11institutetext:  Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, 830011 Urumqi, PR China
11email<EMAIL_ADDRESS>22institutetext: University of Chinese Academy of
Sciences, 100080 Beijing, PR China 33institutetext: Key Laboratory of Radio
Astronomy, Chinese Academy of Sciences, 830011 Urumqi, PR China
44institutetext: Xinjiang Key Laboratory of Radio Astrophysics, Urumqi 830011,
PR China 55institutetext: Max-Planck-Institut für Radioastronomie, Auf dem
Hügel 69, 53121 Bonn, Germany 66institutetext: Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210008, PR China 77institutetext: Max-
Planck-Institut für Extraterrestrische Physik, Giessenbachstr. 1, 85748
Garching bei München, Germany 88institutetext: Shanghai Astronomical
Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, PR
China 99institutetext: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, PR China 1010institutetext: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, PR China
1111institutetext: School of Mathmatics and Physics, Jinggangshan University,
343009 Ji’an, PR China 1212institutetext: School of Astronomy and Space
Science, Nanjing University, 210093 Nanjing, PR China

# Kinetic temperature of massive star-forming molecular clumps measured with
formaldehyde

V. The massive filament DR21

X. Zhao Kinetic temperature of massive star-forming molecular clumps measured
with formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehyde X. D. Tang Kinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehyde C. Henkel Kinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehyde Y. Gong Kinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehyde Y. Lin Kinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehyde D. L. Li Kinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehyde Y. X. He Kinetic temperature of massive star-
forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehyde Y. P. Ao Kinetic temperature of massive star-forming molecular
clumps measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehyde X. Lu Kinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehyde T. Liu Kinetic temperature of massive star-forming molecular
clumps measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehyde Y. Sun Kinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehyde K. Wang Kinetic temperature of massive star-forming molecular
clumps measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehyde X. P. Chen Kinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehyde J. Esimbek Kinetic temperature of massive star-forming molecular
clumps measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehyde J. J. Zhou Kinetic temperature of massive star-
forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehydeKinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehyde J. W. Wu Kinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehyde J. J. Qiu Kinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehyde X. W. Zheng Kinetic temperature of massive star-
forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehyde J. S. Li
Kinetic temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehyde C. S. Luo Kinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehyde Q. Zhao Kinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehyde

The kinetic temperature structure of the massive filament DR21 within the
Cygnus X molecular cloud complex has been mapped using the IRAM 30 m
telescope. This mapping employed the para-H2CO triplet ($J_{\rm K_{a}K_{c}}$ =
303–202, 322–221, and 321–220) on a scale of $\sim$0.1 pc. By modeling the
averaged line ratios of para-H2CO 322–221/303–202 and 321–220/303–202 with
RADEX under non-LTE assumptions, the kinetic temperature of the dense gas was
derived, which ranges from 24 to 114 K, with an average temperature of 48.3
$\pm$ 0.5 K at a density of $n$(H2) = 105 cm-3. In comparison to temperature
measurements using NH3 (1,1)/(2,2) and FIR wavelengths, the para-H2CO (3–2)
lines reveal significantly higher temperatures. The dense clumps in various
regions appear to correlate with the notable kinetic temperature ($T_{\rm
kin}$ $\gtrsim$ 50 K) of the dense gas traced by H2CO. Conversely, the
outskirts of the DR21 filament display lower temperature distributions
($T_{\rm kin}$ $<$ 50 K). Among the four dense cores (N44, N46, N48, and N54),
temperature gradients are observed on a scale of $\sim$0.1-0.3 pc. This
suggests that the warm dense gas traced by H2CO is influenced by internal star
formation activity. With the exception of the dense core N54, the temperature
profiles of these cores were fitted with power-law indices ranging from $-$0.3
to $-$0.5, with a mean value of approximately $-$0.4. This indicates that the
warm dense gas probed by H2CO is heated by radiation emitted from internally
embedded protostar(s) and/or clusters. While there is no direct evidence
supporting the idea that the dense gas is heated by shocks resulting from a
past explosive event in the DR21 region on a scale of $\sim$0.1 pc, our
measurements of H2CO towards the DR21W1 region provide compelling evidence
that the dense gas in this specific area is indeed heated by shocks
originating from the western DR21 flow. Higher temperatures as traced by H2CO
appear to be associated with turbulence on a scale of $\sim$0.1 pc. The
physical parameters of the dense gas as determined from H2CO lines in the DR21
filament exhibit a remarkable similarity to the results obtained in OMC-1 and
N113 albeit on a scale of approximately 0.1-0.4 pc. This may imply that the
physical mechanisms governing the dynamics and thermodynamics of dense gas
traced by H2CO in diverse star formation regions may be dominated by common
underlying principles despite variations in specific environmental conditions.

###### Key Words.:

stars: formation – stars: massive – ISM: clouds – ISM: molecules – radio
lines: ISM

##  1 Introduction

Increasing evidence indicates that the physical properties of the interstellar
medium (ISM) have an impact on the star formation rate, the spatial
distribution and essential properties of the next generation of stars, such as
the elemental composition and initial mass function (e.g., Paumard et al.
2006; Kennicutt 1998a, b; Klessen et al. 2007; Papadopoulos et al. 2011; Zhang
et al. 2018; Tang et al. 2019). High-mass stars exert significant influence on
their surroundings and subsequent star formation through their feedback such
as outflows, winds, and UV radiation. However, the specifics of the high-mass
star formation process and how their feedback affects the initial conditions
of high-mass stars during formation are not yet fully understood. To
investigate the intricacies or star formation within molecular gas, it is
crucial to obtain accurate measurements of its physical properties such as
kinetic temperature and volume density.

###  1.1 DR21 Filament

The Cygnus X molecular cloud complex provides a large sample of star formation
regions that exhibit diverse physical conditions and evolutionary stages,
making it an ideal laboratory for investigating the process of star formation
(e.g., Reipurth & Schneider 2008; Schneider et al. 2010). The DR21 filament at
a distance of $\sim$1.5 kpc (Rygl et al., 2012) is the densest and most
massive region in the entire Cygnus X molecular complex, which is one of the
most active star-forming regions within 2 kpc from the Sun (e.g., Schneider et
al. 2006; Motte et al. 2007; Davis et al. 2007; Kumar et al. 2007; Beerer et
al. 2010; Csengeri et al. 2011). The DR21 filament is an elongated and dense
molecular ridge that spans $\sim$4 pc in length, extending in a North-South
direction. It hosts a mass of $\sim$104 M⊙, which is typically defined by high
column densities, i.e., $N$(H2) ¿ 1023 cm-2 (Schneider et al., 2010; Hennemann
et al., 2012). The DR21 filament encompasses three prominent regions, namely
DR21, DR21(OH), and W75S. Additionally, the DR21W1 region situated to the west
of the DR21 region is associated with the western DR21 flow (e.g., Jaffe et
al. 1989; Lane et al. 1990; Davis et al. 2007; Zapata et al. 2013). The two
well-known star-forming regions DR21 and DR21(OH) are located in the southern
and central parts of the DR21 filament, respectively. DR21 is a relatively
more evolved massive star-forming region, which harbors a group of luminous H
II regions and massive protostars associated with very energetic outflows
(e.g., Harris 1973; Garden et al. 1991a; Russell et al. 1992; Kogan & Slysh
1998; Cyganowski et al. 2003; Rygl et al. 2012). The observed outflows in DR21
exhibit a bipolar structure, characterized by its most prominent molecular
lobes extending in a northeast-southwest direction (e.g., Davis et al. 2007;
Schneider et al. 2010; Skretas et al. 2023). DR21(OH) is an extremely massive
star-forming region with intensive maser activity (e.g., Genzel & Downes 1977;
Norris et al. 1982; Batrla & Menten 1988; Kogan & Slysh 1998; Harvey-Smith et
al. 2008; Schneider et al. 2010). The W75S region, which contains several far-
infrared (FIR) sources, is positioned to the north of the DR21 filament
(Harvey et al., 1986; Wilson & Mauersberger, 1990; Davis et al., 2007; Kumar
et al., 2007; Reipurth & Schneider, 2008).

Extensive investigations of the DR21 filament have been conducted across
various wavelengths, including radio, millimeter, and infrared wavelengths.
The region displays a remarkable complexity, with numerous infrared filaments
oriented perpendicularly to the ridge, indicating potential sites of star
formation as evidenced by observations from the Spitzer and Herschel
telescopes (Marston et al., 2004; Kumar et al., 2007; Beerer et al., 2010;
Hennemann et al., 2012). Dust continuum studies reveal a chain of dusty, dense
cores along the ridge of the DR21 filament (Chandler et al., 1993; Vallée &
Fiege, 2006; Motte et al., 2007; Cao et al., 2019). A large number of
molecular line observations have been performed, such as in CO, CN, CS, HCN,
HCO+, CH3OH, H2CO, NH3, and N2H+ (e.g., Wilson & Mauersberger 1990; Vallée &
Fiege 2006; Schneider et al. 2006, 2010, 2016; Zapata et al. 2012; Dobashi et
al. 2019; Keown et al. 2019; Cao et al. 2022; Bonne et al. 2023; Gong et al.
2023b), which reveal the dynamics and evolution of the dense molecular
structure of the DR21 filament on large scales.

The gas temperature structure of the DR21 filament has been mapped in NH3
(1,1) and (2,2) with the GBT (beam size $\sim$32′′; Keown et al. 2019). The
typical kinetic temperature of the dense gas, derived from NH3 (2,2)/(1,1)
line ratios, fall within the range of 20–30 K. In specific regions, the
typical gas kinetic temperatures are around 20 K in the W75S region,
approximately 22–64 K in the DR21(OH) region, and about 35–60 K in the DR21W1
region. Mangum et al. (1992) presented high-resolution imaging of the NH3
(1,1) and (2,2) lines within the DR21(OH) region, utilizing the Very Large
Array (VLA) with a beam size of $\sim$4′′. By also analyzing the line ratios
of NH3 (2,2)/(1,1) within dense ammonia cores, they derived the gas kinetic
temperatures. These temperatures range from 20 to $>$80 K, with a typical
value of approximately 30 K. The dust temperature structure of the DR21
filament has been derived from spectral energy distribution (SED) fitting to
Herschel continuum data spanning 70–500 $\mu$m (beam size $\sim$40′′;
Hennemann et al. 2012; Cao et al. 2019, 2022). The typical dust temperature
was found to be between 15 and 31 K, with specific regions exhibiting typical
values of 20, 21, 22, 20, 31, 23, 21, 18, and 16 K for W75S FIR 3, W75S FIR 2,
W75S FIR 1, DR21(OH)N, DR21(OH), DR21(OH)S, DR21, DR21W1, and DR21S,
respectively. These studies provide valuable insights into the gas and dust
temperature variations within the DR21 filament. Nevertheless, it is worth
noting that the NH3 (1,1) and (2,2) lines exhibit critical densities on the
order of $n_{\rm crit}$(NH3) $\sim$ 103 cm-3 (Shirley, 2015). This
characteristic suggests that these particular spectral lines possess a
sensitivity suitable for probing the moderate-density gas within molecular
clouds. A more comprehensive database of gas temperature which is related to
star formation ($n$(H2)¿104 cm-3) in the DR21 filament, with high resolution
at a sub-pc scale, is still lacking. Hence, further research and high-
resolution observations of the DR21 filament are necessary to achieve a
comprehensive characterization of the thermal properties of the dense gas.

