Physical Conditions and Kinematics of the Filamentary Structure in Orion Molecular Cloud 1

Observations of the N₂H⁺ J = 3–2, HCO⁺ J = 3–2, and HCN J = 3–2 lines together with the 1.1 mm continuum were carried out with the SMA on 2014 February 14, 20, and 24. A subcompact configuration with six antennas in the array was used, providing baselines ranging from 9.476 to 25.295 m. The shortest and longest uv distances are 5.6 kλ and 23.6 kλ, respectively. The primary beam of the 6 m antennas had a size of 42'' in half-power beam width (HPBW), and the synthesized beam size was 553 × 525. The bandwidth was 4 GHz per sideband, and the frequency resolution was 203 kHz that corresponds to the velocity resolution of ∼0.22 km s⁻¹ at the rest frequency of N₂H⁺ 3–2. Using a 144 pointing mosaic with a Nyquist sampled hexagonal pattern, as shown in Figure 1, the observed area covered $\sim 5^{\prime} \times 7^{\prime}$. In order to obtain uniform uv coverage for 144 pointings, we observed each pointing for 5 s and visited all pointings in a loop. Each of the 144 pointings was visited three times in each observing run, giving a total on-source integration time per pointing of 45 s.

Zoom In Zoom Out Reset image size

The visibility data were calibrated using the MIR/IDL software package.⁴ The gain calibrators were 0501-019 and 0607-085, the flux calibrator was Ganymede on February 14 and Callisto on the other two days, and the bandpass calibrator was 3C279. The image processing was carried out using the MIRIAD package (Sault et al. 1995). The image cube was generated with Briggs weighting with a robust parameter of 0.5, followed by a nonlinear joint deconvolution using the CLEAN-based algorithm, MOSSDI. The final data cube has a rms noise level of ∼0.5 K for a ∼0.22 km s⁻¹ velocity channel. In this paper, we focus on the results and analyses of the N₂H⁺ line. Results of HCO⁺ and HCN will be presented in a forthcoming paper.

We simultaneously observed the N₂H⁺ J = 3–2 and HCO⁺ J = 3–2 lines on 2018 November 16 and 17 using the SMT of the Arizona Radio Observatory. We used the SMT 1.3 mm ALMA band 6 receiver and the filter-bank backend. The beam size was 2845 in HPBW at the frequency of N₂H⁺ 3–2, and the main-beam efficiency is 0.71 ± 0.05. The on-the-fly mode was used in order to cover the mapping area of 6' × 9' centered at R.A. $({\rm{J}}2000)={5}^{{\rm{h}}}{35}^{{\rm{m}}}12\buildrel{\rm{s}}\over{.} 1$ and decl. $({\rm{J}}2000)=-5^\circ 21^{\prime} 15\buildrel{\prime\prime}\over{.} 4$. The data were reduced with CLASS.⁵ The rms noise levels were ∼0.3 K at a spectral resolution of 250 kHz (∼0.27 $\mathrm{km}\ {{\rm{s}}}^{-1}$).

We combined the data cubes of the SMA and SMT by using the MIRIAD task immerge. The method of this task, known as feathering, linearly merges two images with different resolutions in their Fourier domain (i.e., spatial frequency). In the case of combining the single-dish and mosaicing data, immerge gives unit weight to the single-dish data at all spatial frequencies, and tapers the low spatial frequencies of the mosaicing data, so as to produce the Gaussian beam of the combined data equal to that of the mosaicing data. Inputs of immerge include the "CLEANed" SMA image cube, SMT image cube, and the flux calibration factor. After checking the consistency in the flux scale of two input image cubes, we set the flux calibration factor to 1. The integrated intensity (moment 0) map and the intensity-weighted radial velocity (moment 1) map after combination are shown in Figure 2. The angular resolution and the rms noise level of the combined map are ∼54 and $1.0\,{\rm{K}}\,\mathrm{km}\ {{\rm{s}}}^{-1}$, respectively.

Zoom In Zoom Out Reset image size

To analyze the physical properties of the filamentary structure in high resolution, we use the N₂H⁺ 1–0 data cube provided by Hacar et al. (2018), where the ALMA and IRAM 30 m data were combined (Hacar 2018). The ALMA + IRAM 30 m combined image cube has a circular beam with a size of 45 in FWHM, and a rms level of 25 mJy beam⁻¹ at a spectral resolution of 0.1 $\mathrm{km}\ {{\rm{s}}}^{-1}$. The observational details are described in Hacar et al. (2017, 2018).

4.1.1. Filament Identification

Filament identification is done by applying the python package FilFinder (Koch & Rosolowsky 2015) to the SMA + SMT combined moment 0 map. The FilFinder algorithm segments filamentary structure by using adaptive thresholding, which performs thresholding over local neighborhoods and allows for the extraction of structure over a large dynamic range. Input parameters for FilFinder include: (1) global threshold—the minimum intensity for a pixel to be included; (2) adaptive threshold—the expected full width of filaments for adaptive thresholding; (3) smooth size—scale size for removing small noise variations; (4) size threshold—minimum number of pixels for a region to be considered as a filament.

