<figure><img src="image_1.png" /></figure>

# Dense Gas in the Outer Spiral Arm of M51

Hao Chen<sup>1,2,3</sup>, Jonathan Braine<sup>4,5</sup>, Yu Gao<sup>6,7</sup>, Jin Koda<sup>8</sup>, and Qiusheng Gu<sup>1,2,3</sup>

¹School of Astronomy and Space Science, Nanjing University, Nanjing 210093, P. R. China; chenhao198701@yahoo.com, qsgu@nju.edu.cn

²Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Nanjing 210093, P. R. China

³Collaborative Innovation Center of Modern Astronomy and Space Exploration, Nanjing 210093, P. R. China

⁴Univ. Bordeaux, Laboratoire d'Astrophysique de Bordeaux, F-33270, Floirac, France

⁵CNRS, LAB, UMR 5804, F-33270, Floirac, France

⁶Purple Mountain Observatory, Chinese Academy of Sciences, 2 West Beijing Road, Nanjing 210008, P. R. China

⁷Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, P. R. China

⁸Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA

*Received 2016 April 23; revised 2016 November 18; accepted 2016 November 21; published 2017 February 10*

## Abstract

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 (~4", ~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 interferometer. The HCN and HCO$^+$ lines were detected in six giant molecular associations (GMAs), while HNC emission was only detected in the two brightest GMAs. One of the GMAs hosts a powerful H II region, and HCN is stronger than HCO$^+$ there. Comparing observations of GMAs in the disks of M31 and M33 at similar angular resolution (~100 pc), we find that GMAs in the outer disk of M51 are brighter in both the HCN and the 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 µm 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.

**Key words:** galaxies: individual (M51, NGC 5194) – galaxies: ISM – galaxies: star formation – ISM: molecules – radio lines: galaxies

## 1. Introduction

Stars form in the dense cores of giant molecular clouds (GMCs). The dense cores are traced by high-dipole-moment molecules such as HCN, HCO$^+$, and CS (Evans 1999). In cold regions where the density is very high, a commonly used probe in the Galaxy is $N_2H^+$, because it does not deplete onto dust grains. In the warm regions surrounding massive protostars, high-$J$ CO lines are useful probes of the density and temperature (in the submillimeter regime; Lu et al. 2014; Liu et al. 2015a; Zhao et al. 2016). The dense gas tracer HCN ($n_{eff} \sim 10^5$ cm$^{-3}$) exhibits the strongest line in galaxies after CO and $^{13}$CO. In intensely star-forming galaxies, such as ultra-luminous infrared galaxies (ULIRGs), the HCN line can be stronger than the $^{13}$CO (e.g., Baan et al. 2008). A linear relationship between the star formation rate (SFR) traced by infrared luminosity and the dense gas mass traced by HCN luminosity is found in galaxies (Gao & Solomon 2004a; Zhang et al. 2014; Liu et al. 2015b) and Galactic clumps (Wu et al. 2005, 2010). The IR/HCN(1–0) luminosity ratio is a proxy for the ratio of the SFR and dense gas mass, referred to as the star formation efficiency of dense molecular gas, SFE$_{dense}$, and this ratio is almost constant in galaxies. The strong linear relation between HCN intensity and IR emission, even in ULIRGs in which the IR–CO relation becomes nonlinear, suggests that it is the mass of dense gas rather than the molecular gas reservoir (Solomon et al. 1992; Gao et al. 2007) that governs star formation.

From large-scale mapping of the HCN emission in M51, Chen et al. (2015) and Bigiel et al. (2016) showed that the IR/HCN ratio (SFE$_{dense}$) is lower in the central kpc than in the outer disk. The HCN emission is also strong compared to CO in the central kpc of M51. If the SFE$_{dense}$ is in fact not constant, then either HCN(1–0) is not a reliable measure of the dense gas mass throughout the disk, or other factors (e.g., turbulence enhanced by the shear motion; Krumholz & Kruijssen 2015) prevent dense gas from turning into stars. This effect is not limited to M51. The Antennae galaxies (NGC 4038/4039) show that the IR/HCN ratio is on average four times higher in the three overlap regions than in the two nuclei (Bigiel et al. 2015). For 48 HCN detections of 29 nearby disk galaxies in the HERACLES survey, Usero et al. (2015) found that the IR/HCN ratios increase systematically with radius in galaxies 6–8 times lower near galaxy centers than in disks. Longmore et al. (2013) stated that the current SFR in the inner 500 pc of our Galaxy is 10 times lower than the rate predicted from the dense gas.