###  1.2 H2CO as a Temperature Tracer

Formaldehyde (H2CO), a slightly asymmetric rotor, is ubiquitous in
interstellar clouds (e.g., Downes et al. 1980; Hurt et al. 1996; Mühle et al.
2007; Ao et al. 2013; Tang et al. 2013, 2014; Guo et al. 2016; Yan et al.
2019; Brinkmann et al. 2020), exhibits a large number of transitions from
centimeter to millimeter wavelengths and is thought to be sensitive to probe
physical conditions in various molecular clouds (e.g., Henkel et al. 1980,
1983; Mangum & Wootten 1993; Mangum et al. 2008, 2013a, 2019; Ginsburg et al.
2011, 2016; Ao et al. 2013; Tang et al. 2017a, b, 2018a, 2018b, 2021; Immer et
al. 2016). Previous observations of H2CO indicate that variations in the
fractional abundance of H2CO only rarely exceed one order of magnitude at
various stages of star formation (e.g., Mangum et al. 1990, 1993; Caselli et
al. 1993; Johnstone et al. 2003; Gerner et al. 2014; Tang et al. 2017a, b,
2018b; Liu et al. 2020). It is possibly the best of the very few molecular
thermometers that are available for the analysis of dense molecular gas.

Since the relative populations of the $K_{\rm a}$ ladders of H2CO are
predominantly governed by collisions, ratios of H2CO line fluxes involving
different $K_{\rm a}$ ladders can act as good indicators of kinetic
temperature, such as para-H2CO $J_{\rm K_{a}K_{c}}$ = 322–221/303–202,
422–321/404–303, and 523–422/505–404 (Mangum & Wootten, 1993; Tang et al.,
2018b; Mazumdar et al., 2022; Kahle et al., 2023). These H2CO line ratios can
accurately trace dense molecular gas as thermometers and have been used to
measure the temperature of dense gas in Galactic star formation regions (e.g.,
Mangum & Wootten 1993; Hurt et al. 1996; Mangum et al. 1999; Watanabe &
Mitchell 2008; Nagy et al. 2012; Lindberg et al. 2015; Tang et al. 2017a,
2018a, 2018b; Gieser et al. 2021; Izumi et al. 2023), Galactic center clouds
(Qin et al. 2008; Ao et al. 2013; Johnston et al. 2014; Ginsburg et al. 2016;
Immer et al. 2016; Lu et al. 2017), and even external galaxies (Mühle et al.
2007; Tang et al. 2017b, 2021; Mangum et al. 2019).

To understand the feedback from the high-mass star-forming process, it is
critical to investigate the correlation between the gas temperature and star
formation activities in the molecular clouds. The main goal of this paper is
to map the kinetic temperature structure of the massive DR21 filament in the
north of the Cygnus X molecular cloud complex, making use of the para-H2CO
triplet ($J_{\rm K_{a}K_{c}}$ = 303–202, 322–221, and 321–220, hereafter H2CO)
with a critical density on the order of $n_{\rm crit}$(H2CO 303–202) $\sim$ a
few 105 cm-3 (Shirley, 2015) and to investigate the gas heating mechanism
affecting the dense gas in the processes of star formation. In Sects. 2 and 3,
we introduce our observations of the H2CO triplet, data reduction, and
results. We discuss the resulting kinetic temperatures derived from H2CO in
Sect. 4. Our main conclusions are summarized in Sect. 5. This paper is part of
the ”Kinetic temperature of massive star-forming molecular clumps measured
with formaldehyde” series of studies exploiting H2CO as a probe of gas
conditions in a variety of Galactic and extragalactic sources.

##  2 Observations and Data Reduction

Our data presented here were observed with the IRAM 30 m telescope111Based on
observations obtained with the IRAM 30 m telescope. IRAM is supported by
INSU/CNRS (France), MPG (Germany), and IGN (Spain). in 2017 May and 2018
October using the HERA receiver (Schuster et al., 2004). The H2CO ($J_{\rm
K_{a}K_{c}}$ = 303–202, 322–221 and 321–220) transitions have rest frequencies
of 218.222, 218.475, and 218.760 GHz, respectively, and could be measured
simultaneously. At $\sim$218 GHz, the beam size is $\sim$ 12′′ ($\sim$ 0.09 pc
at 1.5 kpc distance; Rygl et al. 2012). The main beam efficiency and the
forward efficiency222https://publicwiki.iram.es/Iram30mEfficiencies are 0.60
and 0.94, respectively. The backend spectrometer has a bandwidth of $\sim$1
GHz and 5377 spectral channels, providing a channel width of $\sim$0.27 km
s-1. The central frequency was set to 218.475 GHz. The on-the-fly observing
mode was used to measure four $\sim$3$\times$3 arcmin2 maps with steps of
3.5′′ in both right ascension and declination. The surveyed area of the DR21
filament is $\sim$4$\times$10 arcmin2 ($\sim$1.7$\times$4.4 pc2), centered on
$\alpha_{2000}$ = 20h39m01$\,.\\!\\!^{s}$0 and $\delta_{2000}$ =
42$\degr$22′49$\,.\\!\\!^{\prime\prime}$8.

Data reduction for spectral lines and maps was performed with
GILDAS333http://www.iram.fr/IRAMFR/GILDAS. To improve the signal-to-noise
ratio (S/N) in individual channels, we smoothed contiguous channels to a
velocity resolution $\sim$0.8 km s-1 and the spectra were resampled in steps
of $\sim$6′′. In total, we acquired 3200 positions corresponding to 3200
spectra for each transition. Nearly 55% of all positions were detected in H2CO
303–202 and $\sim$23% were also detected in H2CO 322–221 and 321–220, with
S/Ns of $\gtrsim$3$\sigma$. We used Gaussian profiles to fit all H2CO spectra
with S/Ns above 3$\sigma$. A typical rms noise level (1$\sigma$) is $\sim$0.19
K ($T_{\rm mb}$ scale) at a channel width of 0.8 km s-1.

Generally, H2CO 322–221 and 321–220 show weaker emissions than H2CO 303–202.
The H2CO 322–221 and 321–220 transitions have the same $K_{\rm a}$ ladder and
almost the same energy of the upper levels, $E_{\rm u}$ $\simeq$ 68 K, above
the ground state. They also show similar distributions and line profiles
(brightness temperature, linewidth, and velocity in our observations) in star
formation regions (e.g., Mangum & Wootten 1993; Tang et al. 2017a, b, 2018a,
2018b, 2021). Furthermore, H2CO 322–221/303–202 and 321–220/303–202 line
ratios exhibit similar behaviors to trace the kinetic temperature of dense gas
(e.g., Mangum & Wootten 1993; Tang et al. 2017a). To further enhance the S/Ns
of H2CO 322–221 and 321–220, we combined the line intensity of two transitions
channel by channel.

In the vicinity of DR21(OH), the H2CO 303–202 spectral lines possess two
distinct velocity components. However, identifying these two distinct velocity
components becomes difficult for the weaker H2CO 322–221 and 321–220 lines. To
address this, we employed a Gaussian method to fit these $\sim$70 positions
that clearly exhibit two velocity components. The two velocity components
measured using H2CO 303–202 are approximately at local standard of rest
velocities ($V_{\rm LSR}$) of $-$4 and $-$2 km s-1, consistent with previous
findings from observations of HCO+, H13CO+, and N2H+ (Schneider et al., 2010;
Cao et al., 2022; Bonne et al., 2023). High-resolution observations revealed
that these two velocity components were attributed to two closely aligned
fibers in the DR21(OH)/W75S region (Cao et al., 2022). Subsequently, we
combined the data from these two velocity components based on the integrated
intensity ratio between each pair of velocity component lines. Hence, the gas
temperatures of these positions in the two velocity components were not
determined individually.

##  3 Results

###  3.1 Overview

The velocity-integrated intensity distributions of the H2CO triplet and the
combined H2CO 322–221 and 321–220 lines in the DR21 filament ($V_{\rm LSR}$ =
$-$10 to 4 km s-1) are shown in Fig. 1. The intensity-weighted velocity field
(moment 1), linewidth (moment 2), and channel maps of H2CO 303–202 are shown
in Figs. 2 and 7. The observed H2CO spectra of sixteen dense cores identified
with 1.2 mm continuum emission by Motte et al. (2007) are shown in Fig. 3. The
locations of these dense cores in the DR21 filament are indicated in Fig. 4
(left panel) and are listed in Table 1. The parameters of the Gaussian fits,
which include velocity-integrated intensity, $\int T_{\rm mb}$ d$v$, local
standard of rest velocity, $V_{\rm LSR}$, full width at half maximum (FWHM)
line width, and peak temperature ($T_{\rm mb}$) of the H2CO spectra in these
dense cores are listed in Table 1. Three dense cores, namely N35, N45, and
N52, situated at the periphery of the DR21 filament, exhibit weak emissions of
H2CO 322–221 and 321–220 with S/Ns below 3$\sigma$, as depicted in Figs. 1 and
4. Consequently, these cores were omitted from Fig. 3.

###  3.2 Distribution of H2CO

The H2CO 303–202 line exhibits an extensive distribution and provides a clear
picture of the dense ridge structure of the DR21 filament (see the left panel
of Fig. 1). It illuminates notable dense molecular structures including the
DR21, DR21(OH), and W75S regions on a linear scale of $\sim$0.1 pc. Obviously,
the strongest emission of H2CO is associated with the two most active star-
forming regions, DR21 and DR21(OH). These results are consistent with prior
observational findings using other dense gas tracers, such as CN, C18O, CS,
HCN, and HCO+ (Schneider et al. 2010; Dobashi et al. 2019). The spatial
distribution of H2CO 303–202 illustrates a close correspondence with the
spatial distribution of the dust emission at wavelengths of 850 $\mu{\rm m}$
and 1.2 mm, including the presence of dense ridge structures and dust emission
peaks within the DR21 filament (Davis et al., 2007; Motte et al., 2007; Ching
et al., 2022), which confirms previous observational results regarding massive
star formation regions (e.g., Tang et al. 2017a, 2018a, 2018b, 2021). In
contrast, H2CO 322–221 and 321–220 indicate less extended distributions and
are only detectable in the densest areas of the DR21 filament, as shown in the
rightmost three panels of Fig. 1.

