Nobeyama 45 m Cygnus-X CO Survey. II. Physical Properties of C¹⁸O Clumps

The C¹⁸O (J = 1–0) data of the Cygnus X region were obtained by the FOREST receiver (Minamidani et al. 2016) mounted on the Nobeyama 45 m radio telescope, along with the ¹²CO (J = 1–0), ¹³CO (J = 1–0), and CN (N = 1–0) data¹¹ (Yamagishi et al. 2018). The observations covered a ∼9 deg² field, which included the main parts of the North and South GMCs, by connecting 1° × 1° patches (FUGIN scan, Umemoto et al. 2017). The angular resolution of the telescope was ∼16'' FWHM at the C¹⁸O band.

To improve the sensitivity, we convolved the cube to a spatial resolution of 46'' FWHM with a pixel grid of 227 and binned to a velocity resolution of 0.25 km s⁻¹. Consequently, the median rms noise level of the final C¹⁸O image T_rms became 0.35 K on the T_mb scale. The observation and data analysis procedures are described in Yamagishi et al. (2018) in detail.

We identify C¹⁸O clumps from the C¹⁸O cube using the astrodendro package, which is based on the dendrogram algorithm (Rosolowsky et al. 2008). A dendogram is used to construct a tree structure consisting of trunks, branches, and leaves. A "trunk" is defined as a set of voxels such that T_mb is larger than T_min and the voxel number, n_vox, is not less than the integer, ${n}_{\mathrm{vox}}^{\min }$. A trunk is split into one or more leaves by a "branch," which is a node of more compact structures (leaf or branch). A "leaf" is defined as a local peak such that its height is higher than T_delta from the skirt of the peak and the voxel number is not less than ${n}_{\mathrm{vox}}^{\min }$. From the definition, leaves are identified as compact clumps that do not have multiple peaks, and therefore, a dendrogram is available as an identification algorithm for molecular clumps and cores. Cheng et al. (2018) used the dendrogram method to identify the dense cores and clumps in G 286.21+0.27 and reported that the dendrogram-identified cores showed a spectral index of the core mass function that was more consistent with the Salpeter-IMF than that of the clumpfind-identified cores. Thus, it is reasonable to adopt the dendrogram algorithm as a core and clump identification procedure.

In the analysis, we adopted T_min = 3T_rms and T_delta = 2T_rms to identify C¹⁸O clumps with a reliable signal-to-noise ratio. We also used ${n}_{\mathrm{vox}}^{\min }$ = 16 to avoid false detection of clumps by picking up random noise.

We estimated C¹⁸O clump properties such as radius R_cl, local thermal equilibrium (LTE) mass M_LTE, FWHM velocity width dv_cl, and virial mass M_vir. We followed the method generally adopted by previous C¹⁸O core studies (Onishi et al. 1996; Ikeda & Kitamura 2009, 2011; Shimajiri et al. 2015) and made further refinements to improve the reliability of the estimated physical properties, motivated by Nishimura et al. (2015).

The radius of a C¹⁸O clump is defined using the pixel number projected on the sky (l–b plane of the galactic coordinate), n_sky, assuming a spherically symmetric clump

$$\begin{eqnarray}&&{R}_{\mathrm{cl}}=D{\theta }_{\mathrm{pix}}\sqrt{\displaystyle \frac{{n}_{\mathrm{sky}}}{\pi }},\end{eqnarray} \tag{ 1 }$$

where θ_pix = 227 is the pixel grid spacing along with the galactic coordinate of the data cube and D is the distance to the clump from the solar system.