The IR-HCN relation in the outer disk at high resolution is not well known because the HCN emission in the outer disk is much weaker than in the center (Curran et al. 2001; Gao & Solomon 2004b; Chen et al. 2015) and there are few high-resolution dense gas observations toward the outer disk, except for the very nearby galaxies M31 (Brouillet et al. 2005) and M33 (Rosolowsky et al. 2011; Buchbender et al. 2013). In this work, we present observations at 150 pc resolution of a region along the northern spiral arm in the outer disk (~5 kpc from the
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Bottom left panel: B band (4350 Å) image from the *Hubble Space Telescope* (Mutchler et al. 2005); bottom middle panel: 70 µm image from the Herschel telescope; bottom right panel: CO 1–0 image from the CARMA array (Koda et al. 2011) with the HCN contours from the IRAM 30M telescope shown (Chen et al. 2015). The area we observed is indicated by red ellipses. The upper panels are details of the areas in the red squares. The white contours in the upper panels show the GMAs defined in Figure 2. The detail of the 70 µm image is shown with black contours representing the Hα intensity (Rand 1992). The beam size of each map is shown at the bottom of each image.</figcaption></figure>

center) of M51 in HCN(1–0), HCO$^+$(1–0), and HNC(1–0), all of which trace dense molecular gas. HCN and HNC are isomers and have similar energy spectra and dipole moments. HCO$^+$ has a slightly lower dipole moment and, although it is an ion, appears to also trace dense gas quite well (Jiang et al. 2011). With these observations, we can fill the gap (between the GMC and the kpc-scale region in the galactic disk, $10^7–10^8 L_\odot$ in IR) in IR–HCN and IR–HCO$^+$ relations.

This region was selected because it is in the outer disk and because of its high signal-to-noise HCN spectrum observed with the IRAM 30M telescope (Chen et al. 2015). The metallicity is solar to within the uncertainties (Bresolin et al. 2004). With these data in a rather small region (in M51), we obtain several independent data points without the potential influence on star formation by the radial (or other) variations of turbulence, metallicity, and pressure (Chen et al. 2015; Usero et al. 2015; Bigiel et al. 2016).

## 2. Observations and Data Reduction

### 2.1. HCN(1–0), HCO$^+$(1–0), HNC(1–0)

The two-field mosaic was observed with the Northern Extended Millimeter Array (NOEMA) interferometer in the C and D configurations during eight short (1–4 hr) sessions between 2014 July and 2015 April. The size and shape of the mosaic is shown as an ellipse in Figure 1. The fields were placed at (8", −8") and (−8", 8") offset from 13:29:55.7, 47:13:43.0 (J2000.0). The standard line and WIDEX correlators were used with spectral resolutions of 1.25 and

2 MHz, respectively. The only lines detected were HCN(1–0), HCO$^+$(1–0), and HNC(1–0).

The calibration of the UV data was done using the GILDAS<fnref n="9" /> software package CLIC and the standard pipeline. J1259+516 and/or J1418+546 served as phase and amplitude calibrators and were observed every 20 minutes between source observations. The flux of J1418+546 was 10% lower for the data observed on 2014 October 17 than for the other days, so we used J1259+516 as the calibrator for the October 17 data. The uncertainty of the absolute flux calibration was about 10% at 3 mm.

