Testing the universality of the star formation efficiency in dense molecular gas

TL;DR

This study investigates whether the star formation efficiency in dense molecular gas is universal across different environments, using new calibrations of dense gas tracers in Galactic clouds and applying them to extragalactic data.

Contribution

It provides a calibration of HCN and HCO+ as dense gas tracers, demonstrating the near-universality of star formation efficiency in dense gas across diverse scales and environments.

Findings

SFE_dense is nearly constant over 8 orders of magnitude in dense gas mass.

HCN and HCO+ trace gas down to moderate densities (Av>2).

The HCN conversion factor depends on the FUV strength G_0.

Abstract

Recent studies have suggested that star formation (SF) in dense gas may be governed by essentially the same law in Galactic clouds and external galaxies. This conclusion remains controversial, however, because different tracers have been used to probe the dense gas mass in Galactic and extragalactic studies. We conducted observations of Aquila, Oph., and Orion B in HCN(1-0), HCO$^+$(1-0), and their isotopomers to calibrate the HCN and HCO$^+$ lines used as dense gas tracers in extragalactic studies and to test the possible universality of the star formation efficiency in dense gas, SFE$_{dense}$. H$^{13}$CO$^+$ and H$^{13}$CN were observed to be good tracers of the filaments detected with Herschel. Comparing the luminosities $L_{HCN}$ and $L_{HCO^+}$ with the reference masses $M_{Herschel}^{Av>8}$, the empirical conversion factors $\alpha_{Herschel-HCN}$ and $\alpha_{Herschel-HCO^+}$ were found to be anti-correlated with the local FUV strength. In agreement with Pety et al. (2017), HCN and HCO$^+$ were also found to trace gas down to Av>2. As a result, published extragalactic HCN studies must be tracing all of the moderate density gas. Estimating the contribution of this moderate density gas from the typical column density PDFs in nearby clouds, we obtained the following $G_0$-dependent HCN conversion factor for external galaxies:$\alpha_{Herschel-HCN}^{fit'}=64\times G_0^{-0.34}$. Re-estimating the dense gas masses in external galaxies with $\alpha_{Herschel-HCN}^{fit'}$, we found that SFE$_{dense}$ is remarkably constant over 8 orders of magnitude in dense gas mass. Our results confirm that SFE$_{dense}$ of galaxies is quasi-universal on a wide range of scales from ~1-10pc to >10kpc. Based on the tight link between SF and filamentary structure found in Herschel studies of nearby clouds, we argue that SFE$_{dense}$ is primarily set by the microphysics of SF in filaments.

Tables

Table 1. Table 1: Observations

Region	Aquila	Ophiuchus	Orion B
Distance	260 pc*	139 pc$†$	400 pc$‡$
Telescope	IRAM 30m	MOPRA 22m	Nobeyama 45m
Receiver	EMIR	3mm	TZ
Correlator	FTS50	MOPS	SAM45
Obs. period	18 – 29 Dec 2014	26 July – 2 Aug 2015	7–21 May 2015
	2–9 Sep 2015
Mapping area	0.42 deg2 ($\sim$8.7 pc2)	0.21 deg2 ($\sim$1.2 pc2)	0.14 deg2 ($\sim$6.8 pc2)
${\theta }_{\mathrm{ant}}$ at 86 GHz	28$″$.6 ($\sim$0.04 pc)	39$″$.0 ($\sim$0.03 pc)	19$″$.1 ($\sim$0.04 pc)
$dV$ at 86 GHz	0.15 km/s	0.10 km/s	0.023 km/s

Table 2. Table 2: Physical parameters derived from H 13 CO + (1–0) observations

Region	$V_{\mathrm{peak}}$ [km s-1]			$d{V}_{\mathrm{FWHM}}$ [km s-1]			$N_{H^{13}{\mathrm{CO}}^{+}}$ [cm-2]			$X_{H^{13}{\mathrm{CO}}^{+}}$
	min	max	mean	min	max	mean	min	max	mean	min	max	mean
Aquila/W40	4.0	8.8	7.0	0.3	3.3	0.7	4.4$\times$1010	1.7$\times$1012	2.9$\times$1010	2.8$\times$10-12	1.2$\times$10-11	5.5$\times$10-11
Aquila/Serp S	3.8	8.9	7.3	0.3	2.3	0.8	4.6$\times$1010	4.1$\times$1012	3.3$\times$1010	3.3$\times$10-12	4.6$\times$10-11	1.5$\times$10-11
Aquila/cold	5.6	8.6	6.5	0.3	1.1	0.5	7.2$\times$1010	5.7$\times$1011	2.4$\times$1011	9.2$\times$10-12	4.1$\times$10-11	2.2$\times$10-11
Oph/main (L1688)	2.5	4.9	3.6	0.3	2.0	0.7	6.4$\times$1011	2.9$\times$1012	1.1$\times$1012	8.0$\times$10-12	2.0$\times$10-10	3.3$\times$10-11
Oph/cold	2.9	3.6	3.3	0.3	0.7	0.4	6.3$\times$1011	1.5$\times$1012	9.6$\times$1011	2.2$\times$10-11	1.2$\times$10-10	5.5$\times$10-11
Orion B/NGC 2023	8.1	11.7	10.0	0.2	2.3	1.0	2.6$\times$1011	37.8$\times$1011	15.2$\times$1011	1.8$\times$10-11	9.0$\times$10-11	5.0$\times$10-11
Orion B/NGC 2024	5.1	13.2	10.9	0.2	2.7	0.9	2.3$\times$1011	37.9$\times$1011	14.3$\times$1011	1.9$\times$10-11	9.7$\times$10-11	5.8$\times$10-11
Orion B/NGC 2068	7.1	14.8	10.9	0.2	1.8	0.6	2.3$\times$1011	22.4$\times$1011	10.2$\times$1011	1.8$\times$10-11	6.9$\times$10-11	4.5$\times$10-11
Orion B/NGC 2071	6.1	12.7	9.5	0.2	2.2	0.9	2.6$\times$1011	42.0$\times$1011	15.6$\times$1011	1.8$\times$10-11	1.1$\times$10-10	4.5$\times$10-11

Table 3. Table 3: Virial masses from H 13 CO + †

Region	$d{V}_{H^{13}{\mathrm{CO}}^{+}}^{\mathrm{detect}}$	$A_{H^{13}{\mathrm{CO}}^{+}}^{\mathrm{detect}}$	$R_{H^{13}{\mathrm{CO}}^{+}}^{\mathrm{detect}}$	$M_{\mathrm{VIR},H^{13}{\mathrm{CO}}^{+}}^{\mathrm{detect}}$	$M_{\mathrm{Herschel}}^{H^{13}{\mathrm{CO}}^{+}-\mathrm{detect}}$	$\frac{M_{\mathrm{VIR},H^{13}{\mathrm{CO}}^{+}}^{\mathrm{detect}}}{M_{\mathrm{Herschel}}^{H^{13}{\mathrm{CO}}^{+}-\mathrm{detect}}}$	$d{V}_{H^{13}{\mathrm{CO}}^{+}}^{A_V>8}$	$A_{H^{13}{\mathrm{CO}}^{+}}^{A_V>8}$	$R_{H^{13}{\mathrm{CO}}^{+}}^{A_V>8}$	$M_{\mathrm{VIR},H^{13}{\mathrm{CO}}^{+}}^{A_V>8}$	$M_{\mathrm{Herschel}}^{A_V>8}$	$\frac{M_{\mathrm{VIR},H^{13}{\mathrm{CO}}^{+}}^{A_V>8}}{M_{\mathrm{Herschel}}^{A_V>8}}$
	[km s-1]	[pc2]	[pc]	[$M_{\odot }$]	[$M_{\odot }$]		[km s-1]	[pc2]	[pc]	[$M_{\odot }$]	[$M_{\odot }$]
Aquila/W40	1.66	0.90	0.54	307.3	339.1	0.9	2.18	2.68	0.92	917.0	748.1	1.2
Aquila/Serp S	1.19	1.53	0.70	205.8	707.4	0.3	1.35	2.51	0.89	339.0	954.1	0.4
Aquila/cold	1.04	0.36	0.34	76.7	74.2	1.0	1.07	0.40	0.36	85.9	88.8	1.0
\hdashlineAquila (total)	—	—	—	589.8	1120.8	0.5	—	—	—	1341.8	1790.9	0.7
Oph/main (L1688)	1.48	0.23	0.27	122.0	167.9	0.7	2.09	0.91	0.54	492.0	416.0	1.2
Oph/cold	0.40	0.02	0.07	2.5	6.5	0.4	0.55	0.06	0.14	8.5	16.2	0.5
\hdashlineOph (total)	—	—	—	124.5	174.4	0.7	—	—	—	500.5	432.1	1.2
Orion B/NGC2023	1.50	0.30	0.31	144.9	175.2	0.8	1.89	0.76	0.49	367.7	300.1	1.2
Orion B/NGC2024	1.97	0.26	0.29	236.3	206.3	1.1	2.57	0.76	0.49	680.8	335.2	2.0
Orion B/NGC2068	0.99	0.23	0.27	55.0	113.7	0.5	1.21	0.52	0.41	125.3	189.6	0.7
Orion B/NGC2071	1.23	0.46	0.38	121.1	262.9	0.5	1.47	0.94	0.55	247.3	398.7	0.6
\hdashlineOrion B (total)	—	—	—	557.3	758.2	0.7	—	—	—	1421.1	1223.5	1.2

