The Dense Gas Mass Fraction and the Relationship to Star Formation in M51

Observations of ¹²CO J = 1 – 0 and HCN J = 1 – 0 emission from M51 were obtained with the 50 m Large Millimeter Telescope (LMT) Alfonso Serrano between 2020 January and March, using the 16 element focal plane array receiver SEQUOIA. The half-power beamwidths (HPBWs) of the telescope at the line rest frequencies for CO (115.2712018 GHz) and HCN (88.630416 GHz) are 12'' and 16'', respectively. The Wide-band Array Roach Enabled Spectrometer (WARES) was used to process the spectral information using the configuration with 800 MHz bandwidth and 391 kHz per spectral channel, which provides a velocity resolution of 1.1 km s⁻¹ for CO and 1.3 km s⁻¹ for HCN.

For ¹²CO, 10 maps covering the same 10' × 10' area were observed using On-the-Fly (OTF) mapping with all scanning along the R.A. axis. HCN imaging comprised multiple OTF maps covering three 7' × 4' overlapping areas centered on the nucleus of M51, 2' north of the nucleus, and 2' south of the nucleus. Data were calibrated by a chopper wheel that allowed switching between the sky and an ambient temperature load. The chopper wheel method introduces a fractional uncertainty of ∼10% to the measured antenna temperatures (Narayanan et al. 2008). Routine pointing and focus measurements were made to ensure positional accuracy and optimal gain.

All data were processed with the LMT spectral line software package, which included zero-order baseline subtraction and the coadding of multiple maps into a final spectral line data cube at the native angular and spectral resolutions. To directly compare the ¹²CO and HCN intensities, we also processed the ¹²CO data to the HCN resolution λ/D = 14'' and λ/2D=7'' sampling, where D is the 50 m diameter of the LMT antenna. At a distance of 8.58 ± 0.10 Mpc (McQuinn et al. 2016), this resolution corresponds to a spatial size of 582 pc. Both CO and HCN data cubes were spectrally smoothed and resampled to 5 km s⁻¹ resolution. A main-beam efficiency of 0.65 is applied to convert the data from the T ${}_{A}^{* }$ temperature scale to main-beam temperatures. The median rms sensitivities in main-beam temperature units at 5 km s⁻¹ spectral resolution are 28 mK for ¹²CO and 8 mK for HCN.

Our analysis requires several ancillary data sets. To measure the unobscured star formation rates, we use far-ultraviolet (FUV) data taken from the GALEX Ultraviolet Atlas of Nearby Galaxies (Gil de Paz et al. 2007). The obscured star formation rate is derived from the Spitzer MIPS 24 μm data from the Spitzer Infrared Nearby Galaxies Survey (SINGS) (Kennicutt & Armus 2003). Models of the galactic structure are derived from the 3.6 μm image from SINGS. The mass of the stellar component of M51 is calculated using the H-band image from 2MASS (Jarrett et al. 2003), supplemented by g- and i-band images from the Sloan Digital Sky Survey (SDSS-III) (Aihara et al. 2011) and produced by Brown et al. (2014). All ancillary data sets were obtained from the NASA/IPAC Extragalactic Database (NED).

Images of velocity-integrated CO J = 1 – 0 emission and HCN J = 1 – 0 emission are displayed in Figure 1. These images are created by masking the spectral channels where CO is detected for each spectrum in the coadded map in order to limit contributions of noise that can occur when integrating over a fixed, wide velocity interval of a rotating galaxy. This masking is achieved by the following steps described by Dame (2011). First, the CO data are smoothed along the angular axes using a Gaussian kernel with an HPBW of 5 pixels (35'') and along the spectral axis using a boxcar kernel with a width of three spectral channels (15 km s⁻¹) while keeping the angular and spectral sampling of 7'' and 5 km s⁻¹. For a given spectrum in the smoothed data cube, any grouping of three or more consecutive channels with brightness temperatures greater than 3.75σ is flagged as active voxels and assigned a value of 1 in the mask data cube, M(x, y, v). All other channels are assigned a value of zero. This signal-to-noise threshold is a result of many trials with varying signal-to-noise ratios to maximize the number of active pixels while minimizing false detections. The integrated intensity image, W, is calculated from the data cube, T_mb(x, y, v) from the expression

$$\begin{eqnarray}&&W(x,y)={\rm{\Delta }}v\sum _{k}{T}_{\mathrm{mb}}(x,y,{v}_{k})M(x,y,{v}_{k}),\end{eqnarray} \tag{ 1 }$$

where the sum is over all spectral channels and Δv is the channel width of 5 km s⁻¹. The statistical error of this integrated intensity is $\sigma {(x,y){\rm{\Delta }}v({N}_{c}(x,y))}^{1/2}$, where σ(x, y) is the rms of channels outside the velocity range 360–800 km s⁻¹ of the unsmoothed data cube and N_c (x, y) is the number of active channels for position x, y.