For imaging and cleaning, we used the GILDAS software package MAPPING. The two pointings are combined to create the dirty map, including the primary beam correction. Natural weighting was used to obtain the best signal-to-noise ratio. After imaging, we ran the CLEAN algorithm using the HOGBOM method on the central part of the field for 10 iterations. Then, for the channels where flux was detected, we cleaned carefully using the CO map to guide the CLEAN algorithm. The region cleaned is shown as a black dashed line in the CO panel of Figure 2, corresponding roughly to the 5σ contour of the CO integrated intensity map. The results for each channel were checked to make sure that the algorithm gave proper results. We chose the best result by comparing with the dirty beam, noise levels, and adjacent channels to identify whether further cleaning was necessary. The noise level was 10% better when we cleaned with more iterations (~2000,

<footnote n="9">⁹ http://www.iram.fr/IRAMFR/GILDAS</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** CO, HCN, HCO$^{+}$, and HNC integrated intensity maps. All the maps are in units of K km s$^{-1}$ and show the restoring beam size to the lower left of the panel and the line name to the upper right. The color wedge for the HCN, HCO$^{+}$, and HNC maps is indicated to the right (−0.8 to 1.6 K km s$^{-1}$). The CO color wedge goes from −40 to 80 K km s$^{-1}$ with the same linear color coding. The $3\sigma$, $5\sigma$, and $7\sigma$ contours are shown in blue for the HCN, HCO$^{+}$, and HNC images, respectively, and the CO image shows the $5\sigma$, $9\sigma$, and $13\sigma$ contours. The black dashed contour in the CO map shows the polygon, defined from the CO image, used for cleaning the HCN, HCO$^{+}$, and HNC data. Six black numbered polygons show the GMAs we used for the calculations and the spectra. The same $H\alpha$ contours as in Figure 1 are shown in white over all the images in order to compare sites of star formation with the molecular gas tracers.</figcaption></figure>

<table><caption>**Table 1**
CO, HCN, HCO$^{+}$, and HNC Observations</caption><thead><tr><th>Line Name</th><th>Velocity Resolution (km s<sup>-1</sup>)</th><th>Velocity Range (km s<sup>-1</sup>)</th><th>Beam Size</th><th>Data Cube rms (mK)</th></tr></thead><tbody><tr><td>CO</td><td>5.1</td><td>380.6–420.4</td><td>3"68 × 2"87</td><td>395.3</td></tr><tr><td>HCN</td><td>4.2</td><td>385.4–421.5</td><td>4"88 × 3"67</td><td>12.0</td></tr><tr><td>HCO<sup>+</sup></td><td>4.2</td><td>386.9–419.8</td><td>4"85 × 3"64</td><td>11.7</td></tr><tr><td>HNC</td><td>6.6</td><td>394.2–412.6</td><td>4"96 × 3"59</td><td>11.3</td></tr></tbody></table>

**Note.** The CO data were obtained by Koda et al. (2011). The Jy K$^{-1}$ conversion factors are 8.71, 8.69, 8.70, and 8.35 for the CO, HCN, HCO$^{+}$, and HNC data cubes, respectively.

reaching the default threshold) than it was with fewer iterations (~100), and the fluxes were stable. The data presented here represent what we estimate as the most reliable reduction, but they underestimate slightly the true flux density for the HCN, HCO$^{+}$, and HNC lines because some residual sidelobes are still present. The Jy K$^{-1}$ conversion factors are 8.71, 8.69, 8.70, and 8.35 for the CO, HCN, HCO$^{+}$, and HNC data cubes, respectively. The spectral resolution, beam size, and rms of the cleaned data cubes are shown in Table 1.

The HCN flux with the IRAM 30M telescope centered at 13:29:66.64, 47:13:58.0 (J2000.0) is 0.48 K km s$^{-1}$, or 2.3 Jy km s$^{-1}$. The flux of the NOEMA observation is 1.1

Jy km s$^{-1}$ in the same region. The difference could be due to missing short spacings, such that larger structures are resolved out, or to the residual interferometric “bowl”, which has not been completely eliminated by the cleaning.

## 2.2. CO and IR Data

The CO $J = 1-0$ data were taken with the CARMA array combined with zero spacings from the 5 × 5 Beam Array Receiver System on the Nobeyama Radio Observatory 45M telescope (Koda et al. 2011). Schinnerer et al. (2013) observed M51 with the Plateau de Bure interferometer at high resolution, but their map does not extend to regions as far out as this one.

