11institutetext:  Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, 830011 Urumqi, PR China
11email<EMAIL_ADDRESS>22institutetext: Key Laboratory of Radio
Astronomy, Chinese Academy of Sciences, 830011 Urumqi, PR China
33institutetext: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
53121 Bonn, Germany
33email<EMAIL_ADDRESS>44institutetext: Astronomy Department, King
Abdulaziz University, PO Box 80203, 21589 Jeddah, Saudi Arabia
55institutetext: Korea Astronomy and Space Science Institute, 776 Daedeok-
daero, Yuseong-gu, Daejeon 34055, Republic of Korea 66institutetext: National
Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
77institutetext: Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210008, PR China 88institutetext: Argelander Institut für Astronomie,
Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany 99institutetext:
Department of Space, Earth and Environment, Chalmers University of Technology,
Onsala Space Observatory, SE-43992 Onsala, Sweden 1010institutetext:
Observatorio de Madrid, OAN-IGN, Alfonso XII, 3, E-28014-Madrid, Spain
1111institutetext: European Southern Observatory, Alonso de Córdova 3107,
Vitacura, Santiago 763-0355, Chile 1212institutetext: Joint ALMA Observatory,
Alonso de Córdova 3107, Vitacura, Santiago 763-0355, Chile 1313institutetext:
Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The
Netherlands 1414institutetext: Department of Physics and Astronomy, University
College London, Gower Street, London, WC1E 6BT, UK 1515institutetext: Ioffe
Physical-Technical Institute, Polytekhnicheskaya Str. 26, 194021 St.
Petersburg, Russia 1616institutetext: Kapteyn Astronomical Institute,
University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands
1717institutetext: National Radio Astronomy Observatory, P.O. Box O, 1003
Lopezville Road, Socorro, NM 87801, USA 1818institutetext: Joint ALMA
Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile
1919institutetext: Department of Physics, Nagoya University, Furo-cho,
Chikusa-ku Nagoya, 464-8601, Japan 2020institutetext: School of Astronomy and
Space Science, Nanjing University, 210093 Nanjing, PR China 2121institutetext:
University of Chinese Academy of Sciences, 100080 Beijing, PR China

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

IV. The ALMA view of N113 and N159W in the LMC

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 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 formaldehydeKinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehyde K. M. Menten Kinetic
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 formaldehyde C. -H. R. Chen
Kinetic 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 formaldehyde M.-Y. Lee 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. G. Mangum Kinetic 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 S. Mühle Kinetic temperature of massive star-
forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehyde S. Aalto
Kinetic temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehyde S. García-Burillo Kinetic temperature of massive
star-forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehyde S. Martín
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 S. Viti 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 S. Muller Kinetic
temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehyde F. Costagliola Kinetic temperature of massive star-
forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehyde H. Asiri
Kinetic temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehyde S. A. Levshakov Kinetic temperature of massive
star-forming molecular clumps measured with formaldehydeKinetic temperature of
massive star-forming molecular clumps measured with formaldehyde M. Spaans
Kinetic temperature of massive star-forming molecular clumps measured with
formaldehydeKinetic temperature of massive star-forming molecular clumps
measured with formaldehyde J. Ott Kinetic temperature of massive star-forming
molecular clumps measured with formaldehydeKinetic temperature of massive
star-forming molecular clumps measured with formaldehyde C. M. V. Impellizzeri
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. Fukui Kinetic
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 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 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 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 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 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

We mapped the kinetic temperature structure of two massive star-forming
regions, N113 and N159W, in the Large Magellanic Cloud (LMC). We have used
$\sim$1$\,.\\!\\!^{\prime\prime}$6 ($\sim$0.4 pc) resolution measurements of
the para-H2CO $J_{\rm K_{a}K_{c}}$ = 303–202, 322–221, and 321–220 transitions
near 218.5 GHz to constrain RADEX non-LTE models of the physical conditions.
The gas kinetic temperatures derived from the para-H2CO line ratios
322–221/303–202 and 321–220/303–202 range from 28 to 105 K in N113 and 29 to
68 K in N159W. Distributions of the dense gas traced by para-H2CO agree with
those of the 1.3 mm dust and _Spitzer_ 8.0 $\mu$m emission, but do not
significantly correlate with the H$\alpha$ emission. The high kinetic
temperatures ($T_{\rm kin}$ $\gtrsim$ 50 K) of the dense gas traced by
para-H2CO appear to be correlated with the embedded infrared sources inside
the clouds and/or YSOs in the N113 and N159W regions. The lower temperatures
($T_{\rm kin}$ $<$ 50 K) are measured at the outskirts of the H2CO-bearing
distributions of both N113 and N159W. It seems that the kinetic temperatures
of the dense gas traced by para-H2CO are weakly affected by the external
sources of the H$\alpha$ emission. The non-thermal velocity dispersions of
para-H2CO are well correlated with the gas kinetic temperatures in the N113
region, implying that the higher kinetic temperature traced by para-H2CO is
related to turbulence on a $\sim$0.4 pc scale. The dense gas heating appears
to be dominated by internal star formation activity, radiation, and/or
turbulence. It seems that the mechanism heating the dense gas of the star-
forming regions in the LMC is consistent with that in Galactic massive star-
forming regions located in the Galactic plane.

###### Key Words.:

stars: formation – Magellanic Clouds – ISM: clouds –ISM: molecules – radio
lines: ISM

##  1 Introduction

Molecular gas is the fuel for star formation. Increasing evidence indicates
that its physical properties affect star formation rate, spatial distribution,
and essential properties of the next generation of stars, like 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). Unfortunately, the most readily accessible molecular
transitions, those of CO, suffer from radiative transfer effects such as high
optical depths or subthermal excitation, and are not a reliable probe of these
physical conditions. Furthermore, many molecules tracing the molecular mass in
external galaxies (e.g., Gao & Solomon 2004a, b; Wu et al. 2005; Zhang et al.
2014; Henkel et al. 2018; Li et al. 2021), including CO, HCN, HCO+, HNC, and
CS, suffer from a coupled sensitivity to kinetic temperature and density,
making them degenerately sensitive to high spatial densities and low kinetic
temperatures or vice versa. Obtaining information about the individual
physical conditions in these regions requires a molecular tracer that
possesses singular sensitivity to kinetic temperature and that can then be
used to also determine, in a second step, the number density of molecular
hydrogen, $n$(H2).

Formaldehyde (H2CO), a slightly asymmetric rotor molecule, is a ubiquitous
molecule in the interstellar medium (e.g., Downes et al. 1980; Bieging et al.
1982; Henkel et al. 1991; Zylka et al. 1992; Liszt et al. 2006; Mangum et al.
2008, 2013, 2019; Ao et al. 2013; Tang et al. 2013, 2014; Ginsburg et al.
2015, 2016; Guo et al. 2016; Yan et al. 2019). H2CO can be formed on the
surface of dust grains by successive hydrogenation of CO
(CO$\rightarrow$HCO$\rightarrow$H2CO) (Watanabe & Kouchi, 2002; Woon, 2002;
Hidaka et al., 2004; Yan et al., 2019). It is then released into the gas phase
by shocks or UV heating and destroyed by photodissociation. Alternatively, gas
phase formation is also possibly contributing to the total H2CO abundance in
Galactic environments such as the dark cloud TMC-1 (e.g., Soma et al. 2018).
CH3OH, which is also the product of hydrogenation of CO on grain surfaces and
is hardly formed in the gas phase, is deficient in the the Large Magellanic
Cloud (LMC) (e.g., Shimonishi et al. 2016). In the LMC where dust is less
abundant than in the Galaxy, gas phase formation of H2CO may thus be relevant.

H2CO has a stable fractional abundance in star formation regions. Variations
in fractional abundance of H2CO 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; Zhu et al. 2020). As further demonstrated in recent studies of
formaldehyde in the LMC, the fractional abundance of para-H2CO in its star-
forming regions has similar values as in Galactic regions of massive star
formation (Tang et al., 2017a, b, 2018b). H2CO has a rich variety of
millimeter and submillimeter transitions which are a reliable probe to trace
physical conditions of molecular clouds (e.g., Henkel et al. 1980, 1983;
Mangum & Wootten 1993; Mühle et al. 2007; Mangum et al. 2008, 2013, 2019;
Ginsburg et al. 2011; Tang et al. 2017a, b, 2018a, 2018b). Since the relative
populations of the $K_{\rm a}$ ladders of H2CO are governed by collisions,
line ratios involving different $K_{\rm a}$ ladders are good tracers of the
kinetic temperature (Mangum & Wootten, 1993). Therefore, these H2CO line
ratios are an ideal thermometer for tracing the temperature of dense gas in
low-metallicity galaxies with strong UV radiation (Tang et al., 2017b).