Table 4. Table 4: Virial masses from H 13 CN †

Region	$d{V}_{H^{13}\mathrm{CN}}^{\mathrm{detect}}$	$A_{H^{13}\mathrm{CN}}^{\mathrm{detect}}$	$R_{H^{13}\mathrm{CN}}^{\mathrm{detect}}$	$M_{\mathrm{VIR},H^{13}\mathrm{CN}}^{\mathrm{detect}}$	$M_{\mathrm{Herschel}}^{H^{13}\mathrm{CN}-\mathrm{detect}}$	$\frac{M_{\mathrm{VIR},H^{13}\mathrm{CN}}^{\mathrm{detect}}}{M_{\mathrm{Herschel}}^{H^{13}\mathrm{CN}-\mathrm{detect}}}$	$d{V}_{H^{13}\mathrm{CN}}^{A_V>8}$	$A_{\mathrm{Herschel}}^{A_V>8}$	$R_{\mathrm{Herschel}}^{A_V>8}$	$M_{\mathrm{VIR},H^{13}\mathrm{CN}}^{A_V>8}$	$M_{\mathrm{Herschel}}^{A_V>8}$	$\frac{M_{\mathrm{VIR},H^{13}\mathrm{CN}}^{A_V>8}}{M_{\mathrm{Herschel}}^{A_V>8}}$
	[km s-1]	[pc2]	[pc]	[$M_{\odot }$]	[$M_{\odot }$]		[km s-1]	[pc2]	[pc]	[$M_{\odot }$]	[$M_{\odot }$]
Aquila/W40	1.40	0.25	0.28	115.0	108.5	1.1	2.54	2.68	0.92	1246.8	748.1	1.7
Aquila/Serp S	1.17	0.77	0.49	140.7	445.0	0.3	1.57	2.51	0.89	460.6	954.1	0.5
Oph/main(L1688)	1.58	0.16	0.23	117.2	90.0	1.3	2.43	0.91	0.54	665.5	416.0	1.6
Orion B/NGC2024	1.86	0.20	0.25	181.5	172.2	1.1	2.62	0.76	0.49	705.5	335.2	2.1
Orion B/NGC2068	1.91	0.17	0.23	179.0	71.8	2.5	2.53	0.52	0.41	546.4	189.6	2.9
Orion B/NGC2071	1.90	0.37	0.34	257.1	203.0	1.3	2.40	0.94	0.55	661.7	398.7	1.7

Table 5. Table 5: HCN (1–0) to HCO + (1–0) line intensity ratios

Region	Median	Standard Deviation
Aquila	1.6	1.5
Oph	1.3	0.9
Orion B/NGC2023	0.7	0.2
Orion B/NGC2024	1.1	0.3
Orion B/NGC2068	1.0	0.3
Orion B/NGC2071	0.9	0.2

Table 6. Table 6: Derived parameters for the dense portions of the target nearby clouds where A V > 8 subscript 𝐴 V 8 A_{\rm V}>8 mag

Region	$M_{\mathrm{Herschel}}^{\mathrm{map}>8mag}$†	$G_0$	$A_{\mathrm{Herschel}}^{\mathrm{map}>8mag}$†	$I_{\mathrm{HCN}}$	$L_{\mathrm{HCN}}$	${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}$†	$I_{{\mathrm{HCO}}^{+}}$	$L_{{\mathrm{HCO}}^{+}}$	${\alpha }_{\mathrm{Herschel}-{\mathrm{HCO}}^{+}}$†
	[$M_{\odot }$]		[pc2]	[K km s-1]	[K km s-1 pc2]	[$M_{\odot }$ (K km s-1 pc2)-1]	[K km s-1]	[K km s-1 pc2]	[$M_{\odot }$ (K km s-1 pc2)-1]
Aquila/W40	748	327	2.68	1.13	3.04	246	0.75	2.01	372
Aquila/Serp S	954	20	2.51	0.91	2.30	416	0.60	1.52	629
Aquila/cold	89	1	0.40	0.48	0.20	454	0.33	0.13	667
Oph/main (L1688)	416	134	0.91	2.68	2.44	170	1.90	1.73	241
Oph/cold	16	5	0.06	0.79	0.05	354	0.73	0.04	386
Orion B/NGC2023	300	226	0.76	7.21	5.49	55	7.53	5.74	52
Orion B/NGC2024	335	4091	0.76	12.57	9.59	35	8.72	6.65	50
Orion B/NGC2068	190	134	0.52	8.32	4.36	44	5.54	2.90	65
Orion B/NGC2071	399	121	0.94	6.79	6.41	62	5.62	5.31	75

Table 7. Table 7: Number of YSOs and star formation rate in each subregion

Region	$N$(YSOs)$†$	SFRYSO [$M_{\odot }$ yr-1]$‡$
Aquila/W40	88–115	(22.0–28.8)$\times$10-6
Aquila/Serp S	57–139	(14.2–34.8)$\times$10-6
Aquila/cold	11–15	(2.8–3.8)$\times$10-6
\hdashlineAquila (total)	156–269	(39.0–67.3)$\times$10-6
Oph/main (L1688)	60–112	(15.0–28.0)$\times$10-6
Oph/cold	0–0	—-
\hdashlineOph (total)	60–112	(15.0–28.0)$\times$10-6
Orion B/NGC2023	9–15	(2.2–3.8)$\times$10-6
Orion B/NGC2024	37–55	(9.2–13.8)$\times$10-6
Orion B/NGC2068	6–12	(1.5–3.0)$\times$10-6
Orion B/NGC2071	26–43	(6.5–10.8)$\times$10-6
\hdashlineOrion B (total)	78–125	(19.5–31.3)$\times$10-6

Table 8. Table 8: Optical depths of the HCN (1–0) and HCO + (1–0) lines †

Region	molecule	$A_V$ ¡ 8	8 $\le$ $A_V$ ¡ 12	12 $\le$ $A_V$ ¡ 16	16 $\le$ $A_V$
Aquial/W40	HCN(1–0)	—‡	—‡	35	¿¿1$SA$
	HCO+(1–0)	—‡	159	¿¿1$SA$	¿¿1$SA$
Aquila/Serp S	HCN(1–0)	—‡	—‡	17	103
	HCO+(1–0)	17	67	102	¿¿1$SA$
Aquila/cold	HCN(1–0)	—‡	—‡	16	135
	HCO+(1–0)	—‡	51	124	¿¿1$SA$
Oph/main (L1688)	HCN(1–0)	—‡	—‡	—‡	14
	HCO+(1–0)	—‡	—‡	80	¿¿1$SA$
Oph/cold	HCN(1–0)	—-‡	—‡	‡‡	—‡
	HCO+(1–0)	—‡	—‡	128	¿¿1$SA$
Orion B/NGC2023	HCN(1–0)	—‡	—‡	112	12
	HCO+(1–0)	—‡	10	8	18
Orion B/NGC2024	HCN(1–0)	—‡	—‡	—‡	6
	HCO+(1–0)	—‡	7	10	14
Orion B/NGC2068	HCN(1–0)	—‡	—‡	—‡	8
	HCO+(1–0)	—‡	11	16	20
Orion B/NGC2071	HCN(1–0)	—‡	—‡	—‡	24
	HCO+(1–0)	—‡	21	29	¿¿1$SA$

