MNRAS 466 49ndash62 (2017) doi101093mnrasstw2996Advance Access publication 2016 November 22

Optical depth estimates and effective critical densities of dense gas tracersin the inner parts of nearby galaxy discs

M J Jimenez-Donaire1lsaquo F Bigiel1lsaquo A K Leroy2lsaquo D Cormier1 M Gallagher2

A Usero3 A Bolatto4 D Colombo5 S Garcıa-Burillo3 A Hughes67 C Kramer8

M R Krumholz9 D S Meier10 E Murphy11 J Pety1213 E Rosolowsky14

E Schinnerer15 A Schruba16 N Tomicic15 and L Zschaechner15

1Institute fur theoretische Astrophysik Zentrum fur Astronomie der Universitat Heidelberg Albert-Ueberle Str 2 D-69120 Heidelberg Germany2Department of Astronomy The Ohio State University 140 W 18th Street Columbus OH 43210 USA3Observatorio Astronomico Nacional Alfonso XII 3 E-28014 Madrid Spain4Department of Astronomy and Laboratory for Millimeter-Wave Astronomy University of Maryland College Park MD 20742 USA5Max-Planck-Institut fur Radioastronomie Auf dem Hugel 69 D-53121 Bonn Germany6CNRS IRAP 9 Av colonel Roche BP 44346 F-31028 Toulouse cedex 4 France7Universite de Toulouse UPS-OMP IRAP F-31028 Toulouse cedex 4 France8Instituto de Radioastronomıa Milimetrica (IRAM) Av Divina Pastora 7 Nucleo Central E-18012 Granada Spain9Research School of Astronomy and Astrophysics Australian National University Canberra ACT 2611 Australia10Department of Physics New Mexico Institute of Mining and Technology 801 Leroy Place Soccoro NM 87801 USA11National Radio Astronomy Observatory 520 Edgemont Road Charlottesville VA 22903 USA12Institut de Radioastronomie Millimetrique (IRAM) 300 Rue de la Piscine F-38406 Saint Martin drsquoHeres France13Observatoire de Paris 61 Avenue de lrsquoObservatoire F-75014 Paris France14Department of Physics University of Alberta Edmonton AB T6G 2E1 Canada15Max-Planck-Institut fur Astronomie Konigstuhl 17 D-69117 Heidelberg Germany16Max-Planck-Institut fur extraterrestrische Physik Giessenbachstrasse 1 D-85748 Garching Germany

Accepted 2016 November 16 Received 2016 November 15 in original form 2016 August 26

ABSTRACTHigh critical density molecular lines like HCN (1-0) or HCO+ (1-0) represent our best tool tostudy currently star-forming dense molecular gas at extragalactic distances The optical depthof these lines is a key ingredient to estimate the effective density required to excite emissionHowever constraints on this quantity are even scarcer in the literature than measurements ofthe high-density tracers themselves Here we combine new observations of HCN HCO+ andHNC (1-0) and their optically thin isotopologues H13CN H13CO+ and HN13C (1-0) to measureisotopologue line ratios We use IRAM 30-m observations from the large programme EMPIREand new Atacama Large Millimetresubmillimetre Array observations which together targetsix nearby star-forming galaxies Using spectral stacking techniques we calculate or placestrong upper limits on the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios in theinner parts of these galaxies Under simple assumptions we use these to estimate the opticaldepths of HCN (1-0) and HCO+ (1-0) to be τ sim 2ndash11 in the active inner regions of our targetsThe critical densities are consequently lowered to values between 5 and 20 times 105 cmminus3 1 and3 times 105 cmminus3 and 9 times 104 cmminus3 for HCN HCO+ and HNC respectively We study the impactof having different beam-filling factors η on these estimates and find that the effective criticaldensities decrease by a factor of η12

η13τ12 A comparison to existing work in NGC 5194 and NGC

253 shows the HCNH13CN and HCO+H13CO+ ratios in agreement with our measurementswithin the uncertainties The same is true for studies in other environments such as the GalacticCentre or nuclear regions of active galactic nucleus dominated nearby galaxies

Key words ISM molecules ndash galaxies abundances ndash galaxies star formation

E-mail mjimenezzahuni-heidelbergde (MJJ-D) bigieluni-heidelbergde (FB) leroy42osuedu (AKL)

1 IN T RO D U C T I O N

Most studies of the molecular gas in nearby galaxies have focusedon the lower-J carbon monoxide (CO) transitions as these are the

Ccopy 2016 The AuthorsPublished by Oxford University Press on behalf of the Royal Astronomical Society

50 M J Jimenez-Donaire et al

brightest molecular lines in other galaxies and are essential to char-acterize their total molecular masses and their dynamical properties(Kennicutt amp Evans 2012) However stars form out of physicallysmall (sim01 minus 1 pc) structures made primarily of dense moleculargas Thus CO spectroscopy alone is not sufficient to constrain thephysical properties of the actual star-forming molecular gas withdensities simnH2 gt 104 cmminus3 observations of tracers with highercritical densities are necessary

These immediately star-forming structures remain too small to re-solve in external galaxies In particular they are most easily studiedusing lines that preferentially select the dense star-forming mate-rial The low-J transitions HCN HCO+ and HNC are among thebrightest of these high critical density lines and can now be studied inother galaxies These kind of observations are very challenging dueto the faintness of these molecules and they have been mostly stud-ied in regions of active star formation in the Milky Way (eg Lada ampLada 2003 Wu et al 2005 Heiderman et al 2010 Lada Lombardiamp Alves 2010 Andre et al 2014 Stephens et al 2016) includingthe Galactic Centre (eg Marr Wright amp Backer 1993 Christopheret al 2005 Riquelme et al 2010 Battisti amp Heyer 2013 Haradaet al 2015) Only a few nearby galaxies have been analysed usingmultiline data sets of dense gas tracers These are often individualpointings on bright galaxy centres (eg Gao amp Solomon 2004Gracia-Carpio et al 2006 2008 Bussmann et al 2008 Kripset al 2008 Garcıa-Burillo et al 2012) which have recentlybeen expanded to cover the discs of star-forming galaxies (egBuchbender et al 2013 Meier et al 2015 Usero et al 2015 Schirmet al 2016)

Emission of common dense gas tracers like H12CN H12CO+

and HN12C is often found to be optically thick This leads to theline trapping effects that allow the lines to be excited at lowerdensities than one would naively assume at high optical depth thecritical density scales linearly with the inverse of the opacity forany optically thick line That is the density of gas traced by densegas tracers depends on their optical depth As a result knowing theoptical depth of lines like HCN (1-0) and HCO+ (1-0) is essential inestimating the masses and density structure of the dense interstellarmedium (ISM)

A good observational way to test the opacity of dense gas tracersis to combine the observations of the main dense gas tracing lineswith those of their rarer isotopologues eg those that include 13Cinstead of 12C These lines eg H13CN and H13CO+ are usuallyoptically thin because the molecules are less abundant by the iso-topic ratio Combined with few simplifying assumptions the ratioof eg H13CN (1ndash0) to H12CN (1ndash0) offers the prospect to test theopacity of dense gas tracers and so to help quantify the density towhich these lines are sensitive

The obstacle to such measurements is the faintness of emissionfrom dense gas tracer isotopologues As a result these have beendetected only within few very nearby galaxies (Chin et al 1998Henkel et al 1998 Wang et al 2004 Aladro et al 2013) Inthis paper we use new data from the IRAM 30-m telescope andthe Atacama Large Millimetresubmillimetre Array (ALMA) tomake new measurements of these line ratios and use them toconstrain the optical depth of dense gas tracers We rely on twomain data sets whole disc mapping of NGC 5194 first presentedin Bigiel et al (2016 see Section 2) and ALMA observations offour nearby star-forming galaxies NGC 3351 NGC 3627 NGC4254 and NGC 4321 presented in detail by Gallagher et al (inpreparation) We supplement these with observations of the nuclearstarburst in NGC 253 (Bolatto et al 2013 see Section 22) Allof these observations targeted both dense gas tracers and optically

thin isotopologues The properties of each target can be found inTable 2

Beyond improving our general knowledge of dense gas tracersa specific motivation of this study is to complement the IRAMlarge programme EMPIRE [lsquoEMIR Multiline Probe of the ISMRegulating Galaxy Evolutionrsquo Bigiel et al (in preparation)] Thissurvey which first results are presented in Bigiel et al (2016) isthe first multiline mapping survey that targets both the dense gas(eg HCN HCO+ HNC) and total gas (13CO C18O 12CO) tracersacross the whole area of a sample of nearby star-forming galaxies

The paper is structured as follows We describe the data inSection 2 including the details of data reduction and uncertaintycalculations We present the methods used to analyse the spectrain Section 3 and the main results are shown in Section 4 and arediscussed in Section 5 Finally we summarize our main conclu-sions in Section 6 Throughout the paper we will refer to the 12Cisotopologues without special notation eg HCN etc and men-tion explicitly if we refer to their isotopologues composed of othercarbon isotopes

2 O B S E RVAT I O N S A N D DATA R E D U C T I O N

21 IRAM 30-m observations of NGC 5194

NGC 5194 or M51 is a nearby spiral galaxy (SAbc) at a distance of76 Mpc (Ciardullo et al 2002) It was mapped with the IRAM 30-mtelescope at Pico Veleta over the course of 75 h spread across sevenruns during 2012 July and August The EMIR (Carter et al 2012)band E090 receiver was used in the dual polarization mode to mapthe emission from several high-density tracers We reached a sen-sitivity value of rms (Tmb) = 23 mK The observations cover thewhole active area of M51 and a frequency range of sim86ndash106 GHzThis range covers HCN (1-0) HCO+ (1-0) and HNC (1-0) as wellas the analogous transitions from their rarer isotopologues H13CN(1-0) H13CO+ (1-0) and HN13C (1-0) More information about theobservations eg detailed frequency settings integration time andsystem noise temperatures can be found in Bigiel et al (2016)

All the scans were combined and exported using the GILDASCLASS

software1 which was also used for further analysis including linearbaseline subtraction smoothing and flux measurements The datawere convolved to a common resolution of 28 arcsec which corre-sponds to a 10 kpc linear resolution at the distance of M51 Thismeans that our observations average together the emission frommany giant molecular clouds The final velocity resolution aftersmoothing is 4 km sminus1 The uncertainties in the fluxes are estimatedby means of the rms per channel of width dv within the signal-freeparts of the spectra and multiplying this value by the square root ofthe number of channels covered by each line We note that the re-ceiver setup for the other eight EMPIRE galaxies is slightly differentto the one used for M51 and did not cover the 13C isotopologues

22 ALMA observations of NGC 3351 NGC 3627 NGC 4254NGC 4321 and NGC 253

We also study four galaxies NGC 3351 NGC 3627 NGC 4254and NGC 4321 which were observed during ALMArsquos Cycle 2campaign (Gallagher et al in preparation) These observations cov-ered the molecular lines listed in Table 1 except for HN13C The

1 httpwwwiramfrIRAMFRGILDAS for more information see Pety(2005)

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 51

Table 1 Properties of observed molecular lines

Name Transition Rest frequency Ek ncrit

(GHz) (K) (cmminus3)

13CO 1-0 11020 529 65 times 102

HCN 1-0 8863 425 50 times 106

HNC 1-0 9066 430 12 times 106

HCO+ 1-0 8919 428 74 times 105

H13CN 1-0 8634 414 97 times 106

H13CO+ 1-0 8675 416 67 times 105

HN13C 1-0 8709 418 12 times 106

Notes The critical densities were computed assuming op-tically thin transition lines for an excitation temperatureof 20 K The collision rates are adapted from the LeidenLAMDA data base van der Tak et al (2007) For H13CN andHN13C the collisional coefficients for HCN and HNC fromDumouchel Faure amp Lique (2010) were used For H13CO+the collisional coefficients of HCO+ from Flower (1999)were used

data were processed using the ALMA pipeline and the CASA soft-ware (McMullin et al 2007) with details presented in Gallagheret al (in preparation) After calibration the data were continuumsubtracted and then imaged into separate data cubes for each lineWe reached a sensitivity value of rms (Tmb) = 15 mK A mild uminus v taper was applied to increase the surface brightness sensitivityand then the data were convolved to have a 5 arcsec circular beam(sim300 pc) and 10 km sminus1 wide channels

We supplement these new observations with data for the nearbystarburst galaxy NGC 253 These data obtained as part of ALMArsquosCycle 0 campaign were originally presented by Meier et al (2015)and Leroy et al (2015) Like the new data above these cover HCNand HCO+ H13CN and H13CO+ The average beam size is sim4 arc-sec (sim70 pc) over a field of view of approximately 15 arcmin(sim15 kpc) A summary of our data sets is given in Tables 1 and 2

3 M E T H O D O L O G Y

Our main goal is to detect emission from the faint isotopo-logues H13CN H13CO+ and HN13C and compute the line ra-tios HCNH13CN HCO+H13CO+ and HNCHN13C The emis-sion from these molecules is faint for individual lines of sight Toovercome this we use spectral stacking Because the line-of-sightvelocity varies with the position in each galaxy due to rotationwe first re-grid velocity axis of each spectrum in order to set thelocal mean velocity of the bulk molecular medium to 0 km sminus1We then co-add the emission from many different lines of sightto produce an average higher signal-to-noise ratio spectrum Thisprocedure resembles the one described by Schruba et al (2011) andCaldu-Primo et al (2013) We use 13CO emission to estimate the

local mean velocity of the molecular gas that we use as a referencefor the stacking In NGC 5194 we use the 13CO map from PAWS(Pety et al 2013 Schinnerer et al 2013) For the ALMA data weuse 13CO maps also observed by ALMA in a separate tuning

We stack spectra across the inner parts of our target galaxiesto maximize the signal-to-noise ratio We defined the active lsquoinnerregionrsquo of our targets by specifying a circular aperture with fixedangular size for the ALMA galaxies and different size for the M5130 m data This leads to modestly varying physical aperture sizefrom galaxy to galaxy The size of the aperture for each galaxy isindicated in Fig 1 and noted in Table 2 Specifically for the ALMAgalaxies we stack spectra inside a central 10 arcsec radius apertureFor M51 we picked a larger aperture of 50 arcsec in radius due tothe lower resolution of the observations

In order to measure spectral line parameters we fit the stackedspectrum of each line with a single Gaussian To perform the fit weuse the MPFIT function in IDL The free parameters we calculatefrom the fit are the line centre velocity the peak intensity andthe velocity dispersion We obtain the integrated HCN H13CNHCO+ and H13CO+ line intensities by integrating the fitted profileWe compute the uncertainties on the integrated intensity from thewidth of the profile and the rms noise measured from the signal-freepart of the spectrum If the peak intensity of the stacked spectrumis below 3σ then we compute an upper limit to the integratedintensity We take the limit to the integrated intensity to be the fluxof a Gaussian profile with the full width at half-maximum of thecorresponding 12C line and a peak three times the noise level for the13C profile This assumes that the 12C and 13C lines are well mixedand although the limit is formally stronger than 3σ this approachmatches what one would identify as an upper limit by eye

4 R ESULTS

Our targets listed in Table 2 are all massive star-forming nearbydisc galaxies These are among the closest and hence best-studiedstar-forming galaxies Our goal is to measure the ratios among densegas tracers and their (presumably) optically thin 13C isotopologuesFig 1 shows the distribution of 13CO (1-0) and HCN (1-0) emissionfrom each targets We use the 13CO emission shown in colourto indicate the distribution of the total molecular gas reservoir Italso serves as the reference for our spectral stacking Bright 13COemission is widespread across our targets and we see it at goodsignal to noise everywhere that we see HCN Red contours show thelocation of significant HCN (1-0) emission tracing higher densitygas Although detected at lower significance than the 13CO HCN isalso widespread across our targets In both lines and all galaxies thebrightest emission comes from the central regions of the target (theNGC 253 data cover only the centre) with emission which extendsalong the spiral arms

Table 2 Galaxy sample

Source RA DEC i PA r25 D Morphology SFR Inner aperture Telescope(20000) (20000) () () (arcmin) (Mpc) (M yrminus1 kpcminus2) (kpc)

NGC 3351 1043577 1142130 112 73 36 42 SBb 52 times 10minus3a 020 ALMANGC 3627 1120150 1259300 62 173 51 94 SABb 77 times 10minus3a 045 ALMANGC 4254 1218500 1424590 32 55 25 144 SAc 18 times 10minus3a 070 ALMANGC 4321 1222550 1549190 30 153 30 143 SABbc 90 times 10minus3a 069 ALMANGC 5194 1329527 4711429 20 172 39 76 Sbc 20 times 10minus3a 184 IRAM-30 mNGC 253 0047331 -2517197 76 55 135 35 SABc 14b 017 ALMA

Notes aLeroy et al (2013) average SFR surface density inside 075r25 bLeroy et al (2015) taking SFR sim1 M yrminus1 within r50

MNRAS 466 49ndash62 (2017)

52 M J Jimenez-Donaire et al

Figure 1 13CO (1-0) integrated intensity maps for our target galaxies The red contours show HCN (1-0) intensity levels between 3σ and 40σ for NGC 5194between 8σ and 25σ in NGC 3351 NGC 3627 NGC 4254 and NGC 4321 and between 30σ and 60σ for NGC 253 The white circle in the inner part of eachtarget shows the selected region for each galaxy to stack the spectra see details in Section 4

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 53

Figure 2 Stacked 13CO HCN and HCO+ spectra for the bright inner regions (radius 10 arcsec) in our target galaxies We also include the stacked HNCemission for NGC 5194 13CO which serves as our reference for the stacking is scaled to match the intensity scale of the other lines The 13CO and dense gastracers are all detected at very high signal-to-noise ratio The stacked dense gas tracers show good agreement with the mean 13CO velocity and all lines showsimilar line widths indicative of being well mixed on the scale of the beams (sim few hundred pc to kpc) for our data Table 3 reports Gaussian fits to the lines

Although we detect dense gas tracers over large areas in eachtarget in most of our targets only the inner bright regions yielduseful stacked constraints on the isotopologue line ratios We ex-perimented with also stacking over the discs of our targets but theemission of the 12C lines over this region already has only modestsignal-to-noise ratio Stacking over the extended discs of galaxiestypically yielded upper limits that barely constrained the 13C linesto be fainter than the 12C lines and so offered little constraint onthe optical depth Thus we concentrate mainly on the bright innerregions for the remainder of this paper As Fig 1 shows these arebright in 13CO HCN and HCO+ Here we detect the 13C lines in

some of the cases and in all the cases place limits comparable totypical 12CO13CO ratios in nearby galaxies (though see discussionbelow)