The Large Magellanic Cloud is the closest star-forming galaxy to our Milky Way
($\sim$50 kpc, e.g., Pietrzyński et al. 2013, 2019). In the LMC, the far-
ultraviolet (FUV) radiation field is stronger than in the Milky Way while
metallicities are lower (Westerlund, 1990; Rolleston et al., 2002). The LMC
provides an ideal laboratory for studying star formation, particularly massive
star formation associated with its numerous stellar clusters in such an active
low-metallicity galaxy. Previous observations of CO $J$ = 1–0 and 3–2 show
that the higher gas temperatures in low/moderate density regions ($n$(H2)
$\sim$ a few $\times$ 103 cm-3 in most cases) are correlated with stronger
H$\alpha$ flux in giant molecular clouds of the LMC (Minamidani et al., 2008,
2011). Further observations of CO $J$ = 4–3 to 12–11 indicate that the gas
temperatures are high ($T_{\rm kin}$ ¿ 150 K) at densities $\sim$103 cm-3 in
active star-forming regions of the LMC (Lee et al., 2016). These suggest that
FUV heating may strongly impact on the molecular gas in low density regions
($n$(H2) $<$ 105 cm-3) of the LMC. In contrast, recent low-resolution (30′′ or
$\sim$7 pc) observations of para-H2CO $J$ = 3–2 show that the kinetic
temperatures of the dense gas ($n$(H2) $\sim$ 105 cm-3) are correlated with
the ongoing massive star formation in the LMC (Tang et al., 2017b). This
suggests that inside the LMC the gas heating may be different between
molecular cloud layers of lower and higher density. Although gas heating in
low density regions of molecular clouds of the LMC can be expected to be due
to the exciting OB stars of photon dominated regions (PDRs) (e.g., Kaufman et
al. 1999; Lee et al. 2019), the detailed relationship between the temperature
of the dense gas and the star formation in the LMC is still under debate.

In this paper, we aim to map the kinetic temperature structure of two massive
star-forming regions, N113 and N159W, in the LMC with the para-H2CO triplet
($J_{\rm K_{a}K_{c}}$ = 303–202, 322–221, and 321–220) near 218.5 GHz and to
investigate the gas heating process affecting the dense gas. In Sects. 2 and
3, we introduce our targets, observations of the para-H2CO triplet, data
reduction, and results. We discuss the resulting kinetic temperatures derived
from para-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 exploring H2CO
as a probe of gas conditions in a variety of Galactic and extragalactic
sources.

##  2 Targets, observations, and data reduction

###  2.1 Targets

The two massive star-forming regions, N113 and N159W, in the LMC were selected
from previous single dish observations with the Atacama Pathfinder EXperiment
telescope (APEX) (beam size $\sim$30′′) in the course of which three
transitions of para-H2CO ($J$ = 3–2) near $\sim$218.5 GHz were detected (Tang
et al., 2017b). The dense gas kinetic temperatures of N113 and N159W derived
from these para-H2CO line ratios are 54 $\pm$ 7 K and 35 $\pm$ 4 K,
respectively, on a scale of $\sim$7 pc (Tang et al., 2017b).

Table 1: Observational parameters. Source | Central frequency | Bandwidth | Channel width | Beam size | Position angle | RMS
---|---|---|---|---|---|---
| MHz | MHz | km s-1 | ”$\times$” | ° | mJy beam-1
N113 | 217104.980 | 937.5 | 1.35 | 1.7$\times$1.3 | 60 |  $\sim$2.0
| 218475.632 | 937.5 | 1.34 | 1.6$\times$1.3 | 61 |  $\sim$2.0
| 230538.000 | 937.5 | 1.27 | 1.6$\times$1.2 | 61 |  $\sim$2.0
N159W | 217104.980 | 937.5 | 1.35 | 1.6$\times$1.3 | 55 |  $\sim$2.0
| 218475.632 | 937.5 | 1.34 | 1.6$\times$1.3 | 56 |  $\sim$2.0
| 230538.000 | 937.5 | 1.27 | 1.5$\times$1.3 | 56 |  $\sim$2.0

Below follows a brief description of the selected targets.
N113, located in the central part of the LMC, contains the most intense H2O
maser of the Magellanic Clouds (Ellingsen et al., 2010), which indicates
active star formation activity at an early stage. A large number of molecular
transitions have been reported in N113 (e.g., Chin et al. 1996, 1997; Heikkilä
et al. 1998; Wang et al. 2009; Paron et al. 2014; Nishimura et al. 2016; Tang
et al. 2017b; Sewiło et al. 2018, 2019), while interferometric data from HCN,
HCO+, HNC, CH3OH, and 1.3 mm continuum (Wong et al., 2006; Seale et al., 2012;
Sewiło et al., 2018, 2019) reveal a filamentary structure elongated roughly
North-South with a length of $\sim$6 pc, including a few dense clumps with
radius $\sim$0.5 pc.
N159W, located at the southwestern tip of 30 Doradus, one of the most intense
star-forming regions in the LMC, contains a large number of O- and B-type
stars, embedded young stellar objects (YSOs), and ultracompact H II regions
(e.g., Jones et al. 2005; Fariña et al. 2009; Chen et al. 2010; Carlson et al.
2012). It shows the brightest single-dish CO ($J$ = 1–0) peak in the LMC
(e.g., Johansson et al. 1994; Fukui et al. 1999; Wong et al. 2011) and has
been frequently observed with a large number of molecular transitions (e.g.,
Johansson et al. 1994; Chin et al. 1996, 1997; Heikkilä et al. 1998, 1999;
Paron et al. 2014; Pineda et al. 2008; Lee et al. 2016; Nishimura et al. 2016;
Tang et al. 2017b). High resolution observations reveal a complex filamentary
distribution of molecular gas (Seale et al., 2012; Fukui et al., 2015). The
gas temperature obtained from the NH3(2,2)/(1,1) line ratio at $\sim$19′′
resolution is $\sim$16 K (Ott et al., 2010), which is only about half the
temperatures derived from low resolution ($\sim$30′′) para-H2CO and the dust
(Tang et al., 2017b). After deriving kinetic temperatures from our high
resolution H2CO data, this difference will be discussed in more detail in
Sect. 4.1.

###  2.2 Observations and data reduction

Our observations were carried out with the 12 m array of the Atacama Large
Millimeter/submillimeter Array (ALMA) at Band 6 (Project: 2013.1.01136.S). Two
Execution Blocks were conducted using 35 and 38 antennas in January 17 and
April 8, 2015, respectively. The projected baseline lengths ranged from 15 to
349 m. The primary beam (FWHM) and the maximum recoverable scale are
$\sim$$29^{\prime\prime}$ and $\sim$$13^{\prime\prime}$, respectively. The
flux, bandpass, and phase calibrators were J0519-454, J0519-4546, and
J0635-7516, respectively. Three spectral windows were centered on SiO (5–4),
H2CO (322–221), and CO (2–1), respectively, with a $\sim$937.5 MHz bandwidth
and 960 channels for each band, yielding a channel width of 976.562 kHz,
corresponding to 1.27–1.35 km s-1. For the para-H2CO triplet the synthesized
beam size is
$\sim$1$\,.\\!\\!^{\prime\prime}$6$\times$1$\,.\\!\\!^{\prime\prime}$3
($\sim$0.4$\times$0.3 pc2 at 50 kpc distance). The surveyed areas of N113 and
N159W are centered on $\alpha_{2000}$ = 05:13:18.2, $\delta_{2000}$ =
–69:22:25.0 and $\alpha_{2000}$ = 05:39:36.0, $\delta_{2000}$ =– 69:45:25.0,
respectively. Detailed observational parameters are listed in Table 1. The
para-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.

The data were calibrated using the CASA111https://casa.nrao.edu version 4.2.2
pipeline and were then imaged using CASA version 5.6.1. We imaged the
calibrated data using a modified script associated with the data release. No
primary beam correction has been applied for Figs. 1, 8, and 9, but it has
been applied for data analysis. Data analysis associated with spectral lines
and images was performed using GILDAS222http://www.iram.fr/IRAMFR/GILDAS. A
typical rms noise level is $\sim$2.0 mJy beam-1 for a 1.34 km s-1 wide channel
(see Table 1).

As mentioned in Sect. 2.1, the para-H2CO (3–2) triplet has been observed
before with the APEX 12 m telescope in N113 and N159W (Tang et al., 2017b). We
integrate all the emission within our ALMA data and compare it to the single-
dish APEX 12 m data (beam size $\sim$30′′). The fluxes recovered by our
interferometer data are $\sim$75%, $\sim$80%, and $\sim$80% for para-H2CO
303–202, 321–220, and 322–221, respectively, in N113. We find that $\sim$45%,
$\sim$50%, and $\sim$50% of the para-H2CO 303–202, 321–220, and 322–221
integrated flux observed by the APEX 12 m telescope is recovered for N159W by
our ALMA data, respectively. These indicate that the para-H2CO (3–2) triplets
have, for a given source, similar missing flux percentages.

We note that our observational field centers of N113 and N159W have $\sim$6”
and $\sim$13” offsets with respect to those used by Tang et al. (2017b),
respectively. This may affect the comparison of our ALMA and single-dish data,
especially for N159W. Our central positions are farther to the east (N113) and
north (N159W) of the bulk of the emission shown in Fig. 1, so that part of the
missing flux in our ALMA data may not be due to a lack of short baselines but
due to the chosen field centers. In the optically thin regime, the critical
density of the para-H2CO 303–202 transition is $\sim$6$\times$105 cm-3 at
kinetic temperature 50 K (Shirley, 2015), so the para-H2CO (3–2) triplet
probes compact emission regions. Considering that the para-H2CO (3–2) triplet
is a dense gas tracer and has similar missing flux percentages within N113 and
N159W, the line ratios of the para-H2CO (3–2) transitions are likely only
weakly influenced by missing flux.

