Star Formation Occurs in Dense Gas, but What Does "Dense" Mean?

In pioneering work, Gao & Solomon (2004) showed that the far-infrared luminosities in starburst galaxies follow a very tight, linear correlation with the luminosities of HCN line emission. Wu et al. (2005) showed that this relationship extended to massive, dense clumps in the Milky Way, arguing that the fundamental unit of massive, clustered star formation is such a massive, dense clump. Subsequent studies have defined a "threshold" surface density of ${A}_{{\rm{V}}}\gt 8$ mag (about 120 M_⊙ pc⁻²) in nearby clouds (Heiderman et al. 2010; Lada et al. 2010, 2012), above which the vast majority of dense cores and YSOs are found. Evans et al. (2014) compared various models of star formation to observations of nearby clouds and found that the mass of dense gas was the best predictor of the star formation rate. Most recently, Vutisalchavakul et al. (2016) showed that a similar result applied to more distant and massive clouds in the Galactic plane, using millimeter continuum emission from the BGPS survey (Ginsburg et al. 2013) to measure the mass of dense gas. Vutisalchavakul et al. (2016) found a substantial dispersion in the star formation rate per mass of dense gas (0.50 dex), but the logarithmic averages of the star formation rate per mass of dense gas were in general agreement for nearby clouds, inner Galaxy clouds, and extragalactic clouds. The dispersion among the averages for these three entities was only 0.19 dex (Figure 11 of Vutisalchavakul et al. 2016), considerably lower than that for total molecular gas probed by CO or ¹³CO, 0.42 dex (Figure 12 of Vutisalchavakul et al. 2016). The star formation rate per unit mass of dense gas is thus remarkably constant over a huge range of scales and conditions. A recent detailed study of HCN emission toward other galaxies (Jiménez-Donaire et al. 2019) also found a small dispersion (0.22 dex) in the star formation rate per mass of dense gas. After accounting for galaxy-to-galaxy variations, the intra-galaxy dispersion was only 0.12 dex. A smaller dispersion for measurements over an entire galaxy is expected if the dispersion among the clouds within a galaxy is due to variations in the evolutionary state of the molecular cloud (e.g., Kruijssen et al. 2018).

While the tight connection between dense parts of molecular clouds and star formation is clear, we must better define what we mean by "dense". The most direct measure exists for the nearby clouds, where surface density can be determined by extinction maps of background stars. For the Galactic plane clouds, continuum emission by dust or line emission by HCN was used. For other galaxies, HCN emission has been the only tracer of dense gas in general use, although ${\mathrm{HCO}}^{+}$ emission has also been explored (Barnes et al. 2011; Privon et al. 2015; Jiménez-Donaire et al. 2019). The extinction maps are strictly sensitive to surface density, while the dust continuum emission is also sensitive to temperature, and the molecular line emission, in addition to surface density and temperature, is sensitive to volume density and abundances. Temperature and molecular abundances can depend on the radiation environment (Pety et al. 2017; Shimajiri et al. 2017). A detailed comparison of these various tracers of "dense" gas can clarify the situation. In this paper, we will compare maps of HCN, ${\mathrm{HCO}}^{+}$, and ¹³CO $J=1\to 0$ emission to the regions selected as dense by extinction or millimeter-wave continuum emission in a sample of Galactic plane clouds. We refer to all of the molecular lines as "line tracers" and the HCN and ${\mathrm{HCO}}^{+}$ lines as "dense line tracers" for convenience, while noting that our purpose is to test the proposition that they trace gas of the density relevant to star formation.

Specifically, we can learn what fraction of the luminosity from the line tracers arises from low-level, extended emission from less dense gas. Stephens et al. (2016) have argued that most of the Galaxy's luminosity of HCN arises from distributed, very sub-thermal, emission rather than from dense gas. Since observations of other galaxies would integrate large areas of low-level emission, their HCN luminosity could be dominated by the same gas that is probed by CO, complicating the connection found by Wu et al. (2005) between dense clumps in the Milky Way and other galaxies. Pioneering work by Helfer & Blitz (1997) showed that HCN emission is very weak compared to CO (${I}_{\mathrm{HCN}}/{I}_{\mathrm{CO}}=0.014\pm 0.020$) averaged over random observations of the Galactic plane. They further showed that maps of HCN $J=1\to 0$ in nearby large molecular clouds were tiny in comparison to the CO maps. This work argues against the idea of Stephens et al. (2016), but improved instrumentation now allows for a much stronger test. Recent work has shown that HCN emission can indeed arise from more extended regions (Pety et al. 2017; Kauffmann et al. 2017; Shimajiri et al. 2017), but the focuses of those studies were all on clouds in the solar neighborhood (out to the distance of the Orion clouds). A recent study extends these results to the M17 cloud Nguyen-Luong et al. (2020).

Maps of the full extent of ¹³CO emission in inner Galaxy clouds allow us to directly test the contribution of more diffuse molecular gas to the HCN luminosity in a very different environment. The simultaneous observations of ${\mathrm{HCO}}^{+}$ provide a direct comparison of these two tracers. One might predict that ${\mathrm{HCO}}^{+}$ $J=1\to 0$ traces more widespread gas of lower mean density than does HCN $J=1\to 0$ because the critical density for ${\mathrm{HCO}}^{+}$ $J=1\to 0$ at ${T}_{{\rm{K}}}=20$ K (${n}_{\mathrm{cr}}\approx 4.5\times {10}^{4}$ cm⁻³) is nearly an order of magnitude less than that for HCN $J=1\to 0$ (${n}_{\mathrm{cr}}\approx 3.0\times {10}^{5}$ cm⁻³; Evans 1999; Shirley 2015). A comparison of these two tracers in a well-defined sample with star formation rates will be useful for evaluating relations seen in extragalactic studies of dense gas. Studies of HCN and ${\mathrm{HCO}}^{+}$ have found some evidence of environmental effects in other galaxies, such as the presence of an active galactic nucleus (see, e.g., Privon et al. 2015 for a discussion of this issue). It is important first to understand the relation between these two tracers in more controlled environments. We also evaluate whether ¹³CO $J=1\to 0$ can trace the relevant gas for star formation; while it has a much lower critical density (${n}_{\mathrm{cr}}\approx 4.8\times {10}^{2}$ cm⁻³), it is easier to observe.

With spectrally resolved maps, we can also assess the balance between gravity and turbulence, most simplistically captured in the virial parameter. One attractive explanation for the low star formation efficiency in molecular clouds is that most clouds are not gravitationally bound; only relatively dense regions within them are bound (Dobbs et al. 2011; Barnes et al. 2016). While studies differ, even the most massive clouds defined by CO emission may be unbound (Nguyen-Luong et al. 2016). By calculating the virial ratio for the structures traced by the different species, we may be able to shed light on this issue.

The target clouds (listed in Table 1) comprise a subset of the Vutisalchavakul et al. (2016) sample, chosen to sample a range of conditions and environments, as well as for suitable size (8–35 pc) and relative lack of confusion. The sample has maps of CO and ¹³CO from the Five College Radio Astronomy Observatory (FCRAO) and millimeter-wave continuum emission from the Bolocam Galactic Plane Survey (BGPS), obtained at the Caltech Submillimeter Observatory (CSO; Aguirre et al. 2011; Ginsburg et al. 2013), and two measures of the star formation rate, radio continuum, and mid-infrared emission (Vutisalchavakul et al. 2016). The CO and ¹³CO data were, respectively, taken from the UMass-Stony Brook Survey (Sanders et al. 1986) and the Boston University-FCRAO Galactic Ring Survey (GRS; Jackson et al. 2006). The FWHM beam sizes were 45'' for CO and 46'' for ¹³CO. The velocity resolutions were 0.65 km s⁻¹, smoothed to 1.0 km s⁻¹, for CO, and 0.21 km s⁻¹ for ¹³CO. The rms noise (in ${T}_{{\rm{A}}}^{* }$) was 0.4 K for CO and 0.13 K for ¹³CO. The BGPS was obtained with a filter centered at 271.4 GHz and a bandwidth of 46 GHz (avoiding the CO $J=2\to 1$ line) at 33'' effective resolution and rms noise ranging from 11 to 53 mJy beam⁻¹ (Aguirre et al. 2011). We use version 2.0 of the BGPS catalog, which has an improved positional accuracy and response to extended structure (Ginsburg et al. 2013). In particular, 95% of the flux was recovered for scales between 33'' and 80'', and emission out to 300'' was partially recovered.

Table 1.  Sample of Clouds

Source	d	KDAR	Size	Log ${M}_{\mathrm{cloud}}$	Log ${M}_{\mathrm{dense}}$	Log SFR	Map Size	13CO Lim	Note
	(kpc)		(pc)	(M⊙)	(M⊙)	(M⊙ Myr−1)	(' × ')	(K)
G034.158+00.147	${3.48}_{-0.32}^{+0.43}$	N	${22.68}_{-2.09}^{+2.80}$	${4.75}_{-0.25}^{+0.17}$	${4.13}_{-0.09}^{+0.10}$	${2.44}_{-0.10}^{+0.10}$	30 × 30	2.2
G034.997+00.330	${10.43}_{-0.41}^{+0.38}$	F	${35.41}_{-1.39}^{+1.29}$	${5.52}_{-0.23}^{+0.15}$	${4.08}_{-0.23}^{+0.15}$	${1.85}_{-0.06}^{+0.05}$	10 × 20	2.6	1
G036.459−00.183	${8.68}_{-0.60}^{+0.56}$	F	${17.21}_{-1.19}^{+1.11}$	${4.96}_{-0.24}^{+0.15}$	${3.49}_{-0.23}^{+0.15}$	${1.36}_{-0.08}^{+0.07}$	14 × 22	2.2	1
G037.677+00.155	${6.60}_{-0.14}^{+0.13}$	T	${27.15}_{-0.58}^{+0.53}$	${5.45}_{-0.23}^{+0.15}$	${3.02}_{-0.25}^{+0.16}$	${2.09}_{-0.05}^{+0.04}$	10 × 26	1.9
G045.825−00.291	${8.31}_{-0.62}^{+0.46}$	F	${25.69}_{-1.92}^{+1.42}$	${5.41}_{-0.24}^{+0.15}$	${3.91}_{-0.24}^{+0.15}$	${1.87}_{-0.09}^{+0.06}$	30 × 30	1.6
G046.495−00.241	${3.71}_{-0.60}^{+0.68}$	N	${8.27}_{-1.34}^{+1.52}$	${4.38}_{-0.32}^{+0.19}$	${2.85}_{-0.32}^{+0.19}$	${1.81}_{-0.18}^{+0.14}$	20 × 10	1.6

Note. (1) Mapped in equatorial coordinates.