We define the integrated intensity of an object as

$$\begin{eqnarray}&&I=\displaystyle \sum _{i=1}^{{n}_{\mathrm{vox}}}{T}_{\mathrm{mb}}^{i}{{dv}}_{\mathrm{spec}}{\left(D{\theta }_{\mathrm{pix}}\right)}^{2}\end{eqnarray} \tag{ 2 }$$

where i is the voxel number of each core, ${T}_{\mathrm{mb}}^{i}$ is the main beam temperature at voxel i, and dv_spec = 0.25 km s⁻¹ is the spectral velocity width. The LTE mass is estimated using the equation

$$\begin{eqnarray}&&{M}_{\mathrm{LTE}}({M}_{\odot })=13.2\displaystyle \frac{{X}_{{{\rm{C}}}^{18}{\rm{O}}}}{5.9\times {10}^{6}}\displaystyle \frac{{T}_{\mathrm{ex}}}{1\,{\rm{K}}}{e}^{\tfrac{5.27{\rm{K}}}{{T}_{\mathrm{ex}}}}\displaystyle \frac{I}{1\,{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}},\end{eqnarray} \tag{ 3 }$$

where T_ex is the excitation temperature of the C¹⁸O molecules. We assumed that ¹²CO was optically thick and the T_ex of ¹²CO and C¹⁸O was the same. Thus,

$$\begin{eqnarray}&&{T}_{\mathrm{ex}}({\rm{K}})=\displaystyle \frac{5.53}{\mathrm{ln}\left(1+\tfrac{5.53}{{T}_{\mathrm{mb}}^{{}^{12}\mathrm{CO}}+0.83}\right)},\end{eqnarray} \tag{ 4 }$$

where ${T}_{\mathrm{mb}}^{{12}_{\mathrm{CO}}}=\max \{{T}_{\mathrm{mb}}^{{12}_{\mathrm{CO}}}$, i_mb; i = 1, 2, ..., n_vox}. We used the isotopic abundance ratio of the C¹⁸O molecules relative to the H₂ molecules, ${X}_{{{\rm{C}}}^{18}{\rm{O}}}$ = 5.9 × 10⁶ (Frerking et al. 1982). Assuming a spherical clump shape, we also defined the mean gas density as

$$\begin{eqnarray}&&{n}_{{{\rm{H}}}_{2}}=\displaystyle \frac{3{M}_{\mathrm{LTE}}}{4\pi 2\mu {m}_{{\rm{p}}}{{R}_{\mathrm{cl}}}^{3}},\end{eqnarray} \tag{ 5 }$$

where μ = 1.36 is the mean molecular weight per proton, and m_p = 1.67 × 10⁻²⁴ g is a proton mass.

To calculate the virial masses, we removed the effect of the velocity width broadening by the limitation of the spectral resolution, dv_spec. The spectral window function obtained by a SAM45 spectrometer can be approximated by a rectangular window function. Thus, we estimated the actual velocity widths of the clumps, dv_cl, by deconvolving the intensity-weighted velocity dispersions, dv_obs, with a rectangular function.

Following Solomon et al. (1987) and Bolatto et al. (2013), we estimated the virial masses of the identified clumps with the radial density profile of ρ(R) ∝ R^−k:

$$\begin{eqnarray}&&{M}_{\mathrm{vir}}=\displaystyle \frac{3(5-2k)}{G(3-k)}{R}_{\mathrm{cl}}{{\sigma }_{\mathrm{cl}}}^{2},\end{eqnarray} \tag{ 6 }$$

where G is the gravitational constant, k is a parameter of the density profile, and ${\sigma }_{\mathrm{cl}}={{dv}}_{\mathrm{cl}}/(2\sqrt{2\,\mathrm{ln}\,2})$, which is assuming a Gaussian line profile. We used k = 0 for the calculation, which assumes a spherically uniform clump that has no external pressure, by following the previous C¹⁸O studies (e.g., Tachihara et al. 2002; Ikeda & Kitamura 2009, 2011; Shimajiri et al. 2015). The possible bias of the virial mass estimate by the selection of k is a factor of 1–1.4 in the possible range of 0 < k < 2 for an isothermal gas sphere (e.g., Shu 1977). We also defined the virial ratio, α_vir ≡ M_vir/M_LTE.

