PRUSSIC II - ALMA imaging of dense-gas tracers in SDP.81 Evidence for low mechanical heating and a sub-solar metallicity in a z=3.04 dusty galaxy

We present deep ALMA Band 3 observations of the HCN, HCO + , and HNC (4–3) emission in SDP.81, a well-studied z = 3 . 042 strongly lensed galaxy. These lines trace the high-density gas, which remains almost entirely unexplored in z ≥ 1 galaxies. Additionally, these dense-gas tracers are potentially powerful diagnostics of the mechanical heating of the interstellar medium. While the HCN(4–3) and HNC(4–3) lines are not detected, the HCO + (4–3) emission is clearly detected and resolved. This is the third detection of this line in a high-redshift star-forming galaxy. We find an unusually high HCO + / HCN intensity ratio of ≥ 2 . 2. Based on the photodissociation region modelling, the most likely explanation for the elevated HCO + / HCN ratio is that SDP.81 has low mechanical heating – less than 10% of the total energy budget – and a sub-solar metallicity, Z ≈ 0 . 5 Z ⊙ . While such conditions might not be representative of the general population of high-redshift dusty galaxies, lower-than-solar metallicity might have a significant impact on gas masses inferred from CO observations. In addition, we report the detection of CO(0–1) absorption from the foreground lensing galaxy and CO(1–0) emission from a massive companion to the lensing galaxy, approximately 50 kpc to the southeast.


Introduction
One of the key questions in astrophysics is understanding the process of star formation, namely, exploring how cold, molecular gas is converted into newborn stars.Extensive surveys have revealed that the star-forming activity of the Universe peaks between redshifts z = 2 − 4. Beyond z ≈ 1, cosmic star formation is dominated by dust-obscured, star-forming galaxies (DS-FGs) with star formation rates (SFRs) of a few hundred to few thousand M ⊙ /yr and far-infrared (FIR) luminosities of ≥ 10 12 L ⊙ (Casey et al. 2014;Zavala et al. 2021).
In this work, we also consider how the immense SFRs of DS-FGs are linked to their molecular gas reservoirs.Current studies of cold gas in high-z DSFGs have focused mainly on the bright CO, [C ii], and [C i] emission lines (see reviews by Carilli & Walter 2013;Hodge & da Cunha 2020).However, these lines trace gas down to densities of 10 2 to 10 3 cm −3 , well below the typical densities of star-forming clouds.Crucially, they provide little to no insight into some of the key ingredients of star formation; namely, the high-density gas (n ≥ 10 4 cm −3 ) and corresponding physical conditions.In particular, due to their intense SFRs, molecular clouds in DSFGs will be exposed to numerous shocks from supernovae, outflows from young stellar objects, and winds from massive stars injecting significant amounts of energy into the gas (Loenen et al. 2008).Indeed, mechanical heating can play a significant role in gas heating in nearby ultraluminous infrared galaxies (ULIRGs) as shown, for example, by Rosenberg et al. (2015).
Assessing the relative contribution of mechanical heating to the energy budget of DSFGs is challenging and limited to indirect probes.Recent studies of lensed DSFGs have found them to be overluminous in high-J CO emission, which has been attributed to mechanical heating (Riechers et al. 2021, however, c.f. Butler et al., in prep.).Similarly, radiative transfer modelling of well-sampled CO ladders of Planck-selected DSFGs has found significant non-thermal excitation (Harrington et al. 2021).However, using high-J CO lines as mechanical heating tracers is fraught with significant uncertainties, as the high CO excitation can be caused by a number of diverse mechanisms, such as cosmic-ray or X-ray heating.