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The typical angular extent of BGPS sources is a few arcmin, so we selected clouds with ¹³CO extents of 10'–20' to fully sample the "diffuse" gas. The boundaries of the ¹³CO emission (column 9 of Table 1) were set in order to separate the cloud from the background; they were substantial and varied from cloud to cloud. The procedure was described in Vutisalchavakul et al. (2016), but generally, the threshold was the minimum needed to distinguish the cloud from background/foreground ¹³CO emission. These thresholds are 5–12 times the rms noise, so the clouds undoubtedly are larger and more massive, but uncertainty in the inner Galaxy limits the region that can be isolated. We favored clouds with at least one BGPS source with a size of at least 3' so that we can clearly compare the morphology of the HCN/${\mathrm{HCO}}^{+}$ emission and the dust emission. We also favored clouds with larger and stronger BGPS sources. We identified six clouds in the sample of Vutisalchavakul et al. (2016) that meet these criteria. The sample spans a good range of cloud mass (2.4 × 10⁴ M_⊙ to 3.3 × 10⁵ M_⊙), dense gas mass (700–1.3 × 10⁴ M_⊙), and star formation rate (23–275 M_⊙ Myr⁻¹). While the clouds range in distance from us (3.5–10.4 kpc), all lie between 5.1 and 6.4 kpc from the Galactic center, in the molecular ring.

The distances used by Vutisalchavakul et al. (2016) were mostly taken from Anderson et al. (2014), who used a variety of sources of information on the velocity and methods for kinematic distance ambiguity resolution (KDAR). We now have velocities of the dense gas traced best by H¹³CO⁺ or ${\mathrm{HCO}}^{+}$ (see later section), and there are newer distance estimators. We recalculated the distances using the tool described in Wenger et al. (2018), which provides a distance probability density function (pdf) and two-sided uncertainties. While we use the two-sided uncertainties, they are in general fairly symmetric. We use the same choice of near, far, or tangent point distances as Anderson et al. (2014; noted by N, F, or T in the KDAR column of Table 1). In particular, G037.677+00.155 was placed at the tangent point because its velocity was within 10 km s⁻¹ of the velocity at the tangent point. We have scaled the sizes, cloud masses, dense gas masses, and star formation rates to the new distances and propagated the uncertainties in the distance to uncertainties in other quantities. The distance uncertainties typically dominate if they enter the calculation of a quantity. We use the star formation rates from the mid-infrared emission. The results are in Table 1. As discussed in detail in the Appendix, we use the kinematic distance to G034.158+00.147 rather than the closer distance from maser parallax. The latter is quite uncertain and would require an unreasonably large (40 km s⁻¹) peculiar motion.

We mapped the six clouds in the HCN (1−0) (88.631847 GHz; DeLucia & Gordy 1969) and HCO⁺ (1−0) (89.188526 GHz; Sastry et al. 1981) lines simultaneously using the SEQUOIA array with 16 pixels in a 4 × 4 array at the 14 m telescope of the Taeduk Radio Astronomy Observatory (TRAO; Roh & Jung 1999; Jeong et al. 2019). The observations were conducted by the on-the-fly (OTF) technique in the absolute position switching mode with OFF positions checked to be free from appreciable emission. The mapped areas were different for the individual clouds (Table 1). G034.997+00.330 and G036.459−00.183 were mapped in equatorial coordinates occasionally between 2017 February and May, while the others were mapped in Galactic coordinates between 2018 January and April. The backend was a 2G FFT spectrometer that can accept 32 signal streams at the same time. Thus, it is possible to observe two transitions simultaneously between 85 and 100 GHz or 100 and 115 GHz. The backend bandwidth is 62.5 MHz with 4096 channels, yielding a velocity resolution of 0.05 km s⁻¹. The observed line temperature was calibrated on the ${T}_{{\rm{A}}}^{* }$ scale by the standard chopper wheel method. We checked the telescope pointing and focus every 3 hr by observing strong SiO maser sources at 86 GHz. The pointing accuracy was better than 10''. The system temperatures depended on the weather conditions and source elevation. They were usually around 200 K during the observing runs. The rms noise levels of the observed spectra were typically about 0.1 K after smoothing to 0.2 km s⁻¹ resolution. We smoothed the data with the boxcar function to a velocity resolution of about 0.2 km s⁻¹ when fitting line profiles, but some analysis of the data cubes was done with the original resolution. To assess optical depth effects, we observed the innermost footprint in the lines of H¹³CN and H¹³CO⁺ with the same method and rms noise.

The TRAO telescope is the same model as the FCRAO telescope. The panels were readjusted and the radome was replaced before our observations were taken. The resulting instrumental properties are described in Jeong et al. (2019). The TRAO has a main-beam size of 58'' and a main-beam efficiency of 46% at 89 GHz. The individual SEQUOIA beams vary in beam size (efficiency) by only a few arcseconds (few percent). The beams are very circular; variation of efficiency with elevation is less than 3%.

Many of our sources are quite extended, so the efficiency on the Moon is more relevant for some. This has not yet been measured for the TRAO, but it should be at least as high as that of the FCRAO, where η_Moon = 0.7. The error beam has not been measured for the TRAO, but it should be no worse than that of the FCRAO. There are two error beams for the FCRAO: one is caused by the radome and the other by small-scale irregularities on the panels. The radome error beam is spread over 4° at −30 dB (0.1%), too weak and diffuse to be an issue here. The panel error beam has a size of 30' at −18 dB (1.6%). We will assess whether this error beam could affect our results in Section 5.2.1.

The most compact presentation of the results is in Figures 1–6. In each figure, all panels present the outermost contour of the velocity integrated intensity of ¹³CO emission with four tracers of dense gas (in color) superimposed: (1) the ${A}_{{\rm{V}}}\gt 8$ mag mask based on the ¹³CO column density, (2) the millimeter-wave continuum emission from BGPS, (3) the HCN integrated intensity, and (4) the ${\mathrm{HCO}}^{+}$ integrated intensity. (The method used to determine the ${A}_{{\rm{V}}}\gt 8$ mask is described in Section 5.1.) Two panels show in color the column density determined from Herschel data and the radio continuum emission.

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Inspection reveals a considerable range of behavior. G034.158+00.147 and G034.997+00.330 show strong emission from both line tracers and BGPS, and all tracers agree qualitatively on the location of the dense gas. At the other extreme, G037.677+00.155 has only very weak emission from either dense line tracer, both concentrated in the upper right corner of the ¹³CO map; G037.677+00.155 has weak ¹³CO emission and no regions with ${A}_{{\rm{V}}}\gt 8$ mag. G036.459−00.183 and G045.825−00.291 have weak emission from the dense line tracers that is more extended and poorly correlated with the BGPS and ${A}_{{\rm{V}}}\gt 8$ mag indicators. G046.495−00.241 has three, often overlapping, velocity components that complicate analysis, but in general, it shows moderate agreement among the various tracers of dense gas. More complete results for each cloud (spectra at peaks, contour diagrams of integrated intensity, etc.) are provided in the Appendix.

5.1. Tracer Comparison

We quantify the results from the visual comparison in Section 4, focusing on the fraction of the luminosity of the line tracers that comes from regions indicated to be dense, based on extinction or millimeter-wave continuum emission. This section is geared toward understanding how the line tracers will behave when clouds are observed with spatial resolutions much greater than 1 pc, as in observations of other galaxies. Thus, we do not separate emission from clumps that differ in spatial or velocity location. That analysis is presented in Section 5.2.

First, we describe the method used to make the maps of column density and the ${A}_{{\rm{V}}}\gt 8$ mag mask. CO (our convention is that the most common isotope is indicated unless otherwise specified) and ¹³CO data were used to define column density maps for each target. The procedure to convert these data into ¹³CO column densities for this paper was largely described by Ripple et al. (2013). In short, we used Equations (1)–(5) of Ripple et al. (2013) to calculate the column density of ¹³CO, assuming low to moderate optical depths of ¹³CO emission and optically thick CO emission. Monte Carlo simulations are computed to derive the excitation temperature and ¹³CO column density and corresponding uncertainties generated by the thermal noise of the data. We then converted to molecular hydrogen by assuming an isotopic ¹²C/¹³C ratio of 45 (Milam et al. 2005) and fractional abundance of 6000 for CO, based on a recent determination of the CO abundance by Lacy et al. (2017). Our conversion to extinction of ${A}_{{\rm{V}}}=1.0\times {10}^{-21}N({{\rm{H}}}_{2})$ is also provided by Lacy et al. (2017). Fractionation could decrease the ratio of CO to ¹³CO, which would cause us to overestimate the column density with our assumed isotopic ratio, but we have no good method to correct for this, and the effect is not large in these relatively warm clouds.