Figs 2 and 3 show the stacked spectra for 13CO the HCN HCO+

HNC lines and their 13C isotopologues for the bright inner regionof each target galaxy Table 3 lists the line parameters derived froma Gaussian fit Table 4 reports key ratios derived from comparingthese fitted parameters

Fig 2 shows that we detect 13CO HCN HCO+ and (in NGC5194) HNC at very high significance over these active regionsThe ordering of intensity is the same across all systems with the

MNRAS 466 49ndash62 (2017)

54 M J Jimenez-Donaire et al

Figure 3 Stacked H13CN and H13CO+ spectra for the bright inner regions of our target galaxies We show scaled versions of the stacked HCN and HCO+spectra from Fig 2 for reference These spectra show the expectation for the 12C-to-13C line ratios of 1-to-5 to 1-to-10 common values for the 12CO-to-13COratio in galaxy discs The red line indicates our working upper limit (dashed) or fit (solid) to the isotopologue spectrum Limits and fit parameters are reportedin Table 3

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 55

Figure 3 ndash continued

intensity of 13CO gt HCN gt HCO+ In NGC 5194 where we alsomeasure HNC this line is the weakest of the four

Fig 3 shows the stacked spectra for the 13C isotopologues of thedense gas tracers Here scaled versions of the 12C lines appear forreference The 13C isotopologues of the dense gas tracers are all farfainter than the 12C lines We do securely detect H13CN in NGC3351 NGC 3627 and NGC 253 In the other targets we place firmupper limits on the intensity of these lines We detect H13CO+ onlyin NGC 253 placing limits in the other systems

41 Line ratios

Table 4 reports the line ratios averaged over the active regionswhich are the main objective of our investigations We measureHCNH13CN integrated intensity ratios ranging from 7to21 (see

Table 4 for more details) and we constrain the ratio to be 6 forthe entire sample For HCO+H13CO+ we constrain the ratio to be7 in our sample and we measure a ratio of 24 in NGC 253 Forcomparison 12CO13CO ratios of sim6ndash10 are typically measuredin the discs of galaxies including the Milky Way and M51 The12C13C ratios for the dense gas tracers are significantly higher thanthose for CO in our targets Possible explanations are lower opticaldepth in the dense gas tracers isotopic abundance variations in theactive parts of galaxies or different filling factors for the 12C and13C lines We will discuss this further in Section 5

How do our measured ratios compare to previous measurementsTable 5 summarizes the literature measurements of isotopologueline ratios from the Milky Way and nearby galaxies including twothat overlap our targets (NGC 253 and NGC 5194) Broadly theseagree with our measurements with ratios often 10 and sometimes

MNRAS 466 49ndash62 (2017)

56 M J Jimenez-Donaire et al

Table 3 Spectral line parameters for the galaxies derived from the stacked spectra in Figs 2 and 3

Galaxy Molecule Integrated intensity Line width Peak rms(K km sminus1) (km sminus1) times 10minus2(K) (mK)

NGC 3351 13CO 120 plusmn 01 62 plusmn 10 194 plusmn 04 11HCN 84 plusmn 01 67 plusmn 11 118 plusmn 06 15

H13CN 04 plusmn 01 59 plusmn 10 07 plusmn 01 09HCO+ 50 plusmn 01 64 plusmn 11 74 plusmn 02 14

H13CO+ lt03 ndash lt04 13NGC 3627 13CO 74 plusmn 01 121 plusmn 14 68 plusmn 04 10

HCN 51 plusmn 01 129 plusmn 14 45 plusmn 01 08H13CN 07 plusmn 01 170 plusmn 17 04 plusmn 01 10HCO+ 49 plusmn 01 128 plusmn 14 35 plusmn 01 10

H13CO+ lt04 ndash lt05 14NGC 4254 13CO 85 plusmn 02 44 plusmn 9 181 plusmn 04 23

HCN 29 plusmn 02 48 plusmn 9 55 plusmn 02 33H13CN lt05 ndash lt13 40HCO+ 20 plusmn 01 44 plusmn 9 38 plusmn 01 18

H13CO+ lt03 ndash lt10 19NGC 4321 13CO 107 plusmn 01 55 plusmn 10 193 plusmn 04 11

HCN 69 plusmn 01 69 plusmn 11 97 plusmn 02 14H13CN lt03 ndash lt10 16HCO+ 46 plusmn 01 63 plusmn 11 68 plusmn 02 14

H13CO+ lt03 ndash lt10 13NGC 5194 13CO 46 plusmn 01 71 plusmn 6 64 plusmn 02 09

HCN 22 plusmn 01 76 plusmn 8 27 plusmn 01 06H13CN lt 02 ndash lt02 07HCO+ 14 plusmn 01 78 plusmn 10 16 plusmn 03 06

H13CO+ lt 02 ndash lt02 07HNC 08 plusmn 01 77 plusmn 8 11 plusmn 02 08

HN13C lt 02 ndash lt02 09NGC 253 13CO 312 plusmn 9 25 plusmn 6 1250 plusmn 40 40

HCN 32 plusmn 2 22 plusmn 4 120 plusmn 4 90H13CN 19 plusmn 04 19 plusmn 5 90 plusmn 03 56HCO+ 29 plusmn 3 21 plusmn 5 109 plusmn 4 92

H13CO+ 12 plusmn 04 18 plusmn 4 61 plusmn 02 80

Table 4 Computed line ratios for the galaxies

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

HCNH13CN 21 plusmn 3 7 plusmn 1 gt6 gt23 gt11 17 plusmn 1HCO+H13CO+ gt17 gt12 gt7 gt15 gt7 24 plusmn 2HNCHN13C ndash ndash ndash ndash gt4 ndashHCNHCO+ 17 plusmn 01 10 plusmn 01 15 plusmn 01 15 plusmn 01 16 plusmn 01 11 plusmn 0113COHCN 14 plusmn 02 15 plusmn 01 30 plusmn 03 16 plusmn 02 21 plusmn 01 10 plusmn 1

Table 5 Integrated intensity ratios from the literature

Target HCNH13CN HCO+H13CO+ Ref

NGC 1068 sim16 sim20 1Orion B ndash sim25 2Sgr A ndash sim15 3Galactic Centre sim11 sim21 4NGC 5194 sim25 sim30 5NGC 253 sim15 sim19 6

References (1) Wang et al (2014) (2) Pety (private com-munication) (3) Riquelme et al (2010) (4) Harada et al(2015) (5) Watanabe et al (2014) (6) Meier et al (2015)

20 In detail Watanabe et al (2014) measured line ratios inNGC 5194 and found HCNH13CN = 27 plusmn 18 HCO+H13CO+

= 34 plusmn 29 and HNCHN13C gt 16 consistent with our resultsfor that system Using the same data that we employ for NGC

253 Meier et al (2015) found similar ratios of HCNH13CN andHCO+H13CO+ (see Leroy et al 2015) Observations of NGC 1068by Wang et al (2014) found HCO+H13CO+ = 20 plusmn 1 HCNH13CN= 16 plusmn 1 and HNCHN13C = 38 plusmn 6 This galaxy is characterizedby a strong active galactic nucleus (AGN) which should influencethe surrounding chemistry Two of our targets also host AGN NGC5194 and NGC 3627 however we do not see any significant differ-ence in the measured line ratios compared to the rest of the sampleThough we note that at the coarse resolution of our observationssuch effects may be difficult to isolate

Studies focused on Milky Way yield a similar picture Riquelmeet al (2010) carried out a large-scale survey of the Galactic Centreregion and found HCO+H13CO+ sim 15 in Sgr A similar to thosefound here Harada et al (2015) observed the Circumnuclear Discof the Galactic Centre and reported ratios of HCO+H13CO+ sim 21and HCNH13CN sim 11 In the Solar Neighbourhood Pety (privatecommunication) found HCO+H13CO+ sim 25 in Orion B

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 57

Figure 4 Comparison between the line ratios measured in our sample of galaxies and the literature values summarized in Table 5 Circles show measurementsfrom the stacked spectra in the inner regions of galaxies whereas the arrows indicate upper limits Note the non-continuous x-axis as the HCN integratedintensity for NGC 253 is much higher than for the other galaxies The red boxes show the regime of values found for the sources in the literature in Table 5agreeing largely with the values and upper limits we derive

Fig 4 plots our measured ratios along with these literature valuesRed boxes highlight an indicative range of values for each ratiosim15ndash25 for HCNH13CN and sim15ndash30 for HCO+H13CO+ Wediscuss the interpretation of these ratios in Section 5

42 Line widths

In addition to line ratios the stacked spectra allow us to compare linewidths among the different molecular lines Potential differences inthe line widths may hint at differences in the distribution of suchemission along the line of sight

Table 3 and Fig 2 show that we do not observe strong differencesamong the line widths of the dense gas tracers and the 13CO emissiontracing lower density gas Thus on the scale of our beam whichranges from a few hundred pc to asymp1 kpc (except for the much higherresolution NGC 253) dense gas tracers appear well mixed with13CO Note that at these scales we expect each beam to encompassmany individual clouds and that we stack beam by beam Thereforethe statement here supports the idea that the inter-cloud line widthfor HCN and HCO+ resembles that for 13CO There does not appearto be a large population of clouds emitting only 13CO that showdifferent velocities than the clouds emitting dense gas tracers

We do observe spatial variations in the line width When ex-ploring stacked emission from discs which we mostly omit fromthis paper we find line widths as much as a factor of asymp2 narrowerthan in galaxy centres (see Figs 2 and 5 ) And in the high spatialresolution data that we use to explore NGC 253 we find narrowerline widths than in our other targets despite the high inclinationand high degree of turbulence in this galaxy In NGC 5194 thedifference between the disc line width (sim48 km sminus1) and the centralline width (sim70 km sminus1) is almost a factor of 2 and is present forall lines The simplest explanation for this difference and a simi-lar difference shown for NGC 3627 is that the broad line widthsin the central region of the galaxy still contain large amounts ofunresolved bulk motion We expect this to mostly be rotation unre-solved by our coarse beam (ie lsquobeam smearingrsquo) but this may alsoinclude streaming motions along the spiral arms and bar or evensome contributions from the AGN (eg see Scoville et al 1998

Figure 5 Stacked spectra across the entire molecular disc for NGC 3627and NGC 5194 excluding the central aperture (Figs 1 and 2) We show spectrafor the main dense gas tracers together with the stacked 13CO spectrum Linewidths are larger by up to a factor of sim2 in the inner parts compared to thediscs which we attribute largely to unresolved bulk motions

MNRAS 466 49ndash62 (2017)

58 M J Jimenez-Donaire et al

Table 6 Carbon isotope ratios from the literature in differentenvironments

Type of Object Galaxy 12C13C Ref

Centre Milky Way sim25 1LMC Magellanic Clouds sim50 2NGC 2024 Milky Way sim65 3Orion A Milky Way sim45 3Starburst NGC 253 sim40 4

sim80 5Starburst M 82 gt40 4

gt130 5Local ULIRGs Mrk 231Arp 220 sim100 4 6ULIRG z = 25 Cloverleaf gt 100 7

Notes (1) Guesten Henkel amp Batrla (1985) (2) Wang et al (2009)(3) Savage et al (2002) (4) Henkel et al (2014) (5) Martın et al(2010) (6) Gonzalez-Alfonso et al (2012) (7) Henkel et al (2010)

Meidt et al 2013 Matsushita et al 2015 Querejeta et al 2016)The narrower line widths in NGC 253 likely reflect the fact that weinclude less large-scale dynamical motion and perhaps that we sam-ple the turbulent cascade at an intermediate scale (eg Caldu-Primoamp Schruba 2016)

5 D ISCUSSION

Fig 4 shows our constraints on isotopologue line ratios for HCNand HCO+ The 12C-to-13C line ratios tend to be higher than thoseobserved for CO molecules but lower than typical isotopic ratiosHow should we interpret these A main motivation for this studywas to constrain the optical depth of the dense gas tracers and soto understand the importance of line trapping In the simple caseof a fixed 12C13C abundance and a cloud with a single density anduniform abundances these ratios can be translated to an opticaldepth The ratios that we observe tend to be lower than commonlyinferred 12C13C abundances implying some optical depth in the12C lines This section steps through key assumptions and resultsfor this simple model We then discuss how a breakdown in thesesimple assumptions may drive our observed line ratios In particularin the case where the 12C lines are optically thick then we wouldexpect the 12C and 13C lines to emerge from distinct regions of theclouds

51 Carbon isotope abundance ratio

Table 6 lists estimate of the 12C13C abundance ratio A ratio of 89is thought to characterize the Solar system at its formation (Wilsonamp Rood 1994) while local clouds show sim40ndash60 the Milky Waycentre shows a low ratio of sim25 and starburst galaxies at low andhigh redshift can reach ratios of sim100 This value varies at leastin part because 12C and 13C are produced by different phases ofstellar evolution 13C is only created primarily as a result of evolvedintermediate-mass stars whereas very massive stars are the onesthat produce 12C at the end of their lives As a consequence the12C13C ratio depends on the recent star formation history tendingto be high in regions of recent star formation and then droppingwith time

We measure ratios for the central regions of spiral galaxies Thesetend to be actively star-forming regions but not as extreme as thosein Ultra Luminous Infrared Galaxies (ULIRGs) Based on the rangeof literature values we adopt 50 as our fiducial 12C13C ratio sinceits determination is not possible with the present data set but we

note that this is likely uncertain by a factor of sim2 We considerthis effect in our EMPIRE and ALMA sample in Jimenez-Donaireet al (2016) Several effects can affect this ratio like selective pho-todissociation or isotopic fractionation While the strong interstellarradiation field emitted from OB stars especially in strong starburstsdoes not dissociate 12CO in large amounts due to self-shielding (egvan Dishoeck amp Black 1988) 13CO may be more severely affected(Bally amp Langer 1982) which may increase the 12C13C ratio Onthe other hand chemical fractionation in cold regions would lead topreferential formation of 13CO Qualitatively fractionation effectswould likely marginally increase the H13CO+ abundance (Langeramp Penzias 1990 Milam et al 2005) whereas H13CN and HN13Care expected to decrease (eg Roueff Loison amp Hickson 2015)

52 Optical depth in the simple case

521 Framework

In the simple case of a cloud with a single density 12C and 13C linesare evenly mixed throughout the cloud and the 12C line is opticallythick then the line ratio implies an optical depth

To estimate this optical depth we begin in the following with thesimplest case of assuming local thermodynamic equilibrium (LTE)and in particular Tex(HCN) = Tex(H13CN) Note that we will dropthis assumption in Section 53 We consider the intensity Jν ofthe 12C and 13C lines in units of brightness temperature but notnecessarily on the RayleighndashJeans tail so that

Jν = hνk

ehνkT minus 1 (1)

where ν is the observed frequency k is Boltzmannrsquos constant h isPlanckrsquos constant and T is the temperature characterizing the sourcefunction For our cloud T will be Tex in LTE Then the equationof radiative transfer gives the observed intensity Tobs in units ofbrightness temperature

Tobs = ηbf

[Jν(Tex) minus Jν(Tbg)

](1 minus exp (minusτ )) (2)

Here ηbf is the beam-filling factor the fraction of the beam area thatis filled by the emitting region Tbg is the background radiation fieldtemperature (271 K) and τ is the optical depth of the transition inquestion Note that ηbf the beam-filling factor is expected to be avery small number bringing our observed intensities from sim10 Kdown to the observed small fraction of a Kelvin

If we make the simplifying assumption that the 12C lines egHCN (1-0) are optically thick then 1 minus exp (minusτ ) asymp 1 and

T 12Cobs = η12C

bf

[Jν12C(Tex) minus Jν12C(Tbg)

] (3)

where Jν 12C refers to equation (1) evaluated at the frequency of the12C line On the other hand the 13C line has unknown optical depthso that

T 13Cobs = η13C

bf

[Jν13C(Tex) minus Jν13C(Tbg)

] (1 minus exp (minusτ 13C)

) (4)

Here we have distinguished the frequency of the 13C line from thatof the 12C line above The equations also allow that the two linesmight have different filling factors and so distinguish η13C

bf fromη12C

bf The optical depth of the 13C line is unknown However ourobserved ratios are far from sim1 which would be expected for bothlines to be optically thick and having matched ηbf Thus it appearsunlikely that both lines are optically thick Therefore we expect(1minusexp minus τ 13C) asymp τ 13C in most cases

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

52 M J Jimenez-Donaire et al

Figure 1 13CO (1-0) integrated intensity maps for our target galaxies The red contours show HCN (1-0) intensity levels between 3σ and 40σ for NGC 5194between 8σ and 25σ in NGC 3351 NGC 3627 NGC 4254 and NGC 4321 and between 30σ and 60σ for NGC 253 The white circle in the inner part of eachtarget shows the selected region for each galaxy to stack the spectra see details in Section 4

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 53

Figure 2 Stacked 13CO HCN and HCO+ spectra for the bright inner regions (radius 10 arcsec) in our target galaxies We also include the stacked HNCemission for NGC 5194 13CO which serves as our reference for the stacking is scaled to match the intensity scale of the other lines The 13CO and dense gastracers are all detected at very high signal-to-noise ratio The stacked dense gas tracers show good agreement with the mean 13CO velocity and all lines showsimilar line widths indicative of being well mixed on the scale of the beams (sim few hundred pc to kpc) for our data Table 3 reports Gaussian fits to the lines

Although we detect dense gas tracers over large areas in eachtarget in most of our targets only the inner bright regions yielduseful stacked constraints on the isotopologue line ratios We ex-perimented with also stacking over the discs of our targets but theemission of the 12C lines over this region already has only modestsignal-to-noise ratio Stacking over the extended discs of galaxiestypically yielded upper limits that barely constrained the 13C linesto be fainter than the 12C lines and so offered little constraint onthe optical depth Thus we concentrate mainly on the bright innerregions for the remainder of this paper As Fig 1 shows these arebright in 13CO HCN and HCO+ Here we detect the 13C lines in

some of the cases and in all the cases place limits comparable totypical 12CO13CO ratios in nearby galaxies (though see discussionbelow)

Figs 2 and 3 show the stacked spectra for 13CO the HCN HCO+

HNC lines and their 13C isotopologues for the bright inner regionof each target galaxy Table 3 lists the line parameters derived froma Gaussian fit Table 4 reports key ratios derived from comparingthese fitted parameters

Fig 2 shows that we detect 13CO HCN HCO+ and (in NGC5194) HNC at very high significance over these active regionsThe ordering of intensity is the same across all systems with the

MNRAS 466 49ndash62 (2017)

