We have analysed Herschel observations of M31, using the PPMAP procedure. The resolution of PPMAP images is sufficient (31 pc on M31) that we can analyse far-IR dust emission on the scale of Giant Molecular Clouds. By comparing PPMAP estimates of the far-IR emission optical depth at 300 microns (tau_300), and the near-IR extinction optical depth at 1.1 microns (tau_1.1) obtained from the reddening of RGB stars, we show that the ratio R_OBS.tau = tau_1.1/tau_300 falls in the range 500 to 1500. Such low values are incompatible with many commonly used theoretical dust models, which predict values of R_MODEL.kappa = kappa_1.1/kappa_300 (where kappa is the dust opacity coefficient) in the range 2500 to 4000. That is, unless a large fraction, at least 60%, of the dust emitting at 300 microns is in such compact sources that they are unlikely to intercept the lines of sight to a distributed population like RGB stars. This is not a new result: variants obtained using different observations and/or different wavelengths have already been reported by other studies. We present two analytic arguments for why it is unlikely that at least 60% of the emitting dust is in sufficiently compact sources. Therefore it may be necessary to explore the possibility that the discrepancy between observed values of R_OBS.tau and theoretical values of R_MODEL.kappa is due to limitations in existing dust models. PPMAP also allows us to derive optical-depth weighted mean values for the emissivity index, beta = - dln(kappa_lambda)/dln(lambda), and the dust temperature, T, denoted betabar and Tbar. We show that, in M31, R_OBS.tau is anti-correlated with betabar according to R_OBS.tau = 2042(+/-24)-557(+/-10)betabar. If confirmed, this provides a challenging constraint on the nature of interstellar dust in M31.
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