Characterizing the line emission from molecular clouds. Stratified random sampling of the Perseus cloud
AAstronomy & Astrophysics manuscript no. 38727ms © ESO 2021January 11, 2021
Characterizing the line emission from molecular clouds. (cid:63)
Stratified random sampling of the Perseus cloud
M. Tafalla , A. Usero , and A. Hacar , Observatorio Astronómico Nacional (IGN), Alfonso XII 3, E-28014 Madrid, Spaine-mail: [email protected], [email protected] Leiden Observatory, Leiden University, PO Box 9513, 2300 Leiden, The Netherlands Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, Austriae-mail: [email protected]
Received 23 June 2020 / Accepted 8 December 2020
ABSTRACT
Context.
The traditional approach to characterize the structure of molecular clouds is to map their line emission.
Aims.
We aim to test and apply a stratified random sampling technique that can characterize the line emission from molecular cloudsmore e ffi ciently than mapping. Methods.
We sampled the molecular emission from the Perseus cloud using the H column density as a proxy. We divided the cloudinto ten logarithmically spaced column density bins, and we randomly selected ten positions from each bin. The resulting 100 cloudpositions were observed with the IRAM 30m telescope, covering the 3mm-wavelength band and parts of the 2mm and 1mm bands. Results.
We focus our analysis on the 11 molecular species (plus isotopologs) detected toward most column density bins. In allcases, the line intensity is tightly correlated with the H column density. For the CO isotopologs, the trend is relatively flat, while forhigh-dipole moment species such as HCN, CS, HCO + , and HNC, the trend is approximately linear. To reproduce this behavior, wedeveloped a cloud model in which the gas density increases with column density, and where most species have abundance profilescharacterized by an outer photodissociation edge and an inner freeze-out drop. With this model, we determine that the intensitybehavior of the high-dipole moment species arises from a combination of excitation e ff ects and molecular freeze out, with somemodulation from optical depth. This quasi-linear dependence with the H column density makes the gas at low column densitiesdominate the cloud-integrated emission. It also makes the emission from most high-dipole moment species proportional to the cloudmass inside the photodissociation edge. Conclusions.
Stratified random sampling is an e ffi cient technique for characterizing the emission from whole molecular clouds.When applied to Perseus, it shows that despite the complex appearance of the cloud, the molecular emission follows a relativelysimple pattern. A comparison with available studies of whole clouds suggests that this emission pattern may be common. Key words.
ISM: abundances – ISM: clouds – ISM: individual objects: Perseus Cloud – ISM: molecules – ISM: structure – Stars:formation
1. Introduction
Molecular clouds are the coldest and densest constituents of theinterstellar medium and harbor in their interiors the sites wherestars are born. Clouds are believed to form and evolve by thecomplex interplay between turbulence, gravity, and magneticfields, although the exact role of each factor is still a matter ofdebate (see Hennebelle & Falgarone 2012; Dobbs et al. 2014for recent reviews). Progress in our understanding of clouds re-quires characterizing their physical and chemical structure andinterpreting this structure in terms of the di ff erent forces actingon the gas. This characterization is usually done by mapping theemission from molecular lines since these lines provide uniqueinformation on the density, temperature, kinematics, and molec-ular composition of the cloud gas (Evans 1999).The large angular size of the nearby clouds makes mappingtheir line emission very time consuming, especially for the weaksubthermally excited lines that are most sensitive to the physical (cid:63) Table A.1. is only available in electronic form at the CDS via anony-mous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http: // cdsweb.u-strasbg.fr / cgi-bin / qcat?J / A + A / conditions of the gas. Due to this limitation, large-scale maps ofclouds are almost exclusively made in the bright and thermalized(often optically thick) lines of the CO isotopologs (e.g., Gold-smith et al. 2008; Buckle et al. 2010; Umemoto et al. 2017),while the mapping of the more informative subthermal lines isrestricted to the densest parts of the clouds (e.g., Sanhueza et al.2012; Jackson et al. 2013). As a result, our view of the lineemission from molecular clouds is spatially limited and oftenrestricted to a small number of molecular species.Over the past several years, a large e ff ort has been made toovercome previous observing limitations and map entire cloudsusing multiple line tracers. This e ff ort has been made possibleby a new generation of heterodyne receivers with large frequencybandwidths (e.g., Carter et al. 2012) and has provided a first mul-tiline view of full or sizable parts of several molecular clouds.An example of this e ff ort is the IRAM Large Program ORION-B (Pety et al. 2017), which has mapped most of the Orion Bcloud in the 3mm-wavelength band, and whose results are cur-rently being published (Orkisz et al. 2017; Gratier et al. 2017;Bron et al. 2018; Orkisz et al. 2019). Using a similar approach,Watanabe et al. (2017) carried out multiline large-scale mapping Article number, page 1 of 27 a r X i v : . [ a s t r o - ph . GA ] J a n & A proofs: manuscript no. 38727ms of the high-mass star-forming region W51 with the Mopra tele-scope. Complementing this e ff ort, Kau ff mann et al. (2017) com-piled multiple observations of the Orion A cloud made with theFCRAO telescope and used the resulting dataset to study howthe emission from the di ff erent molecular species originates indi ff erent parts of the cloud. Although these e ff orts are encour-aging, the large investment of observing time required to mapindividual clouds in multiple lines (often hundreds of hours, seeOrkisz et al. 2019) suggests that full-cloud mapping will remainfor some time a niche approach limited to the study of selectedtargets.While fully mapping clouds is necessary to characterize theiremission in an unbiased way, these same observations show thatclouds tend to present a common underlying behavior despitetheir diverse and chaotic appearance. Examples of this behaviorare the Larson’s relations between global cloud properties suchas mass, size, and velocity dispersion (Larson 1981; Heyer &Brunt 2004), the almost universal fractal dimension found us-ing area-perimeter measurements (Bazell & Desert 1988; Fal-garone et al. 1991), and the common behavior of the probabil-ity distribution function of column densities (Kainulainen et al.2009; Lombardi et al. 2015). These and other trends suggest thatmost clouds have, to first order, a common physical and chem-ical structure that could be described using a small number ofparameters. If this is so, the emission from the clouds will alsolikely present a systematic behavior, of course modulated by thecharacteristics of each individual system.If clouds emit according to some simple and general pattern,it should be possible to characterize this pattern using a limitedset of observations instead of having to map the emission in de-tail. In this paper, we explore this possibility by using a rela-tively sparse sampling technique, which we applied to the nearbyPerseus cloud. This cloud was chosen for having a number of fa-vorable characteristics and a significant amount of previous data(see Bally et al. 2008 for a detailed review of the cloud proper-ties). It is nearby, with a distance that has been variously esti-mated as 234 pc from VLBI VERA observations (VERA collab-oration et al. 2020) and about 300 pc from VLBA and Gaia mea-surements (Ortiz-León et al. 2018; Zucker et al. 2019), and hasan estimated mass of 2 × M (cid:12) (Zari et al. 2016, assuming a dis-tance of 240 pc). It is an active star-forming region that containsthe young cluster IC 348 (Strom et al. 1974; Lada et al. 2006),the embedded cluster NGC 1333 (Strom et al. 1976; Gutermuthet al. 2008), and a more distributed population of young stars andprotostars (Jørgensen et al. 2007; Rebull et al. 2007).The large-scale emission of the di ff erent CO isotopologs inPerseus has been mapped by Bachiller & Cernicharo (1986),Ungerechts & Thaddeus (1987), Ridge et al. (2006), Sun et al.(2006), and Curtis et al. (2010), while the dust component hasbeen mapped using mm, submm, and FIR emission (Hatchellet al. 2005; Enoch et al. 2006; Chen et al. 2016; Pezzuto et al.2020) and optical and IR extinction (Bachiller & Cernicharo1986; Schnee et al. 2008; Lombardi et al. 2010; Zari et al. 2016).In addition to these large-scale studies, a number of observa-tions targeting the dense cores have been presented by Laddet al. (1994), Kirk et al. (2006), Rosolowsky et al. (2008a), andHacar et al. (2017) among others. All these studies (and othersnot mentioned here for brevity) make Perseus one of the beststudied clouds, and therefore an ideal region to test a samplingtechnique.In this paper, we present an emission survey of Perseus withtwo distinct goals: (1) to test whether the cloud emission can becharacterized using a sampling technique and, assuming that theanswer is positive, (2) to use the sampling data to reconstruct the cloud emission and infer the gas physical and chemical prop-erties. Since the focus of our study is the intensity of the lineemission, our sampling technique (described in the next section)is designed to highlight this parameter at the expense of oth-ers. As a result, our analysis will not touch upon other importantcloud properties, such as the gas velocity field, which require adi ff erent approach for their study.
2. Stratified random sampling
The goal of our work is to sample the di ff erent regimes of thecloud molecular emission using a relatively small number of po-sitions, even if the cloud emission properties are not initially wellknown. Choosing positions at random does not seem like a goodstrategy, since a typical cloud has so many more positions withweak emission that the probability of randomly picking brightpositions is negligibly low (Rosolowsky et al. 2008b).A better option is to guide the choice of positions by a proxythat is expected to correlate with the molecular emission. A nat-ural choice for this proxy is the H column density, which hasbeen shown by principal component analysis to have a dominantcontribution to the emission of some clouds (Ungerechts et al.1997; Gratier et al. 2017). The H column density, in addition,can be determined with accuracy over entire molecular cloudsusing a combination of dust extinction and emission measure-ments (e.g., Lombardi et al. 2014). For the Perseus cloud, Zariet al. (2016) have recently determined the distribution of H col-umn density combining dust emission data from the Herschel and
Planck satellites together with NIR dust extinction measure-ments from the 2MASS survey. This determination covers thefull extent of the cloud and has a dynamic range of about twoorders of magnitude. In addition, it has an angular resolution of36 (cid:48)(cid:48) , which is similar to what is currently achievable with single-dish radio telescopes.To sample the cloud using the H column density as a proxy,we need to sample the distribution of H column densities in thecloud so that each column density regime is well represented.Properly sampling this distribution, however, requires some caresince, like the emission, the population of column densities isoverwhelmingly dominated by the positions with the lowest val-ues. Zari et al. (2016) found that the probability distributionfunction of the column density follows a power law with a slopeof −
3, and as a result, that the number of positions at the lowend of the distribution ( N (H ) ≈ cm − ) exceeds the numberof positions at the high end ( N (H ) ≈ cm − ) by about sixorders of magnitude (Fig. 8 in Zari et al. 2016). Sampling the dis-tribution by choosing positions with random column density istherefore impractical since it would require selecting about onemillion samples to ensure that the full range of column densitiesis covered.A more practical approach is to first identify the di ff erentcolumn density regimes in the cloud and then to sample eachregime by choosing a number of cloud positions at random.This sampling approach is an instance of the “stratified randomsampling” technique often used in surveys (Cochran 1977), andwhose name derives from the word “strata” used to denote thedi ff erent subpopulations of the sample, which in our case corre-spond to the column density regimes of the cloud.Since the distribution of column densities in Perseus followsa power law over the approximately two orders of magnitudefor which the extinction measurements are reliable (1 . × < N (H ) < . × cm − , corresponding to 0.2 < A K <
20 mag,Zari et al. 2016), we chose to bin this range using logarithmi-cally spaced column-density intervals. The width of these inter-
Article number, page 2 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds.
Fig. 1.
Sampling of the Perseus cloud.
