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We conclude by briefly discussing the complex dependence of satellite quenching timescales on $\mstar$ from Figure~\ref{fig:quench_times} in the context of the underlying physics of environmental quenching.  At $\mstar\gtrsim10^9\msun$, the long quenching timescales suggests quenching driven by gas depletion in the absence of cosmic accretion after infall (``strangulation''), caused by gravitational tidal stripping and/or ram-pressure stripping of the extended gas around the satellite.  Furthermore, this scenario explains the decline of the quenching timescale with increasing $\mstar$, because higher-mass galaxies generally have lower $\mgas/\mstar$ \citep[in either cold atomic or molecular gas, e.g.,][Bradford et al., submitted]{Schiminovich2010, Saintonge2011a, Huang2012, Boselli2014} submitted]{Schiminovich2010,Saintonge2011a,Huang2012,Boselli2014}  and thus shorter gas depletion timescales in the absence of accretion. Indeed, at $\mstar\sim10^9\msun$, galaxies transition through $\mgas/\mstar\approx1$, with gas depletion timescales comparable to a Hubble time.  In this scenario, the quenching timescales at $\mstar\gtrsim10^9\msun$ do not necessarily require strong additional environmental processes other than the lack of gas accretion to account for quenching in satellites \citep[see related discussions in, e.g.,][]{Wetzel2013, Wheeler2014, McGee2014}. e.g.,][]{Wetzel2013,Wheeler2014,McGee2014}.  However, this scenario cannot explain the rollover in the quenching time at $\mstar\lessim10^9\msun$, because the star-forming dwarf galaxies of the LG also have $\mgas\gtrsim\mstar$ \citep{GrgvichPutman2009}, and thus contain enough gas to fuel star formation for a Hubble time, even absent accretion.  Thus, the rapid decline of the environmental quenching time \emph{requires} an additional process that can remove gas from these dwarf galaxies after infall.  This likely arises from ram-pressure stripping of cold gas in such satellites, whose lower-mass host (sub)halos have shallower potential wells.  Furthermore, the same internal stellar feedback that regulates the low star-formation efficiency in such dwarf galaxies and likely drives significant gas flows to large radii \citep[e.g.,][]{Muratov2015} would strongly assist such environmental stripping to make even more efficient in dwarf galaxies.  In this sense, the rapid environmental quenching timescales for dwarf galaxies likely arise not just from the role of the external environment, but from the non-linear interplay of 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.,][]{Behroozi2013}, 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 satellites to quench more rapidly after infall.