© ESO, 2017

1. Introduction

A close connection between dense molecular gas (with n_H2> 10⁴ cm^-3) and star formation has been established for quite some time on both Galactic and extragalactic scales. On small scales, individual stars of low to intermediate masses are known to form from the collapse of prestellar dense cores (e.g., Myers 1983; Ward-Thompson et al. 1994; André et al. 2000), themselves often embedded in dense cluster-forming gas clumps within molecular clouds (e.g., Lada 1992; Myers 1998). On galaxy-wide scales, the global star-formation rate (SFR) is linearly correlated with the total amount of dense molecular gas as traced by HCN observations (Gao & Solomon 2004a,b), while the correlation between the SFR and the amount of either atomic (HI) or low-density molecular (CO) gas is non linear and not as tight (e.g., Kennicutt 1989). Moreover, as pointed out by Lada et al. (2012), essentially the same relation between SFR and mass of dense gas M_dense is found in nearby Galactic clouds (SFR = 4.6 × 10^-8M_⊙ yr^-1 × (M_dense/M_⊙) – Lada et al. 2010) and external galaxies (SFR = 2 × 10^-8M_⊙ yr^-1 × (M_dense/M_⊙) – Gao & Solomon 2004b). The only exception seems to be the extreme star-forming environment of the central molecular zone (CMZ) of our Galaxy, where a very low star-formation efficiency in dense gas has been reported (Longmore et al. 2013). Investigating the nature and origin of this quasi-universal “star formation law” in the dense molecular gas of galaxies is of fundamental importance for our understanding of star formation and galaxy evolution in the Universe.

Key insight is provided by the results of recent submillimeter imaging surveys of Galactic molecular clouds with the Herschel Space Observatory, which emphasize the role of interstellar filaments in the star formation process (e.g., André et al. 2010; Molinari et al. 2010). The presence of filamentary structures in molecular clouds was already known long before Herschel (e.g., Schneider & Elmegreen 1979; Myers 2009), but Herschel observations now demonstrate that molecular filaments are truly ubiquitous, make up a dominant fraction of the dense gas in molecular clouds, and present a high degree of universality in their properties (e.g., Arzoumanian et al. 2011; Hill et al. 2011; Schisano et al. 2014; Könyves et al. 2015). A detailed analysis of the radial column density profiles of nearby Herschel filaments shows that they are characterized by a narrow distribution of central widths with a typical full width at half maximum (FWHM) value of ~ 0.1 pc and a dispersion of less than a factor of two (Arzoumanian et al. 2011). Another major result from the Herschel Gould Belt survey (HGBS – André et al. 2010) is that the vast majority of prestellar cores are found in dense, “supercritical” filaments for which the mass per unit length exceeds the critical line mass of nearly isothermal, long cylinders (e.g., Inutsuka & Miyama 1997),  pc^-1, where c_s ~ 0.2 km s^-1 is the isothermal sound speed for molecular gas at T ~ 10 K (e.g., Könyves et al. 2015). These Herschel findings in nearby Galactic clouds support a scenario of star formation in two main steps (cf. André et al. 2014): first, large-scale compression of interstellar material in supersonic flows (turbulent or not) generates a quasi-universal web of filaments in the cold interstellar medium (ISM); second, the densest filaments fragment into prestellar cores (and subsequently protostars) by gravitational instability above M_line,crit.

The realization that, at least in nearby clouds, prestellar core formation occurs primarily along dense filaments of roughly constant inner width has potential implications for our understanding of star formation on global galactic scales. Remarkably, the critical line mass of a filament, , depends only on gas temperature (i.e., T ~ 10 K for the bulk of Galactic molecular clouds, away from the immediate vicinity of massive stars). Given the common filament width W_fil ~ 0.1 pc (Arzoumanian et al. 2011) this may set a quasi-universal threshold for core/star formation in the giant molecular clouds (GMCs) of galaxies at M_line,crit ~ 16 M_⊙ pc^-1 in terms of filament mass per unit length, or M_line,crit/W_fil ~ 160 M_⊙ pc^-2 in terms of gas surface density (corresponding to a visual extinction A_V ~ 8), or in terms of gas density (i.e., a number density n_H2 ~ 2.3 × 10⁴ cm^-3). Indeed, independent Spitzer infrared studies of the SFR as a function of gas surface density in nearby cloud complexes (e.g., Heiderman et al. 2010; Lada et al. 2010; Evans et al. 2014) show that the SFR tends to be linearly proportional to the mass of dense gas above a surface density threshold and drops to negligible values below . The observed star formation threshold corresponds to within a factor of ≪ 2 to the line-mass threshold above which interstellar filaments are expected to be gravitationally unstable.

While the observational results summarized above are very encouraging and tentatively point to a unified picture for star formation on GMC scales in both Galactic clouds and external galaxies, there are at least two caveats. First, direct comparison between Galactic (e.g., Lada et al. 2010) and extragalactic (e.g., Gao & Solomon 2004b) studies of the dense gas – star formation connection is difficult at this stage because different tracers have been used to probe dense gas in Galactic and extragalactic situations so far. For instance, Lada et al. (2010) used column density maps from near-infrared extinction and derived the total mass above the extinction/surface density threshold mentioned earlier (i.e., A_V> 8), while Gao & Solomon (2004b) used HCN (1–0) data and estimated the mass of dense gas M_dense above the effective density ~ 3 × 10⁴ cm^-3 of the HCN (1–0) transition from the HCN (1–0) line luminosity, i.e., L_HCN with α_{GS04 − HCN} ~ 10 M_⊙ (K km s^-1 pc²)^-1 (see also Wu et al. 2005). While the effective density of the HCN(1–0) transition (cf. Evans 1999) turns out to be close to the critical threshold density ~ 2.3 × 10⁴ cm^-3 quoted above for ~ 0.1-pc-wide supercritical filaments, the relation between and remains to be properly calibrated in nearby Galactic clouds. Second, significant variations in the apparent star-formation efficiency in dense gas as a function of stellar surface density or galactocentric radius have been found in resolved observations of the disks of several nearby galaxies (Usero et al. 2015; Chen et al. 2015; Leroy et al. 2016; Bigiel et al. 2016). To confirm that there is a universal star formation law converting the dense molecular gas of GMCs into stars, wide-field line mapping observations of Galactic clouds in the same dense gas tracers as used in extragalactic work and at a spatial resolution high enough to resolve ~0.1-pc-wide molecular filaments are crucially needed.

