Dust opacity index β_mm recovered from the simulated maps, as a function of the input τ₃₅₃. The simulated maps in a) have dust emission and CMB, and in b) consist of dust, free-free and CO emission. The black line shows the input β_mm of 1.52 in each case; the red lines give the β_mm–τ₃₅₃ relationship derived in Sect. 5. The colour scale is logarithmic and it represents the density of points. The three contours show the densities for the cumulated fractions, given by the values in each panel, of the data points, from red to yellow.