Figure 1: Velocity-integrated intensity maps ($T_{\rm mb}$ scale; color bar in
units of K km s-1) of H2CO 303–202 (_left_), 322–221 (_center-left_), 321–220
(_center-right_), and averaged 322–221 and 321–220 (_right_), integrated from
$V_{\rm LSR}$ = $-$10 to 4 km s-1 of the DR21 filament. Contour levels are
from 4.8 to 12 K km s-1 with steps of 1.8 K km s-1 and from 12 to 59 K km s-1
with steps of 3.6 K km s-1 for H2CO 303–202, and from 1.7 to 4.4 K km s-1 with
steps of 0.66 K km s-1 and from 4.3 to 21.2 K km s-1 with steps of 1.3 K km
s-1 for H2CO 322–221, 321–220, and combined 322–221 and 321–220. The locations
of source names are taken from Davis et al. (2007), Reipurth & Schneider
(2008), and Ching et al. (2022).

The intensity-weighted velocity (moment 1) and channel maps of H2CO (303–202)
reveal the complex velocity field of the DR21 filament (see the left panels of
Figs. 2 and 7). Notably, multiple velocity gradients traverse the primary DR21
filament as well as the DR21W1 region. The W75S region shows an increasing
gradient of velocity ($-$4.5 to $-$2.5 km s-1) that extends from the northeast
to the southwest, while the DR21(OH) region displays a decreasing northwest-
southeast velocity gradient ($-$2.5 to $-$4.0 km s-1). Utilizing high-
resolution observations of H13CO+, N2H+, and NH2D toward the W75S/DR21(OH)
region (beam size $\sim$ 5′′), it has been observed that three nearly parallel
fibers exist at distinct velocities, primarily aligned in a north-south
orientation (Cao et al., 2022). Such an alignment may give rise to the complex
velocity field found within this region. Furthermore, the DR21 and DR21W1
regions exhibit a comparable increasing northeast-to-southwest velocity
gradient with the mean velocity ranging from $-$3.5 to $-$1.5 km s-1 and from
$-$5.0 to $-$2.0 km s-1, respectively. Our observed H2CO results corroborate
previous observational outcomes obtained from H13CO+, C34S, and N2H+
(Schneider et al., 2010).

As shown in the right panel of Fig. 2, higher linewidths ($\gtrsim$3 km s-1)
are typically associated with dense clumps, such as DR21, DR21W1, DR21(OH),
DR21(OH)S, and DR21(OH)W, in the two most active star-forming regions DR21 and
DR21(OH). Moderate linewidths (2–3 km s-1) are distributed in the dense clumps
of the W75S region, as well as around the dense clumps of DR21 and DR21(OH)
regions, while lower linewidths ($<$2 km s-1) are located on the outskirts of
the DR21 filament. The fitted linewidth of H2CO (303–202) ranges from 1.7 to
10.4 km s-1 with an average of $\sim$3.8$\pm$0.1 km s-1 (errors given here and
elsewhere are standard deviations of the mean).

###  3.3 Line Ratios of H2CO

As mentioned in Sects. 1 and 2, H2CO 322–221/303–202 and 321–220/303–202 line
ratios exhibit similar behaviors in delineating the kinetic temperature of the
dense gas (e.g., Mangum & Wootten 1993; Tang et al. 2017a). The present study
employs the averaged ratio of H2CO 0.5$\times$[(322–221 \+
321–2${}_{20})$/303–202] (obtained from the combined H2CO 322–221 and 321–220
to H2CO 303–202 ratio) as an indicator of the kinetic temperature.
Substantially, higher H2CO line ratios typically correspond to higher kinetic
temperatures (Ao et al., 2013; Ginsburg et al., 2016; Tang et al., 2018a,
2021). Thus, the ratio maps may function as a proxy for the relative kinetic
temperature. The line ratio map of the DR21 filament is shown in the left
panel of Fig. 4. The line ratios are calculated by velocity-integrated
intensities where the combined H2CO 322–221 and 321–220 lines are detected
with S/Ns of $\gtrsim$ 3$\sigma$. The H2CO line ratios range from 0.08 to 0.42
with an average value of 0.22 $\pm$ 0.01 in the DR21 filament. Higher H2CO
line ratios (¿0.35) have been observed in association with dense clumps, such
as W75S FIR 3, DR21(OH), DR21(OH)S, DR21, and DR21W1. Moderate line ratios
ranging from 0.2 to 0.35 are prevalent throughout the vicinity of these dense
clumps. Notably, the outskirts of the DR21 and the area located between
DR21(OH) and W75S exhibit relatively lower line ratios (¡0.2).

###  3.4 Kinetic Temperatures Derived from H2CO Line Ratios

As described in Sects. 2 and 3.3, the determination of the kinetic temperature
involves calculating the averaged ratio of H2CO 0.5$\times$[(322–221 \+
321–2${}_{20})$/303–202]. To model the relationship between the gas kinetic
temperature and the measured average of H2CO 0.5$\times$[(322–221 \+
321–2${}_{20})$/303–202] ratios, we employed the
RADEX444http://var.sron.nl/radex/radex.php non-LTE model (van der Tak et al.,
2007) with collision rates provided by Wiesenfeld & Faure (2013). In Fig. 8,
we adopted a background temperature of 2.73 K, an average measured linewidth
of 3.8 km s-1, and a column density of $N$(para-H2CO) = 6.6$\times$1013 cm-2.
The impact of different column densities of H2CO on the kinetic temperature is
found to be minimal when all lines are optically thin, as shown in Fig. 3 in
Tang et al. (2017b) or Fig. 4 in Tang et al. (2018a). Additionally, opacities
within dense massive star-forming clumps have little effect on the kinetic
temperature (Ginsburg et al., 2016; Immer et al., 2016; Tang et al., 2017b,
2018b). We assume that para-H2CO (3–2) is optically thin in DR21 (see below
for detailed arguments). The measured H2 number densities, $n$(H2), of massive
dense cores in DR21 range from $\sim$ 104 to 106 cm-3 (Schneider et al., 2006;
Jakob et al., 2007; Cao et al., 2019, 2022). Previous observations have
demonstrated that para-H2CO (3–2) is particularly sensitive to the gas density
at $\sim$105 cm-3 (Ginsburg et al., 2016; Immer et al., 2016; Tang et al.,
2017b, 2021). Therefore, we adopt an average spatial gas density of 105 cm-3
for the DR21 filament. By utilizing the methodology outlined in Sect. 3.1 of
Tang et al. (2017b), we can calculate the average column density
$N$(para-H2CO) of the entire DR21 region based on the H2CO (303–202) average
brightness temperature. The resulting value is 6.6$\times$1013 cm-2 at a
volume density of approximately 105 cm-3. Previous observations of para-H2CO
(3–2 and 4–3) toward dense clumps representing various evolutionary stages of
high-mass star formation in the Galactic plane suggest an averaged fractional
abundance of para-H2CO of $\sim$3.9$\times$10-10 (Tang et al., 2018b). The
average column density of H2, $N$(H2), in the DR21 ridge, where the
distribution of $N$(H2) agrees well with that of our H2CO (303–202) lines, is
$\sim$4.2$\times$1023 cm-2 as determined from the Herschel 70–500 $\mu$m
continuum data (Hennemann et al., 2012). Subsequently, we can compute a
typical fractional abundance of para-H2CO of roughly 1.6$\times$10-10 in the
DR21 filament using the above-mentioned average H2CO column density of the
entire DR21 region, which is in agreement with previous observational results
in high-mass star formation regions (Tang et al., 2018b). This supports the
reasonability of our RADEX calculations using the average column density
$N$(para-H2CO) in the DR21 region.

Figure 2: The intensity-weighted velocity field (moment 1, _left_) and
linewidth (moment 2, _right_) of H2CO 303–202. The unit of each colour bar is
km s-1. Contours are H2CO 303–202 integrated intensities (same as in Fig. 1).

Pertaining to kinetic temperatures, it has been observed that subtle
variations occur across different column densities at a given H2 number
density of 105 cm-3 (Tang et al., 2018a). Specifically, we explored this
relationship at column densities of $N$(para-H2CO) = 1.0$\times$1013,
6.6$\times$1013, and 1.0$\times$1014 cm-2, as depicted in Fig. 8. It indicates
that there exists a slight temperature variation corresponding to different
column densities. The temperature differences derived from column densities of
$N$(para-H2CO)=1.0$\times$1013 and 1.0$\times$1014 cm-2, compared to
6.6$\times$1013 cm-2, are found to be smaller than the uncertainties in
temperature determined from the errors associated with observed H2CO line
ratios. Therefore we employed the established correlation between H2CO line
ratios and kinetic temperature at a spatial density of 105 cm-3 and a column
density of 6.6$\times$1013 cm-2 in order to convert the ratio maps into
temperature maps in this work. These transformed temperature maps are
illustrated in Fig. 4. As previously stated, the measured H2 number densities
of massive dense cores within DR21 span from approximately $\sim$ 104 to 106
cm-3. Regarding kinetic temperatures, subtle variations have been noted across
different number densities within the range of $\sim$ 104 to 106 cm-3 (see
Fig. 5 in Tang et al. 2021). It appears that at $n$(H2) = 105 cm-3, $T_{\rm
kin}$ consistently exhibits lower values compared to those at 104 to 106 cm-3
by approximately $\lesssim$35% for $T_{\rm kin}$ $\lesssim$ 100 K in Fig. 4.

Figure 3: Observed H2CO spectra of the dense cores in the DR21 filament
obtained with the IRAM 30 m. Red, green, and blue lines show H2CO 303–202,
321–220, and 322–221, respectively. The dense cores, identified through 1.2 mm
continuum emission as described by Motte et al. (2007), are referenced for
nomenclature, positions, and physical parameters in Table 1.

The applicability of local thermodynamic equilibrium (LTE) as a reliable
approximation for H2CO level populations has been established under certain
conditions, namely when the corresponding transitions are optically thin and
the targets are characterized by high densities (Mangum & Wootten, 1993). By
employing the methodology outlined in Tang et al. (2017b) (refer to their Eq.
2) and utilizing the RADEX model, we conducted a comprehensive analysis to
investigate the relationship between kinetic temperature and the combined H2CO
ratio within the LTE regime in Fig. 8. It indicates a satisfactory agreement
between the temperatures obtained from the LTE model and those derived from
the RADEX model, under the condition of a spatial gas density of $n$(H2) = 105
cm-3, as long as the kinetic temperature remains below 100 K.

The kinetic temperature of the warm dense gas in the DR21 region, as indicated
by the H2CO line ratios, ranges from 24 to 114 K, with an average temperature
of 48.3 $\pm$ 0.5 K at a density of $n$(H2) = 105 cm-3 (see also Table 2). As
shown in Fig. 4 and Table 7, it reveals the inhomogeneous temperature
distribution of the DR21 region across different sub-regions of the molecular
complex. The H2CO line ratio analysis demonstrates that two dense clumps,
DR21(OH) and DR21(OH)S, located within the central parts of the DR21 filament,
exhibit noticeably high temperatures exceeding 100 K. The kinetic temperatures
of the dense clumps W75S FIR 3, DR21, and DR21W1, albeit lower than in
DR21(OH), remain relatively high ranging between 77 to 84 K. Conversely, the
gas temperatures of the dense clumps W75S FIR 1, W75S FIR 2, DR21(OH)N,
DR21(OH)W, and DR21S span the range from 38 to 57 K. Table 1 provides a
summary of the detailed temperature results for all sixteen dense cores
apparent in the DR21 filament. In the immediate vicinity of the dense regions
such as W75S FIR 3, W75S FIR 2, DR21(OH), DR21(OH)S, and DR21, the kinetic
temperatures range between 30 to 45 K. The region located between DR21(OH) and
W75S and the outskirts of the DR21 filament exhibit significantly lower
temperatures below 40 K.

