DR 21 South Filament: A Parsec-sized Dense Gas Accretion Flow onto the DR 21 Massive Young Cluster

In order to derive the mass distribution of DR21SF, we adopt the column density map of Cygnus X complex from Cao19. Herschel PACS (160 μm) and SPIRE (250, 320, and 500 μm) images are employed in the spectral energy distribution (SED) fitting to produce the column density map with an angular resolution of 18''. A detailed description about how the column density map is derived can be found in Cao19. One should notice that the SED fitting in Cao19 did not include the PACS 70 μm data since the emission at this wavelength arises from warm dust and a single-temperature model is no longer applicable. Given that the filament is overall quiescent and at low temperature, the estimate of the column density would not be significantly affected by neglecting the 70 μm data.

We carry out observations of NH₃ (1, 1) and (2, 2) lines toward DR21SF in both on-the-fly (OTF) and point-by-point On–Off modes with the Tianma 65 m telescope to obtain both a position–position–velocity data cube covering the whole filament, and high-resolution spectra of targeted positions. The total bandwidth of 500 MHz (OTF mode) and 187 MHz (On–Off mode) is split into two sub-bands centered at 23694.4955 MHz and 23722.6333 MHz with frequency resolutions of 30.517 kHz (OTF mode) and 5.722 kHz (On–Off mode), corresponding to velocity resolutions of 0.39 km s⁻¹ (OTF mode) and 0.07 km s⁻¹ (On–Off mode). The FWHM beam size is about 48''.

In the OTF observations, we map an area of 45 by 90 covering DR21SF on 2017 November 25 and 2018 February 7 and 9. The average system temperature of the observations in three days is about 70 K. The nominal integration time on each sampled point is about 3 minutes in total. Only the main component of NH₃ (1, 1) is detected in the OTF observations because the filament is overall faint in NH₃ emission and the effective integration time is limited for such a large OTF map. The final data cube is then produced by combining all data with weights inversely proportional to their system temperatures, and re-gridding the image to a pixel size of 16'', approximately one third of the beam size.

The On–Off observations are made with a long integration time of 10 minutes toward each target position. We observe a total of 10 positions along the spine of DR21SF with spacings approximately identical to the beam size, to cover the entire filament. We name these 10 positions P01 to P10 from north to south. We fit all the observed NH₃ (1, 1) profiles with a five-component model, whereas (2, 2) transition line profiles are fit with a single Gaussian model since we fail to detect the satellite lines of this transition.

On the column density map, DR21SF is connected to DR 21 at its northern end, and extends ∼8' to the southeast. Since DR21SF and DR 21 have similar radial velocities (about −3 km s⁻¹), it is reasonable to infer that these two structures are physically associated with each other. We therefore adopt a trigonometric parallax distance of ${1.50}_{-0.07}^{+0.08}$ kpc of DR 21 (Rygl et al. 2012) for DR21SF, which leads to a length of 3.6 pc, given an inclination angle of 18° (see the discussion in Section 4.2). The width of DR21SF is derived by fitting the averaged column density profile perpendicular to the spine of the filament. We only employee the transverse column density profiles at P03 to P09 for the width estimation, since the column densities in the middle segment of the filament are consistent. The asymmetry of the averaged transverse column density profile (see Figure 2) is caused by the clump DR21-10 (identified by Cao19), which is on the east side of DR21SF and about 0.7 pc away from the spine of the filament. According to the cylinder model of an infinite gas filament (e.g., Ostriker 1964; Arzoumanian et al. 2011), whose radial volume density follows

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{{\rho }_{{\rm{c}}}}{{\left[1+{\left(r/{r}_{0}\right)}^{2}\right]}^{p/2}}\end{eqnarray} \tag{ 1 }$$

the transverse column density can be described as a Plummer-like function:

$$\begin{eqnarray}&&N(r)={A}_{{\rm{p}}}\displaystyle \frac{{\rho }_{{\rm{c}}}\sqrt{8}{r}_{0}}{{\left(1+\tfrac{{r}^{2}}{8{r}_{0}^{2}}\right)}^{(p-1)/2}},\end{eqnarray} \tag{ 2 }$$

