Yuchen Xing and Keping Qiu

1 School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China; kpqiu@nju.edu.cn

2 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, China

Received 2022 March 8; revised 2022 May 6; accepted 2022 May 12; published 2022 June 14

Abstract We revisit the mass–size relation of molecular cloud structures based on the column density map of the Cygnus-X molecular cloud complex. We extract 135 column density peaks in Cygnus-X and analyze the column density distributions around these peaks. The averaged column density profiles, N(R), around all the peaks can be well fitted with broken power-laws,which are described by an inner power-law index n,outer power-law index m,and the radius RTP and column density NTP at the transition point. We then explore the M–R relation with different samples of cloud structures by varying the N(R) parameters and the column density threshold, N0, which determines the boundary of a cloud structure. We find that only when N0 has a wide range of values, the M–R relation may largely probe the density distribution,and the fitted power-law index of the M–R relation is related to the power-law index of N(R).On the contrary,with a constant N0,the M–R relation has no direct connection with the density distribution; in this case,the fitted power-law index of the M–R relation is equal to 2(when N0 ≥NTP and n has a narrow range of values), larger than 2 (when N0 ≥NTP and n has a wide range of values), or slightly less than 2 (when N0

Key words: methods: analytical – methods: data analysis – ISM: clouds – ISM: structure

1. Introduction

The density distribution reflects the physical state of a molecular cloud thus is important for understanding star formation. However, both volume and column density distributions are difficult to obtain in large quantities directly.Dust extinctions in optical and near-infrared bands can be used to derive H2distribution at high resolution but cannot probe dense regions (Lada et al. 1994; Lombardi & Alves 2001).Although dust continuum and molecular lines at millimeter and submillimeter wavelengths are free of this problem, they are limited by the low resolution of single-dish radio telescopes and the small dynamic range of interferometers(Kellermann&Moran 2001). Moreover, obtaining the density distributions of a large number of sources across orders of magnitude in density and size is always time-consuming regardless of the observation method used.For decades,the mass–size relation between different structures (hereafter, the M–R relation) has been an important way to explore the density distribution of molecular gas.

An early result of the M–R relation comes from Larson(1981).The famous Larson third law indicated that the density—size relation at 0.1–100 pc is n(H2)∝L?1.10, corresponding to M ∝R1.9. The relation was considered to represent a density distribution of ρ ∝R?1, implying that the structures that they used in obtaining the M–R relation have approximately the same averaged column density.Since then,there have been a number of observational studies deriving a variety of M–R relations from M ∝R1.4to M ∝R3.0, which have been interpreted as ρ ∝R?αdistributions with α=0 ?1.6. The M ∝R2relation is the most commonly seen relation and has been observed in all scales from 10?2pc to 102pc(Larson 1981;Schneider&Brooks 2004;Lada& Dame 2020; Mannfors et al. 2021), while the other indexes are mainly observed at 10?2?101pc(Roman-Duval et al.2010;Traficante et al. 2018; Urquhart et al. 2018; Massi et al. 2019;Lin et al. 2019).

However, how reliable or accurate the M–R relations are probing the density distributions is still a matter of debate.Observational biases, including the sensitivity limit (Kegel 1989; Schneider & Brooks 2004) and the column density selection effects for certain tracers (Scalo 1990; Ballesteros-Paredes & Mac Low 2002), as well as the source extraction methodologies (Kegel 1989;Schneider &Brooks 2004;Heyer et al. 2009), can all play a role in the derived M–R relations,and thus affect the inferred density distributions. In the 2000s,dust continuum surveys brought new opportunities to understand the M–R relation(Enoch et al.2006;Pirogov et al.2007).The advent of the Herschel observatory (Pilbratt et al. 2010)made it possible to map simultaneously extended and compact dust continuum emissions at multi-wavelengths in the farinfrared to submillimeter window. Consequently, the column density profiles (hereafter, N(R) profiles) of dense molecular cloud structures can be derived at moderate angular resolutions(Arzoumanian et al. 2011; Kauffmann et al. 2010; Schneider et al. 2013). The N(R) profiles are found to have different indexes at different scales and their corresponding M(R)profiles may be inconsistent with the M–R relation (Pirogov 2009; Lombardi et al. 2010; Kauffmann et al. 2010;Beaumont et al.2012).Lombardi et al.(2010);Beaumont et al.(2012), and Ballesteros-Paredes et al. (2012) pointed out that measuring the M–R relations based on observations in general implies an effective column density threshold, which in turn would naturally lead to an M ∝R2relation for typical column density probability distribution functions (N-PDFs), such as log-normal (Lombardi et al. 2010; Beaumont et al. 2012),power-law (Ballesteros-Paredes et al. 2012) or log-normal +power-law(Ballesteros-Paredes et al.2012)N-PDFs.However,there has been no study that links real observational M(R)profiles with M–R relations through mathematical calculations.