Table 9. Table 9: Definition of each notation used in the paper

Quantity	Notation	Meaning
Surface area	$A_{\mathrm{mol}}^{\mathrm{detect}}$	Area where the H13CO+ or H13CN emission has been detected.
	$A_{\mathrm{Herschel}}^{A_V>8}$	Area above $A_V$=8 according to the $Herschel$ column density map in each observed cloud
Cloud radius	$R_{\mathrm{mol}}^{\mathrm{detect}}$	Equivalent radius of the area where H13CO+ or H13CN emission has been detected in each cloud.
	$R_{\mathrm{Herschel}}^{A_V>8}$	Equivalent radius of the area above $A_V$=8 in each observed cloud (according to the $Herschel$ column density map).
Velocity width	$d{V}_{\mathrm{FWHM}}$	FWHM velocity width at each pixel
	$d{V}_{\mathrm{mol}}^{\mathrm{detect}}$	FWHM velocity width estimated from the spectrum averaged over the area where H13CO+ or H13CN emission has been detected.
	$d{V}_{\mathrm{mol}}^{A_V>8}$	Scaled FWHM velocity width using $d{V}_{\mathrm{mol}}^{A_V>8}$=$d{V}_{\mathrm{mol}}^{\mathrm{detect}}$ ${\left(\frac{R_{\mathrm{Herschel}}^{A_V>8}}{R_{\mathrm{mol}}^{\mathrm{detect}}}\right)}^{0.5}$
Mass	$M_{\mathrm{VIR},\mathrm{mol}}^{\mathrm{detect}}$	Virial mass of the portion of each cloud where H13CO+ or H13CN emission has been detected.
	$M_{\mathrm{VIR},\mathrm{mol}}^{A_V>8}$	Virial mass of the region above $A_V$=8 in each observed cloud.
	$M_{\mathrm{Herschel}}^{\mathrm{mol}-\mathrm{detect}}$	Dense gas mass estimated from the Herschel column density map of the area where H13CO+ or H13CN emission has been detected.
	$M_{\mathrm{Herschel}}^{A_V>8}$	Dense gas mass estimated from the Herschel column density map for the area above $A_V$=8 in each observed cloud.
	$M_{\mathrm{dense},\mathrm{mol}}$	Dense gas mass estimated from molecular luminosity conversion factor ($M_{\mathrm{dense},\mathrm{mol}}$=${\alpha }_{\mathrm{Herschel}-\mathrm{mol}}^{\mathrm{fit}}$$L_{\mathrm{mol}}$).
Virial ratio	${ℛ}_{\mathrm{VIR},\mathrm{mol}}^{\mathrm{detect},}$	Virial mass ratio of the portion of the cloud where the H13CO+ or H13CN emission has been detected $\left({ℛ}_{\mathrm{VIR},\mathrm{mol}}^{\mathrm{detect}}=M_{\mathrm{VIR},\mathrm{mol}}^{\mathrm{detect}}/M_{\mathrm{Herschel}}^{\mathrm{mol}-\mathrm{detect}}\right)$.
	${ℛ}_{\mathrm{VIR},\mathrm{mol}}^{A_V>8}$	Virial mass ratio of the region above $A_V$=8 in each observed cloud $\left({ℛ}_{\mathrm{VIR},\mathrm{mol}}^{A_V>8}=M_{\mathrm{VIR},\mathrm{mol}}^{A_V>8}/M_{\mathrm{Herschel}}^{A_V>8}\right)$.
Star formation rate	SFRYSO	Star formation rate estimated from the number count of YSOs.
	SFRprestellar	Star formation rate estimated from the number count of prestellar cores.
HCN conversion factor	${\alpha }_{\mathrm{HCN}}$	Conversion factor from HCN(1–0) luminosity to dense gas mass.
	${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}$	Empirical ${\alpha }_{\mathrm{HCN}}$ factor derived for the target nearby clouds using Herschel mass estimates as references (${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}$=$M_{\mathrm{Herschel}}^{A_V>8}$/$L_{\mathrm{HCN}}$).
	${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}^{\mathrm{fit}}$	${\alpha }_{\mathrm{HCN}}$ conversion factor obtained from the relation ${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}^{\mathrm{fit}}$=496$\times$ $G_0^{-0.24}$.
	${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}^{{\mathrm{fit}}^{\prime }}$	${\alpha }_{\mathrm{HCN}}$ conversion factor obtained from the relation ${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}^{{\mathrm{fit}}^{\prime }}$=0.13$\times G_0^{-0.095}\times {\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}^{\mathrm{fit}}$.
	${\alpha }_{\mathrm{GS04}-\mathrm{HCN}}$	${\alpha }_{\mathrm{HCN}}$ conversion factor assumed in Gao & Solomon (2004a).
	${\alpha }_{\mathrm{Wu05}-\mathrm{HCN}}$	${\alpha }_{\mathrm{HCN}}$ conversion factor obtained in Wu et al. (2005).
	${\alpha }_{\mathrm{GB12}-\mathrm{HCN}}$	${\alpha }_{\mathrm{HCN}}$ conversion factor obtained in García-Burillo et al. (2012).
	${\alpha }_{\mathrm{Usero}-\mathrm{HCN}}$	${\alpha }_{\mathrm{HCN}}$ conversion factor obtained in Usero et al. (2015).
${\mathrm{HCO}}^{+}$ conversion factor	${\alpha }_{\mathrm{Herschel}-{\mathrm{HCO}}^{+}}$	${\alpha }_{{\mathrm{HCO}}^{+}}$ conversion factor obtained toward each observed sub region in the present study (${\alpha }_{\mathrm{Herschel}-{\mathrm{HCO}}^{+}}$=$M_{\mathrm{Herschel}}^{A_V>8}$/$L_{{\mathrm{HCO}}^{+}}$).
	${\alpha }_{\mathrm{Herschel}-{\mathrm{HCO}}^{+}}^{\mathrm{fit}}$	${\alpha }_{{\mathrm{HCO}}^{+}}$ conversion factor obtained by the relation ${\alpha }_{\mathrm{Herschel}-{\mathrm{HCO}}^{+}}^{\mathrm{fit}}$=689$\times$ $G_0^{-0.24}$.

Equations

$$N_{{\mathrm{H}}^{13}\mathrm{C}{\mathrm{O}}^{+}}[\mathrm{c}{\mathrm{m}}^{-2}]=5.99\times 10^{10}{T}_{\mathrm{ex}}\int T_{\mathrm{MB}}dv[\mathrm{K}\mathrm{km}{\mathrm{s}}^{-1}].$$

$$\begin{array}{rl}{M}_{\mathrm{VIR},\mathrm{mol}}^{\mathrm{detect}}[M_{\odot }]& =\frac{5R_{\mathrm{mol}}^{\mathrm{detect}}{\sigma }^2}{G}\\ & =209\left(\frac{R_{\mathrm{mol}}^{\mathrm{detect}}}{\mathrm{pc}}\right){\left(\frac{d{V}_{\mathrm{mol}}^{\mathrm{detect}}}{\mathrm{km}{\mathrm{s}}^{-1}}\right)}^2.\end{array}$$

$$d{V}_{\mathrm{mol}}^{A_{\mathrm{V}}>8}[\mathrm{km}{\mathrm{s}}^{-1}]=d{V}_{\mathrm{mol}}^{\mathrm{detect}}{\left(\frac{R_{\mathrm{Herschel}}^{A_{\mathrm{V}}>8}}{R_{\mathrm{mol}}^{\mathrm{detect}}}\right)}^{0.5},$$

$$M_{\mathrm{X}}[M_{\odot }]={\Sigma }_{\mathrm{Herschel}}{A}_{\mathrm{Y}},$$

$${\Sigma }_{\mathrm{Herschel}}=N({\mathrm{H}}_2)m_{\mathrm{H}}{\mu }_{{\mathrm{H}}_2},$$

$${\mathcal{R}}_{\mathrm{VIR},\mathrm{mol}}^{\mathrm{detect}}=\frac{M_{\mathrm{VIR},\mathrm{mol}}^{\mathrm{detect}}}{M_{\mathrm{Herschel}}^{\mathrm{mol}-\mathrm{detect}}},$$

$${\mathcal{R}}_{\mathrm{VIR},\mathrm{mol}}^{A_{\mathrm{V}}>8}=\frac{M_{\mathrm{VIR},\mathrm{mol}}^{A_{\mathrm{V}}>8}}{M_{\mathrm{Herschel}}^{A_{\mathrm{V}}>8}}.$$

$$\frac{T_{\mathrm{peak}}(\mathrm{i})}{T_{\mathrm{peak}}(\mathrm{n})}=\frac{1-e^{-\tau (\mathrm{n})}/R_{\mathrm{i}}}{1-e^{-\tau (\mathrm{n})}},$$

$$G_0=\frac{4\pi I_{\mathrm{FIR}}}{1.6\times 10^{-3}[\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}]}[\mathrm{in}\mathrm{Habing}\mathrm{units}]$$

$$\begin{array}{rl}{I}_{\mathrm{FIR}}& =\left(\frac{F_{70\mu \mathrm{m}}}{[\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}\mathrm{s}{\mathrm{r}}^{-1}]}\times \frac{B_{60\mu \mathrm{m}-80\mu \mathrm{m}}}{[\mathrm{Hz}]}\right)\\ & +\left(\frac{F_{100\mu \mathrm{m}}}{[\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}\mathrm{s}{\mathrm{r}}^{-1}]}\times \frac{B_{80\mu \mathrm{m}-125\mu \mathrm{m}}}{[\mathrm{Hz}]}\right)[\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\mathrm{s}{\mathrm{r}}^{-1}],\end{array}$$

$$\begin{array}{rl}{T}_{\mathrm{d}}& =(8.9\times 10^{-11}{\nu }_0{G}_0\exp (-1.8A_{\mathrm{V}})\\ & +2.7^5+3.4\times 10^{-2}[0.42-\ln (3.5\times 10^{-2}{\tau }_{100}{T}_0])\\ & \times {\tau }_{100}{T}_0^6{)}^{0.2}[\mathrm{K}]\end{array}$$

$$T_0=12.2G_0^{0.2}[\mathrm{K}]$$

$${\tau }_{100}=\frac{2.7\times 10^2}{T_0^5}{G}_0$$

$${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}^{\mathrm{fit}}=(496\pm 94)\times G_0^{-0.24\pm 0.07}[M_{\odot }(\mathrm{K}\mathrm{km}{\mathrm{s}}^{-1}\mathrm{p}{\mathrm{c}}^2{)}^{-1}].$$

$${\alpha }_{\mathrm{Herschel}-\mathrm{HC}{\mathrm{O}}^{+}}^{\mathrm{fit}}=(689\pm 151)\times G_0^{-0.24\pm 0.08}[M_{\odot }(\mathrm{K}\mathrm{km}{\mathrm{s}}^{-1}\mathrm{p}{\mathrm{c}}^2{)}^{-1}].$$

$$\mathrm{SF}{\mathrm{R}}_{\mathrm{YSO}}=0.25\times N(\mathrm{YSOs})\times 10^{-6}[M_{\odot }{\mathrm{yr}}^{-1}].$$

$${\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}^{\mathrm{fi}{\mathrm{t}}^{\prime }}\approx 0.13_{-0.06}^{+0.06}\times G_0^{-0.095\pm 0.02}\times {\alpha }_{\mathrm{Herschel}-\mathrm{HCN}}^{\mathrm{fit}}\propto G_0^{-0.34\pm 0.08}$$

Full text

Aims. * We aimed to calibrate the HCN and HCO+ lines commonly used as dense gas tracers in extragalactic studies and to test the possible universality of the star formation efficiency in dense gas ($\;\buildrel>\over{\sim}\;$104 cm-3*), SFEdense.

*Methods. * We conducted wide-field mapping of the Aquila, Ophiuchus, and Orion B clouds at $\sim$0.04 pc resolution in the $J$=1–0 transition of HCN, HCO+, and their isotopomers. For each cloud, we derived a reference estimate of the dense gas mass $M_{\rm Herschel}^{A_{\rm V}>8}$, as well as the strength of the local far-ultraviolet (FUV) radiation field, using Herschel Gould Belt survey data products, and estimated the star formation rate from direct counting of the number of Spitzer young stellar objects.