where ${A}_{{\rm{p}}}=\tfrac{1}{\cos \theta }\int \tfrac{{du}}{{\left(1+{u}^{2}\right)}^{p/2}}$, θ is the inclination and set to 18°, ρ_c is the central volume density of the filament, and r₀ is the scale radius representing the flat distribution of the density at the spine of the filament. We fit the average transverse column density ${\overline{N}}_{\mathrm{trans}}$ profile with this model. Shown in Figure 2, the best fit, which gives a minimum χ², results in p = 1.7 ± 0.1, ρ_c = (1.6 ± 0.2) × 10⁴ cm⁻³, and r₀ = (4.2 ± 0.6) × 10⁻² pc. The errors are of 1σ uncertainties. The derived p value is in agreement with that found in filaments such as IC 5146 (1.5 < p < 2.5; Arzoumanian et al. 2011) and Serpens South Filament (p = 1.9 ± 0.2; Kirk et al. 2013). The central density ρ_c of DR21SF is significantly lower than that of several other star-forming filaments (see Table 3 of Gong et al. 2018). When p ≈ 2 the FWHM of the radial column density profile of the gas cylinder is about 3 × r₀ (Arzoumanian et al. 2011), thus we estimate the width of DR21SF as 0.13 pc. This result seems in good agreement with the 0.1 pc "universal" width of filaments in molecular clouds in the solar neighborhood proposed by Arzoumanian et al. (2011). However, the resolution of the column density map we used is 18'', which corresponds to 0.13 pc at a distance of 1.5 kpc. We think the width estimation is limited by this resolution, even though the fitting is performed on the column density profile, whose FWHM is significantly larger than the resolution.

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The mass of DR21SF can be estimated from the column density map by

$$\begin{eqnarray}&&M=\sum {N}_{i}\mu {m}_{{\rm{H}}}{A}_{i},\end{eqnarray} \tag{ 3 }$$

where N_i is the column density of the ith pixel on column density map within the area of the filament, μ is the average molecular weight, m_H is the mass of a hydrogen atom, and A_i is the area of the ith pixel at the distance of the source. We calculate the total mass within an area of a stripe along the spine of the filament with a width of 0.75 pc, which is the total width of the best-fit transverse column density profile above the typical background level (1 × 10²² cm⁻²). With μ = 2.8, a total mass of 1048 M_⊙ is derived. Cao19 estimate the average relative uncertainty of column density to be about 40% in the area of DR21SF. Considering the distance uncertainty is about 5%, we claim that the uncertainty of the total mass is 50%.

The mass we estimated is smaller than that (1350 M_⊙) proposed by H12, even though their result is within our uncertainty. The discrepancy mainly comes from the systematic difference in the column density maps produced by H12 and Cao19. As a comparison, the column densities of the densest region in DR 21 and the nearby least dense regions reported by Cao19 are about 1.5 × 10²³ cm⁻² and 7.0 × 10²¹ cm⁻², respectively, whereas H12 present corresponding values of  > 1.0 × 10²⁴ cm⁻² and  ∼ 1.0 × 10²² cm⁻², respectively. The systematic difference can be explained by their data reduction approach. The SED fitting performed by Cao19 is based on images of 160, 250, 350, and 500 μm, whereas H12 do not include the data of 500 μm. The long-wavelength data is essential to the SED fitting of cold gas. For example, with a temperature of 15 K, the SED peaks at a wavelength of 340 μm. Thus, data at 500 μm will certainly help improve the SED fitting of the cold gas in DR21SF. The discrepancy should affect both the mass and the temperature measurements as they are direct results from the SED fitting. We do find a systematic difference of about 3 K in the dust temperature of DR21SF between H12 and Cao19 as well. A column density about 1.2 × 10²³ cm⁻² was independently derived from the archive data of the European Space Agency's Infrared Space Observatory Long-Wavelength Spectrometer by Jakob et al. (2007) toward the densest region in DR 21. This value is in good agreement with that of Cao19. We therefore conclude that the column density map presented by Cao19 is a better estimate than that by H12, and therefore is adopted in this work for the mass calculation. We also noticed that both of these SED-fitting processes did not take into account the 70 μm data. The dust emission at 70 μm mainly traces the warm gas component in the molecular clouds, so may introduce extra uncertainty to the SED if only a single temperature component of cold gas is applied in the fitting. Neglecting the warm gas component traced by shorter-wavelength emission will certainly underestimate the total mass of molecular clouds, especially for those with high star formation rates. However, DR21SF shows very few star formation signposts (see the discussion in Section 4.1); therefore, we think this effect is negligible compared to other sources of uncertainty in the estimation of the total mass.

