ALMA RESOLVES THE SPIRALING ACCRETION FLOW IN THE LUMINOUS OB CLUSTER-FORMING REGION G33.92+0.11

The ALMA 12 m Array (i.e., 12 m dish size) observations were carried out on 2014 May 4, with ∼36 good antennas. The pointing and phase referencing center is R.A. (J2000) = 18^h52^m50272, and decl. (J2000) = 00°55'29 604. We used two 234.4 MHz wide spectral windows (channel spacing 61 kHz, ∼0.085 km s⁻¹), and two 1875.0 MHz wide spectral windows (channel spacing 488 kHz, ∼0.65 km s⁻¹), which tracked the velocity of v_lsr ∼107.6 km s⁻¹ of our target source.

The central frequency of these spectral windows are 231.220690 GHz (¹³CS 5-4), 231.900928 GHz (H30α), 220.679320 GHz (CH₃CN J = 12–11, K = 4), and 217.104980 GHz (SiO 5-4), respectively. The UV sampling range of these observations is 13–430 kλ. The overall on-source time was 43.7 minutes, and the T_sys was in the range of 60–150 K. We observed Titan, J1851 + 0035, and J1751+0939 for absolute flux, gain, and passband calibrations, respectively.

The ALMA Compact Array (ACA; 7 m dishes) observations were carried out on 2014 May 3 and 2014 May 4, with ∼10 available antennas.¹¹ The pointing and phase-referencing center and the spectral setup are the same with the 12 m array observations. These ACA observations cover a UV sampling range of 7.2–47 kλ. The overall on-source time is 24.4 minutes on May 3, and 73.2 minutes on May 4. The T_sys was in the range of 60–150 K. We observed Titan/Neptune, J1851 + 0035, and J1733–1304/J1924–2914 for absolute flux, gain, and passband calibrations, respectively.

The data were calibrated, phase self-calibrated (2 iterations), and jointly imaged using the CASA package (McMullin et al. 2007), version 4.2.0 (release r28322). In the end, we did not implement the phase self-calibration solution to the 12 m Array data because there was no obvious improvement in image quality. We did not self-calibrate the visibility amplitude because of a significant missing flux (≳24%) as compared with previous SMA observations in the same region. The continuum image was constructed by jointly imaging all spectral line-free channels using the multi-frequency imaging algorithm. The Briggs Robust = 0 weighted 12 m+ACA continuum image has a synthesized beam of ${{\theta }_{{\rm maj}}}\times {{\theta }_{{\rm min} }}$ = 0 67 × 0 47 (P.A. = 81°) and a dynamic range-limited rms noise level of 0.2 mJy beam⁻¹. We have corrected the primary beam attenuation, and therefore the images are noisier in the outer region.

The selected ¹³CS 5-4 line was simultaneously covered in a 234.4 MHz spectral window and a 1875.0 MHz spectral window. Due to a correlator failure, we lost the ¹³CS 5-4 data in the 234.4 MHz spectral window. The CH₃CN J = 12–11 K-ladders were observed in a 234.4 MHz spectral window; the DCN 3-2 and SiO 5-4 lines were observed in 1875.0 MHz spectral windows. We performed Briggs Robust = 2 weighted 12 m+ACA spectral line imaging to enhance the sensitivity to diffuse emission, which yielded a synthesized beam of ${{\theta }_{{\rm maj}}}\times {{\theta }_{{\rm min} }}$ = 0 78 × 0 67. The standard one-iteration spectral hanning smoothing procedure, which can better suppress the passband ripple, will yield ∼0.17 km s⁻¹ velocity resolution and ∼1.3 km s⁻¹ velocity resolutions for the 234.4 MHz and the 1875.0 MHz spectral windows, respectively. However, we imaged the ¹³CS 5-4 and the DCN 3-2 lines with ∼0.68 km s⁻¹ channel spacing, intending to better present the kinematics information. The CH₃CN J = 12–11 K-ladders were imaged with the 0.17 km s⁻¹ velocity resolution. We achieved an rms noise level of 1.6 mJy beam⁻¹ (70 mK) and 3.5 mJy beam⁻¹ (170 mK) in the ∼0.65 and 0.17 km s⁻¹ velocity channels, respectively. There is no zero-spacing information available for our 1.3 mm continuum map. We have compared our ALMA observations with the existing SMA observations of the 1.3 mm continuum emission, the CSO SHARC2 bolometric single-dish observations of the 0.35 mm continuum emission (see below), and the VLA observations of the NH₃ lines (Liu et al. 2012b). We confirmed that the resolved morphology by ALMA is not significantly biased by the missing flux on the ≲30'' scale. The ALMA 12 m+ACA Array observations of the dense molecular gas tracers are less biased by missing zero spacing because the emission regions are less extended in the individual velocity channels. Presentation of the ALMA images will be limited to the 15% power of the primary beam width, if not specifically mentioned.

