Do Spectroscopic Dense Gas Fractions Track Molecular Cloud Surface Densities?

EMPIRE HCN Data: EMPIRE (Bigiel et al. 2016) used the IRAM 30 m telescope to map HCN emission from nine galaxies, four of which have high-resolution CO maps suitable for our experiment. The EMPIRE maps cover the whole star-forming disk of each galaxy, but have relatively poor (34'') angular resolution.

M. Jimenez Donaire et al. (2018, in preparation) present a full description of EMPIRE, including the new CO (1−0) maps (see also Cormier et al. 2018), which we pair with the HCN maps to measure the HCN (1−0) to CO (1−0) ratio. Briefly, EMPIRE uses the IRAM 30 m in on-the-fly mapping mode to cover the area of active star formation in each target (see Figure 1). Observations were conducted from 2012 to 2016. The data were calibrated using GILDAS, extracted at 4 km s⁻¹ spectral resolution and then further reduced using an in-house pipeline. The pipeline fits and subtracts a second-order polynomial baseline, avoiding regions of the spectrum known to have bright CO emission. It rejects spectra with measured noise that is significantly larger than that predicted by the radiometer equation. Then it projects the data on to grids with pixel size of 4''. The adopted gridding kernel convolved with the IRAM 30 m beam yields a final angular resolution of 34''. Finally, the pipeline converts the data to main beam temperature units, assuming main beam and forward efficiencies of 0.78 and 0.94 (Carter et al. 2012, IRAM calibration papers).

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ALMA HCN Data: Gallagher et al. (2018) mapped HCN emission from four galaxies and assembled matched-resolution CO (1−0) maps from the literature²⁵ The ALMA maps cover out to r_gal = 3.5–6 kpc, a smaller area than EMPIRE (Figure 1), but they have a sharper 8'' resolution.

We take CO observations from the Berkeley-Illinois-Maryland Association Survey of Nearby Galaxies (BIMA SONG; Helfer et al. 2003) for NGC 3351 and NGC 3627. These cubes include data from both the BIMA interferometer and short spacing data from the NRAO 12 m single-dish telescope on Kitt Peak. We take interferometric CO observations from the Combined Array for Research in Millimeter-wave Astronomy Survey Toward IR-bright Nearby Galaxies (CARMA STING; Rahman et al. 2015) for NGC 4254. We combine this with single-dish data from the CO extension to the IRAM EMPIRE survey (Cormier et al. 2018, M. Jiménez-Donaire et al. 2018, in preparation). We take CO data from the ALMA science verification program for NGC 4321. This includes both main 12 m array and Atacama Compact Array (ACA) short spacing and total power data. We take the CO data from EMPIRE for NGC 5194.

ALMA observed HCN (1−0) using a seven-field mosaic centered on the nucleus of the galaxy. The data were reduced using the CASA (McMullin et al. 2007) package and observatory-provided calibration scripts. They were then imaged using natural weighting with a small u − v taper and a velocity resolution of 10 km s⁻¹. Using the CASA task feather, the ALMA cubes were combined with the IRAM 30 m maps (mostly from EMPIRE) to correct for missing short spacing data. The resulting cubes were convolved to a resolution of 8'', chosen to match archival CO and infrared data (see Gallagher et al. 2018). The statistical noise in each 10 km s⁻¹ channel is ∼5–10 mK.

For each sample, we convolve the HCN (1−0) and CO (1−0) data from the literature for all targets to a common physical resolution set by the physical beam size at the most distant target. This is 650 pc for ALMA and 2770 pc for EMPIRE. At this common resolution, we measure HCN/CO, the ratio of the HCN(1−0) to CO(1−0) integrated intensities. We constructed an integrated intensity map of HCN using a mask defined in position-position–velocity space from the CO (1−0) cube. We then measure HCN/CO everywhere within the fields of view that CO is detected (see Gallagher et al. 2018).

We estimate the cloud-scale surface density from PHANGS-ALMA CO (2−1) and (for NGC 5194) PAWS CO (1−0) data at 120 pc resolution (see Figure 1). PHANGS-ALMA produces CO (2−1) line maps with ∼1''–15 resolution, 2.5 km s⁻¹ velocity resolution, ∼0.1 K noise per channel, and including short spacing and total power information from ALMA's Morita ACA. Data reduction and imaging for PHANGS-ALMA are described in A. K. Leroy et al. (2018, in preparation). Details regarding the creation of moment maps, noise, and completeness for the four targets studied here appear in Sun et al. (2018). Given the distances and angular resolutions of these data, 120 pc represents the common physical resolution for our high-resolution CO data. We convolve all four CO (2−1) maps to share this common physical resolution.