Data from the On–Off observations toward P01 to P10 are used to derive the optical depth and gas temperature of DR21SF. In Figure 3, the line profiles of both NH₃ (1, 1) and (2, 2) transitions at P01 to P10 are presented. To improve the signal-to-noise ratio, we perform Hanning smoothing to the original data, and this results in an effective velocity resolution of 0.15 km s⁻¹. With the assumptions of equal beam-filling factor and equal excitation temperature for each hyperfine transition, and a cosmic background temperature of 2.7 K, we use the CLASS package of the GILAS⁶ software to fit the NH₃ (1, 1) line profiles. We only perform simple Gaussian fitting to the NH₃ (2,2) lines, since we only detect the main component of this transition. Derived parameters are listed in Table 1. We follow Equation (4) in Ho & Townes (1983) to calculate rotational temperature (T_rot). Since NH₃ (1, 1) and (2, 2) transitions are not sensitive to temperatures significantly higher than 30 K (e.g., Lu et al. 2014), the T_rot ≳ 30 K derived with this method can only be treated as a lower limit. Based on T_rot we are able to calculate the kinematic temperature (T_kin), by applying the empirical relationship introduced by Walmsley & Ungerechts (1983):

$$\begin{eqnarray}&&\displaystyle \frac{1}{{T}_{\mathrm{rot}}}=\displaystyle \frac{1}{{T}_{\mathrm{kin}}}+\displaystyle \frac{1}{41.5}\mathrm{ln}\left[1+0.6\exp -\displaystyle \frac{15.7}{{T}_{\mathrm{kin}}}\right].\end{eqnarray} \tag{ 4 }$$

Both T_kin and T_rot for each position are also given in Table 1.

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Table 1.  Physical Properties of NH₃ along the DR 21 South Filament

Position	R.A.	Decl.	Trot	Tkin	τ(1, 1)	Vlsr	ΔVfit	ΔVint	${{ \mathcal M }}_{1D}$
	(deg)	(deg)	(K)	(K)		(km s−1)	(km s−1)	(km s−1)
P01	309.760	42.278	10.9	11.3	0.11	−1.85 ± 0.01	1.98 ± 0.14	1.86	10.18
P02	309.773	42.268	21.4	25.0	0.10	−2.07 ± 0.13	2.63 ± 0.37	2.57	9.45
P03	309.782	42.255	>30.4	>40.6	0.10	−2.31 ± 0.12	1.81 ± 0.42	1.68	4.83
P04	309.791	42.243	12.1	12.7	0.10	−2.63 ± 0.07	1.99 ± 0.16	1.87	9.66
P05	309.798	42.230	10.4	10.8	0.10	−2.68 ± 0.10	1.34 ± 0.27	1.20	6.71
P06	309.803	42.218	9.7	10.0	0.10	−3.05 ± 0.03	1.17 ± 0.08	1.02	5.93
P07	309.809	42.206	>38.3	>58.8	0.10	−3.55 ± 0.03	1.56 ± 0.12	1.42	3.39
P08	309.813	42.192	22.1	26.1	0.10	−3.89 ± 0.03	1.02 ± 0.09	0.83	2.96
P09	309.816	42.179	14.2	15.2	0.10	−4.18 ± 0.04	0.98 ± 0.10	0.78	3.66
P10	309.819	42.165	22.3	26.3	0.10	−4.23 ± 0.15	1.13 ± 0.20	0.97	3.46

Note. The R.A. and decl. are of J2000. T_rot and T_kin are the rotational and kinematic temperature, respectively. τ(1, 1, m) denotes the optical depth of the NH₃ (1, 1) transition. The ΔV_fit and ΔV_int stand for the fitted ("blend") and intrinsic line width, and the one-dimensional Mach is listed in the last column.

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The radial velocities along DR21SF with respect to local standard of rest range from −4.23 to −1.85 km s⁻¹. In the right panel of Figure 1, we present the first moment map of the filament. From both the first moment map and the radial velocities presented in Table 1 one can easily see a gradient in radial velocity along the filament. In Figure 4, we plot the radial velocity as a function of the position along the filament with respect to P01. The radial velocities are well characterized by the linear model, which gives a gradient of (0.8 ± 0.1) km s⁻¹ pc⁻¹. This value is comparable to that in other similar filamentary molecular clouds, e.g., 0.7 km s⁻¹ pc⁻¹ in the Orion Filament (Bally et al. 1987), 0.15–0.6 km s⁻¹ pc⁻¹ in SDC13 (Peretto et al. 2014), and 0.8–2.3 km s⁻¹ pc⁻¹ in DR21(OH) (Schneider et al. 2010). Moreover, a decrease can also be seen in the intrinsic line width along the filament from north to south. We again apply a linear model to give a first-order estimate of such a gradient. The best-fit result reads (0.5 ± 0.1) km s⁻¹ pc⁻¹.

