# Unveiling the formation of the massive DR21 ridge

L. Bonne  SOFIA Science Center, NASA Ames Research Center, Moffett Field, CA
94 045, USA  S. Bontemps  Laboratoire d’Astrophysique de Bordeaux, Université
de Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France
N. Schneider  I. Physikalisches Institut, Universität zu Köln, Zülpicher Str.
77, 50937 Köln  R. Simon  I. Physikalisches Institut, Universität zu Köln,
Zülpicher Str. 77, 50937 Köln  S. D. Clarke  Academia Sinica, Institute of
Astronomy and Astrophysics, Taipei, Taiwan  T. Csengeri  Laboratoire
d’Astrophysique de Bordeaux, Université de Bordeaux, CNRS, B18N, allée
Geoffroy Saint-Hilaire, 33615 Pessac, France  E. Chambers  SOFIA Science
Center, NASA Ames Research Center, Moffett Field, CA 94 045, USA  U. Graf  I.
Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln
J. M. Jackson  SOFIA Science Center, NASA Ames Research Center, Moffett Field,
CA 94 045, USA  Green Bank Observatory, PO Box 2, Green Bank, WV 24 944, USA
R. Klein  SOFIA Science Center, NASA Ames Research Center, Moffett Field, CA
94 045, USA  Y. Okada  I. Physikalisches Institut, Universität zu Köln,
Zülpicher Str. 77, 50937 Köln  A. G. G. M. Tielens  University of Maryland,
Department of Astronomy, College Park, MD 20742-2421, USA  Leiden Observatory,
PO Box 9513, 2300 RA Leiden, The Netherlands  M. Tiwari  University of
Maryland, Department of Astronomy, College Park, MD 20742-2421, USA

###### Abstract

We present new 13CO(1-0), C18O(1-0), HCO+(1-0) and H13CO+(1-0) maps from the
IRAM 30m telescope, and a spectrally-resolved [C II] 158 $\mu$m map observed
with the SOFIA telescope towards the massive DR21 cloud. This traces the
kinematics from low- to high-density gas in the cloud which allows to
constrain the formation scenario of the high-mass star forming DR21 ridge. The
molecular line data reveals that the sub-filaments are systematically
redshifted relative to the dense ridge. We demonstrate that [C II] unveils the
surrounding CO-poor gas of the dense filaments in the DR21 cloud. We also show
that this surrounding gas is organized in a flattened cloud with curved
redshifted dynamics perpendicular to the ridge. The sub-filaments thus form in
this curved and flattened mass reservoir. A virial analysis of the different
lines indicates that self-gravity should drive the evolution of the ridge and
surrounding cloud. Combining all results we propose that bending of the
magnetic field, due to the interaction with a mostly atomic colliding cloud,
explains the velocity field and resulting mass accretion on the ridge. This is
remarkably similar to what was found for at least two nearby low-mass
filaments. We tentatively propose that this scenario might be a widespread
mechanism to initiate star formation in the Milky Way. However, in contrast to
low-mass clouds, gravitational collapse plays a role on the pc scale of the
DR21 ridge because of the higher density. This allows more effective mass
collection at the centers of collapse and should facilitate massive cluster
formation.

ISM: individual objects: DR21 – ISM: kinematics and dynamics – ISM: clouds –
stars: massive – stars: formation

††facilities: SOFIA (upGREAT), IRAM 30m, Herschel, JCMT††software: astropy
(Astropy Collaboration et al., 2013)

##  1 Introduction

Figure 1: Middle left: Herschel column density map (Hennemann et al., 2012) of
the DR21 cloud (ridge and surrounding sub-filaments). The yellow and red
polygons indicate the area covered by the two different backend setups of the
IRAM 30m observations, i.e. setup1 (yellow) and setup2 (red). The southern
sub-filament is only covered by the first setup which contains HCO+(1-0) and
H13CO+(1-0). The crosses indicate the location of the spectra displayed in the
bottom and top panels. The white circles are centered on the DR21(OH) clump
and indicate radii of 0.5, 1.5, 3 and 4 pc at a distance of 1.4 kpc, which
will provide reference for the virial analysis presented in Sect. 4. Middle
right: The red box outlines the area that we use from the available SOFIA [C
II] observations that cover a significantly larger region in Cygnus-X North
(Schneider et al., 2023). The nomenclature indicated for sub-filaments is
adopted from Schneider et al. (2010); Hennemann et al. (2012): N (north), F1N,
F1S, F3N, F3S, SW (south-west), S (south). Bottom and top: 13CO(1-0),
C18O(1-0), HCO+(1-0), H13CO+(1-0) and [C II] spectra from the locations
indicated in the left middle panel.

The formation of massive stars is still a matter of intense debate (e.g.
Zinnecker & Yorke, 2007; Tan et al., 2014; Motte et al., 2018a). The very
short, or potentially non-existing, massive prestellar core phase (e.g. Motte
et al., 2007, 2010; Russeil et al., 2010; Csengeri et al., 2014; Tigé et al.,
2017; Sanhueza et al., 2019) suggests that high-mass stars might not form from
the quasi-static evolution toward collapse of individual prestellar cores.
This opens the question of how massive stars can form. From observations of
molecular clouds that form massive stars it was proposed that some large-scale
dynamics, partly driven by self-gravity, can provide the fast concentration of
mass necessary to form massive stars (e.g. Peretto et al., 2006, 2013, 2014;
Hartmann & Burkert, 2007; Schneider et al., 2010, 2015; Csengeri et al.,
2011a, b; Galván-Madrid et al., 2010; Wyrowski et al., 2012, 2016; Beuther et
al., 2015; Williams et al., 2018; Jackson et al., 2019; Bonne et al., 2022a).
This probable coupling to the molecular cloud dynamics is also put forward by
several theoretical models. In the competitive accretion model (Bonnell et
al., 2001, 2004; Bonnell & Bate, 2006; Wang et al., 2010), the gravitational
pull of low-mass protostellar cores drives the mass accretion from the
surrounding cloud to determine their final mass. The fragmentation-induced
starvation scenario proposes that massive stars form through gravitational
fragmentation at the center of dense filamentary accretion flows (Peters et
al., 2010, 2011; Girichidis et al., 2012), where the fragmentation in the
inflowing filaments sets a limit on the accretion of the most massive stars at
the center of the accretion flow. Another variant proposes that gravitational
collapse on multiple scales after the thermal instability (Vázquez-Semadeni et
al., 2009, 2017, 2019) drives the required dynamics and mass concentration
responsible for the formation of massive stars which is then halted by stellar
feedback. From simulations it was also proposed that compression by the
collision of atomic flows or fully developed molecular clouds at high
velocities can form dense filamentary structures that host massive star
formation (e.g. Dobbs et al., 2012, 2020; Inoue & Fukui, 2013; Wu et al.,
2015; Balfour et al., 2017; Bisbas et al., 2017). The velocity and density of
the H I flows or molecular clouds would then be decisive to determine whether
massive stars can form. Typically, collision velocities $>$10 km s-1 would be
needed to form massive stars (Haworth et al., 2015; Dobbs et al., 2020). Based
on observed bridging with CO lines between separated velocity components in
several clouds, it has been proposed that cloud-cloud collisions (CCCs) might
play a role in high-mass star formation (e.g. Bisbas et al., 2018; Fukui et
al., 2021; Lim et al., 2021). From simulations, it was also proposed that an
oblique shock associated with collision velocities above 7 km s-1 could bend
the magnetic field around the filaments and drive subsequent mass inflow that
enables high-mass star formation (Inoue et al., 2018; Abe et al., 2021).

Analyzing observations of the low-mass Musca filament and the surrounding
Chamaeleon-Musca regions, Bonne et al. (2020a, b) concluded that continuous
mass accretion on the Musca filament was driven by such bending of the
magnetic field due to a 50 pc scale collision at $\sim$ 7 km s-1 between H I
clouds in the Chamaeleon-Musca complex. This mechanism was proposed for the
formation of high-mass star forming filaments (Inoue et al., 2018) but might
also be applicable for low-mass star forming filaments. Specifically, the
Musca filament would be the result of a turbulent overdensity that is
compressed and guided by the bended magnetic field during interaction with
more diffuse gas in the colliding cloud. Interestingly, Faraday rotation
measurements in the radio domain, which trace the magnetic field properties in
the interstellar medium (ISM), unveiled curved magnetic fields around several
nearby low- to intermediate mass star forming regions (Tahani et al., 2019,
2022). It also unveiled a correlation of the cold (CNM) & lukewarm (LNM)
neutral medium with the magnetic field structure in the diffuse ISM (Bracco et
al., 2020). These results would have a straightforward explanation in the
proposed scenario of colliding H I clouds that bend the magnetic field.
Furthermore, recent observations of the massive star forming cloud NGC 6334
found kinematic structure resembling the results in Musca (Arzoumanian et al.,
2022).

Table 1: Summary of the observational datasets.

Instrument | Species | $\lambda$ | $\nu$ |  $\Delta$ v | $\Theta$
---|---|---|---|---|---
|  | [$\mu$m] | [GHz] | [km/s] | [′′]
IRAM |  |  |  |
EMIR |  13CO(1-0) | 2720.4 | 110.20 | 0.2 | 23
EMIR | C18O(1-0) | 2730.8 | 109.78 | 0.2 | 23
EMIR | HCO+(1-0) | 3361.3 | 89.19 | 0.2 | 28
EMIR | H13CO+(1-0) | 3455.7 | 86.75 | 0.2 | 29
SOFIA |  |  |  |
upGREAT | [C II] | 157.74 | 1900.54 | 0.5 | 14
JCMT |  |  |  |
HARP |  12CO 3$\to$2 | 869.0 | 345.80 | 0.42 | 15

For the atomic and molecular lines, the line transition wavelength ($\lambda$)
and frequency ($\nu$) are given in columns 3 and 4, respectively. The
effective velocity resolution is indicated in column 5 and the angular
resolution of the original data is displayed in column 6.

In this paper, we present a multi-wavelength study of the DR21 cloud in
molecular and atomic lines and dust continuum to follow up on the link between
molecular cloud evolution and the formation of dense filamentary structures.
Specifically, after focusing on Musca, we now focus on a region where massive
stars are presently forming. The DR21 cloud is located in the north of the
Cygnus-X molecular cloud complex (Reipurth & Schneider, 2008) at an estimated
distance of $\sim$ 1.4 kpc (Rygl et al., 2012). This cloud hosts the DR21
ridge which is the filamentary structure with a length of $\sim$ 4 pc and a
mass of $\sim$ 104 M⊙ that is typically defined by high column densities,
i.e., $N_{\rm H_{2}}>$ 1023 cm-2 (Schneider et al., 2010; Hennemann et al.,
2012). The DR21 ridge is the densest and most massive filament in the entire
Cygnus-X complex and one of the most active high-mass star forming regions
within 2 kpc from the sun (Motte et al., 2007; Kumar et al., 2007; Bontemps et
al., 2010; Beerer et al., 2010; Csengeri et al., 2011a, b; Duarte-Cabral et
al., 2013, 2014). In this paper, we will use the term ′ridge′ for the inner,
high-column density part ($N_{\rm H_{2}}>$ 1023cm-2) of the molecular cloud
(Hill et al., 2011, 2012), ′sub-filaments′ for the lower density filaments
connected to the ridge and ′cloud′ for the parsec scale surrounding cloud that
includes the ridge.