Top: H column density map of the Perseus molecular cloud from Zari et al. (2016) with squares indicatingthe location of the 100 positions selected to characterize the line emission from the cloud using stratified random sampling. The number ineach square indicates the column density bin to which the position belongs, and ranges from 1 (lowest column density) to 10 (highest). Bottom:
Expanded view of the three regions that concentrate most of the high column density positions, and that correspond to the well-known star-formingregions NGC 1333, B1, and L1448. Contours are in units of N (H ), start at 1 . × cm − , and increase by factors of 2.5. The 10 pc scale barassumes a distance of 300 pc. vals was taken as 0.2 dex, equivalent to a factor of 1.6 in columndensity, with the expectation that the molecular emission will notchange dramatically (more than a factor of 2) between the bins.As shown below, this expectation is satisfied over most of thecloud, although it seems to break down toward the lowest col-umn density bin ( A V ≈ − ff er by multiple orders of magnitude. Asan expedient solution motivated by the limited observing timeavailable, we chose to sample each bin using the same numberof positions, which we took as ten. This number was chosen as acompromise between the competing needs of observing enoughpositions per bin to determine with some accuracy the mean andthe dispersion of the intensity, and the need to make long enoughintegrations to detect the weak emission of the positions in thelowest column density bins. Since the success of this samplingstrategy depends strongly on the intensity and the dispersion ofthe emission inside the bins, which was not known before theobservations, our choice of sampling parameters should be takenonly as a tentative initial approach. A proper optimization of the Article number, page 3 of 27 & A proofs: manuscript no. 38727ms
Fig. 2. H column density as a function of sample number for the 100positions randomly chosen to represent the Perseus molecular cloud(magenta circles). The dashed horizontal lines mark the boundaries ofthe ten column-density bins in which the cloud was divided, and thevertical lines enclose the ten points chosen to sample each bin. The bluecircles indicate the geometrical mean of the H column densities in-side each bin, and their error bars indicate their dispersion. All valuesderived from Zari et al. (2016) data. stratified random sampling technique to characterize molecularclouds is still needed to assess a number of possible weaknessesof our approach, which include the use of limited sampling inthe very extended low column density bins (which could lead tostatistical fluctuations) and the use of H column density as thesole guide to sample the cloud emission to the exclusion of otherparameters, such as the gas temperature or the star-formation ac-tivity. Carrying out this investigation requires having full mapsof clouds in a variety of molecular tracers, which fortunately isnow becoming possible thanks to the e ff orts of dedicated observ-ing programs such as ORION-B (Pety et al. 2017) and LEGO(Barnes et al. 2020).Once the number of sampling positions had been determined,the practical choice of selecting them at random was done byfirst identifying all the pixels in the extinction map of Zari et al.(2016) that belong to each column density bin. These pixels werelisted in a table, and ten of them were selected by repeatedlydrawing random numbers uniformly distributed between one andthe number of pixels in the bin. Table A.1 presents a list with thecoordinates of the resulting 100 cloud positions. Their relativelocation inside the cloud is shown in Fig. 1.To illustrate the two orders of magnitude in H column den-sity covered by the sample, Fig. 2 shows the value determinedusing the prescription from Zari et al. 2016 for each sample po-sition (magenta symbols). Each column density bin is enclosedbetween dashed horizontal lines, and as expected, the points in-side each bin are distributed randomly in column density. Thefigure also shows the geometrical mean and dispersion insideeach bin (blue symbols). We note that the error bars in the indi-vidual column densities are typically smaller than the dispersioninside the bins, although an additional source of uncertainty inthe column density arises from the particular choice of extinc-tion per H atom assumed by Zari et al. (2016), which could shiftthe estimates globally by a factor of up to about 1.5 dependingon the true dust properties (Fig. 2 in Draine 2003).
3. Observations
We observed our sample of Perseus positions using the Institutde Radioastronomie Millimétrique (IRAM) 30m-diameter tele- scope in Pico Veleta (Spain) during three runs in June 2017,September 2017, and January 2018. In the first two runs, the3mm channel of the Eight MIxer Receiver (EMIR, Carter et al.2012) was used to observe the full frequency range from 83.7and 115.8 GHz (telescope FWHM of 29 (cid:48)(cid:48) to 21 (cid:48)(cid:48) ). These ob-servations used the facility fast Fourier Transform Spectrome-ter (FTS, Klein et al. 2012), which was configured to cover thereceiver instantaneous passband with a frequency resolution of200 kHz ( ≈ . − ).For all positions, the integration time was approximately 10minutes after combining the two linear polarizations, except forthe positions in the lower column density bin, which were ob-served twice as long to compensate for their weaker lines. Ad-ditional observations of 33 positions from our sample were car-ried out using simultaneously the 3mm and 2mm channels ofEMIR and covering two narrow frequency ranges centered onHCO + (1–0) and CS(3–2). These observations had the goal ofobtaining high velocity resolution spectra, and used as a back-end the Versatile SPectrometer Array (VESPA) with a frequencyresolution of 20 kHz ( ≈ .
06 km s − ).In the last observing run (January 2018), we used the 1mmchannel of the EMIR receiver to observe four frequency win-dows within the range 213.7-267.7 GHz (telescope FWHM of11 (cid:48)(cid:48) -9 (cid:48)(cid:48) ). These windows were selected for containing higher- J transitions of some 3mm target lines, such as CO(2–1), HCN(3–2), and CS(5–4). Again, the FTS backend was used to coveras much passband as possible with a frequency resolution of200 kHz, equivalent to ≈ .
25 km s − at the frequency of op-eration. Since the 1mm lines are weaker than those at 3mm, thelowest three column density bins of the sample were not fullyobserved, and the integration time per point typically ranged be-tween 10 and 40 minutes (after combining polarizations), de-pending on the strength of the line.All observations were carried out in frequency switchingmode with symmetric o ff sets of ± . , with which repeated observations and over-lapping frequency windows were STITCH ed together. Polyno-mial baselines were used to remove ripples in the passband, andthe intensity was converted to the main beam brightness scaleusing the facility-provided telescope e ffi ciencies. Typical rms inthe spectra range from 6 to 9 mK per 0.6 km s − channel at 100GHz.In most of the following analysis, we rely on the integratedintensity of the di ff erent molecular lines. These intensities wereestimated from the reduced spectra by integrating the emissionover the velocity channels where a visual inspection showed sig-nal. For spectra with no clear signal, the intensity was estimatedby integrating the emission inside the velocity range where the CO(1–0) line was detected since this abundant species wasidentified toward almost all cloud positions, and its velocityrange was found to coincide with that of all the other lines in caseof mutual detection. The uncertainty of each integrated intensitywas estimated from the rms level of the spectrum. Following pre-vious IRAM 30m studies, we added in quadrature a 10% calibra-tion error to include uncertainties in the beam e ffi ciencies andday-to-day variations (Pety et al. 2017; Jiménez-Donaire et al.2019). Table A.1 summarizes the derived intensities. Article number, page 4 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds.
Fig. 3.
Comparison between sampling observations and COMPLETE full maps for CO(1–0) and CO(1–0).
Top panels:
Scatter plots of theintegrated intensity as a function of N (H ). The blue dots represent the results from the COMPLETE maps presented by Ridge et al. (2006) andeach of them contains more than 2 × spectra. The red symbols represent the 100 measurements obtained using stratified random sampling. Thedotted lines mark the 3 σ detection level of the COMPLETE maps. Middle panels:
Ratio between the mean intensity derived using the samplingdata and the COMPLETE maps for each of the ten N (H ) bins in which the cloud was divided. For CO(1–0), the measurement in lowest N (H )bin is not reliable since the sampling data show that the intensity often lies below the COMPLETE detection limit. Bottom panels:
Ratio betweenthe rms derived using the sampling data and the COMPLETE maps. As with the mean values, the CO(1–0) measurement for the lowest N (H )bin lies below the detection limit, so the rms estimate is not reliable. We note how both the mean and rms ratios for the two CO isotopologslie in the vicinity of 1, indicating that the sampling method allows estimating the main emission parameters with an accuracy of about 50-20%depending on the parameter.
4. Survey results
Before analyzing the data, we test how well our sampling ob-servations recover the main properties of the cloud emission.For this, we compare our CO(1–0) and CO(1–0) intensitieswith the results from the Coordinated Molecular Probe Line Ex-tinction and Thermal Emission (COMPLETE) project, whichmapped the entire Perseus cloud in CO(1–0) and CO(1–0)using the FCRAO 14m telescope. These data have been pre-sented by Ridge et al. (2006), and their correlation with the ex-tinction determined using 2MASS data has been studied in detailby Pineda et al. (2008) and Goodman et al. (2009). To compareour survey data with the COMPLETE results, we first convertedthe COMPLETE intensities into the main beam brightness scaleusing the e ffi ciencies recommended by Ridge et al. (2006). Wethen resampled the extinction map from Zari et al. (2016) to thesame spatial grid used by COMPLETE, in order to obtain foreach position of the COMPLETE map an estimate of the H col-umn density N (H ). This estimate was determined from the ex-tinction data using the conversion factors recommended by Zariet al. (2016).In the top panels of Fig. 3, we represent the intensity of the CO(1–0) and CO(1–0) lines as a function of the H columndensity for both the sampling and COMPLETE datasets. This type of plots constitutes the basis of most of our Perseus analysispresented below, so it represents the most adequate tool to per-form the data comparison. The COMPLETE data are representedwith blue symbols and consist of more than 2 × points, one foreach spectrum used to generate each COMPLETE map (Ridgeet al. 2006). Superposed with red circles, we present the resultsfrom our sampling observations, also in main-beam brightnesstemperature scale and with the H column density estimated us-ing the Zari et al. (2016) prescription.As Fig. 3 shows, the distribution of the COMPLETE andsampling points agree both in their correlation with N (H ) andin the amount of dispersion for a given N (H ), indicating that thesampling observations recover the general trends of the emissionfrom the full cloud. There seems to be a slightly better agree-ment between the sampling and the COMPLETE results for the CO(1–0) data, although its cause is unclear since both CO iso-topologs were observed simultaneously by the two surveys. Wenote that small discrepancies between the data are unavoidablegiven their di ff erent calibrations and the factor of two di ff erencein angular resolution between the IRAM 30m telescope used inour sampling survey ( ≈ (cid:48)(cid:48) ) and the FCRAO 14m telescopeused in COMPLETE ( ≈ (cid:48)(cid:48) ).To quantify the comparison between the sampling and theCOMPLETE results, we calculated for each dataset the inten-sity mean and rms inside each of the ten column density bins in Article number, page 5 of 27 & A proofs: manuscript no. 38727ms
Fig. 4.