A more direct tracer of the dense gas and the associated physical conditions is high dipole moment molecules such as HCN, HCO + , and HNC.Surveys of local galaxies have shown that the HCN(1-0) luminosity correlates linearly with the star-formation rate over eight orders of magnitude, from individual molecular clouds to entire galaxies (e.g.Gao & Solomon 2004;Wu et al. 2005;Bigiel et al. 2015;Jiménez-Donaire et al. 2019).Extending these z ≈ 1 studies to high redshift remains challenging, as the HCN emission can be more than 10× fainter than CO.Despite almost two decades of effort, only three z ≥ 1 DSFGs have been detected in HCN(1-0) (Gao et al. 2007;Oteo et al. 2017;Rybak et al. 2022).Indeed, as recently shown by Rybak et al. (2022), DSFGs might have low dense-gas fractions, making the HCN(1-0) emission even harder to detect.
An alternative to observing the ground-state transitions of HCN/HCO + /HNC are the mid-J transitions.The mid-J lines are both intrinsically brighter and, at high redshift, conveniently fall into the easily accessible 3-mm atmospheric window (Wagg et al. 2005).The mid-J lines have been proposed to be better tracers of dense gas than HCN(1-0) (Krips et al. 2008;Viti 2017), which is often associated with densities much lower than its nominal critical density (Kauffmann et al. 2017;Jones et al. 2023), On the other hand, mid-J transitions might be sensitive to, for example, mechanical heating or mid-infrared pumping of the vibrational modes (Aalto et al. 2007;Kazandjian et al. 2012).
To expand the number of high-z galaxies detected in densegas tracers, we launched Prussic2 -a comprehensive census of dense-gas tracers in high-redshift star-forming galaxies.In the first Prussic paper, Rybak et al. (2022) presented the Karl G. Jansky Very Large Array (JVLA) observations of the J upp = 1 HCN, HCO + , and HNC emission in six z ∼ 3 lensed DSFGs, finding low dense-gas fractions and elevated dense-gas star-forming efficiencies.
In this second paper of the Prussic series, we present deep, spatially resolved Atacama Large Millimeter / Sub-millimeter Array (ALMA) observations of the HCN/HCO + /HNC(4-3) emission in SDP.81, a z = 3.042 gravitationally lensed DSFG3 .Using photodissociation region (PDR) modelling, we constrain the range of mechanical heating and metallicity in this galaxy.We also use the high fractional bandwidth of our observations to explore the environment of the foreground lensing galaxy.

ALMA Band 3 observations of SDP.81
The ALMA Band 3 observations of SDP.81 consist of three different programmes, running from 2017 to 2021 and spanning multiple configurations.Table 2 lists individual observations.We now discuss the details of individual observing runs.
The project 2016.1.00663.S (Cycle 4, PI: Rybak) was carried out in a very extended configuration with baselines extending out to 7.5 km.The CO(3-2) line covered by this data was presented by Rybak et al. (2020).In addition to the CO(3-2) line, the observations covered the HCN(4-3) and HCO + (4-3) emission, but not the HNC(4-3) line.The spectral setup consisted of two spectral windows (SPWs) centred at 85.73 and 87.68 GHz that were configured with a spectral resolution of 3.90625 and 7.81250 MHz, with a total width of 1.875 GHz each.The other two SPWs were centred at 97.72 and 99.72 GHz, with a resolution of 15.625 MHz and a total bandwidth of 2.0 GHz.The project 2018.1.00747.S (Cycle 6, PI: Rybak) was carried out in an extended configuration with baselines out to 3.6 km.The observations were taken in two batches, one in the summer of 2019 and another in the summer of 2021.The spectral setup was the same as above.
Unfortunately, the 2021 data were taken in bad weather conditions with copious wet clouds.The data quality was marginal; the phase calibration could not be performed successfully, and the 'check' sources were not detected.Consequently, we have excluded the 2021 observations from our analysis.
Observations for project 2017.1.01694(PI: Oteo) were taken in a compact configuration, with a maximum baseline length of 500 m.The spectral setup differed from the previous two programmes: namely, it used two SPWs centred at 87.988 GHz and 89.688 GHz, with a resolution of 15.625 MHz.This is the only setup that covers the HNC(4-3).