In order to compare with sites of recent star formation, we use the 70 µm image from the Very Nearby Galaxy Survey, (Mentuch et al. 2012) which was accessed through the Herschel Database in Marseille (HeDaM$^{10,11}$). The resolution is only slightly poorer than ours, so we can use the Herschel 70 µm images to estimate the SFRs (Boquien et al. 2011; Galametz et al. 2013).

## 3. Results

Figure 2 shows the integrated intensity maps of the HCN, HCO$^{+}$, and HNC emission lines along with the CO(1–0)

<footnote n="10">10 http://hedam.lam.fr</footnote>

<footnote n="11">11 The Herschel Database in Marseille (HeDaM) is operated by CeSAM and hosted by the Laboratoire d'Astrophysique de Marseille.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 3.** CO (blue), HCN (red), HCO$^+$ (black), and HNC (dotted red) spectra for each GMA. For calculating the spectra, the CO data cube is smoothed to 4"/9 × 3"/6 to match the angular resolution of the HCN, HCO$^+$, and HNC data cubes.</figcaption></figure>

image obtained by Koda et al. (2011). The integrated intensities are measured as $I = \int TdV$, and the emission velocity ranges (V) are defined from the data cube and shown in Table 1. Uncertainties are calculated as $\delta = T_{rms} \sqrt{W\delta_c}$ ($T_{rms}$ is the data

<figure><img src="image_5.png" /><figcaption>**Figure 4.** HCN and HCO$^+$ integrated intensity distributions and line ratio ($I_{CO}/I_{HCN}$, $I_{CO}/I_{HCO^+}$, and $I_{HCN}/I_{HCO^+}$) distributions of the GMAs in M51 (red), M31 (blue), and M33 (yellow).</figcaption></figure>

cube rms, and W and $\delta_c$ are the velocity range and velocity resolution of the integrated intensity map) and are 5.6, 0.15, 0.14, and 0.13 K km s$^{-1}$ for CO, HCN, HCO$^+$, and HNC,
<table><caption>**Table 2**
CO, HCN, HCO$^{+}$, and HNC Intensities</caption><thead><tr><th></th><th>GMA 1</th><th>GMA 2</th><th>GMA 3</th><th>GMA 4</th><th>GMA 5</th><th>GMA 6</th></tr></thead><tbody><tr><td>I<sub>CO</sub> (K km s<sup>-1</sup>)</td><td>46.2 ± 2.8</td><td>47.8 ± 4.0</td><td>35.0 ± 3.9</td><td>28.3 ± 4.0</td><td>30.1 ± 2.7</td><td>34.0 ± 3.2</td></tr><tr><td>I<sub>HCN</sub>(K km s<sup>-1</sup>)</td><td>0.53 ± 0.06</td><td>0.97 ± 0.07</td><td>0.30 ± 0.09</td><td>0.25 ± 0.10</td><td>0.64 ± 0.12</td><td>0.29 ± 0.12</td></tr><tr><td>I<sub>HCO</sub><sup>+</sup> (K km s<sup>-1</sup>)</td><td>0.63 ± 0.06</td><td>0.88 ± 0.09</td><td>0.60 ± 0.11</td><td>0.60 ± 0.11</td><td>0.53 ± 0.08</td><td>0.60 ± 0.10</td></tr><tr><td>I<sub>HNC</sub> (K km s<sup>-1</sup>)</td><td>0.21 ± 0.04</td><td>0.42 ± 0.06</td><td>0.10 ± 0.07</td><td>0.13 ± 0.07</td><td>0.05 ± 0.07</td><td>0.06 ± 0.09</td></tr><tr><td>I<sub>HCN</sub>/I<sub>CO</sub></td><td>0.011 ± 0.002</td><td>0.020 ± 0.003</td><td>0.009 ± 0.004</td><td>0.009 ± 0.005</td><td>0.021 ± 0.006</td><td>0.009 ± 0.004</td></tr><tr><td>I<sub>HCO</sub><sup>+</sup>/I<sub>CO</sub></td><td>0.014 ± 0.002</td><td>0.018 ± 0.003</td><td>0.017 ± 0.005</td><td>0.021 ± 0.007</td><td>0.018 ± 0.004</td><td>0.018 ± 0.005</td></tr><tr><td>I<sub>HCN</sub>/I<sub>HCO</sub><sup>+</sup></td><td>0.84 ± 0.18</td><td>1.10 ± 0.19</td><td>0.50 ± 0.24</td><td>0.42 ± 0.24</td><td>1.21 ± 0.41</td><td>0.48 ± 0.28</td></tr></tbody></table>