Integrated line intensity as a function of H column density for the J = − ) approximately corresponds to one to three times the noise level in the integrated intensity. which we have divided the cloud. Given the large dispersion ofthe data, we operated in logarithmic units and later converted theresults to a linear scale. The results of this calculation are shownin the middle and bottom panels of Fig. 3 in the form of ratiosbetween the estimates derived using the sampling method andthe COMPLETE maps.As expected from the scatter plots, the sam-pling / COMPLETE ratio of the means (middle panels) isclose to unity over the full range of N (H ) for both CO iso-topologs. The largest deviations from unity occur in CO(1–0),but they reach at most a factor of about 1.5 and do not showany systematic pattern of bias. Also as expected from thescatter plots, the CO(1–0) ratios are better behaved, andour estimate indicates an agreement between the samplingand the COMPLETE mean values at the level of 25%. Theonly disagreement between the CO(1–0) sampling andCOMPLETE results occurs in the lowest column density bin( N (H ) < . × cm − ), where some of the line intensitiesmeasured with the sampling technique lie below the three sigmadetection level of the COMPLETE data (dotted line in thescatter plots). This suggests that the COMPLETE data are tooshallow to characterize the CO(1–0) intensity in the lowestcolumn density bin, and as a result, they artificially overestimatethe mean value and underestimate the dispersion.Even better agreement between the sampling and the COM-PLETE results is found for the estimate of the emission rms. Asshown in the bottom panels of Fig. 3, the CO(1–0) data agreebetter than 25%, although the sampling method can barely fol-low the large rms increase in the lowest column density bin seenin the upper panel. In hindsight, having observed additional po-sitions in this bin would have been desirable. For CO(1–0), theagreement between sampling and mapping results is also betterthan 25%, again excluding the lowest column density bin due tothe insu ffi cient sensitivity of the shallower COMPLETE data.To summarize, a comparison with the COMPLETE map-ping data suggests that the stratified random sampling methodcan provide estimates of the intensity mean and dispersion thathave an accuracy of better than a factor of 1.5, and are usu-ally at the 20% level, for the whole range of column densitiescovered by our survey. This comparison is unfortunately limitedto CO data since this is the only species for which large-scalemaps are available. While further testing is needed using di ff er-ent species (and clouds), the results obtained so far support theidea that stratified random sampling is a potentially useful tool toe ffi ciently determine the global properties of the cloud emission. The number of molecular lines detected in each position dependsstrongly on its column density. Toward positions with the high-est column densities (bin number 10, with N (H ) ≥ cm − ),the spectra contain about 50 di ff erent molecular lines in the 3mmband alone. As the column density decreases, the number of de-tected lines decreases rapidly, and in the lowest column densitybin ( N (H ) ≈ cm − ), the detections are often limited to thelines of the abundant CO isotopologs. Since our goal is to studythe dependence of the line intensity with the gas column den-sity, we restrict our analysis to those 3mm lines that are detectedin most positions belonging to at least five column density binsout of the ten in which we divided the cloud. In this section,we focus our discussion on these brighter lines that satisfy ourselection criterion. Table A.1 presents their integrated intensityestimated, as described in Sect. 3, by integrating the emissionover the range of detection, and in case of non detection, by inte-grating the emission over the range at which CO(1–0) was de-tected. Appendix B illustrates the data presenting stacked spectraof all 3mm transitions for each column density bin.For presentation convenience, we have divided these linesinto three di ff erent chemical families. The first chemical familyis that of the CO isotopologs, and consists of CO, CO, C O,and C O. Fig. 4 shows the velocity-integrated intensity of their J = column density. As shownbelow, the results for the J = J = column density overthe two orders of magnitude that this parameter spans across thesample. Since the sample positions in each column density binare located randomly over the cloud, this correlation indicatesthat the H column density is by itself a strong predictor of theCO line intensity, a first hint that our reliance on the H columndensity as a proxy for the line emission is an acceptable choice.As Fig. 4 shows, the brighter CO and CO lines havethe largest dynamic range, and their distribution with H col-umn density presents an abrupt change in slope at around 2 × cm − ( A V ≈ A V <
10 mag (equivalent
Article number, page 6 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds.
Fig. 5.
Integrated line intensity as a function of H column density for the survey species detected in at least five column density bins. Top:
Traditional dense gas tracers (blue symbols).
Bottom:
Additional tracers of dense gas (green symbols). For reference, each panel shows a linearrelation that fits the average intensity of the highest column density bin (dashed line). For HCN(1–0), CN(1–0), N H + (1–0), and C H(1–0), theintensity includes the contribution from all detected hyperfine components. For CH OH(2–1), the intensity includes the contribution from the A + and E components. Depending on the transition, the lower limit of the intensity scale (0.05 K km s − ) approximately corresponds to one to threetimes the noise level in the integrated intensity. to N (H ) < cm − ). It likely results from the photodissocia-tion of the CO molecules in the outer cloud by the UV photonsfrom the external interstellar radiation field (e.g., Tielens & Hol-lenbach 1985; van Dishoeck & Black 1988; Le Bourlot et al.1993; Sternberg & Dalgarno 1995; Visser et al. 2009; Wolfireet al. 2010; Joblin et al. 2018). Inner to the photodissociationedge, the slope of the CO and CO intensities is relatively flatcompared with the linear slope, a trend that will be shown belowto result from saturation e ff ects, in agreement with the previoussuggestion from Pineda et al. (2008).For the C O and C O lines, the photodissociation edge isnot appreciable at low column densities due to insu ffi cient sen-sitivity, although it can be hinted in the bin-averaged data dis-cussed below. As the column density increases, the C O andC O intensities increase almost linearly with N (H ) up to about10 cm − , and then flatten significantly at higher column densi-ties. This flattening is not caused by optical depth e ff ects sincethe C O(1–0) / C O(1–0) ratio has a close-to-constant value of3 . ± .
1, which matches the O / O isotopic ratio found for theISM by Wilson & Rood (1994), indicating that both the C O(1–0) and C O(1–0) lines are optically thin over the entire Perseuscloud. As shown by the cloud model discussed below, the flatten-ing is likely caused by the freeze out of the CO molecules ontothe dust grains at the high densities characteristic of the regionswith high N (H ).The other two families in which we divide the species de-tected in our survey are presented in Fig. 5. The top row of panelsshows species that are commonly used to trace dense gas bothin galactic and extra-galactic studies, such as HCN, CS, HNC,HCO + , and SO (e.g., Evans 1999; Kennicutt & Evans 2012). Wewill refer to these species collectively as “traditional dense gastracers,” with the caveat that their role as true dense gas trac-ers is being reassessed as a result of recent work (Kau ff mannet al. 2017, Pety et al. 2017, Shimajiri et al. 2017, see furtherdiscussion in Sect. 6.2). The bottom row contains a more hetero-geneous mix of species. Most of them are sensitive to dense gas, but they often present strong sensitivity to additional processessuch as shocks, UV radiation, or CO freeze out: CH OH, CN,C H , N H + , and C H (van Dishoeck & Blake 1998; Bergin &Tafalla 2007). We will refer to this group of species as the “ad-ditional tracers” family for lack of a better term.As with the CO isotopologs, the intensity of the traditionaldense gas tracers presents a strong correlation with the H col-umn density over the two orders of magnitude sampled by thesurvey. The lower signal-to-noise ratio of these tracers at low H column density makes it di ffi cult to judge whether they presenta photodissociation edge similar to that of CO, although thereare hints of such an edge at least in the HCN data. Better evi-dence for photodissociation edges in some of these tracers willbe presented in Sect. 4.4 when we analyze the bin-averaged data.Inside the photodissociation edge, all traditional dense gastracers present a clear quasi-linear correlation with N (H ), as in-dicated by the good match between the data points and the lineardashed lines. This systematic behavior of the traditional densegas tracers is somewhat unexpected since common belief amongobservers suggests that a linear correlation with column densitywould only be expected for the thermalized lines of the thin COisotopologs. The traditional dense gas tracers, due to their sub-thermal excitation, are expected to trace volume density, not col-umn density, and therefore present a steeper intensity slope. Un-derstanding this unexpected behavior requires a detailed mod-eling of the cloud physical and chemical properties, which wedefer to Sect. 6.1 below. Here we only stress that the remarkablelinear behavior extends over the two orders of magnitude in H column density covered by the observations.The set of additional tracers, shown in the bottom row ofFig. 5 presents a more diverse behavior in their dependence withthe H column density. This behavior ranges from the close tolinear correlation of CH OH to the significantly nonlinear be-havior of N H + , whose intensity drops precipitously at H col-umn densities below 10 cm − . Less notable, but still signif-icant, are the deviations from linearity seen in CN and C H, Article number, page 7 of 27 & A proofs: manuscript no. 38727ms
Fig. 6.
Dispersion histograms of logarithmic intensity for each column density bin. The panels show the combined dispersion for the family of COisotopologs (left) , the traditional dense gas tracers (middle) , and the additional tracers (right) . In all cases, the lowest column density bin has beenexcluded due to low sensitivity. whose intensities increase slightly but significantly (factors oftwo or three) over the linear trend for column densities lowerthan 10 cm − . The possible origin of all these behaviors is ex-plored below with the help of a radiative-transfer cloud model. As mentioned above, the intensity of each line in the survey cor-relates strongly with the H column density, indicating that de-spite the random location of the observed positions, their lineintensity can be accurately predicted from the value of N (H ).This result is important for the use of the stratified random sam-pling method since, as discussed in Sect. 2, it requires that theH column density is a reliable proxy for the molecular emis-sion. To better quantify how well the H column density predictsthe intensity of each line, we have studied the dispersion of theintensities inside each column density bin.For each transition, we calculated the mean and rms disper-sion of the ten intensity values belonging to each column den-sity bin. Since the observations in the lowest column densitybin often resulted in non detections (likely due to low sensitivityand photodissociation e ff ects), we ignored this bin in our disper-sion calculations. In Fig. 6, we present histograms of the log-scale rms dispersion inside each column density bin for the threechemical families defined before.As can be seen, the rms dispersion of the CO isotopologs andthe traditional dense gas tracers has a well-defined peak near 0.2dex, a result that does not change if we include the data from thelowest column density bins. This level of dispersion is equivalentto a factor of 1.6 in linear scale, and its small value is responsiblefor the tight correlations that characterize the intensity-column-density plots. While low, the dispersion in the intensities is sig-nificantly higher than the dispersion of column densities insideeach bin, which we estimate as 0.05 dex. This indicates that thescatter of intensities is intrinsic to the gas, and not a mere re-flection of the range of column densities contained inside eachbin.As Fig. 6 shows, the family of additional tracers presents awider distribution of dispersions and lacks a clear peak, in agree-ment with the diversity of scatter levels seen in the bottom row ofFig. 5. The mean value of this distribution is 0.3 dex, again inde-pendently of whether the lowest column density bin is included.This value implies that the intensities in this chemical familyhave an rms scatter of a factor of two in linear scale, which isstill much smaller than the two orders of magnitude spanned bythe intensities. This again reflects the significant correlation be-tween line intensities and H column densities. While the low scatter of the intensities supports the use of thecolumn density as a proxy for the stratified random sampling, thefact that the scatter exceeds what would be expected simply fromcolumn density variations indicates that the column density isnot a perfect predictor of the emergent intensity. This should notbe surprising given that the column density is an integrated quan-tity that can be realized by multiple physical and chemical con-ditions along the line of sight. Our data already suggest severalcontributions to the dispersion of intensities associated with asingle column density. Chemical e ff ects, for example, likely playa role in the higher dispersion seen in the lines of C H and C H.These two species are known to present similar sensitivity to thepresence of an external radiation field (Pety et al. 2005), andthis field likely changes significantly across the cloud. Opticaldepth e ff ects, in addition, are likely to contribute to the disper-sion of intensities seen in species like HCO + (1–0), whose scatteris significantly larger toward the high column density bins. Ob-servations of this line at selected positions using high velocityresolution reveal that the HCO + (1–0) lines often su ff er from selfabsorption, a fact confirmed by the data from the optically thin-ner H CO + (1–0), shown in Appendix D, which present a muchlower degree of scatter. Another contribution to the scatter in theline intensities comes from di ff erences in the distribution of den-sities (and possible temperatures) along each line of sight. Thisis suggested by an observed increase in the scatter of the HCNand CS lines as their level energy increases (Sect. 5). Higher J lines have higher critical densities, and are therefore more sensi-tive to density variations along the line of sight. While the aboveexamples show the intrinsic limitation of using column densityas the sole predictor of the emergent intensity, they also illustratehow further understanding of the emission scatter could providenew insights on the internal structure of molecular clouds. column density To quantify how linear the dependence of the intensity with N (H ) is, we made least squares fits to the data points in Figs. 4and 5 (in log-log scale). To avoid any possible e ff ect of the pho-todissociation edge, we excluded the data in the first column den-sity bin, and for N H + , we also excluded the data in the followingthree bins because no emission was detected in them. Table 1presents the fit results (in log-log) together with the Pearson r coe ffi cient for all the transitions.As expected from the diagrams, the CO isotopologs presentslopes that are significantly lower than one, which is the valuethat corresponds to a linear correlation. The lowest slope valuesare those of CO and CO, and reflect the high optical depth of
Article number, page 8 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds.