The data were reduced using the standard ALMA pipeline and Casa versions 4.7 and 5.4 (McMullin et al. 2007).After concatenating all the data, we re-calculate the noise on individual visibilities using casa's statwt task; this ensures that the noise is estimated consistently for all the scheduling blocks.For the frequency range covering the HCN(4-3) and HCO + (4-3) lines, the resulting dataset totals 226 min on-source and provides sensitivity to spatial scales between 0.29 and 46 arcsec at 88.7 GHz.The HNC(4-3) line is only covered by the compact-array observations with 51 min on-source; the array configuration provides sensitivity to spatial scales between 2.1 and 46 arcsec at 88.7 GHz.The primary beam FWHM was 65 arcsec. 4While SDP.81 is detected in the XMM-Newton X-ray observations (Ranalli et al. 2015), this emission is likely associated with the bright AGN in the z = 0.299 lensing galaxy.

Imaging
To image the Band 3 continuum and the three emission lines, we used Casa's tclean task.We first imaged the continuum using the line-free channels (86.0-88.5 GHz and 90.0-101.0GHz) and a manually drawn mask, cleaning down to 1.5σ (Fig. 1).We set the tclean's parameter fastnoise=False to properly recalculate the noise per baseline and channel.
To image the dense-gas tracers, we subtracted the continuum emission using the uvcontsub task, fitting a constant flux to the line-free part of the spectrum.Due to the faintness of the lines, we produced dirty images only (i.e.without any deconvolution).For the HCN(4-3) and HCO + (4-3) lines, the combination of discrepant array configurations produces a dirty beam with very strong sidelobes; we mitigate this effect by using different uvplane tapers with a final beam FWHM of ∼0.85"×0.94".No taper was applied to the HNC(4-3) data as it was taken with a compact configuration.To examine the data in the spectral dimension, we created dirty-image cube of the entire dataset.Figure 2 shows the resulting spectrum extracted from the main lensing arc.

HCN, HCO + , and HNC emission
As shown in Fig. 1, the HCN(4-3) and HNC(4-3) lines are undetected in the dirty-image maps.On the contrary, the HCO + (4-3) line is clearly detected and resolved: the peak signal-to-noise ratio (S/N) is 4.3σ and we see ≥ 3σ excess emission over 4-5 beams.The HCO + (4-3) emission is located at the centre and north of the main Einstein arc; the southern part of the arc and the counter-image (to the West) are not detected.This is only the third detection of the HCO + (4-3) in a high-redshift DSFG, after ACT J2029+0120 (z = 2.64, Roberts-Borsani et al. 2017) and G244.8+54.9(z = 3.01, Cañameras et al. 2021); additionally, HCO + (4-3) was detected in the Cloverleaf quasar (z = 2.56, Riechers et al. 2011b).Compared to the dust continuum, the HCO + (4-3) emission seems to be concentrated to the north; however, given the low S/N of our data, the HCO + (4-3) morphology can not be reliably assessed.The low S/N also precludes a reliable reconstruction of the HCO + (4-3) emission in the source plane.
We extracted the HCO + (4-3) line flux and upper limits for HCN(4-3) and HNC(4-3) from a manually drawn aperture over the main lens arc (note: the contribution from the counterimage is negligible).The aperture area corresponds to ≈6 beams for HCN and HCO + , and 1.5 beams for HNC.For HCN and HNC fluxes, we adopt the corresponding 3σ upper limits.To calculate the line luminosities, we multiplied the measured fluxes by a factor of 1.31 (a ratio between the total line flux and the flux contained within the line FWHM).Table 1 lists the derived skyplane luminosities.
Figure 3 puts our HCN(4-3) and HCO + (4-3) measurements in the context of z = 0 studies (Zhang et al. 2014;Tan et al. 2018) and the predicted L FIR − L ′ relations from Zhang et al. (2014)  5 .The HCO + (4-3) is ≈1σ above the local trend, whereas the HCN(4-3) upper limit falls close to the Zhang et al. (2014) trend.We do not show a similar plot for the HNC(4-3) line due to the limited number of detections at z ∼ 0.