54 M J Jimenez-Donaire et al

Figure 3 Stacked H13CN and H13CO+ spectra for the bright inner regions of our target galaxies We show scaled versions of the stacked HCN and HCO+spectra from Fig 2 for reference These spectra show the expectation for the 12C-to-13C line ratios of 1-to-5 to 1-to-10 common values for the 12CO-to-13COratio in galaxy discs The red line indicates our working upper limit (dashed) or fit (solid) to the isotopologue spectrum Limits and fit parameters are reportedin Table 3

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 55

Figure 3 ndash continued

intensity of 13CO gt HCN gt HCO+ In NGC 5194 where we alsomeasure HNC this line is the weakest of the four

Fig 3 shows the stacked spectra for the 13C isotopologues of thedense gas tracers Here scaled versions of the 12C lines appear forreference The 13C isotopologues of the dense gas tracers are all farfainter than the 12C lines We do securely detect H13CN in NGC3351 NGC 3627 and NGC 253 In the other targets we place firmupper limits on the intensity of these lines We detect H13CO+ onlyin NGC 253 placing limits in the other systems

41 Line ratios

Table 4 reports the line ratios averaged over the active regionswhich are the main objective of our investigations We measureHCNH13CN integrated intensity ratios ranging from 7to21 (see

Table 4 for more details) and we constrain the ratio to be 6 forthe entire sample For HCO+H13CO+ we constrain the ratio to be7 in our sample and we measure a ratio of 24 in NGC 253 Forcomparison 12CO13CO ratios of sim6ndash10 are typically measuredin the discs of galaxies including the Milky Way and M51 The12C13C ratios for the dense gas tracers are significantly higher thanthose for CO in our targets Possible explanations are lower opticaldepth in the dense gas tracers isotopic abundance variations in theactive parts of galaxies or different filling factors for the 12C and13C lines We will discuss this further in Section 5

How do our measured ratios compare to previous measurementsTable 5 summarizes the literature measurements of isotopologueline ratios from the Milky Way and nearby galaxies including twothat overlap our targets (NGC 253 and NGC 5194) Broadly theseagree with our measurements with ratios often 10 and sometimes

MNRAS 466 49ndash62 (2017)

56 M J Jimenez-Donaire et al

Table 3 Spectral line parameters for the galaxies derived from the stacked spectra in Figs 2 and 3

Galaxy Molecule Integrated intensity Line width Peak rms(K km sminus1) (km sminus1) times 10minus2(K) (mK)

NGC 3351 13CO 120 plusmn 01 62 plusmn 10 194 plusmn 04 11HCN 84 plusmn 01 67 plusmn 11 118 plusmn 06 15

H13CN 04 plusmn 01 59 plusmn 10 07 plusmn 01 09HCO+ 50 plusmn 01 64 plusmn 11 74 plusmn 02 14

H13CO+ lt03 ndash lt04 13NGC 3627 13CO 74 plusmn 01 121 plusmn 14 68 plusmn 04 10

HCN 51 plusmn 01 129 plusmn 14 45 plusmn 01 08H13CN 07 plusmn 01 170 plusmn 17 04 plusmn 01 10HCO+ 49 plusmn 01 128 plusmn 14 35 plusmn 01 10

H13CO+ lt04 ndash lt05 14NGC 4254 13CO 85 plusmn 02 44 plusmn 9 181 plusmn 04 23

HCN 29 plusmn 02 48 plusmn 9 55 plusmn 02 33H13CN lt05 ndash lt13 40HCO+ 20 plusmn 01 44 plusmn 9 38 plusmn 01 18

H13CO+ lt03 ndash lt10 19NGC 4321 13CO 107 plusmn 01 55 plusmn 10 193 plusmn 04 11

HCN 69 plusmn 01 69 plusmn 11 97 plusmn 02 14H13CN lt03 ndash lt10 16HCO+ 46 plusmn 01 63 plusmn 11 68 plusmn 02 14

H13CO+ lt03 ndash lt10 13NGC 5194 13CO 46 plusmn 01 71 plusmn 6 64 plusmn 02 09

HCN 22 plusmn 01 76 plusmn 8 27 plusmn 01 06H13CN lt 02 ndash lt02 07HCO+ 14 plusmn 01 78 plusmn 10 16 plusmn 03 06

H13CO+ lt 02 ndash lt02 07HNC 08 plusmn 01 77 plusmn 8 11 plusmn 02 08

HN13C lt 02 ndash lt02 09NGC 253 13CO 312 plusmn 9 25 plusmn 6 1250 plusmn 40 40

HCN 32 plusmn 2 22 plusmn 4 120 plusmn 4 90H13CN 19 plusmn 04 19 plusmn 5 90 plusmn 03 56HCO+ 29 plusmn 3 21 plusmn 5 109 plusmn 4 92

H13CO+ 12 plusmn 04 18 plusmn 4 61 plusmn 02 80

Table 4 Computed line ratios for the galaxies

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

HCNH13CN 21 plusmn 3 7 plusmn 1 gt6 gt23 gt11 17 plusmn 1HCO+H13CO+ gt17 gt12 gt7 gt15 gt7 24 plusmn 2HNCHN13C ndash ndash ndash ndash gt4 ndashHCNHCO+ 17 plusmn 01 10 plusmn 01 15 plusmn 01 15 plusmn 01 16 plusmn 01 11 plusmn 0113COHCN 14 plusmn 02 15 plusmn 01 30 plusmn 03 16 plusmn 02 21 plusmn 01 10 plusmn 1

Table 5 Integrated intensity ratios from the literature

Target HCNH13CN HCO+H13CO+ Ref

NGC 1068 sim16 sim20 1Orion B ndash sim25 2Sgr A ndash sim15 3Galactic Centre sim11 sim21 4NGC 5194 sim25 sim30 5NGC 253 sim15 sim19 6

References (1) Wang et al (2014) (2) Pety (private com-munication) (3) Riquelme et al (2010) (4) Harada et al(2015) (5) Watanabe et al (2014) (6) Meier et al (2015)

20 In detail Watanabe et al (2014) measured line ratios inNGC 5194 and found HCNH13CN = 27 plusmn 18 HCO+H13CO+

= 34 plusmn 29 and HNCHN13C gt 16 consistent with our resultsfor that system Using the same data that we employ for NGC

253 Meier et al (2015) found similar ratios of HCNH13CN andHCO+H13CO+ (see Leroy et al 2015) Observations of NGC 1068by Wang et al (2014) found HCO+H13CO+ = 20 plusmn 1 HCNH13CN= 16 plusmn 1 and HNCHN13C = 38 plusmn 6 This galaxy is characterizedby a strong active galactic nucleus (AGN) which should influencethe surrounding chemistry Two of our targets also host AGN NGC5194 and NGC 3627 however we do not see any significant differ-ence in the measured line ratios compared to the rest of the sampleThough we note that at the coarse resolution of our observationssuch effects may be difficult to isolate

Studies focused on Milky Way yield a similar picture Riquelmeet al (2010) carried out a large-scale survey of the Galactic Centreregion and found HCO+H13CO+ sim 15 in Sgr A similar to thosefound here Harada et al (2015) observed the Circumnuclear Discof the Galactic Centre and reported ratios of HCO+H13CO+ sim 21and HCNH13CN sim 11 In the Solar Neighbourhood Pety (privatecommunication) found HCO+H13CO+ sim 25 in Orion B

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 57

Figure 4 Comparison between the line ratios measured in our sample of galaxies and the literature values summarized in Table 5 Circles show measurementsfrom the stacked spectra in the inner regions of galaxies whereas the arrows indicate upper limits Note the non-continuous x-axis as the HCN integratedintensity for NGC 253 is much higher than for the other galaxies The red boxes show the regime of values found for the sources in the literature in Table 5agreeing largely with the values and upper limits we derive

Fig 4 plots our measured ratios along with these literature valuesRed boxes highlight an indicative range of values for each ratiosim15ndash25 for HCNH13CN and sim15ndash30 for HCO+H13CO+ Wediscuss the interpretation of these ratios in Section 5

42 Line widths

In addition to line ratios the stacked spectra allow us to compare linewidths among the different molecular lines Potential differences inthe line widths may hint at differences in the distribution of suchemission along the line of sight

Table 3 and Fig 2 show that we do not observe strong differencesamong the line widths of the dense gas tracers and the 13CO emissiontracing lower density gas Thus on the scale of our beam whichranges from a few hundred pc to asymp1 kpc (except for the much higherresolution NGC 253) dense gas tracers appear well mixed with13CO Note that at these scales we expect each beam to encompassmany individual clouds and that we stack beam by beam Thereforethe statement here supports the idea that the inter-cloud line widthfor HCN and HCO+ resembles that for 13CO There does not appearto be a large population of clouds emitting only 13CO that showdifferent velocities than the clouds emitting dense gas tracers

We do observe spatial variations in the line width When ex-ploring stacked emission from discs which we mostly omit fromthis paper we find line widths as much as a factor of asymp2 narrowerthan in galaxy centres (see Figs 2 and 5 ) And in the high spatialresolution data that we use to explore NGC 253 we find narrowerline widths than in our other targets despite the high inclinationand high degree of turbulence in this galaxy In NGC 5194 thedifference between the disc line width (sim48 km sminus1) and the centralline width (sim70 km sminus1) is almost a factor of 2 and is present forall lines The simplest explanation for this difference and a simi-lar difference shown for NGC 3627 is that the broad line widthsin the central region of the galaxy still contain large amounts ofunresolved bulk motion We expect this to mostly be rotation unre-solved by our coarse beam (ie lsquobeam smearingrsquo) but this may alsoinclude streaming motions along the spiral arms and bar or evensome contributions from the AGN (eg see Scoville et al 1998

Figure 5 Stacked spectra across the entire molecular disc for NGC 3627and NGC 5194 excluding the central aperture (Figs 1 and 2) We show spectrafor the main dense gas tracers together with the stacked 13CO spectrum Linewidths are larger by up to a factor of sim2 in the inner parts compared to thediscs which we attribute largely to unresolved bulk motions

MNRAS 466 49ndash62 (2017)

58 M J Jimenez-Donaire et al

Table 6 Carbon isotope ratios from the literature in differentenvironments

Type of Object Galaxy 12C13C Ref

Centre Milky Way sim25 1LMC Magellanic Clouds sim50 2NGC 2024 Milky Way sim65 3Orion A Milky Way sim45 3Starburst NGC 253 sim40 4

sim80 5Starburst M 82 gt40 4

gt130 5Local ULIRGs Mrk 231Arp 220 sim100 4 6ULIRG z = 25 Cloverleaf gt 100 7

Notes (1) Guesten Henkel amp Batrla (1985) (2) Wang et al (2009)(3) Savage et al (2002) (4) Henkel et al (2014) (5) Martın et al(2010) (6) Gonzalez-Alfonso et al (2012) (7) Henkel et al (2010)

Meidt et al 2013 Matsushita et al 2015 Querejeta et al 2016)The narrower line widths in NGC 253 likely reflect the fact that weinclude less large-scale dynamical motion and perhaps that we sam-ple the turbulent cascade at an intermediate scale (eg Caldu-Primoamp Schruba 2016)

5 D ISCUSSION

Fig 4 shows our constraints on isotopologue line ratios for HCNand HCO+ The 12C-to-13C line ratios tend to be higher than thoseobserved for CO molecules but lower than typical isotopic ratiosHow should we interpret these A main motivation for this studywas to constrain the optical depth of the dense gas tracers and soto understand the importance of line trapping In the simple caseof a fixed 12C13C abundance and a cloud with a single density anduniform abundances these ratios can be translated to an opticaldepth The ratios that we observe tend to be lower than commonlyinferred 12C13C abundances implying some optical depth in the12C lines This section steps through key assumptions and resultsfor this simple model We then discuss how a breakdown in thesesimple assumptions may drive our observed line ratios In particularin the case where the 12C lines are optically thick then we wouldexpect the 12C and 13C lines to emerge from distinct regions of theclouds

51 Carbon isotope abundance ratio

Table 6 lists estimate of the 12C13C abundance ratio A ratio of 89is thought to characterize the Solar system at its formation (Wilsonamp Rood 1994) while local clouds show sim40ndash60 the Milky Waycentre shows a low ratio of sim25 and starburst galaxies at low andhigh redshift can reach ratios of sim100 This value varies at leastin part because 12C and 13C are produced by different phases ofstellar evolution 13C is only created primarily as a result of evolvedintermediate-mass stars whereas very massive stars are the onesthat produce 12C at the end of their lives As a consequence the12C13C ratio depends on the recent star formation history tendingto be high in regions of recent star formation and then droppingwith time

We measure ratios for the central regions of spiral galaxies Thesetend to be actively star-forming regions but not as extreme as thosein Ultra Luminous Infrared Galaxies (ULIRGs) Based on the rangeof literature values we adopt 50 as our fiducial 12C13C ratio sinceits determination is not possible with the present data set but we

note that this is likely uncertain by a factor of sim2 We considerthis effect in our EMPIRE and ALMA sample in Jimenez-Donaireet al (2016) Several effects can affect this ratio like selective pho-todissociation or isotopic fractionation While the strong interstellarradiation field emitted from OB stars especially in strong starburstsdoes not dissociate 12CO in large amounts due to self-shielding (egvan Dishoeck amp Black 1988) 13CO may be more severely affected(Bally amp Langer 1982) which may increase the 12C13C ratio Onthe other hand chemical fractionation in cold regions would lead topreferential formation of 13CO Qualitatively fractionation effectswould likely marginally increase the H13CO+ abundance (Langeramp Penzias 1990 Milam et al 2005) whereas H13CN and HN13Care expected to decrease (eg Roueff Loison amp Hickson 2015)

52 Optical depth in the simple case

521 Framework

In the simple case of a cloud with a single density 12C and 13C linesare evenly mixed throughout the cloud and the 12C line is opticallythick then the line ratio implies an optical depth

To estimate this optical depth we begin in the following with thesimplest case of assuming local thermodynamic equilibrium (LTE)and in particular Tex(HCN) = Tex(H13CN) Note that we will dropthis assumption in Section 53 We consider the intensity Jν ofthe 12C and 13C lines in units of brightness temperature but notnecessarily on the RayleighndashJeans tail so that

Jν = hνk

ehνkT minus 1 (1)

where ν is the observed frequency k is Boltzmannrsquos constant h isPlanckrsquos constant and T is the temperature characterizing the sourcefunction For our cloud T will be Tex in LTE Then the equationof radiative transfer gives the observed intensity Tobs in units ofbrightness temperature

Tobs = ηbf

[Jν(Tex) minus Jν(Tbg)

](1 minus exp (minusτ )) (2)

Here ηbf is the beam-filling factor the fraction of the beam area thatis filled by the emitting region Tbg is the background radiation fieldtemperature (271 K) and τ is the optical depth of the transition inquestion Note that ηbf the beam-filling factor is expected to be avery small number bringing our observed intensities from sim10 Kdown to the observed small fraction of a Kelvin

If we make the simplifying assumption that the 12C lines egHCN (1-0) are optically thick then 1 minus exp (minusτ ) asymp 1 and

T 12Cobs = η12C

bf

[Jν12C(Tex) minus Jν12C(Tbg)

] (3)

where Jν 12C refers to equation (1) evaluated at the frequency of the12C line On the other hand the 13C line has unknown optical depthso that

T 13Cobs = η13C

bf

[Jν13C(Tex) minus Jν13C(Tbg)

] (1 minus exp (minusτ 13C)

) (4)

Here we have distinguished the frequency of the 13C line from thatof the 12C line above The equations also allow that the two linesmight have different filling factors and so distinguish η13C

bf fromη12C

bf The optical depth of the 13C line is unknown However ourobserved ratios are far from sim1 which would be expected for bothlines to be optically thick and having matched ηbf Thus it appearsunlikely that both lines are optically thick Therefore we expect(1minusexp minus τ 13C) asymp τ 13C in most cases

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 53

Figure 2 Stacked 13CO HCN and HCO+ spectra for the bright inner regions (radius 10 arcsec) in our target galaxies We also include the stacked HNCemission for NGC 5194 13CO which serves as our reference for the stacking is scaled to match the intensity scale of the other lines The 13CO and dense gastracers are all detected at very high signal-to-noise ratio The stacked dense gas tracers show good agreement with the mean 13CO velocity and all lines showsimilar line widths indicative of being well mixed on the scale of the beams (sim few hundred pc to kpc) for our data Table 3 reports Gaussian fits to the lines

Although we detect dense gas tracers over large areas in eachtarget in most of our targets only the inner bright regions yielduseful stacked constraints on the isotopologue line ratios We ex-perimented with also stacking over the discs of our targets but theemission of the 12C lines over this region already has only modestsignal-to-noise ratio Stacking over the extended discs of galaxiestypically yielded upper limits that barely constrained the 13C linesto be fainter than the 12C lines and so offered little constraint onthe optical depth Thus we concentrate mainly on the bright innerregions for the remainder of this paper As Fig 1 shows these arebright in 13CO HCN and HCO+ Here we detect the 13C lines in

some of the cases and in all the cases place limits comparable totypical 12CO13CO ratios in nearby galaxies (though see discussionbelow)

Figs 2 and 3 show the stacked spectra for 13CO the HCN HCO+

HNC lines and their 13C isotopologues for the bright inner regionof each target galaxy Table 3 lists the line parameters derived froma Gaussian fit Table 4 reports key ratios derived from comparingthese fitted parameters

Fig 2 shows that we detect 13CO HCN HCO+ and (in NGC5194) HNC at very high significance over these active regionsThe ordering of intensity is the same across all systems with the

MNRAS 466 49ndash62 (2017)

54 M J Jimenez-Donaire et al

Figure 3 Stacked H13CN and H13CO+ spectra for the bright inner regions of our target galaxies We show scaled versions of the stacked HCN and HCO+spectra from Fig 2 for reference These spectra show the expectation for the 12C-to-13C line ratios of 1-to-5 to 1-to-10 common values for the 12CO-to-13COratio in galaxy discs The red line indicates our working upper limit (dashed) or fit (solid) to the isotopologue spectrum Limits and fit parameters are reportedin Table 3

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 55

Figure 3 ndash continued

intensity of 13CO gt HCN gt HCO+ In NGC 5194 where we alsomeasure HNC this line is the weakest of the four

Fig 3 shows the stacked spectra for the 13C isotopologues of thedense gas tracers Here scaled versions of the 12C lines appear forreference The 13C isotopologues of the dense gas tracers are all farfainter than the 12C lines We do securely detect H13CN in NGC3351 NGC 3627 and NGC 253 In the other targets we place firmupper limits on the intensity of these lines We detect H13CO+ onlyin NGC 253 placing limits in the other systems