We set $9\,{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ as the global threshold. To focus on filaments with lengths $\gtrsim 0.1$ pc, we set the size threshold as 400 square pixels, where the pixel size is 1 arcsec². The adaptive threshold is set to 0.06 pc, which is approximately twice the width of the filaments, and the smooth size is set to 0.03 pc. We find the result matches better with identification by human eyes when the smooth size is set to ∼0.5 times the adaptive threshold. The smaller we set the smooth size, the more short branches would be identified. On the other hand, a larger smooth size tends to make filaments connected, since a larger region of data is smoothed.

In total, 11 filaments have been identified. Figure 4(a) shows the result of filament identification, where the gray-scale image is the combined moment 0 map, and the colored lines are the identified major axes for each filament. The identified filaments and their lengths and widths are listed in Table 1, together with the physical properties described in Section 5.1. Since the hyperfine components in N₂H⁺ 3–2 cannot be separated, the identification is based on the moment 0 map instead of the 3D data cube. We have estimated the typical FWHM of the identified filaments by fitting a Gaussian to the several cuts perpendicular to the filaments. The FWHM of these filaments are 0.02–0.03 pc, which is consistent with those identified in Hacar et al. (2018) based on the ALMA + IRAM 30 m observations in N₂H⁺ 1–0.

Zoom In Zoom Out Reset image size

Table 1.  Properties of the Identified Filaments

Filament	Length	${\mathrm{Width}}_{\mathrm{FWHM}}$	${M}_{\mathrm{lin}}$	Δv	${\rm{\Delta }}{v}_{\mathrm{nt}}/{c}_{s}(T)$	${M}_{\mathrm{crit}}$	${M}_{\mathrm{lin}}/{M}_{\mathrm{crit}}$
	(±0.001 pc)	(±0.001 pc)	(${M}_{\odot }{\mathrm{pc}}^{-1}$)	($\mathrm{km}\ {{\rm{s}}}^{-1}$)		(${M}_{\odot }{\mathrm{pc}}^{-1}$)
Main	0.349	0.025	94.2–101.7	1.0 ± 0.2	1.5	119.7	0.79–0.85
East	0.365	0.023	78.5–85.6	0.9 ± 0.3	1.3	103.8	0.76–0.83
1	0.159	0.024	84.0	0.6 ± 0.2	0.85	66.0	1.27
2	0.100	0.023	76.2	0.7 ± 0.3	1.0	76.9	0.99
3	0.279	0.017	42.5	1.3 ± 0.6	1.9	177.6	0.24
4	0.303	0.020	61.8	1.1 ± 0.3	1.6	137.3	0.45
5	0.155	0.024	84.0	1.0 ± 0.5	1.5	119.7	0.70
6	0.124	0.033	166.3	1.1 ± 0.3	1.6	137.3	1.21
7	0.175	0.022	68.9	1.0 ± 0.4	1.5	119.7	0.58
8	0.237	0.022	81.5–121.0	2.0 ± 0.9	2.95	371.6	0.22–0.33
9	0.159	0.017	42.5	1.0 ± 0.3	1.5	119.7	0.36

Download table as:  ASCIITypeset image

Hacar et al. (2018) has identified the filaments in Orion A with a different algorithm based on the 3D data cube. Since we identified the filaments in 2D using the moment 0 map, the results are different from those of 3D if there are multiple velocity components in the same line of sight. For example, our main filament and filament 6, which can be clearly seen in the moment 0 map, are not identified by Hacar et al. (2018). As shown in Figure 5(a) and (b), these two filaments contain multiple velocity components along the line of sight, which could cause a difference between the results of 2D- and 3D-based algorithms. Apart from these filaments, other OMC1 filaments show a single-velocity component in the spectra. Therefore, there is no significant difference in the results between 2D and 3D identifications.

Zoom In Zoom Out Reset image size

The moment 0 images in ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ reveal three filaments with high-intensity clumpy cores, one of which is in the OMC1-South filament 8. We will refer to the two prominent filaments in the northern region as the main filament and the east filament, which are shown in blue and yellow, respectively, in Figure 4(a). The gas kinematics inside these filaments will be analyzed in Section 4.4.

4.1.2. Core Identification

We identify the high-intensity cores inside the filaments by using the 2D version of Clumpfind (Williams et al. 1994). The Clumpfind algorithm contours input data with the values assigned by users, and then distinguishes each core region by those values. It starts from the highest contour level, and then works down through the lower levels, finding new cores and extending previously defined ones until the lowest contour level is reached. We use the combined SMA + SMT integrated intensity map as the input data, and set the contour levels as 50, 53.6, 57.2, 60.8, and 64.4 ${\rm{K}}\,\mathrm{km}\ {{\rm{s}}}^{-1}$. The spacing of the contour levels are set constantly as ${\rm{\Delta }}T=2{T}_{\mathrm{rms}}=3.6$, which could lower the the percentage of false detection to <2% as suggested in Williams et al. (1994).

Figure 4(b) shows the positions of the 10 cores identified by 2-D Clumpfind. Table 2 lists the properties of these cores including position, peak flux, and effective radius (R_eff), which are all direct outputs of 2-D Clumpfind. Note that the R_eff in Clumpfind is defined as ${R}_{\mathrm{eff}}=\sqrt{A/\pi }$, where A is the area with emission above the criteria. In Table 2, the line width (Δv) is determined by hyperfine spectral fitting (see Section 4.2), the mass (M_core) is derived by using the densities determined from non-local thermodynamic equilibrium (non-LTE) analysis (see Section 4.3), and the virial mass (M_vir) is calculated from the line width using Equation (12) (see Section 5.1).

