11institutetext:  Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany 22institutetext: School of Natural Sciences, University
of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia 33institutetext:
Astronomy Department, Faculty of Science, King Abdulaziz University, PO Box
80203, Jeddah, 21589, Saudi Arabia 44institutetext: Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, Urumqi 830011, PR China
55institutetext: Center for Astrophysics, GuangZhou University, Guangzhou
510006, PR China 66institutetext: Purple Mountain Observatory, Chinese Academy
of Science, Nanjing 210023, PR China

# Redshifted methanol absorption tracing infall motions of high-mass star
formation regions

W. J. Yang 11 K. M. Menten 11 A. Y. Yang 11 F. Wyrowski 11 Y. Gong 11 S. P.
Ellingsen 22 C. Henkel 113344 X. Chen 55 Y. Xu 66<EMAIL_ADDRESS>
(Received date ; accepted date)

###### Abstract

Context. Gravitational collapse is one of the most important processes in
high-mass star formation. Compared with the classic blue-skewed profiles,
redshifted absorption against continuum emission is a more reliable method to
detect inward motions within high-mass star formation regions.

Aims. We aim to test if methanol transitions can be used to trace infall
motions within high-mass star formation regions.

Methods. Using the Effelsberg-100 m, IRAM-30 m, and APEX-12 m telescopes, we
carried out observations of 37 and 16 methanol transitions towards two well-
known collapsing dense clumps, W31C (G10.6$-$0.4) and W3(OH), to search for
redshifted absorption features or inverse P-Cygni profiles.

Results. Redshifted absorption is observed in 14 and 11 methanol transitions
towards W31C and W3(OH), respectively. The infall velocities fitted from a
simple two-layer model agree with previously reported values derived from
other tracers, suggesting that redshifted methanol absorption is a reliable
tracer of infall motions within high-mass star formation regions. Our
observations indicate the presence of large-scale inward motions, and the mass
infall rates are roughly estimated to be $\gtrsim$10-3 $M_{\odot}$ yr-1, which
supports the global hierarchical collapse and clump-fed scenario.

Conclusions. With the aid of bright continuum sources and the overcooling of
methanol transitions leading to enhanced absorption, redshifted methanol
absorption can trace infall motions within high-mass star formation regions
hosting bright HII regions.

###### Key Words.:

star: formation — ISM: kinematics and dynamics — ISM: individual object: W31C
(G10.6$-$0.4) and W3(OH) — ISM: molecules — ISM: structure — radio lines: ISM

††offprints: W. Yang,

##  1 Introduction

Infall motions provide direct evidence of mass accretion and their observation
plays a crucial role in star formation research. The classic blue-skewed
profile is commonly taken as evidence of infall motions (e.g. Leung & Brown
1977; Zhou et al. 1993). However, the interpretation of the blue-skewed
profile as an infall signature can be impaired by kinematic peculiarities
(e.g. outflow and rotation) and chemical abundance variations (Evans 2003).
This situation becomes even more severe when analysing spectra obtained
towards complex high-mass star formation regions by single-dish observations
with a resolution $\gtrsim$10$\arcsec$ (corresponding to 0.24 pc at a distance
of 5 kpc), because they incorporate emission which is statistically located at
farther distances.

Detecting redshifted absorption or inverse P-Cygni profiles towards background
continuum sources turns out to be a more straightforward and reliable method
to identify infall motions associated with high-mass young stellar objects.
Such redshifted absorption has already been detected in lines from a few
molecules (e.g. NH3, Keto et al. 1987a, 1988; Ho & Young 1996; Zhang & Ho
1997; HCO+, Welch et al. 1987; CS, Zhang et al. 1998; H2CO, Di Francesco et
al. 2001) against strong radio- and millimetre-wavelength continuum emission.
However, to date, such absorption has only been found in a couple of sources
with interferometric observations (with typical angular resolutions of
$<$10$\arcsec$). Redshifted absorption in sub-millimetre-wavelength rotational
lines from NH3 and H2O against dust continuum emission has been detected in a
number of objects (Wyrowski et al. 2012, 2016; van der Tak et al. 2019).
However, these transitions are not easily observed using ground-based
telescopes because their frequencies are largely blocked by the Earth’s
atmospheric absorption. Furthermore, these high frequency measurements by
Herschel and SOFIA are rather expensive and only a very limited number of
sources have been studied so far, leaving extensive searches of these
redshifted absorption features towards a larger sample of sources unfeasible.
Therefore, salient tracers that are more accessible by ground-based telescopes
can help to gauge reliable infall motions in a much larger sample of high-mass
star formation regions.

The methanol (CH3OH) molecule has numerous transitions, including many maser
lines, within the centimetre- and millimetre-wavelength ranges and hence
provides a powerful tool to investigate star-forming regions. Early studies
divided CH3OH masers into two categories (Batrla et al. 1987; Menten 1991b)
based on their different observational properties and pumping mechanisms.
Class II CH3OH masers are found in close proximity to infrared sources, OH
masers, and ultracompact Hii (UCHii) regions, and radiative pumping is
believed to be responsible for their pumping (e.g. Menten 1991a; Cragg et al.
2005). In contrast, class I CH3OH masers are thought to trace shocked regions
and are produced by collisional pumping (e.g. Leurini et al. 2016).
Experimentally determined rate coefficients for collisions between CH3OH and
H2 or helium (Lees & Haque 1974; Rabli & Flower 2010) have shown a tendency
for the final state of a collisional excitation process of a CH3OH molecule to
have the same $k$-quantum number as the initial state, that is a propensity
for $\Delta k$ = 0 collisions. This leads, for $E-$type CH3OH111Methanol
exists in two separate (non-interconverting) symmetry species: $E$-type and
$A$-type CH3OH. Its energy levels are described by the total angular momentum
quantum number $J$ and its projection on the near symmetry (O–C) axis, which
for the double degenerate $E-$type levels is lower case $k$ (with $-J\leq
k\leq+J$). This leads to ladders of levels that have the same $k$, but (bottom
to top) an increasing value of $J$. For $A-$type CH3OH, upper case $K$ is used
for the latter quantum number (with $0\leq J$) and, except for the $K=0$
ladder, the levels are split in pairs with opposite parity., to an
overpopulation of the energy levels in the $k=-1$ ladder relative to the
levels in the $k=0$ (or the $k=-2$) ladder that they can radiatively decay to
(see Fig. 1). For transitions originating in the $k=-1$ ladder that have their
upper energy levels in this ladder, that is, those with $J_{\rm upper}>4$
(line numbers 3–8 in Table 1), this results in maser emission, while for $3,2$
and 1, it leads to enhanced absorption (anti-inversion or overcooling), in
particular for the 12.2 GHz $2_{-1}-3_{0}$E line (line number 2). An analogous
situation attains for $A-$type CH3OH, for which the levels in the $K=0$ ladder
are overpopulated relative to those in the $K=1$ ladder (see Fig. 1), causing
maser action for levels with $J_{\rm upper}>7$ (line numbers 27 and 28) and
enhanced absorption in the 6.7 GHz $5_{1}-6_{0}A^{+}$ (line number 26)
transition. In the presence of a strong mid-infrared field, radiative pumping
to torsionally excited states disturbs this simple picture and causes class II
methanol maser action (Sobolev & Deguchi 1994; Sobolev et al. 1997). We note
that the two strongest class II CH3OH lines show, as explained above,
absorption in regions that are conducive to class I maser excitation, which
are often significantly removed from centres of activity marked by strong
compact radio or IR sources (see, e.g. Menten et al. 1986a). The actual
observational picture can be more complex, since in a single dish telescope
beam, the signals from different regions can be merged, which is, among
others, the case for the 12.2 GHz line in W31C, which shows absorption in
addition to maser emission (see Sec. 3.1). These effects successfully explain
the observed bright class I methanol masers (e.g. Leurini et al. 2016) and
overcooling, which enhances the absorption lines’ detectability, in particular
when strong background continuum emission is present, although even absorption
against the cosmic microwave background radiation is observed in the 12.2 GHz
line (Walmsley et al. 1988) and even the 6.7 GHz line, whose lower energy
level is 49 K above the ground (Pandian et al. 2008). Absorption in the 6.7
GHz and 12.2 GHz CH3OH transitions is indeed detected towards several sources,
several of which host I methanol masers (e.g. NGC2264 and DR21-W; Menten
1991a; Ortiz-León et al. 2021), dark clouds (TMC1 and L183; Walmsley et al.
1988), hot corinos (NGC1333; Pandian et al. 2008), molecular clouds and Hii
region complexes (Peng & Whiteoak 1992), and the Galactic Centre regions (Sgr
B2, Sgr A and G1.6$-$0.025; Whiteoak et al. 1988; Peng & Whiteoak 1989;
Whiteoak & Peng 1989; Houghton & Whiteoak 1995). These facts suggest that
methanol transitions that show absorption could have potential to aid in
studying infall motions in high-mass star formation regions.

Two high-mass star formation regions, W31C and W3(OH), have been selected for
this work. W31C, also known as G10.6$-$0.4, is an UCHii region located at a
distance of 4.95${}^{+0.51}_{-0.43}$ kpc (Sanna et al. 2014). Based on
previous dust spectral energy distribution (SED) fitting, this region is found
to have a mass of $\sim$ 2$\times 10^{4}$ M⊙ and a luminosity of $\sim$
3$\times 10^{6}$ L⊙ (e.g. Lin et al. 2016). W31C shows evidence for infall
motions in both molecular gas (e.g. NH3, Ho & Haschick 1986; Keto et al.
1987a, 1988; Sollins & Ho 2005; HCO+, Klaassen & Wilson 2008; high-$J$ CO,
Wyrowski et al. 2006) and ionised gas (in the H66$\alpha$ radio recombination
line, Keto 2002), which is well modelled by the gravitational collapse of a
centrally condensed and centrally heated core (e.g. Keto et al. 1987a; Sollins
et al. 2005). A series of high resolution observations present the overall
picture of a hierarchically collapsing system for W31C (Liu et al. 2010a, b,
2011, 2012; Liu 2017).

The W3(OH) complex is located at a distance of 1.95$\pm$0.04 kpc (Xu et al.
2006), and consists of two high-mass objects W3(OH) and W3(H2O). W3(OH) is
associated with an UCHii region ionised by an O7 star (e.g. Dreher & Welch
1981). In contrast, W3(H2O), located 6$\arcsec$ (0.06 pc) east of W3(OH) (e.g.
Zapata et al. 2011; Qin et al. 2015) is in an earlier stage of massive star
formation and hosts a (double) hot molecular core (Wyrowski et al. 1999;
Ahmadi et al. 2018). On larger ($\leq 1$ pc size) scales, both single-pointing
spectral line observations and mapping reveal infall motions in the W3(OH)
complex (e.g. Wu & Evans 2003; Sun & Gao 2009; Wu et al. 2010). Keto et al.
(1987b) reported that the velocity, density and temperature structure in the
W3(OH) complex is similar to that in W31C. That a blue-skewed profile is
observed in the HCN (3$-$2) line over a region of 110$\arcsec\times
110\arcsec$ ($1.1\times 1.1$ pc), Qin et al. (2016) further supports the
notion that the W3(OH) complex is overall undergoing large-scale collapse like
W31C. These properties make W31C and W3(OH) excellent test beds to examine the
suitability of other spectral lines for tracing infall.