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The left panel in Figure 1 shows widespread detection of ¹²CO emission over the projected area of M51. The brightest emission traces the bulge and inner spiral features identified in previous studies with comparable resolution (Koda et al. 2011; Pety et al. 2013). Signal is also detected within the interarm regions—even within the large, interarm area in the south. Integrating the CO emission over all pixels, we derive a CO luminosity of (2.167 ± 0.003) × 10⁹ K km s⁻¹ pc², assuming a distance of 8.58 Mpc. The quoted uncertainty is based on propagating observational noise rms values through the luminosity calculation. This total CO luminosity includes contributions from NGC 5195. The total molecular mass is (9.32 ± 0.01) × 10⁹ M_⊙ using the H₂-to-CO conversion factor α_CO = 4.3 M_⊙/(Kkms⁻¹pc²) (Bolatto et al. 2013). The CO luminosity of just NGC 5195 is (9.93 ± 0.01) × 10⁷ K km s⁻¹ pc², which corresponds to a molecular mass of (4.27 ± 0.04 ) ×10⁸ M_⊙. The CARMA-NRO CO imaging estimated the total mass of NGC 5194 and NGC 5195 to be 6.0 × 10⁹ M_⊙ applying the same CO–H₂conversion factor and distance as used in this study (Koda et al. 2011). Within calibration uncertainties, masking thresholds, and map coverage, our values are also comparable with those of Pety et al. (2013), who derived L_CO = 1.8 × 10⁹ K km s⁻¹ pc², and molecular mass of 7.9 × 10⁹ M_⊙ from observations with the IRAM 30 m telescope that covered approximately the same area.

The ¹²CO-defined mask is applied to the HCN data from which we produce a velocity-integrated image of HCN emission shown in the right panel of Figure 1. For areas with declinations south of 47°07' and north of 47°15', which include NGC 5195, the HCN noise levels are large due to less data collected near these map edges. These areas are not included in any subsequent analyses presented here. The HCN emission is strong in the central region of M51 but is more patchy and uneven with increasing galactic radius than the CO emission yet still exhibits faint spiral structure in the central 3' × 3' area. The HCN luminosity over the map area is (6.05 ± 0.08) × 10⁷ K km s⁻¹ pc². Adopting the conversion factor ${\alpha }_{\mathrm{HCN}}=10$ M_⊙/(K km s⁻¹ pc²) that assumes self-gravitating dense cores in clouds (Gao & Solomon 2004b), the dense gas mass is (6.05 ± 0.08)×10⁸ M_⊙.

Star formation rates (M_⊙/yr) within M51 are derived from images in the far-ultraviolet (FUV) band measured by GALEX and the MIPS 24 μm band from the Spitzer Space Telescope. In regions of low dust obscuration, massive young stars directly contribute to the observed FUV flux. Newborn stars embedded within regions of moderate to high extinction contribute to the 24 μm flux as their stellar optical and UV radiation heat nearby dust grains that reradiate this energy into the mid- and far-infrared bands. Used together, these bands account for star formation rates from both unobscured and deeply embedded regions of star formation. However, an older stellar population that does not reflect star formation over the last 100 Myr also contributes to the measured luminosities in both the FUV and 24 μm bands. To derive more accurate star formation rates, it is necessary to account for this "cirrus" emission component from the old stellar population in each band.

To evaluate this contribution, we construct three models of the star formation rate. Model 0 is the trivial model, which assumes no contributions to the FUV and 24 μm luminosities from older stars. This model allows a comparison to other studies that also exclude this component (Bigiel et al. 2016; Jiménez-Donaire et al. 2019). The other models consider the distribution of the old stellar component by decomposing the Spitzer image of 3.6 μm surface brightness into galactic structural components using Galfit (Peng et al. 2010). Model 1 follows the example in Peng et al. (2010) for M51 that includes a two-component central bulge and inner and outer spiral arms. We exclude the numerous, bright 3.6 μm emission knots so that each Galfit component more accurately tracks the smooth emission, which limits oversubtractions. Model 2 includes a two-component central bulge and a Sérsic profile that approximates an exponential disk but no spiral-arm structures. In this model, we implicitly assume that the spiral arms are entirely due to young stellar populations, although this assumption is an oversimplification of current results for spiral structures (Kreckel et al. 2016). In calculating the exponential disk, we exclude pixels located in spiral-arm features (see Figure 3) so as to not bias the fit. For both Galfit models, we also simultaneously fit the companion galaxy, M51b, with a two-component central bulge and one spiral-arm component, based on the Peng et al. (2010) models. This avoids biasing the M51a fits in the outer disk and spiral-arm components. The Galfit parameters and the 3.6 μm model images for Models 1 and 2 are shown in the Appendix.

The 3.6 μm Galfit model intensities are scaled to the FUV and 24 μm bands. The respective scaling factors are determined from the ratio of the median observed FUV and 24 μm intensities to the median model 3.6 μm intensity within several 5 × 5 pixel areas located within interarm regions with no star formation activity. The resultant scaled models are then subtracted from the observed images to remove the old stellar population contributions. For this subtraction, we include only model components where we expect a significant contribution to the FUV and 24 μm intensities from the old stellar populations. For 24 μm, we subtract the normalized bulge and both spiral components for Model 1 and the bulge and disk for Model 2. It is clear from the lack of a significant bulge in the FUV image that the unobscured, central stellar population is old and is contributing little UV flux. So for the FUV, we subtract only the two spiral components for Model 1 and the disk for Model 2. Because Model 2 does not account for spiral-arm features in which older stellar populations are enhanced, its subtraction leads to a positive bias of SFR values. All pixels with negative residuals are set to zero to limit artificial model bias in the SFR maps resulting from the image combination. Using the Aniano et al. (2011) kernels, the corrected 24 μm and FUV images are convolved to the resolution (14 '') and pixel size (7'') of the molecular line data.