Fig. 7.
Normalized ratio between integrated intensity and H column density as a function of H column density. The data points represent bin-averaged values and the error bars represent the rms dispersion. The horizontal gray band encloses the region were deviations from unity are withina factor of 2. Table 1.
Linear square fits to the log-
Int vs. log- N (H ) plots. Transition Slope Pearson- r CO isotopologs CO( J = . ± .
04 0.70 CO( J = . ± .
06 0.73C O( J = . ± .
05 0.86C O( J = . ± .
04 0.87Traditional dense gas tracersHCN( J = . ± .
04 0.93CS( J = . ± .
05 0.93HNC( J = . ± .
04 0.94HCO + ( J = . ± .
05 0.88SO( N , J = . ± .
06 0.90Additional tracersCH OH( J k = . ± .
07 0.90CN( N = . ± .
04 0.93C H ( J Ka , Kc = –1 ) 0 . ± .
07 0.81N H + ( J = . ± .
08 0.93C H( N = . ± .
06 0.78these lines, which makes their emission only weakly dependenton the gas column density. The lines from the rare isotopologsC O and C O are optically thin and present higher slope val-ues. Still, their lower-than-one slopes indicate that the lack of a linear correlation between the intensity and the H column den-sity represents an intrinsic property of the CO emission.Also as expected from the scatter plots, the slopes derived forthe traditional dense gas tracers are very close to unity. The slopevalues in Table 1 range between 1.0 to 1.2, and have a scatterat the level of about 5%. In addition, the Pearson r coe ffi cientsof these tracers have similar values of about 0.9 indicative of astrong level of correlation.Most members of the additional tracers group have slope val-ues similar to those of the traditional dense gas tracers, but a fewdeviate noticeably from a linear behavior. The most clear caseis N H + , which has a slope of 1.76 indicative of a preferencefor high column densities. More marginal, but probably still sig-nificant, is the case of CH OH, which has a slope of 1.30. Atthe other end of the scale, C H presents a significantly flat slopeof 0.78, indicative of a slight intensity increase toward the lowcolumn density gas.A more graphic representation of the close-to-linear relationbetween some of the line intensities with N (H ) is presented inFig. 7. This figure shows the ratio between the line intensity andthe H column density for each selected transition. For clarity,the plot shows normalized ratios with error bars that indicate thedata dispersion inside each bin.The top panels in Fig. 7 (red symbols) show how the COisotopologs significantly deviate from the constant ratio that cor-responds to a linear correlation between intensity and columndensity. This deviation is largest in the optically thick lines of CO and CO, whose ratio with the H column density variesby more than one order of magnitude over the cloud range. Thethinner C O and C O lines also present non constant ratios, but
Article number, page 9 of 27 & A proofs: manuscript no. 38727ms the variation of their geometrical mean over the cloud is limitedto a factor of two up and down, a range that is indicated with agray-shaded band. As mentioned above, the emission from thesetwo isotopologs is optically thin, so the curvature in the plots ofFig. 7 likely arises from systematic variations of the molecularabundance inside the cloud, an interpretation that will be furtherexplored with a radiative transfer model in Sect. 5.The middle panels in Fig. 7 (blue symbols) show the behav-ior of the intensity-column density ratio for the traditional densegas tracers. Overall, these tracers behave more linearly than theCO isotopologs, and most of their data points lie inside the gray-shaded band that encloses variations within a factor of 2. Thereis marginal evidence that in most tracers the point from the low-est column density bin lays below the gray-shaded band, withthe possible exception of HCO + . The most likely cause of thisdrop is the photodissociation of molecules by the external UVradiation field.Finally, the bottom panels in Fig. 7 present the results for theadditional tracers (green symbols). Most of these tracers haveratios inside the gray-shaded band, with the clear exception ofN H + and possibly CH OH. The sudden drop of N H + at H col-umn densities lower than 2 × cm − is highly significant sincethe observations had enough sensitivity to detect this species atlow column densities if the intensity had continued the lineartrend. As it will be discussed in the Sect. 5, the drop is con-sistent with the N H + abundance being only significant in thedenser regions of the cloud where CO is frozen out on the grains,as previously indicated by observations of dense cores (Caselliet al. 1999; Bergin et al. 2002; Tafalla et al. 2002).Although the data points of C H remain in the gray-shadedband of Fig. 7, they present a gradual increase by a factor offour from high to low column densities. This increase is againconsistent with the expected abundance enhancement of C H to-ward the outer cloud caused by the external UV radiation field(Pety et al. 2005). The related species CN does not present a no-ticeable increase, although the detailed model of the intensitiesbelow shows that it may be slightly enhanced toward the outercloud.To conclude the discussion, we provide in Table 2 estimatesof the X -factor (defined as the H column density over line inten-sity) for each line of the dense-gas and additional tracers as de-rived from our least-squares fit. For all species except for N H + ,data from all the column density bins but the lowest one wereused in the fit. For N H + , the fit used data only from the high-est four column density bins since the emission of this speciesdrops non linearly at lower column densities. Given the approx-imate linear behavior of all the tracers, the X -factors in the tablecould be used to compare the intensity of the Perseus emissionwith that of other clouds. They could also be used to infer H col-umn densities from the intensity of the observed lines, althoughwithout a proper calibration using other clouds, the result will besubject to a great degree of uncertainty. As mentioned in the Introduction, a recent e ff ort has been madeby di ff erent authors to map the emission of entire or close-to-entire molecular clouds in multiple lines. In this section we com-pare the results from this e ff ort to our observations, both to testthe sampling technique and to study the behavior of the emissionin di ff erent molecular clouds.We first compare our dataset with that of Watanabe et al.(2017), who carried out a multiline study of the high-massstar-forming cloud W51. These authors found that the 3mm- Table 2. X -factors for high-dipole moment species. Transition X a Transition X a HCN(1–0) 1 . × CH OH(2–1) 3 . × CS(2–1) 1 . × CN(1–0) 1 . × HNC(1–0) 2 . × C H (2 –1 ) 1 . × HCO + (1–0) 1 . × N H + (1–0) 1 . × SO(23–12) 2 . × C H(1–0) 2 . × Notes. ( a ) X = N (H ) / Intensity, in units of cm − (K km s − ) − . wavelength emission from W51 is dominated by the lines of theCO isotopologs together with the same traditional dense gas trac-ers found by us in Perseus (HCN, HCO + , HNC, and CS). Lack-ing extinction measurements for W51, Watanabe et al. (2017)used the integrated intensity of CO(1–0) as a proxy for thecloud column density, and found a tight and often close-to-linearcorrelation between this tracer and the integrated intensity ofmost molecular lines. This result is similar to our finding of atight correlation between the intensity of the main molecularspecies and the H column density in Perseus, with the caveatthat in Perseus the CO(1–0) emission does not correlate lin-early with the H column density (although in contrast withPerseus, the CO(1–0) emission in W51 is optically thin, seeWatanabe et al. 2017).A better comparison to our work can be made with the large-scale mapping of Orion B by Pety et al. (2017). These authorsused extinction measurements from Lombardi et al. (2014) tostudy the correlation between the intensity of the di ff erent linesand the extinction, as we have done with the Perseus data. Ingood agreement with our results, these authors find a tight cor-relation that is often close to linear for column densities largerthan a threshold value equivalent to an extinction of A V = H + and CH OH, classified assensitive to gas density, and C H and CN, classified as sensitiveto UV. These species are the same as those identified as peculiarin Perseus (Sect. 4.4), and this common behavior suggests thatdespite their very di ff erent characteristics (Orion B contains sev-eral embedded HII regions and the bright Horsehead PDR), themolecular emission from these two clouds is controlled by thesame main mechanisms.The final dataset with which we compare our Perseus resultsis that presented by Kau ff mann et al. (2017) for Orion A, whichis more limited than our Perseus dataset in number of transitions.Kau ff mann et al. (2017) focus their study of dense gas tracers onHCN, CN, C H, and N H + , which are four of the 11 specieswe studied in Perseus. For the first three species, they find thatthe ratio of the integrated intensity over the gas column densityvaries little over the cloud, and decreases at most by a factor oftwo when the column density increases by one order of magni-tude (their Fig. 2). This behavior is similar to the almost constantintensity-column density ratio found by us in Perseus.Also in agreement with our Perseus results, the N H + emis-sion from Orion A di ff ers from the other tracers by increasing Article number, page 10 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds.
Fig. 8.
Ratio between J = J = column density for the three CO isotopologs for which bothtransitions were observed (red circles). The blue dashed line representsthe prediction from a radiative transfer model of an isothermal cloud at11 K. The dotted lines represent similar models for 15 K (higher curve)and 7.5 K (lower curve). rapidly in the most opaque gas (with the possible exception ofthe densest two bins). As Kau ff mann et al. (2017) discuss, theN H + emission seems to be the only tracer that is sensitive to thedensest component of the cloud.While limited in targets, the above studies and our Perseusdata cover a variety of clouds with di ff erent levels of star-formation activity both in the low and high-mass regimes. De-spite these di ff erences, all clouds present systematic and oftentight correlations between the emission from most moleculartracers and the gas H column density. This correlation indi-cates that the cloud H column density behaves as a reliableproxy of the molecular emission under a variety of cloud con-ditions, which we saw was a requirement for the stratified sam-pling method to work. The data, therefore, support the idea thatthe stratified sampling method could be used to characterize theemission from a large variety of clouds.The data also show that there are significant similarities be-tween the emission from the di ff erent clouds, both in terms ofthe brightest lines and the dependence of their intensity with theH column density. Clearly more work needs to be done in thisarea, especially by comparing clouds using the same observingtechnique. The initial results, however, suggest that despite thelarge di ff erences between the clouds in terms of mass and star-formation activity, the emission that they produce follows a rel-atively simple and similar pattern.
