The Column Density Structure of Orion A Depicted by N-PDF

The observations were made using the PMO-13.7 m millimeter-wavelength telescope located at Delingha in China during 2011 June as a pilot survey of the Milky Way Imaging Scroll Painting project ¹¹1http://www.radioast.nsdc.cn/mwisp.php (Su et al. 2019). The PMO-13.7m telescope is equipped with a nine-beam Superconducting Spectroscopic Array Receiver (SSAR) that works in the 85-115 GHz frequency band (Shan et al. 2012). With a carefully selected local oscillator (LO) frequency, the ¹²CO, ¹³CO, and C¹⁸O $J=1-0$ line emission data can be obtained simultaneously with this instrument. The upper sideband contains the ¹²CO $J=1-0$ and the lower sideband contains the ¹³CO and C¹⁸O $J=1-0$ line emission. For each sideband, a Superconductor-Insulator-Superconductor (SIS) mixer works as the front end of the receiver. The back-end of the receiver is a Fast Fourier Transform Spectrometer (FFTS) with 16,384 frequency channels and a total bandwidth of 1 GHz, providing a velocity resolution of 0.17 km s^-1 at 110 GHz. At 110 GHz and 115 GHz, the half-power beam width (HPBW) of the telescope is about 52 $\arcsec$ and 50$\arcsec$, respectively.

The observations were conducted in the position-switch on-the-fly (OTF) mode. The Orion A GMC is divided into twelve 30$\arcmin\times$30$\arcmin$ cells. For each cell two scannings were made along the directions of right ascension and declination, respectively. The scanning rate of the observation is 50$\arcsec$ per second and the dump time is 0.3 s. The antenna temperature is calibrated according to $\rm{T_{MB}=T^{*}_{A}/\eta_{MB}}$ during the data reduction processes. According to the status report of the PMO-13.7 m telescope, the main beam efficiencies are 0.44 and 0.48 at the ¹²CO $J=1-0$ and the ¹³CO $J=1-0$ wavelengths, respectively. We use the GILDAS/CLASS software package to reduce the data, which includes the subtraction of a second order baseline from each spectrum and the re-griding of the raw data. At the positions where multiple spectra are taken at different times, a combination of the spectra is made using the reciprocal square of the RMS noise level of each spectrum as the weight. The final reduced data cube covers a sky area of $\sim$4.4 deg², which contains the majority CO emission of the GMC as revealed by Bally et al. (1987). However, the spectra at the edge of the surveyed area have low signal-to-noise ratios and they are removed in the analysis of this work. The trimmed edge area, indicated with grey lines in Figure 1, accounts for about twenty percent of the scanned area. The final median RMS noise level of the $\sim$3.5 deg² effective region is 0.61 K per channel at 115 GHz and 0.37 K per channel at 110 GHz.

In this work, we use the DENDROGRAM algorithm to extract “tree” structures from the H₂ column density map and investigate the properties of the N-PDFs of the hierarchical structures and the importance of self-gravity at various spatial scales in Orion A in Section 4. The threshold and the difference between two separate column density peaks for the identification of “leaf” of the tree structure are set to be 2.0 and 3.0 times the median noise level of N${}_{H_{2}}$ (see text in Section 3.1), respectively. The minimum pixel number of a “leaf” structure is set to 200, which physically corresponds to a clump of the size of $\sim$0.8$\times$0.8 pc² at the cloud distance. The outputs of the DENDROGRAM provide masks of the identified structures.

We evaluate the importance of self-gravity in a molecular structure using the virial parameter $\alpha_{vir}$, which measures the ratio between the total kinetic energy and the gravitational energy. The widely used definition of $\alpha_{vir}$ is $5\sigma_{v}^{2}R/GM_{c}$, where $\sigma_{v}$ is the one-dimensional velocity dispersion, G is the gravitational constant, $R$ and $M$ are the effective radius and mass of the structure, respectively. The critical value of $\alpha_{vir}$ for a marginally bound structure is 2 (Kauffmann et al. 2013). The angular size of a structure is defined as the geometric mean of the FWHMs of its major and minor axes which can be obtained in the output table of the DENDROGRAM algorithm, and the LTE mass can be obtained using the total column density within the periphery of the structure. The velocity dispersion of a molecular structure is obtained by fitting a gaussian function to the average spectrum of the ¹³CO emission.

