International audienceContext. The Galactic Cold Cores project has carried out Herschel photometric 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. Previous studies have revealed variations in their dust submillimetre opacity.Aims. The aim is to measure the value of dust opacity spectral index and to understand its variations spatially and with respect to other parameters, such as temperature, column density, and Galactic location.Methods. The dust opacity spectral index β and the dust colour temperature T are derived using Herschel and Planck data. The relation between β and T is examined for the whole sample and inside individual fields. Results. 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 β = 1.84. The values are not correlated with Galactic longitude. We observe a clear T–β anti-correlation. In Herschel observations, constrained at lower resolution by Planck data, the variations follow the column density structure and βFIR can rise to ~2.2 in individual clumps. The highest values are found in starless clumps. The Planck 217 GHz band shows a systematic excess that is not restricted to cold clumps and is thus consistent with a general flattening of the dust emission spectrum at millimetre wavelengths. When fitted separately below and above 700 μm, the median spectral index values are βFIR ~ 1.91 and β(mm) ~ 1.66. Conclusions. The spectral index changes as a function of column density and wavelength. The comparison of different data sets and the examination of possible error sources show that our results are robust. However, β variations are partly masked by temperature gradients and the changes in the intrinsic grain properties may be even greater
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