Independent Core Rotation in Massive Filaments in Orion

High spatial and spectral resolution observations that can resolve filaments (with typical widths of ∼0.1 pc) and cores are powerful tools for studying the dynamic structures of dense gas in massive star formation regions. Herschel images revealed that, in molecular clouds, dense filaments are ubiquitous structures, along which dense cores are commonly found (Molinari et al. 2010; André et al. 2014). Dense cores, presumably the site of current and future star formation, can form within or simultaneously with the filaments (André et al. 2010; Arzoumanian et al. 2011; Chen & Ostriker 2015). Atacama Large Millimeter/submillimeter Array's (ALMA) unprecedented spectral imaging capabilities makes feasible detailed studies of the dynamics of cores and filaments, even in relatively distant massive star-forming regions. In this work, we focus on the angular momentum of cores and its relation with respect to filaments.

For low-mass star-forming regions, earlier studies (e.g., Barranco & Goodman 1998; Caselli et al. 2002; Shinnaga et al. 2004; Punanova et al. 2018) measured the velocity gradients of dense gas at the thermal Jeans scale ∼0.04 pc, such as in L1495, L1521F, TMC-1C, L1251A, and PER4-7. For massive cores, generally more distant, the typical spatial resolution was ∼0.1 pc (see, e.g., Pirogov et al. 2003; Li et al. 2012; Tatematsu et al. 2016). With ALMA, we obtained high spatial resolution (3'') images of N₂H⁺ (1–0) toward filaments in OMC-2 and OMC-3. In conjunction with a sufficient velocity resolution of 0.11 km s⁻¹, a sensitive probe into the angular momentum of cores is now feasible down to the ∼0.05 pc scale, the thermal Jeans scale for these massive star-forming regions.

The Orion Molecular Cloud (OMC), the closest giant molecular cloud with an OB cluster, is an ideal target for studying the relation between dense cores and filaments. OMC-2 and OMC-3 are relatively quiescent (Li 2002; Li et al. 2003) and filamentary clouds in the OMC. We adopt a distance of 400 pc for OMC-2 and OMC-3 following Nutter & Ward-Thompson (2007). Li et al. (2013a) identified 30 massive quiescent cores, which contain no H ii region and no IRAS point sources and are and at least 1 pc away from the OB cluster, in OMC-2 and OMC-3. The core kinetic temperatures range from 13 to 31 K. The majority of cores were found to be gravitationally bound and 14 cores to be supercritical, i.e., the observed thermal and nonthermal gas motion cannot prevent immediate collapse (with a reasonable assumption of magnetic field strength B ≤ 500 μ Gauss).

Widespread and relatively easy to detect, N₂H⁺ is a reliable tracer of cold (T ≃ 10–20 K), dense (n(H₂) >10 × 10⁴ cm⁻³) gas (Bergin & Langer 1997; Tafalla et al. 2004, 2006; Crapsi et al. 2005). Benson et al. (1998) found that most (94%) dense cores in their sample had a detectable rotational transition of N₂H⁺ (J = 1–0). N₂H⁺ is formed through the reaction ${{\rm{N}}}_{2}+{{\rm{H}}}^{+}$ and is mainly destroyed through reaction with CO and electrons (Bergin & Langer 1997; Aikawa et al. 2001; Caselli et al. 2002). Since the abundances of CO and electrons drop in dense gas (Bergin & Langer 1997), the depletion timescale of N₂H⁺ is longer than many other, particularly carbon-bound, molecules (Bergin & Langer 1997; Aikawa et al. 2001), making N₂H⁺ (1–0) especially suitable for tracing dense cores and filaments.

To investigate the alignment between the core rotation and the natal filament, we examine the distribution of the angles between the rational axes and the direction of the filament elongation. Such an angle can potentially discriminate between different mechanisms of core formation. The gravitational fragmentation of a shock-compressed layer (Whitworth et al. 1995), for example, can explain the perpendicular relation between the core angular momentum and filaments. Such a configuration has indeed been found in cores with young-stellar-object-driven outflows (e.g., Anathpindika & Whitworth 2008). In contrast, a near parallel relation can be explained by gravo-turbulent fragmentation (e.g., Banerjee et al. 2006; Offner et al. 2008).

