© ESO, 2017

1. Introduction

The relationship between star formation (SF) and the supply of dense gas is of critical importance for our understanding of cosmic SF. We must develop a detailed picture of the relation between dense gas and SF in galaxies if we wish to explain the structure and evolution of galaxies (e.g., Somerville & Davé 2014). This relation can, for example, be explored in the Milky Way. In molecular clouds within ~ 500 pc from the Sun, one can estimate the star formation rate, Ṁ_⋆, by counting individual young stars. Nearby clouds can be resolved spatially, which also simplifies estimating the mass of gas at high density, M_dg. Recent research suggests defining M_dg as the mass residing at high visual extinctions, A_V ≥ A_V,dg with A_V,dg ≈ 7 mag, resulting in Ṁ_⋆ ∝ M_dg (e.g., Heiderman et al. 2010, Lada et al. 2010).

It is very challenging to study Ṁ_⋆ and M_dg in galaxies. One might, for example, assume that the light of young stars is absorbed and re-emitted by dust. Then the far-infrared luminosity of a galaxy (i.e., at wavelengths of 8 to 1000 μm) characterizes SF via Ṁ_⋆ ∝ L_FIR. Similarly, one might assume that a certain molecular emission line requires elevated densities to be excited. Then M_dg ∝ L_Q for line luminosities of a suitable transition Q. Gao & Solomon (2004b), in particular, introduced the HCN (J = 1–0) transition as a tracer of dense gas in galaxies (i.e., H₂ densities ≫ 10⁴ cm^-3), suggesting that M_dg ∝ L_{HCN (1–0)}.

This raises an important question: is M_dg as derived from A_V equal to M_dg as obtained from L_{HCN (1–0)}? The LEGO project (Molecular Line Emission as a Tool for Galaxy Observations; led by JK) uses wide-field maps to address such questions. Here we summarize key conclusions from a comprehensive study of Orion A (Kauffmann et al., in prep., hereafter Paper II).

Fig. 1 Maps of the peak intensity for transitions near 100 GHz. The left panel gives AV as inferred from Herschel data. Contours at 5 and 30 mag are drawn and repeated in all panels, and the peak intensity is stated for every transition. Line emission maps are smoothed to resolution before filled contours are drawn at signal-to-noise ratios of 3, 5, 10, 30, 50, and 100. Panels are ordered by increasing critical density, ncr.

2. Preparation of observational data

Data on emission lines at ~ 100 GHz frequency (Fig. 1) were obtained with the 14 m telescope of the Five College Radio Astronomy Observatory (FCRAO). Maps of the CCH (N = 1–0, J = 1 / 2–1/2), HCN (J = 1–0), N₂H⁺ (J = 1–0), C¹⁸O (J = 1–0), and CN (N = 1–0, J = 3 / 2–1/2) transitions are taken from Melnick et al. (2011). Data on the (J = 1–0) and ¹³CO (J = 1–0) lines are from Ripple et al. (2013). The full width at half maximum (FWHM) beam size for a given frequency ν is ϑ_beam = 52″·(ν/ 100 GHz)^-1. An efficiency η_mb = 0.47 is used for conversion to the main beam intensity scale, . This paper focuses on the integrated intensities, .

Dust-based estimates of the H₂ column density N(H2) are derived from Herschel observations of Orion at wavelengths of 250 to 500 μm (André et al. 2010) using modified methods from Guzmán et al. (2015) described in Kauffmann et al. (2017). We assume thin ice coatings and dust coagulation for 10⁵ yr at a molecular volume density of 10⁶ cm^-3 to select dust opacities from Ossenkopf & Henning (1994). Paper II describes how we calibrate these data against an extinction-based map from Kainulainen et al. (2011) to predict the visual extinction, A_V/ mag = N(H2) / 9.4 × 20²⁰ cm^-2, at a resolution of 38″.

We fit the filamentary cloud north of −5:14:00 (J2000) with a truncated cylindrical power-law density profile, n(r) = n_R·(r/R)^{− k}, where r is the distance from the filament’s main axis. We then obtain the median density along any line of sight for an offset s from the filament main axis, n_med(s). For given s, half of the mass resides above (and half below) this density, so that n_med(s) can be considered a representative density. Further algebraic operations relate s, n_med(s), and A_V(s) (Fig. 3; see Paper II).

3. Molecules as tracers of cloud material

We seek to explore molecular line emission under conditions that are representative for the Milky Way. We therefore ignore the region south of −5:10:00 declination (J2000). First, much of this region is subject to intense radiation emitted by young stars in the Orion Nebula. This is probably not typical for molecular clouds. Second, the well-shielded southerly regions (with dust temperatures ≤ 22 K) are devoid of embedded stars that are characteristic of SF regions (Megeath et al. 2012). Finally, we ignore pixels where A_V< 2 mag because of observational uncertainties.