Detailed kinematic studies of the DR21 region by Schneider et al. (2010)
demonstrated that the ridge is experiencing large scale gravitational collapse
and mass accretion by sub-filaments that run parallel to the magnetic field
(Vallée & Fiege, 2006; Ching et al., 2022). Figure 1 shows the DR21 cloud and
ridge ($N_{\rm H_{2}}>$1023cm-2) and the sub-filaments defined in Schneider et
al. (2010) and Hennemann et al. (2012). It is also proposed that the DR21
cloud is part of the Cygnus-X complex which forms a single region with
multiple velocity components (Schneider et al., 2006, 2007). This was later
confirmed by Rygl et al. (2012) and provided arguments that DR21 is the result
of a collision between two molecular clouds with velocity components at $\sim$
-3 km s-1 and $\sim$ 9 km s-1 (Dickel et al., 1978; Dobashi et al., 2019).
This collision is revisited and discussed in Schneider et al. (2023) using new
insight from the SOFIA [C II] data.

In this paper, we aim to establish the main processes, i.e. gravity, magnetic
field and turbulence that drive the evolution of this massive star forming
molecular cloud from low-density ($n_{\rm H_{2}}\lesssim$ 103 cm-3) gas in the
surrounding cloud to high-density ($n_{\rm H_{2}}>$ 105 cm-3) gas in the
ridge. In Sect. 2, we give observational details. Section 3 presents the
observational results of the DR21 cloud, followed by the inflow and virial
analysis of these observational results in Sect. 4. Lastly, in Sect. 5 we put
these results into context to propose a scenario for the DR21 cloud evolution
and discuss how it compares to low-mass star forming clouds.

##  2 Observations

The observations tracing different density regimes in the DR21 cloud are
described in the following subsections and are summarized in Table 1.

###  2.1 IRAM 30m observations

Figure 2: Top left: Integrated brightness map of 13CO(1-0) from -8 to 1 km s-1
towards the DR21 cloud. Overlaid on the map are the integrated intensity
contours starting at 2 K km s-1 with increments of 10 K km s-1. The locations
of the DR21 radio continuum source (Downes & Rinehart, 1966), and the DR21(OH)
and DR21-N massive clumps (e.g. Motte et al., 2007) are indicated with black
triangles. The subfilaments in cloud are indicated with their names. Top
right: The same for HCO+(1-0) with increments of 5 K km s-1. The triangles
indicating DR21, DR21(OH) and DR21-N help to compare with the 13CO and C18O
maps, from setup2, which have a different map size. Bottom left: The same for
C18O(1-0), with contour increments of 2 K km s-1. Bottom right: The same for
H13CO+(1-0), with contour increments of 2 K km s-1. Figure 3: Line integrated
emission from -8 to 1 km s-1 for [C II]. The [C II] emission shows that there
is extended emission around the ridge and sub-filaments. The white contours
indicate the Herschel column density map at $N_{\rm H_{2}}$ = 1022 cm-2,
3$\times$1022 cm-2, 1023 cm-2 and 4$\times$1023 cm-2. The red box outlines the
area used to make the position-velocity (PV) diagram perpendicular to the DR21
ridge in Fig. 10. The vertical dashed black line defines the center (r = 0 pc)
used for the PV diagram. The locations of the DR21 radio continuum source
(Downes & Rinehart, 1966), and the DR21(OH) and DR21-N massive clumps (e.g.
Motte et al., 2007) are indicated. Figure 4: Average spectrum of the DR21
ridge (excluding the DR21 continuum source) displaying multiple velocity
components between -10 and 20 km s-1. The focus of this paper is on the
emission between -8 and 1 km s-1 alone, indicated by the dashed vertical
lines, which is the velocity range associated with the DR21 ridge (Schneider
et al., 2006, 2010). Figure 5: 13CO(1-0) channel maps (IRAM) between -6.0 and
-0.4 km s-1 of the DR21 cloud. The grey contours trace the Herschel column
density at $N_{\rm H_{2}}$ = 1022, 2$\times$1022, 5$\times$1022, 1023 and
5$\times$1023 cm-2, to highlight the ridge and sub-filaments. Most sub-
filaments become particularly visible at v${}_{lsr}>$ -3 km s-1.

Observations with the IRAM 30m telescope of the DR21 region (surrounding
cloud, ridge and sub-filaments) were carried out in February 2015. These
observations were performed with two different spectral setups using the FTS50
configuration of the EMIR receiver. Setup 1, which contains HCO+(1-0) and
H13CO+(1-0), covers the spectral ranges $\sim$ 85.2 GHz - 87.2 GHz, 88.5 GHz -
90.5 GHz, 101.0 GHz - 102.8 GHz and 104.2 GHz - 106.0 GHz. Setup 2, which
contains 13CO(1-0) and C18O(1-0), covers the spectral ranges $\sim$ 89.6 GHz -
91.6 GHz, 93.0 GHz - 95.0 GHz, 105.4 GHz - 107.2 GHz and 108.8 GHz - 110.6
GHz. These two setups allow to observe a variety of molecular lines (e.g.
C18O(1-0), N2H+(1-0), HCN(1-0), HCO+(1-0), NH2D[1(1,1)0-1(0,1)0], etc…), which
allows to follow the kinematic and chemical evolution of the dense gas in the
DR21 ridge and the connected sub-filaments. The obtained antenna temperature
(T${}_{A}^{*}$) noise rms within a spectral resolution of 0.2 km s-1 is
$\sim$0.14 K around 90 GHz and $\sim$0.20 K around 110 GHz, respectively. A
main beam efficiency of 0.81 for frequencies around 90 GHz and 0.78 for
frequencies around 110 GHz was used (Kramer et al., 2013). The region on the
sky that is covered by the two different setups is displayed in Fig. 1. The
mapping was performed in the On-the-Fly (OTF) observing mode, and the
resulting data cubes have a spatial resolution between $\sim$23′′ (110 GHz)
and $\sim$29′′ (90 GHz), see Tab. 1. To produce the data cubes, a first order
baseline was fitted to the data with CLASS in
GILDAS111https://www.iram.fr/IRAMFR/GILDAS/ as the baselines are generally
well behaved at the observed frequencies. A full analysis of all the lines
that are covered will be the topic of a future paper. Here, the first results
from 13CO(1-0), C18O(1-0), HCO+(1-0), and H13CO+(1-0) will be presented.

###  2.2 SOFIA [C II] observations with upGREAT

To obtain a more complete view on the kinematics of the DR21 cloud, the IRAM
observations are combined with observations of the [C II] 158 $\mu$m line by
the Stratospheric Observatory for Infrared Astronomy (SOFIA; Young et al.,
2012). The [C II] observations at 158 $\mu$m were taken as part of the SOFIA
Legacy program FEEDBACK222https://feedback.astro.umd.edu, which maps 11
Galactic high-mass star-forming regions (Schneider et al., 2020), including a
part of the Cygnus-X north region, in the [C II] 158 $\mu$m line and the [O I]
63 $\mu$m line. The mapping of Cygnus-X north with FEEDBACK is now
completed333https://astro.uni-koeln.de/index.php?id=18130 and covers the DR21
cloud, the Diamond Ring (Marston et al., 2004) and the W75N molecular cloud.
The data can be found on the IRSA archive with project number
07_0077444https://irsa.ipac.caltech.edu/applications/sofia. We use data that
covers the DR21 cloud, see Fig. 1. The observations were carried out with the
dual-frequency heterodyne array upGREAT receiver (Risacher et al., 2018) in
OTF mapping mode and calibrated with the GREAT pipeline (Guan et al., 2012).
We used an emission-free reference position at RA(2000)=20h39m48.34s,
Dec(2000)=42∘57′39.11′′. The 2$\times$7 pixel LFA array was tuned to the [C
II] 158 $\mu$m line and the 7 pixel high-frequency array (HFA) was tuned to
the [O I] 63 $\mu$m line (data not used here). The beam size at 158 (63)
$\mu$m is 14.1′′ (6′′). Here we employ [C II] data smoothed to an angular
resolution of 20′′ and a velocity resolution of 0.5 km s-1, which results in a
typical noise rms of $\sim$ 1 K per beam. In order to improve the baseline
removal to reduce striping in the data cube we employ the Principal Component
Analysis (PCA) method described in Tiwari et al. (2021); Kabanovic et al.
(2022); Schneider et al. (2023). More details on the SOFIA FEEDBACK
observational scheme and data reduction are found in Schneider et al. (2020).

###  2.3 JCMT observations

With the 15 pixel HARP instrument on the JCMT telescope, a 12CO(3-2) mapping
of the Cygnus-X north and south areas was carried out between 2007 and 2009
within the observing programs M08AU018 (PI N. Schneider) and M07BU019 (PI R.
Simon). The observation method and reduction of this data is the same as the
one described in Gottschalk et al. (2012) for the pilot study and this full
dataset will be presented in more detail in a forthcoming paper. The
observations have a spatial resolution of 15′′ and a spectral resolution of
0.42 km s-1, with a noise rms of $\sim$ 0.25 K.

###  2.4 Herschel observations

We employ Herschel column density maps from the Cygnus-X region that were
taken as part of the HOBYS555Herschel imaging survey of OB young stellar
objects (Motte et al., 2010). program. Low angular resolution (36′′) column
density maps are presented in Schneider et al. (2016a). The procedure for
obtaining these maps is described in Hill et al. (2011, 2012). We here use
column density maps at a higher spatial resolution of 18′′ that makes use of
the method described in Palmeirim et al. (2013). The concept is to employ a
multi-scale decomposition of the flux maps and assume a constant line-of-sight
temperature. The final map is then constructed from the difference maps of the
convolved maps at 500 $\mu$m, 350 $\mu$m, and 250 $\mu$m, and the temperature
information from the color temperature derived from the 160 $\mu$m to 250
$\mu$m ratio.