In this paper,using the Cygnus-X column density map from Cao et al. (2019), we obtain N(R) profiles of 135 dense structures at 0.1?10 pc. It enables us to derive M–R relations from real density profiles,thus deepening the understanding of the M–R relations,density distributions,and the physical states behind them. We present the obtained N(R) profiles and their parameter distributions in Section 2. In Section 3, we study effects of the N(R)profiles and column density threshold N0on the M–R relation. We further discuss the significance of the N(R) profile and the M–R relation from a more realistic perspective in Section 4. The results are summarized in Section 5.

2. N(R) Profiles of Cygnus-X

Cygnus-X is one of the most massive giant molecular clouds in our Galaxy (Motte et al. 2018), and shows rich star formation activities evidenced by numerous H II regions, OB associations, dense molecular gas clumps and cores (Wendker et al. 1991; Uyan?ker et al. 2001; Motte et al. 2007; Cao et al.2019; Wang et al. 2022). It is located at a distance of 1.4 kpc from the Sun (Rygl et al. 2012). Using getsources(Men’Shchikov et al. 2012), Cao et al. (2019) applied SED fittings to the 160, 250, 350, and 500 μm dust continuum images from Herschel, and obtained the temperature map and the column density map of Cygnus-X. The resolution of the column density map was set by the SED fitting of the smallest spatial scale component using the 160 and 250 μm data,and is 18 4 determined by the 250 μm images, corresponding to 0.1 pc at the distance of 1.4 kpc.Using the column density map,Y.Xing et al. (2022, in preparation) obtained the N-PDF of the complex, which shows a log-normal + power-law shape. The turbulence-dominated log-normal component and the gravitydominated power-law component are delimited by a transitional column density at 1.86×1022cm?2(Xing et al.2022,in preparation). We selected all column density peaks above 1.86×1022cm?2for the extraction of density profiles. To avoid the influence of structures that cannot be described by radial density profiles, we check the morphology of every structure within a density threshold of 1.86×1022cm?2, and exclude those with aspect ratios larger than 2. In this way, we eventually obtained 135 peaks which are shown in Figure 1.We divide the area around each column density peak into 24 sectors with the same angular size (i.e., 15°). For each sector,we calculate the distance of each pixel to the column density peak and average all pixels with the same distance to obtain a sectorized radial N(R) profile. Thus for each column density peak we have 24 N(R) profiles extracted from 10 pc down to the resolution at 0.1 pc. We then discard any sectorized profiles that show a column density rise of more than 6.46×1021cm?2, which is the peaking column density of the log-normal part in Cygnus-X’s N-PDF,with the increasing radius to bypass the contamination from nearby sources. We average the remaining sectorized N(R)profiles to obtain a final N(R) profile for each column density peak. The obtained N(R)profiles are shown in Figure 2. In the sections below, to distinguish from the structures used in the M–R relation, we call these 135 structures extending outward from the column density peaks to about 10 pc the 135 Cygnus-X clumps. Note that they are not “clumps” in the usual definition, and have no strict boundaries.

The N(R) profiles are apparently steep in the inner part and flat in the outer part,with the break points roughly around 1 pc(see Figure 2). We then fit the N(R) profiles with a broken power-law distribution as described by

Figure 1.The column density map of Cygnus-X in[cm?2].(a)Yellow circles denote the 1 pc radius around the 135 peaks.White contours outline the regions with column densities higher than 5.0×1021 cm?2.Orange contours outline the regions with column densities higher than 1.86×1022 cm?2.(b)A zoom-in image of the area outlined by the white rectangle in panel (a), to better display the morphology of the column density distribution around the selected density peaks.