The ¹²CO emission is usually optically thick in typical molecular cloud environments, while the ¹³CO emission is often optically thin. Providing the molecular clouds are under the local thermodynamic equilibrium (LTE) conditions and the molecules are uniformly excited, the column density of the clouds can be derived with the ¹²CO and ¹³CO spectra.

Firstly, the excitation temperature can be obtained with the peak brightness temperature of the optically thick ¹²CO line (Li et al. 2018),

where $T_{peak}$ is the peak brightness temperature of the ¹²CO $J=1-0$ line. The optical depth and the column density of the ¹³CO $J=1-0$ emission can be derived through (Pineda et al. 2010; Li et al. 2018)

where $T_{ex}$ is the excitation temperature, and $J(T_{ex})=[exp(5.29/T_{ex})-1]^{-1}$. According to Pineda et al. (2010), when the optical depth of the ¹³CO emission is small, $\tau_{v}^{13}<1$, the integral item $T_{ex}\int{\tau}dv$ can be approximated to $\tau_{0}/(1-e^{-\tau_{0}})\int T_{mb}\ dv$, where $\tau_{0}$ is the peak optical depth of the ¹³CO emission line. Providing that the abundance ratios [¹²C]$/$[¹³C] $=$ 69 (Wilson 1999) and [CO]$/$[H] $\sim$ 10^-4 (Solomon & Klemperer 1972; Herbst & Klemperer 1973), the [$H_{2}/^{13}$CO] ratio is expected to be $7\times 10^{5}$, the column density of molecular hydrogen can be obtained with the following formula (Li et al. 2018)

The peak brightness temperature of ¹²CO emission and the integrated intensity of ¹³CO emission are calculated in the velocity range from 0 to 18 km s^-1. Observationally, molecular clouds are defined as contiguous structures above a fixed brightness temperature in the position-position-velocity space. The molecular line emission from the clouds should have a certain width of velocity dispersion ($\sigma_{v}\sim 1$ km s^-1). To avoid missing weak signals and to ensure an adequate signal-to-noise ratio, we only use the spectra that have at least five contiguous velocity channels with intensities higher than 3 $\sigma_{rms}$ for ${}^{12}CO$ for the calculation of excitation temperature, and 1.5$\sigma_{rms}$ for ${}^{13}CO$ for the calculation of column density. According to the above criteria, the signal-to-noise ratio of selected ¹³CO spectra is above 3.35, which is sufficiently high to exclude the noisy channels. The three-sigma detection limit of the H₂ column density is $\sim$1.5$\times$10²¹ cm^-2 under the criteria.

Figure 2(a) shows the spatial distribution of the excitation temperature derived using Eq. (1). The excitation temperatures of Orion A range from 5 to 107 K. In the ISF and L 1641 N regions, most of the excitation temperatures are above 25 K. In the OMC 1 region, which lies behind the Orion Nebula, the temperatures are $\sim$ 50 K on average and reach the maximum 107 K at the position of Orion-KL. Comparatively, the southern part of the GMC is much colder except for the south-eastern tail. The distribution of the ¹³CO optical depth is shown in Figure 2(b). The optical depths, $\tau^{13}$, range from 0.04 and to 1.35 with a median value of 0.37, and only 23 pixels have optical depths greater than 1, indicating the ¹³CO emission in the region is mostly optically thin.