While different spatial distributions of the tracers might result in differential magnification, we consider this effect to be 5 Note: the correct Zhang et al. ( 2014) L FIR -L ′ HCO + correlation should be log L FIR = (1.12 ± 0.05) × log L ′ HCO + (4−3) + 2.83 ± 0.34 (Z.Zhang, priv.comm.). 6Specifically, we consider the nine galaxies from Israel (2023) which are detected in all HCN J upp = 1 − 5 lines.limited: HCO + and HCN are expected to be co-spatial on kpcscales and previous modelling of dust continuum and CO and [C ii] emission in SDP.81 found magnification factors varying by ≤15% (Rybak et al. 2015a,b;Swinbank et al. 2015;Rybak et al. 2020), which are too small to explain the observed high HCO + /HCN ratio.

Photon-dissociation region modelling
To investigate which physical conditions cause the high HCO + /HCN(4-3) ratio, we used photon-dissociation region (PDR) modelling.Specifically, we explored the impact of varying mechanical heating and gas-phase metallicity7 .We adopted the PDR models of Kazandjian et al. (2012Kazandjian et al. ( , 2015) ) which are an extension of the Leiden PDR-XDR models of Meijerink & Spaans (2005).These PDR models assume a HCN /L ′ CO(1−0) and L ′ HCN /L FIR ratios for highredshift galaxies, with SDP.81 datapoints denoted by stars.We include measurements for individual galaxies (Danielson et al. 2013;Oteo et al. 2017;Béthermin et al. 2018;Cañameras et al. 2021;Rybak et al. 2022, bullet points) and stacks (Reuter et al. 2022;Rybak et al. 2022;Hagimoto et al. 2023).We convert the J upp ≥ 2 luminosities to HCN(1-0) following the z ∼ 0 HCN ladder from Israel (2023).The upper limits on HCN(4-3) emission in SDP.81 are consistent with recent works indicating low HCN/CO and HCN/FIR ratios in DSFGs.semi-infinite parallel-plane slab morphology illuminated by external UV radiation, alongside X-ray and cosmic ray (CR) contributions.The models span a wide range of metallicity (0.1-2.0 Z ⊙ ) and mechanical heating 8 .The latter is parametrised by a factor α, the ratio of the mechanical and photoelectric heating at the surface of the cloud.Therefore, α = 0 corresponds to no mechanical heating, while α = 1 implies that mechanical heating is equal to photoelectric heating.The incident radiation field spectral energy distribution and the corresponding mechanical heating are derived assuming a Salpeter stellar initial mass function.The mechanical heating is assumed to be due to supernovae shock dissipation only and is implemented following the Loenen et al. (2008) prescription, neglecting the contributions from young stellar objects (which are very short-lived) and stellar winds (which contribute ≤6% of the total mechanical heating, Kazandjian et al. 2012); the energy is injected uniformly throughout the cloud volume.Given the lack of evidence for an AGN in SDP.81, we assume all radiation is due to star formation.
For the comparison with models, we used the following dense-gas line ratios: 3, and L ′ HCO + (4−3) /L ′ HCN(1−0) ≥ 3.8 (see Table 1).For further constraints, we also included the CO(5-4) line, which is shown to closely trace the dust continuum emission (Rybak et al. 2015b(Rybak et al. , 2020)).We did not include the CO(8-7) and (10-9) lines, which are concentrated in the northern part of the sources, and the CO(3-2), whose S/N value is too low to be useful for PDR modelling.We conservatively assume 50% errors on the line ratios to account for the flux calibration uncertainties and potential spatial offsets between different tracers.
Figure 6 shows the regions in the G-n space that are consistent with these constraints.The models with Z = 0.1 Z ⊙ and 2.0 Z ⊙ are effectively excluded by the observations.Models with Z = 1 Z ⊙ are consistent with the data for α =5 -10%, but imply unrealistically low gas densities (≤ 10 2 cm −3 ) that are ruled out by the existing mid-and high-J CO detections.