Figure 2. Radial N(R) profiles of the 135 clumps at R=0.1 ?10 pc. The corresponding broken power-law fittings are shown in dashed lines.

3. From N(R) Profiles to M–R Relations

3.1. Obtaining the M–R Relation

By integrating the broken power-law N(R) profile, we can obtain the M(R) profile as

The well known M–R relation is obtained by intercepting a group of M(R)profiles with some column density threshold N0.In real observations, N0is determined by either observational limits,such as the detection limit which is typically a few times the noise level,or by the selection effect of a source extraction algorithm(e.g.,Kegel 1989;Scalo 1990;Ballesteros-Paredes&Mac Low 2002). With the column density threshold N0determined, the mass M and radius R in an M–R relation are defined as

Figure 3.Parameters derived by performing broken power-law fittings to the 135 Cygnus-X clumps.Orange and blue dashed lines show the 1σ(68%)and 2σ(95%)distribution intervals,respectively.(a)The distribution of the inner power-law index n.(b)The distribution of the outer power-law index m.(c)The distribution of the transitional column density NTP. (d) The distribution of RTP, the radius at the transitional point.

Thus the mass and radius of a structure in an M–R relation are determined by both the four N(R) parameters and the column density threshold N0. Further, combining Equation (3) and Equation (4), the M–R relation can be given by

From Equation(5),the M–R relation at N0≥NTPis apparently a function of R2, and is simplified to M ∝R2if all the clumps have the same inner power-law index n and are trimmed at a constant N0.Otherwise,the M–R relation deviates from M ∝R2at a degree depending both on the density profile N(R) and the threshold density N0. We explore the M–R relation in detail in the following subsection.

3.2. Effects of the N(R) Parameters and N0

Figure 4.Gray lines show a set of M(R)profiles equivalent to column density profiles that have n=0.63, m=0.11, andσNRTP= 1. Blue and orange dots mark the data points obtained by intercepting those M(R) profiles with N0=2.5×1022 cm?2 and N0=8.0×1021 cm?2, respectively. The data points are used to fit the M–R relations (solid lines). Blue and orange dashed lines indicate M(R) profiles corresponding to constant column densities of 2.5×1022 cm?2 and 8.0×1021 cm?2, respectively.

Figure 5.Same as Figure 4,but for m=0.11,σNRTP= 0,n evenly distributed from 0 to 1.2, and N0=2.5×1022 cm?2 and N0=5.0×1022 cm?2.

Figure 6. Same as Figure 5, but forσNRTP= 0.5in panel (a) andσNRTP= 1.0 in panel (b).

Figure 8.M–R relations derived from the M(R)profiles of the 135 Cygnus-X clumps,which are intercepted with different choices of N0 as indicated below each panel.Other symbols are the same as those shown in Figures 4–7.

Now we can summarize the effects of the N(R) parameters and the column density threshold N0on the M–R relation as follows: (1) constant N(R) profile power-law index and constant N0give rise to an M ∝R2tendency; (2) the N(R)power-law index with a wide range steepens the M–R relation;(3) NTPand RTPwith wide ranges to some extent weaken the steepening effect due to the variation of the N(R) power-law index; (4) N0with a wide range tend to wash out the M ∝R2relation and get the M–R relation approaching the averaged density profile of the sample sources; (5) at N

3.3. The M–R Relation of the 135 Cygnus-X Clumps

Using the 135 Cygnus-X clumps, we look into the M–R relations from a more realistic perspective. The N(R)parameters of the 135 clumps are described in Section 2. By fixing N0at a certain value, or allowing it to vary within a range,we obtain six samples of the clumps.We then fit the M–R relations with power-law models. Only structures within 0.1–10 pc are included in the fitting. The results are shown in Figure 8.

Cases 1–3 have their N0fixed at a certain value. We adopt N0=7.9×1021cm?2in Case 1. This N0is lower than NTPof most clumps,and the obtained structures mainly fall at 1–10 pc.The M–R relation is similar to that shown in orange in Figure 4:the constant N0favors an M ∝R2relation, while N0

Figure 9. The M–R indexes (left) and the M–R fitting results’ correlation coefficients (right) of the 135 clumps. The column density threshold N0 is defined by log10 ( N0)=log10 (N0,mean)±log10 (N0,half range). The numbers correspond to the cases in Figure 8. White contours show M–R indexes of 2.

