The apparent anti-correlation between the mass opacity of interstellar dust and the surface-density of interstellar gas

Recent analyses of $Herschel$ observations suggest that in nearby disc galaxies the dust mass opacity at $500\mu \mathrm{m}$, ${\kappa }_{500}$, decreases with increasing gas surface density, ${\mathrm{\Sigma }}_{\mathrm{I}\mathrm{S}\mathrm{M}}$ (Clark et al. 2019). This apparent anti-correlation between ${\kappa }_{500}$ and $\Sigma_{\rm ISM}$ is opposite to the behaviour expected from theoretical dust evolution models; in such models, dust in denser, cooler regions (i.e. regions of increased ${\mathrm{\Sigma }}_{\mathrm{I}\mathrm{S}\mathrm{M}}$) tends to grow and therefore to have increased ${\kappa }_{500}$. We show, using a toy model, that the presence of a range of dust temperatures along the line of sight can lead to spuriously low estimated values of ${\kappa }_{500}$. If in regions of higher ${\mathrm{\Sigma }}_{\mathrm{I}\mathrm{S}\mathrm{M}}$ the range of dust temperatures extends to lower values (as seems likely), the magnitude of this effect may be sufficient to explain the apparent anti-correlation between ${\kappa }_{500}$ and ${\mathrm{\Sigma }}_{\mathrm{I}\mathrm{S}\mathrm{M}}$. Therefore there may not be any need for spatial variation in the intrinsic dust properties that run counter to theoretical expectations.