In this work, we present observations of multiple methanol transitions towards
W31C and W3(OH) to test whether methanol transitions can trace infall motions.
These data were collected using the Effelsberg-100 m, the IRAM-30 m, and the
Atacama Pathfinder Experiment (APEX) 12 metre telescopes. We note that both
W3(OH) and W3(H2O) are covered within the same beam of single-dish
observations. Here we use W3(OH) to refer to the W3(OH) complex. Our
observations and data reductions are described in Sec. 2. Sec. 3 presents the
results which are discussed in Sec. 4. A summary of this work is presented in
Sec. 5

##  2 Observations and data reduction

We observed 37 CH3OH transitions towards W31C with rest frequencies ranging
from 6 GHz to 236 GHz. The observations were undertaken between 2011 and 2021
using the Effelsberg-100 m, IRAM-30 m, and APEX-12 m telescopes. Information
on the observed transitions is given in Table 1, and these transitions are
denoted by arrows in the methanol energy level diagram (see Fig. 1). We adopt
the most accurate available rest frequencies, most of which are taken from
Müller et al. (2004). We note that a given error in frequency will lead to a
larger error in velocity for transitions with a lower rest frequency. For all
transitions in Table 1, the corresponding errors in velocity range between
0.002 km s-1 (at 25.294 GHz) and 0.392 km s-1 (at 38.3 GHz). We adopt the
prescription of Shirley (2015) to estimate the optically thin critical density
for each methanol transition. The spectroscopic data for the CH3OH lines are
taken from the Leiden Atomic and Molecular Database
(LAMDA222https://home.strw.leidenuniv.nl/~moldata/) with a data file version
of 2021 April 12. Because the gas kinetic temperatures of W31C and W3(OH) are
about 100 K (Mangum & Wootten 1993; Tang et al. 2018), the optically thin
critical density is calculated by assuming a gas kinetic temperature of 100 K
for this work. The optically thin critical densities are determined to be
2$\times 10^{4}$–6$\times 10^{6}$ cm-3, spanning two orders of magnitude.
Observational parameters are given in Table 2. Four CH3OH transitions, those
at 84, 95, 108 and 111 GHz, were targeted towards an ATLASGAL source
(AGAL010.624$-$00.384, at $\alpha_{\rm J2000}$=18h10m28$\hbox
to0.0pt{.\hss}^{\rm s}$6, $\delta_{\rm J2000}$=$-$19∘55$\arcmin$46$\arcsec$)
near W31C, the other 33 transitions were observed towards the UCHii region
G010.6234$-$0.3837 at $\alpha_{\rm J2000}$=18h10m28$\hbox to0.0pt{.\hss}^{\rm
s}$70, $\delta_{\rm J2000}$=$-$19∘55$\arcmin$49$\hbox to0.0pt{.\hss}\arcsec$8
(Yang et al. 2019, 2021). The angular offset between the two positions is
4$\arcsec$ which is less than one fifth of the IRAM-30 m beam size at 3 mm.

We carried out K-band (18–26 GHz) observations towards W3(OH) in 2021 May 13
using the Effelsberg-100 m. The telescope was pointed towards the position,
$\alpha_{\rm J2000}$=02h27m04$\hbox to0.0pt{.\hss}^{\rm s}$38, $\delta_{\rm
J2000}$=61∘52$\arcmin$20$\hbox to0.0pt{.\hss}\arcsec$5, used by (Menten et al.
1986a), who reported absorption in the 25 GHz $J_{2}-J_{1}E$ CH3OH lines. The
methanol transitions at 95.914, 97.582 and 143.865 GHz were observed in 2019
June 11 using the IRAM-30 m. The telescope was pointed towards the position of
the UCHii region W3(OH), that is $\alpha_{\rm J2000}$=02h27m03$\hbox
to0.0pt{.\hss}^{\rm s}$90, $\delta_{\rm J2000}$=61∘52$\arcmin$24$\hbox
to0.0pt{.\hss}\arcsec$0, which is about 5$\arcsec$ offset (nearly one seventh
of the Effelsberg-100 m beam size at 25 GHz) from the position that we used in
the K-band observations.

Given that the offsets between different positions are much smaller than the
beam sizes of our observations, the position offsets are neglected in the
following analysis. Details of the observations and data reduction for each
telescope are described below.

###  2.1 Effelsberg-100m observations

Using the dual-polarisation S7mm Double Beam RX secondary receiver on the
Effelsberg-100 m telescope333The 100-m telescope at Effelsberg is operated by
the Max-Planck Institut für Radioastronomie (MPIfR) on behalf of the Max-
Planck Gesellschaft (MPG)., we conducted Q band observations towards W31C
($\alpha_{\rm J2000}$=18h10m28$\hbox to0.0pt{.\hss}^{\rm s}$70, $\delta_{\rm
J2000}$=$-$19∘55$\arcmin$49$\hbox to0.0pt{.\hss}\arcsec$8) to observe methanol
transitions ranging from 34.5 to 39.5 GHz on 2020 September 14 and from 40 to
45 GHz on 2020 May 30 (project id: 77-19). Towards W31C, the 6.7 GHz and 12.2
GHz methanol transitions were observed on 2021 January 5 using the S45mm and
S20mm secondary focus receivers, while K-band observations (18 – 26 GHz) were
performed towards both W3(OH) ($\alpha_{\rm J2000}$=02h27m04$\hbox
to0.0pt{.\hss}^{\rm s}$38, $\delta_{\rm J2000}$=61∘52$\arcmin$20$\hbox
to0.0pt{.\hss}\arcsec$5) and W31C on 2021 May 13 and 14 (project id: 17-21).
All these observations were made in the position switching mode with an off-
position offset of 10$\arcmin$ in right ascension. Fast Fourier Transform
Spectrometers (FFTSs) were used as the backend to record signals. For the 6.7
GHz and 12.2 GHz methanol transitions, we used high spectral resolution FFTSs
which provide a channel separation of 3.1 kHz and 4.6 kHz at 6.7 GHz and 12.2
GHz, corresponding to a velocity spacing of 0.14 km s-1 and 0.11 km s-1,
respectively. For other methanol transitions, we adopted broad-bandwidth FFTSs
which cover an instantaneous bandwidth of 2.5 GHz with 65536 channels,
yielding a channel separation of 38.1 kHz and a corresponding velocity spacing
of 0.33 km s-1at 35 GHz (multiply by 1.16 to convert to velocity resolution,
see Klein et al. 2012).

NGC 7027 and 3C 286 were used for pointing, focus and flux density
calibration, and the pointing uncertainty was better than 10$\arcsec$,
5$\arcsec$, 5$\arcsec$, and 10$\arcsec$ at 6.7 GHz, 12.2 GHz, 24 GHz, and 40
GHz, respectively. The observed methanol transitions, corresponding
observation dates, full width at half maximum (FWHM) beam sizes, velocity
resolutions, the 1$\sigma$ rms noise levels at the corresponding velocity
resolutions, main beam efficiencies, $\eta_{\rm mb}$, and the scaling factors
to convert antenna temperature, $T_{\rm A}^{*}$, to flux density, $S_{\nu}$,
are listed in Table 2.

###  2.2 IRAM-30 m observations

We observed multiple methanol transitions within the frequency ranges of
84.6–92.4 GHz and 100.3–108.1 GHz towards W31C ($\alpha_{\rm
J2000}$=18h10m28$\hbox to0.0pt{.\hss}^{\rm s}$70, $\delta_{\rm
J2000}$=$-$19∘55$\arcmin$49$\hbox to0.0pt{.\hss}\arcsec$8) using the Eight
MIxer Receiver (EMIR, Carter et al. 2012) of the IRAM 30m telescope on 2019
June 29 (project id: 043-19). The methanol transitions at 95.914, 97.582 and
143.865 GHz were observed towards W3(OH) ($\alpha_{\rm J2000}$=02h27m03$\hbox
to0.0pt{.\hss}^{\rm s}$90, $\delta_{\rm J2000}$=61∘52$\arcmin$24$\hbox
to0.0pt{.\hss}\arcsec$0) on 2019 June 11 (project id: 045-19) using the EMIR
receivers (E090 and E150). The dual-polarisation EMIR receivers provide nearly
8 GHz bandwidth per sideband, which is connected to two 4 GHz wide FFTSs.
These FFTSs provide a channel width of $\sim$195 kHz, resulting in a velocity
spacing of $\sim$0.6 km s-1 at 95 GHz. The wobbler switching mode, through
small movements of the sub-reflector to change the optical path from antenna
to the receiver, was used for pointing towards the source with a throw of
120$\arcsec$ in azimuth. Additional methanol line data at 84, 95, 108 and 111
GHz towards W31C were taken from an unbiased line survey towards ATLASGAL
sources at $\alpha_{\rm J2000}$=18h10m28$\hbox to0.0pt{.\hss}^{\rm s}$6,
$\delta_{\rm J2000}$=$-$19∘55$\arcmin$46$\arcsec$. The observations were
carried out during 2011 April 8 to 11 (see Csengeri et al. 2016 for details,
project id: 181-10). The observational parameters including the main beam
efficiency, the scaling factor, and the 1$\sigma$ rms noise level of each
transition are listed in Table 2.

###  2.3 APEX-12 m observations

We observed methanol transitions ranging from 209.2 to 220.9 GHz and from 225
to 236.7 GHz towards W31C ($\alpha_{\rm J2000}$=18h10m28$\hbox
to0.0pt{.\hss}^{\rm s}$70, $\delta_{\rm J2000}$=$-$19∘55$\arcmin$49$\hbox
to0.0pt{.\hss}\arcsec$8) with the PI230 receiver at the APEX444This
publication is based on data acquired with the Atacama Pathfinder Experiment
12 m sub-millimetre telescope (APEX). APEX is a collaboration between the Max-
Planck-Institut für Radioastronomie, the European Southern Observatory, and
the Onsala Space Observatory. telescope (Güsten et al. 2006) on 2019 June 1
and June 9 (project id: 0103.F-9516A). The PI230
receiver555https://www.eso.org/public/teles-instr/apex/pi230/ is a dual
polarisation and sideband-separating heterodyne system that covers a frequency
range of 200$-$270 GHz. The FFTS backend provides 65536 channels over 4 GHz
bandwidth, and an overlap of 0.2 GHz between the two backends results in a
total 7.8 GHz coverage for each sideband and polarisation per tuning. The
wobbler switching mode with a throw of 120$\arcsec$ in azimuth was used for
our observations. The pointing was accurate to $\lesssim$2$\arcsec$. The main
beam efficiency, the scaling factor, and the rms noise level after an on-
source integration time of 5 minutes are listed in Table 2.

###  2.4 Data reduction

The CLASS programme that is part of the GILDAS package (Pety 2005; Gildas Team
2013) was used for processing all of our data. For the observations performed
with the 7 mm dual-beam receiver, we only analysed the data for the beam that
tracked the selected targeted positions. Because radio frequency interference
caused a poor baseline for 12.2 GHz spectrum (for W31C), a fourth order
polynomial baseline was subtracted within the baseline parameters determined
over a velocity range of 60 km s-1 around the systemic velocity. A low-order
($<$4) polynomial baseline was subtracted from the other spectra. To improve
the signal-to-noise ratios (S/Ns), Hanning smoothing was performed on the
spectra obtained with the APEX-12 m telescope. Velocities are given with
respect to the local standard of rest (LSR) throughout this work.

Figure 1: Energy level diagram of E-type (left) and A-type (right) methanol
transitions. Transitions from Slysh et al. (1999) belonging to the line series
where absorption features are detected in W31C are also marked, at frequencies
around 133 GHz. Known class I and II methanol masers transitions are marked
with red and blue arrows, respectively. The black arrows represent the
transitions that have no maser detection yet. The dotted arrows are
transitions with detected absorption features in W31C. The green arrows
indicate the thermal transitions of the $(J+1)_{1}-J_{1}A^{\pm}$ series
observed in W3(OH).

##  3 Results

Our observations resulted in the detection of 37 methanol transitions in W31C
and 16 methanol transitions in W3(OH). The observational results for the two
sources are given separately in the following.

###  3.1 Detection of 14 redshifted methanol absorption lines towards W31C

Figures 2–4 present the observed spectra of 37 CH3OH transitions observed
towards W31C, showing diverse line shapes. Gaussian fitting was performed to
characterise the spectral features of each transition. Single-peaked spectra
observed in emission are fitted with a single Gaussian component.