For each model, the star formation rate is calculated using the expression by Liu et al. (2011),

$$\begin{eqnarray}&&\mathrm{SFR}({M}_{\odot }{\mathrm{yr}}^{-1})=4.6\times {10}^{-44}[L(\mathrm{FUV})+6.0L(24\mu {\rm{m}}]),\end{eqnarray} \tag{ 2 }$$

where the luminosities are in erg s⁻¹. This calibration assumes a Kroupa (2001) stellar IMF to account for newly formed low-mass stars that do not significantly contribute to the FUV and 24 μm emission. The uncertainty of SFR is 1.33 × 10⁻⁵ M_⊙ yr⁻¹, which is derived from the standard deviation of the background areas in both the FUV and 24 μm images and propagated through Equation (2).

We acknowledge that our method to quantify the old stellar population contributions to the star formation tracers is complicated and depends on the accuracy of the decomposition of the 3.6 μm image by Galfit as well as the scaling of the model 3.6 μm surface brightness values to the FUV and 24 μm bands. Alternatives to account for the cirrus emission include unsharp masking to identify a diffuse component produced by radiation from the old stellar population (Rahman et al. 2011) and direct modeling of the cirrus emission from radiation fields and dust emissivities (Leroy et al. 2012). Our method offers a more physically based alternative to unsharp masking as Galfit identifies galactic components that are expected to be populated by older stars.

The mass of the stellar component is derived using the method described by Zibetti et al. (2009). In brief, g-, i-, and H-band images are resampled to the 3.6 μm image pixel size and coverage. Background and foreground objects identified in the 3.6 μm image are removed in the g, i, and H images by interpolation. The g and i images are convolved to the 28 resolution of the H-band data, and all images are converted to Jy/pixel surface brightness units. Each image is adaptively smoothed to achieve signal-to-noise ratios greater than 20 using the code of Zibetti (2009). For a given pixel, the largest smoothing element is selected and applied to the remaining images in order to match the spatial resolution. In all cases across the disk of M51, the H-band data required the largest smoothing kernel. The g − i and i − H colors are derived for each pixel and are corrected for galactic foreground extinction assuming A_v =0.096 (Schlafly & Finkbeiner 2011). From the colors, the median mass-to-light ratio at the H band, Ψ_H , is determined based on the look-up tables of Zibetti et al. (2009) that are based on the Bruzual & Charlot (2003) stellar population synthesis models. For pixels with colors outside the look-up table limits, values for Ψ_H are interpolated from valid, neighboring pixel values. H-band luminosities, L_H , are calculated from the adaptively smoothed H-band image using a distance of 8.58 Mpc. We convert these luminosities to "in-band" solar units using L_⊙,H = 1.08 × 10³³ erg s⁻¹, derived from the absolute Vega magnitude of the Sun in the 2MASS H band and the flux zero-point listed in Willmer (2018). The stellar-mass image is L_H Ψ_H . The resultant stellar-mass image is then smoothed and resampled to the pixel size and resolution of the molecular line data.

The primary advantage of using the H band to derive stellar masses is that the method does not require a subtraction of an extended, warm-dust component that is heated by star formation activity as is the case at 3.6μm (Meidt et al. 2012; Querejeta et al. 2015), which can introduce additional uncertainties in the final result. However, owing to the low signal-to-noise ratio of the 2MASS H-band data in the outer regions of the disk, this band is not as useful as the highly sensitive 3.6 μm image for decomposing the galaxy into structural components using Galfit as described in Section 3.2.

The HCN-to-CO ratio can be probed to fainter levels of the HCN emission than the 3σ limits of R shown in Figure 2 by stacking the HCN and ¹²CO spectra selected by a common physical attribute or spatial grouping. The stacked spectra represent the average of the ensemble that satisfy the selection criteria. This spectral stacking method has been applied to HCN data by Bigiel et al. (2016) and Jiménez-Donaire et al. (2019), where HCN spectra are stacked in bins of galactic radius or stellar surface density.

4.1.1. Bulge and Spiral-arm Regions of Interest

The deep gravitational potential of spiral arms provides an environment that could impact the dense gas fractions of the molecular cloud population. To explore this role, we define regions of interest (ROI) in the 3.6 μm image as polygons that circumscribe segments along both spiral arms and interarm regions. We also define the M51 bulge as a distinct ROI that includes the area within a circular radius of 35''. Finally, all CO and HCN spectra within the CO-defined mask (5997 pixels) are stacked to produce globally averaged spectra (ROI 14). The top-left panel of Figure 3 shows the labels for all the ROIs, which are also grouped into separate arms and interarms distinguished by color.