41 Line ratios

Table 4 reports the line ratios averaged over the active regionswhich are the main objective of our investigations We measureHCNH13CN integrated intensity ratios ranging from 7to21 (see

Table 4 for more details) and we constrain the ratio to be 6 forthe entire sample For HCO+H13CO+ we constrain the ratio to be7 in our sample and we measure a ratio of 24 in NGC 253 Forcomparison 12CO13CO ratios of sim6ndash10 are typically measuredin the discs of galaxies including the Milky Way and M51 The12C13C ratios for the dense gas tracers are significantly higher thanthose for CO in our targets Possible explanations are lower opticaldepth in the dense gas tracers isotopic abundance variations in theactive parts of galaxies or different filling factors for the 12C and13C lines We will discuss this further in Section 5

How do our measured ratios compare to previous measurementsTable 5 summarizes the literature measurements of isotopologueline ratios from the Milky Way and nearby galaxies including twothat overlap our targets (NGC 253 and NGC 5194) Broadly theseagree with our measurements with ratios often 10 and sometimes

MNRAS 466 49ndash62 (2017)

56 M J Jimenez-Donaire et al

Table 3 Spectral line parameters for the galaxies derived from the stacked spectra in Figs 2 and 3

Galaxy Molecule Integrated intensity Line width Peak rms(K km sminus1) (km sminus1) times 10minus2(K) (mK)

NGC 3351 13CO 120 plusmn 01 62 plusmn 10 194 plusmn 04 11HCN 84 plusmn 01 67 plusmn 11 118 plusmn 06 15

H13CN 04 plusmn 01 59 plusmn 10 07 plusmn 01 09HCO+ 50 plusmn 01 64 plusmn 11 74 plusmn 02 14

H13CO+ lt03 ndash lt04 13NGC 3627 13CO 74 plusmn 01 121 plusmn 14 68 plusmn 04 10

HCN 51 plusmn 01 129 plusmn 14 45 plusmn 01 08H13CN 07 plusmn 01 170 plusmn 17 04 plusmn 01 10HCO+ 49 plusmn 01 128 plusmn 14 35 plusmn 01 10

H13CO+ lt04 ndash lt05 14NGC 4254 13CO 85 plusmn 02 44 plusmn 9 181 plusmn 04 23

HCN 29 plusmn 02 48 plusmn 9 55 plusmn 02 33H13CN lt05 ndash lt13 40HCO+ 20 plusmn 01 44 plusmn 9 38 plusmn 01 18

H13CO+ lt03 ndash lt10 19NGC 4321 13CO 107 plusmn 01 55 plusmn 10 193 plusmn 04 11

HCN 69 plusmn 01 69 plusmn 11 97 plusmn 02 14H13CN lt03 ndash lt10 16HCO+ 46 plusmn 01 63 plusmn 11 68 plusmn 02 14

H13CO+ lt03 ndash lt10 13NGC 5194 13CO 46 plusmn 01 71 plusmn 6 64 plusmn 02 09

HCN 22 plusmn 01 76 plusmn 8 27 plusmn 01 06H13CN lt 02 ndash lt02 07HCO+ 14 plusmn 01 78 plusmn 10 16 plusmn 03 06

H13CO+ lt 02 ndash lt02 07HNC 08 plusmn 01 77 plusmn 8 11 plusmn 02 08

HN13C lt 02 ndash lt02 09NGC 253 13CO 312 plusmn 9 25 plusmn 6 1250 plusmn 40 40

HCN 32 plusmn 2 22 plusmn 4 120 plusmn 4 90H13CN 19 plusmn 04 19 plusmn 5 90 plusmn 03 56HCO+ 29 plusmn 3 21 plusmn 5 109 plusmn 4 92

H13CO+ 12 plusmn 04 18 plusmn 4 61 plusmn 02 80

Table 4 Computed line ratios for the galaxies

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

HCNH13CN 21 plusmn 3 7 plusmn 1 gt6 gt23 gt11 17 plusmn 1HCO+H13CO+ gt17 gt12 gt7 gt15 gt7 24 plusmn 2HNCHN13C ndash ndash ndash ndash gt4 ndashHCNHCO+ 17 plusmn 01 10 plusmn 01 15 plusmn 01 15 plusmn 01 16 plusmn 01 11 plusmn 0113COHCN 14 plusmn 02 15 plusmn 01 30 plusmn 03 16 plusmn 02 21 plusmn 01 10 plusmn 1

Table 5 Integrated intensity ratios from the literature

Target HCNH13CN HCO+H13CO+ Ref

NGC 1068 sim16 sim20 1Orion B ndash sim25 2Sgr A ndash sim15 3Galactic Centre sim11 sim21 4NGC 5194 sim25 sim30 5NGC 253 sim15 sim19 6

References (1) Wang et al (2014) (2) Pety (private com-munication) (3) Riquelme et al (2010) (4) Harada et al(2015) (5) Watanabe et al (2014) (6) Meier et al (2015)

20 In detail Watanabe et al (2014) measured line ratios inNGC 5194 and found HCNH13CN = 27 plusmn 18 HCO+H13CO+

= 34 plusmn 29 and HNCHN13C gt 16 consistent with our resultsfor that system Using the same data that we employ for NGC

253 Meier et al (2015) found similar ratios of HCNH13CN andHCO+H13CO+ (see Leroy et al 2015) Observations of NGC 1068by Wang et al (2014) found HCO+H13CO+ = 20 plusmn 1 HCNH13CN= 16 plusmn 1 and HNCHN13C = 38 plusmn 6 This galaxy is characterizedby a strong active galactic nucleus (AGN) which should influencethe surrounding chemistry Two of our targets also host AGN NGC5194 and NGC 3627 however we do not see any significant differ-ence in the measured line ratios compared to the rest of the sampleThough we note that at the coarse resolution of our observationssuch effects may be difficult to isolate

Studies focused on Milky Way yield a similar picture Riquelmeet al (2010) carried out a large-scale survey of the Galactic Centreregion and found HCO+H13CO+ sim 15 in Sgr A similar to thosefound here Harada et al (2015) observed the Circumnuclear Discof the Galactic Centre and reported ratios of HCO+H13CO+ sim 21and HCNH13CN sim 11 In the Solar Neighbourhood Pety (privatecommunication) found HCO+H13CO+ sim 25 in Orion B

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 57

Figure 4 Comparison between the line ratios measured in our sample of galaxies and the literature values summarized in Table 5 Circles show measurementsfrom the stacked spectra in the inner regions of galaxies whereas the arrows indicate upper limits Note the non-continuous x-axis as the HCN integratedintensity for NGC 253 is much higher than for the other galaxies The red boxes show the regime of values found for the sources in the literature in Table 5agreeing largely with the values and upper limits we derive

Fig 4 plots our measured ratios along with these literature valuesRed boxes highlight an indicative range of values for each ratiosim15ndash25 for HCNH13CN and sim15ndash30 for HCO+H13CO+ Wediscuss the interpretation of these ratios in Section 5

42 Line widths

In addition to line ratios the stacked spectra allow us to compare linewidths among the different molecular lines Potential differences inthe line widths may hint at differences in the distribution of suchemission along the line of sight

Table 3 and Fig 2 show that we do not observe strong differencesamong the line widths of the dense gas tracers and the 13CO emissiontracing lower density gas Thus on the scale of our beam whichranges from a few hundred pc to asymp1 kpc (except for the much higherresolution NGC 253) dense gas tracers appear well mixed with13CO Note that at these scales we expect each beam to encompassmany individual clouds and that we stack beam by beam Thereforethe statement here supports the idea that the inter-cloud line widthfor HCN and HCO+ resembles that for 13CO There does not appearto be a large population of clouds emitting only 13CO that showdifferent velocities than the clouds emitting dense gas tracers

We do observe spatial variations in the line width When ex-ploring stacked emission from discs which we mostly omit fromthis paper we find line widths as much as a factor of asymp2 narrowerthan in galaxy centres (see Figs 2 and 5 ) And in the high spatialresolution data that we use to explore NGC 253 we find narrowerline widths than in our other targets despite the high inclinationand high degree of turbulence in this galaxy In NGC 5194 thedifference between the disc line width (sim48 km sminus1) and the centralline width (sim70 km sminus1) is almost a factor of 2 and is present forall lines The simplest explanation for this difference and a simi-lar difference shown for NGC 3627 is that the broad line widthsin the central region of the galaxy still contain large amounts ofunresolved bulk motion We expect this to mostly be rotation unre-solved by our coarse beam (ie lsquobeam smearingrsquo) but this may alsoinclude streaming motions along the spiral arms and bar or evensome contributions from the AGN (eg see Scoville et al 1998

Figure 5 Stacked spectra across the entire molecular disc for NGC 3627and NGC 5194 excluding the central aperture (Figs 1 and 2) We show spectrafor the main dense gas tracers together with the stacked 13CO spectrum Linewidths are larger by up to a factor of sim2 in the inner parts compared to thediscs which we attribute largely to unresolved bulk motions

MNRAS 466 49ndash62 (2017)

58 M J Jimenez-Donaire et al

Table 6 Carbon isotope ratios from the literature in differentenvironments

Type of Object Galaxy 12C13C Ref

Centre Milky Way sim25 1LMC Magellanic Clouds sim50 2NGC 2024 Milky Way sim65 3Orion A Milky Way sim45 3Starburst NGC 253 sim40 4

sim80 5Starburst M 82 gt40 4

gt130 5Local ULIRGs Mrk 231Arp 220 sim100 4 6ULIRG z = 25 Cloverleaf gt 100 7

Notes (1) Guesten Henkel amp Batrla (1985) (2) Wang et al (2009)(3) Savage et al (2002) (4) Henkel et al (2014) (5) Martın et al(2010) (6) Gonzalez-Alfonso et al (2012) (7) Henkel et al (2010)

Meidt et al 2013 Matsushita et al 2015 Querejeta et al 2016)The narrower line widths in NGC 253 likely reflect the fact that weinclude less large-scale dynamical motion and perhaps that we sam-ple the turbulent cascade at an intermediate scale (eg Caldu-Primoamp Schruba 2016)

5 D ISCUSSION

Fig 4 shows our constraints on isotopologue line ratios for HCNand HCO+ The 12C-to-13C line ratios tend to be higher than thoseobserved for CO molecules but lower than typical isotopic ratiosHow should we interpret these A main motivation for this studywas to constrain the optical depth of the dense gas tracers and soto understand the importance of line trapping In the simple caseof a fixed 12C13C abundance and a cloud with a single density anduniform abundances these ratios can be translated to an opticaldepth The ratios that we observe tend to be lower than commonlyinferred 12C13C abundances implying some optical depth in the12C lines This section steps through key assumptions and resultsfor this simple model We then discuss how a breakdown in thesesimple assumptions may drive our observed line ratios In particularin the case where the 12C lines are optically thick then we wouldexpect the 12C and 13C lines to emerge from distinct regions of theclouds

51 Carbon isotope abundance ratio

Table 6 lists estimate of the 12C13C abundance ratio A ratio of 89is thought to characterize the Solar system at its formation (Wilsonamp Rood 1994) while local clouds show sim40ndash60 the Milky Waycentre shows a low ratio of sim25 and starburst galaxies at low andhigh redshift can reach ratios of sim100 This value varies at leastin part because 12C and 13C are produced by different phases ofstellar evolution 13C is only created primarily as a result of evolvedintermediate-mass stars whereas very massive stars are the onesthat produce 12C at the end of their lives As a consequence the12C13C ratio depends on the recent star formation history tendingto be high in regions of recent star formation and then droppingwith time

We measure ratios for the central regions of spiral galaxies Thesetend to be actively star-forming regions but not as extreme as thosein Ultra Luminous Infrared Galaxies (ULIRGs) Based on the rangeof literature values we adopt 50 as our fiducial 12C13C ratio sinceits determination is not possible with the present data set but we

note that this is likely uncertain by a factor of sim2 We considerthis effect in our EMPIRE and ALMA sample in Jimenez-Donaireet al (2016) Several effects can affect this ratio like selective pho-todissociation or isotopic fractionation While the strong interstellarradiation field emitted from OB stars especially in strong starburstsdoes not dissociate 12CO in large amounts due to self-shielding (egvan Dishoeck amp Black 1988) 13CO may be more severely affected(Bally amp Langer 1982) which may increase the 12C13C ratio Onthe other hand chemical fractionation in cold regions would lead topreferential formation of 13CO Qualitatively fractionation effectswould likely marginally increase the H13CO+ abundance (Langeramp Penzias 1990 Milam et al 2005) whereas H13CN and HN13Care expected to decrease (eg Roueff Loison amp Hickson 2015)

52 Optical depth in the simple case

521 Framework

In the simple case of a cloud with a single density 12C and 13C linesare evenly mixed throughout the cloud and the 12C line is opticallythick then the line ratio implies an optical depth

To estimate this optical depth we begin in the following with thesimplest case of assuming local thermodynamic equilibrium (LTE)and in particular Tex(HCN) = Tex(H13CN) Note that we will dropthis assumption in Section 53 We consider the intensity Jν ofthe 12C and 13C lines in units of brightness temperature but notnecessarily on the RayleighndashJeans tail so that

Jν = hνk

ehνkT minus 1 (1)

where ν is the observed frequency k is Boltzmannrsquos constant h isPlanckrsquos constant and T is the temperature characterizing the sourcefunction For our cloud T will be Tex in LTE Then the equationof radiative transfer gives the observed intensity Tobs in units ofbrightness temperature

Tobs = ηbf

[Jν(Tex) minus Jν(Tbg)

](1 minus exp (minusτ )) (2)

Here ηbf is the beam-filling factor the fraction of the beam area thatis filled by the emitting region Tbg is the background radiation fieldtemperature (271 K) and τ is the optical depth of the transition inquestion Note that ηbf the beam-filling factor is expected to be avery small number bringing our observed intensities from sim10 Kdown to the observed small fraction of a Kelvin

If we make the simplifying assumption that the 12C lines egHCN (1-0) are optically thick then 1 minus exp (minusτ ) asymp 1 and

T 12Cobs = η12C

bf

[Jν12C(Tex) minus Jν12C(Tbg)

] (3)

where Jν 12C refers to equation (1) evaluated at the frequency of the12C line On the other hand the 13C line has unknown optical depthso that

T 13Cobs = η13C

bf

[Jν13C(Tex) minus Jν13C(Tbg)

] (1 minus exp (minusτ 13C)

) (4)

Here we have distinguished the frequency of the 13C line from thatof the 12C line above The equations also allow that the two linesmight have different filling factors and so distinguish η13C

bf fromη12C

bf The optical depth of the 13C line is unknown However ourobserved ratios are far from sim1 which would be expected for bothlines to be optically thick and having matched ηbf Thus it appearsunlikely that both lines are optically thick Therefore we expect(1minusexp minus τ 13C) asymp τ 13C in most cases

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

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Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

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W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

54 M J Jimenez-Donaire et al

Figure 3 Stacked H13CN and H13CO+ spectra for the bright inner regions of our target galaxies We show scaled versions of the stacked HCN and HCO+spectra from Fig 2 for reference These spectra show the expectation for the 12C-to-13C line ratios of 1-to-5 to 1-to-10 common values for the 12CO-to-13COratio in galaxy discs The red line indicates our working upper limit (dashed) or fit (solid) to the isotopologue spectrum Limits and fit parameters are reportedin Table 3

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 55

Figure 3 ndash continued

intensity of 13CO gt HCN gt HCO+ In NGC 5194 where we alsomeasure HNC this line is the weakest of the four

Fig 3 shows the stacked spectra for the 13C isotopologues of thedense gas tracers Here scaled versions of the 12C lines appear forreference The 13C isotopologues of the dense gas tracers are all farfainter than the 12C lines We do securely detect H13CN in NGC3351 NGC 3627 and NGC 253 In the other targets we place firmupper limits on the intensity of these lines We detect H13CO+ onlyin NGC 253 placing limits in the other systems

41 Line ratios

Table 4 reports the line ratios averaged over the active regionswhich are the main objective of our investigations We measureHCNH13CN integrated intensity ratios ranging from 7to21 (see

Table 4 for more details) and we constrain the ratio to be 6 forthe entire sample For HCO+H13CO+ we constrain the ratio to be7 in our sample and we measure a ratio of 24 in NGC 253 Forcomparison 12CO13CO ratios of sim6ndash10 are typically measuredin the discs of galaxies including the Milky Way and M51 The12C13C ratios for the dense gas tracers are significantly higher thanthose for CO in our targets Possible explanations are lower opticaldepth in the dense gas tracers isotopic abundance variations in theactive parts of galaxies or different filling factors for the 12C and13C lines We will discuss this further in Section 5

How do our measured ratios compare to previous measurementsTable 5 summarizes the literature measurements of isotopologueline ratios from the Milky Way and nearby galaxies including twothat overlap our targets (NGC 253 and NGC 5194) Broadly theseagree with our measurements with ratios often 10 and sometimes

MNRAS 466 49ndash62 (2017)

56 M J Jimenez-Donaire et al

Table 3 Spectral line parameters for the galaxies derived from the stacked spectra in Figs 2 and 3

Galaxy Molecule Integrated intensity Line width Peak rms(K km sminus1) (km sminus1) times 10minus2(K) (mK)

NGC 3351 13CO 120 plusmn 01 62 plusmn 10 194 plusmn 04 11HCN 84 plusmn 01 67 plusmn 11 118 plusmn 06 15

H13CN 04 plusmn 01 59 plusmn 10 07 plusmn 01 09HCO+ 50 plusmn 01 64 plusmn 11 74 plusmn 02 14

H13CO+ lt03 ndash lt04 13NGC 3627 13CO 74 plusmn 01 121 plusmn 14 68 plusmn 04 10

HCN 51 plusmn 01 129 plusmn 14 45 plusmn 01 08H13CN 07 plusmn 01 170 plusmn 17 04 plusmn 01 10HCO+ 49 plusmn 01 128 plusmn 14 35 plusmn 01 10

H13CO+ lt04 ndash lt05 14NGC 4254 13CO 85 plusmn 02 44 plusmn 9 181 plusmn 04 23

HCN 29 plusmn 02 48 plusmn 9 55 plusmn 02 33H13CN lt05 ndash lt13 40HCO+ 20 plusmn 01 44 plusmn 9 38 plusmn 01 18

H13CO+ lt03 ndash lt10 19NGC 4321 13CO 107 plusmn 01 55 plusmn 10 193 plusmn 04 11