The much higher threshold of 1 g cm⁻² proposed by McKee & Tan (2003) to avoid fragmentation and favor the formation of massive stars is not probed by ¹³CO emission, so we used the column density determined from Herschel data (Marsh et al. 2017) and available at the HIGAL site: http://www.astro.cardiff.ac.uk/research/ViaLactea/. We do not use the Herschel data for regions of more modest column density because foreground/background emission strongly contaminates it. It traces only the highest column density regions accurately in the inner Galaxy.

These maps of column density were used to define the regions of column density corresponding to various thresholds used to define "dense" gas. The velocity interval of ¹³CO emission was used to limit the range of velocities of plausible emission from ${\mathrm{HCO}}^{+}$; the range was extended for HCN to account for hyperfine splitting. Then, the data cubes for ¹³CO, BGPS, and Herschel column densities were convolved, resampled, and aligned with the TRAO maps. These were used to measure the luminosity inside and outside the region defined by the criterion described above. This allowed us to determine what fraction of the HCN and ${\mathrm{HCO}}^{+}$ emission arose from "dense" gas, as defined by that criterion. In this section, the main-beam efficiency has been used to correct to the scale of ${T}_{\mathrm{mb}}$. This procedure may overestimate the brightness for very extended emission.

The luminosity inside and outside regions defining various indicators of dense gas are discussed in the following sections. The integration of luminosity is limited to the intersection of the ¹³CO-defined cloud and the mapping box from the TRAO observations for both HCN and ${\mathrm{HCO}}^{+}$ but not for ¹³CO. The equations used to compute the luminosity follow:

$$\begin{eqnarray}&&{L}_{X,\mathrm{in}}={D}^{2}\int {dv}\int d{{\rm{\Omega }}}_{\mathrm{in}}{T}_{X}(l,b,v),\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}&&{L}_{X,\mathrm{out}}={D}^{2}\int {dv}\int d{{\rm{\Omega }}}_{\mathrm{out}}{T}_{X}(l,b,v),\end{eqnarray} \tag{ 2 }$$

where X refers to the tracer, D is the distance in parsecs, and ${{\rm{\Omega }}}_{\mathrm{in}}$ and ${{\rm{\Omega }}}_{\mathrm{out}}$ are the solid angle of pixels that satisfy (in) or do not satisfy (out), respectively, one of the given conditions (column density from ¹³CO, column density from Herschel emission, or overlap with the mask of emission from the BGPS). In practice, a summed spectrum was constructed from all pixels that satisfied the conditions. The uncertainties were calculated as follows: the rms noise of the summed spectrum (${\mathrm{rms}}_{\mathrm{sum}}$) is the quadrature sum of rms noise values of each pixel that satisfies the threshold condition. The uncertainty in the luminosity is then

$$\begin{eqnarray}&&{\mathrm{rms}}_{{\rm{L}}}=\delta v\sqrt{{N}_{\mathrm{ch}}}{\mathrm{rms}}_{\mathrm{sum}}d{\rm{\Omega }}{D}^{2}\end{eqnarray} \tag{ 3 }$$

where $\delta v$ is the channel width, and ${N}_{\mathrm{ch}}$ is the number of channels in the integration range. In most sources, these uncertainties were quite small compared to the luminosity, reflecting the fact that they included only the uncertainties in the summed spectrum. The distance uncertainties (Table 1) were added in quadrature for all luminosities but not for ratios of luminosities, for which the distance cancels out.

5.1.1. Line Tracers versus the Extinction Criterion

How much of the luminosity of the line tracers arises within regions satisfying the extinction criterion (${A}_{{\rm{V}}}\gt 8$ mag)? While our main focus is on HCN and ${\mathrm{HCO}}^{+}$, we consider ¹³CO as well because its emission is stronger and easier to obtain for other galaxies.

We used the procedure described above to determine the luminosity of each tracer for regions with column density above and below that threshold, as measured by the ¹³CO column density. The values for the fraction of pixels inside the ${A}_{{\rm{V}}}$ criterion, the log of the total line luminosity, and the fraction of the total arising inside the ${A}_{{\rm{V}}}\gt 8$ mag region (${f}_{{\rm{L}}}={L}_{\mathrm{in}}/{L}_{\mathrm{tot}}$) are listed for HCN, ${\mathrm{HCO}}^{+}$, and ¹³CO in Table 2. Two-sided errors are given for the logarithmic luminosities, which are dominated by the distance uncertainties. The distance does not enter in the ${f}_{{\rm{L}}}$ values, so the uncertainties are symmetric, much smaller, and given in parentheses. The mean, standard deviation, and median are given for the relevant columns. Since G037.677+00.155 had no pixels above the criterion, its value for ${f}_{{\rm{L}}}$ is zero for all three tracers. The values of ${f}_{{\rm{L}}}$ vary widely, with ${f}_{{\rm{L}}}$ between 0 and 0.54 for HCN, between 0 and 0.56 for ${\mathrm{HCO}}^{+}$, and between 0 and 0.33 for ¹³CO. HCN and ${\mathrm{HCO}}^{+}$ give very similar results for ${f}_{{\rm{L}}}$. In terms of the total line luminosities, the mean of the logarithms of ${L}_{\mathrm{tot}}$ is higher by 0.25 for HCN compared to ${\mathrm{HCO}}^{+}$. So, the two dense line tracers provide similar measures in this sample, but the luminosity of HCN is somewhat higher than that of ${\mathrm{HCO}}^{+}$. ¹³CO provides still higher luminosity but a worse correlation with the extinction criterion for the first two clouds and a similar correlation for the others.

Table 2.  Line Luminosities vs. ${A}_{{\rm{V}}}\gt 8$

Source	$N/{N}_{\mathrm{tot}}$	Log ${L}_{\mathrm{tot}}$	${f}_{{\rm{L}}}$	Log ${L}_{\mathrm{tot}}$	${f}_{{\rm{L}}}$	Log Ltot	${f}_{{\rm{L}}}$
		HCN	HCN	${\mathrm{HCO}}^{+}$	${\mathrm{HCO}}^{+}$	13CO	13CO
G034.158+00.147	0.163	${2.68}_{-0.09}^{+0.10}$	0.539 (0.003)	${2.54}_{-0.09}^{+0.10}$	0.559 (0.003)	${3.75}_{-0.09}^{+0.10}$	0.332 (0.000)
G034.997+00.330	0.079	${3.37}_{-0.04}^{+0.03}$	0.251 (0.002)	${3.09}_{-0.04}^{+0.03}$	0.243 (0.002)	${4.34}_{-0.04}^{+0.03}$	0.125 (0.000)
G036.459−00.183	0.004	${2.95}_{-0.07}^{+0.06}$	0.007 (0.000)	${2.81}_{-0.07}^{+0.05}$	0.010 (0.000)	${3.86}_{-0.06}^{+0.05}$	0.019 (0.000)
G037.677+00.155	0.000	${2.22}_{-0.02}^{+0.02}$	0.000 (0.000)	${2.02}_{-0.02}^{+0.02}$	0.000 (0.000)	${3.59}_{-0.02}^{+0.02}$	0.000 (0.000)
G045.825−00.291	0.054	${3.13}_{-0.07}^{+0.05}$	0.148 (0.003)	${2.56}_{-0.07}^{+0.05}$	0.169 (0.008)	${3.88}_{-0.07}^{+0.05}$	0.113 (0.001)
G046.495−00.241	0.035	${2.36}_{-0.17}^{+0.14}$	0.088 (0.002)	${2.20}_{-0.17}^{+0.14}$	0.098 (0.002)	${3.34}_{-0.17}^{+0.14}$	0.075 (0.000)
Mean	0.056	2.79	0.172	2.54	0.180	3.79	0.111
Std Dev.	0.055	0.41	0.185	0.36	0.190	0.31	0.109
Median	0.045	2.82	0.118	2.55	0.133	3.80	0.094

Note. (1) Units of luminosities are K km s⁻¹ pc².

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Does the fraction of line luminosity arising at high column density correlate with any observable properties? Figure 7 plots ${f}_{{\rm{L}}}$ versus total line luminosity; no trend is apparent. Neither is there a strong trend of increasing ${f}_{{\rm{L}}}$ with SFR (Figure 8). While the highest values of ${f}_{{\rm{L}}}$ correspond to the cloud with the highest star formation rate, the other clouds do not support an overall trend.

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Finally, we comment on the absence of any gas with ${A}_{{\rm{V}}}\gt 8$ mag in G037.677+00.155. It is a large (27 pc) and massive (log ${M}_{\mathrm{cloud}}=5.45$) cloud, with the second highest SFR in the sample. The absence of regions with ${A}_{{\rm{V}}}\gt 8$ mag and only weak, fragmented emission in dense gas tracers is thus surprising, except that the fraction of dense gas indicated by the BGPS emission (3.7 $\times \,{10}^{-3}$) was the lowest in the sample. This cloud is probably more evolved, with the current episode of star formation coming to an end.

5.1.2. Line Tracers versus the 1 g cm⁻² Criterion

McKee & Tan (2003) proposed the criterion of a surface density of 1 g cm⁻² for efficient formation of massive stars. ¹³CO does not trace such high column densities, so we used the column density from the Herschel data, as described in Section 5.1. There is a small region (10 pixels) in G034.158+00.147 and a single pixel in G034.997+00.330 that meet the criterion. The fractions of the HCN luminosity in those regions are 0.034 for G034.158+00.147 and 0.006 in G034.997+00.330. For ${\mathrm{HCO}}^{+}$, the equivalent fractions are 0.046 and 0.006; for ¹³CO, they are 0.019 and 0.0. (For ¹³CO, no pixels were inside that region in G034.997+00.330, presumably because of slightly different sampling.) These numbers are interestingly small; the criterion was originally formulated in the context of the early work on dense clumps identified with strong localized emission by CS $J=7\to 6$, (Plume et al. 1997) similar in spirit to the HCN and ${\mathrm{HCO}}^{+}$ line tracers but much more biased toward very high densities. The sample of Plume et al. (1997) was originally based on sources with water masers. As far as we know, G034.158+00.147 is the only source in our sample with a water maser. A recent study of the M17 cloud found that neither HCN nor ${\mathrm{HCO}}^{+}$ $J=1\to 0$ was particularly good at distinguishing regions above this criterion (Nguyen-Luong et al. 2020). Higher J transitions are needed to probe such regions (Wu et al. 2010).