Figure 1: Velocity-integrated intensity maps (color bars in units of Jy beam-1
km s-1) of para-H2CO 303–202 (_left panels_), 322-221 (_central panels_), and
321–220 (_right panels_) of N113 (Integrated Local Standard of Rest velocity
range: 229–240 km s-1; _top panels_) and N159W (integrated velocity range:
232–244 km s-1; _bottom panels_). The centers of the fields are those also
given in Sect. 2.2, i.e. $\alpha_{2000}$ = 05:13:18.2, $\delta_{2000}$ =
–69:22:25.0 and $\alpha_{2000}$ = 05:39:36.0, $\delta_{2000}$ =– 69:45:25.0.
The contours provide, with respect to the peak intensity, levels from 10% to
100% with steps of 10% ($\sim$10$\sigma$ and $\sim$7$\sigma$) for para-H2CO
303–202 in N113 and N159W, and from 10% to 100% with steps of 10%
($\sim$5$\sigma$) for para-H2CO 322-221 and 321–220. The para-H2CO 303–202
peak intensities are 1.2 and 0.5 Jy beam-1 km s-1, the para-H2CO 322-221 peak
intensities are 0.5 and 0.2 Jy beam-1 km s-1, and the para-H2CO 321–220 peak
intensities are 0.5 and 0.2 Jy beam-1 km s-1 in N113 and N159W, respectively.
The pixel size of each image is
0$\,.\\!\\!^{\prime\prime}$2$\times$0$\,.\\!\\!^{\prime\prime}$2\. The beam
size of each image is shown in the lower left corner. Gray circles show the
half power beam sizes of the N113 and N159W APEX-12m data, reported by Tang et
al. (2017b). The shown fields of view encompass slightly more than our ALMA
primary half power beam width (Sect. 2.2). No primary beam correction has been
applied. Angular scales of 4′′ correspond to a linear scale of $\sim$1 pc.

##  3 Results

###  3.1 Overview

The integrated intensity distributions of the para-H2CO triplet in N113 and
N159W are shown in Fig. 1. The observed spectra from the H2CO peaks in N113
and N159W are shown in Fig. 2. The location of the seven dense H2CO clumps in
each of the two massive star forming regions, selected by eye, are indicated
in Figs. 3 and 4, left panels, and are listed in Table 2.

###  3.2 Distribution of H2CO

In both sources, N113 and N159W, para-H2CO (303–202) shows extended
distributions and clearly traces dense molecular structure, which is
consistent with previous observational results probed with other gas tracers
of e.g. 13CO, HCN, HNC, HCO+ (Wong et al., 2006; Seale et al., 2012; Fukui et
al., 2015; Sewiło et al., 2018). The para-H2CO 322–221 and 321–220 lines are
only detected in the densest regions of N113 and N159W, and show less extended
distributions than the para-H2CO 303–202 transition.

The para-H2CO 303–202 velocity channel maps of N113 and N159W are shown in
Figs. 8 and 9 of Appendix A, respectively. In N113, the filamentary structure,
roughly elongated in North-South direction with a length of $\sim$25′′, is
prominent at low velocities, $\sim$232–236 km s-1, while the northeastern
clump (clump 1 following the nomenclature of Fig. 3 and Table 2) becomes
visible at higher velocities, $\sim$236–238 km s-1. In the N159W region, there
appear to be two velocity components centered at $\sim$236 and $\sim$240 km
s-1 spreading from the northeast to the southwest.

###  3.3 H2CO line ratios

As mentioned in Sect. 1, the intensity ratio of para-H2CO lines involving
different $K_{\rm a}$ ladders yields estimates of the kinetic temperature of
the gas (Mangum & Wootten, 1993). Generally, higher line ratios of para-H2CO
(322–221/303–202 and 321–220/303–202) indicate higher kinetic temperatures
(e.g., Mangum & Wootten 1993; Ao et al. 2013; Ginsburg et al. 2016; Tang et
al. 2017b, 2018a, 2018b; see also Fig. 5 for a more detailed view). Therefore,
the ratio maps can be used as a proxy for relative kinetic temperature. The
upper levels of the para-H2CO 322–221 and 321–220 transitions have almost the
same energy above the ground state ($E_{\rm u}$ $\simeq$ 68 K). They show
similar spatial distributions (see Fig. 1), similar line profiles (for
brightness temperature, linewidth, and velocity in our observations; see Fig.
2 and also Tang et al. 2017a, b, 2018a, 2018b), and are often detected
simultaneously in molecular clouds (e.g., Bergman et al. 2011; Wang et al.
2012; Lindberg & Jørgensen 2012; Ao et al. 2013; Immer et al. 2014; Treviño-
Morales et al. 2014; Ginsburg et al. 2016, 2017; Tang et al. 2017a, b, 2018a,
2018b; Lu et al. 2017, 2021; Mangum et al. 2019). Therefore, we use the
averaged ratio para-H2CO 0.5$\times$[(322–221 \+ 321–2${}_{20})$/303–202]
between para-H2CO 322–221/303–202 and 321–220/303–202 for the first highly
qualitative analysis below.

Averaged para-H2CO (3–2) line ratio maps of N113 and N159W are shown in the
left panels of Figs. 3 and 4, respectively. The line ratios are calculated by
velocity-integrated intensities where the para-H2CO 322–221 and 321–220 lines
are detected with signal-to-noise ratios (S/N) $\gtrsim$ 3$\sigma$. Para-H2CO
line ratios range from 0.10 to 0.38 with an average of 0.22 $\pm$ 0.01 (errors
given here and elsewhere are standard deviations of the mean) in N113 and from
0.10 to 0.28 with an average of 0.20 $\pm$ 0.01 in N159W (see also Table 3).
The lower ratios ($<$0.2) occur in the outskirts of the N113 and N159W H2CO
clouds (see Figs. 3 and 4). Higher ratios ($\gtrsim$0.2) associate with dense
clumps, YSOs (or YSO candidates) (Chen et al., 2010; Carlson et al., 2012),
and/or H2O masers (Ellingsen et al., 2010) in N113 and N159W.

###  3.4 Kinetic temperatures from H2CO line ratios

According to Sect. 3.3, the para-H2CO 322–221 and 321–220 transitions have
almost the same upper-state energies above the ground state and similar
observed line profiles. Para-H2CO 322–221/303–202 and 321–220/303–202 ratios
are both good thermometers to determine kinetic temperatures ($T_{\rm kin}$ ¡
150 K; Mangum & Wootten 1993; Tang et al. 2018b) and show a similar behavior
to kinetic temperature and spatial density at densities $n$(H2) $\gtrsim$ 105
cm-3 (Lindberg et al., 2015; Tang et al., 2017a). Therefore, we continue to
use the averaged para-H2CO ratio 0.5$\times$[(322–221 \+
321–2${}_{20})$/303–202] to directly determine gas kinetic temperatures.

Using the RADEX333http://var.sron.nl/radex/radex.php non-LTE modeling program
(van der Tak et al., 2007) with collisional rate coefficients from Wiesenfeld
& Faure (2013), we modeled the relation between the gas kinetic temperature
and the measured average of para-H2CO 0.5$\times$[(322–221 \+
321–2${}_{20})$/303–202] ratios, adopting a 2.73 K background temperature, an
average observational linewidth of 4.0 km s-1, and column densities
$N$(para-H2CO) = 2.7$\times$1012 and 3.7$\times$1012 cm-2 for N113 and N159W,
respectively. The results are shown in Fig. 5. The values of the para-H2CO
column density were obtained with APEX data (beam size $\sim$ 30′′; Tang et
al. 2017b), which cover similar regions. Different column densities of
para-H2CO only weakly affect derived kinetic temperatures (see Fig. 3 in Tang
et al. 2017b or Fig. 4 in Tang et al. 2018a; this was also shown in Fig. 13
and discussed in Sect. 4.3.1 of Mangum & Wootten 1993) as long as all lines
are optically thin. Considering that the relation between the gas temperature
and the para-H2CO line ratio may vary at different spatial densities (see Fig.
2 in Tang et al. 2017b), we modeled it at spatial densities 104, 105, and 106
cm-3 in Fig. 5. It appears that $T_{\rm kin}$ at $n$(H2) = 105 cm-3 is
consistently lower than values at 104 and 106 cm-3 by $\lesssim$23% and
$\lesssim$34%, respectively, for $T_{\rm kin}$ $\lesssim$ 100 K. Local
thermodynamic equilibrium (LTE) is a good approximation for the H2CO level
populations under optically thin and high-density conditions (Mangum &
Wootten, 1993; Tang et al., 2017a, b, 2018b). Following the method applied by
Tang et al. (2017b) in their Eq. (2), we plot the relation between the LTE
kinetic temperature, $T_{\rm LTE}$, and the para-H2CO (3–2) line ratio in Fig.
5. Apparently, $T_{\rm LTE}$ agrees well with $T_{\rm non-LTE}$ at volume
densities $n$(H2) $\sim$ 105 cm-3 as long as $T_{\rm kin}$ $\lesssim$ 100 K.
Previous observations show that para-H2CO (3–2) is sensitive to gas
temperature at density 105 cm-3 (Ginsburg et al., 2016; Immer et al., 2016;
Tang et al., 2017b). The spatial density measured with para-H2CO (303–202) and
C18O (2–1) in N113 and N159W is $n$(H2) $\sim$ 105 cm-3 on a size of
$\sim$30′′ (Tang et al., 2017b). Therefore, here we adopt 105 cm-3 as an
averaged spatial gas density in the N113 and N159W regions.