To address the impact of H2CO (3–2) line saturation on our temperature
measurements, we employed the RADEX non-LTE model. This model allowed us to
calculate the opacities of H2CO (3–2) lines, adopting an average temperature
of $T_{\rm kin}$ = 48 K, an average measured linewidth of 3.8 km s-1, an H2CO
column density of $N$(para-H2CO) = 6.6$\times$1013 cm-2, and a gas volume
density of $n$(H2) = 105 cm-3. The resulting optical depths for the H2CO
303–202, 322–221, and 321–220 transitions were approximately 1.2, 0.28, and
0.28, respectively. These values suggest that in the location of the dense
cores, opacities in the H2CO 303–202 lines may exhibit slight saturation,
potentially leading to a slight overestimation of kinetic temperatures by
$\sim$20$\%$.

Figure 4: Left two panels: The averaged velocity-integrated intensity ratio
map of H2CO 0.5$\times$[(322–221 \+ 321–2${}_{20})$/303–202] in the DR21
filament (see Sect. 3). The kinetic temperatures derived from the H2CO line
ratios (color bar in units of Kelvin). Right two panels: Maps of non-thermal
linewidth (color bar in units of km s-1) and Mach number. The black stars and
triangles show the locations of protostars (Davis et al., 2007) and the dense
cores (Motte et al., 2007), respectively. The black circle and diamond
illustrate the locations of the explosion center as described by Zapata et al.
(2013) and the infrared-quiet massive dense core as delineated by Cao et al.
(2019), respectively. Black contours show the integrated intensity of H2CO
303–202 at 4.8 K km s-1 (see Fig. 1).

###  3.5 Thermal and Non-thermal Motions Obtained from H2CO

The thermal and non-thermal linewidths (Pan & Padoan, 2009; Dewangan et al.,
2016), represented by $\sigma_{\rm T}$ = $\sqrt{\frac{kT_{\rm kin}}{m_{\rm
H_{2}CO}}}$ and $\sigma_{\rm NT}$ = $\sqrt{\frac{\Delta v^{2}}{8{\rm
ln}2}-\sigma_{\rm T}^{2}}$ $\thickapprox$ $\Delta v/2.355$, respectively, are
determined based on the kinetic temperature derived from the H2CO line ratios.
In this context, $k$ denotes the Boltzmann constant, $T_{\rm kin}$ corresponds
to the kinetic temperature of the gas, $m_{\rm H_{2}CO}$ represents the mass
of a formaldehyde molecule, and $\Delta v$ signifies the measured FWHM
linewidth of H2CO 303–202. As mentioned in Sect. 2, approximately 70 positions
in the DR21(OH) region exhibit two distinct velocity components at about $-$2
and $-$4 km s-1 (see also Fig. 3). Additionally, determining the individual
gas temperatures for these positions in the two velocity components proved to
be challenging. Hence, in this study, we employ the approach of utilizing the
averaged linewidth, which is weighted by the integrated intensity of H2CO
303–202. By analyzing the data, the thermal linewidth ranges from 0.08 to 0.18
km s-1 with an average value of 0.11 $\pm$ 0.01 km s-1, whereas the non-
thermal linewidth spans from 0.74 to 4.40 km s-1 with an average value of 1.60
$\pm$ 0.02 km s-1 (see also Table 2). It is apparent that the non-thermal
linewidth significantly exceeds the thermal linewidth, indicating that the
dense gas traced by H2CO is primarily governed by non-thermal motions. These
findings align with previous studies conducted on star-forming regions in our
Galaxy and the Large Magellanic Cloud using H2CO ($J$=3–2 and 4–3)
observations (Tang et al., 2017a, 2018a, 2018b, 2021).

The non-thermal linewidth distribution of H2CO 303–202 is depicted in Fig. 4,
showing similarities with the H2CO linewidth observed in the W75S and DR21
regions. However, a significant disparity in the distribution of non-thermal
linewidth and H2CO linewidth is observed in the DR21(OH) region, as
illustrated in Fig. 2. As mentioned above, the lower values for the non-
thermal linewidth in the DR21(OH) region can be attributed to the averaging of
linewidths derived from the two distinct velocity components. In the western
region of DR21(OH)N, weak H2CO emission is detected, as depicted in Fig. 2. At
many positions in this region, two velocity components are present, but one of
them exhibits lower S/Ns, making it challenging to identify it as a
significant signal. Consequently, we employed a single Gaussian velocity
component to fit the H2CO lines in this specific area, resulting in higher
non-thermal linewidths in the western region of DR21(OH)N.

Table 1: Parameters of dense cores in DR21 filament.

Dense core | Offset | Transition |  $\int T_{\rm mb}$ d$v$ | $V_{\rm LSR}$ | FWHM | $T_{\rm mb}$ | $\sigma_{\rm T}$ | $\sigma_{\rm NT}$ | $c_{\rm s}$ | $\mathcal{M}$ | $T_{\rm kin}$ | $T_{\rm turb}$
---|---|---|---|---|---|---|---|---|---|---|---|---
| ” , ” |  |  K km s-1 |  km s-1 |  km s-1 | K |  km s-1 |  km s-1 |  km s-1 |  | K | K
N34 | –71.5, –211.5 |  303–202 | 13.0 (0.8) | –2.4 (0.1) | 5.4 (0.5) | 2.3 | 0.15 (0.01) | 2.30 (0.20) | 0.54 (0.05) | 4.3 | 84 (15) | 70
|  |  (321–220+322–221)/2 | 4.6 (0.4) | –2.4 (0.3) | 6.9 (0.7) | 0.6 |  |  |  |  |  |
N36 | –29.5, 40.5 |  303–202 | 10.2 (0.4) | –3.2 (0.1) | 4.0 (0.2) | 2.4 | 0.11 (0.01) | 1.68 (0.07) | 0.40 (0.02) | 4.2 | 46 (5) | 43
|  |  (321–220+322–221)/2 | 2.2 (0.2) | –3.2 (0.2) | 3.8 (0.5) | 0.5 |  |  |  |  |  |
N37 | –29.5, 106.5 |  303–202 | 5.0 (0.4) | –2.8 (0.1) | 3.5 (0.3) | 1.4 | 0.10 (0.01) | 1.47 (0.13) | 0.36 (0.04) | 4.0 | 38 (7) | 36
|  |  (321–220+322–221)/2 | 0.8 (0.2) | –2.9 (0.4) | 3.1 (0.8) | 0.3 |  |  |  |  |  |
N38 | –17.5, –25.5 |  303–202 | 27.9 (2.5) | –2.9 (0.1) | 2.7 (0.2) | 4.8 | 0.13 (0.01) | 1.15 (0.07) | 0.44 (0.02) | 2.6 | 57 (6) | 50
|  |  (321–220+322–221)/2 | 7.5 (0.2) | –2.9 (0.1) | 3.0 (0.2) | 1.2 |  |  |  |  |  |
N40 | –11.5, 52.5 |  303–202 | 9.9 (0.5) | –3.5 (0.1) | 3.1 (0.2) | 3.0 | 0.11 (0.01) | 1.32 (0.08) | 0.39 (0.03) | 3.4 | 44 (7) | 41
|  |  (321–220+322–221)/2 | 2.0 (0.4) | –3.5 (0.2) | 2.8 (0.6) | 0.7 |  |  |  |  |  |
N41 | –11.5, 34.5 |  303–202 | 14.5 (0.5) | –2.9 (0.1) | 3.9 (0.2) | 3.5 | 0.12 (0.01) | 1.64 (0.06) | 0.42 (0.01) | 3.9 | 51 (3) | 47
|  |  (321–220+322–221)/2 | 3.5 (0.2) | –2.9 (0.1) | 3.4 (0.2) | 1.0 |  |  |  |  |  |
N42 | –5.5, –223.5 |  303–202 | 15.8 (0.4) | –2.4 (0.1) | 3.5 (0.1) | 4.3 | 0.12 (0.01) | 1.47 (0.04) | 0.43 (0.02) | 3.5 | 52 (4) | 47
|  |  (321–220+322–221)/2 | 3.9 (0.3) | –2.5 (0.1) | 3.6 (0.3) | 1.0 |  |  |  |  |  |
N43 | –5.5, 106.5 |  303–202 | 8.6 (0.4) | –3.1 (0.1) | 3.8 (0.2) | 2.1 | 0.12 (0.01) | 1.62 (0.08) | 0.42 (0.03) | 3.9 | 50 (8) | 46
|  |  (321–220+322–221)/2 | 2.0 (0.3) | –2.7 (0.3) | 3.9 (0.7) | 0.5 |  |  |  |  |  |
N44 | 0.5, –1.5 |  303–202 | 54.7 (1.4) | –2.9 (0.1) | 3.4 (0.1) | 8.0 | 0.17 (0.01) | 1.43 (0.04) | 0.60 (0.04) | 2.4 | 103 (15) | 78
|  |  (321–220+322–221)/2 | 21.8 (1.6) | –3.0 (0.6) | 3.6 (0.6) | 3.0 |  |  |  |  |  |
N46 | 6.5, –193.5 |  303–202 | 22.4 (0.6) | –2.8 (0.1) | 5.5 (0.2) | 3.9 | 0.15 (0.01) | 2.32 (0.08) | 0.52 (0.02) | 4.5 | 77 (7) | 66
|  |  (321–220 +322–221)/2 | 7.6 (0.4) | –3.2 (0.1) | 5.6 (0.4) | 1.3 |  |  |  |  |  |
N47 | 6.5, –175.5 |  303–202 | 16.2 (0.5) | –2.8 (0.1) | 3.3 (0.1) | 4.7 | 0.13 (0.01) | 1.38 (0.06) | 0.47 (0.02) | 3.0 | 63 (5) | 54
|  |  (321–220+322–221)/2 | 4.7 (0.3) | –3.2 (0.1) | 3.7 (0.3) | 1.2 |  |  |  |  |  |
N48 | 6.5, –49.5 |  303–202 | 26.1 (0.4) | –4.4 (0.1) | 4.1 (0.1) | 6.0 | 0.17 (0.01) | 1.73 (0.03) | 0.59 (0.02) | 2.9 | 100 (5) | 77
|  |  (321–220+322–221)/2 | 10.3 (0.2) | –4.4 (0.1) | 3.6 (0.1) | 2.7 |  |  |  |  |  |
N50 | 6.5, –253.5 |  303–202 | 7.8 (0.4) | –1.8 (0.1) | 2.9 (0.2) | 2.6 | 0.09 (0.01) | 1.21 (0.07) | 0.32 (0.02) | 3.8 | 30 (4) | 29
|  |  (321–220+322–221)/2 | 0.9 (0.2) | –2.2 (0.3) | 3.1 (0.9) | 0.3 |  |  |  |  |  |
N51 | 18.5, 130.5 |  303–202 | 15.2 (0.5) | –4.0 (0.1) | 3.1 (0.1) | 4.6 | 0.12 (0.01) | 1.32 (0.05) | 0.41 (0.02) | 3.2 | 48 (5) | 44
|  |  (321–220+322–221)/2 | 3.5 (0.4) | –4.5 (0.2) | 3.1 (0.3) | 1.1 |  |  |  |  |  |
N53 | 24.5, 178.5 |  303–202 | 14.8 (0.6) | –3.8 (0.1) | 5.0 (0.2) | 2.8 | 0.11 (0.01) | 2.13 (0.10) | 0.40 (0.02) | 5.4 | 45 (5) | 45
|  |  (321–220+322–221)/2 | 3.2 (0.4) | –4.6 (0.1) | 2.6 (0.4) | 1.1 |  |  |  |  |  |
N54 | 30.5, 160.5 |  303–202 | 7.2 (0.4) | –4.0 (0.1) | 3.1 (0.2) | 2.2 | 0.15 (0.01) | 1.32 (0.10) | 0.52 (0.05) | 2.6 | 77 (16) | 63
|  |  (321–220+322–221)/2 | 2.4 (0.3) | –3.9 (0.3) | 4.8 (0.8) | 0.5 |  |  |  |  |  |