5. A simple emission model for the Perseus cloud
The systematic and relatively simple behavior of the line intensi-ties in Perseus suggest that the emission from the cloud must becontrolled by the global properties of its gas, and not by specificdetails of di ff erent parts of the cloud. If this is the case, it shouldbe possible to reproduce the emission behavior using a relativelysimple treatment of the gas physical and chemical properties.In this section, we explore this possibility by building a simplecloud model that reproduces simultaneously the main emissionproperties identified by our Perseus survey. While this model re-sults from a deliberate attempt to reproduce the observations, itshould be not be seen as the product of a systematic search forthe best possible fit, but as an illustration of how the intensitiesobserved in Perseus can be naturally explained as arising fromgas conditions expected to occur in a molecular cloud of its type. Since our data show that the Perseus line intensities depend tofirst order on the H column density, we modeled the cloud us-ing the H column density as the main physical parameter. Allthe other gas properties that contribute to the line intensity, suchas the temperature, density, velocity dispersion, and molecularabundances were modeled as functions of the cloud H columndensity using simple parametric expressions. The form of theseexpressions was determined by fitting the intensity of transi-tions known to be sensitive to specific physical properties. Ta-ble. 3 summarizes our choice for the physical parameters, andAppendix C does the same for the molecular abundances.To determine the cloud gas temperature profile, we used the2–1 / CO, CO, and C O since thesespecies span a large range of optical depths and are easily ther-malized due to their low dipole moment. As Fig. 8 shows, the2–1 / observations ofalmost 200 dense cores. Fig. 8 shows that 11 K indeed providesa reasonable fit to the data (blue dashed lines), and while the fitquality could be slightly improved by adding small ad hoc devi-ations from isothermality, the constant temperature profile waspreferred in the name of simplicity. A constant temperature solution may at first seem surpris-ing since the cloud outer layers are likely warmer than the inte-rior (Wannier et al. 1983). Detailed modeling of molecular cloudsurfaces by Wolfire et al. (2010), however, shows that the CO-emitting gas that we used for the temperature determination is We note that our analysis does not correct for di ff erences in the angu-lar resolution of the J = ff ect the line intensities. Applyingthe standard beam dilution correction, for example, would be inappro-priate because this correction assumes that the emission arises from asmall source located at the beam center, while our observations dealwith extended emission observed with a random sampling. Not apply-ing a beam correction will likely increase the noise in the line ratio, butwill avoid introducing a systematic bias. The good behavior in Fig. 8 ofthe CO line ratio, which is temperature insensitive and therefore can-not be compensated with a special temperature choice, seems to supportour approach. Article number, page 11 of 27 & A proofs: manuscript no. 38727ms
Table 3.
Model physical parameters
Parameter Value T K
11 K n (H ) 2 × cm − ( N (H ) / cm − ) . ∆ V a − ( N (H ) / cm − ) . Notes. ( a ) Linewidths of CO and CO were multiplied by additionalfactors of 4 and 1.75 respectively to match observations. See text andFig. 9. almost insensitive to this outer warming because the same UVphotons that are responsible for the warming are also responsi-ble for photodissociating CO (see also Glover & Clark 2012).The net result of this process is that most of the warm outer gasis CO dark, and most of the CO-emitting gas remains at a close-to-constant temperature of 10 K. This e ff ect can be best seen inFigs. 4 and 6 from Wolfire et al. (2010), which show that the gastemperature remains close to 10 K all the way from the cloudinterior to an A V of about 2, which corresponds to the inner edgeof the lowest column density bin in our sampling. Only the outer1 A V magnitude of the cloud contains CO-emitting gas that iswarmer than 10 K, a fact that our modeling cannot test well dueto the weak signal of the emission. Any temperature increase inthe outer cloud will therefore only a ff ect our modeling of thealready poorly constrained outermost bin.The second cloud parameter that we model is the volumedensity of the gas. Assigning a single volume density to a givencolumn density represents a very strong simplification since anyline of sight through the cloud likely contains densities that varyby more than one order of magnitude. As discussed below, thissimplification limits the quality of the fits, but unfortunately isnecessary if we are to use a simple radiative transfer model. Todetermine the best-fit density profile we used a more indirect ap-proach than for the temperature since no molecular line or lineratio is significantly more sensitive to this density than others.After some experimentation, we chose to fit simultaneously theemission from multiple transitions of the traditional dense gastracers HCN (J = = N (H ) . .While the above density profile represents our favored choiceto fit the observed line emission, it should be considered only asa model parameterization. It represents a not-well-defined line-of-sight average weighted by the emissivity of the di ff erent lines,and it likely spans a limited range of values compared with thetrue range of volume densities in the cloud. This can be seenfrom the fact that if we were to use a similar relation to deter-mine the spatial extent of the gas in the di ff erent density regimesassuming simply that N (H ) = n (H ) × L , a steeper density pro-file would be required to reproduce the observed larger extent ofthe lower-density gas. Unfortunately, no similar global fit of theemission has been carried on other clouds, so it is not possible tocompare our results with previous work. We note however, thata similar (or close to linear) density dependence with columndensity can be seen in the compilation of numerical simulationspresented by Bisbas et al. (2019) (their appendix B).The final physical quantity required by our model is thegas velocity dispersion. We parameterized it using the observed Fig. 9.
Gas internal motions.
Top: C O(2–1) FWHM linewidth as afunction of H column density derived from Gaussian fits to the spectra(red circles). The dashed line represents an analytic approximation usedto model the trend. Bottom:
Linewidth ratios of the two main CO iso-topologs over C O. The dashed lines represent the constant ratios usedin the model (4 for CO and 1.75 for CO). linewidth of C O(2–1) since this line is optically thin and wasobserved with relatively high velocity resolution (0.27 km s − )due to its higher frequency. The top panel of Fig. 9 shows that theC O(2–1) linewidth increases weakly with H column densityin a way that we parameterized as N (H ) . (dashed line). Thislinewidth increase with column density is likely caused by thestar formation activity in the high column density bins. As Fig. 1shows, these bins are concentrated in the main star-forming re-gions of the cloud (NGC 1333, B1, and L1448), and many oftheir CO spectra present wings indicative of outflow contamina-tion. Surveys of both Taurus and Perseus have shown that thelinewidth of the C O lines tends to be larger in dense cores withstars compared to starless cores (Zhou et al. 1994; Kirk et al.2007), and the survey of Taurus cores by Onishi et al. (1996)found a systematic increase in the C O linewidth with H col-umn density similar to the one found by us in Perseus.The lower panels of Fig. 9 show that the J = CO and CO is significantlylarger than that of C O(2–1), by factors of 1.75 and 4, respec-tively. These larger linewidths most likely result from opticaldepth broadening (e.g., Hacar et al. 2016), and since the radiativetransfer model described below does not reproduce this feature,we incorporated them explicitly into the model when predictingthe CO and CO emission.
Once the physical properties of the model cloud were fixed, theonly parameter left to fit the emission of each species was its
Article number, page 12 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds.
Fig. 10.
Comparison between observations and model results for the family of CO isotopologs. The red symbols represent the observed intensitiesof the J = abundance profile. We aimed to describe these profiles using aparameterization that is both simple and consistent with our cur-rent knowledge of the chemical processes occurring in a molec-ular cloud like Perseus. After some exploring, we found that rea-sonable fits can be obtained by using a parameterization contain-ing three terms: X ( N (H )) = X × f out ( N (H )) × f in ( N (H )) , (1)where X represents a constant scaling factor, and f out and f in aretwo normalized factors that represent abundance changes withrespect to X in the outer and inner parts of the cloud (i.e., at lowand high H column densities).Appendix C provides a detailed discussion of the meaningof the di ff erent factors and the analytic expressions used in themodel. In this section, we summarize the main ideas that arerequired to understand the model results presented below.The f out factor represents any abundance change that takesplace in the outer layers of the cloud, likely as a result of theirexposure to the external UV radiation field. It is dominated bythe contribution of molecular photodissociation in a thin outerlayer of a few magnitudes of extinction, a process that has beenmodeled in great detail by previous chemical work (e.g., Tie-lens & Hollenbach 1985; van Dishoeck & Black 1988; Le Bour-lot et al. 1993; Sternberg & Dalgarno 1995; Visser et al. 2009;Wolfire et al. 2010; Joblin et al. 2018). In our model, a simplephotodissociation drop seems enough to fit the observed intensi-ties of most species in the lowest column density bins. For thesespecies, we used an analytic formula based on the realistic PDRmodels of Röllig et al. (2007). This formula is presented in Ap-pendix C, and has as its only free parameter the location of thesharp edge expressed in units of A V . Changes in this parameterallow the model to adjust for the still poorly characterized valueof the UV radiation field in the cloud, which has been previouslyestimated to have a Draine χ parameter (Draine 1978)) between1-3 (Pineda et al. 2008) or 24 (Navarro-Almaida et al. 2020). Itshould be noted, however, that the low column-density intensi-ties for most species are too weak to constrain the location of theedge, so most lines were fitted with a fixed value of A V = H and CN, the data show an intensity enhancementin a layer interior to the photodissociation edge, so we comple-mented the drop term with a more gradual outward abundanceenhancement. This term is based on the detailed modeling of theOrion Bar PDR by Cuadrado et al. (2015), who found evidence for an outer abundance increase in the small hydrocarbons drivenby gas-phase reactions involving C + .The final factor in our abundance parameterization is f in ,which describes possible variations in the cloud interior (i.e., athigh N (H ) values). For all species except N H + , the data requirea systematic abundance decrease with N (H ), likely caused byfreeze out. This is consistent with previous findings toward star-less dense cores in di ff erent environments (Caselli et al. 1999;Bergin et al. 2002; Tafalla et al. 2002), and for this reason, weparameterized f in with an expression used by Tafalla et al. (2002)to describe such systems. This expression has as only free pa-rameter the volume density characteristic of freeze out, whichhas been adjusted for each molecular species. As with the pho-todissociation edge, a narrow range of choices (1-2 × cm − )is enough to fit all the observations.For N H + , the observations require an abundance enhance-ment toward the cloud interior, in agreement with the theoret-ical expectation that this species increases its abundance whenCO freezes out (Bergin & Langer 1997; Aikawa et al. 2001). Toparameterize this e ff ect, we used a simple expression related tothat used for freeze out, and which is further described in Ap-pendix C.As an additional constraint to the model, we required thatthe relative abundances of the isotopologs follow the ratios de-termined for the local ISM by Wilson & Rood (1994). We thusassumed the following isotopic ratios: 77 for C / C, 560 for O / O, 3.2 for O / O, and 22 for S / S. To predict the emergent line intensities from the cloud model,we used a large velocity gradient (LVG) approximation to theradiative transfer (Castor 1970; Scoville & Solomon 1974). Thisapproximation provides a reasonable estimate of the emissiongiven the uncertain geometry of the cloud (White 1977), andthanks to its speed, allows exploring e ffi ciently a large rangeof input cloud parameters. Our LVG code is based on that pre-sented by Bieging & Tafalla (1993), and was complemented withmolecular data compiled by the Leiden Atomic and Molecu-lar Database (LAMDA), which is continually updated with themost recent literature values (Schöier et al. 2005; van der Taket al. 2020). In this section, we present the results of modelingseveral representative species that illustrate the di ff erent emis- https://home.strw.leidenuniv.nl/~moldata/ Article number, page 13 of 27 & A proofs: manuscript no. 38727ms
Fig. 11.
