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We now translate the quiescent fractions in Figure~\ref{fig:quiescent_fraction} into the typical timescales over which environmental processes quench satellites after they fall into a host halo, following the methodology of \citet{Wetzel2013}.
First, motivated by the dearth of \emph{isolated} galaxies at $z\approx0$ with $\mstar<10^9\msun$ that are quiescent (see Introduction), our model assumes that all satellites with $\mstar(z=0)<10^9\msun$ were actively star-forming prior to first infall.
However, because most galaxies with $\mstar(z=0)<10^4\msun$ may have been quenched at high redshift by cosmic reionization
\citep[e.g.,][]{Weisz2014a,Brown2014}, \citep[e.g.,][]{Weisz2014a, Brown2014}, we do not model those masses.
At $\mstar(z=0)=10^{4-5}\msun$, satellites' star-formation histories show a mix of complete quenching by $z\gtrsim3$ (e.g., Bootes I, Leo IV) and signs of star formation at $z\lesssim1$ (e.g., And XI, And XII, And XVI)
\citep{Weisz2014a,Weisz2014c,Brown2014}, \citep{Weisz2014a, Weisz2014c, Brown2014}, so quenching at these masses may
arise come from a mix of reionization and the host-halo environment.
%Leo T had recent star formation, suggesting that galaxies at least down to logM_star ~ 5 can form stars today if not for environment.
That said, the 100\% quiescent fraction for satellites at this $\mstar$ means that if both processes are responsible, both are highly efficient.
Furthermore, if the satellites that were quenched by reionization have a similar infall-time distribution to those that were quenched by the host-halo environment, our modeling approach remains valid.
Thus, we include this $\mstar$ in our results but label it distinctly to emphasize caution in interpretation.
Within each 1-dex bin of $\mstar$, we use
the ELVIS
simulations to compute the distribution of infall times
that for satellites at
$z=0$ experienced. $z=0$.
Assuming that environmental quenching
likelihood correlates with time since infall, we designate those that fell in earliest as having
been quenched, and we adjust the time-since-infall threshold for quenching until we match the observed quiescent fraction in each bin.
Several works have shown that this model successfully describes the dependence of satellite quiescent fractions on host-centric distance \citep[e.g.,][]{Wetzel2013, Wetzel2014, Wheeler2014} because infall time correlates with host-centric distance \citep[e.g.,][]{Wetzel2015}.
However, this This correlation means that we must account for observed
satellite's distances, including satellites' distances in computing their infall times.
%including incompleteness at large
distance distances for fainter satellites, in computing their infall times.
Thus, in
selecting satellites in ELVIS, ELVIS we only
use those select satellites out to the maximum host-centric distance that they are observed
from the MW or M31 in each $\mstar$ bin.
While this This turns out to matters
for the fainter satellites, we find that it is most
important at
$\mstar=10^{8-9}\msun$, because the highest mass bin, where all
such observed satellites (M32, NGC 205, LMC/SMC) reside $<61\kpc$ from the MW or M31.
Figure~\ref{fig:quench_times} shows the inferred environmental quenching
timescales, that is, the timescales (the time duration from first infall to being fully
quenched/gas-poor, as a function of $\mstar$. quenched/gas-poor) versus $\mstar$ (top axis shows corresponding subhalo $\mpeak$)..
Blue circles show the
satellite galaxies satellites in the MW and M31, and we shade the lowest $\mstar$ bin
lighter to highlight caution in interpretation because of
reionization, as explained above. reionization.
We derive error bars from the 68\% uncertainty in the observed quiescent fractions in
Figure~\ref{fig:quiescent_fraction}; these Figure~\ref{fig:quiescent_fraction}.
%these uncertainties are typically larger than the host-to-host scatter in satellites' infall times in ELVIS.
As explored in \citet{Wetzel2015}, many satellites first fell into a another host halo (group), typically of $\mvir=10^{10-12}\msun$, before falling into the MW/M31 halos.
Because the importance
of this environmental preprocessing in lower-mass groups remains unclear, we present quenching timescales both
including and neglecting
(left panel) and including (right panel) such group preprocessing.
Thus, %Thus, the left panel of Figure~\ref{fig:quench_times} uses time since infall into the MW/M31 halos, ignoring group preprocessing, while the right panel uses time since infall into \emph{any} host halo, including group preprocessing.
The latter results in longer quenching timescales, though it primarily shifts the upper 16\% of the distribution.