5.1.3. Line Tracers versus BGPS Emission

Vutisalchavakul et al. (2016) used the millimeter-wave continuum emission from the BGPS survey to estimate the mass of dense gas. To test how well the line tracers correlate with that criterion for "dense", we also computed the line luminosities inside and outside the mask supplied by the BGPS catalog. In the mask file, each pixel is set to zero if no emission was detected or to the catalog number of the source if emission was significant (Ginsburg et al. 2013).

As can be seen from Figure 5, G045.825−00.291 has very little overlap between the BGPS emission and the HCN/${\mathrm{HCO}}^{+}$ emission. Consequently, the values in the table for that source are effectively upper limits. For the remaining four sources, the correlation with BGPS is reasonably strong, as captured in ${f}_{{\rm{L}}}$ for BGPS in Table 3. As for the extinction criterion, the two dense line tracers agree well, but the luminosity of HCN is greater by 0.25 in the log on average. The ${f}_{{\rm{L}}}$ values for ¹³CO are more comparable to the dense line tracers than was the case for the extinction criterion. This reflects the fact that the BGPS emission from distant clouds traces a lower average density than it traces in nearby clouds, so that it begins to trace structures between the scale of clouds and dense clumps beyond distances of 5–10 kpc (Dunham et al. 2011). Thus, ¹³CO, HCN, and ${\mathrm{HCO}}^{+}$ all seem to trace the gas inferred from the BGPS data similarly. Since Vutisalchavakul et al. (2016) used BGPS to measure dense gas mass, we would expect their values of SFR per unit mass to be lower than those for the nearby clouds, where the ${A}_{{\rm{V}}}\gt 8$ mag criterion can be used. Indeed, this was the case.

Table 3.  Line Luminosities vs. BGPS Emission

Source	$N/{N}_{\mathrm{tot}}$	Log ${L}_{\mathrm{tot}}$	${f}_{{\rm{L}}}$	Log ${L}_{\mathrm{tot}}$	${f}_{{\rm{L}}}$	$N/{N}_{\mathrm{tot}}$	Log ${L}_{\mathrm{tot}}$	${f}_{{\rm{L}}}$
	HCN	HCN	HCN	${\mathrm{HCO}}^{+}$	${\mathrm{HCO}}^{+}$	13CO	13CO	13CO
G034.158+00.147	0.625	${2.68}_{-0.09}^{+0.10}$	0.843 (0.006)	${2.54}_{-0.09}^{+0.10}$	0.865 (0.006)	0.612	${3.75}_{-0.09}^{+0.10}$	0.741 (0.001)
G034.997+00.330	0.267	${3.37}_{-0.04}^{+0.03}$	0.454 (0.004)	${3.09}_{-0.04}^{+0.03}$	0.452 (0.004)	0.261	${4.34}_{-0.04}^{+0.03}$	0.344 (0.000)
G036.459−00.183	0.205	${2.95}_{-0.06}^{+0.05}$	0.336 (0.004)	${2.81}_{-0.06}^{+0.05}$	0.401 (0.003)	0.235	${3.86}_{-0.06}^{+0.05}$	0.347 (0.000)
G037.677+00.155	0.189	${2.22}_{-0.03}^{+0.03}$	0.186 (0.008)	${2.02}_{-0.02}^{+0.02}$	0.304 (0.010)	0.230	${3.59}_{-0.02}^{+0.02}$	0.275 (0.001)
G045.825−00.291	0.110	${3.13}_{-0.07}^{+0.05}$	0.142 (0.004)	${2.56}_{-0.08}^{+0.05}$	0.184 (0.010)	0.116	${3.88}_{-0.07}^{+0.05}$	0.141 (0.001)
G046.495−00.241	0.145	${2.36}_{-0.17}^{+0.14}$	0.240 (0.004)	${2.20}_{-0.17}^{+0.14}$	0.252 (0.004)	0.128	${3.34}_{-0.17}^{+0.14}$	0.197 (0.000)
Mean	0.257	2.79	0.367	2.54	0.410	0.264	3.79	0.341
Std Dev.	0.172	0.41	0.236	0.36	0.222	0.165	0.31	0.194
Median	0.197	2.82	0.288	2.55	0.353	0.233	3.80	0.309

Note. (1) Units of luminosities are K km s⁻¹ pc².

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5.1.4. Conversion of Line Tracer Luminosity to Mass of Dense Gas

We also plot the mass of dense gas determined from BGPS versus the luminosity of HCN in Figure 9 for both the total line tracer luminosity and the line tracer luminosity inside the BGPS mask. Both correlate, but the correlation is better for L_in. The values of ${M}_{\mathrm{dense}}$, ${L}_{\mathrm{in}}$, and both conversion factors ${\alpha }_{\mathrm{in}}={M}_{\mathrm{dense}}/{L}_{\mathrm{in}}$ and ${\alpha }_{\mathrm{tot}}={M}_{\mathrm{dense}}/{L}_{\mathrm{tot}}$ are tabulated in Table 4 for both HCN and ${\mathrm{HCO}}^{+}$, along with the means, standard deviations, and medians.

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Table 4.  ${M}_{\mathrm{dense}}$ vs. Luminosities

Source	Log ${M}_{\mathrm{dense}}$	Log ${L}_{\mathrm{in}}$	Log ${\alpha }_{\mathrm{in}}$	Log ${\alpha }_{\mathrm{tot}}$	Log ${L}_{\mathrm{in}}$	Log ${\alpha }_{\mathrm{in}}$	Log ${\alpha }_{\mathrm{tot}}$
	M⊙	HCN	HCN	HCN	${\mathrm{HCO}}^{+}$	${\mathrm{HCO}}^{+}$	${\mathrm{HCO}}^{+}$
G034.158+00.147	${4.13}_{-0.09}^{+0.10}$	${2.61}_{-0.09}^{+0.10}$	${1.52}_{-0.02}^{+0.02}$	${1.45}_{-0.02}^{+0.02}$	${2.48}_{-0.09}^{+0.10}$	${1.66}_{-0.02}^{+0.02}$	${1.59}_{-0.02}^{+0.02}$
G034.997+00.330	${4.08}_{-0.23}^{+0.15}$	${3.03}_{-0.04}^{+0.03}$	${1.05}_{-0.22}^{+0.15}$	${0.71}_{-0.22}^{+0.15}$	${2.74}_{-0.04}^{+0.03}$	${1.34}_{-0.22}^{+0.15}$	${0.99}_{-0.22}^{+0.15}$
G036.459−00.183	${3.49}_{-0.23}^{+0.15}$	${2.48}_{-0.06}^{+0.05}$	${1.01}_{-0.22}^{+0.14}$	${0.54}_{-0.22}^{+0.14}$	${2.41}_{-0.06}^{+0.05}$	${1.08}_{-0.22}^{+0.14}$	${0.68}_{-0.22}^{+0.14}$
G037.677+00.155	${3.02}_{-0.25}^{+0.16}$	${1.49}_{-0.03}^{+0.02}$	${1.53}_{-0.25}^{+0.16}$	${0.80}_{-0.25}^{+0.16}$	${1.51}_{-0.02}^{+0.02}$	${1.51}_{-0.25}^{+0.16}$	${0.99}_{-0.25}^{+0.16}$
G045.825−00.291	${3.91}_{-0.24}^{+0.15}$	${2.28}_{-0.07}^{+0.05}$	${1.63}_{-0.22}^{+0.14}$	${0.78}_{-0.22}^{+0.14}$	${1.82}_{-0.07}^{+0.05}$	${2.09}_{-0.22}^{+0.14}$	${1.35}_{-0.22}^{+0.15}$
G046.495−00.241	${2.85}_{-0.32}^{+0.19}$	${1.74}_{-0.17}^{+0.14}$	${1.11}_{-0.23}^{+0.15}$	${0.49}_{-0.23}^{+0.15}$	${1.60}_{-0.17}^{+0.14}$	${1.25}_{-0.23}^{+0.15}$	${0.65}_{-0.23}^{+0.15}$
Mean	3.579	2.271	1.308	0.793	2.094	1.486	1.044
Std. Dev.	0.505	0.521	0.254	0.315	0.471	0.326	0.339
Median	3.701	2.381	1.314	0.743	2.118	1.423	0.992

Note. (1) Units of luminosities are K km s⁻¹ pc².

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While the BGPS emission is tracing a somewhat lower-density material than the ${A}_{{\rm{V}}}\gt 8$ mag criterion for the distant clouds, it has still been used to trace dense gas, so a conversion factor between the dense line tracers and the mass determined from BGPS is of interest. If we restrict the luminosity of HCN to the region inside the BGPS mask, we get $\langle \mathrm{log}({\alpha }_{\mathrm{in}})\rangle \,=1.308\pm .254$, translating to ${M}_{\mathrm{dense}}={20}_{-9}^{+16}$ $L(\mathrm{HCN})$. Including the luminosity of the entire cloud, the result is $\langle \mathrm{log}({\alpha }_{\mathrm{tot}})\rangle =0.793\pm .315$, translating to ${M}_{\mathrm{dense}}\,={6.2}_{-3.0}^{+6.6}$ $L(\mathrm{HCN})$. The latter value is more appropriate for extragalactic observations, where the luminosity will arise from the whole cloud if one wants to estimate the mass of material with densities similar to that of BGPS sources in the Galaxy. A conversion factor of 20 (appropriate for the luminosity inside the BGPS mask) is the same as the average of 20 ± 5 derived by Wu et al. (2010) while the value of 6.2 is more similar to, but a bit lower than, that used in extragalactic studies (Gao & Solomon 2004; Liu et al. 2015; Jiménez-Donaire et al. 2019).