For NGC 5194, we also convolve the PAWS CO (1−0) moment-zero map (Pety et al. 2013; Schinnerer et al. 2013) to 120 pc resolution. To place these measurements on the same CO (2−1) intensity scale as our other targets, we then multiply the PAWS CO (1−0) intensities by a typical CO (2−1)/CO (1−0) ratio of 0.7 (uncertain by ±0.15 dex, see Leroy et al. 2013).

We measure HCN/CO at coarser resolution than we measure the CO (2−1) intensity, I_CO(2−1). To connect the two measurements, we calculate the intensity-weighted average I_CO(2−1) within each larger HCN beam. This weighted average, $\langle {I}_{\mathrm{CO}(2-1)}\rangle$, measures the mean 120 pc resolution I_CO(2−1) from which CO photons emerge. Formally,

$$\begin{eqnarray}&&\langle {I}_{\mathrm{CO}(2-1)}\rangle =\displaystyle \frac{{({I}_{\mathrm{CO}(2-1)})}^{2}\ast {\rm{\Omega }}}{{I}_{\mathrm{CO}(2-1)}\ast {\rm{\Omega }}}.\end{eqnarray} \tag{ 1 }$$

Here, I_CO(2−1) is the CO map at 120 pc resolution, the asterisk represents convolution, and Ω indicates the Gaussian kernel used to change the resolution of the map from 120 pc to the final resolution (650 pc for the ALMA sample and 2770 pc for the EMPIRE sample).

$\langle {I}_{\mathrm{CO}(2-1)}\rangle$ is the expectation value of CO intensity weighted by itself within each coarser HCN beam. In practice, given some conversion between light and mass (i.e., α_CO), $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ captures the mass-weighted 120 pc resolution surface density of molecular gas inside each larger HCN beam, $\langle {{\rm{\Sigma }}}_{\mathrm{mol}}\rangle$.

The advantage of this approach, which is discussed at length by Leroy et al. (2016; and see Ossenkopf & Mac Low 2002; Utomo et al. 2018) is that it preserves the high-resolution information and down-weights empty regions. Compared to the unweighted average, this intensity-weighted average, $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ is 0.6 dex higher at 650 pc resolution and 1.4 dex higher at 2770 pc resolution. The intensity weighting is not equivalent to smoothing, as it effectively leverages the high-resolution information and yields characteristic surface densities that are ∼4–30 times higher than smoothed maps. The difference reflects beam dilution due to the large amount of empty space in the CO maps, which is also visible from Figure 1.

Uncertainty: We estimate the uncertainty in $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ via a Monte Carlo calculation. We begin with the original CO (2−1) cubes, add randomly generated Gaussian noise with the correct mean amplitude, and then run these noise-added cubes through our full analysis procedure. We repeat this process 100 times, and calculate the standard deviation in $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ over all realizations. The mean error calculated in this way is ∼2 K km s⁻¹. As a result, all $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ values in this Letter have a signal-to-noise ratio (S/N) >5.

The distances to our targets are uncertain by ∼10%–30%. The angular scale corresponding to 120 pc is correspondingly uncertain, adding an additional uncertainty to our calculation. Using the same data that we use here, Sun et al. (2018) showed that changing from a physical resolution of 80–120 pc (i.e., by >30%) alters the mean I_CO(2−1) in a galaxy by ∼0.05–0.1 dex.

Our EMPIRE and ALMA surveys only detect HCN along individual lines of sight at high signal-to-noise in the brightest regions of our targets. However, our data also contain a large amount of information at lower signal-to-noise.

We recover $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ at high signal-to-noise across a wide area. Therefore, to access the fainter HCN emission, we measure the average HCN/CO in bins of $\langle {I}_{\mathrm{CO}(2-1)}\rangle$. We report the integrated HCN divided by the integrated CO in each bin, with the statistical uncertainty on this binned ratio propagated from the original maps following Gallagher et al. (2018). This binning increases the signal-to-noise in HCN/CO via averaging and extends the dynamic range in our measured correlation dramatically.

