Dependence of radiation temperature ratios on gas density for the 1 → 0, 2 → 1, 3 → 2, and 4 → 3 emission lines. Superscripts CC and CS denote that the rate coefficients used in the radiative transfer calculations were obtained using the close-coupling (CC) method and the coupled states’ (CS) approximation, respectively. The dashed line indicates perfect agreement between the methods. All calculations are performed under the LVG formalism as implemented in the RADEX code, assuming a FWHM line width of 1.0 km s⁻¹ and an HCNH⁺ column density of 2.3 × 10¹³ cm⁻² (van der Tak et al. 2007).