However, lowering the metallicity to Z = 0.5 Z ⊙ allows the data to be reproduced over a wide range of G, n, and α.Lowering the metallicity even further to Z = 0.2 Z ⊙ and 0.1 Z ⊙ does not yield any physical solutions.In the lower right panel of Fig. 6, we show the histogram of all α for all Z = 0.5 Z ⊙ models consistent with the observed dense-gas tracers line ratios (grey).If we restrict the parameter space to the G, n range from the 200-pc resolution PDR modelling from Rybak et al. (2020), only models with low mechanical heating (α=0-10%) are left.When we repeated our analysis including the CO(8-7) line (which is concentrated in the northern part of the source), the only feasible models are Z = 0.5 Z ⊙ and α = 0 − 5%.We therefore conclude that the most direct explanation of the high HCO + /HCN ratio in SDP.81 is a combination of a lower-than-solar metallicity and low (or outright negligible) mechanical heating.

Mechanical heating in high-z dusty star-forming galaxies
Several recent studies have argued for a significant mechanical heating in DSFGs due to large cloud-scale turbulence.In particular, Riechers et al. (2021) found that high-z DSFGs are overluminous in CO(9-8) emission compared to z ∼ 0 empirical trends and argued that this excess luminosity is due to mechanical heating.Harrington et al. (2021) claimed the presence of significant mechanical heating in lensed Planck-selected, starburst DSFGs, on the basis of radiative transfer modelling of CO ladders.However, mechanical heating is not the only mechanism that can produce highly excited CO ladders and other large-sample studies have not found DSFGs to be overluminous in CO(9-8) (K.Butler, priv. comm.).
Our PDR modelling shows that the HCO + /HCN/HNC emission in SDP.81 is consistent with little to no mechanical heating.A similar conclusion for this source was reached by Rybak et al. (2020) based on the analysis of dust, [C ii], and CO (3-2) to (10-9) lines.We note that unlike the Riechers et al. (2021) sample, SDP.81 is not overluminous in the high-J CO emission and its L FIR − L ′ CO(10−9) ratio is consistent with local star-forming galaxies (Liu et al. 2015, see also Kamenetzky et al. 2015).Also, compared to the Harrington et al. (2021) sample, which is composed of the most extreme DSFGS with SFRs of few 1000 M ⊙ /yr, SPD.81 has a SFR of 400 M ⊙ /yr, and might be inherently less turbulent as a result.Nevertheless, radiative transfer analysis indicates that the HCN and HCO + ladders are much more sensitive to mechanical heating than the CO emission (Kazandjian et al. 2015).Future studies of the HCN, HCO + , and HNC ladders in large samples of high-redshift galaxies will be necessary to properly assess the role of mechanical heating in their ISM thermodynamics and how it varies with galaxy properties.

Metallicity estimates for high-z dusty galaxies
Gas-phase metallicity measurements in DSFGs are very sparse, as the widely-used rest-frame optical and near-infrared indicators suffer from extreme dust obscuration (Maiolino & Mannucci 2019).Consequently, a different set of tracers (unaffected by the dust) is necessary to measure the metallicity of obscured, submm bright galaxies.
One option is to use emission lines in the rest-frame farinfrared spectrum.The potential of far-infrared lines as a metallicity tracer was first studied by Nagao et al. (2011), who used radiative transfer models of H ii regions to predict the dependence on different fine-structure lines on the gas-phase metallicity.Nagao et al. (2011) found that the combination of the [O iii] 52and 88-µm lines and the [N iii] 57-µm line provides good metallicity diagnostics.Similar conclusions were reached by Pereira-Santaella et al. (2017), who also used the Herschel observations of O and N fine-structure lines to measure metallicity in a sample of highly obscured z ∼ 0 ULIRGs, obtaining values between Z = 0.7 − 1.5 Z ⊙ .