Figure 2: Spectra towards H2CO clumps in N113 (_left panels_) and N159W
(_right panels_ ; see Table 2 and, for the numbering, the left panels of Figs.
3 and 4) taken from the strongest pixel obtained with ALMA. The pixel size is
0$\,.\\!\\!^{\prime\prime}$2$\times$0$\,.\\!\\!^{\prime\prime}$2\. Black:
para-H2CO 303–202, red: para-H2CO 322–221, and blue: para-H2CO 321–220. The
spectra are not smoothed, i.e. the channel widths are those given in Table 1.
Figure 3: _Left two panels_ : The averaged velocity-integrated intensity ratio
map of para-H2CO 0.5$\times$[(322–221 \+ 321–2${}_{20})$/303–202] in N113. The
kinetic temperatures derived from the para-H2CO (3–2) line ratios (color bar
in units of Kelvin). _Right two panels_ : Maps of non-thermal velocity
dispersion (color bar in units of km s-1) and Mach number. Black contours show
levels of integrated intensity of the para-H2CO (303–202) line (see Fig. 1).
Offsets are relative to our reference position for N113 (see Fig. 1). The
pixel size of each image is
0$\,.\\!\\!^{\prime\prime}$2$\times$0$\,.\\!\\!^{\prime\prime}$2\. The beam
size of each image is shown in the lower left corner. Stars and triangles show
the locations of YSOs (or YSO candidates) and H2O masers (Chen et al., 2010;
Ellingsen et al., 2010; Carlson et al., 2012), respectively. Figure 4: Same as
Fig. 3, but for N159W.

We use the relation between kinetic temperature and para-H2CO line ratios at
averaged spatial density 105 cm-3and averaged column densities of N113 and
N159W (shown in Fig. 5) to convert ratio maps to temperature maps in Figs. 3
and 4. The gas kinetic temperatures derived from the para-H2CO line ratios are
warm at density $n$(H2) = 105 cm-3, ranging from 28 to 105 K in N113 and from
29 to 68 K in N159W on a scale of $\sim$0.4 pc (see also Table 3). It seems
that higher temperatures ($T_{\rm kin}$ $\gtrsim$ 50 K) associate with dense
clumps, YSOs (or YSO candidates), and/or H2O masers in N113 and N159W (see
Figs. 3 and 4). Gas with lower temperatures ($T_{\rm kin}$ $<$ 50 K) is
located at the outskirts of the H2CO distributions of both N113 and N159W. The
kinetic temperatures of the dense gas are high ($T_{\rm kin}$ = 44–96 K) in
the dense H2CO clumps (see Table 2), which is consistent with previous
observational results measured with para-H2CO ($J$=3–2 and 4–3) from dense
clumps representing various evolutionary stages of high-mass star formation in
the Galactic plane (Tang et al., 2018b). The gas kinetic temperatures show
apparent gradients in these dense H2CO clumps of N113 and N159W (e.g., clumps
2, 3, 4, 5, and 7 in N113 and clumps 1, 4, and 6 in N159W; see Figs. 3 and 4)
on a clump scale of $\sim$1 pc.

As mentioned in Sect. 2.1, the dense gas kinetic temperatures in N113 and
N159W have been estimated with the same transitions of para-H2CO ($J$=3–2)
observed by the APEX 12-m telescope (beam size $\sim$30′′; Tang et al. 2017b).
For a comparison with the lower resolution APEX data, we have therefore
averaged our data for areas, where the presence of the dense gas shown in
Figs. 3 and 4 could be identified by the para-H2CO 303–202 line. The averaged
temperatures are 48 and 36 K in N113 and N159W, respectively, which are
consistent with previous results in N113 (54 $\pm$ 7 K) and N159W (35 $\pm$ 4
K) obtained with the single-dish observations (Tang et al., 2017b).

Previous observations toward the Galactic central molecular zone (CMZ) clouds
and dense massive star-forming clumps in the Galactic plane show that the
opacities of para-H2CO ($J$=3–2) lines influence only weakly the measurements
of gas kinetic temperature (Ginsburg et al., 2016; Immer et al., 2016; Tang et
al., 2018b). Nevertheless, a major caveat in our discussion are the poorly
known opacities of para-H2CO ($J$=3–2) lines in N113 and N159W. While the
para-H2CO 322–221 and 321–220 lines, being much weaker than the 303–202 line,
appear to be optically thin, this is not necessarily the case for our
strongest emitter, the 303–202 transition. If this line were optically thick
in the molecular cores we have observed, line ratios would be too close to
unity, and we would overestimate, relative to the optically thin case, the
para-H2CO 0.5$\times$[(322–221 \+ 321–2${}_{20})$/303–202] intensity ratios
and kinetic temperatures. This implies that in case of saturation we would
obtain upper limits to the kinetic temperature only. This could easily mimic
the high temperatures in the cores and the lower temperatures in the optically
thin outskirts of the N113 and N159W clouds, we have mentioned above, possibly
providing a good agreement with the low kinetic temperature derived from
ammonia (beam size
18$\,.\\!\\!^{\prime\prime}$6$\times$15$\,.\\!\\!^{\prime\prime}$7; Ott et al.
2010), since the gas traced by ammonia presumably resides in a well shielded
highly obscured cloud core.

A first estimate of optical depth can be attained by a comparison of para-H2CO
303–202 main beam brightness temperatures with the temperatures derived from
the dust and from the line temperature ratios. On our 0.4 pc scale, observed
main beam brightness temperatures should be close to the real brightness
temperatures of the emission (e.g. Tang et al. 2018a), which should reach
values of order of 16 K (the $T_{\rm kin}$ value from the NH3(2,2)/(1,1) line
ratio; Ott et al. 2010) or higher to reveal strong indications of saturation.
Converting with $T_{\rm mb}$ = $S$ $\times$ $\lambda$2/(2.65 $\times$
$\theta_{b}$2) ($T_{\rm mb}$: main beam brightness temperature in units of K;
$S$: flux density in units of Jy; $\lambda$: wavelength in units of cm;
$\theta_{b}$: our beam size in units of arcmin) the flux density scale into
units of main beam brightness temperature and accounting for typical total
linewidths of $\sim$4.0 km s-1 in case of N113 and N159W (see Table 2), we
obtain peak temperatures of only $\sim$2.5 K. This is well below the low
ammonia temperature from Ott et al. (2010) and indicates either highly
subthermal excitation with potentially saturated para-H2CO 303–202 line
emission or a purely optically thin scenario.

Table 2: Parameters of H2CO clumps derived from N113 and N159W.