We used a single Gaussian component to fit the absorption signal for the
transitions at 19.9, 20.1, 23.4, 37.7, 38.3, 38.5, 85.6, 86.6 and 86.9 GHz, in
order to clearly characterise the absorption feature. For other spectra with
multiple emission peaks and absorption dips, we employed multiple Gaussian
components for the fitting. The results of the fitting are listed in Table 3.

Among the observed spectra, we find 12 methanol transitions showing single-
peaked emission profiles. These are the 104, 107, 108, 111, 216, 218, 220,
229, 230, 231, 232, and 234 GHz lines. Although the 104, 107, 108, 216, 218,
229, 231 and 232 GHz lines have been found to show maser action towards other
sources (e.g. Val’tts et al. 1999; Slysh et al. 2002; Salii & Sobolev 2006;
Voronkov et al. 2006; Ellingsen et al. 2012; Hunter et al. 2014), the observed
broad widths of the lines are similar to the widths of lines from other
species, which indicates that they are dominated by thermally excited emission
towards W31C. Therefore, we use the fitted velocities of these 12 transitions
to derive the systemic velocity of W31C. Adopting inverse-variance weighting,
we use the weighted mean velocity as the systemic velocity, and it is
determined to be $V_{\rm LSR}$=$-$3.43$\pm$0.44 km s-1. The systemic velocity
we adopt is consistent with the value of $\sim-$3 km s-1 used in previous
studies (e.g. Keto 1990; Liu et al. 2010a; Kim et al. 2020). In Figure 5, it
can be seen that the velocity centroids of these 12 methanol transitions
become more redshifted with increasing energy level and critical density,
which indicates that there is a velocity gradient from the outer envelope to
the centre.

Absorption features are detected in 14 transitions, the 6.7, 12.2, 19.9, 20.1,
23.1, 23.4, 25.541, 25.878, 37.7, 38.3, 38.5, 85.6, 86.6 and 86.9 GHz lines
(see Column 3 in Table 1). Absorption in five of these transitions (6.7, 12.2,
19.9, 23.1 and 23.4 GHz) towards W31C has been discussed in previous studies
(Wilson et al. 1984, 1985; Menten 1991b; Peng & Whiteoak 1992; Caswell et al.
1995b; Breen et al. 2010, 2014). The remaining nine absorption profiles at
20.1, 25.541, 25.878, 37.7, 38.3, 38.5, 85.6, 86.6 and 86.9 GHz are reported
here for the first time. Furthermore, the CH3OH transitions at 19.9, 25.541,
25.878 and 85.6 GHz show a typical inverse P-Cygni profile. The absorption
features have a large scatter in their LSR velocities, ranging from $-$1.91 km
s-1 to 0.43 km s-1 (see Table 3 and Fig. 5). Nevertheless, all 14 absorption
features are redshifted with respect to the systemic velocity ($V_{\rm
LSR}$=$-$3.43$\pm$0.44 km s-1, see also Fig. 5).

Both the $5_{1}-6_{0}A^{+}$ and $2_{0}-3_{-1}E$ lines at 6.7 (Fig. 3b) and
12.2 GHz (Fig. 2b) show broad absorption and maser features. The absorption
features in these two transitions lie in the velocity range $-$2 to $-$1 km
s-1 and are therefore redshifted with respect to the systemic velocity of the
source. The line width of the absorption feature at 12.2 GHz is the broadest
we have detected for absorption towards W31C. The 6.7 GHz spectral profile
shows absorption sandwiched between two maser features, which is consistent
with previous observations (Menten 1991b; Caswell et al. 1995b). Breen et al.
(2010, 2014) reported the detection of a 12.2 GHz maser. In their spectra
obtained in 2008, the intensity of the absorption feature is about $-$0.5 Jy,
similar to our result. On the other hand, the peak flux density of the maser
feature at 4.6 km s-1 is 1.4 Jy, which is nearly three times our value. This
indicates that the 12.2 GHz maser flux density has declined over the past 13
years.

The 19.9 GHz line (Fig. 4a) shows an inverse P-Cygni profile with an emission
feature at about $-$6.6 km s-1 and an absorption feature at about $-$0.9 km
s-1. The 23.1 GHz line (Fig. 4c) exhibits one prominent absorption feature at
about $-$0.74 km s-1. These two spectra are consistent with previous studies
(Wilson et al. 1984, 1985). We adopted a rest frequency of 23444.778 MHz
(Mehrotra et al. 1985; Voronkov et al. 2011) for the $10_{1}-9_{2}A^{-}$ line,
which is 42 kHz lower than the value used by Menten et al. (1986b). Thus, to
compare with the velocity of the absorption feature observed by Menten et al.
(1986b), a value of $-$0.54 km s-1should be subtracted from the velocity they
reported, which shifts the peak absorption to 0.26 km s-1and reduces the
difference with our result to 0.77 km s-1. The remaining difference may be due
to the noisy spectra and slightly different fitting methods. Despite the low
S/N ratio in the 20.1 GHz line (Fig. 4b), its absorption feature has a similar
velocity as those of the 19.9, 23.1, 23.4 GHz lines, indicating that it is a
reliable detection.

A series of nine $J_{2}-J_{1}$ lines ($J$ from 2 to 10) of $E$-type methanol
near 25 GHz are detected by our observations. Both the $9_{2}-9_{1}E$ line at
25.541 GHz (Fig. 4l) and the $10_{2}-10_{1}E$ line at 25.878 GHz (Fig. 4m)
exhibit typical inverse P-Cygni profiles with the absorption dips at about 0.1
km s-1. We note that such absorption dips are also present in other
$J_{2}-J_{1}$ lines at a nearly identical velocity but blended with emission,
which make them deviate from the typical inverse P-Cygni profile. The dips
appear to become more prominent with increasing $J$. The presence of narrow 25
GHz Class I methanol maser emission also makes the profiles more complex for
the $5_{2}-5_{1}E$ line at 24.959 GHz, the $6_{2}-6_{1}E$ line at 25.018 GHz
and the $7_{2}-7_{1}E$ line at 25.124 GHz (see Figs. 4h–4j). Menten et al.
(1986a) reported the detection of five $J_{2}-J_{1}E$ lines ($J$ = 2, 3, 4, 5,
6) towards W31C, but the maser emission peaks at a location 9$\arcsec$ offset
from our pointing centre. Nevertheless, the masers should be covered by our
Effelsberg beam of $\sim$34$\arcsec$, and they give rise to the observed
narrow spikes between $-$6.0 km s-1 and $-$8.0 km s-1 in our spectra of the
$5_{2}-5_{1}E$, $6_{2}-6_{1}E$ and $7_{2}-7_{1}E$ transitions (see Figs.
4h–4j). On the other hand, likely influenced by our offset position, the flux
densities are much lower than the values reported in Menten et al. (1986a). In
addition to the position difference, our low spectral resolution may be
further reducing the measured peak flux densities.

An absorption feature with a velocity of $-$0.68 km s-1 was detected in the
$7_{-2}-8_{-1}E$ line at 37.7 GHz (see Fig. 2g), and, with a flux density of
$-$0.89 Jy. This is the strongest absorption feature among our detected
absorption lines. Both $6_{2}-5_{1}A^{\mp}$ lines (from levels of different
parity) at 38.3 and 38.5 GHz (see Figs. 3e–3f) have a narrow emission spike
with a velocity of $-$1.9 km s-1 that is superimposed on a broad absorption
feature. Similar to the $9_{2}-9_{1}E$ and the $6_{2}-6_{1}E$ methanol
transitions near 25 GHz in W3(OH) (Menten et al. 1986a), the narrow spike may
arise from a weak maser. The velocities of the absorption features in the 38.3
and 38.5 GHz transitions match that of the 37.7 GHz line. The line widths of
the absorption features in the 38.3 and 38.5 GHz transitions are $\sim$4.8 km
s-1, which is slightly narrower than the line width ($\sim$5.35 km s-1) of the
37.7 GHz line. Ellingsen et al. (2011) also searched for the 37.7, 38.3 and
38.5 GHz CH3OH lines towards W31C but nothing was detected, owing to their rms
noise of $\sim$1.2 Jy with a channel width of 0.27 km s-1. On the other hand,
Ellingsen et al. (2018) discovered an absorption feature in the 37.7, 38.3 and
38.5 GHz transitions towards G337.705$-$0.053, but their velocities are not
redshifted with respect to its systemic velocity traced by NH3 (1,1) (Purcell
et al. 2012). Our observations are therefore the first detection of redshifted
absorption in these three lines.

Absorption features of methanol transitions at 85.6, 86.6 and 86.9 GHz in this
work are detected for the first time. The CH3OH transition at 85.6 GHz (Fig.
2g) clearly shows an inverse P-Cygni profile. The emission feature peaks at
about $-$4 km s-1, and the absorption feature lies at a velocity of about 0.22
km s-1. Despite the low S/N ratios in the CH3OH spectra at 86.6 and 86.9 GHz
(Figs. 3g–3h), their shapes are indicative of an inverse P-Cygni profile. The
velocities and line widths of the absorption features of the 86.6 and 86.9 GHz
lines are similar to that of the 85.6 GHz line, but their absorption line
strengths are much weaker than that of the 85.6 GHz line. This could be due to
a lower excitation temperature of the 85.6 GHz line.

The most common class I CH3OH transitions at 36, 44, 84 and 95 GHz show strong
and narrow maser features, and the velocities of these maser features range
from $-$7 to $-$6 km s-1 (see Table 3). Therefore, all the class I methanol
masers are significantly blueshifted with respect to the systemic velocity
(see Fig. 5). These masers have been already reported by previous studies. The
peak velocity and flux density of the 84 and 95 GHz CH3OH masers from our
observations are consistent with previous observations (Chen et al. 2012;
Breen et al. 2019). However, the flux densities of the 36 and 44 GHz CH3OH
masers of our Effelsberg-100 m observations are about 2–3 times lower than
previous single-dish measurements (Haschick & Baan 1989; Kang et al. 2016;
Breen et al. 2019; Kim et al. 2019). The discrepancy might arise from the
different beam sizes, spectral resolutions, and pointing positions. On the
other hand, maser variability may also contribute to the observed difference.
To our knowledge, there are few published variability studies of class I
methanol masers in the Galaxy. Despite previous claims to the contrary, Menten
et al. (1988), using the Effelsberg 100 m telescope, found no significant
variability in the 25 GHz $J_{2}-J_{1}E$ class I CH3OH maser lines ($J$ = 2 –
8) in the Orion Kleinmann-Low Nebula, neither on a time baseline of 2 years
nor over 13 years, comparing their data with those that Hills et al. (1975)
obtained with the same telescope. Because class I methanol masers are often
distributed over larger scales than the class II masers, it is more difficult
to infer variability from observations using different telescopes with
different spatial and spectral resolutions. For example, in Orion KL the 25
GHz masers arise from different location spread over an area that is
comparable to the Effelsberg beam ($34$″).

