Beam filling peak intensity, $I_{1.3\mathrm{m}\mathrm{m}}^{\text{bf,peak}}$, versus dual-band spectral index, α. The black circles with error bars show the dual-band detections, of which the KDE is shown in gray contours. The self-consistent dust models (Liu 2019) are shown with shaded regions that cover the temperature range of 6–48 K. The gray, pink, and blue colors represent cases for a_max = 0.01, 1, and 10 mm, respectively. The tracks start with the optically thin case from the bottom where $I_{1.3\mathrm{m}\mathrm{m}}^{\mathrm{b}\mathrm{f},\text{peak}}$, peak increases linearly with surface density and α keeps invariant. As it becomes optically thick, $I_{1.3\mathrm{m}\mathrm{m}}^{\mathrm{b}\mathrm{f},\text{peak}}$ becomes more saturated, and α decreases down to ≃ 2 as in the case of blackbody.