555Offsets relative to our reference position for the DR21 filament (see Sect.
2 and Fig. 4). Velocity-integrated intensity, $\int T_{\rm mb}$ d$v$, local
standard of rest velocity, $V_{\rm LSR}$, full width at half maximum line
width (FWHM), and peak temperature ($T_{\rm mb}$) were obtained from Gaussian
fitting using CLASS as part of the GILDAS software package. For the thermal
and non-thermal linewidths, $\sigma_{\rm T}$ and $\sigma_{\rm NT}$, the sound
velocity $c_{\rm s}$, the Mach number $\mathcal{M}$, the kinetic temperature
$T_{\rm kin}$, and the turbulent temperature $T_{\rm turb}$, see Sects. 3 and
4.3.3. Values in parentheses are uncertainties.

Table 2: Thermal and non-thermal parameters derived from DR21, OMC-1, and
N113.

Parameter |  DR21 ($\sim$0.1 pc ) |  |  OMC-1a ($\sim$0.06 pc ) |  |  N113b ($\sim$0.4 pc )
---|---|---|---|---|---
| Range | Median | Mean |  | Range | Median | Mean |  | Range | Median | Mean
H2CO line ratio | 0.08–0.42 | 0.22 |  0.22 $\pm$ 0.01 |  | 0.12–0.61 | 0.21 |  0.28 $\pm$ 0.01 |  | 0.10–0.38 | 0.21 |  0.22 $\pm$ 0.01
$T_{\rm kin}$ /K | 23.9–114.5 | 46.4 |  48.3 $\pm$ 0.5 |  |  30–$>$200 | 48.5 |  62.0 $\pm$ 2.0 |  | 27.6–105.4 | 48.5 |  51.4 $\pm$ 0.4
$\sigma_{\rm T}$ / km s-1 | 0.08–0.18 | 0.11 |  0.11 $\pm$ 0.01 |  | 0.09–0.27 | 0.11 |  0.12 $\pm$ 0.01 |  | 0.08–0.16 | 0.11 |  0.11 $\pm$ 0.01
$\sigma_{\rm NT}$ / km s-1 | 0.74–4.40 | 1.47 |  1.60 $\pm$ 0.02 |  | 0.34–2.78 | 1.64 |  0.98 $\pm$ 0.02 |  | 0.93–2.76 | 1.64 |  1.69 $\pm$ 0.01
$c_{\rm s}$ / km s-1 | 0.29–0.63 | 0.40 |  0.41 $\pm$ 0.01 |  | 0.30–0.97 | 0.40 |  0.44 $\pm$ 0.01 |  | 0.30–0.58 | 0.40 |  0.40 $\pm$ 0.01
$\mathcal{M}$ | 1.9–10.3 | 3.7 |  4.0 $\pm$ 0.1 |  | 0.7–4.3 | 4.2 |  2.3 $\pm$ 0.1 |  | 2.3–6.2 | 4.2 |  4.2 $\pm$ 0.1

666(a) Parameters of OMC-1 taken from Tang et al. (2018a). (b) Parameters of
N113 taken from Tang et al. (2021). The values enclosed in parentheses denote
the linear scale of each respective source. The errors shown in Columns 4 and
7 are the standard deviations of the mean.

Distributions of the Mach number (given as $\mathcal{M}$ = $\sigma_{\rm
NT}/a_{\rm s}$, $a_{\rm s}$ = $\sqrt{\frac{kT_{\rm kin}}{\mu m_{\rm H}}}$,
where $\mu$ = 2.37 is the mean molecular weight for molecular clouds and
$m_{\rm H}$ is the mass of a hydrogen atom) of the DR21 filament is shown in
Fig. 4. The Mach number ranges from 1.9 to 10.3 with an average of 4.0 $\pm$
0.1 (see also Table 2). This suggests that supersonic non-thermal motions
(e.g., turbulence, outflows, shocks, and/or magnetic fields) are dominant in
the dense gas traced by H2CO. The DR21W1 region, linked to the western DR21
flow, exhibits particularly high Mach numbers ($\mathcal{M}$ $\sim$ 10) as
evidenced by previous studies (e.g., Jaffe et al. 1989; Lane et al. 1990;
Davis et al. 2007; Zapata et al. 2013). This association indicates a strong
influence of shocks on the dense gas traced by H2CO. Within the dense core of
the DR21 region, the Mach numbers are approximately $\mathcal{M}$ $\sim$ 7,
while in the W75S FIR 3 region, they are around $\mathcal{M}$ $\sim$ 6,
slightly lower than those observed in the DR21W1 region. This observation of
high Mach numbers suggests that local star formation activities (e.g.,
outflows, shocks; Davis et al. 2007; Motte et al. 2007; Schneider et al. 2010)
in the DR21 and W75S FIR 3 regions may impact the dense gas. In the DR21(OH)
region, the Mach numbers are lower (2–3), whereas in the western region of
DR21(OH)N, they are higher (6–7). These variations are attributed to the two
velocity components mentioned in the above paragraph. Moving towards the
outskirts and extended regions of the DR21 filament, the typical Mach number
ranges from 2 to 4.

##  4 Discussion

###  4.1 Comparison of Temperatures Derived from H2CO, NH3, and CO

Ammonia (NH3) is widely recognized as a standard temperature tracer for dense
molecular gas (e.g., Ho & Townes 1983). Its unique properties make it an
invaluable tool for studying the thermal characteristics of various molecular
environments (e.g., Henkel et al. 1987, 2008; Molinari et al. 1996; Jijina et
al. 1999; Dunham et al. 2011; Wang et al. 2012; Wienen et al. 2012; Mangum et
al. 2013b; Lu et al. 2014, 2017; Ott et al. 2014; Gong et al. 2015a, b;
Krieger et al. 2017; Sokolov et al. 2018; Wang et al. 2023; Zhang et al.
2024). In order to investigate the temperature variations across various
layers of the molecular cloud, we conducted a comparative analysis between our
results of gas temperature derived from H2CO (3–2) (with $n_{\rm crit}$(H2CO
303–202) $\sim$ few 105 cm-3, Shirley 2015) and the previous measurements
using NH3 $(J,K)$ = (1,1) and (2,2) (with $n_{\rm crit}$(NH3) $\sim$ 103 cm-3,
Shirley 2015), specifically within the DR21 filament. As mentioned in Sect.
1.1, the temperature distribution of gas within the DR21 filament has been
charted using NH3 (1,1) and (2,2) with the GBT (beam size $\sim$32′′; Keown et
al. 2019). Analysis of the NH3 (2,2)/(1,1) line ratios has shown that the
kinetic temperature of the dense gas typically ranges between 20 and 30 K.
However, there are variations in temperature across different regions (see
Table 7). Specifically, the W75S region has a typical temperature of about 20
K, while the DR21(OH) region has temperatures ranging from 22 to 64 K, and the
DR21W1 region has temperatures around 35–60 K. High-resolution imaging of the
NH3 (1,1) and (2,2) lines within the DR21(OH) region was presented by Mangum
et al. (1992), using the VLA with a beam size of $\sim$4′′ ($\sim$0.03 pc at
1.5 kpc distance). The gas kinetic temperatures of dense ammonia cores derived
from line ratios of NH3 (2,2)/(1,1) range from 20 to ¿80 K, with a typical
value of $\sim$30 K. On a larger scale of $\sim$0.23 pc, the temperature
distribution derived from our H2CO (3–2) data exhibits similarities to that
obtained from NH3 (1,1) and (2,2). Notably, our analysis of H2CO (3–2)
consistently reveals higher gas temperatures compared to NH3 (1,1) and (2,2),
both within dense clumps and across extended regions of the DR21 filament (see
Sect. 3.4 and Table 1). These findings align with previous observations in
star-forming regions of the Milky Way, the Large Magellanic Cloud, and our
Galactic center clouds (Tang et al., 2017a, b, 2018a, 2018b, 2021; Ginsburg et
al., 2016).

Observations conducted using the KOSMA 3 m telescope have provided insights
into the gas properties within the DR21 filament region through the analysis
of low- to high-$J$ CO lines on a spatial scale of $\sim$0.4 pc (Jakob et al.,
2007). These observations reveal the presence of two distinct gas components
(see Table 7). The cold component exhibits a kinetic temperature ranging from
30 to 40 K, while the warm component displays temperatures ranging from 80 to
150 K in regions such as W75S FIR 1, DR21(OH), DR21, and DR21W1, at densities
of $n$(H2) $\sim$ 104–106 cm-3. Furthermore, in these regions, the gas
temperatures derived from H2CO line ratios range from 50 to 114 K on a spatial
scale of $\sim$0.1 pc. Notably, these values align closely with the
temperatures obtained from the warm gas component identified through the
analysis of CO lines.

The observed disparities likely arise from temperature gradients, as H2CO and
NH3 may probe distinct regions within star-forming regions. The H2CO (3–2)
lines emerge as an interesting tool for detecting the presence of warm and
dense molecular gas within star-forming regions. These transitions are highly
sensitive to environments where the gas temperature is elevated, making it an
ideal tracer of energetic conditions associated with active star formation. In
contrast, the NH3 (1,1) and (2,2) transitions are well-suited for probing the
cooler and less dense molecular gas prevalent in these regions. It would be
intriguing to conduct mapping of the region using higher excited NH3
transitions (e.g., Henkel et al. 1987, 2008; Mangum et al. 2013b; Lu et al.
2017; Gong et al. 2015a, b; Krieger et al. 2017). This approach would also
serve as a valuable tool to verify the authenticity of the high temperatures,
ensuring that they are not a result of slightly saturated H2CO 303–202 lines.

Table 3: Comparison of temperatures and Mach number of different regions.

