We use Planck HFI data combined with ancillary radio data to study the emissivity index of the interstellar dust emission in the frequency range 100.353 GHz, or 3.0.8 mm, in the Galactic plane. We analyse the region l = 20°44° and |b| ≤ 4° where the free-free emission can be estimated from radio recombination line data. We fit the spectra at each sky pixel with a modified blackbody model and two opacity spectral indices, βmm and βFIR, below and above 353 GHz, respectively. We find that βmm is smaller than βFIR, and we detect a correlation between this low frequency power-law index and the dust optical depth at 353 GHz, 353. The opacity spectral index βmm increases from about 1.54 in the more diffuse regions of the Galactic disk, |b| = 3°4° and τ353 ∼ 5 × 10-5, to about 1.66 in the densest regions with an optical depth of more than one order of magnitude higher. We associate this correlation with an evolution of the dust emissivity related to the fraction of molecular gas along the line of sight. This translates into βmm ∼ 1.54 for a medium that is mostly atomic and βmm ∼ 1.66 when the medium is dominated by molecular gas. We find that both the two-level system model and magnetic dipole emission by ferromagnetic particles can explain the results. These results improve our understanding of the physics of interstellar dust and lead towards a complete model of the dust spectrum of the Milky Way from far-infrared to millimetre wavelengths. © 2014 ESO
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