With the advent of sensitive heterodyne receivers and wideband spectrometers on millimeter-wave telescopes, wide-field mapping observations in several lines simultaneously are now feasible at an angular resolution down to ~10–30″. In this paper, we present wide-field imaging data in the HCN (1–0), H¹³CN (1–0), HCO⁺ (1–0), and H¹³CO⁺ (1–0) transitions toward three nearby star-forming clouds, Aquila, Ophiuchus, and Orion B. The paper is organized as follows. In Sect. 2, we describe our IRAM 30 m, MOPRA 22 m, and Nobeyama 45 m observations. In Sect. 3, we present the results of the HCN, H¹³CN, HCO⁺, and H¹³CO⁺ mappings and estimate the far ultraviolet (FUV) field strength from Herschel HGBS 70 μm and 100 μm data. In Sect. 4, we discuss evidence of significant variations in the conversion factor α_HCN between HCN luminosity and mass of dense gas, and, in particular, the dependence of α_HCN on the FUV field strength. We then revisit the question of the universality of the star-formation efficiency in the dense molecular gas of galaxies and propose an interpretation of this universality in terms of the filamentary structure of GMCs. Our conclusions are summarized in Sect. 5.

2. Millimeter line observations

We carried out observations in the HCN (J = 1 − 0, 88.6318473 GHz), H¹³CN (J = 1 − 0, 86.340167 GHz), HCO⁺ (J = 1 − 0, 89.188526 GHz), and H¹³CO⁺ (J = 1 − 0, 86.75433 GHz) transitions toward three nearby star-forming regions, Aquila, Ophiuchus, and Orion B using the IRAM 30 m, MOPRA 22 m, and Nobeyama 45 m telescopes. The effective excitation densities¹ of the HCN (1–0), H¹³CN (1–0), HCO⁺ (1–0), and H¹³CO⁺ (1–0) lines at 10 K are 8.4 × 10³ cm^-3, 3.5 × 10⁵ cm^-3, 9.5 × 10² cm^-3, and 3.9 × 10⁴ cm^-3, respectively (Shirley 2015). Table 1 shows a summary of our molecular line observations. We describe the details of each observation below.

2.1. IRAM 30 m observations toward the Aquila cloud

During two observing runs (18 December 2014–29 December 2014 and 2 September 2015–9 September 2015), we carried out mapping observations toward a 0.4 deg² region in the Aquila cloud, including the three subregions Aquila/W40, Aquila/Serp. South, and Aquila/cold (see Figs. 1 and A.1), with the Eight MIxer Receiver (EMIR) receiver on the IRAM-30 m telescope. All molecular line data were obtained simultaneously. At 86 GHz, the 30 m telescope has a beam size of 28.6″ (HPBW) and the forward and main beam efficiencies (F_eff and B_eff) are 95% and 81%. As backend, we used the FTS50 spectrometer, providing a bandwidth of 1820 MHz and a frequency resolution of 50 kHz. The latter corresponds to a velocity resolution of ~0.15 km s^-1 at 86 GHz. The standard chopper wheel method was used to convert the observed signal to the antenna temperature in units of K, corrected for the atmospheric attenuation. The data are given in terms of the main beam brightness temperature corresponding to . During the observations, the system noise temperatures ranged from 80 K to 330 K. The telescope pointing was checked every hour by observing the quasar source 1741-038 and was found to be better than 3″ throughout the two runs. Our mapping observations were made with the on-the-fly (OTF) mapping technique. We chose the positions (RA_J2000, Dec_J2000) = (16:29:06.4, − 24:26:57.0), (16:25:34.5, − 24:36:44.0), and (16:26:14.5, − 24:02:00.0) as our reference (off) positions. We decomposed the target field into a series of 10′ × 10′ subfields and took pairs of OTF maps toward each subfield using two perpendicular scanning directions (along the RA or Dec axes). Combining such pairs of OTF maps reduces scanning artifacts. We smoothed the data spatially with a Gaussian function resulting in an effective beam size of 40″. The 1σ noise level of the final mosaiced data cube at an effective angular resolution of 40″ and a velocity resolution of ~0.15 km s^-1 is 0.07 K in T_MB.

2.2. MOPRA observations toward the Ophiuchus cloud

Between 26 July 2015 and 2 August 2015, we carried out mapping observations toward a 0.21 deg² region in the Ophiuchus cloud, including the two subregions Oph/main (L1688) and Oph/cold (see Fig. A.2), with the 3 mm receiver installed on the MOPRA-22 m telescope. All molecular line data were obtained simultaneously. At 86 GHz, the telescope has a beam size of 39″ (HPBW) and a main beam efficiency η_MB of 49% (Ladd et al. 2005). As backend, we used the Mopra spectrometer (MOPS) in zoom mode, providing a bandwidth of 137.5 MHz and a frequency resolution of 33.8 kHz. The latter corresponds to a velocity resolution of ~0.1 km s^-1 at 86 GHz. The standard chopper wheel method was used to convert the observed signal to the antenna temperature in units of K, corrected for atmospheric attenuation. The data are given in main beam brightness temperature, /η_MB. During the observations, the system noise temperatures ranged from 240 K to 420 K. The telescope pointing was checked every hour by observing the SiO maser sources AH Sco, VX Sgr, and W Hya, and was better than 5″ throughout the entire run. Our mapping observations were made with the OTF mapping technique. The positions (RA_J2000, Dec_J2000) = (16:29:06.4, − 24:26:57.0), (16:25:34.5, − 24:36:44.0), and (16:26:14.5, − 24:02:00.0) were used as off positions. We obtained a series of OTF maps with two different scanning directions along the RA or Dec axes covering a subfield of 6′ × 6′ each and combined them into a single map to reduce scanning effects as much as possible. We smoothed the data spatially with a Gaussian function of 19.5″ (FWHM), resulting in an effective beam size of 50″. The scanning effects were minimized by combining scans along the RA and Dec directions with the Emerson & Graeve (1988) PLAIT algorithm. The 1σ noise level of the final data at an effective angular resolution of 50″ and a velocity resolution of 0.1 km s^-1 is 0.5 K in T_MB.

2.3. NRO 45 m observations toward the Orion B cloud

Between 7 May and 21 May 2015, we carried out mapping observations toward a 0.14 deg² region in the Orion B cloud, including the four subregions NGC 2023, NGC 2024, NGC 2068, and NGC 2071 (see Figs. A.3–A.6), with the TZ receiver on the Nobeyama 45 m telescope. All molecular line data were obtained simultaneously. At 86 GHz, the telescope has a beam size of 19.1″ (HPBW) and a main beam efficiency η_MB of ~50%. As backend, we used the SAM45 spectrometer which provides a bandwidth of 31 MHz and a frequency resolution of 7.63 kHz. The latter corresponds to a velocity resolution of ~0.02 km s^-1 at 86 GHz. The standard chopper wheel method was used to convert the observed signal to the antenna temperature in units of K, corrected for the atmospheric attenuation. The data are given in terms of the main beam brightness temperature, /η_MB. During the observations, the system noise temperatures ranged from 140 K to 630 K. The telescope pointing was checked every hour by observing the SiO maser source Ori-KL, and was better than 3″ throughout the entire observing run. Our mapping observations were made with the OTF mapping technique. We chose the positions (RA_J2000, Dec_J2000) = (5:39:21.819, − 2:9:8.54) and (5:44:41.15, 0:33:4.98) as off positions. We obtained OTF maps with two different scanning directions along the RA or Dec axes covering a subfield of 6′ × 6′ and combined them into a single map to reduce the scanning effects as much as possible. As a convolution function, we adopted a Gaussian function with a FWHM of half the beam size. The scanning effects were minimized by combining scans along the RA and Dec directions with the Emerson & Graeve (1988) PLAIT algorithm. We also applied spatial smoothing to the data with a Gaussian function resulting in an effective beam size of 30″. The 1σ noise level of the final data at an effective resolution of 30″ and a velocity resolution of 0.1 km s^-1 is 0.28 K in T_MB.