Region | H2CO | NH3 | CO | FIR | Mach number
---|---|---|---|---|---
| K | K | K | K |
W75S | 24–95 | 15–64 | 41(82/37) | 15–25 | 2.1–7.2
DR21(OH) | 26–114 | 22–64 | 31(99/29) | 15–31 | 2.2–7.0
DR21 | 24–87 | 20–58 | 36(148/32) | 15–¿22 | 1.9–6.7
DR21W1 | 33–88 | 35–60 | 54(118/52) | 18–22 | 2.5–10.3
777Column 2: kinetic temperatures derived from H2CO line ratios (see Sect.
3.4). Column 3: kinetic temperatures derived from the NH3 (2,2)/(1,1) line
ratios taken from Keown et al. (2019). Column 4: average, warm, and cold
components temperatures of low– to high–$J$ CO taken from Jakob et al. (2007).
Columns 5: far infrared (FIR) dust temperatures taken from Hennemann et al.
(2012) and Cao et al. (2019). Columns 6: Mach number derived from H2CO (see
Sect. 3.5).

###  4.2 Comparison of Temperatures Obtained from Gas and Dust

Gas temperature refers to the kinetic temperature of the gas particles, while
dust temperature reflects the thermal equilibrium of the dust grains. The
investigation of gas and dust temperatures in molecular clouds is of paramount
importance in understanding the physical conditions within these regions. It
is commonly expected that gas and dust are thermally coupled in dense regions
($n$(H2) $\gtrsim$ 104 cm-3) due to the increased frequency of interactions
between these two components (Goldsmith, 2001). Previous observations with NH3
(1,1) and (2,2) lines have consistently demonstrated that the temperatures
derived from gas and dust (derived at far-infrared wavelengths) measurements
are typically in concordance within the active dense clumps found in Galactic
disk clouds (e.g., Dunham et al. 2010; Giannetti et al. 2013; Battersby et al.
2014; Merello et al. 2019; Tursun et al. 2020; Zhang et al. 2024). However, it
should be noted that gas temperatures obtained from H2CO (3–2 and 4–3) lines
do not exhibit agreement with FIR-derived dust measurements in both our
Galactic star-forming regions (Tang et al., 2017a, 2018a, 2018b) and Galactic
center clouds (Ao et al., 2013; Ginsburg et al., 2016; Immer et al., 2016).

As detailed in Sect. 1.1, the dust temperature distribution within the DR21
filament has been established through analyzing Herschel continuum data
observed at 70–500 $\mu$m (beam size $\sim$40′′), as reported by both
Hennemann et al. (2012) and Cao et al. (2019). The analysis reveals that the
dense clumps within our observed region of the DR21 filament exhibit a range
of dust temperatures spanning from 15 to 35 K (see Table 7). The typical
value, approximately 20 K, is found to be similar to the gas temperature
derived from the NH3 (2,2)/(1,1) line ratios (Mangum et al., 1992; Keown et
al., 2019), albeit lower than the results obtained from the H2CO line ratios
(see Sect. 3.4 and Table 1). These findings align with previous observational
results in star-forming regions within our galaxy (Tang et al., 2017a, 2018a,
2018b).

In order to facilitate a comparison between the temperatures derived from gas
and dust on a larger scale, we have applied a smoothing technique to our H2CO
data, resulting in a beam size of $\sim$40′′. The peak temperatures obtained
from this analysis are approximately 80, 95, 80, and 75 K in the W75S FIR 3,
DR21(OH), DR21(OH)S, and DR21 regions, respectively. It is worth noting that
these values remain higher than the temperatures measured using NH3 and dust
emission at FIR wavelengths. The structure of dust temperature closely
resembles that of the gas temperature obtained through NH3 and H2CO
observations on a scale of $\sim$0.3 pc in the DR21 filament. It is important
to note that the emission of dust at FIR wavelengths primarily originates from
colder dust components which may not be directly associated with the process
of star formation (e.g., Mangum et al. 2013b; Tang et al. 2018a). In
comparison to temperature measurements utilizing NH3 (1,1) and (2,2) and the
FIR continuum, the H2CO lines trace distinctly higher temperatures. This
observation confirms that H2CO may possess sensitivity towards the dense gas
that is intimately linked to active star formation.

###  4.3 Gas Heating

Gas heating in star-forming regions is a multifaceted process that holds
paramount significance in the formation and evolution of stars. Acquiring a
comprehension of the pivotal role played by gas heating in these regions,
measurements of temperature are mandatory to elucidate the underlying
mechanisms that drive star formation and shape the evolutionary trajectory of
galaxies.

####  4.3.1 Radiative Heating

Radiative heating plays a pivotal role in the intricate process of star
formation, particularly during the initial phases when protostars coalesce
within dense molecular clouds. During this period, protostars accrete material
and emit substantial amounts of radiation, which interact with and profoundly
influence the surrounding ISM. Radiation transfers energy to the ISM through
processes such as photoionization, photodissociation, and the generation of
photoelectrons. These energy transfer mechanisms result in the heating of the
ISM and serve as driving forces behind the subsequent stages of star
formation. Previous studies have conducted observations of several molecules,
such as NH3, H2CO, H2CS, CH3CN, CH3CCH, and CH3OH, in regions where massive
stars are forming (e.g., Lu et al. 2014; Giannetti et al. 2017; Tang et al.
2018a, b, 2021; Gieser et al. 2019, 2021, 2022, 2023; Lin et al. 2022; Wang &
Wang 2023). These observations suggest that the dense gas surrounding embedded
infrared sources experiences internal radiative heating on scales of
$\lesssim$0.1–1.0 pc. As stated in Sects. 3.3 and 3.4, high gas temperatures
obtained from H2CO line ratios associate with dense clumps especially in the
W75S FIR 3, DR21(OH), DR21(OH)S, and DR21 regions (see Fig. 4). This indicates
that the warm dense gas traced by H2CO may be influenced by radiation from
internally embedded infrared sources. Furthermore, investigations of CO, [CI],
[CII], and [OI] emissions in the DR21 region have revealed that the gas
experiences significant radiative heating (Lane et al., 1990; Jakob et al.,
2007; Ossenkopf et al., 2010).

Figure 5: Gas kinetic temperatures were determined by analyzing the H2CO (3–2)
line ratios from the central regions of the dense cores N44, N46, N48, and N54
(see Table 1), extending towards their respective edges. The format of the
regression fits is $T_{\rm kin}(\rm cores)=a\times(\frac{R}{\rm
arcsec})^{b}\leavevmode\nobreak\ {\rm K}$ while the fitting method is the
Levenberg-Marquardt method without the error of temperature, and the Reduced
Chi-Squared values for N44, N46, N48, and N54 are 1.7, 3.3, 5.7, and 1.9,
respectively. The average kinetic temperature at each projected distance is
represented by green triangles. The green line represents the connection of
the each average kinetic temperature, while the fitted results for H2CO are
depicted by black lines.

To investigate the influence of radiation heating on the extended dense gas
surrounding massive clumps within the DR21 filament on a scale of
$\sim$0.1–0.3 pc, we conducted a temperature profile analysis of four dense
cores situated in the W75S FIR 3, DR21(OH), DR21(OH)S, and DR21 regions (see
Fig. 5). These cores, namely N44, N46, N48, and N54, chosen from the
respective regions, exhibit distinctly higher kinetic temperatures derived
from H2CO line ratios (see Table 1) than those of the surrounding gas. We note
that Fig. 4 shows the spatial distribution of the dense core N38 to the west
of N44 and N48 ($\sim$32′′, $\sim$0.24 pc), the dense core N42 to the
southwest of N46 ($\sim$32′′, $\sim$0.24 pc), the dense core N47 to the north
of N46 ($\sim$18′′, $\sim$0.14 pc), and the dense core N53 to the north of N54
($\sim$19′′, $\sim$0.14 pc). Due to the limited resolution of our observations
(beam size $\sim$12′′), we cannot rule out that nearby sources may have
influenced the temperature profiles of N44, N46, N48, and N54. To reduce this
influence from nearby dense cores, we masked data at the midpoint between
selected dense cores and other sources. For the N44 dataset, southern data was
masked using a distance threshold exceeding 19′′ along the declination axis.
In the case of the N46, both northern and southern data were masked at
distances greater than 9′′ and 15′′, respectively. For N48, northern data was
masked at a distance exceeding 19′′. Similarly, for N54, northern data was
masked using a distance exceeding 10′′. Additionally, for N44 and N48, data
corresponding to the dense core N38 (with a radius of $\sim$16′′) was also
masked.

In the dense cores N44, N46, and N48, there are significant temperature
gradients observed, as depicted in Fig. 5. However, it is worth mentioning
that a relatively shallow temperature gradient is detected in dense core N54,
which could potentially be attributed to the fact that the dense core is
likely in an earlier evolutionary stage (e.g., Motte et al. 2007; Cao et al.
2019), with the embedded protostars or clusters not yet generating sufficient
heat to significantly warm the surrounding molecular envelope. As noted above,
the dense core N53 is positioned to the north of N54, at a distance of
approximately 19′′ ($\sim$0.14 pc). Previous observations have detected the
presence of CH3OH and H2O masers in N53 (e.g., Motte et al. 2007), suggesting
the occurrence of significant shocks in this area. Additionally, two
protostars were identified in the N53/N54 region by Davis et al. (2007) (see
Fig. 4). The complex temperature structure observed in the N53/N54 region may
be attributed to the impact of outflows and shocks originating from the
embedded protostars or clusters within N53 and N54, as well as the radiation
emitted by nearby protostars. The temperature profile surrounding these dense
clumps can be described by the Stefan-Boltzmann blackbody radiation law,
expressed as $T_{\rm kin}=0.86\times(\frac{L}{{\rm
L}_{\odot}})^{1/4}(\frac{R}{\rm pc})^{-1/2}\leavevmode\nobreak\ {\rm K}$,
where the luminosity $L$ is given in terms of solar luminosity (${\rm
L}_{\odot}$) and the distance $R$ is measured in parsecs. By adjusting the
emissivity of dust grains to be smaller than the characteristic black-body
temperature wavelength, the radiation law is modified to $T_{\rm
kin}=2.7\times(\frac{L}{{\rm L}_{\odot}})^{1/5}(\frac{R}{\rm
pc})^{-2/5}\leavevmode\nobreak\ {\rm K}$ (Wiseman & Ho, 1998; Tang et al.,
2018a). The fitted results of the four dense cores are

$\displaystyle T_{\rm kin}({\rm N44})=(268\pm 30)\times\bigg{(}\frac{R}{\rm
arcsec}\bigg{)}^{-0.54\pm 0.04}\leavevmode\nobreak\ {\rm K},$ (1)
$\displaystyle T_{\rm kin}({\rm N46})=(120\pm 16)\times\bigg{(}\frac{R}{\rm
arcsec}\bigg{)}^{-0.28\pm 0.05}\leavevmode\nobreak\ {\rm K},$ (2)
$\displaystyle T_{\rm kin}({\rm N48})=(176\pm 21)\times\bigg{(}\frac{R}{\rm
arcsec}\bigg{)}^{-0.43\pm 0.04}\leavevmode\nobreak\ {\rm K},$ (3)
$\displaystyle T_{\rm kin}({\rm N54})=(45\pm 11)\times\bigg{(}\frac{R}{\rm
arcsec}\bigg{)}^{-0.03\pm 0.09}\leavevmode\nobreak\ {\rm K},$ (4)

with power-law indices of $-$0.54, $-$0.28, $-$0.43, and $-$0.03 for N44, N46,
N48, and N54, respectively. With the exception of the dense core N54, the
temperature power-law indices obtained from fitting the H2CO line ratios in
our study exhibit consistency with previous observational results with a mean
value of about $-$0.4. These results align with previous studies that have
utilized various molecular tracers such as H2CO, H2CS, CH3CN, and CH3CCH in
the investigation of massive star-forming dense cores (e.g., Gieser et al.
2019, 2021, 2022, 2023; Lin et al. 2022). This indicates that the dense gas
heating of the massive cores on a scale of $\sim$0.1–0.3 pc in the DR21(OH),
DR21(OH)S, and DR21 regions is significantly influenced by internal radiation
emitted by associated protostar(s) and/or stellar clusters. The resolution of
our observation, $\sim$0.1 pc, does not allow for a thorough measurement of
the detailed temperature structures within these dense cores. Consequently, a
future high-resolution observation is necessary to address this limitation.