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Due to the limited spectral resolution, the line width directly derived from the observed line profile is the true line width of the emission (intrinsic line width) convolved with an instrumental spectral resolution and blurred with adjacent hyperfine structures. Also, the radial velocity gradient along the filament will broaden the measured line width when observed with a certain beam size. To eliminate these effects, we first subtract the instrumental spectral resolution and contribution of velocity gradient within the beam size (0.3 km s⁻¹ in our case) quadratically from the measured line width to derive a "blurred" line width. We then follow the empirical relation (Barranco & Goodman 1998) between the intrinsic and the "blurred" line width to estimate the former. The results are listed in Table 1. We evaluate the uncertainty of intrinsic line width to be 12%, since the average 1σ uncertainty of the fitted line width is about 12% and other uncertainties are negligible.

Thanks to the deep and unbiased infrared surveys toward Cygnus X carried out with the Balloon-borne Large Aperture Sub-millimeter Telescope (BLAST; Roy et al. 2011), Herschel Space Telescope, and Spitzer Space Telescope, we are able to search deep embedded young stellar objects (YSOs) and compact clumps which can potentially form stars along the filament. We find signposts of star formation at P03, P07, and P10, in the surveys mentioned above. As shown in Figure 1, two compact mid-infrared objects are detected at P03 by the Spitzer Space Telescope, and later recognized as Class I low-mass YSO candidates by SED fitting using Spitzer, UKIDSS, and 2MASS data (Kryukova et al. 2014). With luminosities about 505 L_⊙ and 86 L_⊙, they are found to be coincident with an "IR-quiet"⁷ massive dense core (MDC, named as DR21-22 by Cao19) with a mass of 36.5 M_⊙. The MDC shows no evidence of high mass star formation according to its IR luminosity. The two objects are likely to be clusters of or individual intermediate-/low-mass forming stars. At P07, a sub-millimeter object, J203914+421221, was detected by Roy et al. (2011) with BLAST. By the SED fitting of sub-millimeter and far-infrared data (500, 350, 250, 100, and 60 μm), the authors infer this object to be a pre-stellar massive (80 M_⊙) clump at its very early stage with a total luminosity of 270 L_⊙ and a temperature 20.2 K. However, in a search of dense clumps with a mass completeness level of 35 M_⊙ by Cao19, no clump is detected at this position. The inconsistency may come from distinct data reduction procedures and spatial resolutions. As argued by Cao19, the source extraction process is highly affected by the background diffuse emission removal, thus different background removal algorithms may result in different detection. Also, the image resolution of Cao19 (20'') is significantly higher than that of Roy et al. (2011) (1'), so the clump recognized by the latter may be resolved into several less massive objects or relatively extended structures. We also found two Class I YSOs with very low luminosities of 0.7 L_⊙ and 1.5 L_⊙ at P10. The less luminous one shows a much flatter SED with spectral index of α = 0.21 than that of the other YSO α = 1.67. Greene et al. (1994) referred to such objects as "flat spectrum" since their evolutionary stages show ambiguity between Class I and Class II, and may statistically infer a later evolutionary status than typical Class I YSOs. Makin & Froebrich (2018) carried out a 2.122 μm molecular hydrogen line survey of jets and outflows from young stars covering the whole Cygnus X complex as part of the UWISH2 survey. The 2.122 μm molecular hydrogen emission line is excited by the shocks that produced the high velocity outflow impacting the surrounding molecular gas. We find no detection of 2.122 μm molecular hydrogen emission toward any of the YSOs identified in DR21SF in this survey. This is an indicator that the YSOs in DR21SF lack collimated jets, and may have only produced large-angle outflows that are not able to generate strong enough shocks in the natal clouds.

