The dust mass absorption coefficient, $kappa_{d}$, is the conversion function used to infer physical dust masses from observations of dust emission. However, it is notoriously poorly constrained, and it is highly uncertain how it varies, either between or within galaxies. Here we present the results of a proof-of concept study, using the DustPedia data for two nearby face-on spiral galaxies M74 (NGC 628) and M83 (NGC 5236), to create the first ever maps of $kappa_{d}$ in galaxies. We determine $kappa_{d}$ using an empirical method that exploits the fact that the dust-to-metals ratio of the interstellar medium is constrained by direct measurements of the depletion of gas-phase metals. We apply this method pixel-by-pixel within M74 and M83, to create maps of $kappa_{d}$. We also demonstrate a novel method of producing metallicity maps for galaxies with irregularly-sampled measurements, using the machine learning technique of Gaussian process regression. We find strong evidence for significant variation in $kappa_{d}$. We find values of $kappa_{d}$ at 500 $mu$m spanning the range 0.11-0.25 ${rm m^{2},kg^{-1}}$ in M74, and 0.15-0.80 ${rm m^{2},kg^{-1}}$ in M83. Surprisingly, we find that $kappa_{d}$ shows a distinct inverse correlation with the local density of the interstellar medium. This inverse correlation is the opposite of what is predicted by standard dust models. However, we find this relationship to be robust against a large range of changes to our method - only the adoption of unphysical or highly unusual assumptions would be able to suppress it.
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