3.1. Line emission per unit cloud mass

The line-to-mass ratio, h_Q = W(Q) /A_V, indicates how the emission from transition Q relates to the mass reservoir characterized by A_V. Given A_V ∝ N(H2), the ratio W(Q) /A_V essentially measures the intensity of line emission per H₂ molecule.

It is plausible to assume that h_Q is a function of n and therefore N(H2). This is, for example, expected if the molecular abundance or the excitation is a function of the density. This is explored in Fig. 2. For this analysis we sort the data into logarithmically spaced bins in A_V, and we then derive the mean of h_Q and its uncertainty from counting statistics in this bin. We see that h_Q is indeed a strong function of A_V, which justifies our ansatz to explore the trend of h_Q versus A_V. We normalize h_Q to a maximum value of 1 in the well-detected bins of Fig. 2 in order to simplify comparisons between molecular species.

The trend of h_Q versus A_V is non-trivial, and it differs between molecules. Most molecules start with a significant value of h_Q at low A_V, their line-to-mass ratio increases towards a maximum at an A_V of 5 to 20 mag, and h_Q steadily decreases with increasing A_V at even higher extinction. One single molecule defies this trend: the line-to-mass ratio of N₂H⁺ begins near or at zero at low A_V, and h_Q then begins to steadily rise at A_V ≳ 10 mag, possibly to level out (or decrease) at A_V ≳ 100 mag.

This is a critical result. This means that the N₂H⁺ (J = 1–0) transition is the only transition among those observed here that selectively traces gas at high (column) density. All other transitions are, by contrast, most sensitive to material at A_V ~ 10 mag. Pety et al. (2017) conclude the same in Orion B, using an argument that relates more to the following section.

Fig. 2 Normalized line-to-mass ratio, hQ, for the reference region. Shading indicates the uncertainty at a confidence level ≈ 68%, while gray dashes indicate the limits of bins. N2H+ is a good tracer of dense gas since hQ increases with increasing AV.

3.2. Characteristic (column) density traced by a line

Figure 2 characterizes whether or not a given molecular emission line traces the cloud material well under given conditions. It would be desirable if this information could be collapsed into a single number. One could, for example, attempt to establish the typical H₂ (column) density of material that is traced by a given transition. We use the line luminosities for this purpose. Integration over the map area at column densities corresponding to gives the luminosity as a function of the cutoff value , (1)where is the area element measured in pc². Let be the total luminosity. We then define the characteristic column density of transition Q to be the column density that contains half of the total line luminosity, (2)We then use the density model to define a characteristic density . We also obtain the characteristic column density for the spatial gas mass distribution, , by replacing W_Q with A_V in Eqs. (1)–(2). Figure 3 shows how and n_med increase towards deeper layers of the cloud.

Fig. 3 Top panel: cumulative fraction of emission for various transitions (and mass for dust) indicated by various colors. Dashed vertical lines indicate where selected transitions achieve . Shaded regions indicate uncertainties as described in Sect. 3.2. Bottom panel: estimated median density. Shading indicates confidence ranges of ± 10% and ± 40% around the median estimate.

This analysis is greatly influenced by the choice of the region analyzed. For example, it is generally known that most of a cloud’s mass resides at low column densities. The field north of −5:10:00 declination considered here does, however, not fulfill this condition. For example, in the area shown in Fig. 1, while holds for the entire Orion A cloud. We therefore use the A_V map from Kainulainen et al. (2011) to correct for this relative lack of lower-density material. Specifically, we interpolate the binned information on h_Q (i.e., W [ Q ] vs. A_V) summarized in Fig. 2 to derive a predicted value of W(Q) for every pixel in the Kainulainen et al. map. We then use these predicted W(Q) in Eqs. (1)–(2) to derive predictions of and for the entire Orion A region. We do not treat CCH because of overly large uncertainties in h_Q.

We use the observed values of h_Q from Fig. 2 if the measurements exceed their uncertainty by a factor ≥ 3. Around this detection limit we use linear fits to h_Q versus lg(A_V) to determine the A_V for which h_Q = 0. We then use linear interpolation in h_Q between the point where the transition is safely detected and the point where the emission is predicted to vanish. The latter point has an uncertainty resulting from the aforementioned linear fit. Variation of this point changes and thereby . Further, one might assume that the actual mass distribution (i.e., dM/ dA_V) might actually deviate from the one derived from the Kainulainen et al. map. Here we explore a scenario in which we vary dM/ dA_V by a factor 2 up and down at A_V = 2 mag, leave dM/ dA_V unchanged at A_V ≥ 10 mag, and interpolate linearly at intermediate A_V. Figure 3 shows how extremes in both these modifications might influence the results for HCN and N₂H⁺.