The NH₃ observations (Table 1) show that the gas temperatures at P02, P03, P07, P08, and P10 are significantly higher than typical temperatures (10–15 K) of cold molecular clouds. This can be explained by the objects we have found at P03, P07, and P10, as the on-going star-forming process will heat up the natal clouds. It is also a direct proof that the YSOs or clumps found in these positions are physically associated, rather than only visually coincident, with the filament. We find no YSO at P07, even though the gas temperature and line width are higher than the rest of the filament. It is likely that the forming star or cluster is in its early evolutionary stage, so it may not be visible as a near-/mid-IR point source.

We then try to determine whether DR21SF is stable, or just a transient phenomenon, by investigating the stability using three timescales: life time, crossing time, and freefall time. In the case of DR21SF, the YSOs identified in the filament are Class I objects. The age τ_i of such objects from pre-stellar to Class I stage is confined by 1 Myr  ≤ τ_i ≤  2 Myr (Kirk et al. 2005; Evans et al. 2009). This will put a 1 Myr lower limit on the life time of the DR21SF. Recent observations toward many molecular cloud filaments appear to favor a turbulence-supported gas cylinder scenario. At a temperature of 15 K, the thermal line width is about 0.03 km s⁻¹ for NH₃, which is negligible compared to the measured line widths in DR21SF. Therefore, we suggest that turbulence is the dominant supporting force in the case of DR21SF. Considering the 3D velocity dispersion ${\sigma }_{3{\rm{D}}}=\sqrt{3}{\sigma }_{\mathrm{aver}}=2.5\pm 0.9$ km s⁻¹, where σ_aver is the average intrinsic line width, the crossing time then can be derived as t_cross = R/σ_3D = 7.8 × 10⁴ yr, in a turbulence-supported case. The freefall time of DR21SF can be estimated as ${t}_{\mathrm{ff}}=\sqrt{3\pi /32G\rho }=1.3\times {10}^{5}\,\mathrm{yr}$, where ρ is the average volume density. Both t_cross and t_ff are about an order of magnitude smaller than the lower limit of the age of the filament. This indicates that DR21SF is a stable structure rather than a transient concentration of the ISM. However, the crossing time is significantly smaller than the freefall time. This is a clear sign that the self-gravity of the filament is not strong enough to overcome the turbulent pressure and the filament would have been dispersed by the strong turbulence. To solve this contradiction, we then have to look into the stability status of the filament.

According to Fiege & Pudritz (2000) the linear virial mass of turbulence-supported filaments is given by

$$\begin{eqnarray}&&{\left(M/l\right)}_{\mathrm{vir}}=2{\sigma }^{2}/G=84{\left(\displaystyle \frac{{\sigma }_{3{\rm{D}}}}{\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}\,{M}_{\odot }\,{\mathrm{pc}}^{-1}.\end{eqnarray} \tag{ 5 }$$

We calculate the virial mass using the equation above and the mass using the same method as the total mass estimation, for each of the 10 segments along the filament at P01 to P10. The virial parameter of each segment listed in Table 2 is the ratio of these two values. The virial parameter along the filament is mostly around 1–2, suggesting that DR21SF is overall in a trans-critical status, that the gravity and turbulent pressure rival each other. At P02, and P04, the virial parameters are significantly larger than 2. Since YSOs have already been identified at this area, the large virial parameters do not really reflect stronger turbulence. Instead, the feedback of the star formation activities, such as large-angle outflows, may introduce perturbation to the surrounding molecular clouds and increase the line width. The large virial parameter at P07 and P10, compared to their neighbor segments, can also be explained by a similar effect. This could be more evidence of the unseen star formation activities at extremely early stage at P07, independent from the temperature excess. We notice that the tail of filament is in a supercritical status, as indicated by the small virial parameters. This is usually a reflection of effective collapse of the cloud under its self-gravity, and can possibly explain the later evolutionary stage of one of the YSOs identified at P10.