For this study, we made use of a 350 μm map of G33.92+0.11 obtained by the Herschel Space Telescope. We used the Level 3 (science grade) processed mosaics of the SPIRE-PACS parallel observations of a much larger map (observation ID 1342207031) of the W49 complex observed in 2010 October. No significant glitches are visible in the region of interest. Level 3 maps are produced from a combination of all available contiguous observations of a program. The Level 3 mosaics have optimal WCS solutions and analysis quality grade with zeropoint calibrations estimated using Planck Observatory maps and archived directly in units of surface brightness (MJy/sr). We applied, however, a simple astrometric check by re-creating the mosaic with Montage ¹² and comparing positions of compact features against a WISE Observatory 3.6 micron map.

High angular resolution continuum observations at 0.35 mm were carried out using the SHARC2 bolometer array, installed on the CSO. The array consists of 12 × 32 pixels (approximately 85% of these pixels work well). The simultaneous field of view (FOV) provided by this array is 259 × 097, and the diffraction-limited beam size is ∼8 8.

The data were acquired on 2014 March 24 (τ_{225 > GHz} ∼0.06). The telescope pointing and focusing were checked every 1.5–2.5 hr. Mars was observed for absolute flux calibration. We used the standard 10' × 10' on-the-fly (OTF) box-scanning pattern, centered on R.A. (J2000)=18^h52^m50272 and decl. (J2000) = 00°55'29 604. The total on-source time was 30 minutes. Data calibration was performed using the CRUSH software package (Kovács 2008). In addition, we used the MIRIAD (Sault et al. 1995) task immerge to perform the weighted sum of the SHARC2 0.35 mm image with the Herschel SPIRE image (see Section 2.2) at a similar wavelength in the fourier domain. This minimizes the missing flux in the ground-based single-dish bolometric observations due to the sky subtraction, which typically causes defects on angular scales larger than the simultaneous FOV (or the angular throw in the nodding observations). The final map was smoothed to an angular resolution of 9 6, and the rms noise level achieved was ∼60 mJy beam⁻¹. In addition, we aligned the final 0.35 mm image with the Herschel, ALMA, and SMA images.

The gas mass is usually estimated with dust thermal continuum emission. While the continuum emission at the 0.35 mm wavelength is dominated by dust, the continuum emission at 1.3 mm can have contributions from both the dust thermal emission and the free–free emission from the ionized gas. To separate these two components, we first estimate the free–free emission based on the observations of the hydrogen recombination H30α line (see Section 3.2). Assuming that the free–free continuum emission at 1.3 mm is optically thin, the peak hydrogen radio recombination line to the free–free continuum flux ratio (T_L/T_C) is given by

$$\begin{eqnarray*}&&\sim 7.0\times {{10}^{3}}{{\left( \frac{\delta v}{{\rm km}\;{{{\rm s}}^{-1}}} \right)}^{-1}}{{\left( \frac{\nu }{{\rm GHz}} \right)}^{1.1}}{{\left( \frac{{{T}_{e}}}{{\rm K}} \right)}^{-1.15}}{{\left[ 1+\frac{N({\rm H}{{{\rm e}}^{+}})}{N({{{\rm H}}^{+}})} \right]}^{-1}},\end{eqnarray*}$$

where δv is the FWHM of the hydrogen radio recombination line, ν is the observing frequency, and T_e is the electron temperature. We adopt the nominal value of T_e ∼8000 K for the electron temperature, and estimate T_L and δ v based on the observed peak brightness and the second moment in the H30α line image cube. The assumed T_e is an upper limit since the continuum is over subtracted on a small scale when using higher values for T_e. The resultant 1.3 mm free–free continuum emission model has a peak flux density of 16.7 mJy beam⁻¹. This free–free emission model is then subtracted from the 1.3 mm continuum image to yield the 1.3 mm dust continuum emission image. We smoothed the 1.3 mm continuum image to the angular resolution of the H30α line image before subtracting the free–free emission model.