Table 2.  Properties of the Identified Cores

Core	R.A.	Decl.	Peak Flux	Reff	Δv	Mcore	Mvir
	(J2000.0)	(J2000.0)	(${\rm{K}}\,\mathrm{km}\ {{\rm{s}}}^{-1}$)	(arcseconds)	($\mathrm{km}\ {{\rm{s}}}^{-1}$)	(${M}_{\odot }$)	(M⊙)
1	05:35:13.10	−5:24:12.4	84.3	8.9	3.5	14.4–45.6	46.1
2	05:35:13.70	−5:23:52.4	69.4	5.6	2.8	3.6–11.3	18.5
3	05:35:14.10	−5:23:41.4	66.5	4.7	1.6	2.0–6.4	5.0
4	05:35:15.37	−5:22:04.4	66.0	5.1	2.0	2.6–8.2	8.5
5	05:35:15.84	−5:20:38.4	63.4	3.5	0.9	0.9–2.8	1.2
6	05:35:16.90	−5:19:27.4	61.9	4.0	⋯	1.3–4.2	⋯
7	05:35:16.70	−5:19:34.4	60.8	4.4	⋯	1.7–5.3	⋯
8	05:35:16.84	−5:20:42.4	59.2	3.6	2.2	0.9–3.0	7.4
9	05:35:17.37	−5:19:17.4	58.7	4.2	⋯	1.5–4.7	⋯
10	05:35:15.77	−5:21:24.4	57.1	2.6	1.1	0.4–1.1	1.3

Download table as:  ASCIITypeset image

To determine the physical parameters, we conduct hyperfine spectral fitting on the combined SMA and SMT data for the regions with signal-to-noise ratio > 5. We fit the V_LSR, line widths (Δv), excitation temperatures (T_ex), and total opacities (τ_tot) under the assumption of LTE. With the relative intensities among 16 main hyperfine components for the N₂H⁺ 3–2 line (Caselli et al. 2002), the opacity of each component is assumed as a Gaussian profile

$\begin{eqnarray}&&{\tau }_{i}(v)={\tau }_{i}\exp \left[-4\mathrm{ln}2{\left(\displaystyle \frac{v-{v}_{i}-{v}_{\mathrm{sys}}}{{\rm{\Delta }}v}\right)}^{2}\right],\end{eqnarray} \tag{ 1 }$

where v_i is the velocity offset from the reference component and v_sys is the systemic velocity. Then, we obtain the optical depths of the multiplets as

$\begin{eqnarray}&&\tau (v)={\tau }_{\mathrm{tot}}\sum _{i=1}^{16}{R}_{i}\exp \left[-4\mathrm{ln}2{\left(\displaystyle \frac{v-{v}_{i}-{v}_{\mathrm{sys}}}{{\rm{\Delta }}v}\right)}^{2}\right],\end{eqnarray} \tag{ 2 }$

where R_i is the relative intensity for the ith hyperfine component, and ${\tau }_{i}={\tau }_{\mathrm{tot}}{R}_{i}$. The brightness temperature at each pixel can be represented as

$\begin{eqnarray}&&{T}_{b}(v)=[J({T}_{\mathrm{ex}})-J({T}_{\mathrm{bg}})][1-\exp (-\tau (v))],\end{eqnarray} \tag{ 3 }$

where T_bg is the cosmic background temperature (2.73 K), and

$\begin{eqnarray}&&J(T)=\displaystyle \frac{\tfrac{h\nu }{k}}{\exp \left(\tfrac{h\nu }{{kT}}\right)-1}.\end{eqnarray} \tag{ 4 }$

Figure 5 presents the observed spectra obtained by averaging the $5^{\prime\prime} \times 5^{\prime\prime}$ regions marked in Figure 4(a). The best-fit single-velocity components (in red) are overlaid on the spectra. Note that the multiple velocity components are only limited to the regions near cores 6, 7, and 9 and the northern part of filament 6, as indicated by the dashed green boxes in Figure 4(a). We applied the single-component fitting to the spectra of these regions, because two-component fitting requires too many parameters. Inside the regions with multiple components, the fitted V_LSR are still dominated by the major components, although the fitted line widths are highly affected by secondary components.

Using the centroid velocities determined from the hyperfine fitting, we illustrate the velocity distribution along R.A. and decl., respectively. Figure 6(a) shows the radial velocities for all the N₂H⁺ 3–2 components fitted in the northern and western regions. The figure reveals a sharp velocity transition near R.A.(J2000) = ${5}^{{\rm{h}}}{35}^{{\rm{m}}}{14}^{{\rm{s}}}$, where a clear boundary can also be seen in Figure 2(b) between the northern and western regions. Figure 6(b) shows the radial velocities of all three regions as a function of decl. While the velocities of the northern region are almost constant, those of the western region decrease from north to south and continuously connected to those of the southern region. Such a velocity decrease from north to south in OMC1 has been reported in Hacar et al. (2017, 2018), and interpreted as the gravitational collapse of OMC1. Figure 6(b) shows that the gas in the western region is accelerated toward OMC1-South.