The magnetic field also plays an important role in shaping filamentary molecular clouds by providing a supporting force against gravity or directing the mass accretion flow (Hennebelle & Inutsuka 2019). Supercritical filaments are usually observed to be perpendicular to the magnetic fields (Lee et al. 2017), whereas the striations that are attached to the main filaments are highly elongated along the B-field (Heyer et al. 2016). In the case of DR 21, the large-scale magnetic field runs almost perpendicular to the north–south main ridge consisting of DR 21 and DR21(OH) (e.g., Kirby 2009). The accretion flow onto DR21(OH) through the sub-filaments F3, on the other hand, has been reported as being parallel to the magnetic field (Schneider et al. 2010). We illustrate the large-scale magnetic field orientation around DR 21 derived by Planck dust continuum polarization data⁸ in Figure 1. DR21SF is mostly aligned along the large-scale magnetic field, and is more consistent with the scenario that DR21SF is overall a large-scale accretion flow. Even though, in theory, the length scale of "sausage" fragmentation is maintained with the presence of a magnetic field parallel to the filament (Nagasawa 1987), we do not see such a fragmentation occurring in DR21SF as it does in many supercritical filaments. At the northern end of DR21SF, where the two YSO candidates are found, the magnetic field bends to form a large angle with the spine of the filament. With the limited resolution of Planck data, it is difficult to tell whether this alteration of the magnetic field orientation is caused by the star formation in DR21SF or by the influence of DR 21, where a strong magnetic field almost perpendicular to the ridge is seen. High-resolution observations of the dust polarization toward DR21SF could help us to obtain a better understanding of the stability status of DR21SF.

How will such a filament dynamically evolve? Gong et al. (2018) compared two star-forming filaments with similar mass and distinct central density, Serpens Filament and Serpens South Filament. They infer that the slightly super-critical filament (Serpens Filament) with lower central density and weaker signature of radial collapse is at an earlier evolutionary stage than the other's (Serpens South Filament), which is consistent with the relatively inactive status of Serpens Filament in star formation. From a theoretical viewpoint Inutsuka & Miyama (1992) proposed that the radial collapse starts in an isothermal gas cylinder when the linear mass is greater than the critical value, and accelerates until the central density increase and the equation of state vary greatly from the isothermal one; then the filament may finally fragment. In other words, in a filament in its early evolutionary stage, radial collapse dominates over fragmentation. DR21SF shows significantly lower central density compared with other star-forming filaments (see Table 3 in Gong et al. 2018), which may indicate an early evolutionary stage. This is confirmed by the sparse detection of YSOs in DR21SF. Thus the theory by Inutsuka & Miyama (1992) can be a possible explanation of the coherent profile of DR21SF. However, direct observational evidence of radial collapse is needed to verify this hypothesis in DR21SF.

DR21SF seems to be quiescent since we find no signpost for massive star formation (e.g., H ii regions or Class II methanol masers) and see only two sites of low-/intermediate-mass YSOs (P03 and P10) along the filament with no energetic feedback (e.g., shocks produced by jets or outflows), whereas the star formation sites in other sub-filaments connecting to DR21(OH) are more active with the detection of outflows (see Figure 1 for a comparison). There are other filamentary objects similar to DR21SF in terms of stability status. In the Galactic central molecular zone (CMZ), filaments are often massive and quiescent at the same time due to intensive turbulence. For example the G0.253+0.016 cloud (the "Brick") is a turbulence-dominated filamentary system located in the CMZ, with a mass of 7.2 × 10⁴M_⊙, and an apparent size of 17 pc², and only two dense core candidates and one H₂O maser are detected (Rathborne et al. 2014; Federrath et al. 2016 and references therein). However, the physical condition of the CMZ is apparently more extreme than that of DR21SF. The gas temperature is up to 100 K and the one-dimensional Mach number in the CMZ is 30 (Rathborne et al. 2014), significantly higher than those in DR21SF. Federrath et al. (2016) claimed that the solenoid driving caused by shearing motions in the CMZ is the dominant turbulence-driving model in G0.253+0.016, which reduces the star formation rate compared to typical clouds in the solar neighborhood. In our case, the strong gravitational field produced by DR 21 may be a potential engine of compressing driving turbulence.

There are in principle three possible interpretations of the radial velocity gradient along filamentary molecular clouds: rotation, outflow, or infall. The rotation scenario is unrealistic since the centrifugal force needed to maintain the 3.6 pc filament is too dramatic. Considering an outflow scenario, we assume that the filament is a mass ejection originating from DR 21. In this case, stellar winds or jets would be the most likely candidate sources for driving the outflow motion. However, DR 21 is already harboring one of the most luminous and largest warm (2000 K) bipolar outflows in the Galaxy (Garden et al. 1986). There is no star formation theory that could provide a feasible interpretation of two such large outflows, one warm and bipolar and the other cold and unipolar, coming from a cluster of UC H ii regions. Since neither the rotation nor the outflow scenario appears to be feasible, we propose that the radial velocity gradient of DR21SF is produced by an accelerating infall along the filament. The gas flows from the south to the north along DR21SF, induced by the gravitational field of DR 21. Arzoumanian et al. (2013) proposed that in this process the radially collapsing filaments will keep a constant width along their spine. This is also in agreement with our observation.

