Relationship between temperature, velocity gradient, and column density required to reproduce the observed intensities. At each point, the density is determined to match the ¹²CO ﹩J=7\rightarrow 6﹩/﹩J=3\rightarrow 2﹩ line ratio; then the observed integrated intensities are compared with the LVG model to determine the column density required. Higher temperatures and lower optical depths require smaller column densities. The constraints based on the ¹³CO/¹²CO ﹩J=3\rightarrow 2﹩ ratio are shown as boxes for ﹩dv/ dz=10﹩, 20, 40, and 80 km s⁻¹ pc⁻¹. The sizes of the boxes are estimates of the uncertainties.