##  3 Results

###  3.1 Spectral line integrated maps

From the large variety of lines detected with the IRAM 30m telescope we
selected a subset to present in this paper, i.e. the 13CO(1-0), C18O(1-0),
HCO+(1-0) and H13CO+(1-0) lines, that trace both the ridge and sub-filaments.
A full analysis of all detected lines with the IRAM 30m telescope is out of
the scope of this paper. Figure 2 displays the velocity integrated maps of
these 4 lines. The overall emission distribution of the DR21 cloud is similar
to what was shown in Schneider et al. (2010), but the new maps are
significantly larger. As a result, they better cover the extent of the dense
sub-filaments which are best visible to the west of the DR21 ridge. Note that
the coverage is not the same for all lines which is due to different setups
for the observations. From Fig. 2 we see that the 13CO(1-0) and HCO+(1-0)
emission remains well detected outside the ridge while the C18O(1-0) and
H13CO+(1-0) lines are mostly detected towards the ridge which indicates a
large concentration of mass.

Figure 3 presents the SOFIA [C II] observations. The critical density of the
[C II] 158 $\mu$m line is typically only a few 103 cm-3 for collisions with
atomic and molecular hydrogen (i.e. H & H2; Goldsmith et al., 2012), and it
predominantly traces the photodissociated layers at the interface between
molecular cloud and ionized phase (Hollenbach & Tielens, 1999). It thus nicely
complements the CO/HCO+ observations as it should trace the lower-density gas.
The [C II] line integrated intensity map in Fig. 3 reveals interesting
differences compared to the IRAM molecular line data. Emission is detected
toward the dense sub-filaments that are also seen in 13CO(1-0), HCO+(1-0) and
12CO(3-2) (see App. A) which have typical critical densities of
$\sim$2$\times$103 cm-3, 5$\times$104 cm-3 and 3$\times$104 cm-3, respectively
(Shirley, 2015). However, [C II] additionally also traces the lower density
gas surrounding the DR21 ridge and dense sub-filaments. The [C II]
observations thus unveil a significant mass reservoir that is not located in
the ridge and the sub-filaments. This gas has a typical Herschel column
density ($N_{\rm H_{2}}$) between 5$\times$1021 cm-2 and 3$\times$1022 cm-2
after correction for a $\sim$5$\times$1021 cm-2 background (Schneider et al.,
2016a, 2022). Even though [C II] is detected towards these higher column
densities ($N_{\rm H_{2}}$ $>$ 1022 cm-2), these also start to be detected in
CO lines, see App. B, which indicates that [C II] does not trace the full
column density range there. Assuming the cloud has a thickness666This is
similar to the size in the plane of the sky or assumes it has a slightly
flattened morphology. of 1-3 pc indicates that [C II] in the DR21 cloud might
trace a density range between $n_{\rm H_{2}}$ = 0.5-1.6$\times$103 cm-3 and
$n_{\rm H_{2}}$ = 0.3-1.0$\times$104 cm-3 toward the DR21 cloud. In a later
paragraph we will further constrain the typical density traced by [C II]. Only
at the location of the DR21 radiocontinuum source there is a strong peak of [C
II] emission in the ridge. DR21 is the most evolved region in the ridge and a
site of massive star formation with several compact H II regions and a number
of possible O-stars, including a prominent outflow source (Marston et al.,
2004). Stellar winds (Cyganowski et al., 2003; Immer et al., 2014) probably
dominate the dynamics of that region. As the [C II] emission in the DR21
source is strongly affected by the local feedback from the embedded O stars,
this region is mostly left out for the work presented in this paper. Lastly,
we also note from Fig. 3 that the [C II] emission unveils filamentary gas to
the east of the ridge that are not detected with Herschel or molecular line
data. This is probably because of the low contrast with the significant
Herschel column density background and because these filamentary structures
are not visible in molecular line data.

###  3.2 Spectra and channel maps

The average spectrum for the entire DR21 ridge is presented in Fig. 4 which
shows multiple velocity components in the velocity range between -10 km s-1
and 20 km s-1. In this paper we will exclusively focus on the emission between
-8 and 1 km s-1, highlighted in Fig. 4, which is the emission originating from
the DR21 cloud (Schneider et al., 2006, 2010). The components of the full
spectrum are discussed in Schneider et al. (2023).
The channel maps of 13CO(1-0) emission, shown in Fig. 5, resolve the velocity
structure in the DR21 cloud. The emission in the velocity range from -5 km s-1
to -2.5 km s-1 is organized in a north-south elongated coherent structure
mostly inside the $N_{\rm H_{2}}$=1023 cm-2 dust column density contour and
forms the ridge. In this velocity range, the channel maps show a north-south
velocity gradient in the ridge. At higher and lower velocities inside the
ridge, the emission is more clumpy and concentrates on the DR21(OH) and DR21
locations. At velocities between -4 and -2.5 km s-1 the sub-filament F1S
becomes visible, and between -3 km s-1 and -1 km s-1 all the other western
sub-filaments become prominent.
Inspecting the 13CO(1-0), C18O(1-0), HCO+(1-0), H13CO+(1-0) and [C II] spectra
over the map in Fig. 1, we find that the spectra show skewed profiles. This is
also the case for H13CO+(1-0) and C18O(1-0) which are optically thin for
excitation temperatures in the DR21 ridge down to 10 K. This suggests the
presence of multiple components which we will address later on. The selected
positions in Fig. 1 represent typical lines in the ridge (positions 1,2,3 & 5)
and in the sub-filaments (positions 4 & 6). Note that the HCO+(1-0) spectra in
the ridge display strong self-absorption centered on the velocities of peak
emission for C18O(1-0) and H13CO+(1-0), see Schneider et al. (2010) for more
discussion. Furthermore, it becomes obvious in Fig. 1 that HCO+(1-0) has high-
velocity wing emission towards DR21 and DR21(OH), tracing the prominent
molecular outflows in these regions.
The [C II] spectra also show flat-topped and skewed line profiles in Fig. 1.
Since some bright [C II] regions were found to experience [C II] self-
absorption and optical depth effects (Graf et al., 2012; Guevara et al., 2020;
Kabanovic et al., 2022), we verified whether this could be the case for the
DR21 ridge. The [C II] spectra do not show a noteworthy dip, seen for
HCO+(1-0) which argues against self-absorption. In App. C, the optical depth
of the [C II] line is calculated using the [13C II] noise level which
indicates a maximal optical depth of $\tau$ = 1.7. In addition, we performed
calculations with RADEX (van der Tak et al., 2007) which indicates that the
optical depth for [C II] typically is below 1 for the DR21 cloud. The [C II]
line shape is thus not caused by optical depth effects but rather by multiple
velocity components around the DR21 ridge. This also fits with the observation
in Fig. 4 that the peak [C II] emission is displaced from the peak molecular
line emission. Therefore, we fit the spectra with multiple Gaussian velocity
components (see App. D) over the full map, and analyze the goodness of the
fit, with the BTS fitting algorithm (Clarke et al., 2018). We find that the [C
II] emission in the range of $-$8 to 1 km s-1 consist of either one or two
velocity components that are typically separated by $\sim$ 2-3 km s-1. The
first component is typically around $-$4.5 km s-1 and the second one around
$-$2.5 km s-1 (see Fig. 21 of App. D). The fitting results in Fig. 19 of App.
D also show that the regions with multiple velocity components are
concentrated in the close vicinity of the DR21 ridge.

Figure 6: Top left: Velocity map of 13CO(1-0) over the DR21 ridge and sub-
filaments, obtained from fitting a single Gaussian to the spectra. The
uncertainty of the centroid velocity is below 0.15 km s-1 over the entire map.
Overplotted on the map are the Herschel column density contours at $N_{\rm
H_{2}}$ = 1022, 2$\times$1022, 5$\times$1022, 1023 and 5$\times$1023 cm-2
which indicate the ridge as well as the surrounding sub-filaments. The sub-
filaments are indicated with their name from Hennemann et al. (2012). Top
right: The same for HCO+(1-0) without fitting inside the DR21 ridge because of
the HCO+(1-0) self-absorption there. Bottom left: The same for C18O(1-0).
Bottom right: The same for H13CO+.

Figure 7: Left: Velocity field observed with [C II]. The black contours
indicate the Herschel column density contours at $N_{\rm{H_{2}}}$ = 1022,
2$\times$1022, 5$\times$1022, 1023 and 5$\times$1023 cm-2. The sub-filaments
are indicated with their name. Right: The same for the [C II] FWHM.

###  3.3 The velocity field of the DR21 cloud

Even though the spectra are slightly more complex than a single Gaussian in
the -8 to 1 km s-1 velocity range, we performed a single Gaussian line fitting
on the data sets to obtain information on the global velocity distribution and
linewidth for the different tracers. A fit was accepted when the brightness
was higher than 3$\times$ the noise rms in the different tracers. Note that
moment maps might be better suited to give a view on the global dynamics in
the region. However, in large regions of the observed map the signal-to-noise
(S/N) of the data is not abundant while excellent S/N is required for
producing trustworthy moment maps (e.g. Teague, 2019). As a result, the
produced first and second moment maps have large uncertainties in the
subfilaments and do not provide a good visualization of the velocity field
there.
The resulting velocity maps for the molecular lines are presented in Fig. 6
and show similar velocity gradients over the ridge as the ones presented in
Schneider et al. (2010) using only N2H+(1-0). We confirm that there is not a
single gradient perpendicular to the ridge, but that the velocity pattern over
the ridge shows organised velocity gradients with altering orientation. These
altering velocity gradients are mixed with a clear north-south velocity
gradient from -4.5 to -2.5 km s-1. Remarkably, the velocities of the sub-
filaments (N, F1N, F3N, F3S and SW) in all lines is redshifted
(v${}_{LSR}\,\sim$-2.5 to -1 km s-1) with respect to the ridge (bulk emission
around -3.5 km s-1). Only the F1S sub-filament at velocities around -3.5 km
s-1 is not significantly redshifted with respect to the ridge but is not
blueshifted either. Lastly, the southern sub-filament appears slightly
redshifted close to the ridge, but more to the south this filament becomes
less redshifted with respect to the ridge. This observation of dominantly
redshifted sub-filaments and a complete lack of a truly blueshifted
subfilament with respect to the ridge is noteworthy and confirms the
observations in the channel maps of 13CO(1-0) in Fig. 5.
The [C II] observations trace the lower density gas kinematics in the cloud,
and thus provide a complementary view on the surrounding gas kinematics in the
cloud off the dense filamentary structures. The velocity field is shown in
Fig. 7. There is, similar to the molecular line observations, mostly
systematically redshifted emission (v$\sim$-3 to -1 km s-1) in the cloud
around the ridge. This redshifted gas becomes increasingly prominent further
away from the ridge. In addition, Apps. D & E show that there is an excellent
correspondence in velocity of the subfilaments and this redshifted [C II]
emission, indicating that the subfilaments are directly embedded in this mass
reservoir seen with [C II]. However, the [C II] velocity field also shows
additional features. In particular, blueshifted velocities down to v=-5 km s-1
are found in localized regions directly east and north-west of the ridge. No
blueshifted gas is found further away from the ridge and the F1S filament.
This fits with the observation in Fig. 4 that the average [C II] emission has
an offset from the molecular line emission and that it is associated with the
presence of two [C II] velocity components (at -4.5 and -2.5 km s-1)
established in the previous section. Lastly, from Fig. 20 in App. D we also
note that these regions with more blueshifted velocities overlap with the
regions that have two fitted velocity components. In these regions the
blueshifted velocity component is thus the dominant source of emission which
affects the observed velocity field.