Figure 2: Spectra of observed $E$-type CH3OH transitions towards W31C,
excluding those in K-band (18 – 26 GHz), which are collected in Fig. 4. The
panels from (a) to (l) are arranged following the order in Table 1. The
quantum numbers and rest frequencies of these transitions are labelled in
their respective panels. Class I and II CH3OH maser transitions are labelled
in red and blue, respectively, and their Gaussian fitting results are plotted
in the corresponding colour. The transitions that have no maser detection yet
are labelled in black, and the Gaussian fitting results are plotted in green.
In panel (k), the $-$30 km s-1 component is caused by the blend of four
CH3OCHO transitions (i.e. $20_{0,20}-19_{0,19}\,E$, $20_{0,20}-19_{0,19}\,A$,
$20_{1,20}-19_{1,19}\,E$, and $20_{1,20}-19_{1,19}\,A$). The vertical magenta
dashed lines represent the systemic velocity of $V_{\rm LSR}$=$-$3.43 km s-1.
Figure 3: Spectra of observed $A$-type CH3OH transitions towards W31C,
excluding those from K-band (18–26 GHz), which are collected in Fig. 4. The
panels from (a) to (l) are arranged following the order in Table 1. The
quantum numbers and rest frequencies of transitions are labelled in their
respective panels. Class I and II CH3OH maser transitions are labelled in red
and blue, respectively, and their Gaussian fitting results are plotted in the
corresponding colour. The transitions that have no maser detection yet are
labelled in black, and the Gaussian fitting results are plotted in green. In
panel (k), the $-$18 km s-1 component arises from CH3OH ($5_{-4}-6_{-3}\,E$).
The vertical magenta dashed lines represent the systemic velocity of $V_{\rm
LSR}$=$-$3.43 km s-1.  Figure 4: Observed spectra of CH3OH transitions in
K-band (18 – 26 GHz) towards W31C. The panels from (a) to (m) are arranged
based on increasing frequency or increasing $J$ for the $J_{2}-J_{1}E$ line
series. The quantum numbers and rest frequencies of transitions are labelled
in their respective panels. Class I and II CH3OH maser transitions are
labelled in red and blue, respectively, and their Gaussian fitting results are
plotted in the corresponding colour. The transitions that have no maser
detection yet are labelled in black, and the Gaussian fitting results are
plotted in green. The vertical magenta dashed lines represent the systemic
velocity of $V_{\rm LSR}$=$-$3.43 km s-1.

Figure 5: LSR velocity as a function of the upper energy level (upper panel)
and critical density (lower panel) for W31C. The red and blue dots indicate
the methanol lines with absorption features and maser features, respectively.
The black dots indicate the methanol emission without a narrow and strong
maser emission feature. We only show the $J_{2}-J_{1}E$ lines with absorption
features or maser features in the panels. Their velocities are derived from
Gaussian fitting to the emission peaks or absorption dips. The error bars
indicate Gaussian fitting errors. The horizontal magenta dashed line
represents the systemic velocity of $V_{\rm LSR}$=$-$3.43 km s-1.

###  3.2 Detection of 11 redshifted methanol absorption lines towards W3(OH)

Figure 6 shows the spectra of 16 methanol transitions observed towards W3(OH).
The 19.9, 23.1, 95.914, 97.582, 143.865 GHz transitions appear to be single-
peaked. The 20.1 and 23.4 GHz lines show absorption features, while the nine
$J_{2}-J_{1}$ lines ($J$ from 2 to 10) of $E$-type methanol near 25 GHz show
inverse P-Cygni profiles. We fitted the emission and absorption features using
two Gaussian components for the 23.4 GHz and the $J_{2}-J_{1}E$ line series
near 25 GHz. One Gaussian component is assumed for the other transitions. The
fitted line parameters are given in Table 4. The systemic velocity of the
W3(OH) complex is derived from three thermal methanol transitions at 95.914,
97.582 and 143.865 GHz. Similar to W31C, the weighted mean velocity is $V_{\rm
LSR}$= $-$46.19$\pm$0.02 km s-1. This velocity is consistent with previous
high angular resolution observations of the W3(OH) region whose systemic
velocity is determined to range from $-$46.5 km s-1 to $-$46.0 km s-1 (Keto et
al. 1995; Zapata et al. 2011). In this work, we simply take $V_{\rm
LSR}$=$-$46.19$\pm$0.02 km s-1 from our single-dish observations as the
systemic velocity for the subsequent analysis.

The nine $J_{2}-J_{1}$ lines ($J$ from 2 to 10) of $E$-type methanol near 25
GHz show inverse P-Cygni profiles. The emission features peak at about $-$47.1
km s-1 for all $J_{2}-J_{1}E$ lines, while the absorption features lie at
about $-$44.6 km s-1. In the nine transitions, the systemic velocity is nearly
at the point where the emission feature transits to the absorption feature.
The intensities of the absorption features show a descending trend with
increasing $J$ (see Fig. 6), which appears to be opposite to the case in W31C
(see Fig. 4). Because the difference between the rest frequencies of these
lines is small, the continuum levels should be almost identical. Hence, the
trend strongly suggests that the excitation temperatures of these transitions
decreases with increasing $J$ in W3(OH). Our results are generally consistent
with the line profiles of the $J_{2}-J_{1}$ ($J$=2, 3, 4, 5, 6, 9) lines in
Menten et al. (1986a) except that different line profiles are found in the
$6_{2}-6_{1}E$ and $9_{2}-9_{1}E$ lines. Their observations revealed double
absorption features in these two transitions, but only one single absorption
component is seen in our observations. The double absorption features were
interpreted as a weak maser at a velocity of $-$44 km s-1 that partially
covers the absorption trough (Menten et al. 1986a). The non-detection of the
double absorption features by our observations may indicate temporal variation
of this weak maser.

The absorption feature in both the 20.1 and 23.4 GHz lines has an LSR velocity
of about $-$44.4 km s-1 with a line width of $\sim$2 km s-1, which is in good
agreement with the results reported by Menten et al. (1986b). Despite the
different intensities that might be caused by different spectral resolutions
and calibrations, the line profiles of the observed masers at 19.9 and 23.1
GHz are also consistent with the spectra obtained more than 30 years ago
(Wilson et al. 1984, 1985). The $4_{0}-3_{1}E$ line at 28.3 GHz and the
$2_{1}-3_{0}E$ maser line at 19.9 GHz belong to the same series, but
absorption features were detected in the former line by Slysh et al. (1992)
and Wilson et al. (1993), and the absorption features appear to be nearly at
the same velocity as those at 20.1 and 23.4 GHz.

Figure 7 shows the line velocity as a function of the upper energy level and
critical density of each transition for W3(OH). We find that all detected
absorption features have a velocity of about $-$44.6 km s-1, which is
redshifted with respect to the systemic velocity. The detected class II
methanol masers at 19.9 and 23.1 GHz are even more redshifted than the
absorption features. The three types of methanol emission indicate the
presence of three distinct groups of lines in Fig. 7, and the velocities of
the three groups seem to be independent of the energy level of the respective
transitions and the critical density.

Figure 6: Observed spectra of 16 CH3OH transitions towards W3(OH). The quantum
numbers and rest frequencies of transitions are labelled in their respective
panels. Class I and II CH3OH maser transitions are labelled in red and blue,
respectively. Their Gaussian fitting results are plotted in the corresponding
colour. The transitions that have no maser detection yet are labelled in
black, and the Gaussian fitting results are plotted in green. The vertical
magenta dashed line represents the systemic velocity of $V_{\rm LSR}$=$-$46.19
km s-1.

Figure 7:  LSR velocity as a function of the upper energy level (upper panel)
and critical density (lower panel) for W3(OH). The red and blue dots indicate
the methanol lines with absorption and maser features, respectively. The black
dots indicate the methanol emission without a narrow and strong maser emission
feature. Their velocities are derived from Gaussian fitting to the emission
peaks or absorption dips. The error bars indicate Gaussian fitting errors. The
horizontal magenta dashed line represents the systemic velocity of $V_{\rm
LSR}$=$-$46.19 km s-1.

##  4 Discussion

###  4.1 Modelling redshifted methanol absorption

Our observations have shown that redshifted methanol absorption is more
prominent in the lower frequency (radio wavelength) transitions than in those
at higher frequencies. This is expected since the two sources host UCHII
regions, whose free-free continuum emission becomes optically thin regime at
higher frequencies, which results in a reduction of the continuum brightness
temperature (e.g. Kurtz 2005; Yang et al. 2021). Furthermore, the observed
absorption is mainly caused by cool gas that lies in the foreground of the
free-free continuum emission. Therefore, low frequency lines that are absorbed
from levels with low energy above the ground state tend to be more prominent.
We also note that the absorption features of methanol lines from transitions
at nearby frequencies can exhibit quite different intensities, indicating
different excitation temperatures. This suggests that their populations
deviate from local thermodynamic equilibrium (LTE). As discussed in Sec. 1,
such non-LTE effects can be easily achieved for various methanol lines by
collisional pumping (e.g. Leurini et al. 2016), and such effects could lead to
very low excitation temperatures (even $<$2.73 K) in methanol transitions.

Our single-dish spectra are likely tracing large-scale inward motions, but the
large-scale density, temperature, and velocity structures might not be well
described by simple analytical solutions. Hence, we use a simple two-layer
model for this study instead of more sophisticated radiative transfer models.
In order to demonstrate the feasibility of using redshifted methanol
absorption to quantify infall motions, we make use of a modified two-layer
model as discussed by Myers et al. (1996); Di Francesco et al. (2001) to fit
the observed redshifted methanol absorption profiles. In this model, the two
layers lie along the line of sight. The “front” layer lies between the
observer and the continuum source with an excitation temperature $T_{\rm f}$,
while the “rear” layer lies behind the source with an excitation temperature
$T_{\rm r}$. The continuum source is assumed to be optically thick with a
blackbody temperature $T_{\rm c}$, and the cosmic background radiation $T_{\rm
b}$ is also included. Each layer has a peak optical depth, $\tau_{0}$,
velocity dispersion, $\sigma$, and infall velocity, $V_{\rm in}$, towards the
source. The Planck-corrected brightness temperature at the rest frequency
$\nu$ is defined as $J\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
T_{0}/{\rm exp}[(T_{0}/T)-1]$, where $T_{0}\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ h\nu/k$, $h$ is Planck’s constant, and $k$ is
Boltzmann’s constant. $J_{\rm f}$, $J_{\rm r}$, $J_{\rm c}$, and $J_{\rm b}$
are the Planck-corrected brightness temperature of the “front” layer, “rear”
layer, central continuum source, and the cosmic background radiation,
respectively. The central continuum source fills a fraction of $\Phi$ of the
observing beam with and $V_{\rm LSR}$ is the systemic LSR velocity of the
source. According to this model (Di Francesco et al. 2001), the observed line
brightness temperature can be expressed as,

$\Delta T_{\rm B}=(J_{\rm f}-J_{\rm cr})[1-{\rm exp}(-\tau_{\rm
f})]+(1-\Phi)(J_{\rm r}-J_{\rm b})[1-{\rm exp}(-\tau_{\rm r}-\tau_{\rm
f})]\,,$ (1)

where

$J_{\rm cr}=\Phi J_{\rm c}+(1-\Phi)J_{\rm r}\,,$ (2)

$\tau_{\rm f}=\tau_{\rm 0}\,{\rm exp}[\frac{-(V-V_{\rm in}-V_{\rm
LSR})^{2}}{2\sigma^{2}}]\,,$ (3)

$\tau_{\rm r}=\tau_{\rm 0}\,{\rm exp}[\frac{-(V+V_{\rm in}-V_{\rm
LSR})^{2}}{2\sigma^{2}}]\,.$ (4)

In order to reduce the number of free parameters, we fixed $V_{\rm
LSR}$=$-$3.43 km s-1 for W31C and $V_{\rm LSR}$=$-$46.19 km s-1 for W3(OH),
while $T_{\rm b}$=2.73 K and $\Phi$=0.9 are adopted for both sources. The
fixed parameter $\Phi$ is arbitrarily selected. However, we tested the model
with $\Phi$=0.1–0.9, and the resulting $V_{\rm in}$, $\tau_{0}$ and $\sigma$
stayed nearly identical, while $T_{\rm f}$ and $T_{\rm r}$ changed
dramatically. For each transition, the model inputs also include the
corresponding rest frequency and the radiation from the central continuum
source ($T_{\rm c}$). The continuum flux densities contributed by the central
UCHii region were obtained from interferometric radio continuum measurements
(see Yang et al. 2021 for W31C, see Wilson et al. 1991 for W3(OH)), and
corresponding $T_{\rm c}$ values are thus calculated by adopting different
frequencies and beam sizes. After fixing these parameters, there are still
five free parameters $V_{\rm in}$, $\tau_{0}$, $\sigma$, $T_{\rm f}$, $T_{\rm
r}$ which will be determined by comparison with our spectra.