Results. * The H13CO+(1–0) and H13*CN(1–0) lines were observed to be good tracers of the dense star-forming filaments detected with Herschel. Comparing the luminosities $L_{\rm HCN}$ and $L_{\rm HCO^{+}}$ measured in the HCN and HCO+ lines with the reference masses $M_{\rm Herschel}^{A_{\rm V}>8}$, the empirical conversion factors $\alpha_{\rm Herschel-HCN}$ (= $M_{\rm Herschel}^{A_{\rm V}>8}$/$L_{\rm HCN}$) and $\alpha_{\rm Herschel-HCO^{+}}$ (= $M_{\rm Herschel}^{A_{\rm V}>8}$/$L_{\rm HCO^{+}}$) were found to be significantly anti-correlated with the local FUV strength. In agreement with Pety et al. (2017), the HCN and HCO+ lines were also found to trace gas down to $A_{\rm V}$ $\;\buildrel>\over{\sim}\;$ 2. As a result, published extragalactic HCN studies must be tracing all of the moderate density gas down to n_{\rm H_{2}}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{3}\,\rm{cm}^{-3}. Estimating the contribution of this moderate density gas from the typical column density PDFs in nearby clouds, we obtained the following $G_{0}$-dependent HCN conversion factor for external galaxies: $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}=64\times G_{0}^{-0.34}$. Re-estimating the dense gas masses in external galaxies with $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}(G_{0})$, we found that SFEdense is remarkably constant, with a scatter of less than 1.5 orders of magnitude around $4.5\times 10^{-8}\,{\rm yr}^{-1}$, over 8 orders of magnitude in dense gas mass.

*Conclusions. * Our results confirm that SFEdense of galaxies is quasi-universal on a wide range of scales from $\sim 1$–10 pc to $>10\,$kpc. Based on the tight link between star formation and filamentary structure found in Herschel studies of nearby clouds, we argue that SFEdense is primarily set by the “microphysics” of core/star formation along filaments.

1 Introduction

A close connection between dense molecular gas (with $n_{\rm H_{2}}>10^{4}\,{\rm cm}^{-3}$) 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 $M_{\rm dense}$ is found in nearby Galactic clouds [${\rm SFR}=4.6\times 10^{-8}\,M_{\odot}\,{\rm yr}^{-1}\,\times\left(M_{\rm dense}/M_{\odot}\right)$ – Lada et al. (2010)] and external galaxies [${\rm SFR}=2\times 10^{-8}\,M_{\odot}\,{\rm yr}^{-1}\,\times\left(M_{\rm dense}/M_{\odot}\right)$ – 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 $\sim 0.1$ pc and a dispersion of less than a factor of 2 (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), $M_{\rm line,crit}=2\,c_{\rm s}^{2}/G\sim 16\,M_{\odot}$ pc*-1*, where $c_{\rm s}\sim 0.2$ km s*-1* is the isothermal sound speed for molecular gas at $T\sim 10$ 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 $M_{\rm line,crit}$.

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, $M_{\rm line,crit}=2\,c_{\rm s}^{2}/G$, depends only on gas temperature (i.e., $T\sim 10$ K for the bulk of Galactic molecular clouds, away from the immediate vicinity of massive stars). Given the common filament width $W_{\rm fil}\sim$ 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 $M_{\rm line,crit}\sim 16\,M_{\odot}\,{\rm pc}^{-1}$ in terms of filament mass per unit length, or $M_{\rm line,crit}/W_{\rm fil}\sim 160\,M_{\odot}\,{\rm pc}^{-2}$ in terms of gas surface density (corresponding to a visual extinction $A_{\rm V}\sim 8$), or $M_{\rm line,crit}/W_{\rm fil}^{2}\sim 1600\,M_{\odot}\,{\rm pc}^{-3}$ in terms of gas density (i.e., a number density $n_{\rm H_{2}}\sim 2.3\times 10^{4}\,{\rm cm}^{-3}$). 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 $\Sigma_{\rm gas}^{\rm th}\sim$ 130 $M_{\odot}\,{\rm pc}^{-2}$ and drops to negligible values below $\Sigma_{\rm gas}^{\rm th}$. The observed star formation threshold $\Sigma_{\rm gas}^{\rm th}$ 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 $M^{A_{\rm V}>8}_{\rm dense}$ above the extinction/surface density threshold mentioned earlier (i.e., $A_{\rm V}$ $>$ 8), while Gao & Solomon (2004b) used HCN (1–0) data and estimated the mass of dense gas $M_{\rm dense}$ above the effective density $\sim$3$\times$104 cm*-3* of the HCN (1–0) transition from the HCN (1–0) line luminosity, i.e., $M^{\rm HCN}_{\rm dense}$ = $\alpha_{\rm GS04-HCN}$ $L_{\rm HCN}$ with $\alpha_{\rm GS04-HCN}$ $\sim$10 $M_{\odot}$ (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 $\sim 2.3\times 10^{4}\,\rm{cm}^{-3}$ quoted above for $\sim 0.1$-pc-wide supercritical filaments, the relation between $M^{A_{\rm V}>8}_{\rm dense}$ and $M^{\rm HCN}_{\rm dense}$ remains to be properly calibrated in nearby Galactic clouds. Second, significant variations in the apparent star formation efficiency in dense gas SFE${}^{\rm extragal}_{\rm dense}\equiv L_{\rm FIR}/L_{\rm HCN}$ 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 $\sim$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 $\sim$10–30$\arcsec$. 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 30m, MOPRA 22m, and Nobeyama 45m observations. In Sect. 3, we present the results of the HCN, H13CN, HCO*+, and H13*CO+ mappings and estimate the far ultraviolet (FUV) field strength from Herschel HGBS 70 $\mu$m and 100 $\mu$m data. In Sect. 4, we discuss evidence of significant variations in the conversion factor $\alpha_{\rm HCN}$ between HCN luminosity and mass of dense gas, and, in particular, the dependence of $\alpha_{\rm HCN}$ 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 30m, MOPRA 22m, and Nobeyama 45m telescopes. The effective excitation densities111The effective excitation density of a molecular transition at a given gas kinetic temperature is defined as the density which results in a spectral line with an integrated intensity of 1 K km s*-1* (Shirley 2015). of the HCN (1–0), H13CN (1–0), HCO+ (1–0), and H13CO+ (1–0) lines at 10 K are 8.4$\times$103 cm*-3*, 3.5$\times$105 cm*-3*, 9.5$\times$102 cm*-3*, and 3.9$\times$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 30m 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 8), with the Eight MIxer Receiver (EMIR) receiver on the IRAM-30m telescope. All molecular line data were obtained simultaneously. At 86 GHz, the 30 m telescope has a beam size of 28.6$\arcsec$ (HPBW) and the forward and main beam efficiencies ($F_{\rm eff}$ and $B_{\rm eff}$) 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 $\sim$0.15 km s*-1* at 86 GHz. The standard chopper wheel method was used to convert the observed signal to the antenna temperature $T_{\rm A}^{*}$ in units of K, corrected for the atmospheric attenuation. The data are given in terms of the main beam brightness temperature corresponding to $T_{\rm MB}=T_{\rm A}^{*}\times F_{\rm eff}/B_{\rm eff}$. 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$\arcsec$ throughout the two runs. Our mapping observations were made with the on-the-fly (OTF) mapping technique. We chose the positions (RAJ2000, DECJ2000) = (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\arcmin$$\times10$\arcmin$ 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$\arcsec$. The 1$\sigma$ noise level of the final mosaiced data cube at an effective angular resolution of 40$\arcsec$ and a velocity resolution of $\sim$0.15 km s*-1* is 0.07 K in $T_{\rm MB}$.

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. 9), with the 3 mm receiver installed on the MOPRA-22m telescope. All molecular line data were obtained simultaneously. At 86 GHz, the telescope has a beam size of 39$\arcsec$ (HPBW) and a main beam efficiency $\eta_{\rm MB}$ 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 $\sim$0.1 km s*-1* at 86 GHz. The standard chopper wheel method was used to convert the observed signal to the antenna temperature $T_{\rm A}^{*}$ in units of K, corrected for atmospheric attenuation. The data are given in main beam brightness temperature, $T_{\rm MB}$ = $T_{\rm A}^{*}$/$\eta_{\rm MB}$. 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$\arcsec$ throughout the entire run. Our mapping observations were made with the OTF mapping technique. The positions (RAJ2000, DECJ2000) = (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\arcmin$$\times6$\arcmin$ 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 $19.5\arcsec$ (FWHM), resulting in an effective beam size of 50$\arcsec$. The scanning effects were minimized by combining scans along the RA and Dec directions with the Emerson & Graeve (1988) PLAIT algorithm. The 1$\sigma$ noise level of the final data at an effective angular resolution of 50$\arcsec$ and a velocity resolution of 0.1 km s*-1* is 0.5 K in $T_{\rm MB}$.

2.3 NRO 45m 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 NGC2023, NGC2024, NGC2068, and NGC2071 (see Figs. 10–13), with the TZ receiver on the Nobeyama 45m telescope. All molecular line data were obtained simultaneously. At 86 GHz, the telescope has a beam size of $19.1\arcsec$ (HPBW) and a main beam efficiency $\eta_{\rm MB}$ of $\sim$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 $\sim$0.02 km s*-1* at 86 GHz. The standard chopper wheel method was used to convert the observed signal to the antenna temperature $T_{\rm A}^{*}$ in units of K, corrected for the atmospheric attenuation. The data are given in terms of the main beam brightness temperature, $T_{\rm MB}$ = $T_{\rm A}^{*}$/$\eta_{\rm MB}$. 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$\arcsec$ throughout the entire observing run. Our mapping observations were made with the OTF mapping technique. We chose the positions (RAJ2000, DECJ2000) = (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\arcmin$$\times6$\arcmin$ 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$\arcsec$. The 1$\sigma$ noise level of the final data at an effective resolution of 30$\arcsec$ and a velocity resolution of 0.1 km s*-1* is 0.28 K in $T_{\rm MB}$.