HCN 69 plusmn 01 69 plusmn 11 97 plusmn 02 14H13CN lt03 ndash lt10 16HCO+ 46 plusmn 01 63 plusmn 11 68 plusmn 02 14

H13CO+ lt03 ndash lt10 13NGC 5194 13CO 46 plusmn 01 71 plusmn 6 64 plusmn 02 09

HCN 22 plusmn 01 76 plusmn 8 27 plusmn 01 06H13CN lt 02 ndash lt02 07HCO+ 14 plusmn 01 78 plusmn 10 16 plusmn 03 06

H13CO+ lt 02 ndash lt02 07HNC 08 plusmn 01 77 plusmn 8 11 plusmn 02 08

HN13C lt 02 ndash lt02 09NGC 253 13CO 312 plusmn 9 25 plusmn 6 1250 plusmn 40 40

HCN 32 plusmn 2 22 plusmn 4 120 plusmn 4 90H13CN 19 plusmn 04 19 plusmn 5 90 plusmn 03 56HCO+ 29 plusmn 3 21 plusmn 5 109 plusmn 4 92

H13CO+ 12 plusmn 04 18 plusmn 4 61 plusmn 02 80

Table 4 Computed line ratios for the galaxies

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

HCNH13CN 21 plusmn 3 7 plusmn 1 gt6 gt23 gt11 17 plusmn 1HCO+H13CO+ gt17 gt12 gt7 gt15 gt7 24 plusmn 2HNCHN13C ndash ndash ndash ndash gt4 ndashHCNHCO+ 17 plusmn 01 10 plusmn 01 15 plusmn 01 15 plusmn 01 16 plusmn 01 11 plusmn 0113COHCN 14 plusmn 02 15 plusmn 01 30 plusmn 03 16 plusmn 02 21 plusmn 01 10 plusmn 1

Table 5 Integrated intensity ratios from the literature

Target HCNH13CN HCO+H13CO+ Ref

NGC 1068 sim16 sim20 1Orion B ndash sim25 2Sgr A ndash sim15 3Galactic Centre sim11 sim21 4NGC 5194 sim25 sim30 5NGC 253 sim15 sim19 6

References (1) Wang et al (2014) (2) Pety (private com-munication) (3) Riquelme et al (2010) (4) Harada et al(2015) (5) Watanabe et al (2014) (6) Meier et al (2015)

20 In detail Watanabe et al (2014) measured line ratios inNGC 5194 and found HCNH13CN = 27 plusmn 18 HCO+H13CO+

= 34 plusmn 29 and HNCHN13C gt 16 consistent with our resultsfor that system Using the same data that we employ for NGC

253 Meier et al (2015) found similar ratios of HCNH13CN andHCO+H13CO+ (see Leroy et al 2015) Observations of NGC 1068by Wang et al (2014) found HCO+H13CO+ = 20 plusmn 1 HCNH13CN= 16 plusmn 1 and HNCHN13C = 38 plusmn 6 This galaxy is characterizedby a strong active galactic nucleus (AGN) which should influencethe surrounding chemistry Two of our targets also host AGN NGC5194 and NGC 3627 however we do not see any significant differ-ence in the measured line ratios compared to the rest of the sampleThough we note that at the coarse resolution of our observationssuch effects may be difficult to isolate

Studies focused on Milky Way yield a similar picture Riquelmeet al (2010) carried out a large-scale survey of the Galactic Centreregion and found HCO+H13CO+ sim 15 in Sgr A similar to thosefound here Harada et al (2015) observed the Circumnuclear Discof the Galactic Centre and reported ratios of HCO+H13CO+ sim 21and HCNH13CN sim 11 In the Solar Neighbourhood Pety (privatecommunication) found HCO+H13CO+ sim 25 in Orion B

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 57

Figure 4 Comparison between the line ratios measured in our sample of galaxies and the literature values summarized in Table 5 Circles show measurementsfrom the stacked spectra in the inner regions of galaxies whereas the arrows indicate upper limits Note the non-continuous x-axis as the HCN integratedintensity for NGC 253 is much higher than for the other galaxies The red boxes show the regime of values found for the sources in the literature in Table 5agreeing largely with the values and upper limits we derive

Fig 4 plots our measured ratios along with these literature valuesRed boxes highlight an indicative range of values for each ratiosim15ndash25 for HCNH13CN and sim15ndash30 for HCO+H13CO+ Wediscuss the interpretation of these ratios in Section 5

42 Line widths

In addition to line ratios the stacked spectra allow us to compare linewidths among the different molecular lines Potential differences inthe line widths may hint at differences in the distribution of suchemission along the line of sight

Table 3 and Fig 2 show that we do not observe strong differencesamong the line widths of the dense gas tracers and the 13CO emissiontracing lower density gas Thus on the scale of our beam whichranges from a few hundred pc to asymp1 kpc (except for the much higherresolution NGC 253) dense gas tracers appear well mixed with13CO Note that at these scales we expect each beam to encompassmany individual clouds and that we stack beam by beam Thereforethe statement here supports the idea that the inter-cloud line widthfor HCN and HCO+ resembles that for 13CO There does not appearto be a large population of clouds emitting only 13CO that showdifferent velocities than the clouds emitting dense gas tracers

We do observe spatial variations in the line width When ex-ploring stacked emission from discs which we mostly omit fromthis paper we find line widths as much as a factor of asymp2 narrowerthan in galaxy centres (see Figs 2 and 5 ) And in the high spatialresolution data that we use to explore NGC 253 we find narrowerline widths than in our other targets despite the high inclinationand high degree of turbulence in this galaxy In NGC 5194 thedifference between the disc line width (sim48 km sminus1) and the centralline width (sim70 km sminus1) is almost a factor of 2 and is present forall lines The simplest explanation for this difference and a simi-lar difference shown for NGC 3627 is that the broad line widthsin the central region of the galaxy still contain large amounts ofunresolved bulk motion We expect this to mostly be rotation unre-solved by our coarse beam (ie lsquobeam smearingrsquo) but this may alsoinclude streaming motions along the spiral arms and bar or evensome contributions from the AGN (eg see Scoville et al 1998

Figure 5 Stacked spectra across the entire molecular disc for NGC 3627and NGC 5194 excluding the central aperture (Figs 1 and 2) We show spectrafor the main dense gas tracers together with the stacked 13CO spectrum Linewidths are larger by up to a factor of sim2 in the inner parts compared to thediscs which we attribute largely to unresolved bulk motions

MNRAS 466 49ndash62 (2017)

58 M J Jimenez-Donaire et al

Table 6 Carbon isotope ratios from the literature in differentenvironments

Type of Object Galaxy 12C13C Ref

Centre Milky Way sim25 1LMC Magellanic Clouds sim50 2NGC 2024 Milky Way sim65 3Orion A Milky Way sim45 3Starburst NGC 253 sim40 4

sim80 5Starburst M 82 gt40 4

gt130 5Local ULIRGs Mrk 231Arp 220 sim100 4 6ULIRG z = 25 Cloverleaf gt 100 7

Notes (1) Guesten Henkel amp Batrla (1985) (2) Wang et al (2009)(3) Savage et al (2002) (4) Henkel et al (2014) (5) Martın et al(2010) (6) Gonzalez-Alfonso et al (2012) (7) Henkel et al (2010)

Meidt et al 2013 Matsushita et al 2015 Querejeta et al 2016)The narrower line widths in NGC 253 likely reflect the fact that weinclude less large-scale dynamical motion and perhaps that we sam-ple the turbulent cascade at an intermediate scale (eg Caldu-Primoamp Schruba 2016)

5 D ISCUSSION

Fig 4 shows our constraints on isotopologue line ratios for HCNand HCO+ The 12C-to-13C line ratios tend to be higher than thoseobserved for CO molecules but lower than typical isotopic ratiosHow should we interpret these A main motivation for this studywas to constrain the optical depth of the dense gas tracers and soto understand the importance of line trapping In the simple caseof a fixed 12C13C abundance and a cloud with a single density anduniform abundances these ratios can be translated to an opticaldepth The ratios that we observe tend to be lower than commonlyinferred 12C13C abundances implying some optical depth in the12C lines This section steps through key assumptions and resultsfor this simple model We then discuss how a breakdown in thesesimple assumptions may drive our observed line ratios In particularin the case where the 12C lines are optically thick then we wouldexpect the 12C and 13C lines to emerge from distinct regions of theclouds

51 Carbon isotope abundance ratio

Table 6 lists estimate of the 12C13C abundance ratio A ratio of 89is thought to characterize the Solar system at its formation (Wilsonamp Rood 1994) while local clouds show sim40ndash60 the Milky Waycentre shows a low ratio of sim25 and starburst galaxies at low andhigh redshift can reach ratios of sim100 This value varies at leastin part because 12C and 13C are produced by different phases ofstellar evolution 13C is only created primarily as a result of evolvedintermediate-mass stars whereas very massive stars are the onesthat produce 12C at the end of their lives As a consequence the12C13C ratio depends on the recent star formation history tendingto be high in regions of recent star formation and then droppingwith time

We measure ratios for the central regions of spiral galaxies Thesetend to be actively star-forming regions but not as extreme as thosein Ultra Luminous Infrared Galaxies (ULIRGs) Based on the rangeof literature values we adopt 50 as our fiducial 12C13C ratio sinceits determination is not possible with the present data set but we

note that this is likely uncertain by a factor of sim2 We considerthis effect in our EMPIRE and ALMA sample in Jimenez-Donaireet al (2016) Several effects can affect this ratio like selective pho-todissociation or isotopic fractionation While the strong interstellarradiation field emitted from OB stars especially in strong starburstsdoes not dissociate 12CO in large amounts due to self-shielding (egvan Dishoeck amp Black 1988) 13CO may be more severely affected(Bally amp Langer 1982) which may increase the 12C13C ratio Onthe other hand chemical fractionation in cold regions would lead topreferential formation of 13CO Qualitatively fractionation effectswould likely marginally increase the H13CO+ abundance (Langeramp Penzias 1990 Milam et al 2005) whereas H13CN and HN13Care expected to decrease (eg Roueff Loison amp Hickson 2015)

52 Optical depth in the simple case

521 Framework

In the simple case of a cloud with a single density 12C and 13C linesare evenly mixed throughout the cloud and the 12C line is opticallythick then the line ratio implies an optical depth

To estimate this optical depth we begin in the following with thesimplest case of assuming local thermodynamic equilibrium (LTE)and in particular Tex(HCN) = Tex(H13CN) Note that we will dropthis assumption in Section 53 We consider the intensity Jν ofthe 12C and 13C lines in units of brightness temperature but notnecessarily on the RayleighndashJeans tail so that

Jν = hνk

ehνkT minus 1 (1)

where ν is the observed frequency k is Boltzmannrsquos constant h isPlanckrsquos constant and T is the temperature characterizing the sourcefunction For our cloud T will be Tex in LTE Then the equationof radiative transfer gives the observed intensity Tobs in units ofbrightness temperature

Tobs = ηbf

[Jν(Tex) minus Jν(Tbg)

](1 minus exp (minusτ )) (2)

Here ηbf is the beam-filling factor the fraction of the beam area thatis filled by the emitting region Tbg is the background radiation fieldtemperature (271 K) and τ is the optical depth of the transition inquestion Note that ηbf the beam-filling factor is expected to be avery small number bringing our observed intensities from sim10 Kdown to the observed small fraction of a Kelvin

If we make the simplifying assumption that the 12C lines egHCN (1-0) are optically thick then 1 minus exp (minusτ ) asymp 1 and

T 12Cobs = η12C

bf

[Jν12C(Tex) minus Jν12C(Tbg)

] (3)

where Jν 12C refers to equation (1) evaluated at the frequency of the12C line On the other hand the 13C line has unknown optical depthso that

T 13Cobs = η13C

bf

[Jν13C(Tex) minus Jν13C(Tbg)

] (1 minus exp (minusτ 13C)

) (4)

Here we have distinguished the frequency of the 13C line from thatof the 12C line above The equations also allow that the two linesmight have different filling factors and so distinguish η13C

bf fromη12C

bf The optical depth of the 13C line is unknown However ourobserved ratios are far from sim1 which would be expected for bothlines to be optically thick and having matched ηbf Thus it appearsunlikely that both lines are optically thick Therefore we expect(1minusexp minus τ 13C) asymp τ 13C in most cases

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 55

Figure 3 ndash continued

intensity of 13CO gt HCN gt HCO+ In NGC 5194 where we alsomeasure HNC this line is the weakest of the four

Fig 3 shows the stacked spectra for the 13C isotopologues of thedense gas tracers Here scaled versions of the 12C lines appear forreference The 13C isotopologues of the dense gas tracers are all farfainter than the 12C lines We do securely detect H13CN in NGC3351 NGC 3627 and NGC 253 In the other targets we place firmupper limits on the intensity of these lines We detect H13CO+ onlyin NGC 253 placing limits in the other systems

41 Line ratios

Table 4 reports the line ratios averaged over the active regionswhich are the main objective of our investigations We measureHCNH13CN integrated intensity ratios ranging from 7to21 (see

Table 4 for more details) and we constrain the ratio to be 6 forthe entire sample For HCO+H13CO+ we constrain the ratio to be7 in our sample and we measure a ratio of 24 in NGC 253 Forcomparison 12CO13CO ratios of sim6ndash10 are typically measuredin the discs of galaxies including the Milky Way and M51 The12C13C ratios for the dense gas tracers are significantly higher thanthose for CO in our targets Possible explanations are lower opticaldepth in the dense gas tracers isotopic abundance variations in theactive parts of galaxies or different filling factors for the 12C and13C lines We will discuss this further in Section 5

How do our measured ratios compare to previous measurementsTable 5 summarizes the literature measurements of isotopologueline ratios from the Milky Way and nearby galaxies including twothat overlap our targets (NGC 253 and NGC 5194) Broadly theseagree with our measurements with ratios often 10 and sometimes

MNRAS 466 49ndash62 (2017)

56 M J Jimenez-Donaire et al

Table 3 Spectral line parameters for the galaxies derived from the stacked spectra in Figs 2 and 3

Galaxy Molecule Integrated intensity Line width Peak rms(K km sminus1) (km sminus1) times 10minus2(K) (mK)

NGC 3351 13CO 120 plusmn 01 62 plusmn 10 194 plusmn 04 11HCN 84 plusmn 01 67 plusmn 11 118 plusmn 06 15

H13CN 04 plusmn 01 59 plusmn 10 07 plusmn 01 09HCO+ 50 plusmn 01 64 plusmn 11 74 plusmn 02 14

H13CO+ lt03 ndash lt04 13NGC 3627 13CO 74 plusmn 01 121 plusmn 14 68 plusmn 04 10

HCN 51 plusmn 01 129 plusmn 14 45 plusmn 01 08H13CN 07 plusmn 01 170 plusmn 17 04 plusmn 01 10HCO+ 49 plusmn 01 128 plusmn 14 35 plusmn 01 10

H13CO+ lt04 ndash lt05 14NGC 4254 13CO 85 plusmn 02 44 plusmn 9 181 plusmn 04 23

HCN 29 plusmn 02 48 plusmn 9 55 plusmn 02 33H13CN lt05 ndash lt13 40HCO+ 20 plusmn 01 44 plusmn 9 38 plusmn 01 18

H13CO+ lt03 ndash lt10 19NGC 4321 13CO 107 plusmn 01 55 plusmn 10 193 plusmn 04 11

HCN 69 plusmn 01 69 plusmn 11 97 plusmn 02 14H13CN lt03 ndash lt10 16HCO+ 46 plusmn 01 63 plusmn 11 68 plusmn 02 14

H13CO+ lt03 ndash lt10 13NGC 5194 13CO 46 plusmn 01 71 plusmn 6 64 plusmn 02 09

HCN 22 plusmn 01 76 plusmn 8 27 plusmn 01 06H13CN lt 02 ndash lt02 07HCO+ 14 plusmn 01 78 plusmn 10 16 plusmn 03 06

H13CO+ lt 02 ndash lt02 07HNC 08 plusmn 01 77 plusmn 8 11 plusmn 02 08

HN13C lt 02 ndash lt02 09NGC 253 13CO 312 plusmn 9 25 plusmn 6 1250 plusmn 40 40

HCN 32 plusmn 2 22 plusmn 4 120 plusmn 4 90H13CN 19 plusmn 04 19 plusmn 5 90 plusmn 03 56HCO+ 29 plusmn 3 21 plusmn 5 109 plusmn 4 92

H13CO+ 12 plusmn 04 18 plusmn 4 61 plusmn 02 80

Table 4 Computed line ratios for the galaxies

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

HCNH13CN 21 plusmn 3 7 plusmn 1 gt6 gt23 gt11 17 plusmn 1HCO+H13CO+ gt17 gt12 gt7 gt15 gt7 24 plusmn 2HNCHN13C ndash ndash ndash ndash gt4 ndashHCNHCO+ 17 plusmn 01 10 plusmn 01 15 plusmn 01 15 plusmn 01 16 plusmn 01 11 plusmn 0113COHCN 14 plusmn 02 15 plusmn 01 30 plusmn 03 16 plusmn 02 21 plusmn 01 10 plusmn 1

Table 5 Integrated intensity ratios from the literature

Target HCNH13CN HCO+H13CO+ Ref

NGC 1068 sim16 sim20 1Orion B ndash sim25 2Sgr A ndash sim15 3Galactic Centre sim11 sim21 4NGC 5194 sim25 sim30 5NGC 253 sim15 sim19 6

References (1) Wang et al (2014) (2) Pety (private com-munication) (3) Riquelme et al (2010) (4) Harada et al(2015) (5) Watanabe et al (2014) (6) Meier et al (2015)

20 In detail Watanabe et al (2014) measured line ratios inNGC 5194 and found HCNH13CN = 27 plusmn 18 HCO+H13CO+

= 34 plusmn 29 and HNCHN13C gt 16 consistent with our resultsfor that system Using the same data that we employ for NGC

253 Meier et al (2015) found similar ratios of HCNH13CN andHCO+H13CO+ (see Leroy et al 2015) Observations of NGC 1068by Wang et al (2014) found HCO+H13CO+ = 20 plusmn 1 HCNH13CN= 16 plusmn 1 and HNCHN13C = 38 plusmn 6 This galaxy is characterizedby a strong active galactic nucleus (AGN) which should influencethe surrounding chemistry Two of our targets also host AGN NGC5194 and NGC 3627 however we do not see any significant differ-ence in the measured line ratios compared to the rest of the sampleThough we note that at the coarse resolution of our observationssuch effects may be difficult to isolate