Figure 8: Left: FWHM map of C18O(1-0) over the DR21 ridge and sub-filaments.
The uncertainty of the FWHM is generally below 0.1 km s-1 and always below 0.4
km s-1 over the entire map. Overplotted on the map are the Herschel column
density contours at $N_{\rm{H_{2}}}$ = 1022, 2$\times$1022, 5$\times$1022,
1023 and 5$\times$1023 cm-2. Right: The same for H13CO+(1-0).

###  3.4 Linewidths

The Gaussian line fitting also provides maps of the FWHM for the spectra. The
results for the optically thinner lines C18O(1-0) and H13CO+(1-0) are
presented in Fig. 8. The C18O(1-0) map indicates, where detected, that the
sub-filaments have a lower FWHM than the ridge, typically $\lesssim$ 1 km s-1.
In the ridge, a FWHM between 1 and 2.5 km s-1 is observed (which typically has
two velocity components, see App. E). Noteworthy is the strong local increase
of the FWHM up to 4 km s-1 at locations where the sub-filaments connect to the
ridge. This behaviour is also observed in the velocity dispersion maps of
N2H+(1-0) (Schneider et al., 2010) and NH3 (Keown et al., 2019). One would
expect that this is the result of overlapping velocity components, associated
to the sub-filament and the ridge, at this connection. However, App. E shows
that these regions are best fitted with a single Gaussian component. This
suggests either that the overlapping velocity components are closely blended
together or that there is a rapid change in the line-of-sight velocity field
at the edge of the ridge for example due to accretion shocks.
Comparing the FWHM maps for the molecular lines in Fig. 8 with those of the [C
II] line, we observe three clear differences. Firstly, the [C II] line has a
significantly larger linewidth (typically $>$ 4-5 km s-1) than the molecular
lines over the full map. Note that the largest FWHMs are associated with the
DR21 outflow which is not the focus of our study. Secondly, the highest [C II]
FWHM values, excluding the DR21 outflow, are found outside of the dense DR21
ridge, mostly in the eastern region without molecular dense sub-filaments.
This east-west asymmetry with respect to the ridge for the [C II] linewidth is
remarkable and seems to be correlated with the lack of dense sub-filaments
east of the ridge. Thirdly, examining the map in more detail, it is also
observed that the [C II] emission directly surrounding the sub-filaments has a
significantly larger linewidth than the emission towards the sub-filaments. In
Fig. 1, it is observed that this broader [C II] linewidth is the result of the
flat-topped spectra which are due to more blueshifted [C II] emission with
respect to the molecular lines. This more blueshifted [C II] emission is the
result of the prominent blueshifted [C II] velocity component found in App. D
and thus is not necessarily the result of higher turbulent support. This
demonstrates that the molecular lines miss an important part of the picture to
understand the full DR21 molecular cloud dynamics and evolution.

##  4 Analysis

###  4.1 The C+ column density and excitation

In Sect. 3.1, it was estimated that [C II] traces gas between 5$\times$102 and
104 cm-3. Here, we will explore the excitation conditions and regions traced
by the [C II] emission in more detail. To determine the typical temperature of
the C+ gas, we consider the heating from FUV photons. From the FUV field map
calculated in Schneider et al. (2016b), we find that the DR21 cloud is located
in an FUV field up to G0 = 200-400 Habing due to heating from local UV source
in the region (see also Schneider et al., 2023). This fits with the typical
line integrated [C II] intensity towards the surrounding cloud of $\sim$ 50 K
km s-1 which is expected for an FUV field of G0 = 200 Habing in the PDR
Toolbox (Kaufman et al., 2006; Pound & Wolfire, 2008, 2023) at densities
between 5$\times$102 and 104 cm-3. For this FUV-field strength and density
range, the PDR Toolbox predicts temperatures of 100-200 K for the [C II]
emitting gas of the photo-dissociation region (PDR).
With these excitation conditions, it is possible to produce the C+ column
density map over the full extent of the FEEDBACK map using the equation from
Goldsmith et al. (2012)

${\scriptstyle\Delta{\rm T_{A}}=3.43\times 10^{-16}\times\left[1+0.5{\rm
e}^{91.25/{\rm T_{kin}}}\left(1+\frac{2.4\times 10^{-6}}{{\rm
C_{ul}}}\right)\right]^{-1}\frac{{\rm N(C^{+})}}{\delta{\rm v}}}$ (1)

with $\Delta$TA the brightness temperature for a uniform source that fills the
beam, Tkin the kinetic temperature, Cul the collisional de-excitation rate,
N(C+) the C+ column density and $\delta$v the linewidth. Cul is given by

${\rm C_{ul}}={\rm n}\times{\rm R_{ul}}$ (2)

where Rul is the de-excitation rate coefficient given by

${\rm R_{ul}}=3.8\times 10^{-10}\,{\rm cm}^{3}\,{\rm s}^{-1}\,({\rm
T_{kin}}/100)^{0.14}$ (3)

Using a Tkin = 100 K and $n_{\rm H_{2}}$ = 5$\times$103 cm-3 then gives C+
column densities N(C+) $\approx$ 0.4-1.0$\times$1018 cm-2 over the extent of
the cloud which is shown in Fig. 9. However, note that these assumed
excitation conditions are not valid for the DR21 radiocontinuum region as it
is a dense and strongly irradiated PDR region. Comparing this C+ column
density map with the deduced H2 column density map from the Herschel data also
allows to estimate the [C+]/[H2] abundance ratio over the map. This shows that
we find a C+ abundance around 10-4 in the outer parts of the cloud around the
ridge and sub-filaments, which is of the order of the elemental abundance
ratio of carbon ($\chi_{C}$) in the ISM (e.g. Glover & Clark, 2012). Basically
all carbon is thus found in the ionized state (C+) in these regions. Towards
the sub-filaments, we find under these conditions that $\sim$ 10% of the gas
is traced by [C II] and towards the ridge this drops even lower to values
between 0.1 and 1%. We do however note that the density and temperature are an
important assumption. A higher temperature up to 200 K can reduce the C+
column density with $\sim$ 50% while a lower temperature can increase the C+
column density by a factor 2. A lower typical density of 103 cm-3 can increase
the column density with an additional factor 2. However, such a N(C+) increase
due to a low density would result in a C+ abundance above the elemental
abundance. This suggest that a significant part of the [C II] emission
originates from regions with a typical density of $n_{\rm H_{2}}\sim$
5$\times$103 cm-3. Combining this density with the Herschel column density for
the surrounding clouds results in a maximal line-of-sight depth of 2 pc. This
implies that the surrounding cloud of the DR21 ridge has a sheet-like
morphology.
The same analysis is done for 13CO(1-0) in App. B. This analysis indicates
that the 13CO abundance peaks in some sub-filaments and at the edges of the
DR21 ridge. The abundance then decreases in the ridge and in the outer cloud.
Both in the ridge and the outer part of the cloud, 13CO(1-0) appears to trace
only a fraction of the total gas mass. Combined, this provides compelling
evidence that [C II] indeed traces most of the surrounding gas in the DR21
cloud. The regions surrounding the ridge and sub-filaments are thus CO poor
gas that lights up in [C II]. In combination with the molecular line data, [C
II] thus provides a global view of the cloud.

Figure 9: Top: C+ column density map between -8 and 1 km s-1 assuming a
kinetic temperature of 100 K and $n_{\rm H_{2}}$ = 5$\times$103 cm-3 for the
excitation conditions. Note that these excitation conditions, which should be
representative for most of the map, are most likely not valid towards the peak
around the DR21 radiocontinuum region in the south of the map as this region
experiences strong internal heating. Bottom: The resulting [C+]/[H2] abundance
ratio deduced from the Herschel column density map over the DR21 cloud on a
log scale.

Figure 10: Left: HCO+(1-0) PV diagram with the green and red contours
indicating the emission from C18O(1-0) and H13CO+(1-0), respectively. The PV
diagram was constructed perpendicular to the filament from the region
indicated by the red box in Fig. 3 by averaging the intensities along
declination axis in this box. The black triangles follow the peak brightness
velocity of 12CO(3-2) as a function of the distance (r) from the center of the
ridge. This indicates that the gas is redshifted at both sides of the ridge
which is at a velocity of $\sim$ -3 km s-1 (indicated by the dashed horizontal
line). At the ridge, there is a large increase in linewidth, associated with
the velocity gradients observed over the ridge in Fig. 6. This fits with the
observed HCO+(1-0) blue asymmetry in the PV diagram due to self-absorption in
a collapsing cloud. Right: The [C II] PV diagram perpendicular to the DR21
ridge over the same region with the blue line following the peak brightness
velocity. On large scales ($|$r$|>$ 1 pc), the [C II] emission shows a
redshifted V-shape followed with a sharp transition to more blueshifted
emission at ($|$r$|<$ 1 pc) which makes the velocity profile deviate from a
pure V-shape. To highlight this, we fitted a V-shape to the PV diagram which
is indicated in magenta. We interpret this to be due to a second more
spatially concentrated velocity component that is associated with the
formation of the DR21 ridge. The Pearson coefficients for the central velocity
as a function of radius are -0.76 and 0.89 at the east and west of the ridge,
respectively. This points to a relatively well organized, but not perfect
correlation, for the central velocity with radius.