5.1.5. HCO⁺ versus HCN versus ¹³CO

The fraction of luminosity coming from the ${A}_{{\rm{V}}}\gt 8$ mag region is plotted for each of the line tracers in Figure 10. The first two clouds (G034.158+00.147 and G034.997+00.330) clearly differ from the other four. The values for ${f}_{{\rm{L}}}$ are higher, and those for HCN or ${\mathrm{HCO}}^{+}$ clearly exceed those for ¹³CO. For the other four, all of the values are low, and the different tracers are in rough agreement. In those clouds, the ¹³CO would be equally good or bad at tracing the dense gas.

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The ratio of total luminosities, $L({\mathrm{HCO}}^{+})/L(\mathrm{HCN})$, averaged in the logs, is 0.56, similar to the 0.7 ratio found in the EMPIRE study (Jiménez-Donaire et al. 2019) and in the CMZ of the Galaxy (Jones et al. 2012) as well as the ratio of 0.73 found in M17 (Nguyen-Luong et al. 2020). However, there are substantial variations from cloud to cloud. Despite the differences in critical density, and probably in chemistry as well, the two lines seem to be tracing a similar material on average in this sample. Both lines trace about the same concentration of emission, as measured by ${f}_{{\rm{L}}}$. For the ${A}_{{\rm{V}}}\gt 8$ mag criterion, the average ratio $\langle {f}_{{\rm{L}}}({\mathrm{HCO}}^{+})/{f}_{{\rm{L}}}(\mathrm{HCN})\rangle =1.14\pm 0.18$. This is not what simplistic predictions based on critical densities would have predicted (see Sections 5.2.1 and 6).

5.2. Clump Analysis

In this section, we search for peaks, separating different velocity components where necessary. This analysis is used to characterize the spectra and sizes of the peaks identified in the line tracers. This analysis shows what can be done with a spatial resolution much less than 1 pc, as is common within the Galaxy.

Our standard procedure is as follows. We examine the data to find all velocity intervals with significant emission. Then, we exclude those regions while removing a second-order baseline, also using only enough of a velocity range to get a good baseline on each end (v_sp) and excluding velocities with emission (${v}_{\mathrm{win}}$). The values of the total velocity range and excluded windows are shown in Table B1. Then, I, ${T}_{{\rm{A}}}^{* }$ integrated over a velocity range (${v}_{I}$) is computed for every mapped position, regardless of whether a line is detected there. These are plotted in contour diagrams, which are used to define the peak position, find regions that should be eliminated, and find the FWHM size. Contour diagrams are made in steps of 2σ, starting at 2σ, where σ is the rms noise in the integrated intensity (I), as listed in Table B1. Spectra at the peak position, or for cases of weak emission, spectra averaged over nearby positions, were used to determine line properties such as integrated intensity, velocity, and linewidth (Table 5). This was often an iterative process in which the line properties were used to refine the velocity range (${v}_{I}$) for integrated intensities, and the area outside the 2σ emission regions was refined to determine better noise levels. The values in Table B1 represent the final values.

Table 5.  Line Fits

Source	Line	Intensity	v	Δv	Rms (I)	Note
		(K km s−1)	(km s−1)	(km s−1)	(K km s−1)
G034.158+00.147	${\mathrm{HCO}}^{+}$	10.60	56.42 (0.02)	3.06 (0.05)	0.48
G034.158+00.147	HCN	9.84	56.60 (0.03)	2.68 (0.05)	0.48
G034.158+00.147	H13CO+	3.61	57.91 (0.09)	4.72 (0.22)	0.28
G034.158+00.147	H13CN	6.99	57.80 (0.11)	4.74 (0.25)	0.53
G034.997+00.330	${\mathrm{HCO}}^{+}$	3.82	53.90 (0.03)	3.34 (0.09)	0.23
G034.997+00.330	HCN	7.19	54.50 (0.02)	3.02 (0.05)	0.30
G034.997+00.330	H13CO+	0.42	53.27 (0.14)	2.20 (0.35)	0.17
G034.997+00.330	H13CN	1.04	53.60 (0.17)	2.95 (0.39)	0.34
G036.459−00.183	${\mathrm{HCO}}^{+}$	1.67	77.91 (0.06)	2.56 (0.15)	0.33
G036.459−00.183	HCN	1.72	78.50 (0.11)	2.94 (0.25)	0.33
G037.677+00.155	${\mathrm{HCO}}^{+}$	1.15	82.95 (0.07)	2.52 (0.10)	0.15
G037.677+00.155	HCN	0.48	82.90 (0.21)	3.93 (0.56)	0.25
G045.825−00.291	${\mathrm{HCO}}^{+}$	1.31	50.45 (0.22)	5.35 (0.56)	0.35
G045.825−00.291	HCN	2.23	50.18 (0.41)	11.1 (0.98)	0.51
G046.495−00.241	${\mathrm{HCO}}^{+}$	1.20	51.35 (0.08)	2.23 (0.20)	0.11	v1
G046.495−00.241	HCN	0.96	51.20 (0.14)	2.75 (0.40)	0.15	v1
G046.495−00.241	${\mathrm{HCO}}^{+}$	0.55	54.48 (0.14)	1.93 (0.34)	0.10	v2
G046.495−00.241	HCN	0.65	54.40 (0.16)	2.27 (0.57)	0.08	v2
G046.495−00.241	${\mathrm{HCO}}^{+}$	1.05	58.54 (0.15)	3.05 (0.45)	0.16	v3
G046.495−00.241	HCN	1.61	58.80 (0.19)	2.21 (0.36)	0.16	v3, hfs

Note. (1) v1 mean velocity component 1. (2) hfs mean fit to hyperfine components.

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5.2.1. What Is the Origin of the Line Tracer Luminosity from Low-density Regions?

With a clear identification of the center of emission, we can explore the origin of the luminosity of dense line tracers outside the region of the dust tracers. G034.997+00.330 provides a good test case because there is a single, clearly defined peak in the dense line tracers. In Figure 11, the integrated intensities are sorted by distance from the peak position, normalized to the peak, and plotted versus distance; smoothed versions are also plotted. While both positive and negative values are found at large separations from the peak, there is a positive bias. In this source, roughly 72% of the dense line tracers' luminosity arising outside the ${A}_{{\rm{V}}}\gt 8$ mag mask does not arise in secondary weak emission peaks but in very widespread emission that is mostly below the detection threshold in individual spectra (${T}_{{\rm{A}}}^{* }\sim 0.3$ K for a 3σ detection).

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Could the error beam discussed above create the illusion of low-level, widespread emission? Smoothly extended emission at a level of 1.6% or less relative to the peaked emission could be caused by the error beam. We estimate the observed ratio by using the data inside and outside the A_V = 8 mag contours from Table 2 and the following argument. The surface brightness, $B\propto L/(N{\rm{\Omega }})$, where N is the number of pixels, and Ω is the pixel solid angle. Then,

$$\begin{eqnarray}\begin{array}{rcl}R & = & B(\mathrm{out})/B(\mathrm{in})=L(\mathrm{out})/L(\mathrm{in})\times N(\mathrm{in})/N(\mathrm{out})\\ & = & {f}_{{\rm{L}}}^{-1}\times N(\mathrm{in})/N(\mathrm{out}).\end{array}\end{eqnarray} \tag{ 4 }$$

Approximating $N(\mathrm{out})$ by $N(\mathrm{tot})$, since $N(\mathrm{out})\gg N(\mathrm{in})$, we use the values in Table 2 to estimate R. The values range from 0.29 to 0.40, much larger than the 0.016 that the error beam could contribute.

The calculation above assumes that the emission is smoothly distributed inside and outside the A_V = 8 mag contours. For some sources, the emission is more sharply peaked, so we also divide the average emission in the region outside the A_V = 8 mag contours by the peak emission (Table 5). These ratios range from 5% to 14%, with the exception of G034.158+00.147, for which the ratio is 0.7%. For all but G034.158+00.147, the ratio is greater than could be explained by an error beam. The value for G034.158+00.147 is considerably less than the predicted contribution from the error beam, a fact that could be explained if the TRAO error beam is actually lower than that of the FCRAO or by the fact that the ¹³CO map did not cover the full error beam. To summarize, the extended emission from G034.158+00.147 could arise from the error beam, but none of the other clouds' extended emission could so arise. Having ruled out error beam contributions, we can use the spectra averaged over the whole region outside the ${A}_{{\rm{V}}}\gt 8$ mag region to determine the properties of the extended emission. For all but G034.997+00.330, the characteristic ${T}_{\mathrm{mb}}$ ranges from 0.01 to 0.03 K. If this emission is truly distributed, we can use RADEX (van der Tak et al. 2007) to determine the density of colliders needed to produce such weak lines. The results are shown in Figure 12. The dependence of ${T}_{\mathrm{mb}}$ on density is linearly dependent on the density of colliders in this optically thin, low-density regime because every collisional excitation leads to emission of a photon (Liszt & Pety 2016). Because the observations of ${\mathrm{HCO}}^{+}$ and HCN show nearly equal line strengths, we adjusted the abundance to produce that result. For ${\mathrm{HCO}}^{+}$, we used $N({\mathrm{HCO}}^{+})=1\times \,{10}^{13}$ cm⁻², and for HCN, $N(\mathrm{HCN})=2\times {10}^{14}$ cm⁻². For total column density, $N=1\times {10}^{21}$ cm⁻², corresponding to about ${A}_{{\rm{V}}}=1$ mag, these would correspond to $X=1\times {10}^{-8}$ for ${\mathrm{HCO}}^{+}$, and $X=2\times {10}^{-7}$ for HCN. Higher total column densities would allow for lower abundances. Such abundances are not unreasonable for extinctions of a few, as indicated by Figure 8 of Goldsmith & Kauffmann (2017). For both, we assumed ${T}_{{\rm{K}}}=20$ K and a linewidth of 1 km s⁻¹. These results illustrate that the low-level, very extended emission that can dominate the total luminosity can arise in gas with densities as low as 50–100 cm⁻³. These densities are similar to the average densities ($\bar{n}$) of the entire cloud, determined by dividing the mass by the volume; for the clouds in this sample, the mean value of this average density, $\langle \bar{n}\rangle =96\pm 64$ cm⁻³ (Vutisalchavakul et al. 2016). This is a clear demonstration that common statements about dense gas tracers are too naive, as discussed further in Section 6.