Studies focused on Milky Way yield a similar picture Riquelmeet al (2010) carried out a large-scale survey of the Galactic Centreregion and found HCO+H13CO+ sim 15 in Sgr A similar to thosefound here Harada et al (2015) observed the Circumnuclear Discof the Galactic Centre and reported ratios of HCO+H13CO+ sim 21and HCNH13CN sim 11 In the Solar Neighbourhood Pety (privatecommunication) found HCO+H13CO+ sim 25 in Orion B

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 57

Figure 4 Comparison between the line ratios measured in our sample of galaxies and the literature values summarized in Table 5 Circles show measurementsfrom the stacked spectra in the inner regions of galaxies whereas the arrows indicate upper limits Note the non-continuous x-axis as the HCN integratedintensity for NGC 253 is much higher than for the other galaxies The red boxes show the regime of values found for the sources in the literature in Table 5agreeing largely with the values and upper limits we derive

Fig 4 plots our measured ratios along with these literature valuesRed boxes highlight an indicative range of values for each ratiosim15ndash25 for HCNH13CN and sim15ndash30 for HCO+H13CO+ Wediscuss the interpretation of these ratios in Section 5

42 Line widths

In addition to line ratios the stacked spectra allow us to compare linewidths among the different molecular lines Potential differences inthe line widths may hint at differences in the distribution of suchemission along the line of sight

Table 3 and Fig 2 show that we do not observe strong differencesamong the line widths of the dense gas tracers and the 13CO emissiontracing lower density gas Thus on the scale of our beam whichranges from a few hundred pc to asymp1 kpc (except for the much higherresolution NGC 253) dense gas tracers appear well mixed with13CO Note that at these scales we expect each beam to encompassmany individual clouds and that we stack beam by beam Thereforethe statement here supports the idea that the inter-cloud line widthfor HCN and HCO+ resembles that for 13CO There does not appearto be a large population of clouds emitting only 13CO that showdifferent velocities than the clouds emitting dense gas tracers

We do observe spatial variations in the line width When ex-ploring stacked emission from discs which we mostly omit fromthis paper we find line widths as much as a factor of asymp2 narrowerthan in galaxy centres (see Figs 2 and 5 ) And in the high spatialresolution data that we use to explore NGC 253 we find narrowerline widths than in our other targets despite the high inclinationand high degree of turbulence in this galaxy In NGC 5194 thedifference between the disc line width (sim48 km sminus1) and the centralline width (sim70 km sminus1) is almost a factor of 2 and is present forall lines The simplest explanation for this difference and a simi-lar difference shown for NGC 3627 is that the broad line widthsin the central region of the galaxy still contain large amounts ofunresolved bulk motion We expect this to mostly be rotation unre-solved by our coarse beam (ie lsquobeam smearingrsquo) but this may alsoinclude streaming motions along the spiral arms and bar or evensome contributions from the AGN (eg see Scoville et al 1998

Figure 5 Stacked spectra across the entire molecular disc for NGC 3627and NGC 5194 excluding the central aperture (Figs 1 and 2) We show spectrafor the main dense gas tracers together with the stacked 13CO spectrum Linewidths are larger by up to a factor of sim2 in the inner parts compared to thediscs which we attribute largely to unresolved bulk motions

MNRAS 466 49ndash62 (2017)

58 M J Jimenez-Donaire et al

Table 6 Carbon isotope ratios from the literature in differentenvironments

Type of Object Galaxy 12C13C Ref

Centre Milky Way sim25 1LMC Magellanic Clouds sim50 2NGC 2024 Milky Way sim65 3Orion A Milky Way sim45 3Starburst NGC 253 sim40 4

sim80 5Starburst M 82 gt40 4

gt130 5Local ULIRGs Mrk 231Arp 220 sim100 4 6ULIRG z = 25 Cloverleaf gt 100 7

Notes (1) Guesten Henkel amp Batrla (1985) (2) Wang et al (2009)(3) Savage et al (2002) (4) Henkel et al (2014) (5) Martın et al(2010) (6) Gonzalez-Alfonso et al (2012) (7) Henkel et al (2010)

Meidt et al 2013 Matsushita et al 2015 Querejeta et al 2016)The narrower line widths in NGC 253 likely reflect the fact that weinclude less large-scale dynamical motion and perhaps that we sam-ple the turbulent cascade at an intermediate scale (eg Caldu-Primoamp Schruba 2016)

5 D ISCUSSION

Fig 4 shows our constraints on isotopologue line ratios for HCNand HCO+ The 12C-to-13C line ratios tend to be higher than thoseobserved for CO molecules but lower than typical isotopic ratiosHow should we interpret these A main motivation for this studywas to constrain the optical depth of the dense gas tracers and soto understand the importance of line trapping In the simple caseof a fixed 12C13C abundance and a cloud with a single density anduniform abundances these ratios can be translated to an opticaldepth The ratios that we observe tend to be lower than commonlyinferred 12C13C abundances implying some optical depth in the12C lines This section steps through key assumptions and resultsfor this simple model We then discuss how a breakdown in thesesimple assumptions may drive our observed line ratios In particularin the case where the 12C lines are optically thick then we wouldexpect the 12C and 13C lines to emerge from distinct regions of theclouds

51 Carbon isotope abundance ratio

Table 6 lists estimate of the 12C13C abundance ratio A ratio of 89is thought to characterize the Solar system at its formation (Wilsonamp Rood 1994) while local clouds show sim40ndash60 the Milky Waycentre shows a low ratio of sim25 and starburst galaxies at low andhigh redshift can reach ratios of sim100 This value varies at leastin part because 12C and 13C are produced by different phases ofstellar evolution 13C is only created primarily as a result of evolvedintermediate-mass stars whereas very massive stars are the onesthat produce 12C at the end of their lives As a consequence the12C13C ratio depends on the recent star formation history tendingto be high in regions of recent star formation and then droppingwith time

We measure ratios for the central regions of spiral galaxies Thesetend to be actively star-forming regions but not as extreme as thosein Ultra Luminous Infrared Galaxies (ULIRGs) Based on the rangeof literature values we adopt 50 as our fiducial 12C13C ratio sinceits determination is not possible with the present data set but we

note that this is likely uncertain by a factor of sim2 We considerthis effect in our EMPIRE and ALMA sample in Jimenez-Donaireet al (2016) Several effects can affect this ratio like selective pho-todissociation or isotopic fractionation While the strong interstellarradiation field emitted from OB stars especially in strong starburstsdoes not dissociate 12CO in large amounts due to self-shielding (egvan Dishoeck amp Black 1988) 13CO may be more severely affected(Bally amp Langer 1982) which may increase the 12C13C ratio Onthe other hand chemical fractionation in cold regions would lead topreferential formation of 13CO Qualitatively fractionation effectswould likely marginally increase the H13CO+ abundance (Langeramp Penzias 1990 Milam et al 2005) whereas H13CN and HN13Care expected to decrease (eg Roueff Loison amp Hickson 2015)

52 Optical depth in the simple case

521 Framework

In the simple case of a cloud with a single density 12C and 13C linesare evenly mixed throughout the cloud and the 12C line is opticallythick then the line ratio implies an optical depth

To estimate this optical depth we begin in the following with thesimplest case of assuming local thermodynamic equilibrium (LTE)and in particular Tex(HCN) = Tex(H13CN) Note that we will dropthis assumption in Section 53 We consider the intensity Jν ofthe 12C and 13C lines in units of brightness temperature but notnecessarily on the RayleighndashJeans tail so that

Jν = hνk

ehνkT minus 1 (1)

where ν is the observed frequency k is Boltzmannrsquos constant h isPlanckrsquos constant and T is the temperature characterizing the sourcefunction For our cloud T will be Tex in LTE Then the equationof radiative transfer gives the observed intensity Tobs in units ofbrightness temperature

Tobs = ηbf

[Jν(Tex) minus Jν(Tbg)

](1 minus exp (minusτ )) (2)

Here ηbf is the beam-filling factor the fraction of the beam area thatis filled by the emitting region Tbg is the background radiation fieldtemperature (271 K) and τ is the optical depth of the transition inquestion Note that ηbf the beam-filling factor is expected to be avery small number bringing our observed intensities from sim10 Kdown to the observed small fraction of a Kelvin

If we make the simplifying assumption that the 12C lines egHCN (1-0) are optically thick then 1 minus exp (minusτ ) asymp 1 and

T 12Cobs = η12C

bf

[Jν12C(Tex) minus Jν12C(Tbg)

] (3)

where Jν 12C refers to equation (1) evaluated at the frequency of the12C line On the other hand the 13C line has unknown optical depthso that

T 13Cobs = η13C

bf

[Jν13C(Tex) minus Jν13C(Tbg)

] (1 minus exp (minusτ 13C)

) (4)

Here we have distinguished the frequency of the 13C line from thatof the 12C line above The equations also allow that the two linesmight have different filling factors and so distinguish η13C

bf fromη12C

bf The optical depth of the 13C line is unknown However ourobserved ratios are far from sim1 which would be expected for bothlines to be optically thick and having matched ηbf Thus it appearsunlikely that both lines are optically thick Therefore we expect(1minusexp minus τ 13C) asymp τ 13C in most cases

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

56 M J Jimenez-Donaire et al

Table 3 Spectral line parameters for the galaxies derived from the stacked spectra in Figs 2 and 3

Galaxy Molecule Integrated intensity Line width Peak rms(K km sminus1) (km sminus1) times 10minus2(K) (mK)

NGC 3351 13CO 120 plusmn 01 62 plusmn 10 194 plusmn 04 11HCN 84 plusmn 01 67 plusmn 11 118 plusmn 06 15

H13CN 04 plusmn 01 59 plusmn 10 07 plusmn 01 09HCO+ 50 plusmn 01 64 plusmn 11 74 plusmn 02 14

H13CO+ lt03 ndash lt04 13NGC 3627 13CO 74 plusmn 01 121 plusmn 14 68 plusmn 04 10

HCN 51 plusmn 01 129 plusmn 14 45 plusmn 01 08H13CN 07 plusmn 01 170 plusmn 17 04 plusmn 01 10HCO+ 49 plusmn 01 128 plusmn 14 35 plusmn 01 10

H13CO+ lt04 ndash lt05 14NGC 4254 13CO 85 plusmn 02 44 plusmn 9 181 plusmn 04 23

HCN 29 plusmn 02 48 plusmn 9 55 plusmn 02 33H13CN lt05 ndash lt13 40HCO+ 20 plusmn 01 44 plusmn 9 38 plusmn 01 18

H13CO+ lt03 ndash lt10 19NGC 4321 13CO 107 plusmn 01 55 plusmn 10 193 plusmn 04 11

HCN 69 plusmn 01 69 plusmn 11 97 plusmn 02 14H13CN lt03 ndash lt10 16HCO+ 46 plusmn 01 63 plusmn 11 68 plusmn 02 14

H13CO+ lt03 ndash lt10 13NGC 5194 13CO 46 plusmn 01 71 plusmn 6 64 plusmn 02 09

HCN 22 plusmn 01 76 plusmn 8 27 plusmn 01 06H13CN lt 02 ndash lt02 07HCO+ 14 plusmn 01 78 plusmn 10 16 plusmn 03 06

H13CO+ lt 02 ndash lt02 07HNC 08 plusmn 01 77 plusmn 8 11 plusmn 02 08

HN13C lt 02 ndash lt02 09NGC 253 13CO 312 plusmn 9 25 plusmn 6 1250 plusmn 40 40

HCN 32 plusmn 2 22 plusmn 4 120 plusmn 4 90H13CN 19 plusmn 04 19 plusmn 5 90 plusmn 03 56HCO+ 29 plusmn 3 21 plusmn 5 109 plusmn 4 92

H13CO+ 12 plusmn 04 18 plusmn 4 61 plusmn 02 80

Table 4 Computed line ratios for the galaxies

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

HCNH13CN 21 plusmn 3 7 plusmn 1 gt6 gt23 gt11 17 plusmn 1HCO+H13CO+ gt17 gt12 gt7 gt15 gt7 24 plusmn 2HNCHN13C ndash ndash ndash ndash gt4 ndashHCNHCO+ 17 plusmn 01 10 plusmn 01 15 plusmn 01 15 plusmn 01 16 plusmn 01 11 plusmn 0113COHCN 14 plusmn 02 15 plusmn 01 30 plusmn 03 16 plusmn 02 21 plusmn 01 10 plusmn 1

Table 5 Integrated intensity ratios from the literature

Target HCNH13CN HCO+H13CO+ Ref

NGC 1068 sim16 sim20 1Orion B ndash sim25 2Sgr A ndash sim15 3Galactic Centre sim11 sim21 4NGC 5194 sim25 sim30 5NGC 253 sim15 sim19 6

References (1) Wang et al (2014) (2) Pety (private com-munication) (3) Riquelme et al (2010) (4) Harada et al(2015) (5) Watanabe et al (2014) (6) Meier et al (2015)

20 In detail Watanabe et al (2014) measured line ratios inNGC 5194 and found HCNH13CN = 27 plusmn 18 HCO+H13CO+

= 34 plusmn 29 and HNCHN13C gt 16 consistent with our resultsfor that system Using the same data that we employ for NGC

253 Meier et al (2015) found similar ratios of HCNH13CN andHCO+H13CO+ (see Leroy et al 2015) Observations of NGC 1068by Wang et al (2014) found HCO+H13CO+ = 20 plusmn 1 HCNH13CN= 16 plusmn 1 and HNCHN13C = 38 plusmn 6 This galaxy is characterizedby a strong active galactic nucleus (AGN) which should influencethe surrounding chemistry Two of our targets also host AGN NGC5194 and NGC 3627 however we do not see any significant differ-ence in the measured line ratios compared to the rest of the sampleThough we note that at the coarse resolution of our observationssuch effects may be difficult to isolate

Studies focused on Milky Way yield a similar picture Riquelmeet al (2010) carried out a large-scale survey of the Galactic Centreregion and found HCO+H13CO+ sim 15 in Sgr A similar to thosefound here Harada et al (2015) observed the Circumnuclear Discof the Galactic Centre and reported ratios of HCO+H13CO+ sim 21and HCNH13CN sim 11 In the Solar Neighbourhood Pety (privatecommunication) found HCO+H13CO+ sim 25 in Orion B

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 57

Figure 4 Comparison between the line ratios measured in our sample of galaxies and the literature values summarized in Table 5 Circles show measurementsfrom the stacked spectra in the inner regions of galaxies whereas the arrows indicate upper limits Note the non-continuous x-axis as the HCN integratedintensity for NGC 253 is much higher than for the other galaxies The red boxes show the regime of values found for the sources in the literature in Table 5agreeing largely with the values and upper limits we derive

Fig 4 plots our measured ratios along with these literature valuesRed boxes highlight an indicative range of values for each ratiosim15ndash25 for HCNH13CN and sim15ndash30 for HCO+H13CO+ Wediscuss the interpretation of these ratios in Section 5

42 Line widths

In addition to line ratios the stacked spectra allow us to compare linewidths among the different molecular lines Potential differences inthe line widths may hint at differences in the distribution of suchemission along the line of sight

Table 3 and Fig 2 show that we do not observe strong differencesamong the line widths of the dense gas tracers and the 13CO emissiontracing lower density gas Thus on the scale of our beam whichranges from a few hundred pc to asymp1 kpc (except for the much higherresolution NGC 253) dense gas tracers appear well mixed with13CO Note that at these scales we expect each beam to encompassmany individual clouds and that we stack beam by beam Thereforethe statement here supports the idea that the inter-cloud line widthfor HCN and HCO+ resembles that for 13CO There does not appearto be a large population of clouds emitting only 13CO that showdifferent velocities than the clouds emitting dense gas tracers

We do observe spatial variations in the line width When ex-ploring stacked emission from discs which we mostly omit fromthis paper we find line widths as much as a factor of asymp2 narrowerthan in galaxy centres (see Figs 2 and 5 ) And in the high spatialresolution data that we use to explore NGC 253 we find narrowerline widths than in our other targets despite the high inclinationand high degree of turbulence in this galaxy In NGC 5194 thedifference between the disc line width (sim48 km sminus1) and the centralline width (sim70 km sminus1) is almost a factor of 2 and is present forall lines The simplest explanation for this difference and a simi-lar difference shown for NGC 3627 is that the broad line widthsin the central region of the galaxy still contain large amounts ofunresolved bulk motion We expect this to mostly be rotation unre-solved by our coarse beam (ie lsquobeam smearingrsquo) but this may alsoinclude streaming motions along the spiral arms and bar or evensome contributions from the AGN (eg see Scoville et al 1998

Figure 5 Stacked spectra across the entire molecular disc for NGC 3627and NGC 5194 excluding the central aperture (Figs 1 and 2) We show spectrafor the main dense gas tracers together with the stacked 13CO spectrum Linewidths are larger by up to a factor of sim2 in the inner parts compared to thediscs which we attribute largely to unresolved bulk motions

MNRAS 466 49ndash62 (2017)

58 M J Jimenez-Donaire et al

Table 6 Carbon isotope ratios from the literature in differentenvironments

Type of Object Galaxy 12C13C Ref

Centre Milky Way sim25 1LMC Magellanic Clouds sim50 2NGC 2024 Milky Way sim65 3Orion A Milky Way sim45 3Starburst NGC 253 sim40 4

sim80 5Starburst M 82 gt40 4

gt130 5Local ULIRGs Mrk 231Arp 220 sim100 4 6ULIRG z = 25 Cloverleaf gt 100 7

Notes (1) Guesten Henkel amp Batrla (1985) (2) Wang et al (2009)(3) Savage et al (2002) (4) Henkel et al (2014) (5) Martın et al(2010) (6) Gonzalez-Alfonso et al (2012) (7) Henkel et al (2010)

Meidt et al 2013 Matsushita et al 2015 Querejeta et al 2016)The narrower line widths in NGC 253 likely reflect the fact that weinclude less large-scale dynamical motion and perhaps that we sam-ple the turbulent cascade at an intermediate scale (eg Caldu-Primoamp Schruba 2016)

5 D ISCUSSION

Fig 4 shows our constraints on isotopologue line ratios for HCNand HCO+ The 12C-to-13C line ratios tend to be higher than thoseobserved for CO molecules but lower than typical isotopic ratiosHow should we interpret these A main motivation for this studywas to constrain the optical depth of the dense gas tracers and soto understand the importance of line trapping In the simple caseof a fixed 12C13C abundance and a cloud with a single density anduniform abundances these ratios can be translated to an opticaldepth The ratios that we observe tend to be lower than commonlyinferred 12C13C abundances implying some optical depth in the12C lines This section steps through key assumptions and resultsfor this simple model We then discuss how a breakdown in thesesimple assumptions may drive our observed line ratios In particularin the case where the 12C lines are optically thick then we wouldexpect the 12C and 13C lines to emerge from distinct regions of theclouds