###  4.2 Position-velocity diagram of the ridge

The position-velocity (PV) diagram perpendicular to the DR21 ridge gives an
additional view on the cloud kinematics with respect to the ridge. Figure 10
shows PV diagrams for the optically thick -HCO+(1-0) & 12CO(3-2)- and thin
-H13CO+(1-0) and C18O(1-0)- molecular lines observed with the IRAM 30m and
JCMT (see the red box in Fig. 3 for the covered region). It confirms that the
emission from subfilaments outside the ridge is at more redshifted velocities
with respect to the densest gas in the ridge. This gives rise to a V-like
shape in the PV diagram, similar to what was observed for the Musca filament
(Bonne et al., 2020a) which has about the same size as the DR21 ridge but has
a two orders of magnitude lower mass. For the Musca filament this was proposed
to be the result of magnetic field bending by the interaction of an
overdensity with a more diffuse region in the colliding H I cloud. As for this
previous study of the low-mass Musca filament, the DR21 ridge is located at
the apex of this V-like shape. However, the V-shape for DR21 in molecular
lines is not as clear as the ones observed in Musca. This can be related to
the rapid linewidth increase within 1 pc of the DR21 ridge, while the Musca
filament is trans-sonic (Hacar et al., 2016; Bonne et al., 2020a). Such a
rapid linewidth increase in the densest regions is predicted to be the result
of pc scale gravitational acceleration in massive clouds (e.g. Peretto et al.,
2006, 2007; Hartmann & Burkert, 2007; Gómez & Vázquez-Semadeni, 2014; Watkins
et al., 2019) and could also be the result of accretion driven turbulence
(Klessen & Hennebelle, 2010). A gravitational collapse scenario is also
supported by the strong blue asymmetry self-absorption of HCO+(1-0) that
covers the same velocity interval as the perpendicular velocity gradients over
the ridge (Schneider et al., 2010).
Figure 10 also presents the [C II] PV diagram which traces the surrounding gas
in the DR21 cloud. The PV diagram shows that the surrounding gas forms a
redshifted V-shape perpendicular to the DR21 ridge on large scales ($|$r$|>$ 1
pc). The Pearson correlation coefficients for the central [C II] velocity as a
function of radius are -0.76 and 0.89 on the east and west side of the ridge,
respectively. With bootstrapping we find that these coefficients are far out
of the 95% intervals for the null hypothesis that there is no correlation,
i.e. [-0.19, 0.20] (east) and [-0.18, 0.17] (west), and thus demonstrate a
clearly organized velocity field perpendicular to the ridge. As the cloud
appears to be organized in a sheet, this points at a curved sheet. However, in
[C II] it is also observed that there are deviations from a pure V-shape due
to blueshifted emission in the vicinity of the ridge at $|$r$|$ $\sim$ 1 pc.
This deviation from the V-shape is particularly visualized in the PV diagram
by the sharp peak velocity transition at $|$r$|$ $\sim$ 1 pc and might explain
why the Pearson correlation coefficients are not equal to one. We propose that
this traces two distinct but converging velocity components at -4.5 and -2.5
km s-1 in the DR21 cloud that drive continuous mass inflow to the DR21 ridge
which is located at the intersection of these flows.

###  4.3 Convergence of blue- and redshifted gas in the DR21 cloud

The molecular line observations demonstrated the existence of an organized
velocity field over the ridge, similar to the results presented in Schneider
et al. (2010). This internal velocity field is proposed to be connected to the
inflowing sub-filaments in the surrounding cloud, and is typically considered
to be tracing inflow. Using 8 $\mu$m observations from the Spitzer Space
Telescope (Hora et al., 2009), we attempt to portray a 3D configuration of the
ridge and sub-filaments. Inspecting the 8 $\mu$m map of the DR21 cloud (Fig.
11), it becomes obvious that heavily reduced 8 $\mu$m emission (’IR-dark’)
traces regions inside the ridge, except for the bright compact sources at DR21
and DR21(OH) which shows internal regions heated by stellar/YSO feedback.
Consequently, this indicates that the gas associated with strong 8 $\mu$m
extinction is located at the front side of the DR21 ridge with $N_{\rm
H_{2}}\,>$ 1023 cm-2, as this is a foreground layer absorbing 8 $\mu$m
emission. The region of the ridge with more extended 8 $\mu$m emission is then
located towards the back of the DR21 ridge. Towards the sub-filaments similar,
but lower contrast, 8 $\mu$m extinction is observed. The overlay of the 8
$\mu$m map with contours of molecular and [C II] line emission at v = -4.6 km
s-1 and -1.6 km s-1 indicates that the most blueshifted molecular line
emission corresponds to the extended 8 $\mu$m emission features, while the
redshifted line emission corresponds to the darkest 8 $\mu$m regions. This
clearly indicates that the redshifted gas is located in front of the
blueshifted gas from our point of view, which confirms that the flows in the
ridge and cloud are indeed converging. The red- and blueshifted velocity
component that make the V-shape in the PV diagram are thus accreted on the
ridge. Note that there are several contaminants (DR21, protostellar
objects,…), a lower contrast and a more complex contour distribution that make
the [C II] map less straightforward to evaluate. Nonetheless, there is a clear
association at several locations between the distribution of the redshifted
gas and 8 $\mu$m extinction in the surrounding cloud that is traced by this
line.

Figure 11: Spitzer 8 $\mu$m image of the DR21 cloud overlaid with velocity
contours of C18O(1-0) (left) and [C II] (right) at -1.6 km s-1 and -4.6 km
s-1, and overlaid with the Herschel column density (middle). The northern and
southern part of the ridge shows 8 $\mu$m extinction, i.e. is ’IR-dark’, while
in the middle there is strong emission at the locations of DR21 and DR21(OH).
Some protostellar sources show up as localized emission spots. Outside of the
ridge, the 8 $\mu$m emission is extended, rather diffuse, with some feather-
like features. The contours of C18O(1-0) indicate that the more blueshifted
emission is located towards the region with IR emission in the ridge, while
the redshifted emission is concentrated towards the strong 8 $\mu$m
extinction. The structure of the [C II] emission is more complex as it more
diffuse and also affected by the feedback in DR21 and local protostellar
sources (e.g. around DR21(OH)). However, there is overall a correspondance on
the large scales of the blueshifted gas to bright Spitzer 8 $\mu$m regions and
redshifted gas to 8 $\mu$m extinction. The figure in the middle indicates the
location of DR21, DR21 (OH) and DR21-N for reference of their location.

###  4.4 Virial analysis of the DR21 cloud

In Schneider et al. (2010) it was found that the DR21 ridge is gravitationally
collapsing. With the newly obtained data sets, we can further investigate the
global stability of the molecular cloud as a function of its radius, centered
on the column density peak in DR21(OH). This is done with the different
tracers by estimating the gravitational potential energy, the turbulent energy
and the magnetic energy. In this analysis, the density profile of the cloud
might affect the results that use the simple equations below which correspond
to a sphere with uniform density. The same is valid for the more sheet-like
morphology that we propose for the surrounding DR21 cloud to maintain a
plausible C+ abundance. However, the impact of these differences from a
uniform sphere are predicted to be relatively small ($\lesssim$ 50% for
typical clump density profiles and a not too flattened ellipsoid cloud based
on Bertoldi & McKee, 1992). Therefore, considering the uncertainties, we think
the analysis below is reasonable.
The thermal and turbulent energy is calculated using

${\rm E_{T}}=\frac{3}{2}{\rm M}\,\sigma^{2}$ (4)

with M the mass in the region determined from the Herschel dust column density
map and $\sigma$ the velocity dispersion of the studied region (which we
approximate here with the observed linewidth as there is no direct measure of
the kinetic temperature). The gravitational energy is determined by

${\rm E_{G}}=-\frac{3}{5}\frac{{\rm GM^{2}}}{{\rm R}}$ (5)

where G is the gravitational constant and R the radius. The magnetic energy is
calculated using

${\rm E_{mag}}=\frac{1}{2}{\rm M\,V_{A}^{2}}$ (6)

with VA the Alfvén speed, which is given by VA = $\frac{{\rm
B}}{\sqrt{\mu_{0}\rho}}$ with $\mu_{0}$ the vacuum permeability, $\rho$ the
density, and B the magnetic field strength.
The evolution of mass as a function of radius from DR21(OH) is shown in Fig.
12. The increase in mass flattens as a function of radius at r $>$ 1.5 pc
which is expected with mass concentration in the ridge. For the error on the
mass, we assume an uncertainty of 15% for the derived column density as this
corresponds to typical differences when creating the column density map with
different methods (e.g. Peretto et al., 2016). To estimate the thermal and
turbulent internal support of the cloud, we assume that the linewidths of the
observed lines, i.e. 13CO(1-0), C18O(1-0), H13CO+(1-0) and [C II], are a good
proxy. This turns out to be a good approximation as their linewidth is
dominated by non-thermal motion. Note, that this approach ignores that a
significant part of the linewidth might be associated with organized
convergent flows driven by gravitational collapse or inertial motion instead
of internal support (Traficante et al., 2018a, b, 2020) and that we have
demonstrated the presence of multiple velocity components associated with
inflow. The linewidth as a function of radius for the three considered
molecules is presented in Fig. 12. It shows an increasing linewidth towards
small radii and a fairly constant linewidth at r $>$ 1.5 pc. The increasing
linewidth towards the ridge can fit with gravitationally driven inflow. This
would also fit with predictions by Traficante et al. (2020) since most of the
mass in the DR21 cloud has column densities $\gg$ 0.1 g cm-2 (i.e. $N_{\rm
H_{2}}\gg$ 2.6$\times$1022 cm-2). The estimated kinetic support based on the
linewidth is thus an upper limit on the turbulent support, and might
significantly overestimate it. We do have to note that the current IRAM 30m do
not entirely cover the studied radii, but the [C II] data showed that these
regions are highly CO-poor.