In order to derive the free parameters in the model together with their
uncertainties, we used the emcee code (Foreman-Mackey et al. 2013) to perform
Monte Carlo Markov chain (MCMC) calculations with the affine-invariant
ensemble sampler (Goodman & Weare 2010). Uniform priors are assumed for the
five free parameters. The likelihood function is assumed to be ${\rm
e}^{-\chi^{2}/2}$, and $\chi^{2}$ is defined as:

$\chi^{2}=\Sigma_{i}(T_{\rm{obs},i}-T_{\rm{mod},i})^{2}/\sigma_{\rm{obs},i}^{2}$
(5)

where $T_{\rm{obs},i}$ and $T_{\rm{mod},i}$ are the observed and modelled
brightness temperatures for each channel, and $\sigma_{\rm{obs},i}$ is the
standard deviation of $T_{\rm{obs},i}$. The posterior distribution of these
parameters is estimated by the product of the prior and likelihood functions.
For the MCMC calculations, we run 20 walkers and 10000 steps after the burn-in
phase. The 1$\sigma$ errors in the fitted parameters are estimated by the 16th
and 84th percentiles of the posterior distribution. In order to help the
convergence, the parameter spaces are set to be 0.5–10 km s-1 for $V_{\rm
in}$, 0.05–30 for $\tau_{0}$, 0.5–3 km s-1 for $\sigma$, 0–150 K for $T_{\rm
f}$ and $T_{\rm r}$. The MCMC fitting to the 37.7 GHz CH3OH line in W31C is
shown in Fig. 8. The same method is applied to all the methanol transitions
showing redshifted absorption, and the fitted spectra are presented in Figs.
9–10 for W31C and W3(OH), respectively, and Table 5 shows the fitting
parameters.

We note that the infall velocity is the key parameter in deriving the infall
rates, which are of central interest for understanding mass accretion in star
formation. We tested the model with different fixed parameters to study how
$V_{\rm in}$ is affected. We find that the derived infall velocity is nearly
independent of the assumed parameters. This is generally in line with previous
studies (Pineda et al. 2012) in which such behaviour was verified using a
different method (i.e. minimised-$\chi^{2}$ fitting). However, we also note
that the presence of maser emission components in the redshifted velocity
range might lead to underestimation of $V_{\rm in}$. Furthermore, the
uncertainties in the assumed systemic velocity will propagate to $V_{\rm in}$,
so there may be additional uncertainties of 0.44 km s-1 and 0.02 km s-1 in
$V_{\rm in}$ for W31C and W3(OH) (see Secs. 3.1 and 3.2). On the other hand,
$\tau_{0}$, $T_{\rm f}$ and $T_{\rm r}$ are coupled (see Eqs. 1–4), so their
values can be significantly affected by varying the fixed parameters.

The $V_{\rm in}$ value of 1.07${}^{+0.23}_{-0.21}$ km s-1 in the 25.541 GHz
transition may be underestimated, because the observed spectrum is not well
fitted by the model (see Fig. 9). The cause of the anomaly may be due to an
inaccurate $V_{\rm LSR}$ for the 25.541 GHz line. If we use a velocity of $-$2
km s-1, around which velocity the emission transits to absorption, we can
reproduce a better fit giving $V_{\rm in}=2.32^{+0.40}_{-0.94}$ km s-1.
$V_{\rm in}$ derived from the 86.6 and 86.9 GHz lines has a large uncertainty
due to poor S/N ratios.

Except for the 25.541, 86.6 and 86.9 GHz transitions, the derived $V_{\rm in}$
in W31C ranges from 2.0 to 3.5 km s-1 at corresponding linear scales of
0.3–1.0 pc, which is generally consistent with the infall velocities derived
by other tracers at a linear scale of 0.2–0.3 pc (e.g. Liu et al. 2013; Liu
2017). Good agreement with other estimates of the infall velocity is also seen
in W3(OH). Our derived $V_{\rm in}$ in W3(OH) lies in the range 1.6–1.8 km s-1
at a linear scale of $\sim$0.4 pc, consistent with previously reported infall
velocities (Keto et al. 1987b; Wilson et al. 1991). These results suggest that
redshifted methanol absorption can reliably estimate the infall velocities.
Further discussion of the relationship between the infall velocity and the
upper level, critical density and velocity dispersion can be found in Appendix
B.

The $V_{\rm in}$ values we obtain are lower than the infall velocities
(4.5–6.5 km s-1) measured on smaller scales of 0.02–0.05 pc in W31C (Keto et
al. 1987a, 1988; Keto 2002; Sollins et al. 2005). As already mentioned, our
beams cover both W3(H2O) and W3(OH). High angular HCO+ (3–2) resolution
observations suggest that the infall velocity of W3(H2O) is 2.7$\pm$0.3 km s-1
at a linear scale of $\sim$0.02 pc (Qin et al. 2016), higher than our values
of 1.5–1.8 km s-1. Both cases suggest that the infall velocities are lower at
larger scales. This is readily explained by gravity-dominated velocity fields
that are commonly assumed to follow a power-law distribution (Keto et al.
1988; Evans 2003; Li 2018), for instance, the free-fall velocity follows
$V_{\rm in}\sim r^{-0.5}$. Therefore, our observations are likely to trace the
large-scale inward motions of the two sources. Our spectra of W31C indicate
that inward motions may be present on scales larger than previously reported.
However, our observations can only loosely constrain the linear scale
($\lesssim$1 pc) with the beam size, so further mapping observations are
needed to better constrain the scale of inward motions.

Assuming a uniform density sphere undergoing spherically symmetric free fall
for both sources (e.g. Pineda et al. 2012), we estimate the mass infall rates
to be (3–15)$\times 10^{-3}$ $M_{\odot}$ yr-1 for W31C and (1–2)$\times
10^{-3}$ $M_{\odot}$ yr-1 for W3(OH), similar to the values estimated for
massive clumps in other studies (e.g, Schneider et al. 2010). Mass infall
rates of this magnitude are able to overcome the radiation pressure due to the
luminosity of the central star during the high-mass star formation process
(McKee & Tan 2003). The large-scale inward motions and the high mass infall
rates favour a global collapse for both sources, consistent with the global
hierarchical collapse and clump-fed scenario (e.g. Motte et al. 2018; Vázquez-
Semadeni et al. 2019) that is different from the low-mass star formation
picture (see also Appendix B).

Figure 8: Left: Posterior probability distributions of infall velocity,
$V_{\rm in}$, peak optical depth, $\tau_{0}$, velocity dispersion, $\sigma$,
excitation temperature of the “front” layer, $T_{\rm f}$, and excitation
temperature of the “rear” layer, $T_{\rm r}$ for the 37.7 GHz CH3OH transition
towards W31C. The maximum posterior possibility point in the parameter space
is shown in orange lines and points. The contours represent the 0.5, 1.0, 1.5,
and 2.0$\sigma$ confidence intervals. Right: Observed and modelled spectra for
the 37.7 GHz CH3OH transition. The blue spectrum represents the observed data,
and the red curve indicates the fitted line from the two-layer model (see Sec.
4.1). The vertical black dashed line represents the systemic velocity of
$V_{\rm LSR}$=$-$3.43 km s-1.

Figure 9: Observed and modelled spectra for the methanol transitions with
absorption features towards W31C. The short line name of each transition is
denoted in the lower left corner of each panel. The blue spectra represent the
observed data, and the red curves indicate the fitted line from the two-layer
model with adopted parameters listed in Table 5. The vertical black dashed
lines represent the systemic velocity of $V_{\rm LSR}$=$-$3.43 km s-1.

Figure 10: Observed and modelled spectra for the methanol transitions with
absorption features towards W3(OH). The short line name of each transition is
denoted in the lower left corner of each panel. The blue spectra represent the
observed data, and the red curves indicate the fitted line from the two-layer
model with adopted parameters listed in Table 5. The vertical black dashed
lines represent the systemic velocity of $V_{\rm LSR}$=$-$46.19 km s-1.

###  4.2 Methanol absorption in different environments

Our observations have revealed redshifted absorption in multiple methanol
transitions towards both W31C and W3(OH). However, absorption features are not
always observed in the same transitions for different sources. For example,
redshifted absorption features are detected in the $2_{1}-3_{0}E$ line at 19.9
GHz and the $9_{2}-10_{1}A^{+}$ line at 23.1 GHz towards W31C, but these two
transitions are found to show (class II) maser emission in W3(OH), and thus
may also affect absorption of W31C, towards which also the 6.7 and 12.1 GHz
transitions show absorption as well as maser features.

Furthermore, towards W3(OH), the nine $J_{2}-J_{1}E$ lines all show (apparent)
inverse P-Cygni profiles towards W3(OH). These resemble very closely the
profiles of the 24/25 GHz $J,K$ inversion transitions of NH3 observed towards
this region (like us) with the Effelsberg 100 m telescope by Wilson et al.
(1978); Mauersberger et al. (1986). Mapping of the $J,K=(5,5)$ line by
Mauersberger et al. (1986) (with the $38^{\prime\prime}$ Effelsberg beam)
showed that the emission component arises from a position $\approx
5^{\prime\prime}$ east of the absorption, which arises from the UCHII region.
Consequently, Mauersberger et al. (1986) persuasively argued that the emission
arises from the hot core(s) discussed in Sec. 1, which actually was one of the
very first hot molecular cores ever identified (by Turner & Welch 1984) at the
site of H2O maser emission in the region (Forster et al. 1977).

Because of this source-by-source variation, it might not be straightforward to
use a single methanol transition to systematically search for infall motions
towards high-mass star formation regions. Abundant methanol transitions that
are accessible by ground-based telescopes cover a broad range of energy levels
and critical densities (see Table 1). This allows us to investigate inward
motions at different scales and for different environments. Therefore,
observations of multiple methanol transitions provide a suitable choice for
such studies.

Based on K-band single-dish observations, NGC7538 IRS1 shows methanol maser
emission and absorption features similar to W3(OH), but the absorption
velocities with respect to their respective systemic velocities are different.
In both sources, absorption features have been detected in the $J_{2}-J_{1}E$
($J$=2,3,4,5,6,9) lines near 25 GHz (Menten et al. 1986a), the
$10_{1}-9_{2}A^{-}$ line at 23.4 GHz, and the $11_{1}-10_{2}A^{+}$ line at
20.1 GHz (Menten et al. 1986b), while maser emission has been detected in the
$2_{1}-3_{0}E$ line at 19.9 GHz (Wilson et al. 1985) and the
$9_{2}-10_{1}A^{+}$ line at 23.1 GHz (Wilson et al. 1984). The systemic
velocity of NGC7538 IRS1 is about $-$57.4 km s-1 derived from the
$2_{K}-1_{K}$ thermal methanol quartet lines near 96 GHz (Minier & Booth
2002). The methanol transitions at 20.1, 23.4 GHz and six $J_{2}-J_{1}E$ lines
near 25 GHz show absorption and the velocities of these absorption features
range from $-$59.9 to $-$59.1 km s-1, revealing that the absorption features
are actually blueshifted with respect to the systemic velocity. This result is
also in line with previous NH3 (1,1), (2,2), (3,3) measurements (e.g. Wilson
et al. 1983; Henkel et al. 1984; Keto 1991). These blueshifted absorption
lines are indicative of expansion in NGC7538 IRS1 which is likely caused by
its associated powerful outflows (e.g. Qiu et al. 2011). This is in stark
contrast to our observed redshifted absorption in W3(OH). This indicates that
these two sources have very different dominant large-scale motions on the
clump-scales probed by single-dish observations. Therefore, our single-dish
methanol observations support that both W31C and W3(OH) host ongoing large-
scale inward motions, although both sources also show outflows on smaller
scales (Liu et al. 2010a; Zapata et al. 2011; Qin et al. 2016). On the other
hand, higher angular resolution ($\sim$0$\hbox to0.0pt{.\hss}\arcsec$2)
observations of the high-energy line HCN (4$-$3) $v_{2}$=1 ($E_{\rm up}$ =
1050 K) shows redshifted absorption towards NGC7538 IRS1, indicating infall
motions in its innermost regions (Beuther et al. 2013). Therefore, one might
need high energy and high critical density methanol transitions to disentangle
the infall motions and outflows in sources exhibiting complex velocity
structures. Nevertheless, the single-dish redshifted methanol absorption
measurements have been demonstrated to be a good tracer of the large-scale
inward motions for high-mass star-formation regions, providing motivation for
follow-up studies with higher angular resolution observations.