Finally, we used the canonical distance, D = 1.4 kpc, from the Sun to the clumps, which was determined by parallax measurements by a very long baseline interferometry observation toward four major star-forming regions, as a representative value of the Cygnus X GMC complex. The exception was the distance to a corresponding object of background star-forming region AFGL 2592 at D = 3.3 kpc (Rygl et al. 2012).

Although both the Cygnus X North and South regions are known as large reservoirs of molecular gases (∼10⁶ M_⊙), the North region shows an extremely filamentary ^12/13CO structure compared to the South region (Schneider et al. 2006, 2011) and contains numerous active star-forming regions represented by DR21 and W75N. The above seem to reflect a difference in the star-formation activity and evolution stages of each molecular cloud complex, and therefore, a difference in the statistical properties of the C¹⁸O clumps of the North and South regions is suggested.

Figure 4 shows the probability densities and histograms of the physical properties of the identified C¹⁸O clumps of the North and South regions. The mean and standard deviation of the physical properties classified by the regions are listed in Table 2. The distributions of the radius and velocity dispersion are very similar for the North and South clumps. By contrast, the C¹⁸O clumps in the North region show a slightly larger LTE mass and higher H₂ density than those in the South region. The p-values of Welch's t-test for the radii, velocity dispersions, LTE masses, and H₂ densities of the North and South cores are 0.959, 0.241, 0.070, and 0.002, respectively. This does not support that the average values of the radii and velocity dispersions of the North and South clumps are significantly different. Contrastingly, the average H₂ density of the C¹⁸O clumps of the North region is significantly higher than that of the C¹⁸O clumps of the South region, with a significance level of 5%. Thus, we can expect that the statistical difference in the H₂ density reflects that of the clump evolution stages, and therefore, the difference in the star-formation activities of the North and South regions, as suggested by Schneider et al. (2006) and Yamagishi et al. (2018).

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Table 2.  Averages and 1σ Deviations of Physical Properties of the C¹⁸O Clumps

	All	North	South	Star-forming	Starless
Samples	133	68 (51.1%)	65 (48.9%)	98 (73.7%)	35 (26.3%)
Rcl(pc)	0.49 ± 0.17	0.49 ± 0.17	0.49 ± 0.17	0.52 ± 0.17	0.42 ± 0.13
dVcl(km s−1)	0.75 ± 0.39	0.79 ± 0.42	0.71 ± 0.37	0.83 ± 0.40	0.54 ± 0.29
MLTE(102 M⊙)	${1.9}_{-1.1}^{+3.0}$	${2.2}_{-1.4}^{+3.6}$	${1.5}_{-0.9}^{+2.3}$	${2.3}_{-1.4}^{+3.6}$	${1.0}_{-0.5}^{+1.1}$
${n}_{{{\rm{H}}}_{2}}$(${10}^{3}\,{\mathrm{cm}}^{-3}$)	${6.6}_{-3.2}^{+6.1}$	${7.9}_{-3.9}^{+7.8}$	${5.5}_{-2.4}^{+4.2}$	${7.1}_{-3.4}^{+6.7}$	${5.4}_{-2.4}^{+4.3}$
αvir	0.30 ± 0.24	0.28 ± 0.21	0.33 ± 0.27	0.32 ± 0.41	0.27 ± 0.29

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We also investigated the quantitative difference in the physical properties of star-forming and starless clumps, which seemed to reflect the evolution sequences of the C¹⁸O clumps.

Figure 5 shows the probability densities and histograms of the physical properties of the identified C¹⁸O clumps of the star-forming and starless clumps. In Table 2, we also list the average and standard deviation classified by the star-formation activity of the clumps. The star-forming clumps show a larger radius, velocity dispersion, LTE mass, and H₂ density than the starless clumps. In fact, the significant difference in the mean values of these properties is strongly supported by Welch's t-test, whose p-values are <0.01, <0.01, <0.01, and 0.01, respectively.