3. Results and analysis

3.1. H¹³CO⁺(1–0) and H¹³CN(1–0) emission

Figures 1 and A.1–A.6 compare the H¹³CO⁺(1–0), and H¹³CN(1–0) integrated intensity maps observed toward Aquila, Ophiuchus, Orion B/NGC 2023, Orion B/NGC 2024, Orion B/NGC 2068, and Orion B/NGC 2071 with the column density maps derived from HGBS data toward the same sub-regions. Above the A_V = 16 contour in the Herschel column density maps (assuming N_H2/A_V = 0.94 × 10²¹ cm^-2, Bohlin et al. 1978), it can be seen that the spatial distribution of the H¹³CO⁺(1–0) emission is closely correlated with the texture of the dense gas (e.g., filamentary structure) seen by Herschel. In particular, the H¹³CO⁺(1–0) emission traces the dense “supercritical” filaments detected by Herschel very well. Likewise, the spatial distribution of the H¹³CN (1–0) emission is closely correlated with the column density distribution above the A_V = 20 level, especially in the Aquila and Orion B/NGC 2024 regions. As apparent in Figs. 1 and A.1–A.6, and in agreement with the effective excitation densities quoted in Sect. 2, the H¹³CN (1–0) emission traces higher column density gas compared to the H¹³CO⁺ (1–0) emission. Figure A.7 displays mean H¹³CN(1–0) spectra, obtained by averaging the data over the detected portion of each region and sub-region.

The 1 K km s^-1 level in the H¹³CO⁺(1–0) integrated intensity map roughly matches the A_V = 30 level in the Herschel column density map (see Fig. 1f), which in turn corresponds to a volume density ~ 8.6 × 10⁴ cm^-3 assuming most of the dense gas is concentrated in ~ 0.1 pc filaments (cf. Sect. 1). This is roughly consistent with the H¹³CO⁺ (1–0) effective excitation density of ~ 4 × 10⁴ cm^-3, suggesting that local thermodynamical equilibrium (LTE) may not be too bad an approximation for H¹³CO⁺(1–0). Under the LTE assumption, the column density of H¹³CO⁺ can be derived as follows (cf. Tsuboi et al. 2011): (1)We further assume that the excitation temperature T_ex of the H¹³CO⁺ (1–0) transition is equal to the dust temperature T_dust derived from the HGBS data. The dust temperature ranges from 11 K to 46 K. Table 2 summarizes the results. The mean H¹³CO⁺ column densities range from 2.9 × 10¹⁰ cm^-2 to 1.6 × 10¹² cm^-2. The H¹³CO⁺ abundances relative to H₂, X_H¹³CO⁺ ≡ N_H¹³CO⁺/N_H2, have mean values in the range (1.5–5.8) × 10^-11, using N_H¹³CO⁺ values estimated from the present data and N_H2 values from HGBS data (André et al. 2010; Könyves et al. 2015). These abundance estimates are consistent within a factor of a few with the findings of previous studies in other regions (1.1 ± 0.1 × 10^-11 in OMC2-FIR4: Shimajiri et al. 2015b; and 1.8 ± 0.4 × 10^-11 in Sagittarius A: Tsuboi et al. 2011).

Assuming spherical shapes and uniform density, the virial masses of the detected clumps and structures can be estimated as (see Ikeda et al. 2007; Shimajiri et al. 2015a), (2)The radius of each clump or cloud was estimated as . The velocity dispersion σ was determined as /, where is the mean FWHM velocity width among pixels where the emission was detected. The derived values , , , and are given in Tables 3 and 4.

We also estimated the total virial mass of the area above A_V = 8 for each cloud by scaling the virial mass derived for the detected subregion using the well-known linewidth-size relation σ_V ∝ L^0.5 (Larson 1981; Heyer et al. 2009). In practice, we assumed (3)and we estimated the cloud radius as [pc] = , where is the projected area of each observed cloud above A_V = 8. The total virial mass, , was then estimated from the scaled velocity width () using Eq. (2).

The mass of each cloud was also estimated from HGBS data as where M_X (either or ) is the mass integrated over the area of the corresponding Herschel column density map where (H¹³CO⁺ or H¹³CN) line emission was detected or where A_V> 8. A_Y is the surface area (either or ), m_H is the hydrogen atom mass, and μ_H2 = 2.8 is the mean molecular weight per H₂ molecule. The uncertainties in and are typically a factor 2, mainly due to uncertainties in the dust opacity (cf. Roy et al. 2014). The total gas masses derived from Herschel, and , range from 6.5 M_⊙ to 707 M_⊙ and from 16 M_⊙ to 954 M_⊙, respectively.

Tables 3 and 4 also include estimates of the virial mass ratios, and , defined as (6)and (7)The and ratios range from ~0.30 to ~1.1 and from ~0.40 to ~2.0, respectively, suggesting that all the dense clumps we observed are gravitationally bound, especially the portions detected in H¹³CO⁺. Figure 2a plots against , and shows that these two estimates of the mass of dense (A_V> 8) gas agree to within better than 50% in each region (see also Table 3).

We also estimated and for the regions where H¹³CN emission was detected, i.e., Aquila/W40, Aquila/Serp. South, Oph/main, Orion B/NGC 2024, Orion B/ NGC 2068, and Orion B/NGC 2071. The virial mass ratios and range from 0.3 to 2.5 and from 0.5 to 2.9, respectively (see also Fig. 2b).

3.2. HCN (1–0) & HCO⁺ (1–0) emission

In contrast to the H¹³CO⁺ emission, the spatial distributions of HCN(1–0) and HCO⁺(1–0) emission differ significantly from the column density distribution derived from Herschel data, as shown in Figs. 1 and A.1–A.6. The HCN(1–0) and HCO⁺(1–0) maps appear to trace more extended regions than the denser filaments traced in the Herschel column density maps. In the Aquila cloud, the HCN(1–0) and HCO⁺ (1–0) integrated intensities are strongest toward the W40 HII subregion, while column density is highest in the Serpens South subregion. The W40 HII region is known to be excited by the luminous stars IRS 1A North of spectral type O9.5 and IRS 1A South of spectral type B1V (Shuping et al. 2012). In the Ophiuchus cloud, the HCN(1–0) and HCO⁺(1–0) integrated intensities tend to be strong around the compact HII region excited by the B3 star S1 (Grasdalen et al. 1973; André et al. 1988). These findings suggest that the HCN(1–0) and HCO⁺(1–0) intensities depend on the strength of the local FUV radiation field.