Note. In calculating the intensities, the CO map has been smoothed to 4''9 × 3''6 to match the angular resolution of the HCN, HCO$^{+}$, and HNC maps. The integrated intensities are measured as $I = \int TdV$ over the emission velocity range. The uncertainties are calculated as $\delta = T_{rms} \sqrt{W\delta_c}$. The rms temperature uncertainty ($T_{rms}$) is calculated from the nonemission channels of the spectra. The line width and channel width in velocity are represented by $W$ and $\delta_c$.

<table><caption>**Table 3**
CO, HCN, HCO$^{+}$, and IR Luminosities</caption><thead><tr><th></th><th>GMA 1</th><th>GMA 2</th><th>GMA 3</th><th>GMA 4</th><th>GMA 5</th><th>GMA 6</th></tr></thead><tbody><tr><td>Ω(arcsec<sup>2</sup>)</td><td>47.16</td><td>26.28</td><td>20.88</td><td>11.52</td><td>37.80</td><td>28.44</td></tr><tr><td>L<sub>CO</sub> (10<sup>5</sup> K km s<sup>-1</sup> pc<sup>-2</sup>)</td><td>26.9 ± 1.7</td><td>15.4 ± 1.7</td><td>8.6 ± 1.6</td><td>3.6 ± 1.8</td><td>13.3 ± 1.2</td><td>11.1 ± 1.4</td></tr><tr><td>L<sub>HCN</sub> (10<sup>3</sup> K km s<sup>-1</sup> pc<sup>-2</sup>)</td><td>33.2 ± 3.1</td><td>29.3 ± 2.8</td><td>8.8 ± 3.5</td><td>4.7 ± 3.1</td><td>30.0 ± 5.6</td><td>9.4 ± 4.3</td></tr><tr><td>L<sub>HCO</sub><sup>+</sup> (10<sup>3</sup> K km s<sup>-1</sup> pc<sup>-2</sup>)</td><td>37.3 ± 2.8</td><td>27.9 ± 3.3</td><td>14.5 ± 3.9</td><td>8.1 ± 4.3</td><td>23.6 ± 3.7</td><td>17.4 ± 3.9</td></tr><tr><td>log(L<sub>IR</sub>) (L<sub>☉</sub>)</td><td>7.26 ± 0.09</td><td>7.30 ± 0.09</td><td>7.05 ± 0.09</td><td>6.99 ± 0.09</td><td>7.71 ± 0.09</td><td>7.25 ± 0.09</td></tr><tr><td>log(L<sub>IR</sub>/L<sub>HCN</sub>) (L<sub>☉</sub>/(K km s<sup>-1</sup> pc<sup>-2</sup>))</td><td>2.74 ± 0.09</td><td>2.83 ± 0.09</td><td>3.11 ± 0.09</td><td>3.32 ± 0.09</td><td>3.23 ± 0.09</td><td>3.28 ± 0.09</td></tr><tr><td>log(L<sub>IR</sub>/L<sub>HCO</sub><sup>+</sup>) [L<sub>☉</sub>/(K km s<sup>-1</sup> pc<sup>-2</sup>)]</td><td>2.69 ± 0.09</td><td>2.85 ± 0.09</td><td>2.89 ± 0.09</td><td>3.08 ± 0.09</td><td>3.33 ± 0.09</td><td>3.01 ± 0.09</td></tr></tbody></table>