At high redshift, Wardlow et al. (2017) used Herschel observations of the [O iii] 52-µm and [N iii] 57-µm lines to infer a metallicity of ≥1 Z ⊙ for a stacked spectrum of 13 lensed DS-FGs9 .Alternatively, Rigopoulou et al. (2018) used the [O iii] 88µm / [N ii] 122-µm line ratios in combination with far-IR flux ratios to estimate the gas-phase metallicity in three z = 2 − 3 galaxies.These latter authors inferred metallicity ranges of Z = 0.6 − 1.0 Z ⊙ for HLSW01 (z = 2.96) and Z = 0.7 − 1.1 Z ⊙ for J02399 (z = 2.80).Their third source is SDP.81, for which they obtained only a weak upper limit Z ≤ 2 Z ⊙ , consistent with our value.
Finally Ultimately, our understanding of the metallicity distribution in the DSFG population will soon be revolutionised by the nearand mid-infrared spectroscopy with the James Webb Space Telescope.
As illustrated in Fig. 7, the Z = 0.5 Z ⊙ metallicity inferred in SDP.81 falls below the metallicities derived by Wardlow et al. (2017), but within the range inferred for ALESS 73.1 Fig. 6.Far-UV radiation (G) and gas density (n) from PDR models, with constraints from the HCN, HCO + , HNC, and CO(5-4) observations.Individual panels show models for Z=0.1, 0.2, 0.5, 1.0, 2.0 Z ⊙ .Different colours denote the different levels of mechanical heating contribution α (0% to 100%); the coloured squares denote models consistent with the observed line ratios within 50%; grey shaded contours show the G and n inferred from high-resolution imaging of SDP.81 by Rybak et al. (2020).Only the Z=0.5 Z ⊙ , α=0, 10% models are consistent with the data.The most direct interpretation is that SDP.81 has sub-solar metallicity and only limited mechanical heating.and SPT0418-47.Further support for the sub-solar metallicity in SDP.81 comes from HCO + /HCN surveys of nearby star-forming regions: for example, the Large Magellanic Cloud (LMC, Z ≈ 0.5 Z ⊙ ) has HCO + (4-3)/HCN(4-3) ratios ≥1 throughout most of its volume, with some sub-regions of 30 Doradus and N159W star-forming regions having HCO+/HCN≥5 (Anderson et al. 2014;Galametz et al. 2020).
We further consider what the impacts of less-than-solar metallicities might be when inferring physical properties of DS-FGs.In particular, one potential issue is related to inferring the molecular gas masses from low-J CO emission.This conversion is dependent on the gas-phase metallicity: the CO-to-H 2 conversion factor α CO can increase rapidly with falling metallicity (see Bolatto et al. 2013); depending on the model, α CO increases by a factor of 2 to 5 (Israel 1997;Narayanan et al. 2012).The value of α CO in high-redshift DSFGs remains controversial, with inferred values ranging from ∼ 0.8 (e.g.Calistro Rivera et al. 2018;Frias Castillo et al. 2023) to ∼6; in fact, it is likely that DSFGs span a wide range of α CO (Harrington et al. 2021).However, as the bulk of DSFG studies assumed dusty galaxies to have (super)solar metallicity and α CO between 0.8 and ≈4, assuming a single α CO value for the DSFG population might cause a significant underestimation of gas masses in Z ≤ 1 Z ⊙ DSFGs.
Finally, we note that in our PDR models, changing metallicity values merely ends up scaling the C, N, O abundances with respect to the solar values.However, this is not strictly physical, as the relative elemental abundances can vary with metallicity and the stellar population of the galaxy.For example, the empirical model of Dopita et al. (2016) (based on Galactic data) gives an N/O abundance ratio of ≈0.06 at 0.5 Z ⊙ and 0.1 for 1.0 Z ⊙ .Similarly, the different production and destruction mechanisms for C, N, and O result in a strong variation of their relative abundances as the initial stellar population ages (e.g.Maiolino & Mannucci 2019) with O/N and O/C abundance ratios decreasing as the starburst ages.The overabundance of O in young starbursts will be even more pronounced for a top-heavy stellar IMF, as has been claimed for DSFGs (e.g.Zhang et al. 2018b,a).However, such a detailed analysis is beyond the scope of this paper.