In Cases 4–6, N0of each structure is randomly generated within a range in logarithmic space.Most structures obtained in Case 4 have radii falling in the range of 1–10 pc.The N0range of 1021.9±0.2cm?2is wide enough to cover the column densities of most clumps below NTP. With the wide N0range,the M–R relation can potentially probe the density profiles in the N

We further obtain the M–R indexes in Figure 9. By simply varying the mean and range of N0,M–R relations with indexes from 1.4 to 2.4 are obtained from the 135 clumps. When N0is fixed at a certain value, we obtain M–R relations with indexes of 1.8–2.4. The M–R relations with N0<1022.1cm?2have indexes around 1.8–1.9.At higher N0,we obtain M–R relations with indexes of 2–2.4. When N0has a sufficiently wide range,the M–R index is always smaller than 2 and decreases with the increase of N0. These indexes manifest the density profiles at the corresponding scales.The decreasing trend comes from the difference between the mean shapes of the two parts of the M(R) profiles, i.e., M(R)∝R1.89for R>RTPand M(R)∝R1.37for R ≤RTP. When N0varies within a range but the range is smaller,the M–R index will be between the two cases where N0is fixed and has a wide range.

4. Discussion

In Section 2,we obtained the column density profiles of 135 clumps in Cygnus-X.Their main features are:1)the profiles all show broken power-law shapes at 0.1 ?10 pc, 2) the parameters of each clump’s profile are different, 3) the transition points of the profiles are around 0.8 pc and 1022cm?2, suggesting density profiles of ρ ∝R?1.63in the inner part and ρ ∝R?1.11in the outer part. These two parts of density profiles are consistent with the circumstances of freefall collapse (ρ ∝R?α, with α=1.5 ?2.0) and turbulence dominated nature (α ≈1.0), respectively. They are also comparable to the log-normal + power-law N-PDF of Cygnus-X. The N-PDF of Cygnus-X has its power-law index at 2.33 (Xing et al. 2022, in preparation), corresponding to a density profile of ρ ∝R?1.86.The transitional column density between the lognormal and power-law parts is at 1.86×1022cm?2. These values are slightly different from the parameters of our density profiles,which is because the power-law index of an N-PDF is more affected by the densest sources, and the transitional column density is related to the proportion of high and low density components. These all suggest that the broken powerlaw column density profiles imply a gravity-dominated dense core + turbulence-dominated diffuse cloud, and the transition point at 0.8 pc and 1022cm?2acts as the division of the two components.

In Section 3 we show how the N(R)profiles and N0affect the shape of the M–R relation. From the observational point of view, N0is often determined by the detection limit or set by a threshold in some source extraction method (Kegel 1989;Scalo 1990; Ballesteros-Paredes & Mac Low 2002), and it is likely to be constant. In such a situation the M–R relation may show the well-known M ∝R2scaling law (Lombardi et al.2010; Ballesteros-Paredes et al. 2012), provided N0≥NTPand the power-law index n is nearly constant; the M–R power-law indexes may be slightly less than 2 if N0is lower than NTP(Figure 8.1); the M–R relation may also appear to be steeper than M ∝R2,as seen in some other studies(Roman-Duval et al.2010;Kainulainen et al.2011),when N0≥NTPand the powerlaw index n varies from source to source.

When many molecular cloud structures are included in an analysis(Larson 1981;Urquhart et al.2014,2018),N0is likely to have a wide distribution, if the cloud structures under investigation were obtained from different observations or extracted from highly varying backgrounds.In these cases,the derived M–R relations may to some extent manifest the density profiles of the cloud structures. But caution should be taken in converting the observed M–R relation to a density profile if the observations significantly suffered from short dynamical ranges(e.g., Scalo 1990; Ballesteros-Paredes & Mac Low 2002;Schneider & Brooks 2004). For molecular cloud structures in Cygnus-X, a M ∝R1.9relation is expected to be obtained with an observational study capable of trimming the structures at large and varying radii (e.g., at R>1 pc and N<1022cm?2),as a consequence of a tight distribution of the power-law indexes for the density profiles in the outer parts (i.e.,ρ ∝R?1.1, see Section 2). Shallower relations can be found at smaller scales and higher densities. They correspond to ρ ∝R?αdensity distributions with α clearly larger than 1.