Source | Clump | Offset | Transition |  $\int$Flux d$v$ | $V_{\rm LSR}$ | FWHM | Flux | $\sigma_{\rm T}$ | $\sigma_{\rm NT}$ | $c_{\rm s}$ | $\mathcal{M}$ | $T_{\rm kin}$ | $T_{\rm turb}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | ” , ” |  |  Jy km s-1 |  km s-1 |  km s-1 | Jy |  km s-1 |  km s-1 |  km s-1 |  | K | K
N113 | 1 | –2.4, 4.0 |  303–202 | 0.530 (0.006) | 236.9 (0.1) | 3.3 (0.1) | 0.150 | 0.11 (0.01) | 1.41 (0.02) | 0.40 (0.02) | 3.5 | 50.4 (5.1) | 50
|  |  |  322–221 | 0.110 (0.006) | 236.9 (0.1) | 3.0 (0.2) | 0.035 |  |  |  |  |  |
|  |  |  321–220 | 0.114 (0.007) | 236.7 (0.1) | 3.4 (0.3) | 0.032 |  |  |  |  |  |
| 2 | –5.4, 3.8 |  303–202 | 1.226 (0.019) | 233.5 (0.1) | 5.8 (0.1) | 0.198 | 0.16 (0.01) | 2.46 (0.04) | 0.56 (0.02) | 4.4 | 96.2 (5.8) | 93
|  |  |  322–221 | 0.429 (0.006) | 233.3 (0.1) | 5.2 (0.1) | 0.077 |  |  |  |  |  |
|  |  |  321–220 | 0.435 (0.008) | 233.4 (0.1) | 5.2 (0.1) | 0.078 |  |  |  |  |  |
| 3 | –4.0, 1.2 |  303–202 | 1.258 (0.005) | 233.7 (0.1) | 5.5 (0.1) | 0.216 | 0.13 (0.01) | 2.32 (0.01) | 0.47 (0.01) | 5.0 | 67.7 (2.6) | 67
|  |  |  322–221 | 0.350 (0.008) | 233.8 (0.1) | 4.8 (0.1) | 0.068 |  |  |  |  |  |
|  |  |  321–220 | 0.345 (0.007) | 233.9 (0.1) | 5.0 (0.1) | 0.065 |  |  |  |  |  |
| 4 | –2.8, –2.0 |  303–202 | 0.886 (0.007) | 232.4 (0.1) | 4.8 (0.1) | 0.173 | 0.12 (0.01) | 2.04 (0.02) | 0.44 (0.01) | 4.6 | 60.7 (2.3) | 60
|  |  |  322–221 | 0.231 (0.006) | 232.3 (0.1) | 4.2 (0.1) | 0.051 |  |  |  |  |  |
|  |  |  321–220 | 0.225 (0.005) | 232.2 (0.1) | 4.4 (0.1) | 0.048 |  |  |  |  |  |
| 5 | –4.2, –5.2 |  303–202 | 0.409 (0.005) | 233.9 (0.1) | 3.6 (0.1) | 0.108 | 0.11 (0.01) | 1.52 (0.02) | 0.38 (0.01) | 4.0 | 44.0 (2.5) | 44
|  |  |  322–221 | 0.090 (0.006) | 234.2 (0.2) | 4.6 (0.4) | 0.019 |  |  |  |  |  |
|  |  |  321–220 | 0.073 (0.006) | 234.2 (0.1) | 3.2 (0.3) | 0.022 |  |  |  |  |  |
| 6 | –3.8, –8.6 |  303–202 | 0.203 (0.005) | 234.9 (0.1) | 3.3 (0.1) | 0.057 | … | … | … | … | … | …
|  |  |  322–221 | … |  |  |  |  |  |  |  |  |
|  |  |  321–220 | … |  |  |  |  |  |  |  |  |
| 7 | –5.4, –13.4 |  303–202 | 0.710 (0.010) | 234.3 (0.1) | 3.0 (0.1) | 0.220 | 0.12 (0.01) | 1.28 (0.02) | 0.41 (0.02) | 3.1 | 52.5 (4.5) | 52
|  |  |  322–221 | 0.159 (0.012) | 234.5 (0.1) | 2.9 (0.3) | 0.052 |  |  |  |  |  |
|  |  |  321–220 | 0.165 (0.017) | 234.3 (0.2) | 3.2 (0.4) | 0.049 |  |  |  |  |  |
N159W | 1 | 0.8, 0.4 |  303–202 | 0.245 (0.003) | 238.4 (0.1) | 1.8 (0.1) | 0.128 | 0.11 (0.01) | 0.79 (0.03) | 0.41 (0.01) | 1.9 | 51.8 (4.4) | 51
|  |  |  322–221 | 0.048 (0.003) | 238.0 (0.1) | 2.2 (0.1) | 0.020 |  |  |  |  |  |
|  |  |  321–220 | 0.045 (0.003) | 238.3 (0.1) | 2.0 (0.2) | 0.021 |  |  |  |  |  |
| 2 | 10.4, 0.4. |  303–202 | 0.114 (0.005) | 239.7 (0.1) | 1.6 (0.4) | 0.066 | … | … | … | … | … | …
|  |  |  322–221 | … |  |  |  |  |  |  |  |  |
|  |  |  321–220 | … |  |  |  |  |  |  |  |  |
| 3 | 7.8, –1.2 |  303–202 | 0.141 (0.006) | 236.0 (0.1) | 3.1 (0.2) | 0.043 | 0.12 (0.01) | 1.31 (0.07) | 0.43 (0.03) | 3.1 | 56.0 (9.2) | 55
|  |  |  322–221 | 0.030 (0.006) | 236.4 (0.3) | 2.6 (0.6) | 0.011 |  |  |  |  |  |
|  |  |  321–220 | 0.035 (0.004) | 236.3 (0.2) | 2.8 (0.4) | 0.012 |  |  |  |  |  |
| 4 | 5.0, –2.0 |  303–202 | 0.309 (0.006) | 238.0 (0.1) | 4.3 (0.1) | 0.068 | 0.13 (0.01) | 1.86 (0.05) | 0.46 (0.01) | 4.0 | 65.5 (5.1) | 65
|  |  |  322–221 | 0.104 (0.005) | 238.5 (0.1) | 5.9 (0.4) | 0.017 |  |  |  |  |  |
|  |  |  321–220 | 0.081 (0.006) | 238.3 (0.2) | 4.5 (0.4) | 0.017 |  |  |  |  |  |
| 5 | 5.2, –10.4 |  303–202 | 0.527 (0.013) | 235.4 (0.1) | 5.4 (0.2) | 0.091 | 0.11 (0.01) | 2.32 (0.08) | 0.38 (0.01) | 6.1 | 45.2 (4.0) | 46
|  |  |  322–221 | 0.101 (0.008) | 234.8 (0.2) | 4.8 (0.4) | 0.020 |  |  |  |  |  |
|  |  |  321–220 | 0.115 (0.008) | 234.8 (0.2) | 4.6 (0.4) | 0.023 |  |  |  |  |  |
| 6 | 3.4, –13.0 |  303–202 | 0.914 (0.009) | 236.9 (0.1) | 4.5 (0.1) | 0.190 | 0.11 (0.01) | 1.94 (0.03) | 0.41 (0.01) | 4.8 | 51.3 (3.3) | 51
|  |  |  322–221 | 0.189 (0.008) | 237.0 (0.1) | 3.6 (0.2) | 0.049 |  |  |  |  |  |
|  |  |  321–220 | 0.200 (0.011) | 237.0 (0.1) | 3.8 (0.3) | 0.049 |  |  |  |  |  |
| 7 | –0.6, –16.4 |  303–202 | 0.272 (0.013) | 235.8 (0.1) | 3.1 (0.2) | 0.081 | … | … | … | … | … | …
|  |  |  322–221 | … |  |  |  |  |  |  |  |  |
|  |  |  321–220 | … |  |  |  |  |  |  |  |  |

444Offsets relative to our reference positions for N113 and N159W (see Figs.
1, 3, and 4). Velocity-integrated flux, $\int$Flux d$v$, local standard of
rest velocity, $V_{\rm LSR}$, full width at half maximum line width (FWHM),
and peak flux (Flux) were obtained from Gaussian fits using CLASS as part of
the GILDAS software. For the thermal and non-thermal velocity dispersions,
$\sigma_{\rm T}$ 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. 3.5 and 4.3. Values in
parenthesis are uncertainties.

Table 3: Thermal and non-thermal parameters derived from N113 and N159W.

Parameter | N113 |  | N159W
---|---|---|---
| Range | Median | Mean |  | Range | Median | Mean
H2CO line ratio | 0.10–0.38 | 0.21 | 0.22 $\pm$ 0.01 |  | 0.10–0.28 | 0.20 | 0.20 $\pm$ 0.01
$T_{\rm gas}$ /K | 27.6–105.4 | 48.5 | 51.4 $\pm$ 0.4 |  | 28.7–67.8 | 47.5 | 47.6 $\pm$ 0.3
$\sigma_{\rm T}$ / km s-1 | 0.08–0.16 | 0.11 | 0.11 $\pm$ 0.01 |  | 0.09–0.13 | 0.11 | 0.11 $\pm$ 0.01
$\sigma_{\rm NT}$ / km s-1 | 0.93–2.76 | 1.64 | 1.69 $\pm$ 0.01 |  | 0.57–2.74 | 1.85 | 1.74 $\pm$ 0.02
$c_{\rm s}$ / km s-1 | 0.30–0.58 | 0.40 | 0.40 $\pm$ 0.01 |  | 0.31–0.47 | 0.39 | 0.39 $\pm$ 0.01
$\mathcal{M}$ | 2.3–6.2 | 4.2 | 4.2 $\pm$ 0.1 |  | 1.6–8.3 | 4.5 | 4.5 $\pm$ 0.1
555For the meaning of the parameters in column 1, see Table 2. The errors
shown in columns 4 and 7 are the standard deviations of the mean.

To discriminate between these two possibilities, we used again the RADEX non-
LTE program (van der Tak et al., 2007). Setting $T_{\rm kin}$ to 16 K, the low
value derived by Ott et al. (2010) from ammonia, and a linewidth of 3 km s-1
we can reproduce the $T_{\rm mb}$ (H2CO 303–202) line intensity of $\sim$2.5 K
with $N$(para-H2CO) = 1.4 $\times$ 1014 cm-2, but the resulting para-H2CO 0.5
$\times$ [(322–221 \+ 321–220)/303–202] line ratio is $\sim$0.04, far below
and thus inconsistent with the obtained data. At $T_{\rm kin}$ $\sim$ 50 K,
however, $N$(para-H2CO) $\sim$5$\times$1013 cm-2 leads to $T_{\rm mb}$ (H2CO
303–202) $\sim$ 2.5 K and line ratios of order 0.23–0.24. In this case the
para-H2CO 303–202 line is slightly optically thick ($\tau$ $\sim$ 1.1), while
the weaker 322–221 and 321–220 transitions are characterized by $\tau$ $\sim$
0.3. Compared to the optically thin case shown by Fig. 5, this leads to a
slight overestimate of $T_{\rm kin}$. Adopting the optically thin scenario,
line ratios of 0.25–0.30 lead to kinetic temperatures of 55 to 70 K instead of
the 50 K adopted in the non-LTE model also accounting for line saturation
effects. Interestingly, even H2CO 303–202 line opacities of order 5–10 do not
lead to larger inaccuracies (see Fig. G. 2 in Immer et al. 2016). While
increasing line saturation has a tendency to yield line intensity ratios
closer to unity, this trend is not seen in this case because excitation
temperatures of the H2CO 322–221 and 321–220 transitions are then
significantly below that of the H2CO 303–202 transition. To summarize: While
our $T_{\rm kin}$ values toward the cloud cores might be overestimated, they
are not grossly overestimated and still clearly favor values well in excess of
the 16 K derived from ammonia.

In Galactic dense massive star-forming clumps the para-H2CO 303–202 line tends
to be optically thin (Tang et al., 2018b). Since dust is less prevalent in the
LMC than in the Galaxy (e.g. Wang et al. 2009) leading to reduced shielding,
we may see denser regions with H2CO in the LMC than in the Galaxy, again not
supporting the highly subthermal low kinetic temperature excitation scenario
mentioned above. On the other hand, N113 and N159W are the most intense star-
forming regions of the LMC (e.g., Wang et al. 2009; Lee et al. 2016), which
may yield particularly large H2CO column densities. The peak column densities
$N$(para-H2CO) obtained from the para-H2CO (303–202) brightness temperatures
of dense H2CO clumps in N113 and N159W (see Table 2) using the method of Tang
et al. (2017a) range from 0.7 to 9.0$\times$1013 cm-2 with an average of
$\sim$4.0$\times$1013 cm-2 at a density of 105 cm-3 and from 0.3 to
3.8$\times$1013 cm-2 with an average of $\sim$1.4$\times$1013 cm-2 at a
density of 106 cm-3, which are a few times higher than averaged results
obtained from entire regions with the APEX telescope (Tang et al., 2017b).