Determining the magnetic field in the cloud as a function of its size is the
most uncertain quantity because there are only a few dust polarization
magnetic field observations in the region. Based on these observations, it was
proposed in Ching et al. (2017) that the magnetic field strength in the DR21
ridge is 0.94 mG. Therefore, we used two approaches to estimate the magnetic
field strength evolution in the cloud. First, we make use of the magnetic
field strength relation from Crutcher et al. (2010) which is given by

${\rm B}=\left\\{\begin{array}[]{ll}0.01\>{\rm
mG\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(n}_{\rm H}<300{\rm\>cm}^{-3})\\\
0.01\>(\frac{{n}_{\rm H}}{300})^{0.65}\>{\rm mG\;\;\;\;(n}_{\rm
H}>300{\rm\>cm}^{-3})\end{array}\right.$ (7)

Secondly, we use B $\propto$ n${}_{\text{H}}^{\text{k}}$ and start from the
constraint that the magnetic field strength is 0.94 mG at the density of the
ridge. To extrapolate the relation, we use two values: k = 0.5 and k = 0.67.
These exponents represent two asymptotic cases: k = 0.67 is generally
considered to describe the case where the magnetic field is dominated by
gravitational collapse, and k = 0.5 the case where the magnetic field plays an
important role in support against gravitational collapse (e.g. Basu, 1997;
Hennebelle et al., 2011). The resulting magnetic field strengths as a function
of radius for the different estimates are shown in Fig. 12. Over the full
cloud there is an uncertainty up to a factor 4 for the magnetic field strength
based on the different relations used. This will thus have a significant
impact on the magnetic energy as will be shown in the next paragraph.
The calculated energy terms are shown in Fig. 13 & Fig. 14, compared to the
gravitational energy. The thermal and kinetic energy, estimated from the
velocity dispersion, is very similar for the different molecules and has
values $<$ 20% of the gravitational energy. In addition, it was pointed out
that these values likely are an upper limit. The relation for [C II], tracing
the lower density gas, is different as it reaches up to 40-60% of the
gravitational energy. This seems to suggest that in the lower density gas,
which is particularly found at r $>$ 2-3 pc, there might be significant
thermal or turbulent support. However, it has to be taken into account that we
found multiple velocity components which do not necessarily contribute to
support against collapse in the region. The magnetic field energy depends on
its assumed field strength evolution. Assuming k = 0.5 indicates that the
cloud can experience some support from the magnetic field on large scales
($\sim$ 3-4 pc), but gradually this magnetic support decreases leading to an
increasing importance of gravitational collapse when approaching the ridge (r
$<$ 3 pc). Assuming k = 0.67, the magnetic field provides little support at
low densities while the higher density regions towards the ridge (r $<$ 1 pc)
experience an increased importance of the magnetic field. Yet, in both cases
the values still indicate that the gravitational energy dominates over the
cloud. This becomes particularly evident in Fig. 14 where the total support
terms over the gravitational terms are plotted. This shows that only
considering k = 0.5 and the velocity dispersion of the [C II] emission allow
for support against gravitational collapse, but this ignores that the [C II]
velocity dispersion might not be associated with mass inflow rather than
turbulent support.

Figure 12: Top: Radial profile of the average density in the DR21 cloud (in
black). The average density within each radius was estimated based on the mass
within the considered radius. The mass within the radius r, is shown at the
axis on the right (in red). Middle: The same for the velocity dispersion of
H13CO+(1-0) (in blue), 13CO(1-0) (in red), C18O(1-0) (in green) and [C II] (in
cyan). The molecular lines show very similar behaviour while the [C II] line
has a significantly higher linewidth. However, note that at the outer radii (r
$\sim$ 3-4 pc) the cloud is not fully covered by the IRAM 30m map (Figs. 1 and
2). Bottom: Estimated characteristic magnetic field strength for the the gas
within a radius for the three different considered profiles. The green curve
shows the predictions by the Crutcher et al. (2010) relation. The red and blue
curves use a magnetic field strength of 0.94 mG at the density of the ridge
and k = 0.5 and 0.67, respectively, in B $\propto$ n${}_{{\rm H}}^{k}$.

Figure 13: Ratio of the magnetic (left) and thermal + kinetic energy (right)
over the gravitational energy as a function of the radius in the DR21 cloud.
The radius is centered on DR21(OH). For the thermal + kinetic energy, the
linewidth of H13CO+(1-0), 13CO(1-0), C18O(1-0) and [C II] are used. It appears
that the regions of cloud traced by [C II] experience more support against
gravitational collapse than the molecular regions. However, the presence of
multiple components has to be kept in mind which makes this an upper limit.
The three approaches to estimate the magnetic field support are displayed on
the left. It shows that the magnetic support is always smaller than the
gravitational energy, but that the importance strongly depends on the
assumptions for the magnetic field strength evolution. Figure 14:  The ratio
of the energy terms that provide support, i.e. Eth+kin and EB, over the
gravitational energy as a function of radius in the DR21 cloud. This is shown
for [C II] (top), which has the largest linewidth, and C18O(1-0) (bottom),
which has the smallest linewidth. Only when considering [C II] and a k=0.5
evolution of the magnetic field do the support terms get larger than the
gravitational energy at r $>$ 3 pc.

##  5 Discussion

Combining the results and analysis from Sects. 3 and 4, we will now discuss
the cloud evolution leading to high-mass star formation in the DR21 ridge.

###  5.1 Magnetic field bending followed by gravitational collapse in the DR21
ridge

From the fitted velocity maps and PV diagrams we found indications of an
accelerating velocity field, based on the rapid linewidth increase in the
ridge and a blueshifted asymmetry in the self-absorbed molecular lines towards
the ridge. Additionally, the observations showed that basically all sub-
filaments and the surrounding cloud appear to be organized in a flattened
structure that gives rise to a redshifted V-shape perpendicular to the ridge.
[C II] also displays a second blueshifted velocity component that is localized
in the vicinity of the ridge and looks like a more localized blueshifted
V-shape.
The accelerating velocity field can be the result of the proposed
gravitationally driven inflow (e.g. Peretto et al., 2006; Hartmann & Burkert,
2007). However, to explain the observed redshifted V-like velocity field in
the surrounding cloud, as well as the systematic redshift of the sub-
filaments, we have to consider the apparent flattened geometry of the
molecular cloud. If the flattened geometry is curved, this can provide a
straightforward explanation for the observed velocity field if this curved
sheet geometry is associated with the magnetic field bending in a collision as
predicted by (Hartmann et al., 2001; Inoue & Fukui, 2013; Vaidya et al., 2013;
Inoue et al., 2018; Abe et al., 2021). The presence of a strong magnetic field
in the compressed sheet might then even initially prevent gravitational
acceleration and significantly affect the velocity field until the massive
ridge is reached. We note that this is the same mechanism that was also
proposed to explain the observations toward the Musca cloud that has a very
similar kinematic V-shape perpendicular to the filament (Bonne et al., 2020b).
There, it was proposed that Musca formed at the apex of a bent magnetic field
as the result of an overdensity interacting with diffuse H I gas in a
colliding flow. In fact, the DR21 region is proposed to be in a high-velocity
(v${}_{col}\gtrsim$ 20 km s-1) collision with the diffuse, mostly atomic,
regions of the other velocity component between vLSR = 9 and 15 km s-1 in the
Cygnus-X region (Schneider et al., 2023). This points to the same formation
mechanism for the DR21 cloud as for the Musca cloud. However, Cygnus-X region
is more than an order of magnitude more massive than the Chamaeleon-Musca
region (e.g. Reipurth & Schneider, 2008) and the collision velocity is higher,
which can explain why the DR21 ridge is forming massive stars (e.g. Dobbs et
al., 2020; Liow & Dobbs, 2020). The second [C II] velocity component we found
in the vicinity of the DR21 ridge at vLSR $\sim$ -4.5 km s-1 is then likely
gas associated with the $\gtrsim$20 km s-1 collision, responsible for the
formation of the DR21 ridge, that leads to accretion on the ridge from the
blueshifted side.
One could consider the scenario that the DR21 cloud is a swept-up shell
created by expansion from a local bubble. However, there is no evident massive
cluster candidate near the DR21 cloud that could drive the expansion of such a
massive region (Maia et al., 2016) and the study of the multiple velocity
components by Schneider et al. (2023) does not unveil a typical expanding
shell morphology in the region.

In summary, we propose that the DR21 ridge was formed as the result of a high-
velocity collision in the Cygnus-X region which bends the magnetic field
around the ridge. This bending is at the origin of the inflow seen in the form
of a kinematic V-shape in the flattened surrounding cloud. However, due to the
high-density in the compressed cloud, gravity increasingly starts to dominate
the curved kinematics driven by the magnetic field bending. This explains the
observed accelerated motion in the ridge and the results from the virial
analysis. The proposed scenario thus provides a comprehensive explanation for
the rich set of observations that trace the density range of the DR21 cloud.
Observations that trace the line-of-sight magnetic field morphology, and thus
the proposed bending, would be invaluable to further test this scenario.
Lastly, we note that gravity did not take over the cloud kinematics in Musca
(Bonne et al., 2020a). This is an important difference because the increasing
importance of gravity on larger scales in the DR21 cloud allows more effective
mass provision to the center(s) of collapse, e.g. the DR21(OH) and DR21-N
clumps (Schneider et al., 2010).

###  5.2 Mass accretion on the DR21 ridge

The mass inflow towards the ridge guided by the sub-filaments can be estimated
using $\dot{\rm M}_{acc}$ = N $\pi$R2vinfn (Schneider et al., 2010; Peretto et
al., 2013), with N the number of inflowing filaments, R the radius, n the
density and vinf the velocity of the infalling filaments. Working with 7
molecular inflowing subfilaments with n = 105 cm-3, R = 0.1 pc (Hennemann et
al., 2012) and an inflow velocity of 1-2 km s-1, we get an estimated mass
accretion rate of 1.3-2.6$\times$10-3 M⊙ yr-1 similar to the estimate of
Schneider et al. (2010). In Schneider et al. (2010), it was also shown that
the F3 sub-filament continues its flow towards the DR21(OH) clump in the ridge
and may provide inflow to multiple MDCs. However, this mass inflow rate is
roughly an order of magnitude lower than the required mass accretion rate
($\sim$ 10-2 M⊙ yr-1) to replenish the DR21 ridge within 1 Myr.
Taking into account that the mass inflow probably also happen off the sub-
filaments, we can calculate this contribution of mass inflow to the ridge. We
consider inflow over the full surface of the ridge for which we assume a
cylindrical geometry with R = 0.36 pc and L = 4.15 pc (Hennemann et al.,
2012). For the inflow, we work with a velocity of $\sim$ 1-2 km s-1 and n =
104 cm-3, which is justified by $N_{\rm{H_{2}}}\sim$ 3$\times$1022 cm-2 within
the 1 pc vicinity of the ridge. This density is at the high range of densities
in the ambient gas, which can be expected if the gas is gravitationally
concentrated when approaching the ridge. These values give an estimated mass
accretion rate of 0.56-1.1$\times$10-2 M⊙ yr-1, which can replenish the DR21
ridge in 1-2 Myr. Note that this might be an overestimate up to a factor
$\sim$2 as the inflow is predominantly sideways in the sheet and not uniform.
1-2 Myr is an expected lifetime for the DR21 ridge when taking the expression
from Clarke & Whitworth (2015) for the timescale of a free-fall longitudinal
collapse for a filament

$\tau_{{\rm ff}}=(0.49+0.26\,{\rm A})({\rm G}\,\rho)^{-\frac{1}{2}}$ (8)

where A (= 5.8) is the initial aspect ratio of the filament half length over
the filament radius and $\rho$ (= 3.9$\times$10-19 g cm-3) the average density
of the ridge. This results in a $\tau_{ff}$ of 0.39 Myr. Considering support
against longitudinal collapse from the magnetic field, which is perpendicular
to the DR21 ridge at the center (Vallée & Fiege, 2006; Matthews et al., 2009;
Ching et al., 2022), 1-2 Myr likely is an appropriate time scale to replenish
the DR21 ridge. We thus propose that the DR21 ridge can be continuously
replenished by mass inflow from the surrounding cloud.
Only when feedback from the newly formed massive stars disperses the dense
ridge and prevents further mass accretion it will end active high-mass star
formation. This dispersal of the ridge by stellar feedback, which is proposed
to happen on relatively short timescales after the first massive stars form
(Hollyhead et al., 2015; Luisi et al., 2021; Chevance et al., 2022; Bonne et
al., 2022b) (i.e. $\lesssim$ 3 Myr), might thus be important to maintain a low
star formation effiency (SFE). Even though stellar feedback might halt mass
provision to the ridge, it is observed around the DR21 radiocontinuum source
that regions of dense gas a bit further out, in particular the nearby sub-
filaments (S & SW), remain present. This might be able to maintain continued
lower mass star formation while stellar feedback is already dispersing the
ridge. For example, low-mass cores were detected in the sub-filament F3
(Hennemann et al., 2012).