Applying different source extraction or identification algorithms to the same molecular cloud, different structures(Schneider & Brooks 2004; Li et al. 2020) and M–R relations(Schneider&Brooks 2004)can be obtained.Without knowing the impact on N0of the source extraction process,it is difficult to make convincing interpretations of the M–R relations. For example, ρ ∝R?1profiles and a constant N0can both result in M ∝R2relations. Stronger line-of-sight contamination for larger cloud structures (Ballesteros-Paredes et al. 2019), an approximately constant volume density for all the cloud structures under investigation (Lada et al. 2008; Li &Zhang 2020), and variation from source to source in the power-law index of the N(R) profiles can all lead to M–R relations steeper than M ∝R2, but only when N0is constant or falls in a narrow range can a steep M–R relation come from the difference in the N(R) index. Therefore, before interpreting an M–R relation, one is suggested to carefully check how the cloud structures in the sample are derived and then to determine if a constant N0, or instead a varying N0, is implicitly being used; only in the latter case, the observed M–R relation could be useful in constraining the averaged density profile of the cloud structures under investigation.From another perspective,an observational experiment optimized for converting an M–R relation to a density profile would require high resolution and high sensitivity to reasonably resolve each source and allow an estimate of the source flux and size free of sensitivity limitation(e.g., estimation based on the peak intensity and FWHM size by 2D Gaussian fitting to the source brightness distribution).By this way one equivalently has N0varying from source to source. High resolution also helps to minimize potential lineof-sight contamination, while high sensitivity observations of optically thin tracers are desirable to increase the dynamical range.

Can M–R relations help to determine whether the cloud structures are in virial equilibrium? Due to the lack of velocity information, M–R relations cannot be directly linked to the virial state. However, having the linewidth—size relation of σv∝R0.5satisfied (Larson 1981; Myers et al. 1983; Solomon et al. 1987; Falgarone et al. 2009), the M ∝R2relation is suggested to imply that the cloud structures are in virial equilibrium (Larson 1981; Solomon et al. 1987). However, as we discussed above, the M ∝R2relation does not necessarily mean a density profile of ρ ∝R?1, and thus cannot be a straightforward indicator of virial equilibrium. When the M ∝R2relation is verified to imply ρ ∝R?1, and if the structures also follow σv∝R0.5, the gravitational to kinetic energy ratio is a constant, and means virial equilibrium if that constant is about 2 (Myers and Goodman 1988; Ballesteros-Paredes 2006).

5. Summary

Using the column density map from Cao et al. (2019), we obtain N(R) profiles of 135 dense structures in Cygnus-X. At 0.1–10 pc, all the structures have broken power-law N(R)profiles, suggesting their dense core + diffuse cloud nature.With the transition at approximately 0.8 pc and 1022cm?2, the N(R) profiles have a power-law index of 0.63±0.59 at small radii, and 0.11±0.20 at large radii.

We explore the M–R relation using the broken power-law N(R) profiles. Both the N(R) profiles and the column density threshold N0determine the shape of the M–R relation: for N0>NTP, we find (1) constant N(R) power-law index and N0lead to M ∝R2, (2) the N(R) index with a wide range steepens the M–R relation, (3)NTPand RTPwith wider ranges make the data points in the M–R plot spread out along loci following M ∝R2, (4) N0with a wide range tend to make the M–R relation follow M(R) profiles. For N0

From the observational perspective, the column density threshold N0in extracting cloud structures plays a crucial role in shaping the M–R relation. With a constant N0, the M–R relation cannot be a probe of the density profile.Its M–R index can be slightly less than 2 (when N0

Acknowledgments

This work was supported by the National Key R&D Program of China No.2017YFA0402600.We acknowledge the support from the National Natural Science Foundation of China(NSFC) through grants U1731237, 11473011, 11590781 and 11629302.