The vast majority of dwarf satellites orbiting the Milky Way and M31 are quenched, while comparable galaxies in the field are gas-rich and star-forming. Assuming that this dichotomy is driven by environmental quenching, we use the ELVIS suite of N-body simulations to constrain the characteristic timescale upon which satellites must quench following infall into the virial volumes of their hosts. The high satellite quenched fraction observed in the Local Group demands an extremely short quenching timescale (~ 2 Gyr) for dwarf satellites in the mass range Mstar ~ 10^6-10^8 Msun. This quenching timescale is significantly shorter than that required to explain the quenched fraction of more massive satellites (~ 8 Gyr), both in the Local Group and in more massive host halos, suggesting a dramatic change in the dominant satellite quenching mechanism at Mstar < 10^8 Msun. Combining our work with the results of complementary analyses in the literature, we conclude that the suppression of star formation in massive satellites (Mstar ~ 10^8 - 10^11 Msun) is broadly consistent with being driven by starvation, such that the satellite quenching timescale corresponds to the cold gas depletion time. Below a critical stellar mass scale of ~ 10^8 Msun, however, the required quenching times are much shorter than the expected cold gas depletion times. Instead, quenching must act on a timescale comparable to the dynamical time of the host halo. We posit that ram-pressure stripping can naturally explain this behavior, with the critical mass (of Mstar ~ 10^8 Msun) corresponding to halos with gravitational restoring forces that are too weak to overcome the drag force encountered when moving through an extended, hot circumgalactic medium.
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