51 Carbon isotope abundance ratio

Table 6 lists estimate of the 12C13C abundance ratio A ratio of 89is thought to characterize the Solar system at its formation (Wilsonamp Rood 1994) while local clouds show sim40ndash60 the Milky Waycentre shows a low ratio of sim25 and starburst galaxies at low andhigh redshift can reach ratios of sim100 This value varies at leastin part because 12C and 13C are produced by different phases ofstellar evolution 13C is only created primarily as a result of evolvedintermediate-mass stars whereas very massive stars are the onesthat produce 12C at the end of their lives As a consequence the12C13C ratio depends on the recent star formation history tendingto be high in regions of recent star formation and then droppingwith time

We measure ratios for the central regions of spiral galaxies Thesetend to be actively star-forming regions but not as extreme as thosein Ultra Luminous Infrared Galaxies (ULIRGs) Based on the rangeof literature values we adopt 50 as our fiducial 12C13C ratio sinceits determination is not possible with the present data set but we

note that this is likely uncertain by a factor of sim2 We considerthis effect in our EMPIRE and ALMA sample in Jimenez-Donaireet al (2016) Several effects can affect this ratio like selective pho-todissociation or isotopic fractionation While the strong interstellarradiation field emitted from OB stars especially in strong starburstsdoes not dissociate 12CO in large amounts due to self-shielding (egvan Dishoeck amp Black 1988) 13CO may be more severely affected(Bally amp Langer 1982) which may increase the 12C13C ratio Onthe other hand chemical fractionation in cold regions would lead topreferential formation of 13CO Qualitatively fractionation effectswould likely marginally increase the H13CO+ abundance (Langeramp Penzias 1990 Milam et al 2005) whereas H13CN and HN13Care expected to decrease (eg Roueff Loison amp Hickson 2015)

52 Optical depth in the simple case

521 Framework

In the simple case of a cloud with a single density 12C and 13C linesare evenly mixed throughout the cloud and the 12C line is opticallythick then the line ratio implies an optical depth

To estimate this optical depth we begin in the following with thesimplest case of assuming local thermodynamic equilibrium (LTE)and in particular Tex(HCN) = Tex(H13CN) Note that we will dropthis assumption in Section 53 We consider the intensity Jν ofthe 12C and 13C lines in units of brightness temperature but notnecessarily on the RayleighndashJeans tail so that

Jν = hνk

ehνkT minus 1 (1)

where ν is the observed frequency k is Boltzmannrsquos constant h isPlanckrsquos constant and T is the temperature characterizing the sourcefunction For our cloud T will be Tex in LTE Then the equationof radiative transfer gives the observed intensity Tobs in units ofbrightness temperature

Tobs = ηbf

[Jν(Tex) minus Jν(Tbg)

](1 minus exp (minusτ )) (2)

Here ηbf is the beam-filling factor the fraction of the beam area thatis filled by the emitting region Tbg is the background radiation fieldtemperature (271 K) and τ is the optical depth of the transition inquestion Note that ηbf the beam-filling factor is expected to be avery small number bringing our observed intensities from sim10 Kdown to the observed small fraction of a Kelvin

If we make the simplifying assumption that the 12C lines egHCN (1-0) are optically thick then 1 minus exp (minusτ ) asymp 1 and

T 12Cobs = η12C

bf

[Jν12C(Tex) minus Jν12C(Tbg)

] (3)

where Jν 12C refers to equation (1) evaluated at the frequency of the12C line On the other hand the 13C line has unknown optical depthso that

T 13Cobs = η13C

bf

[Jν13C(Tex) minus Jν13C(Tbg)

] (1 minus exp (minusτ 13C)

) (4)

Here we have distinguished the frequency of the 13C line from thatof the 12C line above The equations also allow that the two linesmight have different filling factors and so distinguish η13C

bf fromη12C

bf The optical depth of the 13C line is unknown However ourobserved ratios are far from sim1 which would be expected for bothlines to be optically thick and having matched ηbf Thus it appearsunlikely that both lines are optically thick Therefore we expect(1minusexp minus τ 13C) asymp τ 13C in most cases

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 57

Figure 4 Comparison between the line ratios measured in our sample of galaxies and the literature values summarized in Table 5 Circles show measurementsfrom the stacked spectra in the inner regions of galaxies whereas the arrows indicate upper limits Note the non-continuous x-axis as the HCN integratedintensity for NGC 253 is much higher than for the other galaxies The red boxes show the regime of values found for the sources in the literature in Table 5agreeing largely with the values and upper limits we derive

Fig 4 plots our measured ratios along with these literature valuesRed boxes highlight an indicative range of values for each ratiosim15ndash25 for HCNH13CN and sim15ndash30 for HCO+H13CO+ Wediscuss the interpretation of these ratios in Section 5

42 Line widths

In addition to line ratios the stacked spectra allow us to compare linewidths among the different molecular lines Potential differences inthe line widths may hint at differences in the distribution of suchemission along the line of sight

Table 3 and Fig 2 show that we do not observe strong differencesamong the line widths of the dense gas tracers and the 13CO emissiontracing lower density gas Thus on the scale of our beam whichranges from a few hundred pc to asymp1 kpc (except for the much higherresolution NGC 253) dense gas tracers appear well mixed with13CO Note that at these scales we expect each beam to encompassmany individual clouds and that we stack beam by beam Thereforethe statement here supports the idea that the inter-cloud line widthfor HCN and HCO+ resembles that for 13CO There does not appearto be a large population of clouds emitting only 13CO that showdifferent velocities than the clouds emitting dense gas tracers

We do observe spatial variations in the line width When ex-ploring stacked emission from discs which we mostly omit fromthis paper we find line widths as much as a factor of asymp2 narrowerthan in galaxy centres (see Figs 2 and 5 ) And in the high spatialresolution data that we use to explore NGC 253 we find narrowerline widths than in our other targets despite the high inclinationand high degree of turbulence in this galaxy In NGC 5194 thedifference between the disc line width (sim48 km sminus1) and the centralline width (sim70 km sminus1) is almost a factor of 2 and is present forall lines The simplest explanation for this difference and a simi-lar difference shown for NGC 3627 is that the broad line widthsin the central region of the galaxy still contain large amounts ofunresolved bulk motion We expect this to mostly be rotation unre-solved by our coarse beam (ie lsquobeam smearingrsquo) but this may alsoinclude streaming motions along the spiral arms and bar or evensome contributions from the AGN (eg see Scoville et al 1998

Figure 5 Stacked spectra across the entire molecular disc for NGC 3627and NGC 5194 excluding the central aperture (Figs 1 and 2) We show spectrafor the main dense gas tracers together with the stacked 13CO spectrum Linewidths are larger by up to a factor of sim2 in the inner parts compared to thediscs which we attribute largely to unresolved bulk motions

MNRAS 466 49ndash62 (2017)

58 M J Jimenez-Donaire et al

Table 6 Carbon isotope ratios from the literature in differentenvironments

Type of Object Galaxy 12C13C Ref

Centre Milky Way sim25 1LMC Magellanic Clouds sim50 2NGC 2024 Milky Way sim65 3Orion A Milky Way sim45 3Starburst NGC 253 sim40 4

sim80 5Starburst M 82 gt40 4

gt130 5Local ULIRGs Mrk 231Arp 220 sim100 4 6ULIRG z = 25 Cloverleaf gt 100 7

Notes (1) Guesten Henkel amp Batrla (1985) (2) Wang et al (2009)(3) Savage et al (2002) (4) Henkel et al (2014) (5) Martın et al(2010) (6) Gonzalez-Alfonso et al (2012) (7) Henkel et al (2010)

Meidt et al 2013 Matsushita et al 2015 Querejeta et al 2016)The narrower line widths in NGC 253 likely reflect the fact that weinclude less large-scale dynamical motion and perhaps that we sam-ple the turbulent cascade at an intermediate scale (eg Caldu-Primoamp Schruba 2016)

5 D ISCUSSION

Fig 4 shows our constraints on isotopologue line ratios for HCNand HCO+ The 12C-to-13C line ratios tend to be higher than thoseobserved for CO molecules but lower than typical isotopic ratiosHow should we interpret these A main motivation for this studywas to constrain the optical depth of the dense gas tracers and soto understand the importance of line trapping In the simple caseof a fixed 12C13C abundance and a cloud with a single density anduniform abundances these ratios can be translated to an opticaldepth The ratios that we observe tend to be lower than commonlyinferred 12C13C abundances implying some optical depth in the12C lines This section steps through key assumptions and resultsfor this simple model We then discuss how a breakdown in thesesimple assumptions may drive our observed line ratios In particularin the case where the 12C lines are optically thick then we wouldexpect the 12C and 13C lines to emerge from distinct regions of theclouds

51 Carbon isotope abundance ratio

Table 6 lists estimate of the 12C13C abundance ratio A ratio of 89is thought to characterize the Solar system at its formation (Wilsonamp Rood 1994) while local clouds show sim40ndash60 the Milky Waycentre shows a low ratio of sim25 and starburst galaxies at low andhigh redshift can reach ratios of sim100 This value varies at leastin part because 12C and 13C are produced by different phases ofstellar evolution 13C is only created primarily as a result of evolvedintermediate-mass stars whereas very massive stars are the onesthat produce 12C at the end of their lives As a consequence the12C13C ratio depends on the recent star formation history tendingto be high in regions of recent star formation and then droppingwith time

We measure ratios for the central regions of spiral galaxies Thesetend to be actively star-forming regions but not as extreme as thosein Ultra Luminous Infrared Galaxies (ULIRGs) Based on the rangeof literature values we adopt 50 as our fiducial 12C13C ratio sinceits determination is not possible with the present data set but we

note that this is likely uncertain by a factor of sim2 We considerthis effect in our EMPIRE and ALMA sample in Jimenez-Donaireet al (2016) Several effects can affect this ratio like selective pho-todissociation or isotopic fractionation While the strong interstellarradiation field emitted from OB stars especially in strong starburstsdoes not dissociate 12CO in large amounts due to self-shielding (egvan Dishoeck amp Black 1988) 13CO may be more severely affected(Bally amp Langer 1982) which may increase the 12C13C ratio Onthe other hand chemical fractionation in cold regions would lead topreferential formation of 13CO Qualitatively fractionation effectswould likely marginally increase the H13CO+ abundance (Langeramp Penzias 1990 Milam et al 2005) whereas H13CN and HN13Care expected to decrease (eg Roueff Loison amp Hickson 2015)

52 Optical depth in the simple case

521 Framework

In the simple case of a cloud with a single density 12C and 13C linesare evenly mixed throughout the cloud and the 12C line is opticallythick then the line ratio implies an optical depth

To estimate this optical depth we begin in the following with thesimplest case of assuming local thermodynamic equilibrium (LTE)and in particular Tex(HCN) = Tex(H13CN) Note that we will dropthis assumption in Section 53 We consider the intensity Jν ofthe 12C and 13C lines in units of brightness temperature but notnecessarily on the RayleighndashJeans tail so that

Jν = hνk

ehνkT minus 1 (1)

where ν is the observed frequency k is Boltzmannrsquos constant h isPlanckrsquos constant and T is the temperature characterizing the sourcefunction For our cloud T will be Tex in LTE Then the equationof radiative transfer gives the observed intensity Tobs in units ofbrightness temperature

Tobs = ηbf

[Jν(Tex) minus Jν(Tbg)

](1 minus exp (minusτ )) (2)

Here ηbf is the beam-filling factor the fraction of the beam area thatis filled by the emitting region Tbg is the background radiation fieldtemperature (271 K) and τ is the optical depth of the transition inquestion Note that ηbf the beam-filling factor is expected to be avery small number bringing our observed intensities from sim10 Kdown to the observed small fraction of a Kelvin

If we make the simplifying assumption that the 12C lines egHCN (1-0) are optically thick then 1 minus exp (minusτ ) asymp 1 and

T 12Cobs = η12C

bf

[Jν12C(Tex) minus Jν12C(Tbg)

] (3)

where Jν 12C refers to equation (1) evaluated at the frequency of the12C line On the other hand the 13C line has unknown optical depthso that

T 13Cobs = η13C

bf

[Jν13C(Tex) minus Jν13C(Tbg)

] (1 minus exp (minusτ 13C)

) (4)

Here we have distinguished the frequency of the 13C line from thatof the 12C line above The equations also allow that the two linesmight have different filling factors and so distinguish η13C

bf fromη12C

bf The optical depth of the 13C line is unknown However ourobserved ratios are far from sim1 which would be expected for bothlines to be optically thick and having matched ηbf Thus it appearsunlikely that both lines are optically thick Therefore we expect(1minusexp minus τ 13C) asymp τ 13C in most cases

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

58 M J Jimenez-Donaire et al

Table 6 Carbon isotope ratios from the literature in differentenvironments

Type of Object Galaxy 12C13C Ref

Centre Milky Way sim25 1LMC Magellanic Clouds sim50 2NGC 2024 Milky Way sim65 3Orion A Milky Way sim45 3Starburst NGC 253 sim40 4

sim80 5Starburst M 82 gt40 4

gt130 5Local ULIRGs Mrk 231Arp 220 sim100 4 6ULIRG z = 25 Cloverleaf gt 100 7

Notes (1) Guesten Henkel amp Batrla (1985) (2) Wang et al (2009)(3) Savage et al (2002) (4) Henkel et al (2014) (5) Martın et al(2010) (6) Gonzalez-Alfonso et al (2012) (7) Henkel et al (2010)

Meidt et al 2013 Matsushita et al 2015 Querejeta et al 2016)The narrower line widths in NGC 253 likely reflect the fact that weinclude less large-scale dynamical motion and perhaps that we sam-ple the turbulent cascade at an intermediate scale (eg Caldu-Primoamp Schruba 2016)

5 D ISCUSSION

Fig 4 shows our constraints on isotopologue line ratios for HCNand HCO+ The 12C-to-13C line ratios tend to be higher than thoseobserved for CO molecules but lower than typical isotopic ratiosHow should we interpret these A main motivation for this studywas to constrain the optical depth of the dense gas tracers and soto understand the importance of line trapping In the simple caseof a fixed 12C13C abundance and a cloud with a single density anduniform abundances these ratios can be translated to an opticaldepth The ratios that we observe tend to be lower than commonlyinferred 12C13C abundances implying some optical depth in the12C lines This section steps through key assumptions and resultsfor this simple model We then discuss how a breakdown in thesesimple assumptions may drive our observed line ratios In particularin the case where the 12C lines are optically thick then we wouldexpect the 12C and 13C lines to emerge from distinct regions of theclouds

51 Carbon isotope abundance ratio

Table 6 lists estimate of the 12C13C abundance ratio A ratio of 89is thought to characterize the Solar system at its formation (Wilsonamp Rood 1994) while local clouds show sim40ndash60 the Milky Waycentre shows a low ratio of sim25 and starburst galaxies at low andhigh redshift can reach ratios of sim100 This value varies at leastin part because 12C and 13C are produced by different phases ofstellar evolution 13C is only created primarily as a result of evolvedintermediate-mass stars whereas very massive stars are the onesthat produce 12C at the end of their lives As a consequence the12C13C ratio depends on the recent star formation history tendingto be high in regions of recent star formation and then droppingwith time

We measure ratios for the central regions of spiral galaxies Thesetend to be actively star-forming regions but not as extreme as thosein Ultra Luminous Infrared Galaxies (ULIRGs) Based on the rangeof literature values we adopt 50 as our fiducial 12C13C ratio sinceits determination is not possible with the present data set but we

note that this is likely uncertain by a factor of sim2 We considerthis effect in our EMPIRE and ALMA sample in Jimenez-Donaireet al (2016) Several effects can affect this ratio like selective pho-todissociation or isotopic fractionation While the strong interstellarradiation field emitted from OB stars especially in strong starburstsdoes not dissociate 12CO in large amounts due to self-shielding (egvan Dishoeck amp Black 1988) 13CO may be more severely affected(Bally amp Langer 1982) which may increase the 12C13C ratio Onthe other hand chemical fractionation in cold regions would lead topreferential formation of 13CO Qualitatively fractionation effectswould likely marginally increase the H13CO+ abundance (Langeramp Penzias 1990 Milam et al 2005) whereas H13CN and HN13Care expected to decrease (eg Roueff Loison amp Hickson 2015)

52 Optical depth in the simple case

521 Framework

In the simple case of a cloud with a single density 12C and 13C linesare evenly mixed throughout the cloud and the 12C line is opticallythick then the line ratio implies an optical depth

To estimate this optical depth we begin in the following with thesimplest case of assuming local thermodynamic equilibrium (LTE)and in particular Tex(HCN) = Tex(H13CN) Note that we will dropthis assumption in Section 53 We consider the intensity Jν ofthe 12C and 13C lines in units of brightness temperature but notnecessarily on the RayleighndashJeans tail so that

Jν = hνk

ehνkT minus 1 (1)

where ν is the observed frequency k is Boltzmannrsquos constant h isPlanckrsquos constant and T is the temperature characterizing the sourcefunction For our cloud T will be Tex in LTE Then the equationof radiative transfer gives the observed intensity Tobs in units ofbrightness temperature

Tobs = ηbf

[Jν(Tex) minus Jν(Tbg)

](1 minus exp (minusτ )) (2)

Here ηbf is the beam-filling factor the fraction of the beam area thatis filled by the emitting region Tbg is the background radiation fieldtemperature (271 K) and τ is the optical depth of the transition inquestion Note that ηbf the beam-filling factor is expected to be avery small number bringing our observed intensities from sim10 Kdown to the observed small fraction of a Kelvin

If we make the simplifying assumption that the 12C lines egHCN (1-0) are optically thick then 1 minus exp (minusτ ) asymp 1 and

T 12Cobs = η12C

bf

[Jν12C(Tex) minus Jν12C(Tbg)

] (3)

where Jν 12C refers to equation (1) evaluated at the frequency of the12C line On the other hand the 13C line has unknown optical depthso that

T 13Cobs = η13C

bf

[Jν13C(Tex) minus Jν13C(Tbg)

] (1 minus exp (minusτ 13C)

) (4)

Here we have distinguished the frequency of the 13C line from thatof the 12C line above The equations also allow that the two linesmight have different filling factors and so distinguish η13C

bf fromη12C

bf The optical depth of the 13C line is unknown However ourobserved ratios are far from sim1 which would be expected for bothlines to be optically thick and having matched ηbf Thus it appearsunlikely that both lines are optically thick Therefore we expect(1minusexp minus τ 13C) asymp τ 13C in most cases