Both panels show shorter median quenching timescales for less massive satellites: $\sim5\gyr$ at $\mstar=10^{8-9}\msun$, $2-3\gyr$ at $\mstar=10^{7-8}\msun$, and less than $1.5\gyr$ at $\mstar<10^7\msun$, depending on the inclusion of group preprocessing.
Moreover, the median timescale for two of the lowest $\mstar$ bins is $0\gyr$ because 100\% of those satellites are quiescent, which implies
that quenching must occur extremely
rapidly rapid quenching after infall.
We next compare these statistically based quenching timescales to
infall/quenching infall timescales directly measured for satellites of the MW.
The 3-D orbital velocity measured for the LMC/SMC strongly suggests that they are on their first infall and passed inside $\rvir$ of the MW $\approx2\gyr$ ago \citep{Kallivayalil2013}.
Given that both remain star-forming, this places a lower limit to their quenching timescale (gray triangle),
which is consistent with our statistical
timescales at similar mass. timescales.
Similarly, measurements of the 3-D orbital velocity and star-formation history for Leo I ($\mstar=5.5\times10^6\msun$) indicate that it fell into the MW halo $\approx2.3\gyr$ ago and quenched $\approx1\gyr$ ago (near its $\approx90\kpc$ pericentric passage), implying a quenching timescale of $\approx1.3\gyr$ \citep[][gray pentagon]{Sohn2013}, again consistent with our results.
We also compare these timescales for satellites with $\mstar\lesssim10^9\msun$ within the MW/M31 halos with previous studies of more massive satellites
within of other
host halos. hosts.
The red squares in Figure~\ref{fig:quench_times} show the timescales from \citet{Wheeler2014}, who used nearly identical methodology, combining the the galaxy catalog from \citet{Geha2012} with satellite infall times (including group preprocessing) from simulation.
% the Millennium II simulation \citep{BoylanKolchin2009}
They examined satellites with $\mstar\approx10^{8.5}$ and $10^{9.5}\msun$ around hosts with $\mstar>2.5\times10^{10}\msun$, which they found likely spans
$\mvir\approx10^{12.5-14}\msun$.
These are $\mvir\approx10^{12.5-14}\msun$, much higher
masses than the
MW/M31, which MW/M31.
%which could mean that the quenching timescales in \citet{Wheeler2014} are \emph{shorter} than for similar mass satellites of MW/M31-like hosts.
Similarly, the green curves in Figure~\ref{fig:quench_times} show the quenching timescales for more massive satellites
in groups with groups with $\mvir=10^{12-13}\msun$ from \citet{Wetzel2013}, who also used identical methodology, combining
a galaxy
groups group catalog from SDSS \citep{Tinker2011, Wetzel2012} with satellite infall times (including group preprocessing) measured in
mock group catalogs in their cosmological simulation.
We %We show their result for groups with $\mvir=10^{12-13}\msun$, which are most similar to MW/M31 masses.
Altogether, Figure~\ref{fig:quench_times} indicates a complex dependence of the environmental quenching timescale on satellite $\mstar$.
Specifically, the typical timescale for satellites in the MW/M31 halos increases with $\mstar$, from $\lesssim1\gyr$ at $\mstar<10^7\msun$ to $\sim5\gyr$ at $\mstar\approx10^{8.5}\msun$.
\citet{Wheeler2014} indicate that this mass dependence continues, though with a rapid increase ($\sim2\times$) to $\approx9.5\gyr$, and no change from $\mstar\approx10^{8.5}$ to $10^{9.5}\msun$.
This rapid increase implies some tension with our
results, specifically, results based on the two
quiescent satellites of M31, NGC 205 and M32 ($\mstar\approx10^{8.5}\msun$), unless
either (1) both experienced unusually early infall $>9.5\gyr$
ago, ago or
(2) M31
quenches quenched its satellites
particularly rapidly, even compared with much more rapidly than the
massive (more massive) hosts in \citet{Wheeler2014}.
%(\citeauthor{Wheeler2014}'s results are consistent with the star-forming LMC/SMC of the MW.)
Finally, At higher $\mstar$, \citet{Wetzel2013} indicate that the quenching timescale rapidly \emph{decreases}
at $\mstar>5\times10^9\msun$ by $5\times10^9\msun$ and continues to decline with increasing $\mstar$.
Overall, the typical environmental quenching timescales are shortest for the lowest-mass satellites and
is are longest for satellites with
$\mstar\sim10^9\msun$ (roughly $\mstar\sim10^9\msun$, roughly Magellanic-Cloud
mass). mass.