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In nearby (≤200 pc) star-forming regions, Tachihara et al. (2002) reported a similar tendency of the C¹⁸O core radii, velocity dispersions, LTE masses, H₂ densities, and H₂ column densities between the starless, star-forming, and cluster-forming cores. This tendency is naturally expected to arise from difference of gas accretion timescale between starless and star-forming cores. Our result also supports that the general trends of the core properties, which evolve with star formation, are applicable to the C¹⁸O clumps in an extremely active high-mass star-forming region.

Here, we discuss the virial ratio, which is an important indicator to determine whether stars will form cores/clumps, in comparison with previous C¹⁸O studies. As references of the previous C¹⁸O surveys, we use the results of nearby (D ≤ 200 pc) molecular clouds, including the low-mass star-forming regions of Taurus, Ophiuchus, Lupus, Lynds 1333, Corona Australis, Southern Coalsack, and the Pipe nebula, observed by NANTEN (Tachihara et al. 2002). We also refer to the C¹⁸O cores properties observed by the NRO 45 m telescope in Orion A (D = 440 pc, Hirota et al. 2007; Shimajiri et al. 2015), the nearest high-mass star-forming GMC, which has about one-tenth of the total molecular gas mass of the Cygnus X GMC complex (Motte et al. 2018). We also consider the Sharpless 2–140 (S 140) H ii region, a compact high-mass star-forming region located at the edge of the Lynds 1204 molecular cloud (D = 760 pc, Hirota et al. 2008; Ikeda & Kitamura 2011). The above is done to compare with smaller and less active high-mass star-forming regions than Cygnus X. We followed the assumption of a uniform and spherical core structure that had no support of rotation, magnetic field, and external pressure, which is assumed in the previous studies. By considering the most compact clumps (n_vox = 16) with relatively high excitation temperature of 36 K, corresponds to the 2σ value of all samples, the detection limit of the LTE mass is estimated to be ∼55 M_⊙ with 3σ intensity detection, which covers the 95th percentile of the identified clumps. Therefore, we defined 95th percentile as induces of relative sensitivity limit for the Cygnus X, S140, Orion A, and nearby low-mass samples. Thus, the relative sensitivity limits of the LTE masses in the Cygnus X samples are ∼10 and 20 times worse than S 140, and Orion A/nearby studies.

We also consider the systematic bias of α_vir caused by different spatial and velocity resolutions of the data set. From the definition of the LTE and virial masses, we can assume that α_vir is proportional to spatial resolution and inversely proportional to velocity resolution. The spatial and velocity resolutions of the Cygnus X clumps were 3 and 2.5 times worse than previous core studies. Thus, the systematic bias of the virial ratio estimate can be estimated to be a factor of ∼1.2 and would not affect our discussion.

The relation between the LTE and virial masses is shown in Figure 6. Whereas the NANTEN and Orion A C¹⁸O cores are located at α_vir ≳ 1, most of the C¹⁸O core/clumps in Cygnus X and S 140 show a virial ratio of α_vir < 1. The average and standard deviation values of the virial ratio of the star-forming and starless clumps in Cygnus X are 0.32 ± 0.26 and 0.27 ± 0.19, respectively, and the difference is not significant with Welch's t-test (p = 0.32). The average value of both the star-forming and starless cores is 0.30 ± 0.24, which is consistent with the virial ratio of S 140 (α_vir = 0.35 ± 0.23) and, therefore, supports that these C¹⁸O clumps are gravitationally bound. The observing region of S 140 is only 20' × 18', and it will be very biased to the center of the active star-forming region. This result suggests that the C¹⁸O clumps in Cygnus X have very similar properties at the center of the high-mass star-forming region. Thus, the C¹⁸O clumps in Cygnus X trace a dense molecular gas clump that directly connects to an extremely active future and current star-formation activity. In addition, the virial ratios in Cygnus X and S 140 are smaller than those in the nearby molecular clouds (α_vir = 2.8 ± 3.6) and Orion A (α_vir = 2.4 ± 2.2). This might reflect the difference in the star-formation mode: low-mass single star or high-mass cluster formation.