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For each ROI, we identify the set of pixels that fall within the subtended area. In the case where the distance between the pixel position and the nearest ROI boundary is less than 7'', corresponding to half of the HPBW, we do not correct for the signal gathered by the telescope beam that extends beyond the ROI boundary. To stack the data, the velocity axis for each selected spectrum in the ROI is shifted to be centered at 0 km s⁻¹ by subtracting the velocity centroid of the CO spectrum at the same position. This relative spectrum is sampled between −200 and +200 km s⁻¹ with a 5 km s⁻¹ spectral resolution and coadded to the stacked spectrum with a 1/σ² weighting. The bottom subplot in Figure 3 shows the stacked ¹²CO and HCN spectra. The HCN spectra are colored according to the ROI groupings.

The resultant stacked ¹²CO and HCN spectra are integrated over velocities where CO is detected in the stacked spectrum. The rms values of the stacked spectra are derived from the standard deviation of intensities outside of these velocity intervals. The HCN-to-CO ratio is calculated from the ratio of HCN-to-CO integrated intensities. Table 1 summarizes the selected velocity intervals, measured intensities, ratios, and uncertainties for each ROI.

Table 1. ROI Stacked Spectra Properties

ROI	Npix	(v1,v2)	W(HCN)	σ(W(HCN))	W(CO)	σ(W(CO))	R	σ(R)	fDG	σ(fDG)
		(km s−1)	(K km s−1)	(K km s−1)	(K km s−1)	(K km s−1)
1	69	[−125, 125]	2.982	0.025	48.014	0.102	0.062	0.001	0.144	0.084
2	188	[−90, 90]	0.376	0.014	18.500	0.070	0.020	0.001	0.047	0.028
3	158	[−75, 75]	0.100	0.027	8.803	0.074	0.011	0.003	0.026	0.015
4	199	[−50, 50]	0.009	0.016	3.357	0.056	0.003	0.005	0.006	0.004
5	144	[−50, 50]	0.104	0.024	2.306	0.062	0.045	0.010	0.105	0.061
6	262	[−75, 75]	0.366	0.015	15.018	0.064	0.024	0.001	0.057	0.033
7	215	[−70, 70]	0.155	0.021	4.847	0.083	0.032	0.004	0.074	0.043
8	105	[−70, 70]	0.094	0.020	4.704	0.110	0.020	0.004	0.046	0.027
9	61	[−60, 60]	0.160	0.034	8.083	0.128	0.020	0.004	0.046	0.027
10	67	[−65, 65]	0.068	0.023	2.796	0.087	0.024	0.008	0.056	0.033
11	139	[−60, 60]	0.045	0.019	1.897	0.077	0.024	0.010	0.055	0.032
12	96	[−75, 75]	0.293	0.021	5.911	0.118	0.050	0.004	0.115	0.067
13	69	[−80, 80]	0.102	0.029	5.420	0.104	0.019	0.005	0.044	0.026
14	5997	[−125, 125]	0.049	0.001	1.522	0.007	0.032	0.000	0.075	0.044

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¹²CO emission is readily detected in all ROIs that span most of the bulge and disk areas of M51—including all of the interarm regions. The ubiquity of CO in the stacked spectra is expected given that all of the ROI areas fall within the CO-defined mask described in Section 3.1. From extinction features within high-resolution, optical images obtained with the Hubble Space Telescope (HST) and 8 μm emission features from Spitzer, the interarm regions of M51 contain narrow spiral spurs of material that stretch between adjacent spiral-arm segments. These molecular features are not resolved by the ¹²CO data presented in this study but are evident in the Plateau de Bure Interferometer Arcsecond Whirlpool Survey (PAWS) ¹²CO image of M51 (Schinnerer et al. 2013). Such spiral spurs emerge from the gas subjected to increased velocity shear in the interarm regions (Kim & Ostriker 2002). In M51, molecular clouds residing within these spurs are not destroyed by the high-velocity shear of the interarm regions but maintain sufficient column densities for self-shielding to remain in the molecular gas phase throughout their transit through the interarm region to the next spiral arm (Koda et al. 2009). HCN emission is detected with signal to noise greater than 3 within most ROIs. It is not detected at this level in ROIs 4, 10, and 11.

The top-right subplot of Figure 3 displays the R values (left axis) and f_DG values (right axis). The error bars correspond to uncertainties in R. The large values of R and f_DG in the M51 bulge have been previously established in earlier studies (Bigiel et al. 2016; Gallagher et al. 2018; Jiménez-Donaire et al. 2019; Querejeta et al. 2019). Within the uncertainties, there is little variation of f_DG between spiral arms and interarms. The dense gas fraction averaged over all spiral-arm and interarm segments are 0.051 and 0.063, respectively. The comparable values of f_DG are unexpected given the deeper gravitational potential that defines the spiral arms. We discuss this limited range in f_DG values in more detail in Section 6.2.

4.1.2. Stellar-mass Surface Density

The scale height of the molecular ISM in galaxies is small in comparison to the distribution of stars and atomic gas. Consequently, molecular clouds are subject to the effective pressure generated by the weight of the stellar, gas, and dark matter components residing above and below the disk midplane. Such pressure facilitates the transition of warm, neutral, atomic gas into the cold, neutral, atomic gas phase, which is often a precursor to the formation of molecular clouds (Elmegreen 1993). Can such pressure also impact the development of high-density gas within molecular clouds?