Figures 1 and 2 show the high-resolution 0.35 mm continuum image and the 1.3 mm dust continuum image. Both continuum images trace the overall gas column density, but the latter is optically thinner (i.e., a brightness temperature of ${{T}_{{\rm B}}}\ll {{T}_{{\rm dust}}}$), thus can penetrate inside the densest regions. The 0.35 mm continuum image peaks at the location of the ∼10⁴ ${{M}_{\odot }}$ massive molecular gas clump G33.92+0.11 A (Watt & Mundy 1999; Liu et al. 2012b). The peak 0.35 mm flux density is ∼73 Jy beam⁻¹ (∼1.3 K, but can be beam diluted). The 0.35 mm image resolves the massive companion G33.92+0.11 B northeast of G33.92+0.11 A. The dominant large-scale features are the projected ∼5 pc scale elongated structures, connecting from the northeast and southwest (hereafter G33-north and G33-south, respectively) to G33.92+0.11 A and B. There might be gas filaments connecting G33.92 + 0.1 A from the west; however, the individual components are not as clearly separated because of confusion with foreground/background gas structures.

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The 0 6 resolution 1.3 mm dust continuum image (Figure 2) resolved abundant structures within the inner parsec-scale radius. The most prominent ones are the two 100–300 ${{M}_{\odot }}$ massive cores A1 and A2 located at the center and the five molecular arms roughly following the paths of A1$\to$A2$\to$A3 (arm-c1), mini-arm-A2w$\to$A4$\to$A5 (arm-c2; see Figures 4, 6), A4$\to$A6$\to$A11$\to$A12$\to$A13 (arm-c3), A1$\to$A9 (arm-c4), and A1$\to$A12 (arm-c5). The nomenclature here follows Watt & Mundy (1999) and Liu et al. (2012b), if the referred structures have been detected. Core A1 is harboring OB stars, which create a UC H ii region. We refer to Liu et al. (2012b) for a preliminary study of the gas mass in the <0.1 pc scale structures. The molecular arm-c4 appears spiraling out and connected with the parsec-scale molecular arm-S1 and arm-S2. The molecular arm-c2 may be connected with the previously observed molecular arm-N in the north (Liu et al. 2012b), which cannot be imaged well with our ALMA 12 m Array data due to the smaller primary beam size.

Following the convention in Liu et al. (2012b), we refer to the OB stars that are embedded in either core A1 or core A2 as the central OB cluster hereafter. For the rest of localized intermediate- or high-mass (proto)stars and their parent molecular cores in G33.92+0.11 A, we refer to them as the satellite high-mass stars and satellite cores. The previously identified core A4 is now resolved into two components, A4n and A4s. In addition to the aforementioned gas structures, we also resolved the ∼0.1 pc scale gas mini-arms connecting with the cores. The gas mini-arms connecting A2 from the west (mini-arm-A2w) have been resolved in the previous SMA observations (Liu et al. 2012b). There are mini-arms connecting A1 from the north and northwest (mini-arm-A1n, mini-arm-A1nw), which can only be marginally seen in Figure 1 owing to blending with other structures. We will address these gas mini-arms based on the spectral line results in greater detail (Section 3.2).

The high dynamic range of the ALMA observations confirms that the two highest-mass cores, A1 and A2, in the center are not round. Core A1 appears slightly elongated in the southeast–northwest direction. This may imply that it is an inclined flattened structure or that it possesses eccentric gas orbits. However, we cannot rule out the possibility that there is internal fragmentation in core A1 or that there are more than two cores blended in the projected LOS. We cannot resolve the internal structure of all detected cores. Some, or all, of the individual cores may eventually form a cluster of stars. The complex morphology resolved by ALMA has suggested that the accretion flows in the massive molecular clump are highly dynamical/chaotic. In fact, the resolved arm-like features may resemble the scaled-down version of eccentric gas arms/arcs connected with the 2–4 pc scale Galactic circumnuclear disk/ring (Liu et al. 2012d).