Conclusions
We present deep ALMA observations of the mid-J HCN, HCO + , and HNC emission in SDP.81, a well-studied z ∼ 3 lensed dusty galaxy.Combining multi-epoch imaging, we obtained a robust detection of the HCO + (4-3) emission, the third reported detection of this line in a high-redshift dusty galaxy.The upper limits on the HCN(4-3)/CO(1-0) and HCN(4-3)/FIR ratios in SDP.81 are consistent with upper limits derived from HCN(1-0) observations from Rybak et al. (2022).
The simultaneous non-detections of the HCN(4-3) and HNC(4-3) lines imply a significantly elevated HCO + /HCN luminosity ratio, making SDP.81 an outlier among extragalactic sources.Using a grid of PDR models, we find that the HCO + , HCN, HNC, and CO observations of SDP.81 are consistent with a low amount of mechanical heating (0-10% of the total energy input).This contradicts recent estimates based on high-J CO emission studies of high-z DSFGs (Harrington et al. 2021;Riechers et al. 2021); however, the HCN/HCO + /HNC lines are expected to be more direct tracers of mechanical heating (Loenen et al. 2008;Kazandjian et al. 2015).
Our PDR modelling also indicates that SDP.81 has a subsolar metallicity (Z = 0.5 Z ⊙ ).This result is lower than typically assumed for high-redshift dusty galaxies, but within the range spanned by analyses of fine-structure lines for other redshift 2-5 sources.
We stress that the sub-solar metallicity in SDP.81 might not be representative of the DSFG population in general, as observations of mid-J HCN and HCO + lines in high-z galaxies generally imply HCO + /HCN ratios closer to one (Béthermin et al. 2018;Cañameras et al. 2021, Rybak et al., in prep.).Nevertheless, if a fraction of DSFG have sub-solar gas-phase metallicity, this might have implications for the systematics of inferring their gas masses via the α CO conversion factor, among other processes.

Fig. 1 .
Fig. 1.ALMA narrow-band images of SDP.81.Upper left: Rest-frame 847-µm continuum at native resolution; the main Einstein arc and the counter-image are clearly visible.The point-source emission in the centre of the image is from the previously identified AGN in the lensing galaxy at z = 0.299.Upper right and lower panels: Narrow-band images at the systemic frequencies of the HCN(4-3), HCO + (4-3), and HNC(4-3) lines.The images are collapsed over 430 km s −1 bandwidth.Contours start at ±2σ, with a 1σ increment.The significantly larger beam in the HNC image is caused by the lack of long-baseline observations at this frequency.The HCN(4-3) and HNC(4-3) lines are not detected, but HCO + (4-3) is clearly detected and resolved.

Fig. 2 .
Fig.2.ALMA Band 3 spectrum of SDP.81, extracted from the main Einstein arc.We derive the spectrum from dirty-image cubes with a 1-arcsec taper and a spectral resolution of 100 MHz.There is a clear positive excess at the position of the HCO + (4-3) line, as well as potential foreground CO(0-1) absorption (see Appendix A.1). Note: the line fluxes in Table1are extracted from the narrow-band images (Fig.1).
Fig.6.Far-UV radiation (G) and gas density (n) from PDR models, with constraints from the HCN, HCO + , HNC, and CO(5-4) observations.Individual panels show models for Z=0.1, 0.2, 0.5, 1.0, 2.0 Z ⊙ .Different colours denote the different levels of mechanical heating contribution α (0% to 100%); the coloured squares denote models consistent with the observed line ratios within 50%; grey shaded contours show the G and n inferred from high-resolution imaging of SDP.81 byRybak et al. (2020).Only the Z=0.5 Z ⊙ , α=0, 10% models are consistent with the data.The most direct interpretation is that SDP.81 has sub-solar metallicity and only limited mechanical heating.Lower right: Histogram of mechanical heating factor α for the Z = 0.5 Z ⊙ model.All PDR models consistent with the line data are shown in grey; the models consistent with G and n inferred fromRybak et al. (2020) are highlighted in blue.