3 Results and Analysis

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

Figure 1 and Figures 8–13 compare the H13CO+(1–0), and H13CN(1–0) integrated intensity maps observed toward Aquila, Ophiuchus, Orion B/NGC2023, Orion B/NGC2024, Orion B/NGC2068, and Orion B/NGC2071 with the column density maps derived from HGBS data toward the same sub-regions. Above the $A_{\rm V}$ = 16 contour in the Herschel column density maps (assuming $N_{\rm H_{2}}$/$A_{\rm V}$ = 0.94$\times$1021cm*-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 $A_{\rm V}$ = 20 level, especially in the Aquila and Orion B/NGC 2024 regions. As apparent in Fig. 1 and Figs. 8–13, 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 14 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 $A_{\rm V}$=30 level in the Herschel column density map (see Fig. 1f), which in turn corresponds to a volume density $\sim 8.6\times 10^{4}$ cm*-3* assuming most of the dense gas is concentrated in $\sim 0.1\,$pc filaments (cf. Sect. 1). This is roughly consistent with the H13CO*+* (1–0) effective excitation density of $\sim 4\times 10^{4}$ 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):

[TABLE]

We further assume that the excitation temperature $T_{\rm ex}$ of the H13CO+ (1–0) transition is equal to the dust temperature $T_{\rm dust}$ 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$\times$1010 cm*-2* to 1.6$\times$1012 cm*-2*. The H13CO+ abundances relative to H2, $X_{\rm H^{13}CO^{+}}\equiv N_{\rm H^{13}CO^{+}}/N_{\rm H_{2}}$, have mean values in the range (1.5–5.8)$\times$10*-11*, using $N_{\rm H^{13}CO^{+}}$ values estimated from the present data and $N_{\rm H_{2}}$ 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$\pm$0.1$\times$10*-11* in OMC2-FIR4: Shimajiri et al. (2015b), and 1.8$\pm$0.4$\times$10*-11* in Sagittarius A: Tsuboi et al. (2011)).

Assuming spherical shapes and uniform density, the virial masses $M_{\rm VIR,mol}^{\rm detect}$ of the detected clumps and structures can be estimated as (see Ikeda et al. 2007; Shimajiri et al. 2015a),

[TABLE]

The radius $R_{\rm mol}^{\rm detect}$ of each clump or cloud was estimated as $R_{\rm mol}^{\rm detect}{\rm\ [pc]}=\sqrt{A_{\rm mol}^{\rm detect}/\pi}$. The velocity dispersion $\sigma$ was determined as $\sigma$ = $dV_{\rm mol}^{\rm detect}$/$\sqrt{8\ln{2}}$, where $dV_{\rm mol}^{\rm detect}$ is the mean FWHM velocity width among pixels where the emission was detected. The derived values $dV_{\rm mol}^{\rm detect}$, $R_{\rm mol}^{\rm detect}$, $A_{\rm mol}^{\rm detect}$, and $M_{\rm VIR,mol}^{\rm detect}$ are given in Tables 3 and 4.

We also estimated the total virial mass $M_{\rm VIR,mol}^{A_{\rm V}>8}$ of the area above $A_{\rm V}$ = 8 for each cloud by scaling the virial mass derived for the detected subregion using the well-known linewidth-size relation $\sigma_{V}\propto L^{0.5}$ (Larson 1981; Heyer et al. 2009). In practice, we assumed

[TABLE]

and we estimated the cloud radius $R_{\rm Herschel}^{A_{\rm V}>8}$ as $R_{\rm Herschel}^{A_{\rm V}>8}$ [pc] = $\sqrt{A_{\rm Herschel}^{A_{\rm V}>8}/\pi}$, where $A_{\rm Herschel}^{A_{\rm V}>8}$ is the projected area of each observed cloud above $A_{\rm V}$ = 8. The total virial mass, $M_{\rm VIR,H^{13}CO^{+}}^{A_{\rm V}>8}$, was then estimated from the scaled velocity width ($dV_{\rm VIR,mol}^{A_{\rm V}>8}$) using Eq. (2).

The mass of each cloud was also estimated from HGBS data as

[TABLE]

[TABLE]

where $M_{\rm X}$ (either $M_{\rm Herschel}^{\rm mol-detect}$ or $M_{\rm Herschel}^{A_{\rm V}>8}$) is the mass integrated over the area of the corresponding Herschel column density map where (H13CO+ or H13CN) line emission was detected or where $A_{\rm V}>8$. $A_{\rm Y}$ is the surface area (either $A_{\rm mol}^{\rm detect}$ or $A_{\rm Herschel}^{A_{\rm V}>8}$), $m_{\rm H}$ is the hydrogen atom mass, and $\mu_{\rm H_{2}}$ = 2.8 is the mean molecular weight per ${\rm H_{2}}$ molecule. The uncertainties in $M_{\rm Herschel}^{\rm mol-detect}$ and $M_{\rm Herschel}^{A_{\rm V}>8}$ 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, $M_{\rm Herschel}^{\rm H^{13}CO^{+}-detect}$ and $M_{\rm Herschel}^{A_{\rm V}>8}$, range from 6.5 $M_{\odot}$ to 707 $M_{\odot}$ and from 16 $M_{\odot}$ to 954 $M_{\odot}$, respectively.

Table 3 and Table 4 also include estimates of the virial mass ratios, $\mathcal{R}_{\rm VIR,mol}^{\rm detect}$ and $\mathcal{R}_{\rm VIR,mol}^{A_{\rm V}>8}$, defined as

[TABLE]

and

[TABLE]

The $\mathcal{R}_{\rm VIR,H^{13}CO^{+}}^{\rm detect}$ and $\mathcal{R}_{\rm VIR,H^{13}CO^{+}}^{A_{\rm V}>8}$ ratios range from $\sim$0.30 to $\sim$1.1 and from $\sim$0.40 to $\sim$2.0, respectively, suggesting that all the dense clumps we observed are gravitationally bound, especially the portions detected in H13CO+. Figure 2 (a) plots $M_{\rm VIR,H^{13}CO^{+}}^{A_{\rm V}>8}$ against $M_{\rm Herschel}^{A_{\rm V}>8}$, and shows that these two estimates of the mass of dense ($A_{\rm V}>8$) gas agree to within better than 50% in each region (see also Table 3).

We also estimated $M_{\rm VIR,H^{13}CN}^{\rm detect}$ and $M_{\rm VIR,H^{13}CN}^{A_{\rm V}>8}$ for the regions where H13CN emission was detected, i.e., Aquila/W40, Aquila/Serp. South, Oph/main, Orion B/NGC2024, Orion B/NGC2068, and Orion B/NGC2071. The virial mass ratios $\mathcal{R}_{\rm VIR,H^{13}CN}^{\rm detect}$ and $\mathcal{R}_{\rm VIR,H^{13}CN}^{A_{\rm V}>8}$ range from 0.3 to 2.5 and from 0.5 to 2.9, respectively (see also Fig. 2(b)).

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 Fig. 1 and Figs. 8–13. 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 far-UV (FUV) radiation field.

Figure 3 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 $V_{\rm LSR}$ $\sim$7, 4, 10 km s-1* in Aquila/W40, Oph/main, and Orion B/NGC2071, 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/NGC2071 clumps may be undergoing large-scale collapse.

Assuming the same excitation temperature for the two isotopic species and a 12C/13C isotopic ratio $R_{\rm i}=62$ (Langer & Penzias 1993), we also estimate the optical depth of HCN(1–0) and HCO*+*(1–0) as follows:

[TABLE]

where $T_{\rm peak}({\rm i})$ 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. 3), whereas $T_{\rm peak}({\rm n)}$ and $\tau({\rm n})$ 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 8).

Recently, Braine et al. (2016) observed a significant dependence of the $I_{\rm HCN}/I_{\rm HCO^{+}}$ intensity ratio on metallicity, with $I_{\rm HCN}/I_{\rm HCO^{+}}$ increasing from $\sim$1/4 at 0.3 solar metallicity to $\sim$1 at solar metallicity among galaxies of the local group. In the nearby clouds observed here, the median $I_{\rm HCN}/I_{\rm HCO^{+}}$ 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, $G_{0}$, can be derived from Herschel 70 $\mu$m and 100 $\mu$m photometric data using the following equations (Kramer et al. 2008; Schneider et al. 2016):

[TABLE]

[TABLE]

where $I_{\rm FIR}$ is the far-infrared (FIR) intensity, and $B_{\rm 60\mu m-80\mu m}$ and $B_{\rm 80\mu m-125\mu m}$ are the bandwidths of the Herschel/PACS broad-band filters at 70 $\mu$m and 100 $\mu$m, respectively. The $G_{0}$ values are in units of the local interstellar radiation field (Habing 1968).

According to Hollenbach et al. (1991), $G_{0}$ can also be estimated from the dust temperature $T_{\rm dust}$ based on the following relations:

[TABLE]

[TABLE]

[TABLE]

where, $\tau_{100}$, $T_{0}$, and $\nu_{0}$ are the effective optical depth at 100$\mu$m, the dust temperature of the cloud surface, and the frequency at 0.1$\mu$m.

Figure 15 shows the pixel-to-pixel correlation between the $G_{0}$ values estimated in our target clouds from Herschel 70 $\mu$m and 100 $\mu$m data using Eqs. (9–10) and the $G_{0}$ values estimated from the Herschel $T_{\rm dust}$ maps using Eqs. (11–13).

The best-fit results are $G_{0}$($T_{\rm dust}$) = 1.38\times$$G_{0}(70,100$\mu$m) for Aquila, $G_{0}$($T_{\rm dust}$) = 0.62\times$$G_{0}(70,100$\mu$m) for Oph, and $G_{0}$($T_{\rm dust}$) = 2.60\times$$G_{0}(70,100$\mu$m) for Orion B. In Orion B, the highest $G_{0}$ values come from pixels in NGC2024 and affect the best-fit results. The best-fit results for pixels with $G_{0}$(70,100$\mu$m) ¡ 100 is $G_{0}$($T_{\rm dust}$) = 1.04\times$$G_{0}(70,100$\mu$m). In summary, our two estimates of $G_{0}$ 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 $G_{0}$(70,100$\mu$m) value within 30%.