We present the full spectral window spectra generated from the central 10'' field of our ALMA 12 m+ACA observations in Figure 3. The identified line species are listed in Table 1. In this work, we focus on the discussion of the CH₃CN J = 12–11 K-ladders, the ¹³CS 5-4 line, and the DCN 3-2 line. The velocity-integrated intensity maps (i.e., moment 0) of these lines are shown in Figure 4. Figure 5 provides the side-by-side comparison of the blown-up line images and the 1.3 mm dust continuum image in the central parsec-scale area. The intensity-weighted velocity maps (i.e., moment 1) and the intensity-weighted velocity dispersion maps (i.e., moment 2) of these lines are presented in Figures 6 and 7. We present the velocity channel maps of the DCN 3-2 line and of the ¹³CS 5-4 line in Figures 8 and 9. The position–velocity (PV) diagrams of these two lines, generated from the selected slices (see Figure 7), are shown in Figure 10. We present the velocity channel maps of the lowest and the highest transitions of the detected CH₃CN J = 12–11 K-ladders in Figures 11 and 12. The PV diagrams for these CH₃CN transitions are provided in Figures 13 and 14. We introduce the observed intensity distribution and velocity profiles separately in Sections 3.2.1 and 3.2.2.

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Table 1.  Molecular Lines Observed within our Cycle-1 ALMA Data

Species	Transition	Frequency	Eu
		(MHz)	(K)
c-C3H2	3$_{3,0}\to$22,1	216278.76	19.5
CCD	N = 3–2, J = 7/2–5/2, F = 9/2–7/2	216372.83	20.8
	N = 3–2, J = 7/2–5/2, F = 7/2–5/2a	216373.32	20.8
	N = 3–2, J = 5/2–3/2, F = 7/2–5/2b	216428.32	20.8
	N = 3–2, J = 5/2–3/2, F = 3/2–1/2	216428.76	20.8
CH3CHO	11$_{1,10}\to$101,9 E	216581.94	64.9
	11$_{1,10}\to$101,9 A	216630.22	64.8
HDCS	7$_{0,7}\to$60,6	216662.43	41.6
	7$_{2,6}\to$62,5	216931.37	77.6
	7$_{2,5}\to$62,4	217263.69	77.6
H2S	2$_{2,0}\to$21,1	216710.44	84.0
CH3OH	5$_{1,4}\to$42,2	216945.60	55.9
SiO	5$\to$4	217104.98	31.3
DCN	3$\to$2	217238.63	20.9
13CN	N = 2–1, J = 3/2–1/2, F1 = 1–1, F = 2–1	217072.80	15.7
	N = 2–1, J = 3/2–1/2, F1 = 1–1, F = 2–2	217074.24	15.7
	N = 2–1, J = 3/2–1/2, F1 = 1–0, F = 0–1	217264.64	15.7
	N = 2–1, J = 3/2–1/2, F1 = 1–0, F = 1–1	217277.68	15.7
	N = 2–1, J = 5/2–3/2, F1 = 2–1, F = 2–2	217286.80	15.7
	N = 2–1, J = 5/2–3/2, F1 = 2–1, F = 1–1	217290.82	15.7
	N = 2–1, J = 5/2–3/2, F1 = 2–1, F = 1–0	217296.60	15.7
	N = 2–1, J = 3/2–3/2, F1 = 2–2, F = 2–2	217298.94	15.7
	N = 2–1, J = 5/2–3/2, F1 = 2–1, F = 2–1	217301.18	15.7
	N = 2–1, J = 5/2–3/2, F1 = 2–1, F = 3–2	217303.19	15.7
	N = 2–1, J = 3/2–1/2, F1 = 1–0, F = 2–1	217304.93	15.7
	N = 2–1, J = 3/2–3/2, F1 = 2–2, F = 3–3	217306.12	15.7
	N = 2–1, J = 5/2–3/2, F1 = 2–1, F = 3–2	217428.56	15.7
	N = 2–1, J = 5/2–3/2, F1 = 2–1, F = 2–1	217436.35	15.7
	N = 2–1, J = 5/2–3/2, F1 = 2–1, F = 2–2	217437.70	15.7
	N = 2–1, J = 3/2–1/2, F1 = 2–1, F = 2–2	217437.82	15.7
	N = 2–1, J = 3/2–1/2, F1 = 2–1, F = 1–1	217443.72	15.7
	N = 2–1, J = 5/2–3/2, F1 = 3–2, F = 4–3	217467.15	15.7
	N = 2–1, J = 5/2–3/2, F1 = 3–2, F = 2–1	217469.15	15.7
	N = 2–1, J = 5/2–3/2, F1 = 3–2, F = 2–2	217480.56	15.7
	N = 2–1, J = 5/2–3/2, F1 = 3–2, F = 3–3	217483.61	15.7
t-C2H5OH	5$_{3,3}\to$42,2	217803.69	23.9
c-HCCCH	6$_{1,6}\to$50,5c	217822.15	38.6
	5$_{1,4}\to$42,3	217940.05	35.4
CH3CN	12$\to$11 K = 4	220679.29	183.1
	12$\to$11 K = 3	220709.02	133.2
	12$\to$11 K = 2	220730.26	97.4
	12$\to$11 K = 1	220743.01	76.0
	12$\to$11 K = 0	220747.26	68.9
OCS	19$\to$18	231060.99	110.9
13CS	5$\to$4	231220.69	33.3
Hα	n = 30	231900.93	0.0
Heα	n = 30	231995.43	0.0
CH3OCH3	13$_{0,13}\to$121,12 EEd	231987.82	80.9
H2C34S	7$_{1,7}\to$61,6	232754.71	57.9