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5.2.2. Properties of the Dense Clumps Identified by the Line Tracers

In this section, we derive the properties of the dense clumps, as separated in position and velocity in order to compare them to the properties of the dense clumps studied by Wu et al. (2010). This analysis addresses the question of whether we are comparing apples to oranges. For this purpose, we follow the procedure in Wu et al. (2010). This procedure requires the integrated line intensity (I) and linewidth (Δv) at the peak position, the FWHM angular size of the emission, and the distance.

We use the spectrum of the line at the peak, or in the case of weak lines that are fairly uniformly distributed, an average over spectra surrounding the peak, to obtain the intensity and linewidth. We use the linewidth from ${\mathrm{HCO}}^{+}$ or H¹³CO⁺ also for the HCN analysis to avoid issues caused by hyperfine structure of HCN. When the lines of the H¹³CO⁺ were strong enough, we used them to get the linewidth, following Wu et al. (2010). This was possible only for G034.158+00.147 and G034.997+00.330.

We find the angular size of the source at the FWHM of the integrated intensity map. The FWHM source size is determined by plotting the 50% contour and determining the geometric mean of the two dimensions. This involves some judgment for weakly peaked regions, and some "peninsulas" were ignored. The uncertainties in the angular size were assessed by repeating the procedure for the 40% and 60% contours; as a result, we included an uncertainty of 20'' for the angular size in the next steps. The source sizes were generally much larger than the beam size, indicating that beam dilution was a minor effect. The most compact source, G034.158+00.147, had a source size of 87'', much larger than the beam size. While there is undoubtedly structure smaller than our beam, it would have a small effect on the following analysis, which is intended to compare our results to those of Wu et al. (2010), who used the FCRAO for $J=1\to 0$ observations, hence, the same beam size on comparably distant sources. The method for deriving source sizes was chosen to be similar to that used by Wu et al. (2010). While crude and somewhat subjective, it provided the most sensible results.

This angular size of the source, corrected for beam size, determines the properties of the dense gas, as defined by the line tracers themselves. We use Equation (1) in Wu et al. (2010) to determine the line luminosity (${L}_{\mathrm{dense}}$). The fraction of line tracer luminosity inside the region defined by the FWHM of the tracer, extended to a full Gaussian, was determined by dividing ${L}_{\mathrm{dense}}$ by the total luminosity in Table 3.

We use Equation (3) of Wu et al. (2010) to compute the dense gas mass from the virial theorem (${M}_{\mathrm{dv}}$). From ${M}_{\mathrm{dv}}$, the surface density (${{\rm{\Sigma }}}_{\mathrm{dense}}$) and volume-averaged density ($\bar{n}$) of the gas in the region defined by the line tracer are computed, using Equations (5) and (6) from Wu et al. (2010). The clump properties determined from HCN are shown in Table 6, while those determined from ${\mathrm{HCO}}^{+}$ are shown in Table 7. The values in the tables have too many significant digits, but the errors given in the tables clarify how many are truly significant.

Table 6.  Clump Properties for HCN

Source	${r}_{\mathrm{dense}}$	${L}_{\mathrm{dense}}$	${f}_{{\rm{L}}}$	${M}_{\mathrm{dv}}$	Σ	$\bar{n}$	Note
	(pc)	(K km s−1 pc2)		(M⊙)	(M⊙ pc−2)	(cm−3)
G034.158+00.147	${0.86}_{-0.14}^{+0.16}$	${94.1}_{-32.7}^{+36.2}$	${0.20}_{-0.07}^{+0.07}$	${2845}_{-548}^{+596}$	${1237}_{-238}^{+259}$	${18538}_{-6477}^{+7163}$
G034.997+00.330	${2.01}_{-0.37}^{+0.37}$	${438.9}_{-169.8}^{+169.3}$	${0.21}_{-0.08}^{+0.08}$	${1454}_{-533}^{+533}$	${114}_{-42}^{+42}$	${728}_{-352}^{+352}$
G036.459−00.183	${3.61}_{-0.40}^{+0.39}$	${246.4}_{-72.2}^{+71.2}$	${0.28}_{-0.08}^{+0.08}$	${3535}_{-569}^{+563}$	${86}_{-14}^{+14}$	${306}_{-76}^{+75}$
G037.677+00.155	${1.74}_{-0.24}^{+0.24}$	${18.4}_{-7.8}^{+7.8}$	${0.11}_{-0.04}^{+0.04}$	${1652}_{-264}^{+264}$	${173}_{-28}^{+28}$	${1274}_{-368}^{+367}$
G045.825−00.291	${2.49}_{-0.34}^{+0.32}$	${166.0}_{-59.6}^{+57.2}$	${0.12}_{-0.04}^{+0.04}$	${10631}_{-2658}^{+2605}$	${547}_{-137}^{+134}$	${2816}_{-970}^{+928}$
G046.495−00.241	${1.12}_{-0.23}^{+0.25}$	${14.4}_{-6.3}^{+6.7}$	${0.06}_{-0.03}^{+0.03}$	${830}_{-224}^{+235}$	${211}_{-57}^{+60}$	${2421}_{-1069}^{+1148}$	v1
G046.495−00.241	${1.02}_{-0.21}^{+0.23}$	${8.3}_{-3.6}^{+3.9}$	${0.04}_{-0.01}^{+0.02}$	${565}_{-230}^{+235}$	${174}_{-71}^{+73}$	${2199}_{-1187}^{+1247}$	v2
G046.495−00.241	${1.45}_{-0.28}^{+0.31}$	${37.9}_{-15.1}^{+16.4}$	${0.17}_{-0.06}^{+0.07}$	${2020}_{-711}^{+732}$	${304}_{-107}^{+110}$	${2678}_{-1297}^{+1377}$	v3
Mean	1.79	128.1	0.147	2942	356	3870
Std. Dev.	0.86	141.7	0.075	3048	360	5611
Median	1.60	66.0	0.144	1836	193	2310

Note. v1, v2, etc. indicate the velocity component when separated.

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Table 7.  Clump Properties for ${\mathrm{HCO}}^{+}$

Source	${r}_{\mathrm{dense}}$	${L}_{\mathrm{dense}}$	${f}_{{\rm{L}}}$	${M}_{\mathrm{dv}}$	Σ	$\bar{n}$	Note
	(pc)	(K km s−1 pc2)		(M⊙)	(M⊙ pc−2)	(cm−3)
G034.158+00.147	${0.55}_{-0.13}^{+0.14}$	${56.2}_{-28.9}^{+30.3}$	${0.16}_{-0.08}^{+0.09}$	${1819}_{-463}^{+487}$	${1936}_{-492}^{+518}$	${45358}_{-21888}^{+23133}$
G034.997+00.330	${1.82}_{-0.36}^{+0.36}$	${205.7}_{-89.2}^{+89.0}$	${0.17}_{-0.07}^{+0.07}$	${1316}_{-495}^{+494}$	${126}_{-47}^{+47}$	${889}_{-455}^{+454}$
G036.459−00.183	${6.32}_{-0.56}^{+0.54}$	${681.6}_{-180.9}^{+177.7}$	${1.06}_{-0.28}^{+0.26}$	${6184}_{-908}^{+895}$	${49}_{-7}^{+7}$	${100}_{-21}^{+21}$
G037.677+00.155	${2.52}_{-0.24}^{+0.24}$	${81.7}_{-18.8}^{+18.7}$	${0.66}_{-0.15}^{+0.15}$	${2389}_{-294}^{+293}$	${120}_{-15}^{+15}$	${609}_{-124}^{+124}$
G045.825−00.291	${3.21}_{-0.37}^{+0.34}$	${150.7}_{-53.5}^{+51.3}$	${0.42}_{-0.14}^{+0.14}$	${13722}_{-3286}^{+3213}$	${424}_{-101}^{+99}$	${1690}_{-529}^{+501}$
G046.495−00.241	${0.61}_{-0.16}^{+0.17}$	${7.7}_{-4.3}^{+4.5}$	${0.05}_{-0.03}^{+0.03}$	${456}_{-146}^{+151}$	${385}_{-123}^{+127}$	${8045}_{-4495}^{+4704}$	v1
G046.495−00.241	${1.15}_{-0.23}^{+0.25}$	${8.6}_{-3.8}^{+4.1}$	${0.05}_{-0.02}^{+0.02}$	${638}_{-259}^{+265}$	${154}_{-62}^{+64}$	${1722}_{-919}^{+966}$	v2
G046.495−00.241	${0.76}_{-0.18}^{+0.19}$	${8.8}_{-4.5}^{+4.7}$	${0.06}_{-0.03}^{+0.03}$	${1058}_{-400}^{+411}$	${580}_{-220}^{+225}$	${9762}_{-5446}^{+5701}$	v3
Mean	2.12	150.1	0.328	3448	472	8522
Std. Dev.	1.82	211.9	0.341	4241	580	14333
Median	1.48	69.0	0.167	1568	270	1706

Note. v1, v2, etc. indicate the velocity component when separated.

