optically thin case represented by H$_2$$^{13}$CO. The J = 1 H$_2$CO doublet then gets a little depopulated and thus the C-band H$_2$CO/H$_2$$^{13}$CO line intensity ratio becomes smaller than if both H$_2$CO and H$_2$$^{13}$CO lines were all optically thin. Here, the RADEX non-LTE model^\[Radex\]^ was used to correct the ratios for photon trapping. As already mentioned in Sect. \[sec:intro\], the collision rates of H$_2$CO are taken from @Wiesenfeld2013, which are calculated for H$_2$CO in collision with H$_2$ and the high accuracy potential energy surface of Troscompt et al. (). These are significantly different from the scaled rates of H$_2$CO in collision with He (@1991ApJS...76..979; see below). Correction factors $f_{12/13}$ calculated by the new collision rates tend to be larger than those derived by the old collision rates, especially for larger optical depths at 4.8 GHz, as shown in Figure \[fig3\]. The correction factor $f_{12/13}$ is defined as in Henkel et al. (): $$f_{12/13} = 50 \tau^{0}_{4.8}/\tau^{*}_{4.8},$$ where the star indicates the RADEX model and the superscript zero refers to the model with the same H$_2$ density and temperature but with a formaldehyde column density which is a factor 50 times lower. The results are listed in Table \[Table3\]. We ran the 