Example of LVG modeling for P4. Top: reduced χ² distribution (mainly vertical contours) for a single-component LVG model fit to the H₂CO brightness temperatures (black contours, χ²=1,2,4), as well as H₂CO 3₂₂ → 2₂₁/3₀₃ → 2₀₂ line ratios (mainly horizonal contours) as a function of n_H2 and T_kin. The solid lines represent the line ratios: 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4. The red dashed lines show the observed line ratio and its lower and upper limits. The para-H₂CO abundances per velocity gradient, [para-H₂CO]/(dv/dr), for the LVG models are 2 × 10^-10 pc(km s^-1)^-1 (left) and 2 × 10^-11pc(km s^-1)^-1 (right), respectively. In the left panel the lines with a given H₂CO line ratio move downwards (lower T_kin) at high density because the H₂CO lines start to become saturated; this causes intensity ratios for a given T_kin to get closer to unity. Bottom: reduced χ² distribution for the H₂CO 3₂₁ → 2₂₀/3₀₃ → 2₀₂ line ratios. The kinetic temperature is sensitive to the gas density so this line ratio is a less suitable thermometer.