^aBlended with transition CCD N = 3–2, J = 7/2–5/2, F = 5/2–3/2. ^bBlended with transition CCD N = 3–2, J = 5/2–3/2, F = 5/2–3/2. ^cBlended with transition c-HCCCH 6$_{0,6}\to$5_1,5. ^dThe CH₃OCH₃ 13$_{0,13}\to$12_1,12 transitions from the AA, AE, and EA species are blended with this line (frequencies are within ± 0.2 MHz).

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3.2.1. Distribution

3.2.2. Velocity Profile

The gas mass ${{M}_{{{{\rm H}}_{2}}}}$, as well as column density in the extended structures, can be approximated by the optically thin dust emission formula

$$\begin{eqnarray}&&{{M}_{{{{\rm H}}_{2}}}}=\frac{2{{\lambda }^{3}}Ra\rho {{D}^{2}}}{3hcQ(\lambda )J(\lambda ,{{T}_{d}})}S(\lambda ),\end{eqnarray} \tag{ 1 }$$

where R is the gas-to-dust mass ratio, a is the mean grain radius, ρ is the mean grain density, D is the distance to the target, Q(λ) ∝ λ^−β is the grain emissivity, T_d is the dust temperature, and S(λ) is the flux of the dust emission at the given wavelength, $J(\lambda ,{{T}_{d}})=1/[{\rm exp} (hc/\lambda {{k}_{{\rm B}}}{{T}_{d}})-1]$ (Hildebrand 1983). c, h, and k_B are the light speed, the Planck constant, and the Boltzmann constant, respectively. Following Lis et al. (1998), we adopt the standard values R = 100, a = 0.1 μm, ρ = 3 g cm⁻³, and Q(λ = 350 μm) = 1 × 10⁻⁴. Based on the previous extensive NH₃ survey toward the high-mass star-forming regions (Lu et al. 2014) and our previous NH₃ rotational temperature measurements of this region (Liu et al. 2012b), we adopt a constant T_d = 20 K on the large scale and T_d ∼ 30 K in the inner 0.3 pc (∼9'') radius of G33.92+0.11 A.

The gas masses of G33-north and G33-south in the polygon regions in Figure 1 are 9700 and 5300 ${{M}_{\odot }}$, respectively. The gas column densities in these two regions are in the range of ${{N}_{{\rm HI}+{{{\rm H}}_{2}}}}\sim 2.5$–7.1 × 10²² cm⁻², with a mean of ∼4.4 × 10²² cm⁻². Given the assumed gas temperature and column density and assuming that the width of the filaments are ∼1–2 pc, the corresponding thermal Jeans length and Jeans mass are 0.20–0.47 pc and 7.2–17 ${{M}_{\odot }}$, respectively; the local free-fall collapsing timescale t_ff for the filaments is 0.42–1.0 Myr. With the resolution of SHARC2/CSO (∼0.34 pc) and sensitivity (3σ mass sensitivity ∼10.5 ${{M}_{\odot }}$; see Section 2.3), we did not convincingly resolve the regularly spaced massive gas clumps in G33-north and G33-south (e.g., as seen in the more evolved OB cluster-forming region G10.6-0.4; see Liu et al. 2012a). Nevertheless, there are a few localized overintensities at Jeans mass scales. The gas structures west of G33.92+0.11 A are embedded within the localized gas clumps but are difficult to identify with the limited angular resolution of the SHARC2/CSO image. The enclosed gas mass in the central ∼5 pc radius is ∼9.3$_{-3.6}^{+2.7}$ × 10⁴ ${{M}_{\odot }}$, where the error incorporates the uncertainties in dust temperature/opacity, as well as the foreground/background confusion. The global free-fall collapsing timescale for the region inside a 5 pc radius is ∼1.1 Myr. We note that the detected gas mass within the central 5 pc radius in G33.92+0.11 is comparable to that in the Galactic mini-starburst region W49N (Galván-Madrid et al. 2013). Therefore, we think G33.92+0.11 has the potential of forming a very luminous OB cluster in the future. We also note that the central ∼0.3 pc radius massive clump G33.92+0.11 A is comparably massive (≳3500 ${{M}_{\odot }}$) to the ∼5 pc scale filaments G33-north/south, which indicates a high gas concentration in the cloud center.