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The distance uncertainties from Table 1 are propagated to other quantities. The virial mass is determined from the distance and linewidth; for the linewidth, the value and uncertainty from the line fits are propagated. These uncertainties also enter the uncertainties for the surface density (Σ) and average density ($\bar{n}$) of the clump. Even with possible underestimates for the uncertainties in angular size, the propagated uncertainty in the clump properties is substantial, especially for $\bar{n}$, which depends strongly on the size. Distance uncertainties were not included in ratios where the distance cancels out, such as ${M}_{\mathrm{dense}}$/${L}_{\mathrm{dense}}$.

The means, standard deviations, and medians are given at the bottom of each table. The properties vary widely from cloud to cloud, as indicated by the very substantial standard deviations (larger or comparable to the mean values). In particular, G036.459−00.183 has a very large value of ${r}_{\mathrm{dense}}$ for ${\mathrm{HCO}}^{+}$, reflecting the very diffuse emission in that tracer; the nominal fraction of the luminosity inside the Gaussian is greater than unity, while the surface and volume densities are very low. Clearly, ${\mathrm{HCO}}^{+}$ is not tracing dense gas in this source; HCN indicates a smaller size and lower ${f}_{{\rm{L}}}$, leading to larger surface and volume densities but is still more characteristic of clouds than of clumps.

For comparison, Wu et al. (2010) found a median ${r}_{\mathrm{dense}}$ of 0.71, a median ${M}_{\mathrm{dense}}$ of 2.7 × 10³ M_⊙, and the median $\bar{n}$ of 1.6 × 10⁴ cm⁻³. The regions probed by the half-power size of line tracer emission in the current sample are larger in size but similar in mass and, therefore, lower in both surface density and mean density. The sample of Wu et al. (2010) was derived from studies originally selected by the presence of water masers and, subsequently, strong emission from CS $J=7\to 6$, so they probably represented particularly dense regions (Plume et al. 1997).

Comparing ${\mathrm{HCO}}^{+}$ and HCN, the clump properties are broadly similar. Using the full width of the half-power size to define the dense clump produces similar results for the two tracers in the median, though differences can be seen in individual sources (e.g., G036.459−00.183). While chemical differences caused by factors like proximity to ionizing sources may well introduce differences between these two tracers in other sources, the two tracers produce similar results in this sample of Galactic sources.

5.2.3. Virial Parameters

Whether or not a particular region is primed to form stars depends most simply on the relative importance of turbulence versus gravity. This competition is crudely captured by the virial parameter. However, measuring the virial parameter is difficult, especially for substructures within clouds, for which boundaries are somewhat arbitrary and the contribution of the surrounding material is ignored (Mao et al. 2019). It is, however, something observers can estimate.

Table 8 presents the virial parameters, calculated from ${\alpha }_{\mathrm{dv}}={M}_{\mathrm{dv}}/{M}_{\mathrm{dense}}$. For G046.495−00.241, the values of ${M}_{\mathrm{dv}}$ for the three separate velocity components have been added together for comparison to ${M}_{\mathrm{dense}}$, which includes all of the BGPS sources. For some clouds, the BGPS and dense line tracer maps agree well, but for others, the agreement is poor (see Figures 1–6). The calculation of ${\alpha }_{\mathrm{dv}}$ for the latter group of clouds is, at best, a crude indicator.

Table 8.  Virial Parameters

Source	${\alpha }_{\mathrm{dv}}$(HCN)	${\alpha }_{\mathrm{dv}}$(${\mathrm{HCO}}^{+}$)	Note
G034.158+00.147	${0.21}_{-0.03}^{+0.04}$	${0.13}_{-0.02}^{+0.02}$
G034.997+00.330	${0.12}_{-0.06}^{+0.06}$	${0.11}_{-0.06}^{+0.06}$
G036.459−00.183	${1.13}_{-0.47}^{+0.47}$	${2.01}_{-0.82}^{+0.83}$
G037.677+00.155	${1.62}_{-0.72}^{+0.73}$	${2.29}_{-1.01}^{+1.03}$
G045.825−00.291	${1.35}_{-0.61}^{+0.61}$	${1.72}_{-0.78}^{+0.77}$
G046.495−00.241	${4.79}_{-2.11}^{+2.14}$	${3.11}_{-1.37}^{+1.39}$	1
Mean	1.54	1.56
Std. Dev.	1.56	1.10
Median	1.24	1.86

Note. (1) Combination of the three velocity components.

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The first two sources have small values for ${\alpha }_{\mathrm{dv}}$. The other four sources have ${\alpha }_{\mathrm{dv}}\geqslant 1$, consistent within uncertainties with unbound structures and certainly less dominated by gravity. The spread in ${\alpha }_{\mathrm{dv}}$ in this sample is consistent with the range of values found for BGPS sources in general by Svoboda et al. (2016), with the first two sources lying near the lowest 10% point of the distribution (Table 7 of Svoboda et al. 2016), but our median values are higher than those for the full BGPS sample.

6.1. The Concept of a Dense Gas Tracer

6.2. Comparison to Other Work

The main conclusions can be summarized as follows.

• 1.  
  The correlation between different tracers of dense gas (extinction, millimeter-wave continuum emission, HCN, ${\mathrm{HCO}}^{+}$) varies from cloud to cloud.
• 2.  
  Broadly, the clouds divide into two groups: one for which the dense line tracers are strong and concentrated, and the other for which they are weak and distributed.
• 3.  
  In clouds where the dense line tracers are sharply peaked, all of the tracers show a general agreement and are more concentrated than the ¹³CO emission.
• 4.  
  In clouds with only weak, distributed emission from dense line tracers, the agreement is poor, and ¹³CO traces similar material.
• 5.  
  Even when the agreement is good, a substantial fraction of the line luminosity arises outside the dust-based measures of dense gas.
• 6.  
  The agreement of dense line tracers with millimeter-wave continuum emission is better than the agreement for ${A}_{{\rm{V}}}\gt 8$ mag. At the distances of some of these clouds, the millimeter-wave continuum emission from BGPS is typically tracing lower-density gas.
• 7.  
  Measurements of $L(\mathrm{HCN})$ toward other galaxies will likely include a large fraction of emission from relatively low-density gas, unless the other galaxy is a starburst galaxy. This variation may be responsible for some of the observed scatter between galaxies in the study of Jiménez-Donaire et al. (2019).
• 8.  
  The conversion from luminosity to mass of dense gas, as measured by extinction or millimeter-wave continuum emission, is quite variable. For the dense regions, the conversion factor is about 20, while it is closer to 6 if the line luminosity of the whole cloud is included.
• 9.  
  For this sample, HCN and ${\mathrm{HCO}}^{+}$ seem to probe about the same material. They are equally good (or bad) tracers.
• 10.  
  The regions probed in this paper in HCN and ${\mathrm{HCO}}^{+}$ in two clouds in the sample are similar to those originally studied by Wu et al. (2010) but are somewhat larger and less dense. The regions probed in the other clouds are substantially more diffuse and less clearly bound.
• 11.  
  The distributed emission of HCN and ${\mathrm{HCO}}^{+}$ can arise from regions of very low density, n = 50–100 cm⁻³. Because of the large area of most clouds at such low densities, these less dense regions can dominate the total luminosity of line tracers.

We thank the staff of the TRAO for support during the course of these observations. N.J.E. thanks the Department of Astronomy at the University of Texas at Austin for ongoing research support. J.W. is supported by NSFC grant Nos. 11590783 and 11673029 and National Key R&D Program of China No. 2017YFA0402600.

In each section, we discuss the details for each cloud. The data reduction details are contained in Table B1.

Table B1.  Reduction Details

Source	Line	${v}_{\mathrm{sp}}$	${v}_{\mathrm{win}}$	${v}_{I}$	Peak Offset	Range	Notes
		(km s−1)	(km s−1)	(km s−1)	(arcsec)	(arcsec)
G034.158+00.147	${\mathrm{HCO}}^{+}$	20, 100	52, 70	52, 70	27, 10	−200, 160; −220, 140
G034.158+00.147	HCN	20, 100	40, 80	40, 80	20, 4	−300, 280; −280, 240
G034.158+00.147	H13CO+	20, 100	52, 70	52, 70	31, 29	−80, 120; −80, 140
G034.158+00.147	H13CN	20, 100	40, 80	45, 70	27, 20	−80, 100; −80, 100
G034.997+00.330	${\mathrm{HCO}}^{+}$	20, 80	40, 65	40, 65	−20, 110	−220, 140; −160, 220
G034.997+00.330	HCN	20, 80	42, 63	42, 60	−20, 110	−220, 140; −160, 220
G034.997+00.330	H13CO+	20, 80	45, 60	45, 60	−20, 110	−80, 20; 80, 160
G034.997+00.330	H13CN	20, 80	42, 63	42, 63	−20, 110	−80, 20; 80, 160
G036.459−00.183	${\mathrm{HCO}}^{+}$	20, 120	45, 85	68, 95	−61, −32	−160, 80; −200, 380
G036.459−00.183	HCN	20, 120	43, 93	67, 93	−20, −20	−100, 180; −160, 340
G037.677+00.155	${\mathrm{HCO}}^{+}$	20, 120	40, 50; 75, 90	77, 88	−98, −156	−180, 120; −240, 0
G037.677+00.155	HCN	20, 120	40, 50; 73, 93	73, 93	−40, −120	−160, 240; −700, 20
G045.825−00.291	${\mathrm{HCO}}^{+}$	20, 80	45, 65	45, 65	−418, 60	−500, −180; 20, 180
G045.825−00.291	HCN	20, 80	45, 65	40, 65	−400, 80	−480, −220; −20, 220
G046.495−00.241	${\mathrm{HCO}}^{+}$	20, 100	40,70	47, 53	87, 3	−20, 120; −60, 200	1
G046.495−00.241	HCN	20, 100	40,70	47, 53	83, 13	−60, 140; −40, 80	1
G046.495−00.241	${\mathrm{HCO}}^{+}$	20, 100	40,70	53, 56	−63, 74	−140, 340; 0, 120	2
G046.495−00.241	HCN	20, 100	40,70	53, 56	276, 150	−140, 340; −60, 200	2
G046.495−00.241	${\mathrm{HCO}}^{+}$	20, 100	40,70	56, 60	−307,104	−540, −140; −20, 200	3
G046.495−00.241	HCN	20, 100	40,70	56, 65	−292,109	−540, 300; −60, 200	3

Note. (1) Position of center peak, velocity component v1. (2) Position of eastern peak, velocity component v2. (3) Position of western peak.