It is important to note that CH3OH absorption in the 6.7 and 12.1 GHz lines
has also been detected in low-mass star formation regions including TMC1,
L183, and NGC 1333 (Walmsley et al. 1988; Pandian et al. 2008). These
generally do not show centimetre continuum radiation. Instead, the absorption
is produced by overcooling against the cosmic microwave background (see Sec.
1). In these sources the absorption dips are at the systemic velocity, that
is, the absorption is neither redshifted nor blueshifted, although they are
known to show infall motions from other studies (e.g. Di Francesco et al.
2001; Schnee et al. 2007; Kirk et al. 2009). There are two possible
explanations for this. One is that the relatively compact size will cause a
significant dilution in single-dish observations. Unlike the case of high-mass
star formation regions, which can show large-scale collapse, low-mass star
formation regions usually exhibit local collapse, that is, collapse only
becomes prominent towards compact dense cores. For the NGC1333 IRAS4A
observations, the beam size was about 40$\arcsec$ for the 6.7 GHz methanol
absorption observations (Pandian et al. 2008), while the collapsing region is
known to have an angular scale of $\sim$5–7$\arcsec$ from interferometric
observations of H2CO $3_{12}-2_{11}$ and N2H+ (1$-$0) (Di Francesco et al.
2001). This implies that the filling factor was only 2%–3%. On the other hand,
anti-inversion of the 6.7 GHz methanol transition is expected in a more
extended region with densities of $<10^{6}$ cm-3 (Pandian et al. 2008).
Therefore, the redshifted methanol absorption might be largely diluted by the
large beam of the single-dish observations. The second possible explanation is
that the infall velocities of these low-mass star formation regions are
$\lesssim$0.5 km s-1 (e.g. Di Francesco et al. 2001; Schnee et al. 2007),
generally lower than those of their high-mass counterparts. The low infall
velocities will result in small velocity shifts in the spectra which are
difficult to disentangle. For example, the velocity shift is as small as
$\lesssim$0.15 km s-1 in TMC1 (Schnee et al. 2007). Interferometric
observations of CH3OH absorption with high spectral resolution are therefore
required to determine whether redshifted methanol absorption can trace infall
motions towards low-mass star formation.

###  4.3 Redshifted methanol absorption is common or not in high-mass star
formation regions

As discussed in Sec. 4.1, the two studied objects contain UCHII regions with
high free-free continuum flux densities. If bright continuum sources are
required to be able to detect the absorption, then redshifted methanol
absorption can only be observed in a limited number of sources at a late
evolutionary stage where they host a bright UCHII region. In order to roughly
estimate the number of potential sources, we have used the results from the
Coordinated Radio “N” Infrared Survey for High-mass star formation (CORNISH)
project that covers 110 deg2 of the northern sky (10${}^{\circ}<l<65^{\circ}$,
$|b|<$1∘) using the VLA in B and BnA configurations at 5 GHz (Hoare et al.
2012; Purcell et al. 2013). Based on the CORNISH survey, Kalcheva et al.
(2018) compiled a catalogue of 239 UCHii regions. In this catalogue, the flux
density of W31C(G10.6234$-$0.3837) at 5 GHz is measured to be 1952.22 mJy. We
find only 10 out of 239 sources (including W31C) have flux densities of
$\geq$1952.22 mJy and if we adopt that as the threshold criterion, it implies
that redshifted methanol absorption may be detected in only 4% of the UCHii
regions, if we neglect the collisional pumping effects. It is worth noting
that the CORNISH survey is only sensitive to compact UCHii regions due to the
missing short spacing problem. Given the single-dish measurements that could
be pointed against extended Hii regions, the estimated percentage of $\sim$4%
based on the CORNISH survey should be underestimated. On the other hand, the
W31C flux threshold appears to contradict the observations of Peng & Whiteoak
(1992) who reported a detection rate of 66% (38/58) for methanol absorption at
12.2 GHz towards bright southern Hii regions and some selected dark clouds.
This suggests that the criteria of using the flux density of W31C is an
unnecessarily high threshold, however, the detection of methanol absorption
towards sources without UCHII regions suggests that collisional pumping
effects cannot be neglected in at least a handful of cases (e.g. Walmsley et
al. 1988; Pandian et al. 2008). This is also supported by our observations
that the absorption intensities differ for different transitions with similar
rest frequencies. The overcooling of methanol transitions due to collisional
pumping can greatly aid in the detection of methanol absorption.

Based on our current work, we can only give a lower limit of $\sim$4% for the
fraction of potential UCHII regions showing methanol absorption. For many
regions the overcooling of methanol transitions may also enhance absorption,
as is observed towards low-mass sources and dark clouds, but this requires
specific physical conditions (e.g. Walmsley et al. 1988; Pandian et al. 2008;
Leurini et al. 2016). The work undertaken to date suggests that redshifted
methanol absorption may be most suitable for studying infall motions towards
high-mass star formation regions hosting bright HII regions.

##  5 Summary

We have performed observations of multiple methanol transitions towards two
well-known collapsing dense clumps, W31C(G10.6$-$0.4) and W3(OH), with the
Effelsberg-100 m, IRAM-30 m, and APEX-12 m telescopes. We detected 14 and 11
redshifted methanol absorption towards W31C and W3(OH), respectively.
Redshifted methanol absorption in the 20.1, 25.541, 25.878, 37.7, 38.3, 38.5,
85.6, 86.6 and 86.9 GHz transitions are reported towards W31C for the first
time. The infall velocities estimated using a two-layer model agree with
previously reported values derived from other tracers, suggesting that
redshifted methanol absorption is a reliable tracer of infall motions within
high-mass star formation regions hosting bright HII regions. Furthermore, our
observations indicate the presence of large-scale ($\lesssim$1 pc) inward
motions, and the mass infall rates are estimated to be $\gtrsim$10-3
$M_{\odot}$ yr-1, which supports the global hierarchical collapse and clump-
fed scenario.

## ACKNOWLEDGMENTS

We greatly thank Dr. Yuxin Lin for her help on the critical density
calculations and discussions about the multi-layer view of global collapse. We
acknowledge the IRAM-30 m, APEX-12 m, and Effelsberg-100 m staff for their
assistance with our observations. IRAM is supported by INSU/CNRS (France), MPG
(Germany) and IGN (Spain). The research leading to these results has received
funding from the European Union’s Horizon 2020 research and innovation
programme under grant agreement No 730562 [RadioNet]. APEX is a collaboration
between the Max-Planck-Institut für Radioastronomie, the European Southern
Observatory, and the Onsala Space Observatory. This work is also based on
observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für
Radioastronomie) at Effelsberg. This work made use of Python libraries
including Astropy666https://www.astropy.org/ (Astropy Collaboration et al.
2013), NumPy777https://www.numpy.org/ (van der Walt et al. 2011),
SciPy888https://www.scipy.org/ (Jones et al. 2001),
Matplotlib999https://matplotlib.org/ (Hunter 2007), APLpy (Robitaille &
Bressert 2012), emcee101010https://emcee.readthedocs.io/en/stable/ (Foreman-
Mackey et al. 2013), and corner.py (Foreman-Mackey 2016). This paper was
prepared using the Overleaf111111https://www.overleaf.com web application.

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##  Appendix A Tables

Table 1: Methanol transitions.

