Effect of ¹³CO optical depth on CO line ratios under LTE (top) and non-LTE (bottom) conditions. We assume optically thick ¹²CO and optically thin C¹⁸O emission. Top left: variation of R_18/13 with τ_¹³CO for different abundance ratios of the ¹³CO and C¹⁸O species. We do not significantly detect C¹⁸O in the center of the galaxy. The orange shaded region hence shows the region where the line ratio in M101’s central 4 kpc region could fall. Average MW value (solid line; Wouterloot et al. 2008), starburst (dashed line; e.g., Tan et al. 2011), and ULIRGs (dotted; e.g., Greve et al. 2009) is shown. Top right: variation of R_13/12 with τ_¹³CO for different beam filling factor ratios between the ¹²CO and ¹³CO emission, η₁₂ and η₁₃. The blue band shows the range of measured R_13/12 values in M101. Average MW value from Roman-Duval et al. (2016), starburst and ULIRGs from Sliwa & Downes 2017. Bottom panels: we use the line ratio model calculation provided in Leroy et al. (2022). These are based on model calculations with RADEX (van der Tak et al. 2007) and used a lognormal density distributions described in Leroy et al. (2017b). Bottom left: we fix the ratio at R₂₁ = 0.6 and use a lognormal density distribution width of σ = 0.6. The color-coded lines show as a function of H₂ volume density (x-axis) and kinetic temperature T_kin (y-axis) the trends for different total CO column densities per line width (N_CO/Δν), which roughly correlate with the optical depth. Bottom center: the line ratio R₂₁ as a function of kinetic temperature for different mean molecular gas densities. Bottom right: the line ratio R₂₁ as a function of the mean molecular gas density for different kinetic temperatures. The green shaded region shows the 16th to 84th percentile range for the values found in M101. For the computation of R₂₁ as function of temperature and density, we fix σ = 0.6 dex and N_CO/Δv = 10¹⁶ cm⁻² (km s⁻¹)⁻¹. We note that we assume a common excitation (density, temperature) for all species and LTE for the top panels.