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Optical Depth Estimates and Effective Critical Densities of Dense Gas Tracers in the Inner Parts of Nearby Galaxy Discs

High critical density molecular lines like HCN(1-0) or HCO+(1-0) represent our best tool to study currently star-forming, dense molecular gas at extragalactic distances. The optical depth of these lines is a key ingredient to estimate the effective density required to excite emission. However, constraints on this quantity are even scarcer in the literature than measurements of the high density tracers themselves. Here, we combine new observations of HCN, HCO+ and HNC(1-0) and their optically thin isotopologues H13CN, H13CO+ and HN13C(1-0) to measure isotopologue line ratios. We use IRAM 30-m observations from the large program EMPIRE and new ALMA observations, which together target 6 nearby star-forming galaxies. Using spectral stacking techniques, we calculate or place strong upper limits on the HCN/H13CN, HCO+/H13CO+ and HNC/HN13C line ratios in the inner parts of these galaxies. Under simple assumptions, we use these to estimate the optical depths of HCN(1-0) and HCO+(1-0) to be τ~2-11 in the active, inner regions of our targets. The critical densities are consequently lowered to values between 5-20$\times 10^5$, 1-3$\times 10^5$ and 9$\times 10^4$ cm-3 for HCN, HCO+ and HNC, respectively. We study the impact of having different beam-filling factors, $η$, on these estimates and find that the effective critical densities decrease by a factor of $\frac{η_{12}}{η_{13}}\,τ_{12}$. A comparison to existing work in NGC 5194 and NGC 253 shows HCN/H13CN and HCO+/H13CO+ ratios in agreement with our measurements within the uncertainties. The same is true for studies in other environments such as the Galactic Centre or nuclear regions of AGN-dominated nearby galaxies.