Following Gallagher et al. (2018), our HCN integrated intensity maps are created by integrating the cube over the region with bright CO emission, whether or not that region shows HCN emission at high signal-to-noise. As a result, this averaging approach is almost equivalent to spectral stacking using the CO velocity field as a prior (as in Schruba et al. 2011; Jiménez-Donaire et al. 2017). We verify this by comparing the two approaches directly. After shifting the HCN cubes to the local mean CO velocity and averaging, we derive stacked line ratios in bins of $\langle {I}_{\mathrm{CO}}\rangle$. On average, the spectral stacking yields the same results as our mask-and-average approach within ∼10%, with no systematic offset.

We report the observed HCN/CO ratio as a function of $\langle {I}_{\mathrm{CO}(2-1)}\rangle$. These quantities are interesting because they trace the fraction of dense gas and the mean surface density of molecular clouds. Adopting simple translations from observables, we indicate these two physical quantities on the alternative right and top axes of Figures 2 and 3.

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To translate HCN/CO to f_dense, we assume α_{CO(1 − 0)} =4.35 M_⊙ pc⁻² (K km s⁻¹)⁻¹ (Bolatto et al. 2013) and a more uncertain α_HCN ≈ 14 M_⊙ pc⁻² (K km s⁻¹)⁻¹ to convert HCN to the mass of gas above a density of n_H2 ≈ 5 × 10³ cm⁻³ (Onus et al. 2018). For comparison, many previous studies have assumed α_HCN = 10 M_⊙ pc⁻² (K km s⁻¹)⁻¹ for gas above 3 × 10⁴ cm⁻³ (following Gao & Solomon 2004).

Both the density of gas traced by HCN (1−0) and the conversion from HCN emission to a dense gas mass remain uncertain. The effective critical density of HCN changes as a function of temperature and optical depth, which are hard to measure (Jiménez-Donaire et al. 2017). Moreover, gas at densities below the effective critical density still emits HCN, rendering the density traced by HCN a product of the emissivity and density distribution (Leroy et al. 2017). For more discussion, see Gao & Solomon (2004), Usero et al. (2015), Leroy et al. (2017), Onus et al. (2018), and Gallagher et al. (2018), as well as the Milky Way studies by Kauffmann et al. (2017), Mills & Battersby (2017), and Pety et al. (2017).

Similarly, we report $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ as our primary measurement and the 120 pc resolution molecular gas surface density, $\langle {{\rm{\Sigma }}}_{\mathrm{mol}}\rangle$, as an alternative axis. For a typical CO (2−1)/CO (1−0) ratio of 0.7, the Galactic CO-to-H₂ conversion factor is α_{CO (2 − 1)} ≈ 6.2 M_⊙ pc⁻² (K km s⁻¹)⁻¹.

To check our results, we also analyzed the ALMA HCN data at the EMPIRE common physical resolution of 2770 pc. At a fixed $\langle {I}_{\mathrm{CO}(2-1)}\rangle$, the ALMA and EMPIRE data differ by a mean of 15% in HCN/CO. Mostly, this offset reflects the fact that the two data sets cover different area (see Figure 1). When we match the areal coverage (i.e., consider EMPIRE only over the ALMA area), a smaller ∼5% difference remains.

Our CO (1−0) data (used in the denominator of HCN/CO) come from different sources for EMPIRE and the ALMA sample. We estimate the uncertainty associated with our choice of CO map by considering NGC 4321, for which we have ALMA, BIMA, and IRAM 30 m CO (1−0) maps. We only have CARMA data for NGC 4254 so we cannot explore how CARMA compares to our other CO (1−0) sources. We repeat our complete analysis with each NGC 4321 CO (1−0) map. At 650 pc resolution, the HCN/CO ratios measured using the ALMA CO map are ≈0.1 dex lower than those measured using the BIMA map. At 2770 pc resolution, the ALMA map yields HCN/CO ratios ≈0.2 dex lower than BIMA, while the EMPIRE maps yields ratios ∼0.25 dex lower than BIMA. The ALMA and EMPIRE results agree within 0.06 dex. Overall, we have good confidence in the EMPIRE and ALMA measurements, but the two BIMA-based CO maps may have results uncertain by ∼0.1 dex at 650 pc resolution. While there are offsets between the ratio values calculated using different input data, these offsets do not change the nature of the observed trends.

Figures 2 and 3 show binned HCN (1−0)/CO (1−0) as a function of $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ for our two subsamples. HCN (1−0)/CO (1−0) and $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ are strongly positively correlated for all galaxies in both samples. Spearman's rank correlation coefficient, ρ, is high for both individual galaxies (ρ =0.97–1.0) and the entire sample (ρ = 0.77–0.83; Table 2). The corresponding low p values indicate that this correlation is unlikely to be produced by random noise. Our measurements offer strong evidence for a significant underlying relationship between our f_dense (traced by HCN/CO) and cloud-scale surface density (traced by $\langle {I}_{\mathrm{CO}(2-1)}\rangle$).