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Another important feature is that the distribution of the virial or LTE mass of the C¹⁸O clumps in Cygnus X is dispersed widely compared to those in the previous studies. In fact, the mean and standard deviation of LTE masses in the previous studies were ${22}_{-12}^{+25}$, ${11}_{-6}^{+14}$, and ${12}_{-8}^{+23}$ M_⊙ in S 140, Orion A, and nearby star-forming regions, respectively. These values are much smaller than those in the Cygnus X (${190}_{-110}^{+300}$ M_⊙) by more than one order of magnitude. This result is consistent with our estimate of the relative sensitivity limits, which would be attribute to the lower spatial resolution, spectral resolution, and image sensitivity than the previous studies.

While the typical LTE and virial masses are larger than the previous studies, the fact that the clumps in Cygnus X show α_vir < 1 supports that these clumps are gravitationally bound objects, which are directly related to star formation. In particular, some of the massive clumps are assumed to be the formation sites of high-mass stars and stellar clusters. Thus, this feature could be related to the extremely active star formation in Cygnus X.

We also examined the core/clump mass function (CMF) in Cygnus X to reveal the detailed mass properties and relation between the IMF and galactic field stars. Based on the definition of Offner et al. (2014), the IMF and CMF are defined as

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dM}}\propto {M}^{\alpha },\end{eqnarray} \tag{ 7 }$$

where N is the number of stars or cores/clumps, M is the mass of the stars or cores/clumps, and α is the spectral index of the IMF or CMF. As an integral form of the CMF for the case of α < −1, we define a cumulative number,

$$\begin{eqnarray}&&N\left(\gt {M}_{\mathrm{LTE}}\right)\equiv {\int }_{{M}_{\mathrm{LTE}}}^{\infty }\displaystyle \frac{{dN}}{{dM}}{dM}=a{{M}_{\mathrm{LTE}}}^{\alpha +1},\end{eqnarray} \tag{ 8 }$$

where a is the factor of proportionality.

The observational studies of dense dust core surveys using (sub)millimeter dust continuum and dust extinction (e.g., Motte et al. 1998; Enoch et al. 2006; Alves et al. 2007; Nutter & Ward-Thompson 2007) have revealed that multiple spectral index components of the CMF are similar to the α = −1.3 ± 0.5 (0.08 < M_*/M_⊙ < 0.5, M_* is a stellar mass) and α = −2.3 ± 0.3 (0.5 < M_*/M_⊙ < 1) components of the Kroupa IMF (Kroupa 2001). For a C¹⁸O core observation, Tachihara et al. (2002) has also reported multiple spectral index components of the CMF (α = 1.25 and 2.5) corresponding to the Kroupa IMF components at 0.08 < M_*/M_⊙ < 0.5 and 0.5 < M_*/M_⊙ < 1 ranges toward nearby low-mass star-forming regions. Our observation in Cygnus X provides large samples of C¹⁸O clumps in an extremely active cluster-forming region. Lada & Lada (2003) suggest that cluster-formation activity in GMCs is the dominant (70%–90%) supplier of field low-mass stars in the galactic disk. Thus, it is important to investigate the relationship between the IMF of the galactic field stars and CMF obtained by our C¹⁸O clump samples, which are more massive and larger than those in the previous C¹⁸O studies.

Figure 7 shows the cumulative number count of the C¹⁸O clumps and suggests that the spectral index of the CMF changes around log(M_LTE/M_⊙) = 2.15. The fitting parameters are shown in Figure 3. Considering the detection limit of the C¹⁸O clumps, log(M_LTE/M_⊙) = 1.75, we fit with two mass functions with ranges of 1.75 < log(M_LTE/M_⊙) < 2.15 and 2.15 < log(M_LTE/M_⊙). From the least-χ² fittings, which are shown in Figure 7, the cumulative number count of the C¹⁸O clumps fits well in each mass range. We obtained α = −1.39 ± 0.04 for 1.75 < log(M_LTE/M_⊙) < 2.15, and α = −2.07 ± 0.04 for 2.15 < log(M_LTE/M_⊙). The errors (1σ) were estimated by Monte Carlo simulation by considering the random errors of the estimated LTE masses of the C¹⁸O clumps. These spectral indices are consistent with the α = −1.3 and −2.3 components of the Kroupa IMF. This confirms the similarity of the IMF in the galactic field stars and clump-scale CMF in a high-mass star-forming region.