Finally, we summarize this section with two resolved features in common, from large to small scales. First, we detect the brightest sources at the center of the large-scale 350 μm map and of the high angular resolution ALMA 1.3 mm image (Figures 1, 2). Within the 5 pc radius, the 350 μm image shows that ∼3.8% of the gas mass is concentrated within 0.4% of the projected area (i.e., G33.92+0.11 A). Within source A inside a 0.3 pc radius, ALMA resolved that ∼5.7% of the gas mass is concentrated to 0.7% of the projected area (i.e., core A1). Such a (self-)similarity in the fraction of gas mass concentrated in the projected area may infer an anchored gravitational collapse from large to small scales. Although these massive gas structures (i.e., G33.92+0.11 A and core A1) do not dominate the gas mass in the referred areas, locally they can be viewed as the dominant compact gravitational source. Both G33.92+0.11 A and core A1 have companions with comparably lower masses. Second, we resolved the parsec-scale molecular gas arms where the ∼5 pc scale gas streams converge to the massive molecular clump. We also resolved the molecular gas mini-arms, which connect the <0.1 pc dense molecular core with the parsec-scale molecular arms. The length of the observed molecular gas arms and mini-arms are ∼2–3 times the size of G33.92+0.11 A and core A1, respectively.

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To examine whether the observed satellite cores in molecular arms can collapse to form high-mass stars, we derive one-dimensional virial velocity dispersions

$$\begin{eqnarray}&&{{v}_{{\rm vir}}}=\sqrt{\frac{\alpha MG}{5{{R}_{{\rm eff}}}}}\end{eqnarray} \tag{ 2 }$$

for cores A3, A4n, A4s, A5–6, and A9–13 based on the formulation in Williams et al. (1994) and compare with the LOS velocity dispersions (σ_v) traced by DCN 3-2. In Equation (2), parameter α is the geometric factor equal to unity for a uniform density profile and 5/3 for an inverse square profile, M is the gas mass, G is the gravitational constant, and R_eff is the effective radius of the dense clump. We suggest that the DCN 3-2 line may be a particularly good dense gas tracer for this purpose since DCN can survive 10^4–5 yr after the onset of (proto)stellar feedback (Rodgers & Millar 1996). To measure core masses and sizes, we fit two-dimensional Gaussians to the 1.3 mm dust continuum images (Figure 2) using the CASA task imfit and then estimate gas mass based on Equation (1). We assume the dust temperature T_d ∼ 30 K and adopt β = 1. To take into account the effects of ambient gas contamination and the missing flux, we perform Gaussian fittings for both the ALMA 12 m+ACA Array 1.3 mm dust continuum image and the dust continuum image generated from the 12 m Array data alone. For each of the cores, we extract the DCN line spectrum in the area defined by the two-dimensional Gaussian fitting and then perform one-dimensional Gaussian fitting to the extracted spectrum to measure σ_v. The results are presented in Figure 16.

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We found that the derived v_vir is in general larger than σ_v. This may imply that (1) the core masses are significantly overestimated by a factor of ∼2.5 or more or (2) the gas motions do not provide sufficient support against the local gravitational collapse, although we cannot rule out the case where (3) all cores are rotationally flattened and the dominant gas motions are perpendicular to the line of sight. Case (1) may be true if large dust grains are present in the satellite cores, such that locally the value of β is close to 0. Another option is that the actual gas-to-dust mass ratio is a few times lower than the assumed value (100). Our data do not allow discriminating these cases. The values of β in satellite cores can be measured by performing dust continuum observations at other wavelengths (e.g., 0.85 mm or 3 mm). Case (3) needs to be tested by future higher angular resolution observations. We tentatively favor case (2), or a combination of cases (2) and (3) for some cores, since there were localized high-velocity CO 2-1 outflow features detected in previous observations (Liu et al. 2012b). In any case, the evidence makes it unlikely that the supersonic gas motions in the satellite cores are much faster than the actual v_vir, such that these cores are not stable.