Figure 4 shows comparisons of the mean H¹³CO⁺(1–0), HCO⁺(1–0) and HCN(1–0) spectra observed toward each region. Clear dips in the HCN(1–0) and HCO⁺(1–0) spectra can be seen at V_LSR ~ 7, 4, 10 km s^-1 in Aquila/W40, Oph/main, and Orion B/NGC 2071, respectively. Furthermore, the velocities of these dips coincide with the peak velocities of the H¹³CO⁺(1–0) spectra. This suggests that the HCN(1–0) and HCO⁺(1–0) spectra are strongly affected by self-absorption effects. In these subregions, the blueshifted components of the HCN(1–0) and HCO⁺(1–0) spectra are stronger than the redshifted components. This type of asymmetric spectral shape, known as blue-skewed asymmetry, suggests the presence of infalling motions in the clouds (cf. Myers et al. 1996; Schneider et al. 2010). Thus, the Aquila/W40, Oph/main, and Orion B/NGC 2071 clumps may be undergoing large-scale collapse.

Assuming the same excitation temperature for the two isotopic species and a ¹²C/¹³C isotopic ratio R_i = 62 (Langer & Penzias 1993), we also estimate the optical depth of HCN(1–0) and HCO⁺(1–0) as follows: (8)where T_peak(i) is the peak intensity of the rare isotopic species [H¹³CN(1–0) or H¹³CO⁺(1–0)] derived from mean spectra averaged over the observed area (see Fig. 4), whereas T_peak(n) and τ(n) are the peak intensity and optical depth of the normal species [HCN(1–0) or HCO⁺(1–0)] at the peak velocity of the rare isotopic species in the averaged spectra. In all observed regions, the HCN(1–0) and HCO⁺(1–0) lines are optically thick (see also Table A.1).

Recently, Braine et al. (2017) observed a significant dependence of the I_HCN/I_HCO⁺ intensity ratio on metallicity, with I_HCN/I_HCO⁺ increasing from ~1/4 at 0.3 solar metallicity to ~1 at solar metallicity among galaxies of the local group. In the nearby clouds observed here, the median I_HCN/I_HCO⁺ intensity ratio ranges from 0.7 to 1.6 (see Table 5), consistent with the solar or near solar metallicity of these clouds.

3.3. Estimating the strength of the FUV radiation field

The FUV field strength, G₀, can be derived from Herschel 70 μm and 100 μm photometric data using the following equations (Kramer et al. 2008; Schneider et al. 2016): where I_FIR is the far-infrared (FIR) intensity, and B_{60 μm − 80 μm} and B_{80 μm − 125 μm} are the bandwidths of the Herschel/PACS broad-band filters at 70 μm and 100 μm, respectively. The G₀ values are in units of the local interstellar radiation field (Habing 1968).

According to Hollenbach et al. (1991), G₀ can also be estimated from the dust temperature T_dust based on the following relations: where, τ₁₀₀, T₀, and ν₀ are the effective optical depth at 100 μm, the dust temperature of the cloud surface, and the frequency at 0.1 μm.

Figure A.8 shows the pixel-to-pixel correlation between the G₀ values estimated in our target clouds from Herschel 70 μm and 100 μm data using Eqs. ((9), (10)) and the G₀ values estimated from the Herschel T_dust maps using Eqs. ((11)–(13)). The best-fit results are G₀(T_dust) = 1.38 × G₀(70,100 μm) for Aquila, G₀(T_dust) = 0.62 × G₀(70,100 μm) for Oph, and G₀(T_dust) = 2.60 × G₀(70,100 μm) for Orion B. In Orion B, the highest G₀ values come from pixels in NGC 2024 and affect the best-fit results. The best-fit results for pixels with G₀(70,100 μm) < 100 is G₀(T_dust) = 1.04 × G₀(70,100 μm). In summary, our two estimates of G₀ generally agree to within a factor 2 to 3. Pety et al. (2017) also estimated the strength of the FUV radiation field toward NGC 2023/2024 in Orion B using Eq. (12). Their estimate agrees with our G₀(70,100 μm) value within 30%.

4. Discussion

4.1. Evidence of large variations in the α_HCN and α_HCO⁺ conversion factors

In many extragalactic studies (e.g., Gao & Solomon 2004a), the mass of dense gas M_dense is estimated from the HCN(1–0) luminosity L_HCN using the relation M_dense = α_HCNL_HCN and assuming a fixed conversion factor α_HCN. Gao & Solomon derived a simple formula for the conversion factor, namely /T_b = 10 M_⊙ (K km s^-1 pc²)^-1, under the assumption that the HCN(1–0) emission originates from gravitationally-bound “cores” or clumps with volume-averaged density n(H₂) ~ 3 × 10⁴ cm^-3 and brightness temperature T_b ~ 35 K. On this basis, they adopted the single value α_{GS04 − HCN} = 10 M_⊙ (K km s^-1 pc²)^-1 in their seminal HCN study of galaxies (Gao & Solomon 2004a). Clearly, however, if the brightness temperature of the HCN emitting clumps is larger than 35 K or if their volume-averaged density is less than 3 × 10⁴ cm^-3, the α_{GS04 − HCN} factor can become smaller than 10 M_⊙ (K km s^-1 pc²)^-1. Calibrating the conversion factor α_HCN in Galactic clouds and assessing its potential variations is thus of crucial importance. For a sample of massive Galactic clumps, Wu et al. (2005) investigated the relationship between virial mass (estimated from C³⁴S observations) and HCN luminosity and found a logarithmic mean value α_{Wu05 − HCN} = 7 ± 2 M_⊙ (K km s^-1 pc²)^-1 for the conversion factor. The fact that the α_{Wu05 − HCN} and α_{GS04 − HCN} values differ by only ~ 30% is very encouraging for extragalactic studies, but the HCN excitation conditions in the massive clumps studied by Wu et al. (2005) are not necessarily representative of the bulk of the HCN-emitting dense gas in galaxies.

Here, we have both high-resolution HCN data and independent estimates of the mass of dense gas (from Herschel data) for a sample of nearby clouds/clumps spanning a broad range of radiation-field conditions, and are thus in a good position to calibrate the α_HCN factor. To do so, we used the mass estimates derived from the Herschel column density maps, , and the HCN(1–0) luminosities from the present observations to compute a factor for each cloud in our sample. As explained in Sect. 1, because most of the dense gas is distributed in filaments of ~ 0.1 pc width, the A_V> 8 level in Herschel column density maps of nearby molecular clouds is an excellent tracer of the gas denser than n_H2 ~ 2.3 × 10⁴ cm^-3, corresponding to supercritical, star-forming filaments (cf. André et al. 2014). The H₂ volume density of 2 × 10⁴ cm^-3 is also very close to the typical gas density of ≳ 3 × 10⁴/τ cm^-3 traced by the HCN(1–0) line in normal spiral galaxies (Gao & Solomon 2004a, where τ ≳ 1 is the optical depth of the line) and lies between the effective excitation density (8.4 × 10³ cm^-3) and the critical density (≳ 3 × 10⁵ cm^-3) of HCN(1–0) (Shirley 2015). The masses derived from Herschel data therefore provide good reference estimates of the mass of dense gas in nearby clouds. The estimated values of α_{Herschel − HCN} range from ~35 to ~454 M_⊙ (K km s^-1 pc²)^-1 (see Table 6). Clearly, large variations in α_{Herschel − HCN} are present.