Zoom In Zoom Out Reset image size

Comparing the fitting results with the observed spectra, we found that the observed satellite components are much brighter than those predicted by the LTE assumption (e.g., Figure 5(c)–(e)). In addition, when the line is optically thin (${\tau }_{\mathrm{tot}}\ll 1$), the fitting cannot constrain T_ex and τ_tot, because Equation (3) will be reduced to

$\begin{eqnarray}&&{T}_{b}(v)\simeq {\tau }_{\mathrm{tot}}\cdot {T}_{\mathrm{ex}},\end{eqnarray} \tag{ 5 }$

indicating that the ${T}_{\mathrm{ex}}$ and τ_tot can be determined arbitrarily. Therefore, we conduct non-LTE analysis in Section 4.3 to derive the physical conditions.

Using RADEX (Van der Tak et al. 2007), a non-LTE radiative transfer code, we construct spectra models in N₂H⁺ 3–2 and 1–0. The synthetic spectra are constructed with the equation

$\begin{eqnarray}&&{T}_{b}(v)={\rm{\Psi }}\left(\displaystyle \frac{\sum J({T}_{\mathrm{ex}}^{i}){\tau }_{i}(v)}{\sum {\tau }_{i}(v)}-J({T}_{\mathrm{bg}})\right)\left(1-{e}^{-\sum {\tau }_{i}(v)}\right),\end{eqnarray} \tag{ 6 }$

where ${T}_{\mathrm{ex}}^{i}$ and τ_i(v) represent the excitation temperature and optical depth for all hyperfine components in the 3–2 or 1–0 transitions, and Ψ is the beam filling factor. In our models, the beam filling factors are assumed to be unity. We also construct an intensity ratio model by dividing the integrated spectra model in N₂H⁺ 3–2 with that in N₂H⁺ 1–0.

The constructed models can be represented by a three-dimensional grid, where the three axes are H₂ density (${n}_{{{\rm{H}}}_{2}}$) ranging from 10⁴–10⁹ cm⁻³, kinetic temperature (T_kin) ranging from 8 K–60 K, and the ratio of N₂H⁺ column density to line width ($N({{\rm{N}}}_{2}{{\rm{H}}}^{+})$/Δv) ranging from 5 × 10⁷–5 × 10⁸ s cm⁻³. The step sizes of the grid are 1 K for T_kin and 0.5 in decimal log scale for both $N({{\rm{N}}}_{2}{{\rm{H}}}^{+})$ and ${n}_{{{\rm{H}}}_{2}}$. As $N({{\rm{N}}}_{2}{{\rm{H}}}^{+})$/Δv is the input parameter in RADEX, the estimation of $N({{\rm{N}}}_{2}{{\rm{H}}}^{+})$ varies among regions with different line widths. Based on our fitting results, the line width in OMC1 could vary from ∼1 to 3 km s⁻¹.

Figure 7 presents the constructed non-LTE model, where the gray-scale background shows the integrated spectra model in N₂H⁺ 3–2, and the contour levels show the 3–2/1–0 ratio model. By applying the observed integrated intensity and line ratio in this model, physical parameters (${n}_{{{\rm{H}}}_{2}}$, T_kin, and $N({{\rm{N}}}_{2}{{\rm{H}}}^{+})$/${\rm{\Delta }}v$) in different subregions can be constrained. For instance, the two bars on the first panel of Figure 7 indicate the two possible solutions that satisfy the conditions with an intensity of 7–15 ${\rm{K}}\,\mathrm{km}\ {{\rm{s}}}^{-1}$ and a line ratio of 2.2 ± 0.4. The corresponding physical conditions of the left and right bars are ${n}_{{{\rm{H}}}_{2}}={10}^{6}$ cm⁻³ and T_kin = 45–60 K, and ${n}_{{{\rm{H}}}_{2}}=3\,\times {10}^{6}$ cm⁻³ and T_kin = 21–30 K, respectively.

Zoom In Zoom Out Reset image size

We use our SMA + SMT data in N₂H⁺ 3–2 together with the ALMA + IRAM 30 m data in N₂H⁺ 1–0 for the non-LTE analysis, where the scale size is 54 (∼0.01 pc). Based on the 3–2 moment 0 map, we define three regions for analysis. The first region is defined as the core regions with integrated intensity 50–60 ${\rm{K}}\,\mathrm{km}\ {{\rm{s}}}^{-1}$ and a 3–2/1–0 ratio of 1 ± 0.3. The second region is the lower intensity regions inside the filaments (20–40 ${\rm{K}}\,\mathrm{km}\ {{\rm{s}}}^{-1}$) with a similar line ratio of 1 ± 0.3, and the third region is the nonfilament region with low intensities (7–15 ${\rm{K}}\,\mathrm{km}\ {{\rm{s}}}^{-1}$) and a higher line ratio of 2.2 ± 0.4. The spectra averaged in these subregions are compared with the model spectra. The derived physical parameters for the non-LTE analysis together with the criteria of the three subregions are listed in Table 3.