###  5.3 The link with high-mass star formation

The above scenario implies that high-mass star formation might only occur over
an interval of the total star formation time in a cloud. In the DR21 cloud,
the formation of high-mass stars would be directly related to the lifetime of
the massive ridge. This potentially shorter phase of high-mass star formation
might fit with the observed shallow slope of the core mass function (CMF) in
high-mass star forming regions (Motte et al., 2018b; Liu et al., 2018; Pouteau
et al., 2022) compared to the power law tail of the initial mass function, as
well as the slightly shallower initial mass functions (IMF) in young clusters
($<$ 4 Myr) of Cygnus-X (Maia et al., 2016). However, further work with e.g.
the ALMA-IMF sample (Motte et al., 2022), will be needed to better understand
the evolution of these top-heavy CMFs in high-mass star forming regions and
how it connects with the IMF of young stellar clusters as well as the
evolution of high-mass star forming clouds and filaments (Pouteau et al.,
2022).

##  6 Conclusion

We have presented new molecular line data of 13CO(1-0), C18O(1-0), HCO+(1-0)
and H13CO+(1-0) from the IRAM 30m telescope and a [C II] map from the FEEDBACK
SOFIA legacy survey of the DR21 cloud. In this study, we exclusively focus on
the emission between -8 km s-1 and 1 km s-1, which is associated with the DR21
cloud, for all lines.
The molecular lines trace the dense ridge ($N_{\rm H_{2}}>$1023 cm-2) and sub-
filaments while [C II] traces the surrounding molecular cloud (i.e. $N_{\rm
H_{2}}\lesssim$ 1022 cm-2) that embeds these filamentary structures. We
demonstrate that this surrounding cloud seen with [C II] is mostly CO poor.
Both in the molecular and [C II] spectra we find indications of more than one
velocity component within the studied velocity range. We also note there is an
offset for the peak velocity in the cloud between the [C II] and molecular
lines, confirming they trace different regions of the cloud. Inside the ridge,
the molecular lines confirm the results from Schneider et al. (2010) while the
observations of the sub-filaments show they are systematically redshifted with
respect to the ridge. The [C II] kinematics show a redshifted V-like shape,
between -3 and -1 km s-1, perpendicular to the ridge that intersects with a
more blueshifted velocity component (at vLSR $\sim$ -5 km s-1) at the location
of the ridge. We confirm that these two intersecting velocity components
converge, and that there is continuous mass accretion on the DR21 ridge. The
V-shape velocity field embeds all the redshifted sub-filaments, and density
requirements to explain the observed [C II] emission indicate that the V-shape
has a flattened morphology. Tracing the CO poor gas, seen in [C II] towards
the DR21 cloud, is thus essential for a comprehensive view on cloud evolution
and filament formation.
Performing a virial analysis, including estimates for the magnetic field
support, we find that the cloud should be gravitationally collapsing on pc
scales. The surrounding gas traced by [C II] appears to have more support
against gravitational collapse, but this does not take into account the
organised velocity field and presence of two velocity components that are not
associated with turbulent support.

We propose that there is continuous mass accretion onto the DR21 ridge. This
accretion is initiated by the bending of the magnetic field around the ridge
due to a large-scale collision. In this collision, an overdensity is
compressed into a dense filament/ridge during the interaction with diffuse
atomic regions of the colliding cloud. However, because of the high density in
the DR21 cloud, gravity is taking over the cloud dynamics which facilitates
rapid mass provision to a (few) center(s) of collapse where massive clusters
are forming. We found that the continuous mass inflow, visible thanks to [C
II], can replenish the ridge. Therefore we propose that the DR21 ridge will
remain present until stellar feedback will disperse it and halt massive star
formation
Lastly, we note that this scenario is the same as proposed for the low-mass
Musca cloud (Bonne et al., 2020a), except for the increasing importance of
gravitational collapse in the massive DR21 cloud. This hints that the same
scenario might be at the origin of filament formation in both a low- and high-
mass star forming region. Furthermore, recent magnetic field and velocity
observations of some nearby low- and high-mass star forming regions (e.g.
Tahani, 2022; Arzoumanian et al., 2022) appear to have a straightforward
explanation in this scenario. This leads us to tentatively propose that these
types of collisions are a widespread mechanism in the Milky Way to initiate a
wide variety of star formation activity in the Milky Way.

We thank W. Lim and F. Comeron for insightful comments on an early draft of
the paper. Based in part on observations made with the NASA/DLR Stratospheric
Observatory for Infrared Astronomy (SOFIA). SOFIA is jointly operated by the
Universities Space Research Association, Inc. (USRA), under NASA contract
NAS2-97001, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK
0901 to the University of Stuttgart. This work was supported by the Agence
National de Recherche (ANR/France) and the Deutsche Forschungsgemeinschaft
(DFG/Germany) through the project ”GENESIS” (ANR-16-CE92-0035-01/DFG1591/2-1)
and the DFG project number SFB 956. The FEEDBACK project is supported by the
Federal Ministry of Economics and Energy (BMWI) via DLR, Projekt Number 50 OR
2217 (FEEDBACK-plus). Financial support for the SOFIA Legacy Program,
FEEDBACK, at the University of Maryland was provided by NASA through award
SOFO070077 issued by USRA.

##  Appendix A [C II] and JCMT 12CO(3-2) observations

In Fig. 15 we present a comparison of the [C II] integrated intensity and JCMT
12CO(3-2) integrated intensity. This JCMT data over the full Cygnus-X region
will be the topic of a forthcoming paper. Because of the larger column density
than for 13CO, the 12CO emission is detected further into the cloud compared
to 13CO (see Fig. 2). However, 12CO still is particularly sensitive to the
massive ridge and surrounding subfilaments and has a clearly distinct
morphology compared to [C II]. This confirms that [C II] provides a unique
view on the lower-column density gas that is not accessible with molecular
lines.

Figure 15: The [C II] integrated intensity map from -8 to 1 km s-1 as in Fig.
3. Here, the contours indicate the JCMT 12CO(3-2) emission starting at 20 K km
s-1 with incremental steps of 20 K km s-1.

##  Appendix B The 13CO abundance

The calculation of the 13CO column density map allows to estimate 13CO
abundance based on the background subtracted Herschel column density map
(Schneider et al., 2016a). This then allows to explore which regions of the
cloud are traced by the observed 13CO(1-0) transition. The calculation of the
13CO column density follows the calculations presented in (Mangum & Shirley,
2015). Below we recapitulate the equations and values used. Note that these
equations assume that the molecule is optically thin. This assumption is not
necessarily valid for 13CO(1-0) over the full map, in particular at the ridge.
However, due to the high density in the ridge, which will result in freeze-out
of the molecule, the opacity should not be excessive even in these regions of
very high column density. When studying the outer regions of the ridge and the
ambient gas, C18O is barely detected which implies that the 13CO opacity will
not be significantly above 1 for standard isotopologue ratios. Nonetheless, it
has to be kept in mind this assumption is a limitation of the calculations
below.
The total column density is then calculated from

${\rm N}[{\rm cm}^{-2}]={\rm f(T_{ex})}\int{\rm T_{mb}dv}$ (B1)

Here Tmb is the main beam brightness in Kelvin, v the velocity in km s-1 and
f(Tex) is given by

${\rm f(T_{ex})}=\frac{3{\rm hZ}}{8\pi^{3}\mu^{2}{\rm J_{t}}}\frac{{\rm
exp(h}\nu/{\rm kT_{ex})}}{[1-{\rm exp(-h}\nu/{\rm kT_{ex})}]({\rm
J(T_{ex})-J(T_{BG})})}$ (B2)

In this equation, Z is the partition function

${\rm Z}=\frac{2{\rm kT_{ex}}}{{\rm h}\nu}+1/3$ (B3)

with k the Boltzmann constant, h the Planck constant, $\nu$ the transition
frequency (for 13CO(1-0) this is 110.201 GHz) and Tex the excitation
temperature for which we assume a value of 18 K, see also Schneider et al.
(2010).
Further in the equation of f(Tex), $\mu$ is the dipole moment (for 13CO(1-0)
this is 0.112 Debye), Jt is the upper value of the rotational quantum number
(i.e. 1) and J(Tex) is given by

${\rm J(T_{ex})}=\frac{{\rm h}\nu}{{\rm k[exp(h}\nu/{\rm kT_{ex}})-1]}$ (B4)

For the background (BG), a temperature of 2.73 K is assumed based on the
cosmic microwave background (CMB).

The estimated 13CO abundance map, based on the Herschel column density map, is
presented in Fig. 16. Using the assumptions above, 13CO is most abundant at
the edges of the ridge and in the sub-filaments (F3N, F3S and SW in
particular). Inside the ridge, there is a significant decrease with increasing
column density, see also Fig. 17, which is expected because of CO depletion at
densities $n_{\rm H_{2}}>$ 104 cm-3. To trace the DR21 ridge kinematics, one
thus needs to rely on tracers like HCO+, N2H+ and deuterated molecules to
probe the very dense gas of the ridge. In the surrounding cloud, at $N_{\rm
H_{2}}<$ 1.5$\times$1022 cm-2 where 13CO is detected, there is also an
observed lower 13CO abundance. Indeed, Fig. 17 shows that the 13CO abundance
starts decreasing again towards lower column densities. This is the result of
an increasing CO dark gas fraction in the DR21 cloud below these column
densities. From Fig. 17, it is also observed that at $N_{\rm H_{2}}<$
1.5$\times$1022 cm-2 the [C II] abundance continues to increase up to the
typical carbon abundance in the ISM. This indicates that [C II] allows to
trace the CO dark gas in the surrounding DR21 cloud.