As described in Sect. 3.2, the HCN emission tends to be strong in areas where the FUV radiation field is strong. Meijerink et al. (2007) demonstrated that the HCN emission is stronger in photon-dominated regions (PDRs) by a factor of two for densities larger than 10⁵ cm^-3. Therefore, the variations we observe in α_{Herschel − HCN} may be due to variations in the strength of the FUV field among the sub-regions.

The blue filled circles in Fig. 5 show a correlation plot between α_{Herschel − HCN} and the mean FUV radiation field strength, G₀, estimated from Herschel 70 μm and 100 μm data (cf. Sect. 3.3). The correlation coefficient between the two variables is − 0.82, showing the presence of a clear anti-correlation. Quantitatively, α_{Herschel − HCN} decreases as G₀ increases according to the following empirical relation: (14)Figure 6a plots the mass of dense gas M_dense,HCN estimated from HCN for each cloud in our sample, using both the standard extragalactic conversion factor α_{GS04 − HCN} [=10 (K km s^-1 pc²)^-1] (black open squares) and the conversion factor from Eq. (14) (red filled circles), as a function of the reference mass estimate . As can be seen, the M_dense,HCN values obtained with the α_{GS04 − HCN} conversion factor underestimate the reference masses by an order of magnitude on average in nearby clouds. In contrast, the M_dense,HCN estimates using the G₀-dependent conversion factor agree well with the reference dense gas mass estimates .

The HCO⁺(1–0) line is another tracer of dense gas which can be used in extragalactic studies (Braine et al. 2017). Like the HCN emission, the HCO⁺ emission tends to be strong in areas where the FUV radiation field is strong (see Sect. 3.2). The red filled circles in Fig. 5 show a correlation plot between α_{Herschel − HCO⁺} and G₀. Here again, a clear anti-correlation is present with a correlation coefficient of − 0.77. The values are slightly larger than the values, but the basic trend as a function of G₀ is the same. The α_{Herschel − HCO⁺} conversion factor decreases as G₀ increases according to the following best-fit relation: (15)Figure 6b plots the mass of dense gas M_dense,HCO⁺ derived from the HCO⁺ luminosity using the conversion factor from Eq. (15) as a function of the reference mass estimate . As can be seen, the M_dense,HCO⁺ estimates using agree well with the reference dense gas mass estimates .

We conclude that both the HCN(1–0) luminosity and the HCO⁺(1–0) luminosity can be used as reasonably good tracers of the total mass of dense gas down to molecular cloud scales ≳ 1 pc, provided that appropriate G₀-dependent conversion factors are employed (and the strength of the radiation field can be estimated). This is true despite the fact that the HCN(1–0) and HCO⁺(1–0) lines are optically thick and do not trace well the details of the filamentary structure of the dense molecular gas (cf. Sect. 3.2).

4.2. Relationship between star-formation rate and dense gas mass

In this subsection, we use our results on nearby clouds, e.g., our finding of a G₀-dependent conversion factor α_Herschel(G₀), to make a new detailed comparison between the star-formation efficiency within dense molecular gas found in nearby star-forming complexes on one hand and in external galaxies on the other hand.

4.2.1. Estimates of the star-formation rate and star-formation efficiency in the observed clouds

We first estimated the SFR in each of our target regions/sub-regions from direct counting of young stellar objects (YSOs) using the available Spitzer census of YSOs in nearby clouds (Evans et al. 2009). Under the assumption that the median lifetime of Class II YSOs is 2 Myr (Evans et al. 2009; Covey et al. 2010; Lada et al. 2010; Dunham et al. 2015) and that their mean mass is 0.5 M_⊙ (Muench et al. 2007), the SFR can be derived from the number of YSOs observed with Spitzer as follows: (16)To evaluate the number of YSOs in each subregion observed here, we used the Spitzer catalog of Dunham et al. (2015) for Aquila and Ophiuchus, and the catalog of Megeath et al. (2012) for Orion B. To count YSOs at each evolutionary stage from these catalogs, we selected objects with an infrared spectral index²a_IR in the ranges 0.3 ≤ a_IR for Class 0/I YSOs, − 0.3 ≤ a_IR< 0.3 for Flat Spectrum sources, and − 1.6 ≤ a_IR< − 0.3 for Class II objects in agreement with standard YSO classification criteria (Greene et al. 1994).

Dunham et al. (2015) used 2MASS and Spitzer data between 2 μm and 24 μm. Megeath et al. (2012) used Spitzer data between 4.5 μm and 24 μm. While Dunham et al. (2015) published a_IR values both before and after correction for extinction, Megeath et al. (2012) used uncorrected a_IR values. For the sake of consistency, we used uncorrected a_IR values to select Class II YSOs in all regions. The resulting YSO number counts and corresponding SFRs are reported in Table 7 for each region/sub-region observed in molecular lines. The blue filled circles in Fig. 8a show the correlation between SFR_YSO and for the clouds of our sample, which can be expressed as /M_⊙). This relationship is in good agreement with previous studies of nearby Galactic clouds using extinction maps to estimate the mass of dense gas (Lada et al. 2010, 2012; Evans et al. 2014).

4.2.2. Calibration of the dense gas mass in external galaxies

In Sect. 4.1, we showed that reliable HCN-based estimates of the dense gas mass in nearby clouds could be obtained using the G₀-dependent conversion factor given by Eq. (14). In this subsection, we try to account for this G₀ dependence in galaxies with published HCN data in an effort to improve the current estimates of their dense gas masses. Before doing so, it is important to notice that the values derived for the nearby clouds of our sample are a factor of ≳ 3 to ~ 50 higher than the standard extragalactic conversion factor α_{GS04 − HCN} = 10 M_⊙ (K km s^-1 pc²)^-1 (see Fig. 5). Since the mean G₀ value in a typical galaxy cannot be much higher than the highest G₀ found in our regions (G₀ ~ 4000 for NGC 2024), another effect besides the G₀ dependence must explain this large difference in conversion factor.

