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In the Local Group, nearly all of the dwarf galaxies
at $\mstar\lesssim10^9\msun$ ($\mstar\lesssim10^9\msun$) that are satellites within $300\kpc$ (the virial radius) of the Milky Way (MW) and Andromeda (M31) have quiescent star formation and little-to-no cold gas.
This contrasts strongly with comparatively isolated dwarf galaxies, which are almost all actively star-forming and gas-rich.
This near dichotomy implies
that environmental processes within a \emph{rapid} transformation after falling into the
halos halo of the
MW and M31 \emph{rapidly} remove gas and quench star formation in satellite dwarfs after infall. MW/M31.
We combine the observed quiescent fractions for satellites of the MW/M31 with the virial-infall times of satellites from the ELVIS suite of cosmological simulations
of MW/M31-like halos to determine the typical timescales over which environmental processes
remove gas and quench
star formation in satellite dwarf galaxies.
The quenching timescales at $\mstar<10^8\msun$ are short,
$<2-3\gyr$, depending somewhat on whether environmental preprocessing in lower-mass groups is important, $\lesssim2\gyr$, and
decrease with decreasing satellite mass.
%and are comparable to a virial-radius crossing time.
We compare These timescales can increase $1-2\gyr$ if environmental preprocessing in lower-mass groups prior to MW/M31 infall is important.
Comparing with timescales for more massive satellites from
the literature, which implies previous works indicates that
the environmental quenching timescale increases rapidly with satellite mass
to $\approx9.5\gyr$ and peak at
$\approx9.5\gyr$ for $\mstar\approx10^9\msun$,
but it then rapidly decreases
with at higher satellite mass to less than $5\gyr$ at
$\mstar>5\times 10^9\msun$. $\mstar>5\times10^9\msun$.
Thus, satellites with masses similar to the Magellanic Clouds exhibit the longest environmental quenching timescales.