Previous studies of HCN and CO in nearby galaxies have already demonstrated a relationship between f_DG and the mass surface density of stars as well as the full pressure component, P ∝ Σ_gasΣ_* (Gallagher et al. 2018; Jiménez-Donaire et al. 2019). Here, we examine the relationship of the dense gas fraction with stellar-mass surface density using spectral stacking. Figure 4 shows the stacked HCN and CO spectra for each log(Σ_*) bin. Table 2 summarizes the integrated intensities and ratios derived from the stacked spectra. Within the disk, where log(Σ_*) <2.5, which contains 85% of the area over all bins, f_DG is ∼0.05 with little variation over the log(Σ_*) range of 1.8 to 2.4. For log(Σ_*) > 2.5, corresponding to much of the stellar bulge component, the dense gas mass fraction rapidly rises. Evaluating the dense gas fraction within 1 kpc wide bins of radius shows a similar constant value of ∼0.05 for radii between 2 and 8 kpc and a steep rise to 0.18 in the central 2 kpc. Both Bigiel et al. (2016) and Jiménez-Donaire et al. (2019) find similar profiles of f_DG with galactic radius.

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Table 2. Stellar-mass Surface Density Stacked Spectra Properties

log($\tfrac{{{\rm{\Sigma }}}_{* }}{{M}_{\odot }{{pc}}^{-2}}$)	Npix	(v1,v2)	W(HCN)	σ(W(HCN))	W(CO)	σ(W(CO))	R	σ(R)	fDG	σ(fDG)
		(km s−1)	(K km s−1)	(K km s−1)	(K km s−1)	(K km s−1)
1.8	891	[−80,80]	0.096	0.009	4.794	0.051	0.020	0.002	0.047	0.027
2.0	661	[−80,80]	0.184	0.008	7.805	0.050	0.024	0.001	0.055	0.032
2.2	413	[−80,80]	0.244	0.014	11.742	0.046	0.021	0.001	0.048	0.028
2.4	290	[−80,80]	0.398	0.016	18.408	0.068	0.022	0.001	0.050	0.029
2.6	179	[−90,90]	0.860	0.019	29.466	0.087	0.029	0.001	0.068	0.040
2.8	91	[−100,100]	1.890	0.029	42.737	0.158	0.044	0.001	0.103	0.060
3.0	52	[−105,105]	3.249	0.039	53.789	0.216	0.060	0.001	0.140	0.082
3.2	28	[−140,140]	4.091	0.074	55.390	0.212	0.074	0.001	0.172	0.100
3.4	23	[−140,140]	4.427	0.096	50.800	0.397	0.087	0.002	0.203	0.118
3.6	16	[−150,150]	5.380	0.091	51.246	0.285	0.105	0.002	0.244	0.142

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M51 has been a primary target of studies exploring the resolved KS relationship between Σ_SFR and Σ_mol with varying results (Kennicutt et al. 2007; Bigiel et al. 2008; Liu et al. 2011; Chen et al. 2015; Bigiel et al. 2016; Leroy et al. 2017). These differences arise in part due to angular resolutions and whether the analysis accounts for contributions from an older (>100 Myr) stellar population to the star formation tracer (Liu et al. 2011). In the top row of Figure 5, we show the KS relationship between Σ_SFR and Σ_mol = α_CO∫dvT(CO) M_⊙ pc⁻² derived for the three SFR models presented in Section 3.2. Data with log(Σ_SFR) less than −9.5 are excluded as the conversion to the star formation rate for these faint FUV and 24 μm luminosities is less reliable (Leroy et al. 2012). We fit for the parameters A and N in the expression, log(Σ_SFR) = A + Nlog(Σ_mol) using bisector least squares in the Python module bces (Nemmen et al. 2012) that is based on Akritas & Bershady (1996). We also calculate the posterior probability distribution for the set of regression parameters (denoted A_B , N_B ) using the emcee package (Foreman-Mackey et al. 2013). Table 3 lists the best-fit values for A and N, uncertainties, and scatter derived from the bisector fit and the ±2σ range from the posterior distributions of A_B and N_B . The bisector fits show the same trend identified by Liu et al. (2011) in which the power-law index is near unity for SFR not corrected for contributions from the older stellar population as in Model 0. For corrected SFR values (Model 1 and Model 2), the index steepens to 1.2. The Bayes' regression shows values of the slope posterior distributions to be linear or marginally sublinear for all three SFR models. Shetty et al. (2013) applied hierarchical Bayesian regression on surface densities of star formation rates and molecular gas for a set of galaxies, including M51, for which they found a strongly sublinear slope of 0.72 with a ±2σ range [0.62, 0.83] over a narrower range of Σ_mol and fewer points than the data used in this study.