The dendrogram of the tree structure of the Orion A GMC is given in Figure 11. The observed region contains two “trunk” structures, as shown in Figure 11. The right trunk is the main cloud, while the left trunk is the small cloud in the top-left corner of Figure 3(a). To investigate the properties of structures at various scales, we only use the right trunk in the following analysis. There are 23 individual structures in the right trunk, among which twelve are “leaves”, ten are “branches”, and one is the trunk itself. The sizes of the leaves span a large range and there is no favorable scale for the leaf structure. The leaves contain both large filaments and small clumps. The “genealogy” diagrams of the spatial distribution of all the structures are given in Figure 12. As shown in Figure 12, the ridge of the Orion A GMC is composed of two main structures, branch 11 that contains the entire ISF and the L1641 N regions, and branch 5 that contains the southern tail of Orion A GMC. Comparatively, the density structures are richer in the northern branch than in the southern branch, and there are more levels in branch 11 than branch 5. Downward from branch 3, the northern and southern structures of the GMC merge into one cloud.

The significance of self-gravity for the dynamics of dense cores on scales of $\sim$0.1 pc is well-known (Larson 2003; di Francesco et al. 2007). However, turbulent simulations without self-gravity successfully yield density and velocity distributions of molecular clouds that resemble observations (Vázquez-Semadeni & García 2001; Padoan & Nordlund 2002; Federrath et al. 2009). Goodman et al. (2009b) have made a “dendrogram” analysis of the L 1448 cloud using ¹³CO observations and found that self-gravity is important on the full range of scales of the structures in that cloud, from subparsec to $\sim$ 1 pc. We present the genealogy diagram of the derived virial parameters and N-PDFs of the structures identified in this work in Figure 13. The panels in Figure 13 that correspond to gravitationally bound structures are filled with light blue. Except for leaf 1 that corresponds to a clump at the southernmost end of the GMC, all structures, on scales from 0.3 pc (leaf 20) to 4.3 pc (trunk 2), are gravitationally bound. This result suggests that self-gravity is also important on much larger scales, i.e., on the clump- to cloud- scales. Our result is consistent with the results of Goodman et al. (2009b) and further highlights the necessity of including self-gravity in simulations of molecular clouds on pc scales.

The N-PDFs of the dendrogram structures shown in Figure 13 are uniformly normalized by the mean column density of the Orion A GMC, which is 1.5$\times$10²² cm^-2. We have performed log-normal or power-law fittings on the N-PDFs as appropriate. The results are indicated with dotted and dashed lines in Figure 13. We note that for most of the dendrogram structures in Figure 13, the adopted uniform mean column density for the normalization of N-PDF is lower than the actual mean column density of the individual structures, which translates the N-PDFs shown in Figure 13 to the right along the $\ln N/<N>$ axis and therefore increases the value of the fitted $\mu$ parameter. Due to this reason, most of the fitted $\mu$ parameters have positive values.

The leaves of branch 11 can be divided into three categories, leaves 18 and 12 with pure log-normal N-PDFs, leaves 21 and 14 that have log-normal N-PDFs with high-density excesses, and leaves 20 and 15 that have power-law N-PDFs at the high-density end. As shown in Figure 12, leaf 18 is located to the south of the ISF structure, and corresponds to the center of sub-region 3 in Section 3.4. A relatively low number of protostars are associated with leaf 18, indicating inactive star formation within leaf 18. Leaf 12 is a relatively faint filament located at the southernmost end of branch 11, and this filament is also quiescent in star formation as indicated by the few associated Class II YSOs in Figure 8. Leaves 21 and 14 correspond to the OMC-1/2/3 regions, which contain a cluster of intermediate-mass protostars and the NGC 1999 region. The high-density excesses in leaves 21 and 14 are concentrated in regions with s$\sim$ 2.3$-$3.2 ($N_{H_{2}}\sim 1.5-3.7\times 10^{23}$ cm^-2) and s$\sim$ 1.2$-$1.5 ( $N_{H_{2}}\sim 5.0-6.7\times 10^{22}$ cm^-2), respectively. The high-density regions account for 12.8% of the mass and 4.2% of the area for leaf 21. For leaf 14, the high-density mass and area fractions are 23.7% and 16.6%, respectively. Leaves 20 and 15 correspond to OMC-4, which includes a dense star-forming region and the H II region L 1641 N. Leaf 20 is possibly on a later evolutionary stage than leaf 21, as indicated by the pure power-law PDF and the lower protostar fraction as discussed in Section 3.4.2. The N-PDFs of branches 19, 17, 16, 13, and 11 can be well-fitted with log-normal functions below $s=2.3$. The high-density excesses in the N-PDFs of these branches are caused by the pixels of density above $N_{H_{2}}\sim 1.5\times 10^{23}$ cm^-2 which is confined to the Orion KL region in leaf 21 and is shown with the white contour in Figure 3a. The high-density pixels with $s>2.3$ only account for a small mass and area proportion of the branch N-PDFs below leaf 21. The mass proportion decreases from 12.8% of leaf 21 to 2.6% of trunk 2, while the corresponding area proportion decreases from 4.2% to 0.2%. Therefore, the majority of the structures of branch 11 are dominated by log-normal N-PDFs and departures from log-normal N-PDFs at high-densities only occur in active star-forming regions.