Note. When calculating luminosities, all maps have been smoothed to the angular resolution of 70 µm (5''6). The line luminosities are calculated with the function of $L$ [K km s$^{-1}$ pc$^{-2}$] = 23.5Ω[arcsec$^2$]d$^2_L$ [Mpc] $I$ [K km s$^{-1}$], where Ω is the solid angle of the GMA, $d_L$ is the luminosity distance of M51 (7.6 Mpc; Ciardullo et al. 2002), and $I$ is the integrated intensity calculated the same way as in Table 2. The uncertainty in $L_{IR}$ comes from the scatter in the conversion of 70 µm to IR (0.09 dex; Galametz et al. 2013). This uncertainty dominates the measurement errors in 70 µm (0.01–0.05 dex).

respectively. Because the HCN, HCO$^{+}$, and HNC emissions show similar but clearly different morphologies, we choose not to bias our results on any of these tracers and instead use the higher signal-to-noise ratio CO map with slightly better angular resolution to define the giant molecular associations (GMAs). A total of six GMAs have been identified (black numbered polygons in Figure 2) by the ClumpFind$^{12}$ algorithm (Williams et al. 1994). The HCN and HCO$^{+}$ emission peaks are similar at about 1.2 and 1.4 K km s$^{-1}$, and the peak positions are consistent with each other at the center of GMA 2. The HNC line is only detected in GMAs 1 and 2, and the emission region is displaced to the south compared to the other lines. GMAs 3, 4, and 6 are weak in HCN but strong in HCO$^{+}$ (and CO). The HCO$^{+}$ distribution is generally broader than that of CO, HCN, and HNC. Figure 2 also shows the Hα emission regions (white contours) in order to allow a comparison with the sites of massive star formation. As can be seen from the detail in Figure 1, the 70 µm emission distribution is quite similar to that of Hα, so the white contours provide a good reference for the sites of star formation in these GMAs. GMA 5 clearly shows strong star formation, and its HCN emission is stronger than the HCO$^{+}$ and more centered on the Hα than the CO. This may point toward HCN as being more linked to star formation than other dense gas tracers, at least in high (~solar) metallicity environments. Clearly, more high-resolution studies are required to clarify this.

Spectra for each line integrated over the area of each GMA are shown in Figure 3, and the integrated intensities and

uncertainties are provided in Table 2. As can be seen, the lines are all at the same velocity and have similar line widths. Because several clouds are probably included within one GMA, the hyperfine structure of the HCN line is not visible, and the broadening (due to the presence of three components) is not detectable.

## 4. Analysis and Discussion

### 4.1. HCN and HCO$^{+}$ Brightness and Line Ratios

M31 (Brouillet et al. 2005) and M33 (Buchbender et al. 2013) have been observed at virtually the same linear resolution as our M51 observations. However, comparison of the M31, M33, and M51 disk GMAs in Figure 4 shows that the integrated intensities of the M51 GMAs are much higher in the HCN and HCO$^{+}$ lines.

The dense gas fractions, as traced by $I_{HCN}/I_{CO}$ and $I_{HCO^{+}}/I_{CO}$, are compared in the lower part of Figure 4. Interestingly, the $I_{HCN}/I_{CO}$ and $I_{HCO^{+}}/I_{CO}$ ratios are similar for the three galactic disks. In M51, the $I_{HCN}/I_{CO}$ ratios vary from 0.007 to 0.021, the latter of which is similar to the ratio of the Galactic disk (~0.026; Helfer & Blitz 1997).

With the spectra in Figure 3 and data in Table 2, we can estimate the CO-based total molecular gas mass (using a galactic conversion factor of $N_{H_2}/I_{CO} = 2 \times 10^{20} \text{ cm}^{-2}/(\text{K km s}^{-1})$) and the HCN-based dense gas mass (using Equation (8) from Gao & Solomon 2004b). With these numbers, we obtain dense gas mass fractions from 2% to 7% for the GMAs in M51. Thus, these observations suggest that a few percent of the mass in a GMC (or a GMA, presumably a