Comparison between observations and model results for HCN (top) and CS (bottom) . Each row presents the intensity of the di ff erenttransitions observed for the species (main and rare isotopologs) as a function of H column density (blue symbols) together with the predictionsfrom the model (dashed red lines). The CS(3–2) dataset has fewer points due to the limited observations carried out in the 2mm-wavelength band.Several H CN(1–0) points can be seen above the plot lower limit at very low column densities. A visual inspection of their spectra suggests thatthey represent noise or baseline residuals and not true molecular emission. sion behaviors identified in the cloud. Modeling results for therest of the species and a table summarizing the abundance pa-rameters derived from the fits are presented in Appendix D.Fig. 10 shows the model results for the J = J = / X value of the di ff erentCO isotopologs was set by fixing the value of C O to the deter-mination by Frerking et al. (1982) ( = . × − ) and using thealready-mentioned isotopic ratios from Wilson & Rood (1994).In addition, the model assumed a photodissociation edge of A V = cm − for all COisotopologs.As can be seen from Fig. 10, the cloud model, although notperfect, fits reasonably well the emission. This is remarkablegiven the simplicity of the model and the small number of freeparameters used to adjust the fit since once the cloud physicalparameters have been fixed, only three free parameters ( X , the A V value of the photodissociation edge, and the freeze-out den-sity) are left to reproduce the intensity of the four CO isotopologsover two orders of magnitude in H column density. Accordingto the model, the CO lines are optically thick and thermalizedeverywhere inside the photodissociation edge, and the slight in-tensity increase with the H column density arises from the sim-ilar increase in the linewidth. The CO line, on the other hand,only becomes optically thick at H column densities higher than2 × cm − . As expected, the C O and C O lines are opti-cally thin everywhere inside the cloud.A number of shortcomings in the model can be attributed toour simplified treatment of the abundance of the di ff erent COisotopologs. Our model assumes that they are scaled versions ofeach other, while in reality the CO abundance is expected to in- crease with respect to CO near the cloud edge due to isotopicfractionation, and the C O and C O abundances are expectedto decrease in the same region due to isotope-selective photodis-sociation (Bally & Langer 1982; van Dishoeck & Black 1988).These two e ff ects would likely enhance the CO intensity anddecrease the C O and C O intensities in the vicinity of the pho-todissociation edge, improving the fit results.To explore the ability of the cloud model to fit the inten-sity of the traditional dense gas tracers, we focus here on HCNand CS, which are the most widely used members of this fam-ily. Appendix D presents the results for the remaining members.For HCN and CS, our observations provided intensities for allsample positions of the transitions in the 1 and 3mm wavelengthbands, and for CS, a limited number of positions were also ob-served in the 2mm wavelength band. In addition, the 3mm tran-sitions of H CN and C S were also detected and included inthe model.Fig. 11 compares all the available HCN and CS data (bluesymbols) with the predictions from our cloud model (dashedred lines). The model assumes the same abundance factor X ( = × − ) and photodissociation edge ( = × cm − forHCN and 10 cm − for CS. As with the CO isotopologs, a sim-ple three-parameter model approximately fits simultaneously allthe observed line intensities over the full range of cloud columndensities.While the model intensities remain within the scatter of thedata points, and are therefore consistent with the observations,Fig. 11 shows that the model does not provide the best possi-ble fit. The HCN model slightly overestimates the J = CN. For CS, the model reproduces well the J = Article number, page 14 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds.
Fig. 12.
Comparison between observations and model results for twospecies that require special abundance profiles: N H + (top) and C H (bottom) . In both panels, the green symbols represent the observed in-tensity, previously shown in Fig. 5. In the N H + panel, the dashed redline represents a constant abundance model that fits the intensity at highcolumn densities. The blue dashed line represents the best-fit model,and includes an internal abundance enhancement. In the C H panel, thedashed red line represents a model with no external enhancement, whilethe blue line represents the externally enhanced best-fit model. transition of both the main and rare isotopologs together withthe 3–2 intensity, but it is close to the lower boundary of the J = + (Appendix D), likely resultfrom the use of a single density value to represent the complexcloud structure along any given line of sight. In a real cloud,the density along any line of sight likely increases toward theinterior. As a result, an optically thick line like HCN(1–0) willsample lower densities than an optically thin line like H CN(1–0). This e ff ect will decrease the HCN(1–0) intensity and increasethe H CN(1–0) intensity compared to our model, in agreementwith the observations. Also, a high critical density line, likeCS(5–4), will be sensitive to the highest density gas along theline of sight, an e ff ect that is again missed by our single-densityassumption. As mentioned above, fixing these fitting imperfec-tions would require having a realistic description of the multipledensity regimes present along any line of sight. Lacking such adescription, we consider that our single-density model providesa reasonably good fit to the observed intensities.As a final illustration of our modeling, we present in Fig. 12the solutions for N H + and C H, the two species that, together with CN, require special abundance profiles. The top panel ofthe figure compares the N H + data with the prediction fromtwo abundance models. The dashed red line corresponds to aconstant-abundance model set to fit the observed intensities athigh N (H ). As can be seen, the model significantly overesti-mates the intensities in the outer cloud, indicating the need ofan abundance drop at low N (H ). The second model (dashedblue line) corresponds to the profile described in Appendix C,and has a significant abundance drop at low H column den-sities, as expected from the destruction of N H + by CO whenthe latter species is abundant in the gas phase (Bergin & Langer1997; Aikawa et al. 2001). This modified model reproducesthe emission both at high and low H column densities, and istherefore in better agreement with the observations. It should benoted, however, that our data sampling in the transitional region( N (H ) ≈ cm − ) is not fine enough to constrain well theabundance drop, and that the drop could be sharper than sug-gested by the model. Additional observations of the N H + emis-sion in this transitional region and at lower column densities areneeded to fully characterize the distribution of this unique densegas tracer through the entire cloud.The bottom panel of Fig. 12 shows the model results for theC H(1–0) line. Again, the panel compares two abundance mod-els with the survey data. The dashed red line represents a stan-dard abundance model that has both outer photodissociation andinner freeze-out contributions. This model fits the emission inthe inner cloud, but fails to reproduce the observations at verylow column densities. The dashed blue line corresponds to anabundance profile that has an outer enhancement next to the pho-todissociation edge, and that is inspired by the PDR model of theC H abundance from Cuadrado et al. (2015) (see Appendix Cfor a full description). As can be seen, this modified model re-produces better the emission enhancement near the outer edge ofthe cloud in addition to the cloud interior. A similar model, butwith a smaller outer enhancement, is also needed to fit the CNemission, as shown in Appendix D.We summarize our modeling results by saying that thePerseus data suggest that the shape of the abundance profile ofany species is mostly controlled by how the species reacts to twoagents: (i) the UV ISRF at low column densities and (ii) dustcollisions at high column densities. All our survey species ex-cept N H + , C H, and CN behave passively with respect to theseagents, in the sense that they are photodissociated by the UV ra-diation and they freeze out onto the dust grains. As a result, theabundance of these species decreases both toward the outer edgeand inside of the cloud in a manner illustrated by the top panelof Fig. C.2.The three exceptions we found to the above abundance pat-tern result from some type of active reaction to either the UVradiation or to the collisions with the dust. In the case of C Hand CN, these species are enhanced near the cloud edge as aresult of the UV ISRF, and in the case of N H + , this moleculethrives when CO starts to freeze out. These two behaviors areillustrated in the middle and bottom panels of Fig. C.2. Whentaken into account, the complete set of observed lines in Perseuscan be reproduced with a relatively simple model.The fact that a simple model like the one presented herecan reproduce the line intensities of so many species and transi-tions suggests that the main properties of the line emission in thePerseus cloud are controlled by a small number of processes thatcan be simulated with a few model parameters. Whether this be-havior is peculiar to Perseus or common to other clouds requiresfurther investigation, and will be explored in future work. Article number, page 15 of 27 & A proofs: manuscript no. 38727ms
6. Discussion
Since our cloud model reproduces the main features of thePerseus emission, we can use it to investigate the origin of thedi ff erent correlations between intensity and H column densityfound for di ff erent species. Of particular interest is the com-parison between the correlation of the CO isotopologs, whichpresent a rather flat dependence with N (H ), and the correlationof the traditional dense gas tracers HCN and CS, which approx-imately follow a linear dependence with N (H ).To investigate the origin of these di ff erent correlations, welook separately at the two factors that contribute to the inten-sity in the solution of the equation of radiative transfer: J ( T ex ) − J ( T bg ) and (1 − e − τ ), which we will refer to as the “excitation”factor and the “optical depth” factor, respectively. In Fig. 13 wepresent their value as a function of N (H ) for the 3mm lines ofthe main CO, HCN, and CS isotopologs (solid lines) and the rareisotopologs C O, H CN, and C S (dashed lines).As Fig. 13 shows, the excitation factor for both the main andrare CO isotopologs (top panel, solid and dashed red lines) hasan approximately constant value close to the gas kinetic temper-ature minus the background temperature, as expected for speciesthat are thermalized over most of the cloud. The optical depthfactor, on the other hand, is di ff erent for the main and rare COisotopologs (bottom panel, red lines). For the main isotopolog,the optical depth factor has a close-to-constant value of 1 overmost of the cloud due to the extremely high optical depth of theemission. For the less abundant C O, the optical depth factor re-flects closely the C O column density, which is not linear with N (H ), but curves downward at high values due to the increasinge ff ect of freeze out. When the excitation and optical depth fac-tors are multiplied to obtain the intensity, the result is an approx-imately constant function for CO and a slightly curved intensitylaw for C O, as observed in the data.For the HCN and CS isotopologs, the excitation factor dif-fers sharply from that of CO. As can be seen from the green andblue lines in the top panel of Fig. 13, while the excitation factorof CO is flat, that of HCN and CS increases rapidly with N (H ).This increase is a consequence of the HCN and CS lines beingsignificantly subthermal, and therefore having an excitation thatincreases rapidly as the volume density increases with N (H ).For the main HCN and CS isotopologs (solid lines), the excita-tion has an additional contribution from photon trapping due tothe high optical depth of the lines, and this enhances their factorover that of the rare isotopologs, for which trapping is insignifi-cant (dashed lines). Independently of this extra contribution, theexcitation factor of all the HCN and CS isotopologs increases byabout two orders of magnitude over the H column density rangeof the cloud.In contrast with the excitation factor, the optical depth factorof HCN and CS behaves like that of CO (bottom panel). For themain isotopologs, the factor has an almost constant value of oneinside the photodissociation edge, while for the less abundantisotopologs, it presents a less-than-linear increase with N (H )due to the e ff ect of freeze out. This similarity with CO is notsurprising since all the species have similar abundance profiles,and the main isotopologs are very optically thick while the rareones are thin.The similar behavior of the optical depth factors of CO andthe traditional dense gas tracers indicates that the di ff erent de-pendence of the intensity of HCN-CS and CO with N (H ) in thecloud interior arises mainly from a di ff erence in the excitation ofthe molecules. For the CO isotopologs, the combination of flat Fig. 13.