The observed gas motions in G33.92+0.11 A, in particular, in arm-c1, arm-c2, arm-c4, and arm-c5, may be explained by rotational and free-fall motions, accelerated by the enclosed matter distribution. For example, we consider that the enclosed mass M(r) within radius r can be approximated by $M(r)={{M}_{0}}\cdot {{r}^{n}}$. Surrounding the ∼0.3 pc scale flattened rotating massive molecular clump, the value of n may be in between 1 and 3 (cf. the solution for a collapsing slowly rotating isothermal cloud given by Terebey et al. 1984). The gas velocity can be estimated by ${{v}_{{\rm free}-{\rm fall}}}\sim \sqrt{2G{{M}_{0}}}\cdot {{r}^{(n-1)/2}}$. Based on the observed increasing velocity offset at larger radii, the value of n may be approximately in between 2 and 3, which is expected in the region where the large-scale gas streams are converging to the flattened rotating structure. PV "cut c4a" does not show a clear velocity gradient in the angular offsets of −3''–0 5 (Figure 10), which is likely because the orientation of this PV cut is perpendicular to this velocity gradient of the rotating clump (cf. Figures 6 and 7). At a larger angular offset, "cut c4a" detects the exterior redshifted infall (more below). We note that this analysis is preliminary because we do not know exactly how the inclination of the individual molecular arms vary with position. Defining the matter distribution as M(r) on the <0.3 pc scale and deprojecting the observed LOS velocity are not possible for this reason. Due to the complex morphology, the matter distribution may also be poorly approximated by M(r). The center of mass of G33.92+0.11 A is likely located in between cores A1 and A2 but cannot be precisely determined at this moment. In addition, our observations do not allow us to separate the rotational motion from infall. Nevertheless, we qualitatively think our interpretation is physically plausible and has to be considered in other case studies.

On smaller scales, the observed projected velocity surrounding core A1 is $\sim \pm$1.00 km s⁻¹ relative to its systemic velocity v_lsr = 108.47 km s⁻¹ at the 6.8 ± 0.8 × 10⁻³ pc projected radius. Assuming that the embedded 100–300 ${{M}_{\odot }}$ gas mass (Liu et al. 2012b) in core A1 is dominant and assuming that the surrounding gas in the <0.01 pc radius follows the nearly circular orbits, the rotating speed ${{v}_{R}}(0.0068\;{\rm pc})$ can be estimated by ${{v}_{R}}={{\left[ (G\cdot (100-300{{M}_{\odot }}))/(0.0068{\rm pc}) \right]}^{1/2}}=7.7$–13 km s⁻¹. We therefore can interpret the southeast–northwest velocity gradient around A1 by the rotation inclined 4°−7° along the position angle of ∼45°. The CH₃CN velocity field surrounding core A2 is more complicated, in particular, in mini-arm-Aw. Nevertheless, we found that at the 0.035–0.05 pc position offset from core A2, the observed ∼0.4 km s⁻¹ velocity offset of mini-arm-A2w from the centroid velocity of core A2 can be consistently interpreted by the gravitationally accelerated motion by core A2 (and A1), projected to a similar inclination angle to that of core A1.

If the velocity gradient on the >0.3 pc scale is consistent with that on the <0.3 pc as well as the southeast–northwest velocity gradient within core A1, we could expect the entire molecular arm-S1 and arm-S2 (i.e., farther south than decl. (J2000)∼00°55'23'') to be blueshifted. We present PV "cut c4b," "cut c4c," and "cut c4d" in Figure 10, which trace the gas motions in molecular arm-S1 and arm-S2. We instead found that the LOS motion of these two arms converge smoothly from the redshifted velocity in the north to the blueshifted velocity in the south and southeast. These results are consistent with the previous 0.6 km s⁻¹ velocity resolution observations of the NH₃ (1,1) hyperfine inversion lines (Liu et al. 2012b). In the marginally spatially resolved system, this approximately linear velocity gradient is most commonly interpreted by the rotational motion of a toroid. In the particular case of G33.92+0.11, we are motivated by the resolved matter distribution and the morphology of the arms to alternatively consider that the parsec-scale velocity gradient in the north–south direction can be the free-fall dominant motion following the eccentric orbits. The resolved velocity gradient from molecular arm-c4 to arms-S1 and arms-S2 is qualitatively similar to the numerical simulation for the infalling filament connected with the low-mass cluster-forming region in Bonnell et al. (2008). The southern molecular arm may be gravitationally accelerated by the massive core A1, and therefore turns into the redshifted velocity in regions close to A1. From both the average velocity map and the PV diagrams of ¹³CS 5-4 (Figures 6, 10), we observe the smaller velocity scale outside the 0.3 pc radius of G33.92+0.11 A than the velocity of the gas arms interior to it. This can be understood if the observed gas structures within the 0.3 pc radius of G33.92+0.11 A dominate the enclosed mass within the ∼1 pc radius.

