Mean microwave dust SED obtained by cross-correlating the Planck and WMAP data with the Hi and 408 MHz templates (blacks dots for Planck and blue squares for WMAP). Top: model 1 in Table 4, with two emission components combining AME and greybody thermal dust emission with β = 1.52. The AME is fitted with the analytical model in Eq. (10). Bottom: spectral fit for model 3, where the AME is fitted with two SPDUST spectra peaking at different frequencies.