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The variation of the SFR with the dense gas mass is also an important relation as it probes the localized volumes of gas more spatially connected to star formation over the ensemble of clouds within the resolution element of the observations. The increased angular resolution of the LMT relative to other single-dish telescopes provides sensitivity to the range of HCN luminosities 5 × 10⁴–10⁶ K km s⁻¹ pc², which is not well populated from previous studies. The LMT data complement the recent Northern Extended Millimeter Array (NOEMA) measurements that are also sensitive to this HCN luminosity range (Querejeta et al. 2019).

The bottom row of Figure 5 shows the relationship between Σ_SFR and ${{\rm{\Sigma }}}_{\mathrm{dense}}={\alpha }_{\mathrm{HCN}}\int {dvT}(\mathrm{HCN})$ M_⊙ pc⁻² for each SFR model in the form of a two-dimensional histogram. Only points with Σ_SFR greater than −9.5 are considered. We apply both bisector and Bayes' regression to the expression log(Σ_*) = A + Nlog(Σ_dense) and summarize the results in Table 3. Sublinear relationships are derived for SFR Models 1 and 2 with slopes ∼0.65 (bisector) and a 2σ posterior distributions range between 0.50 and 0.68 from the Bayes' regression.

One of the key results to emerge from the EMPIRE survey is the anticorrelation of the star formation efficiency of dense gas, SFE_dense = Σ_SFR/Σ_dense, with the local environment properties: stellar-mass surface density, molecular gas surface density, molecular gas fraction, and the midplane pressure (Gallagher et al. 2018; Jiménez-Donaire et al. 2019). From three NOEMA fields of M51 with a high angular resolution, Querejeta et al. (2019) also identify an anticorrelation but with a large scatter of points for a given Σ_*. The authors connect this trend to the Central Molecular Zone (CMZ) of the Milky Way, where there is a large reservoir of dense gas and a large dense gas fraction (Jackson et al. 1996; Jones et al. 2012) but a suppressed star formation rate relative to the disk (Longmore et al. 2013; Kruijssen et al. 2014; Barnes et al. 2017; Lu et al. 2019).

To examine this dependence in our data, we apply spectral stacking within logarithmic bins of Σ_* as in Figure 4 to evaluate Σ_SFR, Σ_dense, and Σ_mol. For a given log(Σ_*) bin, the star formation efficiencies are

$$\begin{eqnarray}&&{\mathrm{SFE}}_{\mathrm{dense}}({\mathrm{Myr}}^{-1})={10}^{6}\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{SFR}}({M}_{\odot }{{pc}}^{-2}{\mathrm{yr}}^{-1})}{{\alpha }_{\mathrm{HCN}}\int {dv}\langle T(\mathrm{HCN})\rangle )}\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}&&{\mathrm{SFE}}_{\mathrm{mol}}({\mathrm{Myr}}^{-1})={10}^{6}\frac{{{\rm{\Sigma }}}_{\mathrm{SFR}}({M}_{\odot }{{pc}}^{-2}{\mathrm{yr}}^{-1})}{{\alpha }_{\mathrm{CO}}\int {dv} \langle T(\mathrm{CO}) \rangle )},\end{eqnarray} \tag{ 6 }$$

where ${{\rm{\Sigma }}}_{\mathrm{SFR}}={\sum }_{k=1}^{{N}_{b}}{\mathrm{SFR}}_{k}/{\sum }_{k=1}^{{N}_{b}}{A}_{k}$, A_k is the area per pixel in pc², N_b is the number of pixels within the log(Σ_*) bin, and $\langle T\left({\rm{HCN}}\right) \rangle$ and 〈T(CO)〉 are the stacked HCN and CO spectra, respectively.

Figure 6 shows the variation of the dense gas (red points) and molecular (blue points) star formation efficiencies with log(Σ_*) and the R calculated within each log(Σ_*) bin for the 3 SFR models. As found for many galaxies in the EMPIRE sample including M51, there is an overall decreasing relationship of SFE_dense with increasing Σ_* and R. However, these profiles are not well described by a power law but, rather, exhibit a broad inflection point over the log(Σ_*) range 2.5–3.0. A similar inflection point is seen in the data presented by Gallagher et al. (2018) for M51. This inflection point is coincident with the stellar-mass surface density at which f_DG increases (see Figure 4). SFE_mol shows a mostly flat (Model 0) or slight decreasing (Models 1 and 2) profile with increasing Σ_* and f_DG that is consistent with a linear or slightly sublinear relationship between Σ_SFR and Σ_mol.

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It is useful to compare these results on M51 with similar measures of f_DG in the Milky Way where the improved spatial resolution enables one to pinpoint the areas within a cloud responsible for the HCN emission. Jackson et al. (1996) and Jones et al. (2012) imaged the HCN and CO J = 1 – 0 emission from the CMZ of the Galaxy. The well-known radio features, Sgr A and Sgr B, are bright in HCN emission but there is extended HCN emission throughout the CMZ that is distributed over a large velocity interval of ±200 km s⁻¹. Averaged over the 500 pc aperture, comparable to the spatial resolution in this study, Jackson et al. (1996) derive an HCN-to-CO ratio of 0.06 corresponding to f_DG = 0.14. Integrating over the entire CMZ, Jones et al. (2012) measure a higher value of R = 0.095, which scales to f_DG = 0.22. Figure 2 shows HCN/CO = 0.12, f_DG = 0.28 in the nucleus of M51, and HCN/CO >0.056 (f_DG ≥ 0.14) in the central bulge volume (see Figures 3 and 4).

