Recent studies of galaxies in the local Universe, including those in the Local Group, find that the efficiency of environmental (or satellite) quenching increases dramatically at satellite stellar masses below ~ $10^8 {rm M}_{odot}$. This suggests a physical scale where quenching transitions from a slow starvation mode to a rapid stripping mode at low masses. We investigate the plausibility of this scenario using observed HI surface density profiles for a sample of 66 nearby galaxies as inputs to analytic calculations of ram-pressure and viscous stripping. Across a broad range of host properties, we find that stripping becomes increasingly effective at $M_{*} < 10^{8-9} {rm M}_{odot}$, reproducing the critical mass scale observed. However, for canonical values of the circumgalactic medium density ($n_{rm halo} < 10^{-3.5}$ ${rm cm}^{-3}$), we find that stripping is not fully effective; infalling satellites are, on average, stripped of < 40 - 70% of their cold gas reservoir, which is insufficient to match observations. By including a host halo gas distribution that is clumpy and therefore contains regions of higher density, we are able to reproduce the observed HI gas fractions (and thus the high quenched fraction and short quenching timescale) of Local Group satellites, suggesting that a host halo with clumpy gas may be crucial for quenching low-mass systems in Local Group-like (and more massive) host halos.
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