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High critical density molecular lines like HCN(1-0) or HCO+(1-0) representour best tool to study currently star-forming, dense molecular gas atextragalactic distances. The optical depth of these lines is a key ingredientto estimate the effective density required to excite emission. However,constraints on this quantity are even scarcer in the literature thanmeasurements of the high density tracers themselves. Here, we combine newobservations of HCN, HCO+ and HNC(1-0) and their optically thin isotopologuesH13CN, H13CO+ and HN13C(1-0) to measure isotopologue line ratios. We use IRAM30-m observations from the large program EMPIRE and new ALMA observations,which together target 6 nearby star-forming galaxies. Using spectral stackingtechniques, 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 ofHCN(1-0) and HCO+(1-0) to be \tau ~2-11 in the active, inner regions of ourtargets. The critical densities are consequently lowered to values between5-20$\times 10^5$, 1-3$\times 10^5$ and 9$\times 10^4$ cm-3 for HCN, HCO+ andHNC, respectively. We study the impact of having different beam-fillingfactors, $\eta$, on these estimates and find that the effective criticaldensities decrease by a factor of $\frac{\eta_{12}}{\eta_{13}}\,\tau_{12}$. Acomparison to existing work in NGC 5194 and NGC 253 shows HCN/H13CN andHCO+/H13CO+ ratios in agreement with our measurements within the uncertainties.The same is true for studies in other environments such as the Galactic Centreor nuclear regions of AGN-dominated nearby galaxies.

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