Observed (markers) and modelled (lines) intensity ratios between the HCN J  =  1  →  0 and J  =  3  →  2 lines as a function of a density measure (Ṁ/v_e) of the CSE. Observations of carbon stars are indicated by blue circles, M-type AGB stars by red squares, and S-type AGB stars by green triangles. Model predictions with HCN abundances of 1 × 10^-5 (solid line) and 1 × 10^-7 (dashed line) are shown for comparison for a circumstellar model as described in the text.