4 Discussion

4.1 Evidence of large variations in the $\alpha_{\rm HCN}$ and $\alpha_{\rm HCO^{+}}$ conversion factors

In many extragalactic studies (e.g. Gao & Solomon 2004a), the mass of dense gas $M_{\rm dense}$ is estimated from the HCN(1–0) luminosity $L_{\rm HCN}$ using the relation $M_{\rm dense}$ = \alpha_{\rm HCN}$$L_{\rm HCN} and assuming a fixed conversion factor $\alpha_{\rm HCN}$. Gao & Solomon derived a simple formula for the conversion factor, namely $\alpha_{\rm GS04-HCN}$ = 2.1 $\sqrt{n({\rm H_{2}})}$/$T_{\rm b}$ = 10 $M_{\odot}$ (K km s*-1* pc2)-1, under the assumption that the HCN(1–0) emission originates from gravitationally-bound “cores” or clumps with volume-averaged density $n({\rm H_{2}}$)$\sim$3$\times$104 cm*-3* and brightness temperature T_{\rm b}$$\sim35 K. On this basis, they adopted the single value $\alpha_{\rm GS04-HCN}$ = 10 $M_{\odot}$ (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$\times$104 cm*-3*, the $\alpha_{\rm GS04-HCN}$ factor can become smaller than 10 $M_{\odot}$ (K km s*-1* pc2)-1. Calibrating the conversion factor $\alpha_{\rm HCN}$ 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 $\alpha_{\rm Wu05-HCN}$ = 7$\pm$2 $M_{\odot}$ (K km s*-1* pc2)-1 for the conversion factor. The fact that the $\alpha_{\rm Wu05-HCN}$ and $\alpha_{\rm GS04-HCN}$ values differ by only $\sim 30\%$ 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 $\alpha_{\rm HCN}$ factor. To do so, we used the mass estimates derived from the Herschel column density maps, $M_{\rm Herschel}^{A_{\rm V}>8}$, and the HCN(1–0) luminosities from the present observations to compute a $\alpha_{\rm Herschel-HCN}\equiv M_{\rm Herschel}^{A_{\rm V}>8}/L_{\rm HCN}$ factor for each cloud in our sample. As explained in Sect. 1, because most of the dense gas is distributed in filaments of $\sim 0.1\,$pc width, the $A_{\rm V}>8$ level in Herschel column density maps of nearby molecular clouds is an excellent tracer of the gas denser than $n_{\rm H_{2}}\sim 2.3\times 10^{4}\,\rm{cm}^{-3}$, corresponding to supercritical, star-forming filaments (cf. André et al. 2014). The ${\rm H_{2}}$ volume density of $2\times 10^{4}\,\rm{cm}^{-3}$ is also very close to the typical gas density of \lower 2.15277pt\hbox{;\buildrel>\over{\sim};}3\times 10^{4}/\tau\,\rm{cm}^{-3} traced by the HCN(1–0) line in normal spiral galaxies (Gao & Solomon 2004a, where \tau\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}1 is the optical depth of the line) and lies between the effective excitation density (8.4$\times$103 cm*-3*) and the critical density (\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}3\times 105 cm*-3*) of HCN(1–0) (Shirley 2015). The masses $M_{\rm Herschel}^{A_{\rm V}>8}$ derived from Herschel data therefore provide good reference estimates of the mass of dense gas in nearby clouds. The estimated values of $\alpha_{\rm Herschel-HCN}$ range from $\sim$35 to $\sim$454 $M_{\odot}$ (K km s*-1* pc2)-1 (see Table 6). Clearly, large variations in $\alpha_{\rm Herschel-HCN}$ 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 $\alpha_{\rm Herschel-HCN}$ may be due to variations in the strength of the FUV field among the sub-regions.

The blue filled circles in Fig. 4 show a correlation plot between $\alpha_{\rm Herschel-HCN}$ and the mean FUV radiation field strength, $G_{0}$, estimated from Herschel 70 $\mu$m and 100 $\mu$m data (cf. Sect. 3.3). The correlation coefficient between the two variables is $-0.82$, showing the presence of a clear anti-correlation. Quantitatively, $\alpha_{\rm Herschel-HCN}$ decreases as $G_{0}$ increases according to the following empirical relation:

[TABLE]

Figure 5(a) plots the mass of dense gas $M_{\rm dense,HCN}$ estimated from HCN for each cloud in our sample, using both the standard extragalactic conversion factor $\alpha_{\rm GS04-HCN}$ [= 10 (K km s*-1* pc2)-1] (black open squares) and the conversion factor $\alpha_{\rm Herschel-HCN}^{\rm fit}$ from Eq. (14) (red filled circles), as a function of the reference mass estimate $M_{\rm Herschel}^{A_{\rm V}>8}$. As can be seen, the $M_{\rm dense,HCN}$ values obtained with the $\alpha_{\rm GS04-HCN}$ conversion factor underestimate the reference masses $M_{\rm Herschel}^{A_{\rm V}>8}$ by an order of magnitude on average in nearby clouds. In contrast, the $M_{\rm dense,HCN}$ estimates using the $G_{0}$-dependent conversion factor $\alpha_{\rm Herschel-HCN}^{\rm fit}$ agree well with the reference dense gas mass estimates $M_{\rm Herschel}^{A_{\rm V}>8}$.

The HCO*+(1–0) line is another tracer of dense gas which can be used in extragalactic studies (Braine et al. 2016). 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. 4 show a correlation plot between $\alpha_{\rm Herschel-HCO^{+}}$ and $G_{0}$. Here again, a clear anti-correlation is present with a correlation coefficient of $-0.77$. The $\alpha_{\rm Herschel-HCO^{+}}^{\rm fit}$ values are slightly larger than the $\alpha_{\rm Herschel-HCN}^{\rm fit}$ values, but the basic trend as a function of $G_{0}$ is the same. The $\alpha_{\rm Herschel-HCO^{+}}$ conversion factor decreases as $G_{0}$ increases according to the following best-fit relation:

[TABLE]

Figure 5 (b) plots the mass of dense gas $M_{\rm dense,HCO^{+}}$ derived from the HCO+ luminosity using the conversion factor $\alpha_{\rm Herschel-HCO^{+}}^{\rm fit}$ from Eq. (15) as a function of the reference mass estimate $M_{\rm Herschel}^{A_{\rm V}>8}$. As can be seen, the $M_{\rm dense,HCO^{+}}$ estimates using $\alpha_{\rm Herschel-HCO^{+}}^{\rm fit}$ agree well with the reference dense gas mass estimates $M_{\rm Herschel}^{A_{\rm V}>8}$.

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 \lower 2.15277pt\hbox{;\buildrel>\over{\sim};}1\,pc, provided that appropriate $G_{0}$-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 $G_{0}$-dependent conversion factor $\alpha_{\rm Herschel}(G_{0})$, 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 star formation rate (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 $M_{\odot}$ (Muench et al. 2007), the SFR can be derived from the number of YSOs observed with Spitzer as follows:

[TABLE]

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 index222The spectral index $a_{\rm IR}$ is defined as the slope of the near-/mid-infrared spectral energy distribution (SED), $a_{\rm IR}$ = $d\log(\lambda S_{\lambda})/d\log(\lambda)$, where $\lambda$ and $S_{\lambda}$ denote wavelength and flux density at that wavelength, respectively. $a_{\rm IR}$ in the ranges $0.3\leq a_{\rm IR}$ for Class 0/I YSOs, $-0.3\leq a_{\rm IR}<0.3$ for Flat Spectrum sources, and $-1.6\leq a_{\rm IR}<-0.3$ 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 $\mu$m and 24 $\mu$m. Megeath et al. (2012) used Spitzer data between 4.5 $\mu$m and 24 $\mu$m. While Dunham et al. (2015) published $a_{\rm IR}$ values both before and after correction for extinction, Megeath et al. (2012) used uncorrected $a_{\rm IR}$ values. For the sake of consistency, we used uncorrected $a_{\rm IR}$ 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. 7 (a) show the correlation between SFRYSO and $M_{\rm Herschel}^{A_{\rm V}>8}$ for the clouds of our sample, which can be expressed as SFRYSO = (2.1–5.0)$\times$10*-8* $M_{\odot}$ yr*-1* $\times$($M_{\rm Herschel}^{A_{\rm V}>8}$/$M_{\odot}$). 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 $G_{0}$-dependent conversion factor $\alpha_{\rm Herschel-HCN}^{\rm fit}(G_{0})$ given by Eq. (14). In this subsection, we try to account for this $G_{0}$ 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 $\alpha_{\rm Herschel-HCN}^{\rm fit}(G_{0})$ values derived for the nearby clouds of our sample are a factor of \lower 2.15277pt\hbox{;\buildrel>\over{\sim};}3 to $\sim 50$ higher than the standard extragalactic conversion factor $\alpha_{\rm GS04-HCN}$ = 10 $M_{\odot}$ (K km s*-1* pc2)-1 (see Fig. 4). Since the mean $G_{0}$ value in a typical galaxy cannot be much higher than the highest $G_{0}$ found in our regions ($G_{0}\sim 4000$ for NGC2024), another effect besides the $G_{0}$ dependence must explain this large difference in conversion factor.

