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Carbon monoxide (CO) provides crucial information about the molecular gasproperties of galaxies. While $^{12}$CO has been targeted extensively,isotopologues such as $^{13}$CO have the advantage of being less opticallythick and observations have recently become accessible across full galaxydiscs. We present a comprehensive new dataset of $^{13}$CO(1-0) observationswith the IRAM 30-m telescope of the full discs of 9 nearby spiral galaxies fromthe EMPIRE survey at a spatial resolution of $\sim$1.5kpc. $^{13}$CO(1-0) ismapped out to $0.7-1r_{25}$ and detected at high signal-to-noise throughout ourmaps. We analyse the $^{12}$CO(1-0)-to-$^{13}$CO(1-0) ratio ($\Re$) as afunction of galactocentric radius and other parameters such as the$^{12}$CO(2-1)-to-$^{12}$CO(1-0) intensity ratio, the 70-to-160$\mu$m fluxdensity ratio, the star-formation rate surface density, the star-formationefficiency, and the CO-to-H$_2$ conversion factor. We find that $\Re$ varies bya factor of 2 at most within and amongst galaxies, with a median value of 11and larger variations in the galaxy centres than in the discs. We argue thatoptical depth effects, most likely due to changes in the mixture ofdiffuse/dense gas, are favored explanations for the observed $\Re$ variations,while abundance changes may also be at play. We calculate a spatially-resolved$^{13}$CO(1-0)-to-H$_2$ conversion factor and find an average value of$1.0\times10^{21}$ cm$^{-2}$ (K.km/s)$^{-1}$ over our sample with a standarddeviation of a factor of 2. We find that $^{13}$CO(1-0) does not appear to be agood predictor of the bulk molecular gas mass in normal galaxy discs due to thepresence of a large diffuse phase, but it may be a better tracer of the massthan $^{12}$CO(1-0) in the galaxy centres where the fraction of dense gas islarger.

Full-disc $^{13}$CO(1-0) mapping across nearby galaxies of the EMPIRE survey and the CO-to-H$_2$ conversion factor