In this Letter, we characterize the orientations of the filaments in OMC-2 and OMC-3 and measure the velocity gradients of 30 cores there based on ALMA N₂H⁺ (1–0) images. ALMA provided a previously unattainable spatial dynamic range, namely, the size of the longer dimension of dense structures divided by that of the resolution element. The high resolution of ALMA makes measuring velocity gradients of dense gas feasible at the Jeans scale, in young OB-cluster-hosting regions, such as the OMC.

This Letter is organized as follows. Observation and data are described in Section 2. Section 3 presents our measurements of the filament orientation and the velocity gradients. In Section 4, we calculate and analyze the alignment (or lack thereof) between core angular momentum and its natal filament. We also compute the ratio between the rotational energy and gravitational energy. Sections 5 and 6 are the discussion and summary, respectively.

We mapped OMC-2 and OMC-3 in N₂H⁺ (1–0) with ALMA in 2014 November and 2015 August in Band 3. The frequency resolution of N₂H⁺ is 35 kHz, which corresponds to a velocity resolution of 0.11 km s⁻¹ at 93 GHz. We used the 12 m main array and the Atacama Compact Array (ACA) to mosaic OMC-3 with 11 pointings, and OMC-2 with 7 pointings. The baselines of the 12 m main array and the ACA observations are at a range of 13.6–340.0 m and 6.8–87.4 m, respectively.

The N₂H⁺ (1–0) images have an angular resolution of ∼3'', which allows us to investigate the alignment between the rotation axes (perpendicular to the direction of velocity gradient) of cores and their filaments in OMC-2 and OMC-3. This 3'' resolution corresponds to ∼0.06 pc, smaller than the characteristic scale of filaments and cores (André et al. 2010, 2014; Molinari et al. 2010; Arzoumanian et al. 2011; Chen & Ostriker 2015). This value is comparable to the Jeans scale ($\sqrt{15{k}_{B}T/(4\pi {Gm}\rho )}$) of ∼0.05 pc for gas of n ∼10⁶ cm⁻³ at 15 K, which are typical conditions for the dense filaments and core envelopes in Orion.

The single-dish data of N₂H⁺ (1–0) were taken from the Nobeyama Radio Observatory Star Formation Legacy Project (Nakamura et al. 2019). The Nobeyama N₂H⁺ observations were executed with the newly developed 100 GHz Band 4 beam dual-polarization receiver (FOREST) during 2017 January and March. The Nobeyama velocity resolution is 0.1 km s⁻¹. The system temperature and noise level are at a range of 150–200 K and 0.26–0.30 K, respectively.

We combined the N₂H⁺ data from ALMA and Nobeyama to recover missing fluxes in the interferometric data. The combination was performed using Common Astronomy Software Applications (CASA; McMullin et al. 2007). The integrated intensity of the combined N₂H⁺ (1–0) images and the overlaid cores, labeled 1–30, are shown in Figure 1(a).

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3.1. Orientations of the Filaments

3.2. Fitting Spectral Lines and Velocity Gradient

Though N₂H⁺ (1–0) has seven closely spaced hyperfine components, there is an isolated one (the shaded region in Figure 1(c)). For measuring the velocity gradient, we compared fitting all hyperfine components and fitting just the isolated one with a single Gaussian. No significant difference was found. Therefore, we relied on the single-Gaussian fit in all subsequent calculations.

To securely measure the velocity gradient, we require the spectral lines to have peak intensity greater than three times that of the rms noise and fitting errors in the cores. We then fit the velocity field as ${v}_{\mathrm{LSR}}={v}_{0}+a{\rm{\Delta }}\alpha +b{\rm{\Delta }}\delta$, and measure the velocity gradients following the method described in Goodman et al. (1993). Δα and Δδ are the offsets from the center position (5^h:35^m:210, −5°:00':00'') of the combined N₂H⁺ images in the R.A. and decl. in radians, respectively. a and b are the projections of the gradient per radian onto the α and δ axes, respectively. The magnitude of the velocity gradient is given by

$$\begin{eqnarray}&&{ \mathcal G }=| {\boldsymbol{\nabla }}{v}_{\mathrm{LSR}}| =\displaystyle \frac{{({a}^{2}+{b}^{2})}^{1/2}}{D},\end{eqnarray} \tag{ 1 }$$

where D denotes the distance to the cores. The gradient direction is given by

$$\begin{eqnarray}&&{\theta }_{G}=\tan \displaystyle \frac{a}{b}.\end{eqnarray} \tag{ 2 }$$

The fitted ${ \mathcal G }$ and θ_G are listed in Table 1. The black arrows (Figure 1(a)) with varying length illustrate the velocity gradients.