While our HCN/HCO+ observations were specifically focused on the densest ($A_{\rm V}$ ¿ 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 $A_{\rm V}=8$ fiducial limit. In a recent independent molecular line study of the southern part of the Orion B cloud (including NGC2023 and NGC2024) with the IRAM 30m telescope, Pety et al. (2017) found that 36% of the HCN(1–0) line flux was emitted from low extinction (2 $\leq$ $A_{\rm V}$ $<$ 6) areas, with $>98\%$ of the line flux coming from $A_{\rm V}$ $\geq$ 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 $\sim 10^{3}\,{\rm cm}^{-3}$ 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 $\alpha_{\rm Herschel-HCN}^{\rm A_{V}-bin}\equiv M_{\rm Herschel}^{\rm A_{V}-bin}\times L_{\rm HCN}^{\rm A_{V}-bin}$ with column density (see Fig. 6). Significant variations in $\alpha_{\rm Herschel-HCN}^{\rm A_{V}-bin}$ from region to region can be seen, in agreement with the dependence of $\alpha_{\rm Herschel-HCN}$ on the strength of the FUV radiation field discussed in Sect. 4.1. Figure 6 nevertheless suggests that, in any given sub-region (i.e., for a given $G_{0}$ value), the conversion between gas mass and HCN(1–0) luminosity remains approximately constant, irrespective of gas (column) density or $A_{\rm V}$, down to $A_{\rm V}\sim 2$. This means that, in addition to the dense molecular gas n_{\rm H_{2}}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}2\times 10^{4}\,\rm{cm}^{-3} (corresponding to A_{\rm V}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}8 in resolved nearby clouds), extragalactic HCN(1–0) studies with spatial resolutions from $\sim 8\,$pc to $\sim 36\,$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 $2<A_{\rm V}<8$ in the observed galaxies. Using typical column density probability density functions (PDFs) observed toward Galactic molecular clouds (e.g. Schneider et al. 2013), we can estimate the relative amount of molecular gas at $A_{\rm V}>8$ and $2<A_{\rm V}\leq 8$ in GMCs, $M(A_{\rm V}>2)/M(A_{\rm V}>8)$. 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, $dN/d{\rm log}N_{\rm H_{2}}\propto N_{\rm H_{2}}^{-2.7\pm 0.2}$ (Könyves et al. 2015, ; Könyves et al., in prep.), as our template. Such a column density PDF implies that $M(A_{\rm V}>2)/M(A_{\rm V}>8)=(2/8)^{-1.7\pm 0.2}\approx 10$, 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 $A_{\rm V}\sim 2$, then the effective $\alpha_{\rm HCN}$ conversion factor for these measurements must be an order of magnitude lower than the $\alpha_{\rm Herschel-HCN}^{\rm fit}(G_{0})$ factor given by Eq. (14) and thus within a factor of a few of the standard extragalactic conversion factor $\alpha_{\rm GS04-HCN}$ = 10 $M_{\odot}$ (K km s-1* pc2)-1.

To go further and to improve current extragalactic $M_{\rm dense,HCN}$ estimates for radiation-field effects, we also need to account for the fact that the minimum column density or extinction $A_{V,{\rm min}}$(HCN) down to which significant HCN(1–0) line flux is emitted itself depends on $G_{0}$. The values of the effective excitation density of HCN(1–0) calculated by Shirley (2015) scale approximately as $n_{\rm eff}^{\rm HCN}\propto T_{\rm gas}^{-0.65}$, where $T_{\rm gas}$ is the gas kinetic temperature. Assuming $T_{\rm gas}\approx T_{\rm dust}$ and using the simplified relation between $T_{\rm dust}$ and $G_{0}$ given by Eq. (12), this means that $n_{\rm eff}^{\rm HCN}\propto G_{0}^{-0.13}$. For both a spherical and a cylindrical cloud with a density gradient $\rho\propto r^{-\alpha}$, column density scales as $r^{1-\alpha}$ or $\rho^{1-\frac{1}{\alpha}}$. Adopting $\alpha=1.75\pm 0.25$, 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 $A_{V,{\rm min}}({\rm HCN})\propto(n_{\rm eff}^{\rm HCN})^{0.43}\propto G_{0}^{-0.056\pm 0.012}$. Normalizing this scaling relation using the fact that $A_{V,{\rm min}}({\rm HCN})\approx 2$ for NGC2023/2024 in Orion B where $G_{0}\approx 20$ (Pety et al. 2017), we obtain the following $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}$ conversion factor for external galaxies, under the assumption that extragalactic HCN observations sample all of the molecular gas down to $A_{V,{\rm min}}$(HCN):

[TABLE]

Using the total infrared luminosities $L_{\rm IR}$ listed in the Gao & Solomon (2004b) paper and adopting $L_{\rm FIR}$ = $L_{\rm IR}/1.3$ (Graciá-Carpio et al. 2008), we can derive relevant $G_{0}$ values from Eq. (9) and $I_{\rm FIR}=L_{\rm FIR}/4\pi d^{2}(\frac{\pi\theta_{\rm beam}^{2}}{4\ln 2})$, where $d$ and $\theta_{\rm beam}$ 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 $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}(G_{0})$ for each galaxy in the Gao & Solomon sample. The resulting $G_{0}$ and $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}$ values range from 44 to 3.9$\times$104 (mean: 1.7$\times$103) and from 1.8 $M_{\odot}$ (K km s*-1* pc2)-1 to 17.7 $M_{\odot}$ (K km s*-1* pc2)-1 (mean: 10.2 $M_{\odot}$ (K km s*-1* pc2)-1), respectively. Using these $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}$ values for the HCN conversion factor, the corrected masses of dense gas range from 4.3$\times$107 $M_{\odot}$ to 4.9$\times$1010 $M_{\odot}$. Under the assumption that the average gas densities of normal galaxies and extreme starbursts are 2$\times$102 cm*-3* and 1$\times$105 cm*-3*, respectively, García-Burillo et al. (2012) advocated revised conversion factors \alpha_{\rm GB12-HCN}$$\sim3–4 $M_{\odot}$(K km s*-1* pc2)-1 for normal galaxies ($L_{\rm IR}$ ¡ 1011 $L_{\odot}$ corresponding to SFR $<$ 2$\times$10 $M_{\odot}$yr*-1*) and \alpha_{\rm GB12-HCN}$$\sim1–2 $M_{\odot}$(K km s*-1* pc2)-1 for extreme starbursts ($L_{\rm IR}$ ¿ 1011 $L_{\odot}$ corresponding to SFR $>$ 2$\times$10 $M_{\odot}$yr*-1*). The revised conversion factors $\alpha_{\rm GB12-HCN}$ of García-Burillo et al. (2012) are roughly consistent with our suggested $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}$ values of 1.8–17.7 $M_{\odot}$(K km s*-1* pc2)-1 for the galaxies sampled by Gao & Solomon (2004b) (L_{\rm IR}$$\sim1010-12 $L_{\odot}$). This agreement suggests that the dependence of the $\alpha_{\rm Herschel-HCN}$ conversion factor on the strength of the FUV radiation field is not restricted to Galactic clouds and also applies to external galaxies. Figure 7 (a) compiles observations of the SFR–$M_{\rm dense}$ relation from the present study, Lada et al. (2010), and Gao & Solomon (2004b). The SFR-$M_{\rm dense}$ 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-$M_{\rm dense}$ relation from Gao & Solomon (2004b) lies a factor of $\sim\,$2–3 below the best-fit relation found for nearby star-forming regions, namely SFR = 4.6$\times$10*-8* $M_{\rm dense}$ (Lada et al. 2010). Conversely, the corrected SFR-$M_{\rm dense}$ relation for the Gao & Solomon (2004b) sample lies above the nearby-cloud relation by a factor of $\sim$3. We also compiled data points for nearby galaxies of the local group, namely the Small Magellanic Cloud (SMC), the Large Magellanic Cloud (LMC), M31, M33, and M51 from Chin et al. (1997, 1998), Brouillet et al. (2005), Buchbender et al. (2013), and Chen et al. (2015, 2017). The SFR for each galaxy was estimated using SFR = 2.0$\times$10*-10*($L_{\rm IR}$/$L_{\odot}$) $M_{\odot}$yr*-1* following Gao & Solomon (2004b). As can be seen in Fig. 7 (a), the observed trend is basically a linear relation between star formation rate and dense gas mass, and the nearby-cloud relation provides a good overall fit to most data points.

Figure 7 (b) provides a blow-up view of the scatter around the nearby-cloud star formation law by plotting the star formation efficiency SFEdense $\equiv$ SFR/$M_{\rm dense}$ against $M_{\rm dense}$ for the same objects as in Fig. 7 (a). As can be seen, SFEdense remains roughly constant within a scatter of less than 1.5 orders of magnitude over 8 orders of magnitude in $M_{\rm dense}$ from $\sim 10^{2}\,M_{\odot}$ to $10^{10}\,M_{\odot}$. This scatter in SFEdense is significantly smaller than the scatter in SFEtotal, defined as SFR divided by total molecular gas mass $M_{\rm total}$, 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. 7b (including the CMZ point) are $\langle$log SFEdense (yr*-1*)$\rangle$=-7.85$\pm$0.31. This is in excellent agreement with the results of Vutisalchavakul et al. (2016) who found $\langle$log SFEdense (yr*-1*)$\rangle$ =-7.74$\pm$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. 7b can almost entirely be attributed to uncertainties in the $\alpha_{\rm HCN}$ 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 SFEdense is less than a factor of $\sim 3$. 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 $\sim 1$–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. 7 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${}_{\rm dense}\equiv{\rm SFR}/M_{\rm dense}$, 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”), $f_{\rm pre}$, the typical lifetime of prestellar cores, $t_{\rm pre}$, and the efficiency of the conversion from prestellar core mass to stellar mass (or stellar system mass), $\epsilon_{\rm core}$, according to the simple relation: SFE${}_{\rm dense}=f_{\rm pre}\times\epsilon_{\rm core}/t_{\rm pre}$. This relation assumes that, in steady state, each supercritical filament has converted a fraction $f_{\rm pre}$ of its mass into prestellar cores, and that each prestellar core converts a fraction $\epsilon_{\rm core}$ of its mass into either a single star or a small stellar system on a timescale $t_{\rm pre}$. 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 $f_{\rm pre}$, $t_{\rm pre}$, $\epsilon_{\rm core}$ have reasonably constant values with little variations from cloud to cloud, at least in Gould Belt regions. Based on the results of the Herschel Gould Belt survey in the Aquila, Ophiuchus, Orion B, and Taurus/L1495 clouds, the prestellar core formation efficiency in the dense ($A_{\rm V}>8$) gas of supercritical filaments is estimated to be $f_{\rm pre}=15\%^{+5\%}_{-5\%}$ (e.g. Könyves et al. 2015; Könyves et al. in prep.; Ladjelate al. in prep.). The typical lifetime of low- to intermediate-mass prestellar cores is known to be $t_{\rm pre}=1^{+0.5}_{-0.3}\,$Myr (e.g. Lee & Myers 1999; Jessop & Ward-Thompson 2000; Könyves et al. 2015). In reality, $t_{\rm pre}$ 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 $M_{\rm line,crit}$. 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 $\epsilon_{\rm core}=30\%^{+20\%}_{-10\%}$ (e.g. Alves et al. 2007; Nutter & Ward-Thompson 2007; Könyves et al. 2015). Combining these estimates leads to the following “prediction” for SFEdense from the “microphysics” of star formation in filaments:

SFE${}_{\rm dense}^{\rm pre}\equiv f_{\rm pre}\times\epsilon_{\rm core}/t_{\rm pre}=(4.5\pm 2.5)\times 10^{-8}\,{\rm yr}^{-1}$,

which is plotted as a solid straight line/strip in Fig. 7. As can be seen, SFE${}_{\rm dense}^{\rm pre}$ provides a reasonably good fit to both the Galactic and extragalactic data points of Fig. 7 (b).