Observing HCN emission throughout the disk of the Milky Way is considerably more challenging given the required coverage and sensitivity. Helfer & Blitz (1997) constructed an unbiased, undersampled survey of HCN and CO emission along the Galactic plane between longitudes 155 and 555, spaced by 1° intervals and b = 0° with the NRAO 12 m telescope. Their HCN-to-CO ratio shows a flat profile with Galactic radius with an average value of 0.026 (f_DG = 0.06). From this low value of R with respect to HCN/CO values observed toward resolved, active star-forming regions (0.1 – 0.3) within Galactic molecular clouds, they inferred that the HCN emission was originating from the more extended, lower-density (10³ cm⁻³) components of the clouds. Evans et al. (2020) imaged HCN J = 1 – 0 emission from six molecular clouds in the first quadrant of the Milky Way. They determined HCN-to-¹³CO ratios between 0.04 and 0.12 but only a fraction (0%–57%) of the HCN luminosity comes from regions with A_v > 8, which is assumed to reflect high gas volume density where stars form. Battisti & Heyer (2014) estimated the dense gas mass fraction for 344 clouds in the first quadrant of the Milky Way using ground-based thermal 1.1 mm dust continuum emission as a measure of dense gas mass and ¹³CO J = 1 – 0 emission for the total cloud mass. While optically thin dust emission is a direct tracer of gas column density, data-processing methods used for ground-based millimeter continuum measurements to account for atmospheric contributions also remove extended emission from the larger clouds depending on the size of the cloud relative to the field of view of the bolometer array (Ginsburg et al. 2013). The resultant image identifies regions of overdensity within a cloud that typically correspond to dense clumps and filaments. Battisti & Heyer (2014) found f_DG values of 0.11 for all dust emission and 0.07 for subregions with mass surface densities greater than 200 M_⊙ pc⁻², which likely represents higher volume densities. The fractions are found to be independent of cloud mass and mass surface density.

Figures 3 and 4 show f_DG values within the disk of M51 with a mean value of 0.05. This mean value is comparable to Milky Way values found by Helfer & Blitz (1997) and inferred from Evans et al. (2020) that point to a lower-density (∼10³ cm⁻³) origin where the HCN rotational energy levels are subthermally excited, resulting in weaker emission. The density of this component is still larger than the mean density of molecular clouds that is inferred from ¹²CO and ¹³CO emission.

A surprising result from our stacking analysis is the near-constant value of f_DG ∼ 0.05 between spiral-arm and interarm regions and throughout the disk component of M51. The actions of a spiral density wave are gas compression and increased interstellar turbulence (Kim & Ostriker 2002) that contribute to the development of dense regions within molecular clouds. So, one might expect to find elevated values of f_DG in the spiral arms relative to those derived in the interarm regions.

The expected higher f_DG values in spiral arms can be resolved with our data if the conversion factor between CO luminosity and H₂ mass is different in these respective domains. There is evidence that the CO conversion factor is ∼2 times larger in the interarm regions than in spiral arms in M51 and M83 (Wall et al. 2016). Adopting this arm–interarm dependence of the conversion factor would lead to f_DG values two times higher in the spiral arms than in the interarm regions.

Alternatively, the constant dense gas mass fraction throughout the disk could result from the density thresholds required to excite the HCN and CO lines within a population of stratified, self-gravitating clouds (Elmegreen 2018). For each spectral line, there is a unique cloud radius, r_c , at which the volume number density is equal to the line's effective critical density, n_c . For a 1/r² density profile, the mass within this radius is proportional to r_c . Because r_c scales with ${n}_{c}^{-1/2}$, then the dense gas fraction derived from HCN and CO measurements is ${({n}_{\mathrm{CO}}/{n}_{\mathrm{HCN}})}^{1/2}$, where n_CO and n_HCN are the critical densities for CO and HCN, respectively. For critical densities of 100 cm⁻³ for CO and 10⁴ cm⁻³ for HCN, f_DG = 0.1, which is a reasonable estimate for the M51 values in the disk given the approximations of the effective critical densities described in Section 4.

Figure 4 shows the dense gas fraction begins to increase for log(Σ_*) > 2.5. In this domain, the pressure, P_e , from the weight of stellar and gas components may exceed the gravitational energy density of a cloud, ${U}_{G}=\tfrac{\pi G}{2}{{\rm{\Sigma }}}_{\mathrm{cloud}}^{2}$, where Σ_cloud is the molecular gas surface density of an individual cloud. Clouds that satisfy the condition P_e > U_G are labeled as diffuse clouds (Elmegreen 2018). In Figure 7 we show the variation of U_G with log(Σ_*) for giant molecular clouds and cloud complexes in M51 residing in the central region, spiral arms, and interarm regions from the catalog constructed by Colombo et al. (2014). The Σ_* value for each cloud is taken from the stellar-mass surface density at the position of the cloud. The total midplane pressure can be approximated as