Table 3.  Parameters for Non-LTE Analysis

	Core Regions	Low-intensity	Nonfilament
	(>50 K km s−1)	Regions	Regions
${n}_{{{\rm{H}}}_{2}}$ (cm−3)	107 or $3\times {10}^{7}$	3 × 106 or 107	106 or 3 × 106
Tkin (K)	19–23 or 18–20	17–22 or 13–16	> 45 or 21–30
$N({{\rm{N}}}_{2{\rm{H}}}^{+})/{\rm{\Delta }}v$ (${\rm{s}}\ {\mathrm{cm}}^{-3}$)	5 × 108	1.5 × 108	5 × 107
Intensity (${\rm{K}}\,\mathrm{km}\ {{\rm{s}}}^{-1}$)	50–60	20–40	7–15
Typical ratio	1 ± 0.3	1 ± 0.3	2.2 ± 0.4

Download table as:  ASCIITypeset image

By comparing between the filament and nonfilament regions, we find that the filament regions have a higher density of ∼10⁷ cm⁻³ and a lower temperature of ∼15–20 K than the nonfilament regions. As the major heating sources may come from outside the filaments, it is likely that the dense gas in the filaments could block the outer radiation, leading to a lower temperature in the filaments (see Section 5.2 for further discussion). Inside the filaments, the core regions have a higher N₂H⁺ column density than the low-intensity regions, if we assume similar line widths in these regions. Also, the volume density of the core regions are generally higher than the low-intensity regions. On the other hand, there is no significant difference in temperature between the core and the low-intensity regions, although the derived T_kin is slightly higher in the core regions.

Using the volume densities determined from the non-LTE analysis and the filament widths, we have estimated the masses of the cores and the line masses of the filaments under the assumption of uniform cylindrical filaments. We adopted this method instead of using the column density of N₂H⁺ because the fractional abundance of N₂H⁺ is highly uncertain in the regions close to Orion KL and because the inclination of each filament is also uncertain. The density is assumed to be ${n}_{{{\rm{H}}}_{2}}=3\times {10}^{6}$ cm⁻³ for the filaments without cores. For those with cores, i.e., main filament, east filament, and filament 8, we use two possible core densities (i.e., 10⁷ or $3\times {10}^{7}\ {\mathrm{cm}}^{-3}$) determined in this section for the core regions and ${n}_{{{\rm{H}}}_{2}}=3\times {10}^{6}$ cm⁻³ for the regions outside the cores. If we adopt the higher density solution, i.e., 10⁷ cm⁻³, for the low-intensity regions of the filaments, the line masses of the filaments could be higher by a factor of ∼3. On the other hand, if we adopt the radial density profile of the isothermal cylinder with the same central density (e.g., Ostriker 1964), the line mass included in the radius of 0.01–0.015 pc (i.e., half of the filament width) becomes lower by a factor of two. Properties of the identified filaments and cores are summarized in Tables 1 and 2, respectively, and will be discussed in Section 5.1.

Characterizing the gas motion inside the filaments requires the studies of velocity structure. We investigate the radial velocity fields along both the major and minor axes of the filaments in OMC1, and compare our results with existing filament formation model and core formation model. Since the main filament and the east filament are identified with core fragmentation, we focus on the analyses of these two filaments.

4.4.1. Minor-axis Analysis

Systematic velocity gradients perpendicular to the filaments have been observed in the filaments of both low- and high-mass star-forming regions (e.g., Schneider et al. 2010; Beuther et al. 2015; Dhabal et al. 2018). Such velocity gradients across the filaments can be explained by the projection of gas accretion toward the filament axes (Dhabal et al. 2018) or the rotation of filaments (Olmi & Testi 2002). In order to assess whether the OMC1 filaments show similar features, we analyze the velocity fields along the minor axes.

In the main and the east filaments, there is no systematic velocity gradient along the minor axes. However, local velocity gradients of ∼0.3 km s⁻¹ are observed in cores 4 and 8 in the east filament. On the other hand, there is no significant velocity gradient in cores 5 and 10. The velocity gradients in cores 6, 7, and 9 in the northern part of the main filament are unclear due to the secondary velocity component along the line of sight. Figure 8 shows P–V diagrams across core 8 and core 4. The directions of velocity gradients across these cores are not consistent; the velocity increases from east to west in core 8, while it decreases in core 4. Such velocity gradients with different directions along the same filament have also been observed along the massive DR21 filament, although its origin is still unclear (Schneider et al. 2010).

Zoom In Zoom Out Reset image size

Using the effective radii determined in Section 4.1.2, i.e., ${R}_{\mathrm{eff}}=5\buildrel{\prime\prime}\over{.} 1$ and 36 for core 4 and core 8, respectively, the velocities at R_eff were determined from the spectral fitting. Then, the velocity gradient (∇V) along the east–west direction was estimated using the velocity difference at the core boundaries and the effective diameter. Assuming a linear velocity variation, core 8 and core 4 have ${\rm{\nabla }}V=17.2$ $\mathrm{km}\ {{\rm{s}}}^{-1}\ {\mathrm{pc}}^{-1}$ and ${\rm{\nabla }}V=-11.3$ $\mathrm{km}\ {{\rm{s}}}^{-1}\ {\mathrm{pc}}^{-1}$, respectively. The origin of the observed velocity gradients with different directions are likely to be the local effects such as rotating motions or unresolved multiple components. If the observed velocity fields come from rotations, the rotational energy E_rot can be estimated under the assumption of solid-body rotation as (Belloche 2013)