Model results for the two factors of the radiative transfer so-lution: excitation term (top) and optical depth term (bottom) . The solidlines represent the results for the main isotopologs of CO (red), HCN(green), and CS (blue). The dashed lines represent equivalent results forthe rare species (C O, H CN, and C S). excitation factors due to thermalization with flat or not too steepoptical depth factors results in relatively flat intensity correla-tions with N (H ). For HCN and CS, the steep excitation factorsresulting from subthermal excitation dominate the dependenceof the emergent intensities and are ultimately responsible for thestrong correlation of the intensities with N (H ).To understand why the intensity of the main and rare iso-topologs of HCN and CS present similar dependence with N (H ), we need to consider now the combined e ff ects of exci-tation and optical depth. As seen in the top panel of Fig. 13, themain isotopologs (solid lines) have significantly steeper excita-tion factors than the rare isotopologs (dashed lines) due to theadditional contribution from photon trapping. These factors haveslopes that, although not constant, are close to linear, as can beseen from a comparison with the dotted line shown in the panels.When these factors are multiplied by the almost constant opticaldepth factors, the resulting intensities retain the close-to-linearslope.The rare isotopologs, on the other hand, present slightly flat-ter excitation factors due to the missing trapping contribution.When these factors are multiplied by the optical depth factors,which have a non-negligible slope over most of the N (H ) range,the resulting emergent intensity approaches the linear slope of Article number, page 16 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds. the thick main isotopologs. It seems therefore that the similarbehavior of the thin and thick traditional dense gas tracers asa function of H column density results from the approximatecompensation of their di ff erent excitation and optical depth fac-tors. While somewhat fortuitous, this behavior seems to be veryrobust since it is displayed by most observed species in multi-ple transitions, and ultimately gives rise to the systematic quasi-linear emission pattern found by most dense gas tracers in oursurvey. The almost linear dependence with H column density of theemission from species like HCN and CS forces us to reevalu-ate the traditional distinction between dense gas tracers and col-umn density tracers used to classify molecules. The correlationsfound in Perseus imply that, strictly speaking, only N H + de-serves the term dense gas tracer since it is the only species trulyselective of the dense gas: its emission remains undetected be-low around 10 cm − (corresponding in our model to a densityof 2 × cm − ), and then rises nonlinearly when N (H ) exceedsthe threshold. This unique behavior of N H + has been previouslyseen at core scales (Caselli et al. 1999; Bergin et al. 2002; Tafallaet al. 2002) and at cloud scales (Kau ff mann et al. 2017; Pety et al.2017), and results from the sensitivity of N H + to the presenceof CO in the gas phase. Although no other molecule in our sur-vey presents a similar behavior, it is likely that NH may do sosince NH shares with N H + a resilience to freeze out in addi-tion to being a late-time chemical species (Suzuki et al. 1992;Tafalla et al. 2002). Indeed, observations of N H + and NH to-ward Perseus dense cores have found a strong correlation be-tween the emission of these two species (Johnstone et al. 2010;Hacar et al. 2017). Extending our Perseus observations to in-clude the cm-wavelength NH lines would be highly desirable totest this prediction.Although HCN, CS, and the other traditional dense gas trac-ers do not display the selectivity to the dense gas of N H + , theyare still sensitive to the cloud dense material in an indirect way.This is so because their emission follows the column density,which we have seen is itself an indirect indicator of the gas vol-ume density. These tracers, therefore, are brighter toward thehigher column density gas, which results from the presence ofdense gas along the line of sight. For this reason, tracers likeHCN and CS are still useful to identify the densest parts ofmolecular clouds in spatially resolved maps, as proven by nu-merous previous studies (e.g., Plume et al. 1997; Beuther et al.2002; Wu et al. 2010).The lack of selectivity to the dense gas of HCN, CS, and sim-ilar tracers is a more serious problem in unresolved observationsof molecular clouds, as those commonly used in extragalacticwork. In this type of work, interpreting the emission from thedi ff erent tracers is not straightforward since this emission repre-sents a weighted average to which all the density regimes in thecloud contribute. The relevant parameter in this case is the prod-uct of the column density probability distribution function (PDF)times the line intensity as a function of column density. As wehave seen, the Perseus data show that for HCN, CS, and othertracers, the intensity is approximately linear with N (H ), whilethe column density PDF depends much more steeply on the pa-rameter, as N (H ) − (Zari et al. 2016). As a result, the integratedemission will be dominated by the contribution from the lowestcolumn density positions, whose nonlinear overabundance morethan compensates for the approximately linear decrease in theirintensity. A similar dominant contribution of the low-density gas in thecloud-integrated emission from the traditional dense gas tracershas also been found by several studies, and likely represents ageneral trend among molecular clouds (Kau ff mann et al. 2017;Pety et al. 2017; Watanabe et al. 2017; Shimajiri et al. 2017;Nishimura et al. 2017; Evans II et al. 2020). This finding shouldbe not surprising if in most clouds, as in Perseus, the PDF hasa very steep negative dependence with N (H ) (Lombardi et al.2015), while the line intensities depend almost linearly with N (H ) (e.g., Pety et al. 2017). These two behaviors naturallycombine to make the contribution of the low column density po-sitions increasingly more important in the cloud-integrated in-tensity.Even if the integrated emission of the traditional dense gastracers is dominated by the low-density gas, it still has a rele-vant physical interpretation. We have seen that in Perseus theemission follows quasi-linearly the column density in the regionabove the photodissociation threshold. As a result, the cloud-wide integrated intensity, which is obtained integrating the prod-uct of the PDR times the intensity as a function of the columndensity, will be proportional to the mass of the cloud above thephotodissociation threshold. Since our models indicate a thresh-old of about A V = A V .While no other cloud has been characterized using the sam-pling method we used for Perseus, we have seen that an ap-proximately linear dependence of the intensity with the columndensity is a likely feature of several other clouds (Sect. 4.5). Ifthis is so in general, the same relation between cloud-integratedintensity and mass above the photodissociation threshold of acloud will apply. Such a possibility, which clearly requires fur-ther investigation, is of interest for extragalactic work, wherethere is evidence for a linear correlation between the emissionof tracers such as HCN(1–0) and the rate of star formation (Gao& Solomon 2004; Kennicutt & Evans 2012; Usero et al. 2015;Jiménez-Donaire et al. 2019). Interpreting the observed line in-tensities as estimates of the cloud mass inside a certain columndensity threshold could potentially help explain the origin of thecorrelation.Given the above, it is worth noting that Lada et al. (2010)have found that for local clouds (including Perseus), the massabove a column density threshold of A V ≈ . A K ≈ . ff erent from(and much smaller than) the mass inferred from the HCN(1–0)intensity . It is therefore unclear whether the Lada et al. (2010)relation can help connect the cloud-integrated HCN(1–0) inten-sity with the star-formation rate as found by extragalactic ob-servations. One possibility is that the masses set by the 2 and7.3 mag thresholds are linearly correlated, something that is pos-sible given the quasi-universal power-law nature of the columndensity PDFs Lombardi et al. (2015).The above speculations clearly show that an observational ef-fort is needed to characterize the emission from multiple cloudsand test whether our Perseus results reflect a more general trend. The 7.3 mag mass will be close to that inferred from the N H + inten-sity, which becomes observable at column densities equivalent to about A V =
10 mag, and is in fact a favored tracer of star-forming sites in localclouds. Article number, page 17 of 27 & A proofs: manuscript no. 38727ms
For such an e ff ort, the stratified random sampling method pre-sented here appears to be an ideal tool.
7. Conclusions
We used stratified random sampling to select 100 target positionsthat represent the di ff erent regimes of the H column density inthe Perseus molecular cloud. We observed these positions withthe IRAM 30m telescope covering the 3mm-wavelength bandand selected parts of the 2mm and 1mm bands. We studied thecorrelation of the observed line intensities with the H columndensity, and we developed a simple cloud model to reproducethe main features of the emission. The main conclusions fromthis work are the following.1. A comparison of our sampling results for CO(1–0) and CO(1–0) with the mapping data from the COMPLETE projectshows that stratified random sampling can be used to estimatethe mean intensity and the intensity dispersion of these two linesas a function of column density within a factor of 1.5 or better.2. The intensity of the molecular lines in Perseus stronglycorrelates with the H column density over the two orders ofmagnitude that this quantity spans through the cloud. The linesof the CO isotopologs and the traditional dense gas tracers (CS,HCN, HCO + , etc.) present a level of dispersion in the intensity- N (H ) relation of only about 0.2 dex. This tight correlation be-tween line intensities and H column density supports the useof column density as a proxy of the intensity for the stratifiedrandom sampling, as initially motivated by principal componentanalysis (Ungerechts et al. 1997; Gratier et al. 2017).3. The intensity of the CO isotopologs increases slower thanlinearly with N (H ), while the intensity of most dense gas tracersincreases approximately linear with N (H ).4. Several dense gas tracers present significant deviationsfrom a linear dependence with N (H ). The largest deviation isthat of N H + (1–0), which presents a very rapid transition fromundetected to relatively bright near N (H ) = cm − . C H(1–0) and (to smaller extent) CN(1–0) present significant enhance-ments in their intensity at low column densities ( < cm − ).5. The main emission trends identified in the Perseus dataare similar to those recently reported from multiline mappingof entire clouds (e.g., Kau ff mann et al. 2017; Pety et al. 2017;Watanabe et al. 2017). This similarity suggests that the samplingtechnique can truly characterize e ffi ciently the emission from acloud. It also points to a general behavior of the emission inclouds that span a relatively large range of star-formation con-ditions.6. A simple cloud model can reproduce the main emis-sion features of Perseus. For most species, the model requiresa molecular abundance that has a sharp drop at low column den-sities (likely due to photodissociation) and a more gradual de-crease at high column densities (likely caused by freeze out).The only species that require di ff erent abundance distributionsare N H + , which is enhanced when CO freezes out, and C Hand CN, which are enhanced by the external UV field. The dif-ferent abundance behaviors of these species are consistent withour current understanding of molecular cloud chemistry.7. Our cloud model suggests that the flat dependence with N (H ) of the CO isotopologs results from a combination of op-tical depth e ff ects and molecular freeze out under thermalizedconditions. The quasi-linear behavior of most traditional densegas tracers and their isotopologs results from the increase in ex-citation with column density combined with molecular freezeout toward the densest gas. A lucky compensation between ex-citation in the optically thick species and column density e ff ects in the rare isotopologs is responsible for their similar behaviorwith N (H ).8. The cloud-integrated emission of the traditional dense gastracers is dominated by the contribution from the lowest col-umn density positions because their nonlinear overabundance(described by the PDF) more than compensates for the approxi-mately linear decrease in their intensity. The quasi-linear depen-dence of their intensity with N (H ) makes the integrated inten-sity of these tracers approximately proportional to the mass ofthe gas interior to the photodissociation edge. This property mayhelp provide a physical meaning to the integrated intensity ofthese tracers in unresolved observations. Acknowledgements.
We thank an anonymous referee for a thorough review ofthe manuscript and for numerous comments and suggestions that have helpedimprove this work. We thank the IRAM sta ff for their excellent support duringthe 30m telescope observations. We acknowledge support from grant AYA2016-79006-P from MINECO, which is partly funded through FEDER, and from grantPID2019-108765GB-I00. AU acknowledges support from PGC2018-094671-B-I00 (MCIU / AEI / FEDER). AH acknowledges support from VENI project639.041.644, which is (partly) financed by the Netherlands Organisation for Sci-entific Research (NWO). This work is based on IRAM 30m-telescope observa-tions carried out under project numbers 034-17 and 104-17. IRAM is supportedby INSU / CNRS (France), MPG (Germany), and IGN (Spain). This research hasmade use of NASA’s Astrophysics Data System Bibliographic Services and theSIMBAD database, operated at CDS, Strasbourg, France.
References
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Appendix A: Sample positions and 3mm lineintensities
Table A.1 presents the coordinates, column density estimates,and 3mm line intensities for the 100 sample positions used in thePerseus survey. A full version of the table is available on-line.
