Dense Gas in the Outer Spiral Arm of M51

There is a linear relation between the mass of dense gas, traced by the HCN(1-0) luminosity, and the star formation rate (SFR), traced by the far-infrared luminosity. Recent observations of galactic disks have shown some systematic variations. In order to explore the SFR-dense gas link at high resolution ($\sim 4"$, $\sim 150$ pc) in the outer disk of an external galaxy, we have mapped a region about 5 kpc from the center along the northern spiral arm of M51 in the HCN(1-0), HCO$^+$(1-0) and HNC(1-0) emission lines using the Northern Extended Millimeter Array (NOEMA) interferometer. The HCN and HCO$^+$ lines were detected in 6 giant molecular associations (GMAs) while HNC emission was only detected in the two brightest GMAs. One of the GMAs hosts a powerful HII region and HCN is stronger than HCO$^+$ there. Comparing with observations of GMAs in the disks of M31 and M33 at similar angular resolution ($\sim 100$ pc), we find that GMAs in the outer disk of M51 are brighter in both HCN and HCO$^+$ lines by a factor of 3 on average. However, the $I_{HCN}/I_{CO}$ and $I_{HCO^+}/I_{CO}$ ratios are similar to the ratios in nearby galactic disks and the Galactic plane. Using the Herschel 70 $\mu$m data to trace the total IR luminosity at the resolution of the GMAs, we find that both the L$_{IR}$-L$_{HCN}$ and L$_{IR}$-L$_{HCO^+}$ relations in the outer disk GMAs are consistent with the proportionality between the L$_{IR}$ and the dense gas mass established globally in galaxies within the scatter. The IR/HCN and IR/HCO$^+$ ratios of the GMAs vary by a factor of 3, probably depending on whether massive stars are forming or not