Table 1.  Dynamic Parameters of Orion Cores

Core	R.A.	Decl.	Mass	Massvir	Major	Minor	Major Axis Orientation	${ \mathcal G }$	θG	J/M	β	$| {\theta }_{f}-{\theta }_{G}|$
	(J2000)	(J2000)	(M⊙ )	(M⊙)	''	''	(deg E of N)	(km s−1 pc−1)	(deg E of N)	(pc km s−1)		(deg)
1	5:35:16.1	−5:00:00	12.4	6.6	16.0	13.0	3	2.2 ± 0.3	47 ± 6	1.2 × 10−3 ± 1.7 × 10−4	1.7 × 10−3 ± 3.3 × 10−5	81 ± 12
2	5:35:18.2	−5:00:21	25.9	5.2	19.0	14.0	12	1.6 ± 0.3	59 ± 8	1.9 × 10−3 ± 1.8 × 10−4	1.8 × 10−3 ± 1.5 × 10−5	38 ± 16
3	5:35:18.2	−5:01:47	3.1	3.0	21.0	18.0	52	10.3 ± 2.6	29 ± 9	9.5 × 10−3 ± 1.6 × 10−3	3.2 × 10−2 ± 8.7 × 10−4	63 ± 6
4	5:35:18.3	−5:13:38	0.3	0.4	13.0	8.0	35	9.1 ± 1.1	−86 ± 3	3.8 × 10−3 ± 2.2 × 10−4	7.8 × 10−2 ± 2.8 × 10−4	79 ± 12
5	5:35:18.9	−5:14:12	0.5	1.2	16.0	10.0	55	5.1 ± 0.8	35 ± 6	1.9 × 10−3 ± 2.8 × 10−4	1.1 × 10−2 ± 2.5 × 10−4	22 ± 3
6	5:35:19.6	−5:15:35	5.8	5.7	27.0	17.0	3	11.1 ± 0.3	139 ± 1	1.1 × 10−2 ± 2.7 × 10−4	1.8 × 10−2 ± 1.1 × 10−5	54 ± 19
7	5:35:20.7	−5:00:53	30.0	1.8	17.0	13.0	14	2.6 ± 0.2	−14 ± 3	1.3 × 10−3 ± 1.3 × 10−4	8.0 × 10−4 ± 8.0 × 10−6	19 ± 22
8	5:35:21.4	−5:14:58	4.0	2.1	24.0	14.0	26	5.6 ± 0.3	124 ± 2	3.4 × 10−3 ± 2.3 × 10−4	2.9 × 10−2 ± 1.3 × 10−4	69 ± 6
9	5:35:21.6	−5:10:39	1.6	1.4	19.0	12.0	47	4.7 ± 0.4	3 ± 3	1.7 × 10−3 ± 2.1 × 10−4	2.4 × 10−2 ± 3.6 × 10−4	6 ± 9
10	5:35:21.7	−5:13:12	9.2	6.8	18.0	18.0	19	16.2 ± 0.4	70 ± 1	8.3 × 10−3 ± 2.6 × 10−4	9.5 × 10−2 ± 9.4 × 10−5	55 ± 8
11	5:35:21.8	−5:14:22	5.7	4.0	20.0	19.0	55	10.2 ± 0.2	82 ± 1	8.5 × 10−3 ± 1.9 × 10−4	1.4 × 10−2 ± 7.6 × 10−5	69 ± 22
12	5:35:22.4	−5:01:14	15.6	1.2	13.0	10.0	1	3.1 ± 0.7	136 ± 9	6.0 × 10−4 ± 2.0 × 10−4	4.1 × 10−4 ± 4.6 × 10−5	11 ± 16
13	5:35:22.5	−5:10:14	2.5	5.4	17.0	13.0	37	5.9 ± 0.3	−15 ± 2	3.2 × 10−3 ± 5.8 × 10−5	1.1 × 10−2 ± 3.6 × 10−5	12 ± 10
14	5:35:22.9	−5:12:40	26.3	6.7	30.0	18.0	37	5.2 ± 0.4	121 ± 3	9.3 × 10−3 ± 3.4 × 10−4	2.6 × 10−2 ± 3.5 × 10−5	74 ± 21