No. |  CH3OH | Rest frequency | Line | Class | $E_{\rm up}$ | $n_{\rm crit}$ | W31C | Note
---|---|---|---|---|---|---|---|---
| Transition | (MHz) | (GHz) |  | (K) |  (cm-3) | Reference |
|  $J_{0}-(J+1)_{-1}E$ series |  |  |  |  |  |  |
1 | $0_{0}-1_{-1}E$ |  108 893.963(7)∗ | 108 | II | 13.1 |  2.28$\times$104 | this work | maser?/thermal
2 | $2_{0}-3_{-1}E$ |  12 178.597(4)∗ | 12.2 | II | 20.1 |  7.58$\times$104 |  this work,(1)$-$(3) | maser & absorption
3 | $4_{-1}-3_{0}E$ |  36 169.265(30)∗ | 36 | I | 28.8 |  5.01$\times$104 | this work,(4),(5) | maser
4 | $5_{-1}-4_{0}E$ |  84 521.169(10)∗ | 84 | I | 40.4 |  1.04$\times$105 | this work,(5) | maser
5 | $6_{-1}-5_{0}E$ |  132 890.692(10)∗ | 132.9 | I | 54.3 |  1.88$\times$105 | (6) | maser?/thermal
6 | $8_{-1}-7_{0}E$ |  229 758.760(50)∗ | 229 | I | 89.1 |  4.92$\times$105 | this work | maser?/thermal
|  $J_{-2}-(J+1)_{-1}E$ series |  |  |  |  |  |  |
7 | $3_{-2}-4_{-1}E$ |  230 027.047(11)α | 230 | - | 39.8 |  2.93$\times$106 | this work | thermal
8 | $5_{-2}-6_{-1}E$ |  133 605.439(11)α | 133.6 | II | 60.7 |  3.19$\times$106 | (6) | maser?/thermal
9 | $6_{-2}-7_{-1}E$ |  85 568.084(10)∗ | 85.6 | II | 74.7 |  3.54$\times$106 | this work | inverse P-Cygni
10 | $7_{-2}-8_{-1}E$ |  37 703.700(30)∗ | 37.7 | II | 90.9 |  3.98$\times$106 | this work | absorption
11 | $11_{-1}-10_{-2}E$ |  104 300.414(7)∗ | 104 | I | 158.6 |  1.49$\times$106 | this work | maser?/thermal
|  $J_{1}-(J+1)_{0}E$ series |  |  |  |  |  |  |
12 | $2_{1}-3_{0}E$ |  19 967.3961(2)∗ | 19.9 | II | 28.0 |  1.53$\times$105 | this work,(7) |  inverse P-Cygni(a)
13 | $8_{0}-7_{1}E$ |  220 078.561(8)α | 220 | - | 96.6 |  1.25$\times$106 | this work | thermal
|  $J_{2}-J_{1}E$ @25 GHz series |  |  |  |  |  |  |
14 | $2_{2}-2_{1}E$ |  24 934.3801(16)β | 24.934 | I | 29.2 |  2.85$\times$104 | this work,(8) |  maser?/thermal(b)
15 | $3_{2}-3_{1}E$ |  24 928.6994(8)β | 24.928 | I | 36.3 |  6.76$\times$104 | this work,(8) |  maser?/thermal(b)
16 | $4_{2}-4_{1}E$ |  24 933.4693(8)β | 24.933 | I | 45.5 |  1.31$\times$105 | this work,(8) |  maser?/thermal(b)
17 | $5_{2}-5_{1}E$ |  24 959.0789(4)∗ | 24.959 | I | 57.1 |  2.29$\times$105 | this work,(8) |  maser(b)
18 | $6_{2}-6_{1}E$ |  25 018.1225(4)∗ | 25.018 | I | 71.0 |  3.71$\times$105 | this work,(8) |  maser(b)
19 | $7_{2}-7_{1}E$ |  25 124.8719(4)∗ | 25.124 | I | 87.3 |  5.69$\times$105 | this work |  maser(b)
20 | $8_{2}-8_{1}E$ |  25 294.4165(2)∗ | 25.294 | I | 105.8 |  8.14$\times$105 | this work |  maser(b)
21 | $9_{2}-9_{1}E$ |  25 541.3979(4)∗ | 25.541 | I | 126.7 |  1.12$\times$106 | this work |  inverse P-Cygni(b)
22 | $10_{2}-10_{1}E$ |  25 878.2661(4)∗ | 25.878 | I | 150.0 |  1.53$\times$106 | this work |  inverse P-Cygni(b)
23 | $5_{1}-4_{2}E$ |  216 945.521(12)α | 216 | II | 55.9 |  4.84$\times$105 | this work | maser?/thermal
24 | $4_{2}-3_{1}E$ |  218 440.063(13)α | 218 | I | 45.5 |  1.31$\times$105 | this work | maser?/thermal
|  $J_{1}-(J+1)_{0}A^{+}$ series |  |  |  |  |  |  |
25 | $3_{1}-4_{0}A^{+}$ |  107 013.803(5)∗ | 107 | II | 28.3 |  9.17$\times$105 | this work | maser?/thermal
26 | $5_{1}-6_{0}A^{+}$ |  6 668.5188(4)β | 6.7 | II | 49.1 |  1.66$\times$106 |  this work,(9)$-$(12) | maser & absorption
27 | $7_{0}-6_{1}A^{+}$ |  44 069.410(10)∗ | 44 | I | 65.0 |  2.80$\times$105 |  this work,(13)$-$(15) | maser
28 | $8_{0}-7_{1}A^{+}$ |  95 169.463(10)∗ | 95 | I | 83.5 |  4.29$\times$105 | this work,(16) | maser
|  $(J+1)_{2}-J_{3}A^{\mp}$ series |  |  |  |  |  |  |
29 | $6_{2}-5_{3}A^{-}$ |  38 293.268(50)∗ | 38.3 | II | 86.5 |  3.76$\times$106 | this work | absorption
30 | $6_{2}-5_{3}A^{+}$ |  38 452.677(50)∗ | 38.5 | II | 86.5 |  3.81$\times$106 | this work | absorption
31 | $7_{2}-6_{3}A^{-}$ |  86 615.600(5)∗ | 86.6 | II | 102.7 |  4.30$\times$106 | this work | absorption
32 | $7_{2}-6_{3}A^{+}$ |  86 902.949(5)∗ | 86.9 | II | 102.7 |  4.26$\times$106 | this work | absorption
33 | $10_{2}-9_{3}A^{-}$ |  231 281.100(12)α | 231 | II | 165.3 |  6.22$\times$106 | this work | maser?/thermal
34 | $10_{2}-9_{3}A^{+}$ |  232 418.521(12)α | 232 | II | 165.4 |  6.36$\times$106 | this work | maser?/thermal
|  $J_{2}-(J+1)_{1}A^{-}$ series |  |  |  |  |  |  |
35 | $4_{2}-5_{1}A^{-}$ |  234 683.370(12)α | 234 | - | 60.9 |  2.99$\times$106 | this work | thermal
36 | $6_{2}-7_{1}A^{-}$ |  132 621.824(12)α | 132.6 | II | 86.5 |  3.76$\times$106 | (6) | maser?/thermal
37 | $10_{1}-9_{2}A^{-}$ |  23 444.778(2)γ | 23.4 | I | 143.3 |  1.44$\times$106 | this work,(17) |  absorption(c)
|  $J_{2}-(J+1)_{1}A^{+}$ series |  |  |  |  |  |  |
38 | $7_{2}-8_{1}A^{+}$ |  111 289.550(10)∗ | 111 | - | 102.7 |  4.26$\times$106 | this work | thermal
39 | $9_{2}-10_{1}A^{+}$ |  23 121.0242(5)∗ | 23.1 | II | 142.2 |  5.68$\times$106 | this work,(18) |  absorption(a)
40 | $11_{1}-10_{2}A^{+}$ |  20 171.089(2)γ | 20.1 | - | 166.4 |  6.50$\times$106 | this work |  absorption(c)
|  $(J+1)_{1}-J_{1}A^{\pm}$ series |  |  |  |  |  |  |
41 | $2_{1}-1_{1}A^{+}$ |  95 914.309(5)∗ | 95.914 | - | 21.4 |  7.04$\times$105 | - |  -(d)
42 | $2_{1}-1_{1}A^{-}$ |  97 582.804(7)∗ | 97.582 | - | 21.6 |  5.14$\times$105 | - |  -(d)
43 | $3_{1}-2_{1}A^{+}$ |  143 865.801(10)∗ | 143.865 | - | 28.3 |  9.17$\times$105 | - |  -(d)

121212Column 1 gives the serial number of each transition (for reference in
the text). Columns 2–4 give the transition, rest frequency and short line name
of each methanol transition, respectively. The frequencies of methanol
transitions are adopted from Müller et al. (2004), marked by an asterisk, and
references for the other methanol transitions are: α the CDMS database (Müller
et al. 2005); β Coudert et al. (2015); γ Mehrotra et al. (1985). The frequency
uncertainties are given in parentheses in units of the least significant
figure. Column 5 shows the classification of methanol maser transitions.
Column 6 gives the upper level energy of each transition. We note that the
$E$-type ground state (1-1) level is at 7.9 K. Column 7 gives the critical
density of each transition which is calculated by assuming optically thin
transitions and a gas kinetic temperature of 100 K. The corresponding code is
available from https://github.com/yxlinaqua/molecule_basic. Column 8 lists the
reference for W31C: (1) Caswell et al. (1995a); (2) Breen et al. (2010); (3)
Breen et al. (2014); (4) Haschick & Baan (1989); (5) Breen et al. (2019); (6)
Slysh et al. (1999); (7) Wilson et al. (1985); (8) Menten et al. (1986a); (9)
Menten (1991b); (10) Caswell et al. (1995b); (11) Caswell (2009); (12) Green
et al. (2010); (13) Forster et al. (1990); (14) Kurtz et al. (2004); (15) Kim
et al. (2019); (16) Chen et al. (2012); (17) Menten et al. (1986b); (18)
Wilson et al. (1984). Column 9 gives the note for W31C, and the note for the
corresponding transitions towards W3(OH) can be found here: (a) maser; (b) all
observed $J_{2}-J_{1}E$ lines near 25 GHz are shown inverse P-Cygni profile;
(c) absorption; (d) thermal emission.

Table 2: Observational parameters for W31C and W3(OH).

Line | Obs. date | beam size | $V_{\rm res}$ | rms | $\eta_{\rm mb}$ | Scaling factor
---|---|---|---|---|---|---
(GHz) | (yyyymmdd) | ($\arcsec$) | (km s-1) | (Jy) |  | (Jy/K)
W31C
Effelsberg-100 m observations |  |  |  |  |
6.7 | 20210105 | 109 | 0.16 | 0.02 | 0.69 | 0.66
12.2 | 20210105 | 62 | 0.13 | 0.02 | 0.67 | 0.70
19.9 | 20210514 | 40 | 0.66 | 0.01 | 0.62 | 0.83
20.1 | 20210514 | 40 | 0.66 | 0.01 | 0.62 | 0.83
23.1 | 20210514 | 36 | 0.57 | 0.02 | 0.60 | 0.97
23.4 | 20210514 | 36 | 0.57 | 0.02 | 0.60 | 0.97
25 series | 20210514 | 34 | 0.53 | 0.02 | 0.58 | 1.02
36 | 20200914 | 24 | 0.37 | 0.04 | 0.47 | 1.19
37.7 | 20200914 | 23 | 0.35 | 0.04 | 0.44 | 1.27
38.3 | 20200914 | 23 | 0.35 | 0.04 | 0.44 | 1.27
38.5 | 20200914 | 23 | 0.34 | 0.05 | 0.44 | 1.27
44 | 20200530 | 20 | 0.30 | 0.04 | 0.39 | 1.47
IRAM-30 m observations |  |  |  |  |
84 | 20110408 | 29 | 0.80 | 0.26 | 0.81 | 5.82
85.6 | 20200629 | 29 | 0.79 | 0.08 | 0.81 | 5.82
86.6 | 20200629 | 28 | 0.78 | 0.09 | 0.81 | 5.82
86.9 | 20200629 | 28 | 0.78 | 0.08 | 0.81 | 5.82
95 | 20110409 | 26 | 0.71 | 0.16 | 0.81 | 5.82
104 | 20200629 | 24 | 0.65 | 0.10 | 0.78 | 5.98
107 | 20200629 | 23 | 0.61 | 0.13 | 0.78 | 5.98
108 | 20110411 | 23 | 0.62 | 0.20 | 0.78 | 5.98
111 | 20110411 | 22 | 0.61 | 0.29 | 0.78 | 5.98
APEX-12 m observations |  |  |  |  |
216 | 20190612 | 29 | 0.17 | 0.60 | 0.72 | 40
218 | 20190612 | 29 | 0.17 | 0.71 | 0.72 | 40
220 | 20190612 | 28 | 0.17 | 0.64 | 0.72 | 40
229 | 20190601 | 27 | 0.16 | 0.74 | 0.72 | 40
230 | 20190601 | 27 | 0.16 | 0.72 | 0.72 | 40
231 | 20190601 | 27 | 0.16 | 1.14 | 0.72 | 40
232 | 20190601 | 27 | 0.16 | 0.73 | 0.72 | 40
234 | 20190601 | 27 | 0.16 | 0.88 | 0.72 | 40
W3(OH)
Effelsberg-100 m observations |  |  |  |  |
19.9 | 20210513 | 40 | 0.66 | 0.01 | 0.62 | 0.83
20.1 | 20210513 | 40 | 0.66 | 0.01 | 0.62 | 0.83
23.1 | 20210513 | 36 | 0.57 | 0.01 | 0.60 | 0.97
23.4 | 20210513 | 36 | 0.57 | 0.01 | 0.60 | 0.97
25 series | 20210513 | 34 | 0.53 | 0.01 | 0.58 | 1.02
IRAM-30 m observations |  |  |  |  |
95.914 | 20190629 | 26 | 0.71 | 0.03 | 0.81 | 5.82
97.582 | 20190629 | 25 | 0.70 | 0.02 | 0.81 | 5.82
143.865 | 20190629 | 17 | 0.47 | 0.03 | 0.73 | 6.32
131313 The scaling factor is used for the conversion between antenna
temperature and flux density. For APEX observations, the velocity resolution
is given after once Hanning smoothing. For information on the Effelsberg-100
m: https://eff100mwiki.mpifr-
bonn.mpg.de/doku.php?id=information_for_astronomers:rx_list, for the IRAM-30
m: https://publicwiki.iram.es/Iram30mEfficiencies, for the APEX telescope:
http://www.apex-
telescope.org/telescope/efficiency/index.php?orderBy=mJyK&planetBy=all&yearBy=2019&sortAs=DESC

Table 3: Observational results of CH3OH transitions in W31C.

