Multi-line modelling in nearby galaxies: The link between dense gas and star formation in different environments

. The J = 1–0 lines of HCN and HCO + have become the default tracers of dense molecular gas in external galaxies. To study the relation between HCN and HCO + emission and density, we have mapped their J = 3–2 lines with the ALMA Compact Array (ACA) in the nearby star-forming galaxies NGC3351, NGC3627, and NGC4321. Combined with archival line maps of HCN(1–0), HCO + (1–0), CO(1–0), and CO(2–1), we use this data set to: (1) compare the excitation of dense gas tracers in normal and starburst / AGN galaxies; (2) explore how line ratios vary across galaxy disks; (3) infer the density distribution and other cloud properties on a pixel-by-pixel basis through a physically-motivated radiative transfer model.


Introduction -The HCN and HCO + J=3-2 observations in context
The J=1-0 lines of HCN and HCO + have become the default dense gas tracers in external galaxies [1][2][3], but in some Galactic clouds their luminosities have a sizeable contribution from gas at low column densities [4,5].Thanks to their higher critical densities, the J=3-2 lines should be less a↵ected by low density gas.We have mapped with the ACA the HCN(3-2) and HCO + (3-2) emission in the inner ⇠2-4 kpc disks of the nearby star-forming galaxies NGC 3351, NGC 3627, and NGC 4321.The HCN  images are shown in Fig. 1-bottom.
We combine these data with archival interferometer images of the HCN(1-0), HCO + (1-0), CO(1-0), and CO(2-1) emission in each galaxy ( [6,7]).Fig. 2 compares our measurements of four HCN and HCO + line ratios (histograms) with the extragalactic literature (black circles), which for the J=3-2 lines is dominated by starburst and AGN samples.In the two leftmost panels, the significant overlap between the two data sets reminds us that HCN-to-HCO + ratios alone provide ambiguous diagnostics of AGN/starburst activity.In the two rightmost panels, our 3-2/1-0 ratios cluster in the lower range of the literature, as expected if the dense gas tracers were less excited in normal galaxies than in more luminous systems.

Empirical relations and radiative transfer modelling
In Fig. 3, we study the systematic variations across the targets of our two HCN/CO proxies for the dense gas fraction and the HCN 3-2/1-0 ratio.To first order, all panels suggest ) integrated intensity maps of our targets from PHANGS-ALMA [6] at ⇠1.5 00 resolution.The footprint of our ACA observations is shown in white.Bottom: ACA HCN(3-2) integrated intensity maps at 6 00 resolution (grey ellipses).The S/N≥5 threshold that we adopt for analysis is shown in white.Black contours represent the CO(2-1) integrated intensity at matching resolution.
that the dense gas fraction/excitation is enhanced when ⌃ ⇤ (an environment property; left) or hI CO i 1 00 440pc (an averaged, cloud-scale property; right) increase.The overall HCN(1-0)/CO(2-1) trends, in particular, are consistent with the fits by [2,3,7,17].In comparison, the HCN(3-2)/CO(2-1) relations are & 2 times steeper, which could make them useful to diagnose changes in cloud properties, such as density.Finally, the negligible galaxy-to-galaxy o↵sets in the hI CO i 1 00 440pc -vs-HCN(3-2)/HCN(1-0) panel (bottom-right) yield a neat relation between line ratios measured at ⇠400-pc scales and the sub-beam properties of the unresolved molecular clouds (also [17]).To get insights into the trends shown in Fig. 3 (also [2,3]), we fit our suite of maps with a physically-motivated radiative transfer model based on [18].We assume that, in each pixel: (1) the observed emission is a sum over a set of density layers that we treat as independent Large Velocity Gradient zones; (2) the layers are weighted by a lognormal+power-law density PDF inspired by ISM models and measurements of Galactic clouds [19]; (3) all layers share the same temperature and abundance per velocity gradient.The model yields maps of the PDF parameters (mean density, Mach number, index of the power-law high density tail), the gas temperature, and the chemical abundances.Fig. 4 shows some examples for NGC 4321.

Conclusions
Compared with those of their J=1-0 counterparts, the extragalactic measurements of HCN(3-2) and HCO + (3-2) are still scarce and dominated by starburst and AGN samples.Our new ACA observations are an example of how current mm-wavelength facilities can map these lines in the fainter but more representative population of main-sequence galaxies.
Multiline interferometer observations open a new opportunity to resolve the excitation con- .Maps of some gas properties inferred from the model in NGC 4321.We sample each beam (grey ellipse) with several pixels to assess model uncertainties (not discussed here).Contours correspond to 1.3 and 1.8 for the power-law index and to 20, 40, 60, and 80% of the maximum elsewhere.ditions of the (dense) gas across galaxy disks.Once fed into physically-motivated radiative transfer models, these data sets can help us constrain key physical and chemical parameters of the molecular clouds that are relevant to many ISM and star formation theories.

Figure 1 .
Figure 1.Top: CO(2-1) integrated intensity maps of our targets from PHANGS-ALMA [6] at ⇠1.5 00 resolution.The footprint of our ACA observations is shown in white.Bottom: ACA HCN(3-2) integrated intensity maps at 6 00 resolution (grey ellipses).The S/N≥5 threshold that we adopt for analysis is shown in white.Black contours represent the CO(2-1) integrated intensity at matching resolution.

Figure 1 .
Figure 1.Top: CO(2-1) integrated intensity maps of our targets from PHANGS-ALMA [6] at ⇠1.5 00 resolution.The footprint of our ACA observations is shown in white.Bottom: ACA HCN(3-2) integrated intensity maps at 6 00 resolution (grey ellipses).The S/N≥5 threshold that we adopt for analysis is shown in white.Black contours represent the CO(2-1) integrated intensity at matching resolution.

Figure 3 .
Figure 3. Line ratios at a common 440-pc resolution plotted on a pixel-by-pixel basis as a function of ⌃ ⇤ (stellar surface density; left) and hI CO i 1 00 440pc (PHANGS CO(2-1) intensity at ⇠ 1 00 resolution, weighted by itself and averaged over the 440 pc beam; right).Coloured symbols, bars, and lines show binned average trends per galaxy.Star symbols indicate bins that might be a↵ected by a completeness bias.

Figure 4
Figure 4. Maps of some gas properties inferred from the model in NGC 4321.We sample each beam (grey ellipse) with several pixels to assess model uncertainties (not discussed here).Contours correspond to 1.3 and 1.8 for the power-law index and to 20, 40, 60, and 80% of the maximum elsewhere.