Appendix B: Stacked spectra
Figs. B.1, B.2, and B.3 present stacked spectra for the 3 mm-wavelength transitions discussed in Sect. 4.2. The stacking wasdone by averaging the ten spectra of each column density binafter having shifted the lines to zero velocity using a Gaussianfit to the CO(1–0) line. To ease inter-comparing the spectra,each plot uses the same intensity scale (in units of T mb ). Mostspectra have been scaled by factors of 1 . n , where n depends onthe bin number, to approximately compensate for the change inH column density between the bins. Appendix C: Modeling abundance profiles
As mentioned in Sect. 5.2, our cloud model aims to provide a de-scription of the abundance profiles that is both simple and con-sistent with the chemical processes known to occur in a cloud. Tothis end, we found convenient to write the abundance profile ofeach species (with respect to H ) as the product of three factors, X ( N (H )) = X × f out ( N (H )) × f in ( N (H )) , where X is a constant scaling, f out represents a departure fromconstant near the outer edge of the cloud, likely caused by theUV photons from the interstellar radiation field, and f in repre-sents a departure from constant at high column densities, likelyrelated to molecular freeze out.For all species but C H and CN, the f out factor needed tofit the data represents a sharp abundance drop that we associatewith the photodissociation of the molecules by the external UVradiation field. Although the detailed shape of this drop is notvery critical for our modeling due to the lack of column-densityresolution near the cloud edge, we used a realistic profile basedon the photon dominated region (PDR) models of Röllig et al.(2007). As a representative profile, we selected the CO abun-dance curve estimated by these authors in their F1 model ( n (H ) = × cm − , and χ = χ pa-rameter (Draine 1978) that is intermediate between the valuesfavored by Pineda et al. (2008) from their fit of the Perseus COlines ( χ = χ = f out ( N (H )) = (cid:40) [ e ( A V − A ) ] if A V ≤ A A V > A , where A V is the H column density expressed in units of visualextinction ( N (H ) = . × A V cm − , Bohlin et al. 1978). A is a free parameter that describes the location of the photodis-sociation edge also in units of visual extinction and allows forchanges in the value of χ with respect to the original model.Fig. C.1 compares a set of abundance values extracted fromFig. 4b in Röllig et al. (2007) (red circles) with our analytic ex-pression for a choice of A = . A values needed to fit the Perseus data range between 1 and 2 mag, in good agreement with the theoret-ical expectation from the PDR model. It should be noted, how-ever, that for most species, our observations cannot constrain theexact location of the photodissociation edge due to the limitedsignal to noise ratio of the intensities at low column densitiesand the coarseness of the column density bins. For these species,we used a single value of 2 mag based on the results from theCO data. This choice provides a reasonable match to the data,but it should not be considered as a well-defined best fit.Since the C H and CN intensities require a significant abun-dance enhancement in the vicinity of the photodissociation edge,we multiplied the f out factor of these species by the additionalterm f (cid:48) out ( N (H )) = + α e − A V / , where the free parameter α describes the magnitude of the outerabundance enhancement. This additional factor was inspired bythe detailed modeling of the C H abundance in the Orion BarPDR by Cuadrado et al. (2015), who found that the species un-dergoes an abundance enhancement of several orders of magni-tude in the vicinity of the cloud edge (their Fig. 17).The last factor in our abundance parameterization is f in ,which describes deviations from a constant value in the cloudinterior. For all species but N H + , the data requires a significantabundance drop at high column densities that is likely caused byfreeze out onto the dust grains. Following previous freeze-outmodeling by Tafalla et al. (2002), we described this abundancedrop with the simple analytic expression f in ( N (H )) = e − n (H ) / n fr , where n (H ) is the gas density (related to N (H ) by the cloud pa-rameterization described in Sect. 5.1), and n fr is a free parameterthat describes the characteristic freeze-out density of the species.As shown in Table D.1, all species in the sample can be fittedwith n fr values in the range (1 − × cm − , characteristic ofmolecular freeze out.For the freeze-out resistant N H + molecule, the above ex-pression was substituted by f (cid:48) in ( N (H )) = − e − ( n (H ) / n fr ) , which represents an abundance enhancement, and where n fr isagain a free parameter that describes the characteristic density atwhich the N H + enhancement occurs.Fig. C.2 presents the three types of abundance profiles gener-ated using our three-factor parameterization that were needed tofit the variety of intensity behaviors identified in our Perseus sur-vey. The top panel presents the abundance profile used to fit allspecies excluding N H + , C H, and CN. This profile is character-ized by a sharp photodissociation edge at low column densities, aclose-to-constant abundance value at intermediate column den-sities, and a gradual abundance drop due to freeze out at highcolumn densities. We refer to this profile as the “standard” one.The middle panel shows the abundance profile needed to fit theN H + emission. It presents an opposite behavior to the standardprofile at high column densities: an initial rapid abundance in-crease followed by a close-to-constant value. Finally, the bottompanel presents the abundance profile used to fit the C H emis-sion (a similar one was used for CN). This profile presents asignificant enhancement at low column densities followed by anexternal photodissociation edge.
Article number, page 20 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds. T a b l e A . . S a m p l e po s iti on s a nd li n e i n t e n s iti e s . P o s iti on a R A ( J ) D ec ( J ) N ( H ) C O ( ) C O ( ) C O ( ) C O ( ) H C N ( ) C S ( ) HN C ( ) H C O + ( ) ( h m s )( ◦ (cid:48)(cid:48)(cid:48) )( c m − )( K k m s − )( K k m s − )( K k m s − )( K k m s − )( K k m s − )( K k m s − )( K k m s − )( K k m s − ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) ( ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) ( ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) ( ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) ( ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) P E R S - . + . . ( . ) × ( ) ( ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) N o t e s . A f u ll v e r s i ono f t h i s t a b l e i s a v a il a b l e on - li n e . ( a ) T h e fi r s t nu m b e r i n t h e po s iti onn a m e i nd i ca t e s t h ec o l u m nd e n s it yb i n a nd t h e s ec ondon e i nd i ca t e s t h e o r d e r i nou r s a m p li ng s e qu e n ce . Article number, page 21 of 27 & A proofs: manuscript no. 38727ms
Fig. B.1.
Stacked spectra of the J = ff erent CO isotopologs Each spectrum represents the average of the ten spectra taken in thecolumn-density bin indicated in the right label. The mean column density of these bins decreases downward by factors of 1.6, and the spectra havebeen scaled up by di ff erent factors to approximately maintain the same physical size.Article number, page 22 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds. Fig. B.2.
Same as Fig. B.1 but for the traditional dense gas tracers: HCN(1–0), CS(2–1), HNC(1–0), HCO + (1–0), and SO(23–12).Article number, page 23 of 27 & A proofs: manuscript no. 38727ms
Fig. B.3.
Same as Fig. B.1 but for the additional gas tracers: CH OH(2–1), CN(1–0), C H (2 –1 ), N H + (1–0), and C H(1–0). For CN(1–0) andC H(1–0), only a subset of the hyperfine components is shown due to the limited velocity window. The negative CN(1–0) feature near -7 km s − is an artifact from the frequency switching observing mode.Article number, page 24 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds. Fig. C.1.
Abundance factor used to model the cloud photodissociationedge. The red symbols represent values of the CO abundance predictedby the F1 PDR model of Röllig et al. (2007) and extracted from theirFig. 4b. The solid blue line is the analytic expression described in thetext for a choice of A = . Fig. C.2.
Three types of abundance profiles generated by a simplemodel and used to fit the emission from di ff erent species in Perseus. Toppanel:
Standard abundance profile consisting of an outer photodissocia-tion edge and an inner freeze-out drop. It was used to fit most moleculardata and is illustrated here with CO.
Middle panel:
Profile with an innerenhancement required to fit the N H + data. Bottom panel:
Profile withan outer abundance enhancement required to fit the C H and CN data(illustrated here with C H).
Appendix D: Fitting results for remaining species
In Sect. 5.3 we presented radiative transfer results for a represen-tative group of molecular species that cover the di ff erent emis-sion behaviors found in Perseus: the family of CO isotopologs,CS and HCN (as examples of traditional dense gas tracers withstandard abundance profile), and N H + and C H, two specieswith inner and outer abundance enhanced profiles, respectively.In this Appendix we present the model results for the remainingspecies of the survey.As mentioned in Sect. 5.3, our model aims to fit simultane-ously for each species all the transitions of the main and rareisotopologs detected in the 1mm, 2mm, and 3mm wavelengthwindows covered by the survey. We assume that the abundancesof the di ff erent isotopologs are related by the ratios estimated byWilson & Rood (1994) in the solar vicinity.Fig. D.1 presents the fit results for the tracers not presentedin Sect. 5.3: the traditional dense gas tracers HNC, HCO + , andSO (blue symbols), and the additional tracers CH OH, CN, andC H (green symbols). For each species, all observed transitionsand isotopologs are presented in the same row, except for CN andC H , for which only one transition was detected, and which arepresented together in the bottom row for space reasons. As can be seen in the figure, the model predictions (dashedlines) generally lie within the cloud of observed data points, in-dicating that the model provides a reasonable fit to the obser-vations. Some deviations from the expected best fit are seen inspecies where the lines span a large range of optical depths, suchas HNC and HCO + . These deviations are similar to those foundwhen fitting the CS and HCN data, in the sense that our modelhas di ffi culty reproducing simultaneously the J = ffi cients used tomodel the excitation. As mentioned in Sect. 5.3, these coe ffi -cients, together with any additional molecular parameters, wereinput into the radiative transfer code using the files provided bythe LAMDA database (Schöier et al. 2005; van der Tak et al.2020).A significant result from Table D.1 is that the abundancevalues derived for Perseus match closely abundance estimatesmade for other star-forming regions. As mentioned before, the A values of the photodissociation edge and the characteris-tic freeze-out density n fr agree with previous observations andchemical model predictions for typical clouds (e.g., Bergin &Langer 1997; Caselli et al. 1999; Bergin et al. 2002; Tafallaet al. 2002; Röllig et al. 2007; Pineda et al. 2008). In addition,the X values in Table D.1 for all species with the exception ofC H agree within a factor of three with the geometrical meanof the undepleted abundances in the Taurus-Auriga L1498 andL1517B dense cores estimated by Tafalla et al. (2006) (the dif-ference for C H is a factor of 6). Considering that these twoestimates used very di ff erent radiative transfer assumptions andeven di ff erent collision rates, the agreement is notable. It sug-gests once again that the gas properties derived from our Perseusanalysis are likely representative of the gas properties in otherstar-forming regions. Article number, page 25 of 27 & A proofs: manuscript no. 38727ms
Fig. D.1.
Comparison between observations and model results for the species not presented in the main text. The data points represent theobservations (traditional dense gas tracers in blue and additional tracers in green), and the red dashed lines represent the model results. Each panelrow contains all observed transitions and isotopologs of any given molecular species, except for the last row that shows together the only observedlines of CN and C H . Several SO and C H points can be seen above the plot lower limit at very low column densities. A visual inspection oftheir spectra suggests that they represent noise or baseline residuals, and not true molecular emission.Article number, page 26 of 27. Tafalla et al.: Characterizing the line emission from molecular clouds. Table D.1.
Best-fit abundance parameters and collision-rate coe ffi cients Species X A α n fr Coll. Coe ff .(mag) (cm − ) ReferenceCO 9 . − − − − + . − − OH 1 . − . − H − H + . − H 2 10 − Notes.
See Appendix C for a full description of the abundance parame-ters.