Density dependence of the HCNH⁺ brightness temperature ratio for the j₁ = 1 → 0, 2 → 1, and 3 → 2 lines. Solid and dashed lines were obtained using the collisional rate coefficients of H₂ and He, respectively. The dash-dotted line refers to thermalization. The line width was set to 1.0 km s⁻¹. Thanks to the optically thin regime, these ratios are valid for column densities lower than 5 × 10¹³ cm⁻².