While our HCN/HCO⁺ observations were specifically focused on the densest (A_V> 8) portions of the target clouds, they also cover a small fraction of the lower density gas in these clouds, and our data clearly show that the HCN(1–0) and HCO⁺(1–0) lines are tracing molecular gas down to much lower column densities than the A_V = 8 fiducial limit. In a recent independent molecular line study of the southern part of the Orion B cloud (including NGC 2023 and NGC 2024) with the IRAM 30 m telescope, Pety et al. (2017) found that 36% of the HCN(1–0) line flux was emitted from low extinction (2 ≤ A_V< 6) areas, with > 98% of the line flux coming from A_V ≥ 2 areas. Likewise, the results of the CHaMP census of molecular clumps in the southern Milky Way with the MOPRA telescope (e.g., Barnes et al. 2016) demonstrate that the HCO⁺(1–0) line emission is generally tracing gas down to volume densities of ~ 10³ cm^-3 or less. Dividing the regions and sub-regions observed in our study in various column density bins, we can investigate possible variations of the empirical conversion factor with column density (see Fig. 7). Significant variations in from region to region can be seen, in agreement with the dependence of α_{Herschel − HCN} on the strength of the FUV radiation field discussed in Sect. 4.1. Figure 7 nevertheless suggests that, in any given sub-region (i.e., for a given G₀ value), the conversion between gas mass and HCN(1–0) luminosity remains approximately constant, irrespective of gas (column) density or A_V, down to A_V ~ 2. This means that, in addition to the dense molecular gas n_H2 ≳ 2 × 10⁴ cm^-3 (corresponding to A_V ≳ 8 in resolved nearby clouds), extragalactic HCN(1–0) studies with spatial resolutions from ~ 8 pc to ~ 36 kpc (Chin et al. 1997, 1998; Gao & Solomon 2004b; Brouillet et al. 2005; Buchbender et al. 2013; Chen et al. 2015, 2017) must also probe essentially all of the molecular gas corresponding to 2 <A_V< 8 in the observed galaxies. Using typical column density probability distribution functions (PDFs) observed toward Galactic molecular clouds (e.g., Schneider et al. 2013), we can estimate the relative amount of molecular gas at A_V> 8 and 2 <A_V ≤ 8 in GMCs, M(A_V> 2) /M(A_V> 8). In practice, we adopt the combined column density PDF derived from Herschel HGBS data in the Aquila and Orion B complexes, which is well described by a power law, (Könyves et al. 2015, and in prep.), as our template. Such a column density PDF implies that M(A_V> 2) /M(A_V> 8) = (2 / 8)^{− 1.7 ± 0.2} ≈ 10, at least in molecular clouds such as Aquila and Orion B. Clearly, if extragalactic HCN measurements of dense gas include low-density gas down to A_V ~ 2, then the effective α_HCN conversion factor for these measurements must be an order of magnitude lower than the factor given by Eq. (14) and thus within a factor of a few of the standard extragalactic conversion factor α_{GS04 − HCN} = 10 M_⊙ (K km s^-1 pc²)^-1.

To go further and to improve current extragalactic M_dense,HCN estimates for radiation-field effects, we also need to account for the fact that the minimum column density or extinction A_V,min(HCN) down to which significant HCN(1–0) line flux is emitted itself depends on G₀. The values of the effective excitation density of HCN(1–0) calculated by Shirley (2015) scale approximately as , where T_gas is the gas kinetic temperature. Assuming T_gas ≈ T_dust and using the simplified relation between T_dust and G₀ given by Eq. (12), this means that . For both a spherical and a cylindrical cloud with a density gradient ρ ∝ r^{− α}, column density scales as r^{1 − α} or . Adopting α = 1.75 ± 0.25, a value intermediate between 1.5 (free-falling cloud onto a point mass) and 2 (isothermal spheres) which is also consistent with the logarithmic slope of − 2.7 for the column density PDF (cf. Könyves et al. 2015), we may thus expect the minimum column density probed by HCN(1–0) to scale as . Normalizing this scaling relation using the fact that A_V,min(HCN) ≈ 2 for NGC 2023/2024 in Orion B where G₀ ≈ 20 (Pety et al. 2017), we obtain the following conversion factor for external galaxies, under the assumption that extragalactic HCN observations sample all of the molecular gas down to A_V,min(HCN): (17)Using the total infrared luminosities L_IR listed in the Gao & Solomon (2004b) paper and adopting L_FIR = L_IR/ 1.3 (Graciá-Carpio et al. 2008), we can derive relevant G₀ values from Eq. (9) and , where d and θ_beam are the distance to each galaxy and the telescope beam size of the corresponding HCN measurement respectively (cf. Buchbender et al. 2013). We can then estimate an effective conversion factor for each galaxy in the Gao & Solomon sample. The resulting G₀ and values range from 44 to 3.9 × 10⁴ (mean: 1.7 × 10³) and from 1.8 M_⊙ (K km s^-1 pc²)^-1 to 17.7 M_⊙ (K km s^-1 pc²)^-1 (mean: 10.2 M_⊙ (K km s^-1 pc²)^-1), respectively. Using these values for the HCN conversion factor, the corrected masses of dense gas range from 4.3 × 10⁷M_⊙ to 4.9 × 10¹⁰M_⊙. Under the assumption that the average gas densities of normal galaxies and extreme starbursts are 2 × 10² cm^-3 and 1 × 10⁵ cm^-3, respectively, García-Burillo et al. (2012) advocated revised conversion factors α_{GB12 − HCN} ~ 3 − 4 M_⊙ (K km s^-1 pc²)^-1 for normal galaxies (L_IR< 10¹¹L_⊙ corresponding to SFR< 2 × 10 M_⊙ yr^-1) and α_{GB12 − HCN} ~ 1 − 2 M_⊙ (K km s^-1 pc²)^-1 for extreme starbursts (L_IR> 10¹¹L_⊙ corresponding to SFR> 2 × 10 M_⊙ yr^-1). The revised conversion factors α_{GB12 − HCN} of García-Burillo et al. (2012) are roughly consistent with our suggested values of 1.8–17.7 M_⊙ (K km s^-1 pc²)^-1 for the galaxies sampled by Gao & Solomon (2004b) (L_IR ~ 10^{10 − 12}L_⊙). This agreement suggests that the dependence of the α_{Herschel − HCN} conversion factor on the strength of the FUV radiation field is not restricted to Galactic clouds and also applies to external galaxies. Figure 8a compiles observations of the SFR–M_dense relation from the present study, Lada et al. (2010), and Gao & Solomon (2004b). The SFR-M_dense relation found in our study is consistent with that of Lada et al. (2010), as expected since both are focused on nearby Galactic clouds. The initial SFR-M_dense relation from Gao & Solomon (2004b) lies a factor of ~2–3 below the best-fit relation found for nearby star-forming regions, namely SFR = 4.6 × 10^-8M_dense (Lada et al. 2010). Conversely, the corrected SFR-M_dense relation for the Gao & Solomon (2004b) sample lies above the nearby-cloud relation by a factor of ~3. We also compiled data points for nearby galaxies of the local group, namely the Small Magellanic Cloud (SMC), the Large Magellanic Cloud (LMC), M31, M33, and M51 from Chin et al. (1997, 1998), Brouillet et al. (2005), Buchbender et al. (2013), and Chen et al. (2015, 2017). The SFR for each galaxy was estimated using SFR = 2.0 × 10^-10 (L_IR/L_⊙) M_⊙ yr^-1 following Gao & Solomon (2004b). As can be seen in Fig. 8a, the observed trend is basically a linear relation between SFR and dense gas mass, and the nearby-cloud relation provides a good overall fit to most data points.