####  4.3.2 Explosive Heating

Explosive heating in molecular clouds refers to the process by which a large
amount of energy is rapidly released, leading to a sudden increase in
temperature within these clouds. The increased temperature can trigger
chemical reactions, influence the stability of the cloud, and even initiate
the formation of new stars. It may play a crucial role in the dynamics and
evolution of molecular clouds. Understanding the processes and effects of
explosive heating is essential for studying the complex interplay between gas
dynamics, star formation, and the overall evolution of molecular clouds.
Previous observations of H2CO and H2CS in the Orion KL region have provided
evidence that the dense gas in this area experiences heating due to shocks
induced by a mysterious explosive event that took place several hundred years
ago (e.g., Tang et al. 2018a; Li et al. 2020). This event had a significant
impact on the energetics of the Orion KL region. Zapata et al. (2013) proposed
that the enigmatic northeast–southwest outflow observed in the DR21 region was
likely the result of an explosive event that transpired $\sim$10,000 years
ago. This event bears resemblance to the disintegration of a massive stellar
system, akin to the occurrence witnessed in Orion KL 500 years ago, albeit
with approximately 20 times greater energy output. The accurate location of
this explosive event in the DR21 region was identified at $\alpha_{2000}$ =
20h39m01$\,.\\!\\!^{s}$1 and $\delta_{2000}$ =
42$\degr$19′37$\,.\\!\\!^{\prime\prime}$9 by Zapata et al. (2013). At the
location of the DR21 outflow origin, there is no presence of a young star in
the radio, submillimeter, or infrared wavelengths. This absence suggests that
the potential ”source” of the outflow may no longer be situated there, similar
to the explosive outflow observed in Orion KL (Zapata et al., 2009, 2013).

The explosive event is observed to occur at the center of an expanding
cometary H II region in DR21 (see Fig. 3 in Zapata et al. 2013). As previously
mentioned, the heating of dense gas in the DR21 region is greatly influenced
by the internal radiation emitted by the associated protostar(s) and/or
stellar clusters. In the star formation region of DR21, differentiating
between radiation and explosive heating can be challenging but can be done by
considering the detailed spatial distribution of $T_{\rm kin}$. Radiation
heating is usually more evenly distributed throughout a star-forming region,
as it originates from the young stars themselves. Explosive heating, however,
may exhibit localized regions of enhanced heating, corresponding to the
locations of energetic events (e.g., Tang et al. 2018a; Li et al. 2020).
Previous observations suggest that H2CO could serve as a promising indicator
for detecting outflows or shocks (e.g., Tang et al. 2017b; Lu et al. 2021;
Izumi et al. 2023). As stated in Sect. 3.5, the DR21 region exhibits
substantial non-thermal linewidths ($\sigma_{\rm NT}$ $\sim$ 2.3 km s-1) and
Mach numbers ($\mathcal{M}$ $\sim$ 4.5), as depicted in Fig. 4. Their
correlation suggests that the dense gas, which is traced by H2CO, may be
influenced by the shocks resulting from explosive flows and/or expanding H II
regions. To investigate the impact of the explosive event on the heating of
dense gas in the DR21 region, we conducted an analysis of the temperature
structure within the dense core N46 (see Figs. 4 and 5) which is located at
the putative position of the supposed explosive event (Zapata et al., 2013).
The analysis of H2CO data reveals that the gas temperature within the dense
core N46 of the DR21 region is measured to be 77 K. This temperature is lower
than that observed in the dense cores N44 and N48 of the DR21(OH) and
DR21(OH)S regions. Furthermore, the temperature gradient observed in the DR21
region is flatter when compared to the temperature gradients observed in the
DR21(OH) and DR21(OH)S regions (see Fig. 5). As previously noted, the dense
gas may exhibit localized regions of increased heat when subjected to heating
by an explosive event. However, a relatively uniform temperature gradient in
the DR21 region is inconsistent with this pattern of explosive heating. As
delineated in the preceding section, the temperature profile derived from the
dense core N46 aligns with the results anticipated from radiation heating. The
heating of the dense gas in the DR21 region may be partly attributed to the
shocks generated by an explosive event that occurred a long time ago. However,
our temperature measurements with H2CO do not provide direct evidence to
support the hypothesis that the dense gas is heated by shocks resulting from a
past explosive event on a scale of $\sim$0.1 pc in the DR21 region.

Zapata et al. (2013) proposed that the DR21 northeast–southwest outflow was
likely generated by a powerful explosion occurring $\sim$10,000 years ago.
However, the genesis of the DR21 outflow remains a subject of ongoing
investigation (e.g., Skretas et al. 2023). Previous observations suggest that
the DR21W1 region is correlated with the western DR21 flow (e.g., Jaffe et al.
1989; Lane et al. 1990; Wilson & Mauersberger 1990; Garden et al. 1991a, b;
Garden & Carlstrom 1992; Davis et al. 2007). It appears that a high
temperature ($\sim$84 K) associates with the dense core N34 in the DR21W1
region (see Fig. 4). A flat temperature structure, analogous to that of the
DR21 region, was detected in the DR21W1 region. The highest non-thermal
linewidths ($\sigma_{\rm NT}$ $\sim$ 4.4 km s-1) and Mach numbers
($\mathcal{M}$ $\sim$ 10) of the entire DR21 filament are found in the
northeastern and southern region of DR21W1 (see Fig. 4). These confirm that
the dense gas in the DR21W1 region is strongly influenced by shocks which may
be caused by explosive flows. In the DR21W1 region, Cao et al. (2019)
discovered an infrared-quiet massive dense core with a luminosity of $\sim$350
${\rm L}_{\odot}$. The absence of strong 8 $\mu$m emission in this region
(refer to Fig. 1 in Kumar et al. 2007) suggests that the dense core is in an
early evolutionary stage. In the study conducted, the gas temperature in the
DR21W1 region, which is influenced by young stellar objects (YSOs), was
determined using the modified Stefan-Boltzmann blackbody radiation law (refer
to Sect. 4.3.1). It was assumed that the embedded YSOs (or YSO candidates)
serve as the primary sources of energy in the DR21W1 region, with a luminosity
of $\sim$350 ${\rm L}_{\odot}$ as reported by Cao et al. (2019). The derived
gas temperature from the modified Stefan-Boltzmann blackbody radiation law was
found to be $\sim$22 K at radii of 0.1 pc, which is significantly lower than
the typical temperature of 70–80 K obtained from the measurements of H2CO line
ratios. This indicates that the radiation heating from internal embedded YSOs
(or YSO candidates) only affects the local region. The high temperatures of
dense gas in the DR21W1 region should be dominated by the shocks from the
western DR21 flow. Our measurements of H2CO towards the DR21W1 region provide
compelling evidence that the dense gas in this area is heated by shocks which
may arise from explosive flows.

####  4.3.3 Turbulent Heating

Turbulence is a prevalent phenomenon observed in various phases of the
interstellar medium (Mac Low & Klessen, 2004; Elmegreen & Scalo, 2004; Scalo &
Elmegreen, 2004; Sokolov et al., 2018; Wang & Wang, 2023; Wang et al., 2023).
Recent observations conducted on Galactic Central Molecular Zone clouds using
H2CO (3–2 and 4–3) indicate that the warm dense gas is primarily heated by
turbulence occurring on a scale of $\sim$1 pc (Ao et al., 2013; Ginsburg et
al., 2016; Immer et al., 2016). In Galactic massive star-forming regions, it
appears that molecular gas heated by turbulence is commonly encountered (Tang
et al., 2018a, b, 2021). Notably, multiple observations utilizing molecular
tracers, including H2CO, NH3, and CH3CCH, in star formation regions, indicate
a significant correlation between linewidth and gas kinetic temperature
(Wouterloot et al., 1988; Molinari et al., 1996; Jijina et al., 1999; Wu et
al., 2006; Urquhart et al., 2011, 2015; Wienen et al., 2012; Lu et al., 2014;
Giannetti et al., 2017; Tang et al., 2017a, 2018a, 2018b, 2021). As a result,
higher temperatures inferred from various tracers align with elevated
turbulence levels.

We investigate the potential relationship between turbulence and temperature
on a scale of $\sim$0.1 pc within the DR21 filament. To assess turbulence, we
utilize the non-thermal linewidth ($\sigma_{\rm NT}$) of H2CO, which serves as
a reliable indicator. Additionally, we determine the gas kinetic temperature
using the H2CO line ratio. Specifically, we focus on positions near dense
regions that exhibit strong non-thermal motions ($\mathcal{M}$ $\gtrsim$ 3.0).
As stated in Sect 3.5, the dense gas in locations characterized by higher Mach
numbers ($\mathcal{M}$ $\gtrsim$ 3.0) is significantly impacted by non-thermal
motions. Therefore, for our analysis, we specifically choose positions in
close proximity to W75S FIR 3, W75S FIR 2, DR21(OH)S, DR21, and DR21W1. In
Fig. 6, we present the correlation between the non-thermal linewidth and gas
kinetic temperature for both the H2CO 303–202 line and the combined H2CO
322–221 and 321–220 lines. A power law relationship is fitted, with $T_{\rm
kin}\propto\sigma_{\rm NT}^{0.42\pm 0.04}$ for H2CO 303–202 and $T_{\rm
kin}\propto\sigma_{\rm NT}^{0.41\pm 0.03}$ for the combined H2CO 322–221 and
321–220 lines. The correlation coefficients, $R$, are determined to be 0.45
and 0.55, respectively. These results suggest that the elevated temperature
observed in regions of strong non-thermal motion within the DR21 filament, as
indicated by H2CO, can be attributed to turbulence occurring at a scale of
$\sim$0.1 pc, which aligns with previous observational studies conducted on
massive star-forming clumps (ATLASGAL sample, Tang et al. 2018b) and the Orion
molecular cloud 1 (OMC-1, Tang et al. 2018a), where similar temperature
variations were observed on a scale ranging from approximately 0.06 to 2 pc.
One should note that this agreement pertains solely to the intercept and not
the power law index (see Fig. 6). The power law index fitted in the DR21
filament is approximately 0.41-0.42, which is lower than the previous
observational results ($T_{\rm kin}\propto\sigma_{\rm NT}^{0.76-1.26}$) in the
ATLASGAL sample and OMC-1 (Tang et al., 2018a, b). As previously mentioned,
the dense gas in the DR21 filament is also influenced by various factors, such
as outflows, shocks, and/or radiation from regions of massive star formation
in W75S, DR21(OH), and DR21. The variation in the power law index could be
attributed to the disparity in the contribution fractions of gas heating
mechanisms to the gas temperature between the DR21 filament and the ATLASGAL
sample, as well as OMC-1.

