of
short-period comet 67P/Churyumov-Gerasimenko. While no unique interpretation is
possible, a simple self shadowing model shows that sublimation from a pit with
depth to diameter ratio near 1/2 matches the observed rate of decline of the
activity, while deeper and shallower pits do not. We estimate the nucleus
radius to be 700$\pm$100 m (geometric albedo 0.05 assumed). Measurements of the
spatial distribution of the dust were obtained from different viewing
geometries. They show that dust was ejected continuously not impulsively, that
the effective particle size is large, $\sim$50 $\mu m$, and that the ejection
speed is $\sim$2.5 m s$^{-1}$. The total dust mass ejected is $\sim$10$^7$ kg,
corresponding to $\sim$10$^{-5}$ of the nucleus mass. The observations are
consistent with partially shadowed sublimation from $\sim$10$^4$ m$^2$ of ice,
corresponding to $\sim$0.2\% of the nucleus surface. For ice to survive in 313P
for billion-year timescales requires that the duty cycle for sublimation be
$\lesssim$10$^{-3}$.