The system consisting of DR21SF and DR 21 seems to be quite similar to the "head–tail" system reported by Tachihara et al. (2002), in which the "head" is an active massive star-forming region with a high column density and the much less dense "tail" often shows only signposts of low-mass star formation. Tachihara et al. claimed that in the "head–tail" scenario, stellar winds from other OB stars in the opposite position of the "tail" are likely to be the trigger of the formation of the cometary structure. We believe in DR21SF this is not likely to be the case, since the massive star-forming site to the north, DR21(OH), is still at an early evolutionary stage and the radiation and winds from embedded high-mass protostars not yet expelling surrounding molecular clouds. Therefore they are not expected to produce a "head–tail" scenario at DR 21. Thus the DR21SF-DR 21 system looks similar to a "head–tail" picture, but the system is not induced by nearby OB stars, and the "tail" is an accretion flow onto the "head."

Because of the conservation of the total energy during the accelerating infall, one can estimate the inclination angle of the filament by analyzing its energy budget, assuming a freefall gas cylinder model. With an inclination angle θ, with respect to the plane of sky, the first-order estimate of the total energy of the infall gas in a unit volume reads

$$\begin{eqnarray}&&{E}_{\mathrm{prot}}+{E}_{\mathrm{kin}}+{E}_{\mathrm{turb}}+{E}_{\mathrm{therm}}=C,\end{eqnarray} \tag{ 6 }$$

where ${E}_{\mathrm{prot}}=\tfrac{{{Gm}}_{i}{m}_{\mathrm{DR}21}\cos \theta }{{r}_{i}}$ is the gravitational energy of gas with mass M_i at a projected distance r_i to DR 21 in the plane of the sky, ${E}_{\mathrm{kin}}={m}_{i}{\tfrac{{v}_{\mathrm{LSR}}}{\sin \theta }}^{2}$ is the kinetic energy with relative radial velocity v_LSR in respect of the southern end of DR21SF, and E_turb + E_therm = m_iΔv² is the internal energy (including thermal and turbulent components) with Δv being the intrinsic line width of the gas. The local mass is defined as ${m}_{i}={P}_{i}\cos \theta /w$, where P_i is the local column density, and w = 0.13 pc is the width of DR21SF. We again use the column density, radial velocity, and intrinsic line width from P03 to P09 to fit the model. The best-fit model is illustrated in Figure 5, which gives an estimate of the inclination angle of 18°. Thus the projection effect is insignificant in the analysis of DR21SF. Kirby (2009) estimated the inclination angle of the DR 21 main ridge is about 20° to the plane of the sky by fitting the line width ratio of emissions from neutral molecules and ions in the magnetic field. This result is in good agreement with ours.

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We then give an estimate of the mass infall rate of DR21SF with the assumption of a homogeneous cylinder (Peretto et al. 2014):

$$\begin{eqnarray}&&\dot{M}=\pi {\left(w/2\right)}^{2}\rho V=\displaystyle \frac{M}{L}V,\end{eqnarray} \tag{ 7 }$$

in which w is the width of the filament, ρ is the average density, V is the infall velocity at the northern end of the filament, where we adopt the velocity of P01, M is the total mass, and L is the length of the filament. The infall velocity can be estimated by radial velocities as ${V}_{\mathrm{infall}}=({V}_{{\rm{P}}01}-{V}_{\mathrm{DR}21})/\sin \theta =3.7$ km s⁻¹, where V_DR21 = − 3.0 km s⁻¹. Thus the mass infall rate is $\dot{M}=1.1\times {10}^{-3}$M_⊙ yr⁻¹. Our estimate is about two orders of magnitude larger than that of the mass infall rate of the SDC13 Filament (2.5 × 10⁻⁵M_⊙ yr⁻¹; Peretto et al. 2014), and is similar to that of the F3 Filament onto the DR21(OH) clump (1.9 × 10⁻³M_⊙ yr⁻¹, Schneider et al. 2010), which is believed to contribute about one third of the total mass infall rate of the clump. Since there is no detected gas flow toward DR 21 along the DR21 ridge (Schneider et al. 2010), the gas infall along the DR21SF is possibly the most important, if not only, accretion flow that is feeding the DR 21 massive star-forming region.