$\begin{eqnarray}&&{E}_{\mathrm{rot}}=\displaystyle \frac{1}{2}I{{\rm{\Omega }}}^{2}=\displaystyle \frac{1}{2}\left[\displaystyle \frac{2}{3}{{MR}}^{2}\left(\displaystyle \frac{3-\alpha }{5-\alpha }\right)\right]{\left(\displaystyle \frac{| {\rm{\nabla }}V| }{\sin i}\right)}^{2}\end{eqnarray} \tag{ 7 }$

where i is the inclination angle of the rotational axis to the line of sight, and α indicates a power-law density profile of $\rho \propto {r}^{-\alpha }$. The gravitational energy can also be derived as

$\begin{eqnarray}&&{E}_{\mathrm{grav}}=-\displaystyle \frac{{{GM}}^{2}}{R}\left(\displaystyle \frac{3-\alpha }{5-2\alpha }\right).\end{eqnarray} \tag{ 8 }$

To estimate the ratio of rotational to gravitational energy (β_rot) for these cores, we assume a uniform density profile, i.e., α = 0, and a random-averaged inclination angle of $\langle \sin i\rangle =\pi /4$. It turns out that core 8 has β_rot ranging from 0.04–0.11, and core 4 has ${\beta }_{\mathrm{rot}}$ ranging from 0.02–0.06. This means that rotations of these cores are significant but not dominant in the energetics.

4.4.2. Major-axis Analysis

The velocity fields along the major axes of the filaments are also analyzed. Using the central velocity (V_LSR) and the peak intensity determined from the hyperfine spectral fitting (see Section 4.2), we plot the intensity and velocity variations along the filament major axis in Figure 9, where the horizontal axes represent the offset (from north to south) in arcseconds, and the vertical axes show the peak intensity on the left and the V_LSR on the right.

Zoom In Zoom Out Reset image size

Analysis of the east filament (see Figure 9(a)) reveals the oscillations in both intensity and velocity. It is found that there are positional shifts between the intensity and velocity peaks toward cores 8 and 4. This is similar to the feature observed in two of the filaments in L1517 (Hacar & Tafalla 2011). The velocity oscillation along the filament can be related to the core-forming motions. If the gas flow is converging to the center of the core, its velocity with respect to the one at the core center is positive in one side and negative in the other side of the density peak. According to the kinematic model proposed in Hacar & Tafalla (2011), where sinusoidal perturbations were assumed for both density and velocity, a λ/4 phase shift between the two distributions is predicted. In Figure 9(a), we show the sinusoidal fits to the velocity oscillation in the east filament. In addition, the locations of the intensity peaks toward cores 8 and 4 match well with the λ/4 shift from the two corresponding sinusoidal peaks.

Figure 9(b) shows the intensity and velocity plot for the main filament. Although there is a secondary velocity component toward part of this filament, the velocities determined from the fitting represent the ones of the major component. In the main filament, the relations between the intensity and velocity variations are unclear. This is probably because of the evolutionary stage of the main filament. Previous 1.3 mm observations with the SMA (Teixeira et al. 2016) revealed CO molecular outflows associated with some cores in the main filament, including core 5. It is therefore likely that some of the cores in the main filament have already harbor young protostars in the Class 0 stage.

In contrast, no evidence of protostars have been observed in the east filament, and the positional shift between its intensity and velocity peaks may indicate that core formation is still ongoing in the east filament. As the evolutionary stage of different filaments can vary (Myers 2017), it is possible that the east filament is in an earlier evolutionary phase than the main filament with star formation signature.

As shown in Table 1, all three filaments with core fragmentation have line masses ≳80 ${M}_{\odot }\,{{\rm{p}}{\rm{c}}}^{-1}$. In contrast, the filaments with lower line masses such as filaments 3, 4, and 9 do not contain cores. However, three filaments having the line masses close to $80\,{M}_{\odot }\,{{\rm{p}}{\rm{c}}}^{-1}$ (filaments 1, 2, and 5) and filament 6 with the largest line mass do not contain any cores. Therefore, although line masses are often used as an indicator of the star formation stage of a filament (Heitsch 2013; Palmeirim et al. 2013; Li et al. 2014), it may not be a conclusive discriminator, which is also stated in Dhabal et al. (2018).

To investigate the internal dynamics of the filaments, we derive the nonthermal velocity dispersion (${\rm{\Delta }}{v}_{\mathrm{nt}}$) by using the equation

$\begin{eqnarray}&&{\rm{\Delta }}{v}_{\mathrm{nt}}=\sqrt{\displaystyle \frac{{({\rm{\Delta }}v)}^{2}}{8\mathrm{ln}2}-\displaystyle \frac{k\cdot {T}_{\mathrm{kin}}}{\mu ({{\rm{N}}}_{2}{{\rm{H}}}^{+})}}\end{eqnarray} \tag{ 9 }$

where Δv is the line width in FWHM obtained from the hyperfine fitting, and μ is the molecular mass. The nonthermal velocity dispersion can be compared with the thermal sound speed

$\begin{eqnarray}&&{c}_{s}(T)=\sqrt{\displaystyle \frac{k\cdot {T}_{\mathrm{kin}}}{\mu ({{\rm{H}}}_{2})}}.\end{eqnarray} \tag{ 10 }$

Then, the critical line mass for an infinite filament in hydrostatic equilibrium can be calculated as (Stodolkiewicz 1963; Ostriker 1964)

$\begin{eqnarray}&&{M}_{\mathrm{crit}}=\displaystyle \frac{2{({\rm{\Delta }}{v}_{\mathrm{eff}})}^{2}}{G}=\displaystyle \frac{2}{G}\left[{c}_{s}{(T)}^{2}+{({\rm{\Delta }}{v}_{\mathrm{nt}})}^{2}\right]\end{eqnarray} \tag{ 11 }$

where ${\rm{\Delta }}{v}_{\mathrm{eff}}$ is defined as the effective velocity dispersion considering both thermal and nonthermal effects.