Figure 5: RADEX non-LTE (_red and blue lines_) and LTE (_black line_) modeling
of the relation between kinetic temperature and average ratio of para-H2CO
322–221/303–202 and 321–220/303–202 with spatial densities of $n$(H2) = 104,
105, and 106 cm-3 (_dashed, solid, and dotted lines, respectively_), an
averaged observed linewidth of 4.0 km s-1, and column densities of
$N$(para-H2CO) = 2.7$\times$1012 and 3.7$\times$1012 cm-2 for N113 (_red
lines_) and N159W (_blue lines_) (Tang et al., 2017b), respectively. $T_{\rm
kin}$-Ratio(H2CO) relations for N113 and N159W at density 104 cm-3 are nearly
the same, which is similar to the situation at 106 cm-3.

###  3.5 Thermal and non-thermal motions

Using the kinetic temperatures derived from the para-H2CO line ratios, we can
derive thermal and non-thermal linewidths i.e. $\sigma_{\rm
T}=\sqrt{\frac{kT_{\rm kin}}{m_{\rm H_{2}CO}}}$ and $\sigma_{\rm
NT}=\sqrt{\frac{\Delta v^{2}}{8{\rm ln}2}-\sigma^{2}_{\rm
T}}\thickapprox\Delta v/2.355$, where $k$ is the Boltzmann constant, $T_{\rm
kin}$ is the kinetic temperature of the gas, $m_{\rm H_{2}CO}$ is the mass of
the formaldehyde molecule, and $\Delta v$ is the measured FWHM linewidth of
para-H2CO 303–202 (Pan & Padoan, 2009). Thermal and non-thermal linewidth
ranges are listed in Table 3. Thermal and non-thermal linewidths of the
individual dense clumps are given in Table 2. The thermal linewidth is
significantly smaller than the non-thermal linewidth in N113 and N159W, which
indicates that the dense gas traced by para-H2CO is dominated by non-thermal
motions. This agrees with previous measurements derived from para-H2CO (3–2
and 4–3) in the Galactic star-forming region OMC-1 and massive clumps of the
Galactic disk (Tang et al., 2017a, 2018a, 2018b). Distributions of the non-
thermal linewidth in N113 and N159W are shown in Figs. 3 and 4, respectively.
They may suggest that the higher non-thermal linewidths ($\sigma_{\rm NT}$ ¿
1.5 km s-1) associate with the dense clumps and the lower non-thermal
linewidths ($\sigma_{\rm NT}$ $\lesssim$ 1.5) are located in the outskirts of
the dense clumps in our two massive star-forming regions, indicating that the
dense gas traced by para-H2CO is strongly influenced by star-forming activity.

Figure 6: Para-H2CO (303–202) with integrated intensity contours (same as Fig.
1) overlaid on H$\alpha$ (_left panels_ ; Smith & MCELS Team 1998), _Spitzer_
8.0 $\mu$m (_middle panels_ ; Meixner et al. 2006), and 1.3 mm continuum
(_right panels_ ; observed with ALMA Band 6; Fukui et al. 2015; Sewiło et al.
2018) images of N113 (_top panels_) and N159W (_bottom panels_). Stars and
triangles show the locations of the YSOs (or YSO candidates) and H2O masers
(Chen et al., 2010; Ellingsen et al., 2010; Carlson et al., 2012),
respectively.

Distributions of the Mach number ($\mathcal{M}=\sigma_{\rm NT}/c_{\rm s}$,
where $c_{\rm s}$=$\sqrt{\frac{kT_{\rm kin}}{\mu m_{\rm H}}}$ is the sound
speed, $\mu$ = 2.37 is the mean molecular weight for molecular clouds, and
$m_{\rm H}$ is the mass of the hydrogen atom) in N113 and N159W are shown in
Figs. 3 and 4, right panels, respectively. The Mach numbers of the individual
dense clumps in N113 and N159W are listed in Table 2. The Mach numbers range
from 2.3 to 6.2 with an average of 4.2 $\pm$ 0.1 in N113 and from 1.6 to 8.3
with an average of 4.5 $\pm$ 0.1 in N159W (see also Table 3). 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.
Locations that are situated in the outskirts of the clumps of N113 and in the
outer regions of the northern clumps of N159W (see Figs. 3 and 4), have low
Mach numbers ($\mathcal{M}$ $\lesssim$ 3.5). The distribution of Mach numbers
may suggest apparent gradients from southeast to northwest along clumps 5 and
6 of N159W (see Fig. 4). Meanwhile, there is a YSO (or YSO candidate) located
in the eastern region of these clumps. This may imply that dense gas probed by
para-H2CO may be influenced by this YSO (or YSO candidate). Higher Mach
numbers ($\mathcal{M}$ ¿ 3.5) are associated with the dense clumps in N113 and
N159W, again suggesting that dense gas traced by H2CO is strongly influenced
by non-thermal motions (e.g., outflows and shocks) in these regions. The
averaged Mach numbers in N113 and N159W are consistent with previous
observational results in Galactic massive clumps on scales of $\sim$0.1–1.8 pc
(mean value $\sim$4.2 obtained from para-H2CO (3–2 and 4–3); Tang et al.
2018b), but are higher than the values measured with the same transitions of
para-H2CO (3–2) in OMC-1 on a scale of $\sim$0.06 pc (mean value $\sim$2.3;
Tang et al. 2018a). We have smoothed the OMC-1 para-H2CO (3–2) data to a
linear resolution of 0.4 pc. The derived Mach numbers are 3.0 and 3.5 in the
Orion KL and Orion South regions, respectively, which is still slightly lower
than the values in N113 and N159W.

One should note that para-H2CO (303–202) linewidths in the northern edges of
clump 1 and 2 of N113 and in the outskirts of clump 1 and 4 of N159W are
narrow ¡3.0 km s-1 (see Figs. 2, 3, and 4), which is corresponding to only 2–3
channels of our H2CO data. Furthermore, there are two velocity components
around clump 1 of N113 and in the northern region of the clump 4 of N159W (see
Figs. 8 and 9). For two velocity components the typical linewidths are
$\sim$2–3 and $\sim$4–6 km s-1 in these regions of N113 and N159W. Our
observational set can not well identify the velocity components with narrow
linewidths using Gaussian profiles in these regions. Therefore, the non-
thermal linewidths and the Mach numbers are likely overestimated in these
regions. For the locations with two velocity components in N113 and N159W, the
non-thermal linewidths and the Mach number are weighted with para-H2CO
(303–202) integrated intensity in Figs. 3 and 4.

Table 4: Gas temperatures at different radius around the young stellar objects
or candidates.

| YSO |  |  $R$ = 0.2 pc |  |  $R$ = 0.4 pc |  |  $R$ = 1.0 pc
---|---|---|---|---|---|---|---
Source | R.A.(J2000) | DEC.(J2000) | Luminosity |  |  $T_{\rm kin}$(Rad) |  $T_{\rm kin}$(H2CO) |  |  $T_{\rm kin}$(Rad) |  $T_{\rm kin}$(H2CO) |  |  $T_{\rm kin}$(Rad) |  $T_{\rm kin}$(H2CO)
|  h m s | ° ′ ″ |  $\times$105 ${\rm L}_{\odot}$ |  | K |  | K |  | K
N113 | 05:13:17.28 | –69:22:36.7 | 0.16 |  | 36 | 35 |  | 27 | 45 |  | 19 | 34
| 05:13:17.69 | –69:22:25.0 | 1.26 |  | 54 | 48 |  | 41 | 57 |  | 29 | 53
N159W | 05:39:37.09 | –69:45:36.7 | 0.82 |  | 50 | 56 |  | 37 | 44 |  | 26 | 72
| 05:39:37.56 | –69:45:25.3 | 2.40 |  | 61 | 69 |  | 46 | 54 |  | 32 | 41
666The young stellar objects or candidates and their luminosity are taken from
Chen et al. (2010), Sewiło et al. (2010), and Carlson et al. (2012).

##  4 Discussion

###  4.1 Comparison of temperatures obtained from H2CO, CO, and NH3

Observations of CO $J$ = 1–0 to 3–2 indicate that the molecular gas traced by
CO in lower density regions ($n$(H2) $<$ 105 cm-3) may be strongly influenced
by the external FUV emission irradiating giant molecular clouds of the LMC on
a scale of $\sim$10 pc (Minamidani et al., 2008, 2011). The physical and
chemical processes in these regions such as formation and destruction of
molecules as well as ionization are dominated by the intense FUV field, which
has been modeled as PDRs (e.g., Kaufman et al. 1999; Minamidani et al. 2011;
Lee et al. 2016, 2019). The gas temperature is $T_{\rm kin}$ ¿ 150 K at
densities $\sim$103 cm-3 derived from multitransition data of CO in N159W on a
scale of $\sim$10 pc (Lee et al., 2016). Our results determined from para-H2CO
line ratios range from 29 to 68 K with an average of 47.6 $\pm$ 0.3 K at
density 105 cm-3 on a scale of 0.4 pc (see Table 3), which are much lower than
values obtained from CO at lower density and even may have to be slightly
modified to the lower side to account for moderate saturation effects (see
Sect. 3.4). This indicates that para-H2CO and CO trace different kinetic
temperature layers in N159W. There is no available gas temperature derived
from multitransition data of CO in N113, so we can not compare it with our
para-H2CO results.