Figure 3 recovers the trends already seen in Fig. 2: most transitions trace lower-density material and have , where for HCN (J = 1–0). The only exception is N₂H⁺ (J = 1–0) with . This transition is the only true tracer of higher column densities. Pety et al. (2017) studied in Orion B how cloud sectors at different A_V contribute to L_Q. They do not calculate , but their results seem broadly consistent with ours.

4. Tracing the dense gas in star forming galaxies

4.1. HCN as a tracer of moderately dense gas

The constraints on h_Q = W(Q) /A_V and are of critical importance for the study of star-forming galaxies. For example, Gao & Solomon (2004b) speculate that gas at densities ≳ 3 × 10⁴ cm^-3 is traced by emission in the HCN (J = 1–0) transition. More recently, Usero et al. (2015) assumed threshold densities as large as 3 × 10⁵ cm^-3 (they deem 10^4to5 cm^-3 likely), while Jimenez-Donaire et al. (2016) estimate threshold densities ≥ 5 × 10⁵ cm^-3 from H¹³CN–to–H¹²CN line ratios in galaxies. More generally, it is often argued that the high critical density of the HCN (1–0) line, n_cr = 1 × 10⁶ cm^-3, implies that this transition traces gas of very high density. However, the analysis presented here shows that , which suggests that . This does not fundamentally question the interpretation of trends like the Gao & Solomon relation, but it critically affects the detailed analysis of data.

The low value of is not entirely surprising. Evans (1999) and Shirley (2015; also see Linke et al. 1977) point out that HCN should become detectable at “effective” densities n_eff ≈ (1 to 3) × 10⁴ cm^-3 for gas at 10 K and regular abundances, for which n_eff ≪ n_cr. Further, HCN can be excited by electrons at H₂ densities ≪ n_cr if fractional electron abundances X(e−) > 10^-5 prevail (Goldsmith & Kauffmann 2017, following a suggestion by S. Glover). Here we provide solid observational evidence supporting such work.

Critical densities simply do not control how line emission couples to dense gas. This is already evident from Fig. 1.

The low characteristic density ≈ 870 cm^-3 for the HCN (1–0) line has important implications for modeling. Theoretical studies often relate Ṁ_⋆ and M_dg via the free-fall time at density , , and a SF efficiency, ε_SF ≤ 1, via Ṁ_⋆ = ε_SF·M_dg/τ_ff. A landmark paper by Krumholz & Tan (2007), for example, assumes , infers ε_SF ≈ 0.01, and concludes that SF is “slow” in regions sampled by HCN (1–0). Our measurements indicate that is a factor ≈ 70 smaller, τ_ff a factor ≈ 70^{1 / 2} ≈ 8 larger, and SF thus by a similar factor more efficient and “faster”. Determinations of for HCN and other molecules are thus of essential importance for SF theory.

Fig. 4 Star formation in the Milky Way and galaxies. The reference relation for the Milky Way does not describe galaxies. This might hint at unknown reservoirs of HCN emission.

4.2. Galactic vs. extragalactic star formation relations

We initially set out to investigate whether M_dg as derived from A_V is equal to M_dg as obtained from L_{HCN (1–0)}. We now return to this question. Lada et al. (2010) argue that Ṁ_⋆ ∝ M_dg if M_dg is calculated as the cloud mass residing at A_V ≳ A_V,dg = 7 mag. In Sect. 3.2 we show that for the HCN (1–0) transition, similar to A_V,dg. This suggests that about half of L_HCN(1–0) originates directly in the dense star-forming gas of galaxies. The remaining fraction of L_HCN(1–0) does not directly trace M_dg. Still, this emission might be an efficient probe of the gas surrounding and shaping M_dg. From this perspective one might postulate (3)where α_HCN(1–0) is a constant. Gao & Solomon (2004b), for example, suggested that α_HCN(1–0) ≈ 10 M_⊙/ (K km s^-1 pc2), based on simple models. But α_HCN(1–0) has never been estimated using observations, in particular not down to densities < 10³ cm^-3 that we suggest are traced by the HCN (1–0) line.