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 59

Table 7 Optical depths and derived effective critical densities assuming matching beam-filling factors and a commonexcitation temperature for the 12C and 13C isotopologues Critical densities are in units of cmminus3

Galaxy NGC 3351 NGC 3627 NGC 4254 NGC 4321 NGC 5194 NGC 253

τ (H13CN) 004 plusmn 002 015 plusmn 007 lt02 lt005 lt007 007 plusmn 001τ (HCN) 24 plusmn 04 42 plusmn 05 lt11 lt23 lt35 25 plusmn 06τ (H13CO+) lt005 lt01 lt01 lt006 lt01 005 plusmn 002τ (HCO+) lt25 lt52 lt68 lt28 lt61 18 plusmn 09τ (HN13C) ndash ndash ndash ndash lt03 ndashτ (HNC) ndash ndash ndash ndash lt11 ndashnthick(HCN) (20 plusmn 02) times 106 gt86 times 105 gt48 times 105 gt20 times 106 gt14 times 106 (20 plusmn 02) times 106

nthick(HCO+) gt29 times 105 gt15 times 105 gt11 times 105 gt25 times 105 gt12 times 105 (42 plusmn 03) times 105

nthick(HNC) ndash ndash ndash ndash gt86 times 104 ndash

Then generally for LTE and an optically thick 12C line we have

T 13Cobs

T 12Cobs

= η13Cbf

η12Cbf

Jν13C(Tex) minus Jν13C(Tbg)

Jν12C(Tex) minus Jν12C(Tbg)

(1 minus eminusτ13C

) (5)

The differences between the 12C and 13C frequencies are smallso that

T 13Cobs

T 12Cobs

asymp η13Cbf

η12Cbf

(1 minus eminusτ13C

) (6)

In the simple case we further assume that the 13C line is opticallythin and that the beam-filling factors of the two lines match (anassumption we will drop in the next section) so that

τ 13C = minus ln

(1 minus T 13C

obs

T 12Cobs

)asymp T 13C

obs

T 12Cobs

(7)

This exercise yields the optical depth of the 13C line Because theopacity is proportional to the column of molecules we can relateτ 13C to τ 12C Again ignoring minor differences between the linesthe ratio of optical depths will simply be the ratio in abundancesbetween the two molecules

τ 12C = τ 13C [H12CN]

[H13CN]asymp τ 13C [12C]

[13C] (8)

522 Results

Using the line ratios tabulated in Table 4 equations (7) and (8)and adopting a Carbon isotope ratio of 12C13C = 50 we estimateτ 13C and τ 12C for each pair of isotopologues We report these inTable 7 Because they use equation (7) these estimates all assumematched beam-filling factors and neglect any differences betweenthe frequencies of the two transitions

In Section 3 we place upper limits on the 13C12C ratio in anumber of cases These translate into upper limits on τ 12C and τ 13CBroadly the 13C lines are consistent with being optically thin In thissimple case all of these optical depths are 03 The correspondingoptical depths for the 12C lines are all consistent with τ 12C 1 inagreement with our assumption of thick 12C emission In the caseof HCN emission for NGC 3351 NGC 3627 and NGC 253 andHCO+ for NGC 253 these are all consistent with τ 12C sim 3 (seealso Meier et al 2015) However in the other cases these resultsare upper limits For 13C upper limits we can mainly conclude thatthe ratios do not rule out optically thick 12C emission

Thus in the simple case the line ratios strongly suggest thin 13Cemission They are consistent with moderate optical depth in the12C lines

53 Non-LTE effects and variable beam-filling factors

In the previous section we assumed that η12Cbf = η13C

bf so that bothlines emerge from the same region In fact the 12C line can beoptically thick while the 13C line is optically thin In this caseline trapping effects will lower the effective density for emis-sion for the 12C line but not the 13C line Then the lower den-sity gas can emit strongly in the 12C line but more weakly in the13C line This should yield η12C

bf gt η13Cbf That is the two lines

will show distinct distributions within a cloud with the 13C line(which has no line trapping) confined to the denser parts of thecloud

The same effect may lead to different excitation temperatures forthe two lines If the 12C line is optically thick then line trappingwill lower its effective critical density If the collider (H2) volumedensity in the galaxy is high compared to the critical density of the12C line but low compared to the critical density of the 13C linethen the 13C line may be sub-thermally excited while the 12C lineapproaches LTE In this case T 13C

ex lt T 12Cex

The strength of non-LTE effects depends on the exact densitydistribution within the emitting region (see Leroy et al 2016) If theregion has mostly gas above the critical density of both lines thenthe LTE case discussed above will apply This might often be thecase eg for 12CO and 13CO

On the other hand if the density of the region is small comparedto the effective critical density of the 12C line then neither line willemit effectively Most collisional excitations will be balanced byradiative de-excitations and the line ratio will drop In the extremelimit the line ratio will reach the abundance ratio

In this paper we observe mostly the central bright areas ofgalaxies focusing on regions with bright HCN emission We donot expect these central zones to be dominated by extremely lowvolume density gas However if a large amount of nH2 sim 1035

minus 104 cmminus3 gas contributes to the emission then these non-LTEeffects may still be important This is the range of densities wherenH2 might be above the effective critical density for an optically thick12C line but not an optically thin 13C line This range of densitiesis also well within the range commonly observed inside molecularclouds in nuclear star-forming regions

Leroy et al (submitted) explore these effects for realistic densitydistributions They calculate the amount by which the 12C to 13C lineratio will be suppressed for different lines given some optical depthof the 12C line and an adopted density distribution The magnitudeof the effect depends on the line the optical depth and the densitydistribution For a lognormal distribution with a power-law tail theyfound that the 12C13C line ratio could be enhanced relative to theLTE expectation by a factor of sim2minus3 for HCN or HCO+ over arange of plausible densities

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

60 M J Jimenez-Donaire et al

Table 8 Optical depths and effective critical densities when considering η12 sim 2 η13 and under the assumption of acommon Tex Critical densities are in units of cmminus3

Target τHCN nHCNcrit τHCO+ nHCO+

crit τHNC nHNCcrit

NGC 3351 50 plusmn 04 (10 plusmn 04) times 106 lt53 gt14 times 105 ndash ndashNGC 3627 88 plusmn 07 (56 plusmn 01) times 105 lt106 gt70 times 104 ndash ndashNGC 4254 lt258 gt12 times 105 lt160 gt45 times 104 ndash ndashNGC 4321 lt53 gt94 times 105 lt58 gt13 times 105 ndash ndashNGC 5194 lt80 gt62 times 105 lt143 gt51 times 104 lt31 gt18 times 104

NGC 253 63 plusmn 05 (79 plusmn 01) times 105 43 plusmn 06 (17 plusmn 01) times 105 ndash ndash

This enhanced line ratio can be captured to first order by apply-ing different filling factors to the two lines Thus to explore theimplication of these effects we revisit our LTE calculations butnow assume that η12C

bf η13Cbf asymp 2 To do this we return to equation

(6) We no longer set the beam-filling factors equal between thetwo lines but still set the temperatures and frequencies equal andassume LTE with a single common temperature Then

τ 13Cobs minus ln

(1 minus η12C

bf

η13Cbf

T 13Cobs

T 12Cobs

) (9)

This is still an approximation but one that captures some of theeffect of differential excitation of the 12C and 13C lines in the non-LTE case We refer the reader to the non-LTE treatment in Leroyet al (submitted) for more details and a tabulation of a variety ofcases

Equation (9) shows that for η12Cbf gt η13C

bf our simple estimates ofτ in the previous section will be underestimated We report revisedvalues in Table 8 These are higher than the corresponding LTEvalues in Table 7

Even accounting for differential filling factors we still derive τ 13C

consistent with optically thin 13C emission These non-LTE effectstend to make the limits on optical depth less stringent because theycan push the line ratio towards lsquooptically thinrsquo values even while the12C line remains thick Accounting for this effect our observationsmay still accommodate substantial thickness in the 12C line withτ 12C potentially as high as sim10 for our HCN detections

The exact magnitude of any non-LTE correction will depend onthe sub-beam density distribution the true temperature of the gasand the optical depth of the lines in question Our best guess isthat a moderate correction does indeed apply and the factor of sim2correction applied in Table 8 may represent a reasonable estimateIn fact if most of the emission that we see comes from denseclouds this may be an overcorrection while if no power-law tailis present the correction for non-LTE effects may be even higherFuture multiline work and isotopologue studies of Galactic Centreclouds will help refine these estimates further

54 Implications for dense gas tracers

Our simple calculations imply moderate optical depth for the densegas tracers (the 12C lines) and this depth increases in the case ofdifferential beam filling Optical thickness in these lines means thatradiative trapping will lead to emitted photons being re-absorbedThis lowers the effective critical density and changes the density atwhich these lines emit most effectively This correction is likely tobe significant for commonly used dense gas tracers It will affectboth the density required for effective emission and the conversionbetween line brightness and dense gas mass the dense gas conver-sion factor

In a review of this topic Shirley (2015) note the effective criticaldensity in the presence of line trapping as

nthickcrit = β Ajksum

i =j γji

(10)

Here β is the solid angle-averaged escape fraction Ajk are theEinstein A coefficients and γ ji are the collision rates (cm3 sminus1) outof level j into another level i The difference with the normal criticaldensity calculation is that β = 1 so that not every spontaneousemission escapes the cloud In the simple case of spherical geometryand large optical depth (τ gt 1) the effective spontaneous transitionrate becomes β Ajk sim Ajkτ so that the critical density is depressedrelative to its nominal (optically thin) value by a factor equal to theoptical depth

Tables 7 and 8 report the effect of applying corrections based onthe estimated optical depths to each line The effect is to reducethe critical density by a factor of sim2ndash6 yielding values that aretypically sim7 times 105 cmminus3 for HCN sim1 times 105 cmminus3 for HCO+ andgt18 times 104 cmminus3 for HNC These are of course as uncertain asthe optical depths but give some guide to the density of gas tracedby these lines

6 SU M M A RY A N D C O N C L U S I O N S

We present observations of the 1-0 transitions of the dense molecu-lar gas tracers HCN HCO+ HNC and their isotopologues H13CNH13CO+ and HC13N across the discs of six nearby galaxies Theseinclude IRAM 30 m observations of NGC 5194 (M51) as part ofthe IRAM large programme EMPIRE ALMA observations of NGC3351 NGC 3627 NGC 4254 NGC 4321 and NGC 253 Given thefaint nature of the 13C isotopologues we focus our analysis on theinner (sim50 arcsec for M51 and sim10 arcsec for the ALMA galaxies)high surface density regions of these galaxies and stack all spectrain this region for each galaxy to improve the significance of ourmeasurements We use this data set to constrain the optical depthof these lines and study the implications for the effective criticaldensities

(i) We detect HCN HCO+ HNC (1-0) with high significance foreach galaxy (gt5σ ) in the stacked spectra Emission from these linesis in fact well detected for individual lines of sight and resolvedacross the discs of our sample and in particular the HCO+ andHNC emissions are distributed very similarly to the HCN emissionin NGC 5194 (Bigiel et al 2016) The distribution of these densegas tracers follows well the bulk molecular medium one as tracedby CO emission it is brightest in the inner parts and follows thespiral structure (where present) in CO

(ii) The H13CN H13CO+ and HN13C (1-0) lines however aremuch fainter in the stacked spectra from the inner regions of oursample we detect H13CN (1-0) in NGC 3351 NGC 3627 andNGC 253 and H13CO+ (1-0) in NGC 253 NGC 4254 NGC 4321

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

R E F E R E N C E S

Aladro R et al 2013 AampA 549 A39Andre P Di Francesco J Ward-Thompson D Inutsuka S-I Pudritz R

E Pineda J E 2014 Protostars and Planets VI Univ Arizona PressTucson p 27

Bally J Langer W D 1982 ApJ 255 143Battisti A Heyer M H 2013 American Astronomical Society Meeting

Abstracts 221 p 34909Bigiel F et al 2016 ApJ 822 L26Bolatto A D et al 2013 Nature 499 450Buchbender C et al 2013 AampA 549 A17Bussmann R S et al 2008 ApJ 681 L73Caldu-Primo A Schruba A 2016 AJ 151 34Caldu-Primo A Schruba A Walter F Leroy A Sandstrom K de Blok

W J G Ianjamasimanana R Mogotsi K M 2013 AJ 146 150Carter M et al 2012 AampA 538 A89Chin Y-N Henkel C Millar T J Whiteoak J B Marx-Zimmer M 1998

AampA 330 901Christopher M H Scoville N Z Stolovy S R Yun M S 2005 ApJ 622

346Ciardullo R Feldmeier J J Jacoby G H Kuzio de Naray R Laychak M

B Durrell P R 2002 ApJ 577 31Dumouchel F Faure A Lique F 2010 MNRAS 406 2488Flower D R 1999 MNRAS 305 651Gao Y Solomon P M 2004 ApJ 606 271Garcıa-Burillo S Usero A Alonso-Herrero A Gracia-Carpio J Pereira-

Santaella M Colina L Planesas P Arribas S 2012 AampA 539A8

Gonzalez-Alfonso E et al 2012 AampA 541 A4Gracia-Carpio J Garcıa-Burillo S Planesas P Colina L 2006 ApJ 640

L135Gracia-Carpio J Garcıa-Burillo S Planesas P Fuente A Usero A 2008

AampA 479 703Guesten R Henkel C Batrla W 1985 AampA 149 195Harada N et al 2015 AampA 584 A102Heiderman A Evans N J II Allen L E Huard T Heyer M 2010 ApJ

723 1019Henkel C Chin Y-N Mauersberger R Whiteoak J B 1998 AampA 329

443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

J Eckart A 2008 ApJ 677 262Lada C J Lada E A 2003 ARAampA 41 57Lada C J Lombardi M Alves J F 2010 ApJ 724 687Langer W D Penzias A A 1990 ApJ 357 477Leroy A K et al 2013 AJ 146 19Leroy A K et al 2015 ApJ 801 25Leroy A K et al 2016 Astrophys J preprint (arXiv161109864)Marr J M Wright M C H Backer D C 1993 ApJ 411 667Martın S Aladro R Martın-Pintado J Mauersberger R 2010 AampA 522

A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

MNRAS 466 49ndash62 (2017)

Dense gas optical depths in nearby galaxies 61

and NGC 5194 show no isotopologue detection in their inner partTherefore we derive stringent upper limits for those cases whichwe use for our analysis

(iii) We use the optically thin isotopologues to compute or con-strain optical depths for each molecular lines in the inner parts ofour target galaxies and where detected also in selected CO andHCN bright disc positions We find that HCN and HCO+ have op-tical depths in the range sim2ndash11 in the inner parts of the galaxiesanalysed (assuming a value of 50 for the 12C13C abundance ratio)HN13C data are only available for NGC 5194 where we derived anupper limit to the opacity for HCN in its inner region τ lt 11 Theoptical depth for H13CN H13CO+ and HN13C shows a larger degreeof variation ranging from 004 to 03 in the inner part of NGC 3351NGC 3627 and NGC 253 We conclude that HCN HCO+ and HNC(1-0) emissions are largely moderately optically thick with opticaldepths of typically a few whereas the H13CN H13CO+ and HN13C(1-0) lines are largely optically thin even in the inner parts of ourgalaxies on the sim05ndash2 kpc scales we probe

(iv) Given their non-negligible optical thickness the critical den-sities of the HCN HCO+ and HNC are reduced by a factor of 2ndash6which implies that these lines are sensitive to molecular gas at lowerdensities We also study the non-LTE conditions and the influence ofvariable beam-filling factors for the different isotopologues giventheir optical thickness and thus sensitivity to lower density gasone may expect larger beam-filling factors for the 12C moleculesThis would further increase optical depths and therefore lead to afurther decrease in the effective critical densities for the opticallythick dense gas tracers

(v) We compare the HCNH13CN HCO+H13CO+ and HNCHN13C line ratios measured in our sample to those compiled fromthe literature There is a good agreement between the values we findfor NGC 5194 and NGC 253 with previous studies The work donein the Galactic Centre also shows compatible values with those wefind in the inner parts of our sample within the uncertainties

AC K N OW L E D G E M E N T S

The authors thank Daniel Harsono Ralf Klessen Fabian WalterSimon Glover and Kazimierz Sliwa for discussions that have im-proved this manuscript We thank the referee for detailed and helpfulcomments FB MJJD and DC acknowledge support from DFG grantBI 15461-1 AKL thanks the NRAOUniversity of Virginia star for-mation group for helpful discussions The National Radio Astron-omy Observatory is a facility of the National Science Foundationoperated under cooperative agreement by Associated UniversitiesInc DC is supported by the Deutsche ForschungsgemeinschaftDFG through project number SFB956C SGB acknowledges sup-port from Spanish grants ESP2015-68964-P and AYA2013-42227-P AH acknowledges support from the Centre National drsquoEtudesSpatiales (CNES) MRK acknowledges support from ARC grantDP160100695 ER is supported by a Discovery Grant from NSERCof Canada

AU acknowledges support from Spanish MINECO grantsFIS2012-32096 and ESP2015-68964 Based on observations car-ried out under project number 206-14 with the IRAM 30m Tele-scope IRAM is supported by INSUCNRS (France) MPG (Ger-many) and IGN (Spain) This paper makes use of the followingALMA data ADSJAOALMA2011000004SV ALMA is a part-nership of ESO (representing its member states) NSF (USA) andNINS (Japan) together with NRC (Canada) NSC and ASIAA(Taiwan) and KASI (Republic of Korea) in cooperation with the

Republic of Chile The Joint ALMA Observatory is operated byESO AUINRAO and NAOJ

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443Henkel C Downes D Weiszlig A Riechers D Walter F 2010 AampA 516

A111Henkel C et al 2014 AampA 565 A3Jimenez-Donaire M J et al 2016 preprint (arXive-prints)Kennicutt R C Evans N J 2012 ARAampA 50 531Krips M Neri R Garcıa-Burillo S Martın S Combes F Gracia-Carpio

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A62Matsushita S Trung D-V Boone F Krips M Lim J Muller S 2015

Publ Korean Astron Soc 30 439McMullin J P Waters B Schiebel D Young W Golap K 2007 in Shaw

R A Hill F Bell D J eds ASP Conf Ser Vol 376 Astronomical DataAnalysis Software and Systems XVI Astron Soc Pac San Franciscop 127

Meidt S E et al 2013 ApJ 779 45Meier D S et al 2015 ApJ 801 63

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This paper has been typeset from a TEXLATEX file prepared by the author

MNRAS 466 49ndash62 (2017)