$$\begin{eqnarray}&&{P}_{\mathrm{tot}}=\displaystyle \frac{\pi }{2}G{{\rm{\Sigma }}}_{\mathrm{gas}}\left({{\rm{\Sigma }}}_{\mathrm{gas}}+{{\rm{\Sigma }}}_{* }\displaystyle \frac{{c}_{g,\mathrm{tot}}}{{c}_{s}}\right),\end{eqnarray} \tag{ 7 }$$

where c_s and c_g,tot are the star and gas velocity dispersions (Elmegreen 1989). At the cloud boundary, the kinematic pressure, P_e = P_tot/(1 + α_◦ + β_◦), where α_◦ = 0.46 and β_◦ =0.30 are the relative cosmic-ray and magnetic field energy densities (Draine 2010). The measured gas velocity dispersion also requires a correction for cosmic rays and the interstellar magnetic field such that ${c}_{g,\mathrm{tot}}={c}_{g}/{(1+{\alpha }_{\circ }+{\beta }_{\circ })}^{1/2}$. From averages of Σ_mol and Σ_* over 1 kpc wide annular radii, we derive the relationship ${{\rm{\Sigma }}}_{\mathrm{mol}}=0.35{{\rm{\Sigma }}}_{* }^{0.95}$. Assuming fully molecular gas (Σ_gas = Σ_mol), c_s = 20 km s⁻¹ and c_g =10 km s⁻¹, we construct the profile of pressure with Σ_* from Equation (7). This profile of pressure with log(Σ_*) is shown as the solid black line in Figure 7. For low values of log(Σ_*) corresponding to the M51 disk (magenta and blue points), most clouds are above the pressure line, consistent with self-gravitating objects. In the central bulge, most of the defined objects (red points) lie below the pressure profile and are considered diffuse clouds.

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From this information, the fraction of CO luminosity contributed by clouds with P_e > U_G for each Σ_* bin is calculated (see the right panel of Figure 7). For bins of stellar-mass surface densities greater than 500 M_⊙ pc⁻², >45% of the molecular cloud mass resides within diffuse clouds. These fractions are also derived for the set of clouds in each region described by Colombo et al. (2014). The mass fraction of diffuse clouds is 91% in the central region, 14% in spiral arms, and 32% in the interarm regions.

The increasing fraction of molecular cloud mass in the diffuse state can account for the rise of the dense gas mass fraction at high stellar-mass surface densities. In the central 2 kpc, the neutral gas is already fully molecular (Jiménez-Donaire et al. 2019) so there is no available atomic hydrogen to compress and contribute to the molecular mass. The external pressure can act upon the molecular gas to increase the mean density of the now diffuse molecular clouds. This compression leads to an increase in the dense gas mass fraction that is seen in Figures 3 and 4. At even higher stellar-mass surface densities within the central bulge, the pressure may even exceed the self-gravity energy density of the dense gas component. We note that for such conditions, our adopted value of ${\alpha }_{\mathrm{HCN}}$ is an upper limit (Gao & Solomon 2004b), which may increase SFE_dense values in the highest stellar-mass surface density bins and flatten the SFE_dense profile.

The profiles of f_DG and the star formation efficiency with increasing stellar-mass surface density are clues for understanding the role of the galactic environment in the dynamical state of molecular clouds and the impact on the formation of new stars in galaxies. In M51, we find higher star formation efficiencies of dense gas in the disk, where the stellar-mass surface density and midplane pressure are low, and smaller values of SFE_dense in the central bulge, where Σ_* and pressures are high. Similar results have been established by earlier studies of M51 and other galaxies (Bigiel et al. 2016; Gallagher et al. 2018; Jiménez-Donaire et al. 2019). The transition between these regimes occurs between log(Σ_*) ∼ 2.5–3.0, where an increasingly larger fraction of the molecular mass resides within the diffuse cloud population (Figure 7).

Gallagher et al. (2018) and Jiménez-Donaire et al. (2019) propose that the lower dense gas star formation efficiencies in the high-pressure environment are a consequence of a higher mean density of clouds such that star formation occurs in regions of even higher volume densities than is traced by the HCN J = 1 – 0 line. The smaller volume and mass fraction of such overdense regions imply a larger amount of lower-density material that is contributing to the HCN J = 1 – 0 luminosity but is not actively forming stars, which leads to lower values of SFE_dense.

The proposed shift of star formation activity to higher-density regions can be understood if one considers the self-gravity of the dense protocluster clumps and protostellar cores as a fundamental requirement for the production of new stars. The formation of stars occurs when such structures collapse under their own self-gravity rather than implode from an external pressure disturbance. For high-pressure environments like the central bulge of M51, the condition U_G > P_e is realized at much higher surface and volume densities than are probed by the HCN J = 1 – 0 line.

A simple test for this description is to measure the variation of the star formation efficiency in dense gas using a tracer with a much higher critical density than the HCN J = 1 – 0 transition used in this and other studies. These data would directly trace the higher volume density and higher surface density regions that remain self-gravitating even under large external pressures. With a tracer of very high densities such that U_G > P_e , one would expect to see the SFE_dense profile to be flat or rising with increasing Σ_* and pressure.