We have modelled the near-infrared to radio images of the Crab Nebula with a Bayesian SED model to simultaneously fit its synchrotron, interstellar and supernova dust emission. We infer an interstellar dust extinction map with an average $A_{text{V}}$=1.08$pm$0.38 mag, consistent with a small contribution (<22%) to the Crabs overall infrared emission. The Crabs supernova dust mass is estimated to be between 0.032 and 0.049 M$_{odot}$ (for amorphous carbon grains) with an average dust temperature $T_{text{dust}}$=41$pm$3K, corresponding to a dust condensation efficiency of 8-12%. This revised dust mass is up to an order of magnitude lower than some previous estimates, which can be attributed to our different interstellar dust corrections, lower SPIRE flux densities, and higher dust temperature than were used in previous studies. The dust within the Crab is predominantly found in dense filaments south of the pulsar, with an average V-band dust extinction of $A_{text{V}}$=0.20-0.39 mag, consistent with recent optical dust extinction studies. The modelled synchrotron power-law spectrum is consistent with a radio spectral index $alpha_{text{radio}}$=0.297$pm$0.009 and an infrared spectral index $alpha_{text{IR}}$=0.429$pm$0.021. We have identified a millimetre excess emission in the Crabs central regions, and argue that it most likely results from two distinct populations of synchrotron emitting particles. We conclude that the Crabs efficient dust condensation (8-12%) provides further evidence for a scenario where supernovae can provide substantial contributions to the interstellar dust budgets in galaxies.
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