15	5:35:23.4	−5:12:05	12.8	3.2	21.0	16.0	2	6.2 ± 0.6	61 ± 4	2.6 × 10−3 ± 3.1 × 10−4	5.8 × 10−3 ± 8.3 × 10−5	46 ± 5
16	5:35:23.5	−5:01:32	23.1	5.7	21.0	16.0	3	6.6 ± 0.4	−24 ± 2	4.0 × 10−3 ± 2.5 × 10−4	7.6 × 10−3 ± 2.9 × 10−5	9 ± 13
17	5:35:23.5	−5:07:34	7.6	3.0	15.0	13.0	39	10.0 ± 0.5	28 ± 2	4.6 × 10−3 ± 2.2 × 10−4	4.2 × 10−2 ± 9.2 × 10−5	42 ± 20
18	5:35:23.6	−5:07:11	6.1	3.4	16.0	13.0	32	16.6 ± 0.5	−55 ± 1	8.5 × 10−3 ± 2.3 × 10−4	1.7 × 10−2 ± 1.2 × 10−4	9 ± 4
19	5:35:24.5	−5:07:54	11.7	3.1	21.0	13.0	17	9.1 ± 0.5	70 ± 3	6.0 × 10−3 ± 3.2 × 10−4	3.5 × 10−2 ± 9.7 × 10−5	84 ± 2
20	5:35:24.5	−5:08:32	9.7	3.7	26.0	16.0	3	4.2 ± 0.2	80 ± 2	2.4 × 10−3 ± 1.9 × 10−4	5.7 × 10−3 ± 3.6 × 10−5	86 ± 7
21	5:35:25.5	−5:02:37	6.8	3.5	27.0	15.0	63	1.6 ± 0.2	−10 ± 4	1.7 × 10−3 ± 2.1 × 10−4	4.0 × 10−3 ± 5.9 × 10−5	23 ± 11
22	5:35:25.8	−5:05:51	13.0	4.3	38.0	22.0	7	2.0 ± 0.1	−118 ± 3	5.5 × 10−3 ± 2.1 × 10−4	1.5 × 10−2 ± 2.1 × 10−5	66 ± 6
23	5:35:26.1	−5:01:26	3.8	0.7	26.0	12.0	39	9.8 ± 1.0	−71 ± 5	8.8 × 10−3 ± 5.1 × 10−4	1.8 × 10−2 ± 7.0 × 10−4	37 ± 10
24	5:35:26.4	−5:08:30	3.8	5.9	16.0	14.0	19	4.4 ± 0.6	−107 ± 7	2.9 × 10−3 ± 2.8 × 10−4	3.2 × 10−2 ± 2.9 × 10−4	87 ± 7
25	5:35:26.5	−5:03:56	13.2	12.8	37.0	28.0	2	4.1 ± 0.1	−70 ± 1	6.8 × 10−3 ± 2.0 × 10−4	2.1 × 10−2 ± 1.9 × 10−5	55 ± 4
26	5:35:26.5	−5:10:11	33.5	8.3	31.0	22.0	10	10.8 ± 0.2	14 ± 1	1.6 × 10−2 ± 2.8 × 10−4	5.8 × 10−2 ± 1.8 × 10−5	17 ± 15
27	5:35:27.1	−5:11:39	2.3	4.5	23.0	14.0	38	11.9 ± 0.7	37 ± 2	2.5 × 10−3 ± 5.1 × 10−4	2.7 × 10−2 ± 1.2 × 10−5	22 ± 7
28	5:35:27.4	−5:05:11	20.1	3.2	32.0	23.0	16	1.3 ± 0.1	108 ± 3	2.5 × 10−3 ± 1.5 × 10−4	2.2 × 10−4 ± 8.1 × 10−6	79 ± 4
29	5:35:27.6	−5:09:35	12.9	12.8	23.0	17.0	1	16.7 ± 0.5	95 ± 1	9.9 × 10−3 ± 3.6 × 10−4	7.6 × 10−2 ± 1.0 × 10−4	41 ± 21
30	5:35:27.9	−5:10:25	6.4	3.3	11.0	10.0	16	12.8 ± 1.1	−41 ± 4	3.9 × 10−3 ± 2.4 × 10−4	5.0 × 10−3 ± 1.8 × 10−4	38 ± 8