We did not find any evidence for the low $T_{\rm kin}$ value ($\sim$16 K)
determined from the NH3 (2,2)/(1,1) line ratio by Ott et al. (2010). This is
interpreted in terms of a highly embedded extremely obscured dense molecular
gas component, which we do not see in H2CO. A similar situation has also been
encountered in M 82 (Weiß et al., 2001; Mühle et al., 2007), where ammonia
suggests $T_{\rm kin}$ $\sim$ 60 K, while H2CO represents gas with
temperatures well in excess of 100 K. We get moderately high $T_{\rm kin}$
values derived from H2CO in the cores but Ott et al. (2010) obtain a low
$T_{\rm kin}$ value determined from NH3, presumably also from a core. Ammonia
has a particularly low energy threshold for photodissociation ($\sim$4.1 eV;
Weiß et al. 2001). Low metallicity environments with low nitrogen abundance
and strong UV radiation (e.g., Westerlund 1990; Chin et al. 1997; Rolleston et
al. 2002; Wang et al. 2009; Ott et al. 2010) in the LMC may allow ammonia to
survive only in the most UV-shielded regions, leading to a low fractional NH3
abundance and a low kinetic temperature of the NH3 emitting gas (Tang et al.,
2017b). It could be explained in the sense, that NH3 is only there, where no
protostar has yet heated its environment. In the outskirts, $T_{\rm kin}$
values from H2CO are low, but CO suggests that there temperatures are high.
This might be explained by the idea that even the H2CO outskirts are far
inside the volume of the CO emitting gas.

###  4.2 Radiative heating

We compare para-H2CO (303–202) integrated intensity distributions with the
H$\alpha$ emission (observed with the UM/CTIO Curtis Schmidt telescope; Smith
& MCELS Team 1998), _Spitzer_ 8.0 $\mu$m (Meixner et al., 2006), and 1.3 mm
continuum (observed with ALMA Band 6; Fukui et al. 2015; Sewiło et al. 2018)
emission of N113 and N159W in Fig. 6. Obviously, the spatial distributions of
para-H2CO (303–202) and H$\alpha$ emission are not well correlated in the two
massive star-forming regions. This suggests that dense gas traced by para-H2CO
may only be weakly affected by the stars exciting the H$\alpha$ emission in
N113 and N159W. Such offsets between molecular and H$\alpha$ emissions are
seen in evolved systems (e.g., Fukui & Kawamura 2010), where newly formed
massive stars have already dispersed a part of the original molecular gas. The
para-H2CO (303–202) integrated intensity distributions agree much better with
the _Spitzer_ 8.0 $\mu$m emission excited by young massive stars in the N113
and N159W regions (see Fig. 6). The para-H2CO (303–202) integrated intensity
distribution agrees best with the 1.3 mm continuum emission in N113 and N159W,
including the dense molecular structure and dust emission peaks (see Fig. 6),
which implies that H2CO associates with the dense gas traced by the dust
emission. This is consistent with previous observational results in our
Galactic massive star-forming regions at various evolutionary stages (Immer et
al., 2014; Tang et al., 2017a, 2018a, 2018b).

As mentioned, there is no spatial correlation between the dense gas traced by
para-H2CO and the H$\alpha$ distributions in N113 and N159W (see Fig. 6).
There also appears to be no significant correlation between the gas
temperature structures derived from para-H2CO line ratios and the H$\alpha$
emission distribution in these two regions (Figs. 3, 4, and 6). These findings
indicate that dense gas traced by H2CO is weakly affected by the external FUV
radiation in N113 and N159W. The dense molecular gas ($\gtrsim$ 105 cm-3) may
be effectively shielded by the lower density molecular gas envelopes in the
massive star-forming regions of the LMC.

Previous observations of e.g. H2CO, NH3, CH3CN, CH3CCH, or CH3OH in Galactic
massive star-forming regions (Lu et al., 2014; Giannetti et al., 2017; Tang et
al., 2018a, b; Gieser et al., 2021) suggest internal radiative heating of
embedded infrared sources of their surrounding dense gas. Present observations
of H2CO in massive star-forming regions of the LMC imply that the kinetic
temperatures traced by para-H2CO (3–2) transitions in dense regions ($\sim$105
cm-3) are correlated with the ongoing massive star formation (Tang et al.,
2017b). As mentioned in Sect. 3.4, it appears that high gas temperatures
obtained from para-H2CO line ratios associate with dense clumps, YSOs (or YSO
candidates), and/or H2O masers in N113 and N159W regions (see Figs. 3 and 4).
This indicates that the warm dense gas traced by para-H2CO is influenced by
radiation from internal embedded infrared sources and/or nearby YSOs in the
low-metallicity environment of the LMC.

We determine the gas temperature in regions heated by YSOs following the
modified Stefan-Boltzmann blackbody radiation law, adjusting the emissivity of
dust grains smaller than the wavelength at the characteristic blackbody
temperature (Wiseman & Ho, 1998; Tang et al., 2018a),

$\displaystyle T_{\rm kin}=2.7\times(\frac{L}{{\rm
L}_{\odot}})^{1/5}(\frac{R}{\rm pc})^{-2/5}\leavevmode\nobreak\ {\rm K},$ (1)

where the luminosity $L$ is in ${\rm L}_{\odot}$ and the distance $R$ is in
parsecs. Four YSOs (or YSO candidates) with luminosities of 0.16, 0.82, 1.26,
and 2.40 $\times$ 105 ${\rm L}_{\odot}$ are found in N113 and N159W (Chen et
al., 2010; Sewiło et al., 2010; Carlson et al., 2012). Assuming these YSOs (or
YSO candidates) are the dominant sources at their locations, the derived gas
temperatures at radii of 0.2, 0.4, and 1.0 pc from Eq. (1) are listed in Table
4. Measured gas temperatures around the four young stellar objects or
candidates derived from para-H2CO line ratios are also listed in Table 4,
which agrees well with the results obtained from the radiation law at a radius
of 0.2 pc. This indicates that dense gas traced by H2CO is significantly
influenced by radiation from YSOs (or YSO candidates). One should note that
the distances among YSOs (or YSO candidates), H2O masers, and 1.3 mm dust
emission peaks are approximately $\sim$1 pc in the N113 and N159W regions (see
Figs. 3, 4, and 6). It also shows that the YSO’s radiation heating could
contribute to a gas temperature (e.g. $T_{\rm kin}$ $\sim$ 19 K for the lowest
luminosity YSO (or candidate) with $L$ = 1.6 $\times$ 104 ${\rm L}_{\odot}$;
see Table 4) even far out, at a radius of 1 pc. This implies that the complex
temperature structure of the dense gas in N113 and N159W regions may be also
affected by multiple YSOs and/or embedded infrared sources. For large areas of
N113 and N159W the H$\alpha$ radiation regions and the general stellar
radiation field may also contribute to the temperature of the dense gas.

Due to the limited spatial resolution ($\sim$0.4 pc) of the ALMA data and our
limited 2-dimensional perspective, the relation between gas temperature and
distance from YSOs is not perfectly revealed. A follow-up sub-arcsecond
angular resolution (corresponding to $\lesssim$ 0.1 pc) mapping study of the
massive star-forming clump’s temperature structure with H2CO in the LMC will
be indispensable and meaningful in the future.

###  4.3 Turbulent heating

Turbulent heating seems to be widespread in Galactic massive star-forming
regions on a $\sim$0.1–1.8 pc scale (Tang et al., 2018a, b). It may play an
important role in heating the dense gas in star-forming regions. Present
observations toward Galactic CMZ clouds with para-H2CO (3–2 and 4–3) show that
the warm dense gas is heated most likely by turbulence on a scale of $\sim$1
pc (Ao et al., 2013; Ginsburg et al., 2016; Immer et al., 2016). Observations
of, e.g. H2CO, NH3, and CH3CCH in Galactic star formation regions suggest that
the linewidth is correlated with 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; Tang et al., 2017a,
2018a, 2018b; Giannetti et al., 2017). Correlations between the kinetic
temperature and linewidth are expected in the case of conversion of turbulent
energy into heat (Güsten et al., 1985; Molinari et al., 1996; Ginsburg et al.,
2016).

