Metallicity Distribution Function of the Eridanus~II Ultra-Faint Dwarf
  Galaxy from Hubble Space Telescope Narrow-band Imaging
  We use deep narrowband Ca H&K ($F395N$) imaging taken with the Hubble Space
Telescope (HST) to construct the metallicity distribution function (MDF) of
Local Group (LG) ultra-faint dwarf (UFD) galaxy Eridanus II (Eri II). When
combined with archival $F475W$ and $F814W$ data, we measure metallicities for
60 resolved red giant branch stars as faint as $m_{F475W}\sim24$ mag, a factor
of $\sim4$x more stars than current spectroscopic MDF determinations. We find
that Eri II has a mean metallicity of [Fe/H]$=$-2.50$^{+0.07}_{-0.07}$ and a
dispersion of $\sigma_{\mbox{[Fe/H]}}=0.42^{+0.06}_{-0.06}$, which are
consistent with spectroscopic MDFs, though more precisely constrained owing to
a larger sample. We identify a handful of extremely metal-poor star candidates
(EMP; [Fe/H] $< -3$) that are marginally bright enough for spectroscopic follow
up. Eri II's MDF appears well-described by a leaky box chemical evolution
model. We also compute an updated orbital history for Eri II using Gaia eDR3
proper motions, and find that it is likely on first infall into the Milky Way.
Our findings suggest that Eri II underwent an evolutionary history similar to
that of an isolated galaxy. Compared to MDFs for select cosmological
simulations of similar mass galaxies, we find that Eri II has a lower fraction
of stars with [Fe/H] $< -3$, though such comparisons should currently be
treated with caution due to a paucity of simulations, selection effects, and
known limitations of Ca H&K for EMPs. This study demonstrates the power of deep
HST CaHK imaging for measuring the MDFs of UFDs.