Note. The values of Mass and Mass_vir are from Li et al. (2013a). Major and Minor axes are from Nutter & Ward-Thompson (2007). Major axis orientation is the major axis direction of cores.

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In this section, we calculate and analyze the distribution of the angles between filaments and velocity gradients, based on the fitted angular differences $| {\theta }_{f}-{\theta }_{G}|$. We also derive the specific angular momentum J/M and the ratio β between the rotational energy and gravitational energy, based on the measured velocity gradients.

4.1. The Distribution of $| {\theta }_{f}-{\theta }_{G}|$

To quantify the alignment between the velocity gradients of cores and their filaments, $| {\theta }_{f}-{\theta }_{G}|$ is derived as

$$\begin{eqnarray}&&| {\theta }_{f}-{\theta }_{G}| =\mathrm{MIN}\{| {\theta }_{f}-{\theta }_{G}| ,180-| {\theta }_{f}-{\theta }_{G}| \},\end{eqnarray} \tag{ 3 }$$

where θ_f and θ_G are measured clockwise from the east, and "MIN" refers to the minimum angular difference between θ_f and θ_G. The derived $| {\theta }_{f}-{\theta }_{G}|$ values are listed in Table 1 and plotted in Figure 2(a). We find that the gradient directions of seven cores (∼23%) are essentially parallel to their filament orientations (0° ± 10°; masked as the dusty blue strip in Figure 2(a)), while the gradient directions of eight cores (∼27%) are essentially perpendicular to their filaments (90° ± 10°; the gray strip in Figure 2(a)).

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To investigate the distribution of $| {\theta }_{f}-{\theta }_{G}|$, Monte Carlo simulations are performed in three-dimensional (3D) space. We generate two random unit vectors within a unit sphere in 3D, and measure the angle (θ_3D) between the two vectors. 10⁶ pairs of unit vectors are generated to produce 10⁶ angles of θ_3D, constrained to a range of 0°–90°. If θ_3D is larger than 90°, the 180°–θ_3D values are adopted (Equation (3)). For θ_3D in the range of 0°–20°, we define it as parallel, while 20°–70° is defined as random, and 70°–90° as perpendicular. Then we project the angles of θ_3D onto a two-dimensional (2D) space. Figure 2(b) plots the cumulative distribution function of our $| {\theta }_{f}-{\theta }_{G}|$ and the projected θ_3D. The details of the Monte Carlo simulations can be found in Appendix A of Stephens et al. (2017). The p-values of the Anderson–Darling (AD) test are used to indicate whether two distributions are consistent. p-values near 1 imply that the two distributions are likely consistent, while p-values near 0 imply that they are not consistent. For the distribution of our $| {\theta }_{f}-{\theta }_{G}|$ and the simulated random distribution, the AD test gives a p-value of 0.97, consistent with being random. This result indicates no correlation between the gradient direction of a core and its filament orientation. This is consistent with the scenario that the rotation axis of a core is independent of the orientation of the large-scale filament in which it resides.