Figure 16: Map of the 13CO abundance on a log-scale over the DR21 cloud. This
is deduced from the 13CO(1-0) intensity, a kinetic temperature of 18 K and the
Herschel column density map. The white contours indicate the Herschel column
density map at N = 1022 cm-2, 3$\times$1022 cm-2, 1023 cm-2 and 4$\times$1023
cm-2.

Figure 17: Top: 13CO abundance as a function of the Herschel column density
for all pixels in the map. Bottom: The same for the C+ abundance.

##  Appendix C The [C II] optical depth

In order to constrain the [C II] optical depth towards the DR21 cloud, and in
particular the ridge, we work with the average spectrum over the ridge in
order to reduce the noise rms. This decrease in noise rms is important in
order to try and detect the [13C II] hyperfine structure lines that are
located at -65.2 km s-1 (F = 1-0), 11.7 km s-1 (F = 2-1) and 62.4 km s-1 (F =
1-1) with respect to the [C II] line. The hyperfine structure lines have a
relative strength of 0.25 (F = 1-0), 0.625 (F = 2-1) and 0.125 (F = 1-1).
Ideally one thus works with the F = 2-1 transition, but this line is very
close to the actual [C II] emission and contaminated by the observed bridging
with a higher velocity component (Schneider et al., 2023). At the location of
the F = 2-1 transition no indication of additional [13C II] emission is found,
see Fig. 18 but we will not work with this transition because of the confusion
from the actual [C II] emission. Therefore we focus on the second brightest
transition (F = 1-0) at -65.2 km s-1. Fig. 18 shows that this transition is
not detected at a noise rms of 7.6$\times$10-1 K within 0.5 km s-1 for the
averaged spectrum. Although not detected, we still try to obtain an idea of
the optical depth associated with this noise level. This can be done using the
equation

$\frac{\rm T_{mb,12}}{\rm T_{mb,13}}=\frac{1-{\rm e}^{-\tau}}{\tau}\alpha$
(C1)

With $\tau$ the optical depth and and $\alpha$ the local 12C/13C abundace for
which we take a value of 60 (Wilson & Rood, 1994) and a correction factor of
0.25 for the relative strength of the F = 1-0 transition. Then using the noise
rms value (7.6$\times$10-1 K) results in an optical depth of $\tau=$ 1.7.
Since the F = 1-0 transition is not detected, this indicates $\tau=$ 1.7
rather is an upper limit which indicates that the [C II] line is optically
thin or marginally optically thick at best towards the DR21 ridge.
To further investigate the potential optical depth of the [C II] line, we also
run RADEX simulations (van der Tak et al., 2007) using input values obtained
in this paper for the [C II] emitting region towards the DR21 ridge. These
calculations are presented in Tab. 2 which shows that the predicted optical
depth typically is below unity. Note that the highest obtained optical depths,
which are still only of the order of unity and occur at T${}_{kin}\approx$ 30
K, have a too low corresponding line brightness such that they likely are not
representative for the observed emission. This is in agreement with the
results from Sec. 4.1 which indicates that the kinetic temperature in the [C
II] emitting gas is likely of the order of 100 K (for which we obtain lower
optical depths).

Figure 18: Left: [13C II] F = 1-0 transition shifted to the same velocity as the [C II] line. There is no detected counterpart for this [13C II] transition at the velocities of the DR21 ridge component. Middle: The same for the [13C II] F = 1-1 transition. Right: The same for the [13C II] F = 2-1 transition. For this transition, there is confusion from the higher velocity components (Schneider et al., 2023). Therefore, we do not use this transition here. N(C+) (1017 cm-2) | Tkin (K) |  $n_{\rm H_{2}}$ (cm-3) | FWHM (km s-1) | $\tau$ | TR (K)
---|---|---|---|---|---
5.0 | 30 | 104 | 4.0 | 7.7$\times$10-1 | 1.6
5.0 | 50 | 104 | 4.0 | 6.1$\times$10-1 | 5.2
5.0 | 75 | 104 | 4.0 | 4.7$\times$10-1 | 9.0
5.0 | 100 | 104 | 4.0 | 3.9$\times$10-1 | 1.2$\times$101
8.0 | 30 | 104 | 4.0 | 1.2 | 2.2
8.0 | 50 | 104 | 4.0 | 9.7$\times$10-1 | 7.4
5.0 | 30 | 103 | 4.0 | 8.2$\times$10-1 | 4.3$\times$10-1
Table 2: Results produced with RADEX for conditions (N(C+), $n_{\rm H_{2}}$,
Tkin, FWHM) that could be representative of the [C II] emitting gas. TR is the
radiation temperature, which is equivalent to the main beam temperature under
the assumption that the beam filling is unity (this should be a reasonable
assumption seeing the extended [C II] emission). The optical depth ($\tau$)
predictions are basically always below unity which suggests relatively
optically thin [C II] emission.

##  Appendix D [C II] multi-component fit

As the [C II] emission likely is not optically thick, we examine here whether
more than one Gaussian component is required to explain the observed [C II]
line spectral line profiles. To fit all spectra over the full map within the
-8 to 1 km s-1 velocity range, we employ the Beyond The Spectrum (BTS) fitting
tool which is updated with the Akaike Information Criterion (AIC) (Clarke et
al., 2018, 2019, in prep.). However, we do exclude the DR21 radiocontinuum
source and associated outflow from the analysis. As the [C II] noise rms is
still relatively high at the native resolution, which can mask multiple
velocity components, we smoothed the data to a spatial resolution of 30′′.
This value was taken to balance significantly reducing the noise rms while
avoiding the inclusion of large velocity gradients in a single beam which
would give rise to additional velocity components. Note that the 30′′ data is
also used in Schneider et al. (2023). In the fitting iteration procedure, when
a fit has been done with n velocity peaks the fit will be repeated with n-1
and n+1 velocity peaks and the corrected AIC is calculated for each new fit.
Then the fit with the smallest number of velocity peaks is chosen unless a
higher number of velocity peaks is smaller by $\Delta_{AIC}>$ 10 (see e.g.
Clarke et al. (2019) for the equations). This process is iterated until the
number of peaks does not change anymore. The defined $\Delta_{AIC}$ parameter
was found to be robust not to overfit noisy spectra, in particular for large
spectral cubes (Clarke et al. in prep.).
The resulting number of velocity components over the map in the -8 to 1 km s-1
velocity range are presented in Fig. 19. This demonstrates that the regions
with two velocity components are in particular located at the edge of the DR21
ridge. To test this result, we also fitted the [C II] data cube between -8 and
1 km s-1 at the 20′′ resolution. This demonstrated a similar spatial
distribution of the two velocity components. Only, because of the higher noise
rms, the procedure found less pixels with two velocity components. Comparing
the number of velocity components with the [C II] velocity field in Fig. 20,
we find that regions with two velocity components have a close spatial
correlation with the regions that have a blueshifted velocity in [C II]. This
confirms that the location of the DR21 ridge is associated with the
intersection of the proposed V-shape accretion flow with an additional
velocity component. Lastly, Fig. 21 shows the velocity distribution of the two
fitted velocity components, while Fig. 22 shows the associated maps for the
velocity and linewidth. The more blueshifted velocity component is typically
found within the range of -6 km s-1 and -4 km s-1 while the redshifted
component varies between -4 and -1 km s-1. The typical velocity difference of
the two converging components is between 2 and 3 km s-1.

Figure 19: Number of fitted velocity components within the -8 to 1 km s-1
velocity range. The white box indicates the region around the DR21 radio
continuum source and its associated outflow which artificially increases the
number of fitted velocity components. The white contours indicate the Herschel
column density at $N_{\rm{H_{2}}}$ = 1022, 3$\times$1022, 1023 and
4$\times$1023 cm-2.

Figure 20: Left: [C II] velocity field from Fig. 7 with the black contours
indicating the regions that are fitted with two velocity components between
vLSR -8 and 1 km s-1. Right: The same for the [C II] FWHM map. Figure 21: Peak
velocity distribution of the blueshifted (blue) and redshifted (red) velocity
components of the double Gaussian fit to the data.

Figure 22: Top: (left) The [C II] velocity field of the DR21 cloud when taking
into account all emission between -8 and 1 km s-1. The black contours indicate
the Herschel column density at $N_{\rm{H_{2}}}$ = 1022, 3$\times$1022, 1023
and 4$\times$1023 cm-2. (middle left) The [C II] velocity field in the pixels
that have only one velocity component. (middle right) The [C II] velocity
field of the blueshifted component in the pixels that were fitted with two
Gaussian components. (right) The [C II] velocity field of the redshifted
component in the pixels that were fitted with two Gaussian components. Bottom:
The same as above for the FWHM.

##  Appendix E C18O(1-0) and H13CO+(1-0) multi-component fit

We also performed a multi-component fit with the BTS fitting tool to the
C18O(1-0) and H13CO+(1-0) molecular lines between -8 and 1 km s-1 as they are
not affected by strong opacity or self-absorption effects. This shows that
there are significant regions that host two resolved velocity components in
both lines, see Figs. 23 & 26. These regions are particularly in and at the
edge of the DR21 ridge. It is noteworthy when inspecting these figures that
the regions where the subfilaments connect to the ridge, which is associated
with a high linewidth (FWHM $>$3 km s-1), are fitted with a single velocity
component. Either this suggests that several velocity components blend closely
together there or that there might be a rapid reorientation in the line-of-
sight velocity of the inflow or accretion shocks. From the velocity field
maps, it can be observed that these regions are associated with a significant
changes in velocity, see Figs. 25 & 28. Inspecting the velocity distribution
for the regions with two fitted components, see Figs. 24 & 27, it is found
that the interval for the redshifted velocity component is very similar for [C
II] and the molecular lines. This tends to confirm that the redshifted
filaments are embedded in the redshifted inflowing mass reservoir. However,
for the blueshifted velocity components, it appears that [C II] emission is
shifted towards more blueshifted velocities with respect to the emission in
the molecular lines. This shift for the blueshifted [C II] velocity component
fits with the fact that there are no blueshifted subfilaments and thus that
the blueshifted envelope of the DR21 ridge has a different morphology than the
redshifted envelope.

Figure 23: Left: The same as Fig. 19 for C18O(1-0) with the column density
contours in black. Middle & right: Same as Fig. 20 for C18O(1-0). Figure 24:
The same as Fig. 21 for C18O(1-0). The grey histogram indicates the peak
velocity distribution for the pixels with a single fitted velocity component.

Figure 25: The same as Fig. 22 for C18O(1-0).

Figure 26: The same as Fig. 23 for H13CO+(1-0) Figure 27: The same as Fig. 24
for H13CO+(1-0)

Figure 28: The same as Fig. 22 for H13CO+(1-0).

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