<footnote n="12">12 http://www.ifa.hawaii.edu/users/jpw/clumpfind.shtml</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 5.** (Upper left) The correlation between $L_{IR}$ tracing SFR and $L_{HCN}$ tracing dense gas mass for clumps in the Milky Way (Wu et al. 2010), GMCs in the Magellanic Clouds (Chin et al. 1997, 1998), GMAs in nearby galaxies (Brouillet et al. 2005; Buchbender et al. 2013; this work), a nearby galactic disk (Kepley et al. 2014; Bigiel et al. 2015; Chen et al. 2015; Usero et al. 2015), and whole galaxies (Gao & Solomon 2004a; Gao et al. 2007; Graciá-Carpio et al. 2008; Krips et al. 2008; Juneau et al. 2009; García-Burillo et al. 2012; Privon et al. 2015). (Lower left) The correlation between $L_{IR}$ tracing SFR and $L_{HCO}^+$ tracing dense gas mass for clumps in the Milky Way (Ma et al. 2013), GMCs in the Magellanic Clouds (Chin et al. 1997, 1998), GMAs in nearby galaxies (Brouillet et al. 2005; Buchbender et al. 2013; this work), a nearby galactic disk (Kepley et al. 2014; Bigiel et al. 2015), and whole galaxies (Graciá-Carpio et al. 2008; Krips et al. 2008; Juneau et al. 2009; García-Burillo et al. 2012; Privon et al. 2015). (Right panels) The variations of $L_{IR}/L_{HCN}$ and $L_{IR}/L_{HCO}^+$ tracing the star formation efficiency of dense gas, along with $L_{IR}$ tracing the SFR, are shown for the same data as in the left panels. The insets show the zoom-in plots of the data in this work. Only the uncertainties of HCN and $HCO^+$ for the GMAs in M51 are shown in the plots. Solid lines show the averaged ratios of IR/HCN and IR/$HCO^+$ for the Gao & Solomon (2004a) and Graciá-Carpio et al. (2008) samples.</figcaption></figure>

collection of neighboring GMCs) is in the form of dense gas. The simulations by Kroupa et al. (2001) find that 30% of dense gas mass is turned into stars, such that 0.6%–2.1% of the total cloud mass (2%–7% × 0.3) forms stars in our sample. This is in reasonable agreement with the estimate of Zuckerman & Evans (1974) that about 1% of the mass of a molecular cloud is converted into stars.

The $I_{HCN}/I_{HCO^+}$ line ratios are also compared for the GMAs in M51, M31, and M33. All but two of the GMAs show similar $I_{HCN}/I_{HCO^+}$ ratios ranging from 0.4 to 1.0. The two GMAs in M51 and M31 hosting powerful H II regions (star-forming regions) show high $I_{HCN}/I_{HCO^+}$ ratios (1.21 and 1.72).

## 4.2. SFR versus Dense Gas Mass

To compare the dense gas mass with the SFR of the GMAs in the outer spiral arm of M51, we use the HCN and $HCO^+$ luminosities to trace the dense gas mass and the IR luminosity to trace the SFR. The line luminosities are calculated as $L\,[K\,km\,s^{-1}\,pc^2] = 23.5\,\Omega\,[arcsec^2]\,d_L^2\,[Mpc]\,I\,[K\,km\,s^{-1}]$, where $\Omega$ is the solid angle of the GMA, $d_L$ is the luminosity

distance of M51 (7.6 Mpc; Ciardullo et al. 2002), and $I$ is the integrated intensity calculated as in Table 2. The line luminosity uncertainties are determined from the same formula, substituting $I_{rms}$ for $I$ and taking $\Omega$ as either the solid angle of the GMA or the $5''$ beam size, whichever is larger. The IR luminosities are derived from Herschel 70 $\mu$m data following the conversion function shown in Table 2 of Galametz et al. (2013). The uncertainty in $L_{IR}$ comes from the scatter in the conversion of 70 $\mu$m to IR (0.09 dex; Galametz et al. 2013). This uncertainty dominates the measurement error at 70 $\mu$m (0.01–0.05 dex). When calculating luminosities, all the maps have been smoothed to the angular resolution of the 70 $\mu$m data (5''6), and the results are listed in Table 3.