4.2. J/M and β Calculations

The J/M and β calculations were performed under the assumption that cores have a power-law profile density ($\rho \propto {R}^{-1.6}$; Bonnor 1956). We used the following equations to calculate J/M and β:

$$\begin{eqnarray}&&J/M=0.059{ \mathcal G }{R}^{2},\end{eqnarray} \tag{ 4 }$$

and

$$\begin{eqnarray}&&\beta =\displaystyle \frac{0.34{{ \mathcal G }}^{2}{R}^{3}}{{GM}}.\end{eqnarray} \tag{ 5 }$$

M is the mass within a radius R. The calculated J/M and β are listed in Table 1, and plotted in Figures 3(a) and (b), respectively. We find that J/M increases with increasing R, following a power law $J/M\,\propto \,{R}^{1.52\pm 0.14}$. The slope is consistent with the index value of 1.6 found by Goodman et al. (1993) for a sample of dark clouds. β ranges from 0.00041 to 0.094, with no significant correlation with R. As investigated in previous studies (Goldsmith & Arquilla 1985; Goodman et al. 1993; Caselli et al. 2002; Curtis & Richer 2011; Tatematsu et al. 2016), these small values of β indicate that rotation alone is not enough to support the core from collapsing due to gravity.

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Magnetic fields play an important role during the collapse and fragmentation of massive molecular clumps, as well as the formation of dense cores. Zhang et al. (2014) found that the magnetic fields at dense core scales are either aligned with or perpendicular to the parsec-scale magnetic fields. One may reasonably expect such bimodality for the angles between core rotation and filament elongation. Anathpindika & Whitworth (2008) reported that the rotational axes of prestellar cores are perpendicular to their filaments. Stephens et al. (2017) showed that a mix of parallel and perpendicular angles exist between the rotation axes of protostellar cores and their filaments. In our sample, however, we find no correlation between the rotational direction and the natal filament's elongation. The distribution is found to be consistent with being random. Similar results were also seen in earlier observations (e.g., Heyer 1988; Davis et al. 2009; Lee et al. 2016; Offner et al. 2016; Punanova et al. 2018).

Considerable differences exist among the analyses mentioned above, in terms of sample sizes, definition of filament and cores, and the fitting of rotation and elongation. Anathpindika & Whitworth (2008) and Stephens et al. (2017) regarded the outflow orientations as the rotation axes of prestellar/protostellar cores, while the directions of velocity gradients trace the rotation axes of cores in Punanova et al. (2018), as well as in our sample. Stephens et al. (2017) identified filaments using FILFINDER (Koch & Rosolowsky 2015) and SExtractor (Bertin & Arnouts 1996), while some studies (Arzoumanian et al. 2018; Bresnahan et al. 2018; Gong et al. 2018; Tanimura et al. 2019) employed DisPerSE (Sousbie 2011) to trace filaments. Although our experience seems to suggest that these technical differences are unlikely to explain the diversity in the relative orientations between rotation and filament, high spectral and spatial images of a much larger sample are needed to tackle the issue systematically.

We mapped N₂H⁺ (1–0) in OMC-2 and OMC-3 with both ALMA and Nobeyama at 93 GHz. The combined single-dish and interferometric data offer a rare opportunity to study the relative orientation between the rotational axes of the massive quiescent cores and the elongation of their natal filaments, down to the thermal Jeans scale of dense gas in a massive star-forming region. Our results are summarized as follows.

1. The angle ($| {\theta }_{f}-{\theta }_{G}|$) between the orientation of the filaments (θ_f) and the direction of core velocity gradient θ_f was found to be random, based on a Monte Carlo simulation. The core rotation seems to have disengaged itself from its natal filament and, by association, the large-scale magnetic field.

2. The measured velocity gradients of 30 cores range from 1.3 to 16.7 km s⁻¹ pc⁻¹. The measured specific angular momentum (J/M) ranges from 0.0012 to 0.016 pc km s⁻¹. A power-law scaling was found between the specific angular momentum and the core size as $J/M\,\propto \,{R}^{1.52\pm 0.14}$.

3. The ratio β between the rotational and gravitational energy ranges from 0.00041 to 0.094, indicating that rotation cannot stop gravitational collapse in these dense cores.

This work is supported by the National Natural Science Foundation of China grant No. 11988101, No. 11725313, No. 11721303 and the International Partnership Program of Chinese Academy of Sciences grant No. 114A11KYSB20160008. This Letter makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00662.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.