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Table 2 reports power-law fits (fit via the bisector method) that offer a reasonable description of each sample. We plot these as black lines in Figures 2 and 3.

These fits offer a good first-order description of the observed trends. However, we do observe substantial galaxy-to-galaxy variations in HCN/CO at fixed $\langle {I}_{\mathrm{CO}(2-1)}\rangle$. The standard deviation in HCN/CO at fixed $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ is 0.11 dex for the ALMA sample and 0.08 dex for the EMPIRE sample. Because of our binning approach, this reflects only the galaxy-to-galaxy scatter. If we were in a position to measure the cloud-to-cloud or region-to-region scatter within each $\langle {I}_{\mathrm{CO}(2-1)}\rangle$, we would expect to find more variation.

NGC 3627 and NGC 4321 in the EMPIRE data (Figure 3) also exhibit different behavior at high and low $\langle {I}_{\mathrm{CO}(2-1)}\rangle$. For these galaxies, the slope relating HCN/CO to $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ steepens near ${\mathrm{log}}_{10}\langle {I}_{\mathrm{CO}(2-1)}\rangle \sim 1.75$ (∼55 K km s⁻¹ ∼350 M_⊙) and these galaxies show higher ${\mathrm{log}}_{10}\langle {I}_{\mathrm{CO}(2-1)}\rangle$ and lower HCN/CO compared to the other two targets. Though we do not find similar curvature, the same two targets show a similar offset from NGC 4254 to high ${\mathrm{log}}_{10}\langle {I}_{\mathrm{CO}(2-1)}\rangle$ and lower HCN/CO in the ALMA observations (Figure 2). Below, we suggest that "starburst"-like conversion factors in the centers of these galaxies offer a likely explanation for this behavior.

The correlation between HCN/CO and $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ supports the idea that both quantities trace the density distribution of molecular gas. This suggests that the mean surface density of a molecular cloud and its dense gas content both reflect an underlying, environment-dependent gas density distribution.

Below, we show that this would be expected from simple models as long as cloud-scale mean surface density traces cloud-scale mean volume density. In fact, as discussed in (Leroy et al. 2017, see their Figure 5), surface and volume density do correlate in recent molecular cloud catalogs. Moreover, the molecular gas scale height in the Milky Way appears relatively constant over the inner ∼8 kpc (Heyer & Dame 2015). In short, current evidence appears to support the idea that at high resolution molecular gas surface density tracks molecular gas volume density to first order (see also Utomo et al. 2018).

Connection to Environment: The regions of galaxies with high gas and stellar surface densities and high interstellar gas pressure also tend to have high dense gas fractions, f_dense (Usero et al. 2015; Bigiel et al. 2016; Gallagher et al. 2018). These also tend to be in the inner parts of galaxies, so the binned results in Figures 2 and 3 also roughly map to radius and stellar surface density, with the central regions of each galaxy mostly contributing to the top-right part of the relationship.

At the same time, many recent studies have shown an environmental dependence of the cloud-scale properties of molecular gas (e.g., Hughes et al. 2013; Colombo et al. 2014; Leroy et al. 2016; Faesi et al. 2018; Sun et al. 2018). Broadly, these results have the same sense as those for f_dense. The internal pressure of molecular clouds appears to correlate with large-scale environmental gas pressure, radius, and the stellar mass of the host galaxy (Hughes et al. 2013; Schruba et al. 2018; Sun et al. 2018).

Our results directly connect these two lines of evidence. The correlation that we observe suggests that the properties of the bulk molecular gas (e.g., Hughes et al. 2013; Colombo et al. 2014; Leroy et al. 2016; Faesi et al. 2018; Schruba et al. 2018; Sun et al. 2018) and f_dense (Gao & Solomon 2004; Usero et al. 2015; Bigiel et al. 2016; Gallagher et al. 2018) reflect different aspects of the same environment-dependent density distribution. The sense of this correlation should broadly be that high-pressure, high-surface-density, inner parts of galaxies have both high f_dense and high cloud-scale mean surface density.