Our observations of the six GMAs fill the gap in IR–HCN and IR–$HCO^+$ relations between the large-scale observations, kpc or larger, and the Galactic measurements (see Figure 5). Both the $L_{IR}-L_{HCN}$ and $L_{IR}-L_{HCO^+}$ relations in the outer disk GMAs are consistent with the proportionality between $L_{IR}$ and dense gas mass established globally in galaxies. The observations of the GMAs presented here are quite close to the average
IR-HCN relation with no obvious shift or trend, unlike some of the Galactic observations or observations of galactic centers, which tend to have low IR/HCN flux ratios.

There is still some scatter in the IR/HCN and IR/HCO⁺ ratios in the GMAs of the outer disk, as shown by the insets in Figure 5. Both the IR/HCN and the IR/HCO⁺ ratios of GMA 5 are 3 times higher than those of GMAs 1 and 2. This could be because massive stars are forming in GMA 5, which hosts an H II region. The IR/HCN ratio of GMAs 3, 4, and 6 is almost the same as that of GMA 5, but their IR/HCO⁺ ratio is half that of GMA 5. It is consistent with the lower HCN/HCO⁺ ratio of GMAs 3, 4, and 6 (about 0.5) rather than that of GMAs 1, 2, and 5 (about 1.0). The results do not change when we take extinction-corrected Hα (Calzetti et al. 2007) or far-UV (Liu et al. 2011) instead of IR to trace SFR.

Figure 5 shows not only the extragalactic data points sampling objects with IR luminosities above $10^5 L_\odot$ at a variety of scales but also many Galactic observations, thus enabling sampling down to very low luminosities. While some of the apparent scatter in Figure 5 could simply be due to noise, Wu et al. (2005) and Ma et al. (2013) show that the IR luminosity at the weak end appears to be lower than the prediction from the HCN (or HCO⁺) luminosity by the IR-HCN (or IR-HCO⁺) linear relation. At very small scales or low luminosities, this nonlinearity could be introduced by the incomplete sampling of the stellar initial mass function (IMF; e.g., Kennicutt & Evans 2012) and thus poor measurement of the SFR, which depends strongly on the high stellar mass sampling. Kennicutt & Evans (2012) suggest that the SFR should be larger than 0.001 ~ 0.01 $M_\odot$ yr⁻¹ to completely sample the stellar IMF. The SFR of our GMAs is about 0.002 ~ 0.01 $M_\odot$ yr⁻¹ (derived from IR with Equation (9) in Gao & Solomon 2004a, although mainly adopted/used globally in star-forming galaxies), so it is not clear whether the scatter in IR/HCN and IR/HCO⁺ ratios in these GMAs could be due to incomplete sampling.

## 5. Summary

We mapped a selected region on the outer spiral arm of M51 in HCN(1–0), HCO⁺(1–0), and HNC(1–0) using the NOEMA interferometer with an angular resolution of 4" (~150 pc).

(1) We detected bright emission of HCN and HCO⁺ in six GMAs defined by CO(1–0) data, while HNC emission was only detected in the two brightest GMAs.

(2) The HCO⁺ spatial distribution is generally broader than that of HCN and HNC. One of the GMAs hosts a powerful H II region, and HCN is stronger than HCO⁺ there.

(3) The GMAs in M51 are brighter in both HCN and HCO⁺ than the GMAs in M31 and M33, but the ratios of CO/HCN, CO/HCO⁺, and HCN/HCO⁺ are similar for the three objects.

(4) Combined with Herschel 70 µm data, we find that both the $L_{IR}-L_{HCN}$ and $L_{IR}-L_{HCO^+}$ relations in the GMAs of M51 follow the proportionality between the $L_{IR}$ and the dense gas mass established globally in galaxies within the scatter.

(5) The IR/HCN and IR/HCO⁺ ratios of the GMAs vary by a factor of 3, probably depending on whether massive stars are forming.

We appreciate the generous support of the IRAM staff and GILDAS team during the observations and data reduction. This work is supported by the program for outstanding PhD candidates of Nanjing University, the National Natural Science Foundation of China (grants 11173059, 11390373, 11420101002, 11273015, and 11133001), the CAS pilot-b program (No. XDB09000000), and the National Basic Research Program (973 Program No. 2013CB834905). We are very grateful to Campus France for Xu Guangqi grant 34454YG, which helped fund the travel necessary for this work.

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