Based on the non-LTE results in Section 4.3, we take T_kin = 20 K for estimation, which leads to ${c}_{s}(T)=0.287$ $\mathrm{km}\ {{\rm{s}}}^{-1}$. In the case of the purely thermally-supported filament, ${M}_{\mathrm{crit}}=38.4\ {M}_{\odot }{\mathrm{pc}}^{-1}$. Due to the rather large nonthermal velocity dispersion, the critical line masses listed in the OMC1 filaments are ∼2–4 times larger than the purely thermally-supported case. Table 1 reveals that 7 of the 11 filaments have $0.5\leqslant {M}_{\mathrm{lin}}/{M}_{\mathrm{crit}}\leqslant 1.5$, suggesting that most of the filaments are gravitationally bound. Filaments 3, 4, and 9 having low line masses are found to have ${M}_{\mathrm{lin}}/{M}_{\mathrm{crit}}\lt 0.5$ and thus may be gravitationally unbound. On the other hand, filament 8 has ${M}_{\mathrm{lin}}/{M}_{\mathrm{crit}}\lt 0.5$ even though this filament contains cores. This filament resides in OMC1-South, where a cluster of young stellar objects (YSOs) have already been formed (Zapata et al. 2004, 2006). Due to the powerful outflows from these YSOs, the gas in this region is highly turbulent. Thus, filament 8 has the largest line width of ∼3 $\mathrm{km}\ {{\rm{s}}}^{-1}$ among all the filaments, leading to a high ${M}_{\mathrm{crit}}$ and a low ${M}_{\mathrm{lin}}/{M}_{\mathrm{crit}}$ ratio. The low ${M}_{\mathrm{lin}}/{M}_{\mathrm{crit}}$ value could imply that filament 8 is in the phase of disruption. Another possibility is the high external pressure in this region. If the filament is confined by external pressure, the filament is prone to fragmentation even though its line mass is smaller than the critical line mass (e.g., Fischera & Martin 2012). Since filament 8 in OMC1-South is adjacent to the Orion Nebula Cluster, the high external pressure from hot and diffuse gas in the cluster could lead the core formation in this filament.

Our results show that the filaments in OMC1 have ${M}_{\mathrm{lin}}/{M}_{\mathrm{crit}}\lt 1.5$, which is consistent with that of Hacar et al. (2018) based on their N₂H⁺ 1–0 data. However, even though the derived ${M}_{\mathrm{lin}}/{M}_{\mathrm{crit}}$ ratios are similar, both the M_lin and ${M}_{\mathrm{crit}}$ determined from our data are higher than those of Hacar et al. (2018) by a factor of a few. While they used N₂H⁺ intensities to estimate column densities of H₂, we use the volume densities of H₂ derived from the non-LTE analysis. The empirical relation between N₂H⁺ intensities to H₂ column densities used by Hacar et al. (2018) has large scatter. The inclination of each filament is also uncertain. On the other hand, the uncertainty in inclination does not affect our estimation. However, the M_lin derived under our assumption of uniform density inside the filaments (excluding the cores) might be overestimated if the filaments have radial density profiles. The difference in ${M}_{\mathrm{crit}}$ likely results from the difference between the line widths observed in N₂H⁺ 3–2 (∼1.0 km s⁻¹) and 1–0. This is partly because of the difference in the beam size of our measurement (54) and that of Hacar et al. (2018) (45); the spectra observed with the larger beam contain the nonthermal motion in the larger area. The hyperfine components of 3–2, which are much more complicated than those of 1–0, introduce additional uncertainty in the derived line width if the relative intensities of the hyperfine components are different from the LTE values.

Table 2 shows that most of the cores have masses (M_core) ranging from 1–10 M_⊙. Both the sizes and the masses of these cores are larger than those determined by Teixeira et al. (2016), because multiple smaller-scale cores were revealed in their 1.3 mm continuum data using the SMA. These masses can also be compared with the masses inferred by the virial theorem. By assuming a uniform density in the cores, the virial mass (M_vir) can be derived as

$\begin{eqnarray}&&{M}_{\mathrm{vir}}=\displaystyle \frac{5\ {R}_{\mathrm{eff}}{({\rm{\Delta }}v)}^{2}}{8\ G\mathrm{ln}2},\end{eqnarray} \tag{ 12 }$

where ${R}_{\mathrm{eff}}$ is an effective radius of the core, and Δv is a typical line width. The line width Δv was determined from the hyperfine fitting except for cores 6, 7, and 9 having multiple velocity components along the line of sight.

