The Galactic Cold Cores project has made Herschel observations of 116 fields where the Planck survey has found signs of cold dust emission. The fields contain sources in different environments and different phases of star formation. The dust opacity spectral index beta and the dust colour temperature T are derived using Herschel and Planck data. The relation between beta and T is examined for the whole sample and inside individual fields. Based on IRAS and Planck data, the fields are characterised by a median colour temperature of 16.1 K and a median opacity spectral index of beta=1.84. We observe a clear T-beta anti-correlation. In Herschel observations, constrained at lower resolution by Planck data, the variations follow the column density structure and beta(FIR) can rise to ~2.2 in individual clumps. The Planck 217 GHz band shows a systematic excess that is consistent with a general flattening of the dust emission spectrum at millimetre wavelengths. When fitted separately below and above 700 um, the median spectral index values are beta(FIR) ~ 1.91 and beta(mm) ~ 1.66. The spectral index changes as a function of column density and wavelength. Beta variations are partly masked by temperature gradients and the changes in the intrinsic grain properties may be even greater.
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