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We also investigate the difference in the spectral indices of the star-forming and starless clumps. The mass function of the star-forming clumps well fits the two components of the spectral indices: α = −1.30 ± 0.04 for 1.75 < log(M_LTE/M_⊙) < 2.15 and α = −2.00 ± 0.04 for 2.15 < log(M_LTE/M_⊙), which are also consistent with the IMF. However, for the starless clumps, we can fit the CMF with a single spectral index of α = −1.94 ± 0.06 at the mass range of 1.75 < log(M_LTE/M_⊙), and the index of the starless clumps is consistent with the spectral index of a star-forming clump at a high mass range. Thus, we can assume that the starless clumps will evolve into star-forming clumps with further gas mass accretion. This is also supported by the fact that ∼2 times lowers the average mass of the starless clumps more than that of the star-forming ones, as can be seen from Table 2.

We can estimate the molecular gas mass fraction that contribute to the stellar mass in a clump, which is called the star-formation efficiency (SFE), from the boundary gas mass that changes the spectral index. Here, we assume that the clumps with a boundary mass of log(M_LTE) = 2.15 (i.e., M_LTE ≃ 140 M_⊙) evolve into a single star that has a boundary mass of the Kroupa IMF of 0.5 M_⊙ or into a cluster that has a maximal stellar mass of 0.5 M_⊙. In case cluster formation, using the relation between maximal stellar mass M_*,max and cluster mass M_cluster:M_*,max = 0.39 ${M}_{\mathrm{cluster}}^{2/3}$, assuming the hierarchical cluster-formation model (Bonnell et al. 2003, 2004; Weidner & Kroupa 2006; Weidner et al. 2010), the total cluster stellar mass of the cluster is expected to be 1.5 M_⊙. Thus, the SFE of the typical C¹⁸O clumps is expected to be 0.3%–1%. This is very unlikely because the SFE is excessively lower than that estimated for the low-mass star-forming regions observed by NANTEN from the comparison of the CMF and IMF, with the assumption of a single star formation in the C¹⁸O cores (∼10%, Tachihara et al. 2002).

It is known that some studies of massive clumps also reveal a high SFE (∼10%, e.g., Lada & Lada 2003) by comparing the gas amount with the stellar content in GMCs. Assuming an SFE of 10%, the ∼10 C¹⁸O clumps that have gas masses of ≳10³ M_⊙ will evolve into open clusters having a total stellar mass of ≳100 M_⊙ and containing one or more high-mass stars (>8 M_⊙). This scenario is consistent with a high-resolution interferometry study of massive dense cores in Cygnus X North (Bontemps et al. 2010), which revealed numerous fragmentary structures inside massive dense cores.

The discrepancy between the SFEs of the NANTEN C¹⁸O cores and our samples could be explained in terms of the physical spatial resolution of our data set of Cygnus X (∼0.3 pc) being worse than those of the NANTEN observations (∼0.1 pc). This is because the identified C¹⁸O clumps in Cygnus X are larger than in the NANTEN study, and therefore, the mass of these clumps is higher than of those in the latter study. Thus, we can also expect that most of the C¹⁸O clumps in Cygnus X have an internal structure, and our predicted SFE using the relation of the IMF and CMF might be underestimated. Further high-resolution, high-sensitivity, and wide-field surveys of C¹⁸O and other dense gas tracers toward high-mass star-forming regions are important to understand the complete mechanism of star formation across a GMC.