Figure 8b provides a blow-up view of the scatter around the nearby-cloud star formation law by plotting the star-formation efficiency SFE_dense≡SFR/M_dense against M_dense for the same objects as in Fig. 8a. As can be seen, SFE_dense remains roughly constant within a scatter of less than 1.5 orders of magnitude over 8 orders of magnitude in M_dense from ~ 10²M_⊙ to 10¹⁰M_⊙. This scatter in SFE_dense is significantly smaller than the scatter in SFE_total, defined as SFR divided by total molecular gas mass M_total, observed among nearby clouds and massive Galactic clouds, which exceeds 2 to 3 orders of magnitude over only 4 orders of magnitude in gas mass (cf. Lada et al. 2010; Vutisalchavakul et al. 2016). The mean and standard deviation of (the logs of) all the data points in Fig. 8b (including the CMZ point) are ⟨ log SFE_dense (yr^-1)⟩ = − 7.85 ± 0.31. This is in excellent agreement with the results of Vutisalchavakul et al. (2016) who found ⟨ log SFE_dense (yr^-1)⟩ = − 7.74 ± 0.50 toward star-forming regions in the Galactic plane using masses derived from the Bolocam Galactic Plane Survey (BGPS) data at 1.1 mm and SFR values estimated from WISE mid-infrared data.

The vertical scatter in Fig. 8b can almost entirely be attributed to uncertainties in the α_HCN conversion factor used to estimate the dense gas mass for galaxies. In nearby clouds, for which assumptions about this conversion factor are not needed, the observed scatter in SFE_dense is less than a factor of ~ 3. To conclude, our results support the view that the relationship between star-formation rate and dense gas mass, i.e., the star-formation efficiency in the dense molecular gas of galaxies, is quasi-universal on a wide range of scales from ~ 1–10 pc to >10 kpc.

4.3. Origin of the quasi-universal star-formation efficiency in dense molecular gas

On the grounds that filaments dominate the mass budget of dense molecular gas in GMCs and that most, if not all, prestellar cores form in filaments (see Sect. 1), André et al. (2014) proposed that the quasi-universal star-formation efficiency in dense gas discussed above and summarized in Fig. 8 originates from the physics of filament evolution and core formation along filaments. Following André et al. (2014), we suggest that the star-formation efficiency in dense gas, SFE_dense ≡ SFR /M_dense, is primarily set by three parameters characterizing dense cores along filaments, namely the typical fraction of dense filament gas in the form of prestellar cores (or “core formation efficiency”), f_pre, the typical lifetime of prestellar cores, t_pre, and the efficiency of the conversion from prestellar core mass to stellar mass (or stellar system mass), ϵ_core, according to the simple relation: SFE_dense = f_pre × ϵ_core/t_pre. This relation assumes that, in steady state, each supercritical filament has converted a fraction f_pre of its mass into prestellar cores, and that each prestellar core converts a fraction ϵ_core of its mass into either a single star or a small stellar system on a timescale t_pre. The latter assumption is supported by the similarity between the prestellar core mass function (CMF) and the stellar initial mass function (IMF; e.g., Motte et al. 1998; Enoch et al. 2008; André et al. 2010). Observationally, it seems that the three parameters f_pre, t_pre, ϵ_core have reasonably constant values with little variations from cloud to cloud, at least in Gould Belt regions. Based on the results of the HGBS in the Aquila, Ophiuchus, Orion B, and Taurus/L1495 clouds, the prestellar core formation efficiency in the dense (A_V> 8) gas of supercritical filaments is estimated to be (e.g., Könyves et al. 2015; Könyves et al., in prep.; Ladjelate et al., in prep.). The typical lifetime of low- to intermediate-mass prestellar cores is known to be Myr (e.g., Lee & Myers 1999; Jessop & Ward-Thompson 2000; Könyves et al. 2015). In reality, t_pre likely depends on both core density (Jessop & Ward-Thompson 2000) and core mass (Hatchell & Fuller 2008), but what matters here is the characteristic lifetime of the bulk of prestellar cores near the peak of the CMF and forming in filaments just above the critical line mass M_line,crit. The efficiency of the conversion process from core mass to stellar mass, as estimated from the global shift between the CMF and the IMF is believed to be (e.g., Alves et al. 2007; Nutter & Ward-Thompson 2007; Könyves et al. 2015). Combining these estimates leads to the following “prediction” for SFE_dense from the microphysics of star formation in filaments: (18)which is plotted as a solid straight line and strip in Fig. 8. As can be seen, SFE provides a reasonably good fit to both the Galactic and extragalactic data points of Fig. 8b.

4.4. Comments on apparent SFE variations in resolved galactic disks

Recently, Usero et al. (2015), Chen et al. (2015), and Bigiel et al. (2016) reported significant variations in the apparent star-formation efficiency in dense gas, , across the spatially-resolved disks of several nearby galaxies such as M51. In particular, the L_FIR/L_HCN ratio was observed to decrease as L_FIR increases from the outskirts to the center of the M51 disk (see for instance the map of the L_IR/L_HCN ratio presented by Chen et al. 2015). One plausible interpretation of this trend was a decrease in the star-formation efficiency SFE_dense from the outer disk to the center of the M51 galaxy (e.g., Bigiel et al. 2016). We note, however, that since the FUV radiation field is significantly stronger near the center of the disk compared to the outer parts of the galaxy, and since L_IR/L_HCN scales as , the trend observed in M51 can at least partly originate from the expected decrease in the α_HCN factor toward the center of the disk according to Eq. (14), with the efficiency SFE_dense remaining approximately constant.

We can further quantify this statement using the detailed results published by Usero et al. (2015) for 29 nearby disk galaxies. These authors found an anti-correlation between the apparent local star-formation efficiency in dense gas SFE_{Usero − dense} ≡ I_FIR/I_HCN and the mass surface density of stars Σ_star, expressed as SFE_{Usero − dense} (Myr^-1) = 10^{−0.78 ± 0.30} (M_⊙ pc^-2) (Fig. 2 of Usero et al. 2015). Usero et al. (2015) evaluated the mass surface density of dense gas Σ_dense as α_{Usero − HCN} × I_HCN and the mass surface density of old stars Σ_{Usero − star} values from the 3.6 μm intensity I_{3.6 μm} (with Σ_{Usero − star} ∝ I_{3.6 μm}). Since the I_{3.6 μm} intensity is directly proportional to the FUV intensity I_FUV across the disk of nearby galaxies (e.g., Ford et al. 2013), the anti-correlation can be rewritten as . The dependence of α_HCN on G₀ found in Sects. 4.1 and 4.2.2 and expressed by Eq. (17), i.e., , can largely account for the Usero et al. (2015) anti-correlation, and suggests that SFE_dense actually depends at most very weakly on Σ_⋆ as .