As shown in Table 2, the measured core masses are similar to the calculated virial masses, indicating that most of the cores are gravitationally bound. One of the cores, core 5, already shows the signatures of star formation; toward this core, there is an infrared source and a clear bipolar outflow in CO (Teixeira et al. 2016). The velocity dispersion as well as M_vir is large in the southern cores i.e., cores 1 and 2, because of the strong turbulence from the stellar activities in OMC1-South (Zapata et al. 2006; Palau et al. 2018).

The N₂H⁺ 3–2/1–0 ratio map presented in Figure 3 shows higher ratios in the eastern part of OMC1. From the non-LTE model shown in Figure 7, regions with a higher intensity ratio generally indicate a higher kinetic temperature. This suggests that the eastern OMC1 has higher temperatures compared with the remaining area.

We find that the overall distribution of the high-ratio gas is similar to that of the CN and C₂H molecules presented in Ungerechts et al. (1997) and Melnick et al. (2011). Since CN and C₂H are sensitive to the presence of UV radiation (Fuente et al. 1993; Stauber et al. 2004), it is likely that the higher temperatures in the eastern OMC1 are caused by the UV heating from the high-mass stars in M42. For example, UV photons from the ${\theta }^{1}$ Ori C at (R.A., decl.) = (${5}^{{\rm{h}}}{35}^{{\rm{m}}}16\buildrel{\rm{s}}\over{.} 5$, $-5^\circ 23^{\prime} 22\buildrel{\prime\prime}\over{.} 8$), which is southeast of the Orion KL, could be one of the major heating sources. In addition, the [C i]/CO intensity ratio map in Shimajiri et al. (2013) shows a ratio peak of ∼0.17 around the position (R.A., decl.) = (${5}^{{\rm{h}}}{35}^{{\rm{m}}}{20}^{{\rm{s}}}$, $-5^\circ 18^{\prime} 30^{\prime\prime}$), implying that UV radiation also contributes to the heating of this region. Possible heating sources include the exciting star of M43—NU Ori at (R.A., decl.) = (${5}^{{\rm{h}}}{35}^{{\rm{m}}}31\buildrel{\rm{s}}\over{.} 0$, $-5^\circ 16^{\prime} 12^{\prime\prime}$), which is northeast of the OMC1 region.

The external heating scenario also explains the 3–2/1–0 ratios inside and outside the filaments. As shown in Figure 3, heating features are seen only outside the filament regions, while temperatures inside the filaments remain significantly lower. Wiseman & Ho (1998) reported the variation of NH₃ $(J,K)$ = (2, 2) to (1, 1) line intensity ratio in the filaments; the higher ratio (i.e., higher temperature) gas appears between the emission peaks with a lower ratio (i.e., lower temperature). Such a patchy heating pattern in the filaments does not appear in the N₂H⁺. There is no significant difference in the N₂H⁺ 3–2/1–0 ratio between the core regions and the low-intensity regions in the filaments. This is probably because the N₂H⁺ lines have higher critical densities than the NH₃ trace in the inner and denser part of the filaments where the gas is well shielded from the external radiation.

Apart from external UV heating, local heating from Orion KL has been discussed on the basis of previous observations (Wiseman & Ho 1998; Zapata et al. 2011; Bally et al. 2017; Tang et al. 2018). However, since N₂H⁺ emission is missing toward Orion KL, local heating around the KL core (∼100 K) is not observed in our data. The 3–2/1–0 ratio at the position of core 4, ∼2.0, is significantly higher than other regions of the east filament. This could be the effect of the local heating from Orion KL.

Recent studies have shown that large-scale dynamical collapse is important in high-mass star-forming regions (e.g., Hartmann & Burkert 2007; Hacar et al. 2017). The snapshots from the magnetohydrodynamic (MHD) simulation of globally collapsing clouds presented in Schneider et al. (2010) and Peretto et al. (2013) reveal multiple filaments and sharp boundaries of radial velocity changes that separate the cloud into several regions, and these regions and filaments converge toward the massive core at the center. The radial velocity distribution of OMC1 shows the trimodal pattern centered near Orion KL with the sharp boundaries between the northern and western regions (Figure 6(a)) and the northern and southern regions (Figure 6(b)). In addition, the radial velocities in the western region monotonically decrease from north to south, and continue to the velocity gradient in the southern region (Figure 6(b)). This velocity gradient corresponds to the part of the V-shaped velocity structure centered around the OMC1-South, which is interpreted as the presence of accelerated gas motion inflowing toward the Orion Nebula Cluster (Hacar et al. 2017). Our results have revealed that the gas in the western and southern regions contributes to this inflow.

Such a global collapse picture is also supported by the morphology of the magnetic field in OMC1. The magnetic field in OMC1 revealed by the B-fields in Star-forming Region Observations survey using the James Clerk Maxwell Telescope shows a well-ordered U-shaped structure, which can be explained by the distortion of an initially cylindrically symmetric magnetic field due to large-scale gravitational collapse (Pattle et al. 2017). Interestingly, the filaments in the northern and southern regions are almost perpendicular to the local magnetic field direction, while those in the western region such as filaments 3, 4, and 5 are aligned along the magnetic field. This implies that the filaments in the western region are feeding material along the magnetic field lines to the Orion Nebula Cluster, as predicted from the MHD simulation of global collapse (Schneider et al. 2010).