Results from excitation analysis of HCN multi-transition single-dish observations assuming a Gaussian abundance distribution (see Eq. (2)). Top panel – derived initial fractional abundance (f₀) as a function of envelope density (Ṁ/v_e) for M- (squares) and S-type AGB stars (triangles). Bottom panel – derived envelope sizes (r_e) as a function of envelope density (Ṁ/v_e) for M- (squares) and S-type AGB stars (triangles). The solid line is a least-squares fit to the data points as described by Eq. (3). The dashed and dotted lines show the results from a simple photochemical model (Sect. 4.1.1). The dashed line is calculated for h = 0.2 and the dotted line for h = 0.5 (see text for details). The carbon star R For (dot) is a also shown in the plots.