We summarize our working hypothesis to interpret the resolved structures and kinematics in the schematics model in Figure 15. We think the molecular gas streams are converging from a larger than 1 pc radius toward the center of G33.92+0.11 A due to gravitational infall acceleration. The gas motions become dominated by the rotational motion on the ∼0.3 pc scale. Within the 0.3 pc scale radius, we detect the gas mini-arms connecting to the central cores A1 and A2, which are rotating about each other. These gas mini-arms show ≲1 km s⁻¹ velocity offsets from the ambient dense gas, which can be explained by the gravitational acceleration and the tidal interaction of the dense core. However, we cannot yet rule out that the motion of the gas mini-arms is powered by the expansional motion of the ionized gas or the molecular wind/jet, which needs to be examined in higher-resolution observations.

The magnetic field in rapidly collapsing molecular clumps is presumably weak. The initial turbulence is rapidly dissipated in such dense regions, although can be replenished by the protostellar turbulence in a later evolutionary stage (Wang et al. 2010). The rotational motion can play an important role in supporting collapsing molecular clumps, which permits localized fragmentation and star formation in longer timescales. It is not yet clear how to transport angular momentum outward efficiently on a spatial scale of a fraction of a parsec. The excess of angular momentum may leave its footprint on the morphology of the molecular clumps and may therefore bias the upper end of the core/stellar mass function.

Observationally, we have found clumpy rotating toroids and spiral-like interacting gas features toward the low-mass cluster-forming region L1287 (C. Juaréz et al. 2015, in preparation), the OB cluster-forming molecular clumps (e.g., G10.6-0.4: Liu et al. 2010; G35: Qiu et al. 2013; Sánchez-Monge et al. 2013; NGC 6334 V: Juaréz et al. 2015, in preparation), and the Galactic mini-starburst region W49N (Welch et al. 1987; Galván-Madrid et al. 2013, etc). These results imply that this physical problem is spatially scalable, and the previously spatially resolved systems are the large-scale examples seen in a closer to face-on projection. However, toroids may also be the projected central OB cluster-forming cores surrounded by satellite cores. The critical spatial scales to look at are the centrifugal radius (or radii), where the centrifugal force marginally balances the gravitational force from the enclosed molecular and stellar mass. We expect more of these systems to be discovered or be spatially resolved in high angular resolution ALMA observations. We hypothesize that the inward migration of satellite cores can explain the formation of the extremely concentrated dense condensations, similar to what was resolved in NH₃ (3,3) satellite hyperfine line absorption against the UC H ii region G10.6-0.4 (Sollins & Ho 2005; Liu et al. 2010).

Finally, we hypothesize that the spiraling arm-like asymmetry is essential in the gravitationally dominated accretion flow and needs to be considered in modeling frameworks. These structures may either be induced at regions where the eccentric accretion streams collide with each other or be created by tidal interactions. We note that tidal accretion of eccentric accretion streams, or tidally interacting hot cores, may deposit the hot molecular gas tracers from the hot cores to the entire parsec-scale massive molecular clump.

We would like to emphasize that the location of the objects in the region discussed favor the idea of the convergence of parsec-scale gas accretion flows onto a central clump. Our high spectral resolution DCN 3-2 and ¹³CS 5-4 line observations suggest that this massive molecular clump is fed by free-falling exterior gas streams (or arms). Based on the resolved images, we found that accretion flows in this region are highly dynamical and spatially non-uniform. Individual accretion gas streams likely carry different amounts of specific angular momentum. Even in the case where the averaged specific angular momentum on a large scale is negligibly small, it remains uncertain how the specific angular momentum carried by individual gas streams will alter the gas accretion and fragmentation. High angular resolution molecular line observations to study a large ensemble may be required to elucidate this issue.