Except for leaf 7 that corresponds to the H II region L 1641 S, all leaves in branch 5 have log-normal N-PDFs without high-density excesses. The N-PDF of leaf 4 is well fitted with double log-normal functions, indicating the possible existence of mixed components along the line-of-sight. We can see from the last two panels in Figure 6 that there are two separate velocity components at the location of leaf 4, one of which is in the range from 5 to 8 km s^-1, the same as the EC counterpart. However, it is most likely that the velocity separation in the leaf 4 region is a local dynamic property and not caused by the EC gas. The N-PDFs of branches 9 and 5 are fitted with log-normal functions, while the N-PDF of branch 6 that contains the H II region L 1641 S is fitted with power-law at its high density tail.

Generally, the $N_{H_{2}}$ dynamical ranges of the structures above branch 5 are smaller than those of the structures above branch 11. From leaf 21 to branch 11, the $\mu$ of the fitted log-normal function decreases from $\sim$1.5 to $\sim$0.3, and the $\sigma_{s}$ increases from $\sim$0.3 to $\sim$0.70. From leaf 8 to branch 5, the $\mu$ and the $\sigma_{s}$ of the fitted log-normal functions vary gradually from $\sim$0.4 to $\sim$0.1 and $\sim$0.1 to $\sim$0.3, respectively. The ranges of $\mu$ and $\sigma_{s}$ of structures in branch 5 are much smaller than those in branch 11, indicating the southern filament (branch 5) of the Orion A GMC is at an earlier evolutionary stage than the northern filament (branch 11), which is consistent with the results in Section 3.4. We also plot the fitted log-normal N-PDFs of branches 5 and 11, and their summation in Figure 3. The N-PDFs of the two filaments account for the high-density part of the column density distribution of Orion A, supporting that the Orion A molecular cloud is mainly composed of these two filaments.

Below branches 11 and 5, there are two leaves and three branches. Leaves 1 and 22 correspond to a southernmost and a northernmost clump, respectively, both inactive in star formation. The N-PDF of leaf 1 is fitted with a log-normal function while the N-PDF of leaf 22 has an irregular shape. Branch 3 is mainly the combination of filaments 5 and 11, and is approximately the main ridge of the Orion A GMC. The N-PDFs of branches 3, 0, and 2 are fitted with log-normal functions and getting closer to the overall N-PDF of the whole cloud successively. The high-density excesses in the N-PDFs of these three branches are caused by leaf 21, while the low-density excess of branch 2 is caused by the EC gas, as discussed in Section 3.3.2.

In summary, regardless the sub-structure is in the northern or southern part of the GMC, power-laws and deviations from log-normal distributions at the high-density tail of N-PDFs only exist in active star-forming regions. The majority of the structures in Orion A GMC have log-normal dominated N-PDFs on scales from $R\sim 0.4$ to 4 pc. The predominant existence of log-normal N-PDFs in structures on such a broad range of scales suggests a dominant role of turbulence on these spatial scales.