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Our map was centered, not on the source name position, but on l = 34.250, b = 0.150, near the H ii region, G34.26+0.15. This cloud has been studied extensively under other names. The clumps to the "northeast" in our maps are associated with the IRDC 34.43+0.24, in which Rathborne et al. (2006) identified 9 mm continuum sources. These are blended into two clumps in the HCN/${\mathrm{HCO}}^{+}$ maps. G34.26−0.15 is a well-known star-forming region with a water maser (Hofner & Churchwell 1996) and a methanol maser (Breen et al. 2015; Kim et al. 2019).

The near kinematic distance is 3.7 kpc (Anderson et al. 2014). A VERA parallax measurement (Kurayama et al. 2011) of a water maser found by Wang et al. (2006) gives the distance as ${1.56}_{-0.11}^{+0.12}$ kpc, but Kurayama et al. (2011) note that the cloud would then have a peculiar velocity of about 40 km s⁻¹. Because this seems unlikely, we follow other recent work in using the kinematic distance. IRDC G034.43+00.24 has a polarization map (Soam et al. 2019).

The BGPS millimeter continuum emission is shown in Figure 1. The maps of integrated intensity peak about 25'' east (higher l) of the reference position, essentially on top of the H ii region. Both HCN and ${\mathrm{HCO}}^{+}$ show self-absorption and even absorption below zero around 60 km s⁻¹, while the H¹³CO⁺ spectrum shows a clear peak at somewhat higher velocities than those of the main lines (Figure A1). The isotopologues peak slightly north of the main lines. The HCN line is particularly badly affected by absorption. We attribute this effect to the actual absorption of the continuum from the H ii region, which has been subtracted out by baseline process. Similar continuum absorption has been seen in ${\mathrm{HCO}}^{+}$ $J=1\to 0$ and HCN $J=3\to 2$ by Liu et al. (2013), who interpret the line profiles as a signature of infall at about 3 km s⁻¹. Mookerjea et al. (2007) measured a continuum of 6.7 Jy at 2.8 mm in a source size of 16 by 14. This would produce a continuum temperature in our 58'' beam of $3.5{(2.8/3.4)}^{a}$ K, where a is the spectral index between the two wavelengths. Because both free–free and dust continuum emission are contributing in this spectral region, the value of a is uncertain, but the wavelengths are close enough that it makes little difference. There is clearly sufficient continuum emission to explain the absorption that we observe. For extragalactic observations, these issues would not be recognized. The contour diagrams for the main isotopologues are shown in Figure A2, and those for the rarer isotopologues are shown in Figure A3.

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To determine core properties, we used the peak integrated intensity from the fit to the emission and linewidths from fits to the isotopic lines. For the HCN line properties (Table 5), we used the area within the window of 43–70 km s⁻¹, because the hyperfine components, the velocity structure, and the absorption made fits impossible. The H¹³CN line for peak 1 was fitted with hyperfine components.

This cloud was mapped in equatorial coordinates with center position ${\alpha }_{2000}={18}^{{\rm{h}}}{54}^{{\rm{m}}}01\buildrel{\rm{s}}\over{.} 83$, ${\delta }_{2000}=01^\circ 59^{\prime} 18\buildrel{\prime\prime}\over{.} 0$. The BGPS millimeter continuum emission is shown in Figure 2. The spectra at the peaks indicated in Table B1 are shown in Figure B1. There is a single velocity component at about 57 km s⁻¹. The HCN line is well fitted with hyperfine components. The H¹³CO⁺ and H¹³CN lines are about 10% of the main lines, indicating only a modest optical depth. Nonetheless, we use the H¹³CO⁺ line averaged over its detected region for the linewidth in computing virial mass, etc.

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The contour diagrams of the integrated intensity are shown in Figure B2. The BGPS, HCN, and ${\mathrm{HCO}}^{+}$ emission regions correspond well. However, there is significant emission at large distances from the peak of the emission. The plots of the integrated intensity of ${\mathrm{HCO}}^{+}$ and HCN versus distance from the peak are shown in Figure 11. The right-most panel shows the smoothed ${\mathrm{HCO}}^{+}$ along with that from ¹³CO.

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This cloud was mapped in equatorial coordinates with center position ${\alpha }_{2000}={18}^{{\rm{h}}}{58}^{{\rm{m}}}31\buildrel{\rm{s}}\over{.} 78$, ${\delta }_{2000}=03^\circ 03^{\prime} 18\buildrel{\prime\prime}\over{.} 2$. The BGPS millimeter continuum emission is shown in Figure 3. The spectra are shown in Figure C1. Neither the H¹³CO⁺ nor the H¹³CN lines was detected at any position. This cloud has a primary component at about 73 km s⁻¹, which is associated with the H ii region, based on the recombination line velocity. A secondary component at about 55 km s⁻¹ appears in the spectrum, but it is not related to this cloud, so we do not analyze it. The two components nearly overlap but are separable at about 67–68 km s⁻¹. Hyperfine structure was used to fit the HCN lines, but the integrated intensity was integrated over all hyperfine components.

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The maps of ${\mathrm{HCO}}^{+}$ and HCN are shown in Figure C2. The ${\mathrm{HCO}}^{+}$ and HCN integrated intensity generally map similarly, though there are clearly differences in detail. The emission is not well peaked, and the half-power contour is nearly as large as the region of the detected emission, so the properties are poorly defined.

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This source was mapped in galactic coordinates centered on the source name position, l = 37.677, b = 0.155. The BGPS millimeter continuum emission is shown in Figure 4. There are at least two velocity components: one near 45 km s⁻¹ and one near 83 km s⁻¹. The second one is near the velocity of the radio recombination line for this H ii region, so we focus on that. Figure D1 shows the spectra. Nine positions around a nominal peak were averaged for HCN to produce the spectrum. The data were not well fitted with hyperfine structure, so we isolated the velocities around 83 km s⁻¹ to be fitted.

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The ${\mathrm{HCO}}^{+}$ and HCN integrated intensity maps (Figure D2) show extended weak emission, so the peaks are not well defined. The peaks of the two lines differ. The weak emission is surprising because the cloud is relatively massive and has a substantial star formation rate and mass of dense gas, based on the submillimeter continuum data. However, the cloud was difficult to define, as it exists in a region of considerable line uncertainty in ¹³CO.

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This source was mapped in galactic coordinates centered on the source name position, l = 45.825, $b=-0.291$. The BGPS millimeter continuum emission is shown in Figure 5. There are several peaks in the BGPS data. Only the "northwest" peak shows up in the ${\mathrm{HCO}}^{+}$/HCN maps (Figure E2).

This source has two velocity components that are barely separable in ${\mathrm{HCO}}^{+}$ and difficult to separate in HCN. Both of the ${\mathrm{HCO}}^{+}$ and HCN lines are weak, but HCN is a bit stronger in integrated intensity. Figure E1 shows the spectra. We average over the nine positions around the peak to improve the noise since the lines are very similar. Since both velocity features peak in the same area, we include both in the maps of intensity. The isotopologue lines were not detected and did not provide useful constraints on optical depth. We plot the contours of emission for both main species (Figure E2); the peaks are slightly different for the two species but generally consistent. For ${\mathrm{HCO}}^{+}$ and HCN, we use the velocity width and area of the 50.5 km s⁻¹ feature to compute virial mass, etc. However, the HCN is made uncertain by hyperfine structure.

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This source was mapped in galactic coordinates centered, not on the source name, but on l = 46.400, $b=-0.241$, because the source name was that of a millimeter-wave continuum source in the eastern part of the cloud. The BGPS millimeter continuum emission is shown in Figure 6. There are three regions of continuum emission, which we refer to as west, central, and east, in order of increasing longitude. Both ${\mathrm{HCO}}^{+}$ and HCN appear to have several velocity components (Figure F1) and three separated emission regions when the emission is integrated over all velocities, which correspond roughly to the continuum peaks. There were no detections of the ¹³C isotopologues, so we concluded that the velocity structure was unlikely to be caused by self-absorption.

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It was possible to separate the three velocity components in ${\mathrm{HCO}}^{+}$ and to make contour maps. The maps of the integrated intensity are shown in Figure F2. The spectra at the peaks of each component are shown in Figure F1. The lowest velocity component, v1, peaks most strongly on the central peak but has secondary peaks to the north and west that do not correspond exactly to the peaks of the other components but overlap with them. The middle velocity component (v2) emits over an extended region, but weakly, with no strong peaks. We picked the most central and largest peak, which is southeast of the peak of v1. The highest velocity component, denoted v3, peaks most strongly on the western (lower l) peak but has secondary maxima near the other peaks. The other lines are largely absent from the western peak. Because the individual velocity components are narrow, the rms in the integrated intensity is small, so it is possible to draw more contours. The secondary peaks and extended plateaus are reflected in the secondary maxima, but the main peak is reasonably well defined. The separation for HCN is more difficult because of hyperfine structure, so we used the velocity intervals from the ${\mathrm{HCO}}^{+}$ analysis for components v1 and v2. Hyperfine structure was more visible and separate for v3, so we fit that component. The separation into velocity components would not be possible in the observations of other galaxies. The contour maps for each species and component in Figure F3 indicate the complexity of overlapping regions.

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