Grids of column density, N, versus kinetic temperature, T_k, for the case of HCOOCH₃ A when excitation temperature, T_ex, was calculated at each point for the 8_1,7 - 7_1,6 A transition (top) and the 3_0,3 - 2_0,2 A transition (middle). Overlaid on the grids are minimization contours for these two ‘A’ state transitions, where 8_1,7 - 7_1,6 A is in purple and 3_0,3 - 2_0,2 A is in cyan. The volume density is set to 7.72 × 10⁵ cm⁻³ and 9.75 × 10⁵ cm⁻³ for the 8_1,7 - 7_1,6 A and 3_0,3 - 2_0,2 A lines, respectively. In the case of ‘Minimization 1’, black arrows denote when the two modeled transitions are both best matched to the observations and overlap in N and T_k space. In the case of ‘Minimization 2’, black arrows denote where in parameter space we would expect to be if only the 3_0,3 - 2_0,2 transition was considered. Minimization curves at the bottom of the figure are also extracted at slices of the grid where T_k = 14 K, and T_k = 27 K to further illustrate how a best-fit was determined.