Does α_CO Drive Galaxy-to-galaxy Variations? Sandstrom et al. (2013) found that the CO-to-H₂ conversion factor (α_CO) is often lower than the standard Galactic value in the inner regions of galaxies with dense, bar-fed centers. Specifically, they found 2–3 times lower α_CO in the center of NGC 3627 and NGC 4321 compared to the disks. NGC 4254 (Sandstrom et al. 2013) and NGC 5194 Leroy et al. (2017) do not show central α_CO depressions, and NGC 4254 may even show a central rise in α_CO. An arrow in Figure 3 indicates the effect of lowering α_CO (but not α_HCN) by a factor of 2. Adjusting the inner (high $\langle {I}_{\mathrm{CO}(2-1)}\rangle$) NGC 3627 and NGC 4321 points in this way would bring the different galaxies into better agreement. If α_CO and α_HCN change in the same way, then the points only move horizontally.

Though not a unique explanation, a low α_CO due to bright diffuse (but not dense) molecular gas offers a feasible explanation for some of the offset among our targets. This change in α_CO as a function of $\langle {I}_{\mathrm{CO}(2-1)}\rangle$) could also explain the curvature in the EMPIRE results for NGC 3627 and NGC 4321.

Indirect Evidence that Both Axes Trace Density: A number of recent studies have raised concerns about the ability of HCN to trace dense gas (e.g., Rathborne et al. 2015; Kauffmann et al. 2017; Leroy et al. 2017; Pety et al. 2017). The unknown abundance of the molecule, its uncertain opacity (Jiménez-Donaire et al. 2017), and possible excitation effects (Shimajiri et al. 2017) all are likely to affect the HCN-to-dense gas conversion factor. Similarly, $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ at 120 pc resolution may suffer from excitation effects (Koda et al. 2012). Variations in the line-of-sight depth could also introduce scatter into the relationship between mean volume density and mean surface density.

Despite these concerns, these two observables remain among our most practical tracers of f_dense and mean cloud-scale surface density for observations of external galaxies. Our finding that they track each other is a powerful, though still indirect, evidence that both the HCN/CO ratio and the cloud-scale CO intensity are meaningful tracers of gas density. Moreover, even if HCN traces lower density gas than is commonly assumed, the contrast between the HCN and CO lines still captures the shape of the density distribution to some extent.

Our observed correlation would be expected if (1) HCN/CO traces the fraction of gas above some threshold density, (2) $\langle {I}_{\mathrm{CO}(2-1)}\rangle$ traces the mean surface density of molecular clouds, and (3) the mean surface density of a molecular cloud traces its mean volume density. Here we illustrate this by integrating over some simple models of gas density distributions.

We calculate f_dense as a function of mean density for three model gas volume density distributions²⁶ : (1) a "bottom-heavy" power law with a slope of −2.5, such that dp/dn  ∝ n^−2.5; (2) a power law with dp/dn  ∝ n⁻² and a width of 2 dex; such a distribution has equal mass per logarithmic bin (i.e., a "top hat" in mass); and (3) a lognormal distribution with 1σ width of 1.0 dex. Power-law and lognormal distributions are currently the most popular ways to represent volume and column density distributions (e.g., Padoan & Nordlund 2002; Kainulainen et al. 2009; Lombardi et al. 2015).

Figure 4 shows the shape of each model distribution for several mean densities. To calculate the dense gas mass fraction, we take the ratio of mass above two thresholds, one at low density (teal) and one at high density (light green). These thresholds correspond to the n_eff of the two lines. Dividing the mass above the high-density threshold by the mass above the low-density threshold we construct a model "dense gas fraction" similar to what we expect to find using the HCN/CO ratio.

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To predict how f_dense depends on mean density, we repeat the exercise for many distributions. We leave the width (when applicable) and slope of each distribution fixed, keep the CO and HCN density thresholds fixed, and vary only the mean density of the cloud. The lines in the bottom-right panel of Figure 4 shows the resulting f_dense as a function of mean density (the points indicate the specific cases illustrated in the other panels).

For all distributions, Figure 4 shows that as the mean density increases, a larger and larger fraction of the gas sits above the effective density of HCN (1−0) (the light green region). As a result, we expect a correlation between mean density and f_dense. At intermediate densities, where the mean density lies between the low- (CO) and high- (HCN) density threshold, power-law-like scaling relations between mean density and f_dense are common. If our observed $\langle {I}_{\mathrm{CO}(2-1)}\rangle$, tracing the mean cloud-scale surface density, also traces the mean cloud-scale volume density, then our observations would match the expectation from these simple models.