We explore the quenching of low-mass galaxies (10^4 < Mstar < 10^8 Msun) as a function of lookback time using the star formation histories (SFHs) of 38 Local Group dwarf galaxies. The SFHs were derived from analyzing color-magnitude diagrams of resolved stellar populations in archival Hubble Space Telescope/Wide Field Planetary Camera 2 imaging. We find: (1) Lower mass galaxies quench earlier than higher mass galaxies; (2) Inside of virial radius there is no correlation between a satellites current proximity to a massive host and its quenching epoch; (3) There are hints of systematic differences in quenching times of M31 and Milky Way (MW) satellites, although the sample sample size and uncertainties in the SFHs of M31 dwarfs prohibit definitive conclusions. Combined with literature results, we qualitatively consider the redshift evolution (z=0-1) of the quenched galaxy fraction over ~7 dex in stellar mass (10^4 < Mstar < 10^11.5 Msun). The quenched fraction of all galaxies generally increases toward the present, with both the lowest and highest mass systems exhibiting the largest quenched fractions at all redshifts. In contrast, galaxies between Mstar ~ 10^8-10^10 Msun have the lowest quenched fractions. We suggest that such intermediate-mass galaxies are the least efficient at quenching. Finally, we compare our quenching times with predictions for infall times of low-mass galaxies associated with the MW. We find that some of the lowest-mass satellites (e.g., CVn II, Leo IV) may have been quenched before infall while higher mass satellites (e.g., Leo I, Fornax) typically quench ~1-4 Gyr after infall.
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