Assuming that the dominant factor heating the gas is turbulence, we employ the
methodology outlined by Tang et al. (2018a) in their Equation (2) to compute
the gas kinetic temperature, denoted as $T_{\rm turb}$, resulting from
turbulence in the DR21 filament. Our calculations are based on the following
assumptions: a gas density of $n$(H2) = 105 cm-3, a typical velocity gradient
of 1 km s-1 pc-1, the non-thermal linewidth values of H2CO provided in Table
1, a cloud size of $\sim$1 pc, and a dust temperature equal to the gas
temperature derived from H2CO. The computed $T_{\rm turb}$ values are
presented in Table 1, revealing a notable correlation between the derived
$T_{\rm turb}$ value and the kinetic temperature obtained from the H2CO line
ratio. These results suggest that turbulence plays a significant role in
determining the dense gas kinetic temperature on a scale of $\sim$0.1 pc in
the DR21 filament.

Figure 6: The relationship between the non-thermal linewidth ($\sigma_{\rm
NT}$) and gas kinetic temperature, as determined from the H2CO line ratio, is
examined in those parts of the DR21 filament, which are characterized by a
Mach number $\mathcal{M}$ $\gtrsim$ 3.0 (see Sect. 4.3.3). A linear fit is
applied to the H2CO 303–202 and the combined H2CO 322–221 and 321–220 lines,
yielding the following results: log$T_{\rm kin}=(0.42\pm 0.04)\times{\rm
log}\sigma_{\rm NT}+(1.60\pm 0.01)$ and log$T_{\rm kin}=(0.41\pm
0.03)\times{\rm log}\sigma_{\rm NT}+(1.63\pm 0.01)$, respectively. In this
fitting, the linear fit was performed on the unweighted data using the least
squares method. The Pearson correlation coefficients, $R$, for these fits (red
lines) are 0.45 and 0.55, respectively.

###  4.4 Comparison with the OMC-1 and N113

The Orion molecular cloud 1 is a well-known target for studying the physical
and chemical properties of molecular clouds and their role in star formation.
Its complex gas heating has been revealed by observations made with the APEX
12 m telescope over a region of size $\sim$1.1$\times$1.7 pc2, where the same
H2CO (3–2) lines were mapped on a scale of $\sim$0.06 pc (Tang et al., 2018a).
The complex gas heating in the OMC-1 is likely due to multiple processes
including star formation activity, radiation, and turbulence. The N113 region,
which is located in the Large Magellanic Cloud, is a well-studied site for
massive star formation and shows strong molecular line emission (e.g., Tang et
al. 2017a, 2021; Gong et al. 2023a). The complex gas heating in N113 was also
studied using an ALMA survey over a relatively large area of N113
($\sim$2.4$\times$6.1 pc2), where the transitions of the same H2CO (3–2) lines
were detected on a linear scale of $\sim$0.4 pc (Tang et al., 2018b). The gas
heating process in N113 appears to be as intricate as that of its counterpart
in OMC-1.

As discussed above, our observations in the DR21 filament
($\sim$1.4$\times$4.4 pc2) on a scale of $\sim$0.1 pc also indicate the
presence of similar gas heating mechanisms as in OMC-1 and N113. In addition,
we compare the detailed physical parameters of dense gas, H2CO line ratio,
$T_{\rm kin}$, $\sigma_{\rm T}$, $\sigma_{\rm NT}$, $c_{\rm s}$, and
$\mathcal{M}$, derived from H2CO (3–2) lines in the DR21 filament, OMC-1, and
N113 in Table 2. The median values of these physical parameters in the dense
gas within the DR21 filament appear to be nearly identical to the results
obtained in OMC-1 and N113 on a scale of $\sim$0.1–0.4 pc (see Table 2). It
should be noted that the mean values of non-thermal linewidth and Mach number
in OMC-1 are marginally lower than those observed in DR21 and N113. This
discrepancy may be attributed to the superior resolution of OMC-1 (beam size
$\sim$0.06 pc) and/or the inherent turbulence fluctuations across various
molecular clouds. These similarities in physical parameters suggest that the
processes governing the dynamics and thermodynamics of the dense gas traced by
H2CO in different star formation regions may share common underlying
principles, despite variations in specific environmental conditions. The dense
gas in star formation regions is influenced by similar environmental factors,
such as gravitational collapse, turbulent motions, and thermal processes,
which contribute to the consistency in these parameters across various regions
and evolutionary stages. The physical conditions in star-forming regions
appear to be shaped by similar mechanisms, leading to the convergence of these
parameters. This can include processes such as supersonic turbulence, thermal
pressure, and the interaction between magnetic fields and turbulent motions,
all of which contribute to the observed similarities in the physical
parameters.

##  5 Summary

The kinetic temperature structure of the massive filament DR21 in the Cygnus X
molecular cloud complex has been mapped using the H2CO 218 GHz line triplet
observed with the IRAM 30 m telescope. The main results are the following:

  1. 1.

The H2CO 303–202 emission exhibits an extensive distribution and provides a
clear measurement of the dense ridge structure of the DR21 filament.

  2. 2.

By employing the RADEX non-LTE model, we have determined the gas temperature
through the modeling of the measured H2CO 0.5$\times$[(322–221 \+
321–2${}_{20})$/303–202] line ratios. The derived gas kinetic temperatures
exhibit a range, varying from 24 to 114 K, with an average of 48.3$\pm$0.5 K
at a spatial density of 105 cm-3. In comparison to temperature measurements
utilizing NH3 (1,1)/(2,2) and FIR wavelengths, the H2CO lines trace distinctly
higher temperatures.

  3. 3.

The dense gas, traced by H2CO, exhibiting kinetic temperatures in excess of 50
K appears to be correlated with the dense clumps found in various regions,
namely W75S FIR 3, DR21(OH), DR21(OH)W, DR21(OH)S, DR21, and DR21W1. On the
other hand, the outskirts of the DR21 filament show lower temperature
distributions ($T_{\rm kin}$ $<$ 50 K).

  4. 4.

Four dense cores, N44, N46, N48, and N54, located in the DR21(OH), DR21,
DR21(OH)S, and W75S FIR 3 regions, respectively, exhibit temperature gradients
on a scale of approximately 0.1–0.3 pc. This suggests that the warm dense gas
traced by H2CO is influenced by internal star formation activity. Unlike the
dense core N54, the temperature profiles of these dense cores could be fitted
with power-law indices ranging from $-$0.3 to $-$0.5, with a mean value of
$\sim-$0.4, indicating that the warm dense gas probed by H2CO is heated by
radiation emitted from internally embedded protostar(s) and/or stellar
clusters.

  5. 5.

In the DR21 region, our temperature measurements towards the dense core N46,
located at the putative position of a supposed major explosive event, provide
no direct evidence supporting this notion on a scale of $\sim$0.1 pc.
Nevertheless, we find compelling evidence that the dense gas in the DR21W1
region is indeed heated by shocks originating from the western DR21 flow.

  6. 6.

The non-thermal linewidths of H2CO exhibit a correlation with the gas kinetic
temperatures within the DR21 filament. This suggests that higher temperatures,
as traced by H2CO, are associated with turbulence on a scale of $\sim$0.1 pc.

  7. 7.

The physical parameters of the dense gas, as determined from H2CO lines in the
DR21 filament, exhibit a striking similarity to the results obtained in OMC-1
and N113 on a scale of approximately 0.1-0.4 pc. This may imply that the
mechanisms governing the dynamics and thermodynamics of dense gas traced by
H2CO in diverse star formation regions may be governed by common underlying
principles, despite variations in specific environmental conditions.

###### Acknowledgements.

We thank the staff of the IRAM telescope for their assistance in observations.
This work acknowledges the support of the National Key R&D Program of China
under grant No. 2023YFA1608002, the Chinese Academy of Sciences (CAS) “Light
of West China” Program under grant No. xbzg-zdsys-202212, the Tianshan Talent
Program of Xinjiang Uygur Autonomous Region under grant No. 2022TSYCLJ0005,
and the Natural Science Foundation of Xinjiang Uygur Autonomous Region under
grant No. 2022D01E06. It was also partially supported by the National Key R&D
Program of China under grant No. 2022YFA1603103, the CAS “Light of West China”
Program under grant No. 2020-XBQNXZ-017, the National Natural Science
Foundation of China under grant Nos. 12173075 and 12373029, and the Youth
Innovation Promotion Association CAS. T. Liu, K. Wang, X. P. Chen, and J. W.
Wu acknowledge support by the Tianchi Talent Program of Xinjiang Uygur
Autonomous Region. C. Henkel acknowledges support by the Chinese Academy of
Sciences President’s International Fellowship Initiative under grant Nos.
2023VMA0031 and 2025PVA0048. This research has used NASA’s Astrophysical Data
System (ADS).

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##  Appendix A H2CO Velocity Channel Maps

Figure 7: Channel maps of the H2CO (303–202) transition. The contour levels
are running from 0.5 to 2.0 K in steps of 0.5 K and from 2 to 10 K in steps of
1 K ($T_{\rm mb}$ scale; color bar in units of K).

##  Appendix B The LTE and Non-LTE Models for H2CO

Figure 8: RADEX non-LTE modeling of the relation between the kinetic
temperature and the average ratio of H2CO 322–221/303–202 and 321–220/303–202
with an assumed density of $n$(H2) =105 cm-3 and column densities
$N$(para-H2CO) =1.0$\times$1013, 6.6$\times$1013 and 1.0$\times$1014 cm-2
(red, green, and blue), and the LTE (cyan), and an averaged linewidth of 3.8
km s-1. The LTE kinetic temperature ($T_{\rm LTE}$) is determined using the
methodology described in Mangum & Wootten (1993) and Tang et al. (2017b).
$T_{\rm LTE}$ is calculated as $T_{\rm LTE}\,=\,\frac{47.1}{{\rm
ln}(0.556/{\rm Ratio})}\leavevmode\nobreak\ {\rm K}$, where Ratio represents
the average ratio of H2CO 322–221/303–202 and 321–220/303–202. The black
points are derived from our observed H2CO line ratios from the DR21 filament
for a column density $N$(para-H2CO) = 6.6$\times$1013 cm-2. The temperature
uncertainties are obtained from observed H2CO line ratio errors.