χ²-map showing the quality of the fit to available HCN multi-transition single-dish data for the AGB stars R For (left panel), W Aql (middle panel), and TX Cam (right panel) when varying the adjustable parameters, f₀ and r_e, in the model. Contours are drawn at the 1, 2, and 3σ levels. Indicated in the upper right corner is the reduced χ² of the best-fit model and the number of observational constraints used, N.