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\section{Discussion}  We conclude by briefly discussing thecomplex  dependence of satellite quenching timescales on $\mstar$ from Figure~\ref{fig:quench_times} in the context of the underlying physics of environmental quenching. physical drivers.  At $\mstar\gtrsim10^9\msun$, the long quenching timescales suggests quenching driven by gas depletion in the absence of cosmic accretion after infall (``strangulation''), accretion,  caused by gravitational tidal stripping and/or ram-pressure the  stripping ofthe  extended gas around the satellite.  Furthermore, this satellite, after infall (``strangulation'').  This  scenario also  explains the decline of the quenching timescale with increasing $\mstar$, because higher-mass higher-$\mstar$ (non-satellite)  galaxies generally have lower $\mgas/\mstar$ \citep[in either cold atomic or molecular gas, e.g.,][Bradford et al., submitted]{Schiminovich2010,Huang2012,Boselli2014} submitted]{Schiminovich2010, Huang2012, Boselli2014}  and thus shorter gas depletion timescales in the absence of accretion. Indeed, Conversely,  at $\mstar\sim10^9\msun$, galaxies transition through have  $\mgas/\mstar\approx1$, with gas depletion timescales comparable to a Hubble time. In this scenario, the Thus, satellite  quenching timescales at $\mstar\gtrsim10^9\msun$ do not necessarily require \emph{require}  strongadditional  environmental processes other than the lack of beyond truncated  gas accretionto account for quenching in satellites  \citep[see related also  discussions in, e.g.,][]{Wetzel2013,Wheeler2014,McGee2014}. in][]{Wetzel2013, Wheeler2014, Phillips2014, McGee2014}.  However, this scenario strangulation  cannot explain the rollover in the satellite  quenching time times  at $\mstar\lesssim10^9\msun$, because the star-forming gas-rich  dwarf galaxies of the LG also have $\mgas\gtrsim\mstar$ \citep{GrcevichPutman2009}, \citep{GrcevichPutman2009}  and thus contain enough cold  gas to fuel star formation for a Hubble time, time  even absent accretion. Thus, the rapid decline of the environmental quenching time at lower $\mstar$  \emph{requires} an additional process that can process(es) to  remove gas from these satellite  dwarf galaxies after infall. This likely arises from the increased efficiency of  ram-pressure stripping of in removing  cold gas in from  such satellites, whose lower-mass host (sub)halos have satellites with  shallower potential wells. Furthermore, Moreover, for dwarf galaxies,  the same internal stellar feedback that regulates the their  low star-formation efficiencyin such dwarf galaxies  and likely drives heats/drives  significant cold  gasflows  to large radii \citep[e.g.,][]{Muratov2015} wouldstrongly  assist such environmental stripping to make become  even more efficient in dwarf galaxies.  In this sense, efficient.  Thus,  the rapid environmental quenching timescales for dwarf galaxies likely may  arisenot just from the role of the external environment, but  from the non-linear interplay of both  internal feedback and external stripping \citep[e.g.,][]{NicholsBlandHawthorn2011,BaheMcCarthy2015}. \citep[e.g.,][]{NicholsBlandHawthorn2011, BaheMcCarthy2015}.  The above scenario may also help to explain the curious similarity of Figure~\ref{fig:quench_times} with the mass dependence of the underlying galaxy-halo $\mstar/\mvir$ relation \citep[e.g.,][]{Behroozi2013c}, which peaks at $\mstar\sim10^{10}\msun$ (higher but similar mass).  In particular, at high $\mstar$, the same process that lowers $\mstar/\mvir$ with increases mass also lowers the underlying gas fraction, which in turn causes more massive Overall,  satellites with $\mstar\sim10^9\msun$ (similar  toquench more rapidly after infall.  At low $\mstar$,  the same shallower potential wells that allow Magellanic Clouds) represent the transition between these effects, and no quenching mechanism (either  internal feedback to lower $\mstar/\mvir$ also allows external stripping or external) appears  to occur more easily. operate efficiently near this mass \citep[see also][]{Weisz2015}.  Overall, satellites Finally, we note that the above scenario may explain the curious, though qualitative, similarity of Figure~\ref{fig:quench_times}  with$\mstar\sim10^9\msun$ (near  the Magellanic Clouds) represent mass dependence of  the transition between these effects, underlying galaxy-halo $\mstar/\mvir$ relation, which is low at both high and low $\mstar$  and peaks at $\mstar\sim10^{10}\msun$ \citep[e.g.,][]{Behroozi2013c}.  In particular, at high $\mstar$, the same physical process(es) that lowers $\mstar/\mvir$ also lowers a galaxy's cold gas fraction, which  in general, there appears turn causes more massive satellites  to be no quenching mechanism (either internal or external) quench more rapidly.  At low $\mstar$, the same shallower potential wells  that operates efficiently at such masses \citep[see also][]{Weisz2015}. allow internal feedback to lower $\mstar/\mvir$ also allows external stripping to occur more easily and quenching to occur more rapidly.  %This analysis represents a statistical approach, but in future work we will combine the measured SFHs with the orbtal phase-space coordinates of each satellites to pursue a similar but more rigorous analysis on a satellite-by-satellite basis.