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 nH2 and Tkin. 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 Tkin) at high density because the H2CO lines start to become saturated; this causes intensity ratios for a given Tkin 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.
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