The Metallicity Dependence of the CO-to-H$_2$ Conversion Factor from Observations of Local Group Galaxies

High-resolution CO maps of 9 molecular clouds in IC 10 are combined with the new measurement of the distance to this nearby metal poor galaxy to measure accurately the CO-to-H2 conversion factor. The result for IC 10 is combined with published data for four other Local Group galaxies (M31, M33, NGC 6822, the SMC) to trace the dependence of the CO-to-H2 conversion factor on oxygen abundance. These data show conclusively that the CO-to-H$_2$ conversion factor increases as the metallicity of the host galaxy decreases, with the conversion factor increasing by a factor of 4.6 for a factor of 10 decrease in metallicity.