We examine whether there is a relationship between the turbulence and the gas
temperature derived from the para-H2CO line ratio on a $\sim$0.4 pc scale in
N113. As discussed in Sect. 3.5, our observational sets have low channel width
($\sim$1.34 km s-1) for para-H2CO data, which strongly affects the estimation
of para-H2CO linewidths in the N159W region, but weakly influences the N113
region which has wider linewidths. Therefore, we did not examine this
relationship in N159W. We adopt the non-thermal velocity dispersion
($\sigma_{\rm NT}$) of para-H2CO (303–202) in good approximation as proxy for
the turbulence, and the kinetic temperature derived from the para-H2CO line
ratio as the gas kinetic temperature. We select positions with strong non-
thermal motions ($\mathcal{M}$ $\gtrsim$ 3.5) located near the dense clumps
(see Figs. 3 and 4). The relation between kinetic temperature and non-thermal
velocity dispersion is shown in Fig. 7. It indicates that the non-thermal
velocity dispersion of para-H2CO is positively correlated with the gas kinetic
temperatures, by a power-law of the form $T_{\rm kin}\,\propto\,\sigma_{\rm
NT}^{1.03\pm 0.03}$ with a correlation coefficients $R$ of 0.78, which agrees
with results from Galactic massive star-forming regions ($T_{\rm
kin}\propto\sigma_{\rm NT}^{0.66-1.26}$; gas kinetic temperature measured with
para-H2CO 3–2 and 4–3 line ratios; Tang et al. 2018a, b). The intercept and
absolute values of the $T_{\rm kin}$–$\sigma_{\rm NT}$(H2CO) relation in N113
are also consistent with previous results from Galactic massive star-forming
regions on scales of $\sim$0.1–1.8 pc (Tang et al., 2018a, b). These imply
that the higher temperature traced by para-H2CO is related to higher
turbulence on a scale of $\sim$0.4 pc. To our knowledge, this is the first
evidence for dense gas heated in a star-forming region of the LMC by
turbulence. Our $T_{\rm kin}$–$\sigma_{\rm NT}$(H2CO) relation agrees with
previous observational results ($T_{\rm kin}\propto\sigma_{\rm NT}^{0.8-1.0}$;
gas kinetic temperature measured with NH3 and H2CO) in molecular clouds of the
Galactic center (Güsten et al., 1985; Mauersberger et al., 1987; Immer et al.,
2016), only in terms of slope, not of intercept and absolute value.

We derive the gas kinetic temperature, $T_{\rm turb}$, assuming turbulent
heating is the dominant heating process of the molecular gas in N113 and
N159W, following the method applied by Tang et al. (2018a) in their Eq. (2),

$\displaystyle 3.3\times 10^{-27}\leavevmode\nobreak\ n\leavevmode\nobreak\
\sigma_{\rm NT}^{3}\leavevmode\nobreak\ L^{-1}=4\times
10^{-33}\leavevmode\nobreak\ n^{2}\leavevmode\nobreak\ T_{\rm
turb}^{1/2}(T_{\rm turb}-T_{\rm dust})$ $\displaystyle+\leavevmode\nobreak\
6\times 10^{-29}\leavevmode\nobreak\ n^{1/2}\leavevmode\nobreak\ T_{\rm
turb}^{3}\leavevmode\nobreak\ {\rm d}v/{\rm d}r.$ (2)

The gas density $n$ is in units of cm-3, the velocity gradient d$v$/d$r$ is in
units of km s-1pc-1, the one-dimensional non-thermal velocity dispersion
$\sigma_{\rm NT}$ is in units of km s-1, and the cloud size $L$ is in units of
pc, and the temperatures $T_{\rm turb}$ and $T_{\rm dust}$ are in K. We assume
a cloud size of $\sim$1 pc (see Fig. 1), a typical velocity gradient of 1 km
s-1 pc-1, a measured para-H2CO (303–202) non-thermal velocity dispersion (see
Table 2, column 5), and an averaged gas spatial density $n$(H2) = 105 cm-3
(Tang et al., 2017b). Previous observations have shown that the gas kinetic
temperatures derived from para-H2CO (3–2) line ratios and the dust
temperatures obtained from _Herschel_ data are in good agreement in the dense
star-forming regions of the LMC (Tang et al., 2017b), so here we adopt the gas
temperatures obtained from our H2CO line ratios as the dust temperatures. The
derived gas kinetic temperatures $T_{\rm turb}$ are listed in Table 2. It
seems that the derived $T_{\rm turb}$ values are reasonably well consistent
with the gas kinetic temperatures $T_{\rm kin}$(H2CO) obtained from our H2CO
line ratios. This indicates that turbulent heating contributes to the gas
temperature on a scale of $\sim$0.4 pc. As discussed in Sect. 4.2, gas in
these dense clumps is also affected by internal star formation activity and/or
nearby YSOs. Therefore, turbulent heating may play an important role in
heating the dense gas in the star-forming regions of N113 and N159W.

Figure 7: Non-thermal velocity dispersion ($\sigma_{\rm NT}$) vs. gas kinetic
temperature derived from the para-H2CO line ratio for locations (pixels) with
the Mach number $\mathcal{M}$ $\gtrsim$ 3.5 in N113. The blue line is the
result from an unweighed linear fit, log$T_{\rm kin}=(1.03\pm 0.02)\times{\rm
log}\sigma_{\rm NT}+(1.45\pm 0.01)$, with a correlation coefficient, $R$, of
0.78.

Observations of the transitions of the para-H2CO (3–2) lines with the APEX 12
m telescope in the Galactic star-forming region OMC-1 ($\sim$1.1$\times$1.7
pc2) on a scale of $\sim$0.06 pc suggest complex gas heating which is most
likely due to a number of different processes, e.g. star formation activity,
radiation, and turbulence (Tang et al., 2018a). Further 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 on scales of
$\sim$0.1–1.8 pc indicate similar gas heating processes for the dense gas
(Tang et al., 2018b). Our observations reveal also similar gas heating
mechanisms in a relatively large area of N113 ($\sim$2.4$\times$6.1 pc2) and
N159W ($\sim$3.6$\times$5.1 pc2) on a scale of $\sim$0.4 pc in the LMC. These
imply that the gas heating for the dense gas of star-forming regions in low
metallicity environments with strong UV radiation in the LMC may be dominated
by internal star formation activity, radiation, and/or turbulence. The
physical and chemical processes of dense gas on a scale of $\sim$0.4 pc in
star-forming regions of the LMC might be different from those in low density
regions ($n$(H2) $<$ 105 cm-3), so the very dense gas clouds should not be
simply modeled as PDRs in low metallicity environments.

##  5 Summary

We have mapped the kinetic temperature structures of two star formation
regions, N113 and N159W, in the LMC in the para-H2CO $J_{\rm K_{a}K_{c}}$ =
303–202, 322–221, and 321–220 lines using ALMA Band 6. The main results are
the following:

  1. 1.

There is no significant correlation between the spatial distributions of the
dense gas traced by para-H2CO and H$\alpha$ emission in N113 and N159W.
However, the integrated intensity distributions of para-H2CO are similar to
those of the 1.3 mm dust emission in N113 and N159W, indicating that the H2CO
emission associates well with dense gas traced by the emission of the cold
dust. There is also some agreement with the _Spitzer_ 8.0 $\mu$m emission in
N113 and N159W, which suggests that dense gas traced by para-H2CO associates
with massive star-forming regions.

  2. 2.

Using the RADEX non-LTE program, we derive the gas kinetic temperature by
modeling the measured para-H2CO 0.5$\times$[(322–221+321–220)/303–202] line
ratios. The gas kinetic temperatures derived from para-H2CO line ratios are
warm at a spatial density of 105 cm-3, ranging from 28 to 105 K in N113 and 29
to 68 K in N159W. The aforementioned upper bounds to the kinetic temperature,
arising from the cores, may be affected by moderate line saturation and thus
may be slightly overestimated.

  3. 3.

The high kinetic temperatures ($T_{\rm kin}$ $\gtrsim$ 50 K) of the dense gas
traced by para-H2CO appear to be correlated with the internal embedded
infrared sources and/or YSOs in the N113 and N159W regions. The lower
temperatures ($T_{\rm kin}$ $<$ 50 K) are located at the outskirts of the H2CO
distributions of both N113 and N159W. The gas kinetic temperatures show
apparent gradients in some dense H2CO clumps of N113 and N159W.

  4. 4.

The kinetic temperatures of the dense gas traced by para-H2CO appear to be
only weakly affected by the external FUV radiation.

  5. 5.

The non-thermal velocity dispersions of para-H2CO are correlated with the gas
kinetic temperatures in N113, implying that higher temperatures traced by
para-H2CO are related to turbulence on a $\sim$0.4 pc scale.

  6. 6.

In the LMC, the gas heating for the dense gas may be dominated by internal
star formation activity, radiation, and/or turbulence in star-forming regions
with low metallicity environments, which is consistent with our Galactic
massive star-forming regions located in the Galactic plane.

###### Acknowledgements.

The authors thank the anonymous referee for helpful comments. This work
acknowledges support by the Heaven Lake Hundred-Talent Program of Xinjiang
Uygur Autonomous Region of China, the National Natural Science Foundation of
China under Grant 11903070, 11433008, and 11973076, the ”TianShan Youth Plan”
under Grant 2018Q084, the CAS “Light of West China” Program under Grant Nos.
2018-XBQNXZ-B-024 and 2020-XBQNXZ-017, and the Collaborative Research Council
956, subproject A6, funded by the Deutsche Forschungsgemeinschaft (DFG). C. H.
acknowledges support by Chinese Academy of Sciences President’s International
Fellowship Initiative under Grant No. 2021VMA0009. This paper makes use of the
following ALMA data: ADS/JAO.ALMA#2012.1.00554.S, 2013.1.01136.S, and
2015.1.01388.S. ALMA is a partnership of ESO (representing its member states),
NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA
(Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of
Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This
research has used NASA’s Astrophysical Data System (ADS).

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##  Appendix A H2CO velocity channel maps

Figure 8: Para-H2CO 303–202 channel maps of N113. The contour levels are from
10% to 100% with steps of 10% of the peak intensity (0.25 Jy beam-1; color bar
in units of Jy beam-1). The centers of the fields are given in Fig. 1. Angular
scales of 4′′ correspond to a linear scale of $\sim$1 pc. No primary beam
correction has been applied. Figure 9: Same as Fig. 8, but for N159W.