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, SFE${}^{\rm extragal}_{\rm dense}\equiv L_{\rm FIR}/L_{\rm HCN}$, across the spatially-resolved disks of several nearby galaxies such as M51. In particular, the $L_{\rm FIR}$/$L_{\rm HCN}$ ratio was observed to decrease as $L_{\rm FIR}$ increases from the outskirts to the center of the M51 disk (see for instance the map of the $L_{\rm IR}$/$L_{\rm HCN}$ ratio presented by Chen et al. 2015). One plausible interpretation of this trend was a decrease in the star formation efficiency SFEdense 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 $L_{\rm IR}/L_{\rm HCN}$ scales as $\frac{\rm SFR}{M_{\rm dense}}\times\alpha_{\rm HCN}={\rm SFE_{dense}}\times\alpha_{\rm HCN}$, the trend observed in M51 can at least partly originate from the expected decrease in the $\alpha_{\rm HCN}$ factor toward the center of the disk according to Eq. (14), with the efficiency ${\rm SFE_{dense}}$ 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${}_{\rm Usero-dense}\equiv I_{\rm FIR}/I_{\rm HCN}$ and the mass surface density of stars $\Sigma_{\rm star}$, expressed as SFEUsero-dense (Myr*-1*) = 10*-0.78±0.30* $\Sigma_{\rm Usero-star}^{-0.37\pm 0.11}$($M_{\odot}$pc*-2*) (Fig. 2 of Usero et al. 2015). Usero et al. (2015) evaluated the mass surface density of dense gas $\Sigma_{\rm dense}$ as $\alpha_{\rm Usero-HCN}\times I_{\rm HCN}$ and the mass surface density of old stars $\Sigma_{\rm Usero-star}$ values from the $3.6\mu$m intensity $I_{\rm 3.6\mu m}$ (with $\Sigma_{\rm Usero-star}\propto I_{\rm 3.6\mu m}$). Since the $I_{\rm 3.6\mu m}$ intensity is directly proportional to the FUV intensity $I_{\rm FUV}$ across the disk of nearby galaxies (e.g. Ford et al. 2013), the anti-correlation SFE${}_{\rm Usero-dense}\propto\Sigma_{\rm Usero-star}^{-0.37\pm 0.11}$ can be rewritten as SFE${}_{\rm dense}\times\alpha_{\rm Usero-HCN}\propto I_{\rm FUV}^{-0.37\pm 0.11}$. The dependence of $\alpha_{\rm HCN}$ on $G_{0}$ found in Sect. 4.1 and 4.2.2 and expressed by Eq. (17), i.e., $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}\propto G_{0}^{-0.34\pm 0.08}$, can largely account for the Usero et al. (2015) anti-correlation, and suggests that SFEdense actually depends at most very weakly on $\Sigma_{\star}$ as SFE${}_{\rm dense}\propto\Sigma_{\star}^{-0.03\pm 0.13}$.

Significant variations in SFEdense 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. 7) 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. 7 (b) suggest that SFEdense 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 $\sim$0.04 pc in HCN ($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 22m, IRAM 30m, and Nobeyama 45m telescopes. Our main results can be summarized as follows:

The spatial distributions of the H13CO+(1–0) and H13CN(1–0) emission are tightly correlated with the filamentary texture of the dense gas seen in Herschel column density maps, showing that the H13CO+(1–0) and H13CN(1–0) lines are good tracers of the densest (“supercritical”) filaments seen in Herschel submillimeter continuum images. Quantitatively, H13CO+(1–0) and H13CN(1–0) trace Herschel filaments very well above $A_{\rm V}>16$ (i.e., M_{\rm line}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}30\,M_{\odot}\,{\rm pc}^{-1}) and $A_{\rm V}>20$ (i.e., M_{\rm line}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}40\,M_{\odot}\,{\rm pc}^{-1}) respectively. Moreover, the virial mass estimates derived from the H13CO+(1–0) and H13CN(1–0) velocity dispersions agree well with the dense gas mass estimates derived from Herschel data for the same sub-regions. 2. 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 H${\rm II}$ regions. Based on a detailed comparison of the HCN(1–0) and HCO+(1–0) integrated intensity maps with the Herschel 70/100 $\mu$m 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 ($G_{0}$). 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 $G_{0}$-dependent conversion factors $\alpha_{\rm Herschel-HCN}(G_{0})$ and $\alpha_{\rm Herschel-HCO^{+}}(G_{0})$ are adopted when converting $L_{\rm HCN}$ and $L_{\rm HCO^{+}}$ to $M_{\rm dense}$. 3. 3.

Using the masses $M_{\rm Herschel}^{A_{\rm V}>8}$ derived from Herschel column density maps above $A_{\rm V}>8$ as reference estimates of the mass of dense gas in each nearby cloud, we found that the conversion factors $\alpha_{\rm HCN}$ and $\alpha_{\rm HCO^{+}}$ are anti-correlated with the strength of the local FUV radiation field according to the following best-fit empirical relations: $\alpha_{\rm Herschel-HCN}^{\rm fit}$ = (496\pm 94)$$\times$$G_{0}-0.24±0.07 $M_{\odot}$ (K km s*-1* pc2)-1 and $\alpha_{\rm Herschel-HCO^{+}}^{\rm fit}$ = (689\pm 151)$$\times$$G_{0}-0.24±0.08 $M_{\odot}$ (K km s*-1* pc2)-1. 4. 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 $A_{\rm V}>8$, for the nearby clouds/clumps of our sample can be expressed as SFRYSO = (2.1–5.0)$\times$10*-8* $M_{\odot}$ yr*-1*$\times$($M_{\rm Herschel}^{A_{\rm V}>8}$/$M_{\odot}$). 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$\pm$2.6)$\times$10*-8* $M_{\odot}$ yr*-1*$\times$($M_{\rm dense}$/$M_{\odot}$), Lada et al. 2010, 2012). 5. 5.

In nearby molecular clouds, the optically thick HCN(1–0) and HCO+(1–0) lines are tracing both moderate column density areas ($2\leq A_{\rm V}\leq 8$) and high column density areas ($A_{\rm V}$ ¿ 8) (see also Pety et al. 2017). Therefore, published extragalactic HCN(1–0) studies, which have spatial resolutions between $\sim 10\,$pc and $\sim 50\,$kpc, must be tracing all of the moderate density gas down to n_{\rm H_{2}}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{3}\,\rm{cm}^{-3} in the observed galaxies, in addition to the dense gas with $n_{\rm H_{2}}>10^{4}\,\rm{cm}^{-3}$. 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: $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}=0.13^{+0.06}_{-0.06}\times$ G_{0}^{-0.095\pm 0.02}$$\times$$\alpha_{\rm Herschel-HCN}^{\rm fit}(G_{0}) $\propto$ $G_{0}^{-0.34\pm 0.08}$. 6. 6.

Using this $G_{0}$-dependent conversion factor $\alpha_{\rm Herschel-HCN}^{\rm fit^{\prime}}(G_{0})$ 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${}_{\rm dense}\equiv$ SFR/$M_{\rm dense}$, is remarkably constant, with a scatter of less than 1.5 orders of magnitude around the nearby-cloud value of $4.5\times 10^{-8}\,{\rm yr}^{-1}$, over 8 orders of magnitude in $M_{\rm dense}$ from $\sim 10^{2}\,M_{\odot}$ to $10^{10}\,M_{\odot}$. This suggests that the star formation efficiency in the dense molecular gas of galaxies is quasi-universal on a wide range of scales from $\sim 1$–10 pc to $>10\,$kpc. 7. 7.

Following André et al. (2014), we suggest that SFEdense 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, $f_{\rm pre}=15\%^{+10\%}_{-5\%}$, the typical lifetime of solar-type prestellar cores, $t_{\rm pre}=1^{+0.5}_{-0.3}\,$Myr, and the efficiency of the conversion from prestellar core mass to stellar mass, $\epsilon_{\rm core}=30\%^{+20\%}_{-10\%}$, according to the simple relation: SFE${}_{\rm dense}^{\rm pre}=f_{\rm pre}\times\epsilon_{\rm core}/t_{\rm pre}=(4.5\pm 2.5)\times 10^{-8}\,{\rm yr}^{-1}$.

Acknowledgements.

We thank the referee for useful suggestions which 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. Based on observations carried out with the IRAM 30m Telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). 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.

Appendix A Complementary figures and tables

Figures 8 – 16 are complementary figures. The optical depths of the HCN (1–0) and HCO+ (1–0) lines in each $A_{\rm V}$ bin are listed in Table 8. Table 9 summarizes the definition of each notation used in this paper.