We estimate , following the procedure described in Sect. 3.2. Recall that this is a lower limit since we cannot predict W_Q for A_V< 2 mag (Sect. 3). We further derive M_dg ≈ 1.6 × 10⁴M_⊙ by evaluating the mass of material residing at A_V ≳ 7 mag in the Kainulainen et al. (2011) extinction map. We thus find α_HCN(1–0) ≲ 20 M_⊙/ (K km s^-1 pc2). This is in good agreement with modeling by Gao & Solomon (2004b); but for the wrong reasons, given their models essentially assume , which exceeds the true value by a factor ≈ 30. Shimajiri et al. (2017) estimate α_HCN(1–0) ≈ 10 M_⊙/ (K km s^-1 pc2) from observations of Aquila, Ophiuchus, and Orion B. Their work assumes a scaling factor to include gas at A_V< 8 mag. Our work differs from theirs in that we actually measure this factor (Fig. 3) while Shimajiri et al. implement a sophisticated treatment of interstellar radiation fields.

In Fig. 4 we use our new observational determination of α_HCN(1–0) to compare SF in the Gao & Solomon (2004a) galaxies to SF in molecular clouds near the Sun (Lada et al. 2010) and in the Galactic Center (GC; Longmore et al. 2013; Kauffmann et al. 2017). For the galaxies we adopt Ṁ_⋆ = β_FIR·L_FIR with (Eq. (4) and the offset from Fig. 3 of Murphy et al. 2011). Lada et al. (2010) suggest a reference relation describing SF rates in clouds within ~ 500 pc of the Sun, Ṁ_⋆,MW = (4.6 ± 2.6) × 10^-8M_⊙ yr^-1·(M_dg/M_⊙), from which GC clouds appear to deviate by a factor ~ 10.

Figure 4 also shows that galaxies deviate from Ṁ_⋆,MW by an average factor ⟨ Ṁ_⋆,MW/Ṁ_⋆ ⟩ ≲ 4.5. Given that ⟨ Ṁ_⋆,MW/Ṁ_⋆ ⟩ ∝ α_HCN(1–0), could α_HCN(1–0) in galaxies be smaller than estimated here? Significant contributions to L_HCN(1–0) from reservoirs outside those considered here, for example from diffuse cloud envelopes, could indeed reduce α_HCN(1–0) = M_dg/L_HCN(1–0).

5. Summary

We study the relationship between various emission lines and dense gas. This analysis is based on observations of various molecules at frequencies near 100 GHz (, , C¹⁸O, CN, CCH, HCN, and N₂H⁺). We focus on the HCN (1–0) transition, for which we find that it typically traces gas at , corresponding to a characteristic H₂ density ≈ 870 cm^-3 (Sect. 3.2). The only molecular transition clearly connected to dense gas is the N₂H⁺ (1–0) transition, characteristic of A_V ≈ 16 mag and densities ≈ 4000 cm^-3. The low characteristic densities derived for the HCN (1–0) line are about two orders of magnitude below values commonly adopted in extragalactic research (Sect. 4.1). This impacts theoretical discussions of SF trends in galaxies. We use this new knowledge on the emission from HCN to compare SF in galaxies to SF in the Milky Way (Sect. 4.2). The comparisons indicate that galaxies either deviate from SF relations holding in the Milky Way, or hitherto unknown reservoirs of emission contribute to L_HCN(1–0).

Acknowledgments

We thank a knowledgeable anonymous referee for helping to significantly improve the paper. This research was conducted in part at the Jet Propulsion Laboratory, which is operated by the California Institute of Technology under contract with the National Aeronautics and Space Administration (NASA). A.G. acknowledges support from Fondecyt under grant 3150570.

References

All Figures

Fig. 1 Maps of the peak intensity for transitions near 100 GHz. The left panel gives AV as inferred from Herschel data. Contours at 5 and 30 mag are drawn and repeated in all panels, and the peak intensity is stated for every transition. Line emission maps are smoothed to resolution before filled contours are drawn at signal-to-noise ratios of 3, 5, 10, 30, 50, and 100. Panels are ordered by increasing critical density, ncr.

In the text

	Fig. 2 Normalized line-to-mass ratio, hQ, for the reference region. Shading indicates the uncertainty at a confidence level ≈ 68%, while gray dashes indicate the limits of bins. N2H+ is a good tracer of dense gas since hQ increases with increasing AV.
In the text

	Fig. 3 Top panel: cumulative fraction of emission for various transitions (and mass for dust) indicated by various colors. Dashed vertical lines indicate where selected transitions achieve . Shaded regions indicate uncertainties as described in Sect. 3.2. Bottom panel: estimated median density. Shading indicates confidence ranges of ± 10% and ± 40% around the median estimate.
In the text

	Fig. 4 Star formation in the Milky Way and galaxies. The reference relation for the Milky Way does not describe galaxies. This might hint at unknown reservoirs of HCN emission.
In the text