Significant variations in SFE_dense may still exist in the most extreme star-forming environments like the central molecular zone (CMZ) of our Milky Way (e.g., Longmore et al. 2013, see also Fig. 8) or extreme starburst galaxies (e.g., García-Burillo et al. 2012), especially at high redshift (e.g., Gao et al. 2007). By and large, however, the results summarized in Fig. 8b suggest that SFE_dense is remarkably constant over a wide range of scales and environments.

5. Summary and conclusions

In an effort to calibrate dense gas tracers commonly used in extragalactic studies, we carried out wide-field mapping observations at a spatial resolution of ~0.04 pc in HCN (J = 1 − 0), H¹³CN (J = 1 − 0), HCO⁺ (J = 1 − 0), and H¹³CO⁺ (J = 1 − 0) toward the nearby molecular clouds in Ophiuchus, Aquila, and Orion B using the MOPRA 22 m, IRAM 30 m, and Nobeyama 45 m telescopes. Our main results can be summarized as follows:

• 1.

  The spatial distributions of the H¹³CO⁺(1–0) and H¹³CN(1–0)emission are tightly correlated with the filamentary texture of thedense gas seen in Herschel column density maps, showing that theH¹³CO⁺(1–0) and H¹³CN(1–0) lines are good tracers of the densest(“supercritical”) filaments seen in Herschel submillimetercontinuum images. Quantitatively, H¹³CO⁺(1–0) and H¹³CN(1–0)trace Herschel filaments very well above A_V> 16 (i.e., M_line ≳ 30 M_⊙ pc^-1) and A_V> 20 (i.e., M_line ≳ 40 M_⊙ pc^-1)respectively. Moreover, the virial mass estimates derived fromthe H¹³CO⁺(1–0) and H¹³CN(1–0) velocity dispersions agree wellwith the dense gas mass estimates derived from Herschel data forthe same sub-regions.

• 2.

  In contrast, the spatial distributions of the HCN(1–0) and HCO⁺(1–0) emission differ significantly from the column density distribution derived from Herschel data. The HCN(1–0) and HCO⁺ (1–0) lines are only poor tracers of the supercritical filaments seen with Herschel and tend to be stronger around HII regions. Based on a detailed comparison of the HCN(1–0)and HCO⁺(1–0) integrated intensity maps with the Herschel 70/100 μm and dust temperature maps, it appears that the HCN(1–0) and HCO⁺(1–0) integrated intensities are strongly correlated with the strength of the local FUV radiation field (G₀). The luminosities of the HCN and HCO⁺ lines can nevertheless be used to derive reasonably good estimates of the masses of dense gas in the nearby clouds we observed, provided that appropriate G₀-dependent conversion factors α_{Herschel − HCN}(G₀) and α_{Herschel − HCO⁺}(G₀) are adopted when converting L_HCN and L_HCO⁺ to M_dense.

• 3.

  Using the masses derived from Herschel column density maps above A_V> 8 as reference estimates of the mass of dense gas in each nearby cloud, we found that the conversion factors α_HCN and α_HCO⁺ are anti-correlated with the strength of the local FUV radiation field according to the following best-fit empirical relations: M_⊙ (K km s^-1 pc²)^-1 and M_⊙ (K km s^-1 pc²)^-1.

• 4.

  The relation between the star-formation rate, estimated from direct counting of Spitzer YSOs, and the mass of dense gas, derived from Herschel data above A_V> 8, for the nearby clouds/clumps of our sample can be expressed as SFR_YSO = (2.1 − 5.0) × 10^-8M_⊙ yr/M_⊙). This is consistent within errors with the relation found by Lada and collaborators for another and broader sample of nearby star-forming clouds based on Spitzer and near-infrared extinction data (SFR = (4.6 ± 2.6) × 10^-8M_⊙ yr^-1× (M_dense/M_⊙), Lada et al. 2010, 2012).

• 5.

  In nearby molecular clouds, the optically thick HCN(1–0) and HCO⁺(1–0) lines are tracing both moderate column density areas (2 ≤ A_V ≤ 8) and high column density areas (A_V> 8) (see also Pety et al. 2017). Therefore, published extragalactic HCN(1–0) studies, which have spatial resolutions between ~ 10 pc and ~ 50 kpc, must be tracing all of the moderate density gas down to n_H2 ≲ 10³ cm^-3 in the observed galaxies, in addition to the dense gas with n_H2> 10⁴ cm^-3. Estimating the contribution of this moderate density gas from the typical column density PDFs observed in nearby clouds, we obtained the following effective HCN conversion factor for external galaxies: .

• 6.

  Using this G₀-dependent conversion factor to improve the dense gas mass estimates of external galaxies with published HCN data, we found that the star-formation efficiency in dense molecular gas, SFE_dense ≡ SFR/M_dense, is remarkably constant, with a scatter of less than 1.5 orders of magnitude around the nearby-cloud value of 4.5 × 10^-8 yr^-1, over 8 orders of magnitude in M_dense from ~ 10²M_⊙ to 10¹⁰M_⊙. This suggests that the star-formation efficiency in the dense molecular gas of galaxies is quasi-universal on a wide range of scales from ~ 1–10 pc to > 10 kpc.

• 7.

  Following André et al. (2014), we suggest that SFE_dense is primarily set by three parameters characterizing the “microphysics” of core/star formation along molecular filaments, namely the typical fraction of dense filament gas in the form of prestellar cores, , the typical lifetime of solar-type prestellar cores, Myr, and the efficiency of the conversion from prestellar core mass to stellar mass, , according to the simple relation: .

---

Acknowledgments

We thank the referee for useful suggestions that improved the clarity of the paper. The Mopra radio telescope is part of the Australia Telescope National Facility which is funded by the Australian Government for operation as a National Facility managed by CSIRO. The 45-m radio telescope is operated by Nobeyama Radio Observatory, a branch of National Astronomical Observatory of Japan. This research has made use of data from the Herschel Gould Belt survey (HGBS) project (http://gouldbelt-herschel.cea.fr). The HGBS is a Herschel Key Programme jointly carried out by SPIRE Specialist Astronomy Group 3 (SAG 3), scientists of several institutes in the PACS Consortium (CEA Saclay, INAF-IFSI Rome and INAF-Arcetri, KU Leuven, MPIA Heidelberg), and scientists of the Herschel Science Center (HSC). Our work was supported by the ANR-11-BS56-010 project “STARFICH” and the European Research Council under the European Union’s Seventh Framework Programme (ERC Advanced Grant Agreement No. 291294 – “ORISTARS”). N.S. acknowledges support from the DFG through project number Os 177/2-1 and 177/2-2, and central funds of the DFG-priority program 1573 (ISM-SPP). Y.G. acknowledges support from the National Natural Science Foundation of China (grants 11390373 and 11420101002) and the Chinese Academy of Sciences’ Key Research Program of Frontier Sciences.

References

Appendix A: Complementary figures and tables

Figures A.1–A.9 are complementary figures. The optical depths of the HCN (1–0) and HCO⁺ (1–0) lines in each A_V bin are listed in Table A.1. Table A.2 summarizes the definition of each notation used in this paper.

All Tables

All Figures