Line | $V_{\rm lsr}$ | $\Delta V$ | $S_{\rm pk}$ | $\int S{\rm d}V$
---|---|---|---|---
(GHz) |  (km s-1) |  (km s-1) | (Jy) |  (Jy km s-1)
6.7 |  $-$8.19 (0.01) | 0.59 (0.03) | 1.42 (0.21) | 0.89 (0.04)
6.7 |  $-$1.93 (0.07) | 3.34 (0.16) |  $-$0.65 (0.21) |  $-$2.31 (0.09)
6.7 | 4.78 (0.01) | 0.88 (0.02) | 3.70 (0.21) | 3.45 (0.05)
12.2 |  $-$1.23 (0.03) | 5.40 (0.05) |  $-$0.42 (0.05) |  $-$2.39 (0.02)
12.2 | 4.84 (0.01) | 0.96 (0.02) | 0.53 (0.05) | 0.55 (0.01)
19.9 |  $-$0.90 (0.11) | 3.99 (0.21) |  $-$0.11 (0.01) |  $-$0.46 (0.03)
20.1 |  $-$0.40 (0.31) | 4.18 (0.63) |  $-$0.06 (0.02) |  $-$0.27 (0.03)
23.1 |  $-$0.74 (0.01) | 4.69 (0.16) |  $-$0.23 (0.02) |  $-$1.15 (0.04)
23.4 |  $-$0.51 (0.22) | 3.87 (0.49) |  $-$0.13 (0.03) |  $-$0.54 (0.05)
24.928 |  $-$9.25 (0.13) | 1.34 (0.36) | 0.04 (0.01) | 0.06 (0.02)
24.928 |  $-$5.18 (0.03) | 4.88 (0.31) | 0.24 (0.01) | 1.26 (0.07)
24.928 |  $-$1.90 (0.07) | 2.37 (0.15) | 0.18 (0.01) | 0.46 (0.04)
24.928 | 2.08 (0.13) | 3.31 (0.32) | 0.11 (0.01) | 0.37 (0.03)
24.933 |  $-$8.04 (0.01) | 2.79 (0.29) | 0.15 (0.01) | 0.44 (0.03)
24.933 |  $-$5.40 (0.04) | 2.71 (0.18) | 0.23 (0.01) | 0.66 (0.04)
24.933 |  $-$2.90 (0.01) | 2.19 (0.13) | 0.25 (0.01) | 0.59 (0.04)
24.933 |  $-$0.52 (0.05) | 2.74 (0.48) | 0.15 (0.01) | 0.45 (0.07)
24.933 | 2.64 (0.26) | 1.85 (0.40) | 0.07 (0.01) | 0.14 (0.04)
24.934 |  $-$4.77 (0.14) | 5.85 (0.31) | 0.16 (0.01) | 0.98 (0.05)
24.934 |  $-$1.99 (0.12) | 1.03 (0.46) | 0.06 (0.01) | 0.06 (0.04)
24.959 |  $-$8.02 (0.11) | 1.57 (0.18) | 0.13 (0.01) | 0.22 (0.01)
24.959 |  $-$5.98 (0.04) | 1.89 (0.20) | 0.28 (0.01) | 0.56 (0.07)
24.959 |  $-$2.80 (0.15) | 3.74 (0.17) | 0.28 (0.01) | 1.11 (0.03)
24.959 | 2.09 (0.14) | 5.36 (1.01) | 0.08 (0.01) | 0.48 (0.07)
25.018 |  $-$8.00 (0.13) | 2.20 (0.32) | 0.12 (0.01) | 0.28 (0.03)
25.018 |  $-$5.97 (0.06) | 1.12 (0.15) | 0.30 (0.01) | 0.35 (0.07)
25.018 |  $-$4.44 (0.11) | 1.67 (0.42) | 0.19 (0.01) | 0.33 (0.12)
25.018 |  $-$2.18 (0.14) | 2.49 (0.32) | 0.21 (0.01) | 0.56 (0.09)
25.018 | 2.21 (0.24) | 3.04 (0.62) | 0.07 (0.01) | 0.22 (0.04)
25.124 |  $-$7.78 (0.12) | 1.69 (0.75) | 0.08 (0.01) | 0.15 (0.04)
25.124 |  $-$5.98 (0.05) | 1.08 (0.17) | 0.20 (0.01) | 0.23 (0.05)
25.124 |  $-$2.92 (0.11) | 4.14 (0.33) | 0.19 (0.01) | 0.82 (0.05)
25.124 | 3.10 (0.17) | 2.59 (0.46) | 0.08 (0.01) | 0.22 (0.03)
25.294 |  $-$4.78 (0.14) | 5.30 (0.27) | 0.12 (0.02) | 0.67 (0.03)
25.541 |  $-$4.93 (0.18) | 4.83 (0.60) | 0.10 (0.01) | 0.49 (0.05)
25.541 | 0.20 (0.18) | 2.21 (0.32) |  $-$0.07 (0.01) |  $-$0.17 (0.04)
25.878 |  $-$5.97 (0.17) | 3.10 (0.34) | 0.07 (0.01) | 0.24 (0.03)
25.878 | 0.10 (0.14) | 2.73 (0.31) |  $-$0.09 (0.01) |  $-$0.26 (0.03)
36 |  $-$7.35 (0.04) | 2.64 (0.03) | 3.52 (0.41) | 9.91 (0.25)
36 |  $-$6.19 (0.03) | 1.43 (0.01) | 12.70 (0.41) | 19.31 (0.31)
36 |  $-$2.84 (0.03) | 7.59 (0.15) | 1.69 (0.41) | 13.67 (0.21)
37.7 |  $-$0.68 (0.05) | 5.35 (0.11) |  $-$0.89 (0.05) |  $-$5.05 (0.09)
38.3 |  $-$0.74 (0.10) | 4.77 (0.22) |  $-$0.38 (0.04) |  $-$1.94 (0.08)
38.5 |  $-$0.65 (0.14) | 4.81 (0.28) |  $-$0.30 (0.06) |  $-$1.54 (0.08)
44 |  $-$7.26 (0.03) | 2.86 (0.05) | 7.58 (0.69) | 23.11 (0.03)
44 |  $-$6.65 (0.00) | 0.76 (0.01) | 44.87 (0.69) | 36.51 (0.33)
84 |  $-$6.94 (0.03) | 2.36 (0.09) | 9.05 (0.39) | 22.70 (1.16)
84 |  $-$3.82 (0.08) | 8.41 (0.13) | 9.77 (0.39) | 87.38 (1.64)
85.6 | 0.22 (0.19) | 3.80 (0.35) |  $-$0.55 (0.08) |  $-$2.22 (0.21)
86.6 | 0.20 (0.56) | 3.79 (1.12) |  $-$0.17 (0.10) |  $-$0.70 (0.20)
86.9 | 0.43 (0.31) | 3.72 (0.88) |  $-$0.29 (0.07) |  $-$1.17 (0.21)
95 |  $-$7.00 (0.01) | 2.33 (0.03) | 13.87 (0.70) | 34.32 (0.55)
95 |  $-$3.65 (0.07) | 9.29 (0.09) | 6.13 (0.70) | 60.54 (0.80)
104 |  $-$2.88 (0.18) | 5.81 (0.51) | 0.75 (0.12) | 4.65 (0.33)
107 |  $-$3.95 (0.04) | 7.00 (0.11) | 6.03 (0.32) | 44.98 (0.58)
108 |  $-$4.00 (0.05) | 7.83 (0.12) | 6.41 (0.28) | 53.43 (0.64)
111 |  $-$4.13 (0.26) | 4.23 (0.68) | 1.32 (0.24) | 5.96 (0.75)
216 |  $-$3.26 (0.05) | 7.26 (0.12) | 14.10 (0.79) | 108.72 (1.47)
218 |  $-$3.33 (0.02) | 7.54 (0.02) | 82.92 (1.70) | 665.33 (1.36)
220 |  $-$2.84 (0.04) | 7.56 (0.09) | 14.30 (0.86) | 114.85 (1.15)
229 |  $-$3.47 (0.01) | 8.18 (0.03) | 43.30 (1.09) | 376.80 (1.24)
230 |  $-$3.21 (0.05) | 6.96 (0.12) | 10.00 (0.78) | 74.41 (1.14)
231 |  $-$2.35 (0.23) | 6.43 (0.52) | 3.37 (0.91) | 23.12 (1.72)
232 |  $-$2.83 (0.17) | 6.87 (0.47) | 3.45 (0.74) | 25.19 (1.35)
234 |  $-$2.92 (0.06) | 7.03 (0.15) | 11.41 (0.99) | 84.74 (1.38)
141414Focusing on the absorption, we fitted single Gaussian profile to only to
the absorption component of 19.9, 20.1, 23.4, 37.7, 38.3, 38.5, 85.6, 86.6 and
86.9 GHz lines.

Table 4: Observational results of CH3OH transitions in W3(OH). Line | $V_{\rm lsr}$ | $\Delta V$ | $S_{\rm pk}$ | $\int S{\rm d}V$
---|---|---|---|---
(GHz) |  (km s-1) |  (km s-1) | (Jy) |  (Jy km s-1)
19.9 |  $-$43.53 (0.00) | 1.61 (0.01) | 30.16 (0.96) | 51.65 (0.10)
20.1 |  $-$44.40 (0.03) | 2.31 (0.07) |  $-$0.21 (0.01) |  $-$0.51 (0.01)
23.1 |  $-$43.36 (0.00) | 1.45 (0.01) | 2.86 (0.15) | 4.42 (0.03)
23.4 |  $-$46.41 (0.47) | 19.84 (1.02) | 0.07 (0.01) | 1.53 (0.07)
23.4 |  $-$44.47 (0.03) | 2.16 (0.08) |  $-$0.36 (0.01) |  $-$0.83 (0.03)
24.928 |  $-$46.43 (0.24) | 4.56 (0.34) | 0.19 (0.02) | 0.93 (0.11)
24.928 |  $-$44.62 (0.04) | 2.44 (0.09) |  $-$0.49 (0.02) |  $-$1.27 (0.11)
24.933 |  $-$46.78 (0.19) | 4.87 (0.28) | 0.16 (0.03) | 0.84 (0.07)
24.933 |  $-$44.62 (0.02) | 2.26 (0.07) |  $-$0.46 (0.03) |  $-$1.10 (0.06)
24.934 |  $-$45.20 (0.18) | 6.26 (0.50) | 0.14 (0.02) | 0.95 (0.09)
24.934 |  $-$44.51 (0.03) | 2.51 (0.08) |  $-$0.53 (0.02) |  $-$1.42 (0.08)
24.959 |  $-$47.50 (0.26) | 6.51 (0.41) | 0.14 (0.02) | 0.94 (0.07)
24.959 |  $-$44.59 (0.03) | 2.12 (0.09) |  $-$0.40 (0.02) |  $-$0.90 (0.06)
25.018 |  $-$47.24 (0.21) | 6.58 (0.34) | 0.14 (0.02) | 1.00 (0.06)
25.018 |  $-$44.65 (0.03) | 2.21 (0.09) |  $-$0.37 (0.02) |  $-$0.87 (0.05)
25.124 |  $-$47.66 (0.30) | 7.45 (0.57) | 0.10 (0.01) | 0.82 (0.07)
25.124 |  $-$44.63 (0.04) | 2.14 (0.10) |  $-$0.29 (0.01) |  $-$0.66 (0.04)
25.294 |  $-$47.36 (0.42) | 6.66 (0.53) | 0.09 (0.02) | 0.66 (0.08)
25.294 |  $-$44.64 (0.04) | 2.14 (0.13) |  $-$0.30 (0.02) |  $-$0.69 (0.06)
25.541 |  $-$47.90 (0.36) | 7.94 (0.56) | 0.07 (0.02) | 0.58 (0.05)
25.541 |  $-$44.62 (0.03) | 2.17 (0.09) |  $-$0.28 (0.02) |  $-$0.65 (0.03)
25.878 |  $-$48.12 (0.39) | 7.39 (0.59) | 0.08 (0.01) | 0.64 (0.06)
25.878 |  $-$44.57 (0.04) | 2.15 (0.10) |  $-$0.29 (0.01) |  $-$0.66 (0.05)
95.914 |  $-$46.19 (0.01) | 4.81 (0.03) | 4.72 (0.39) | 24.13 (0.13)
97.865 |  $-$46.23 (0.01) | 4.76 (0.03) | 5.01 (0.42) | 25.39 (0.11)
143.865 |  $-$46.15 (0.01) | 4.81 (0.02) | 11.13 (0.73) | 56.91 (0.15)
Table 5: Two-layer modelling results for the methanol transitions with
redshifted absorption feature towards W31C and W3(OH).

Line | Fixed parameters | Output parameters
---|---|---
| $T_{\rm c}$ | $V_{\rm in}$ | $\tau_{0}$ | $\sigma$ | $T_{\rm f}$ | $T_{\rm r}$
(GHz) | (K) | (km s-1) |  | (km s-1) | (K) | (K)
W31C
6.7 | 6.83 | 1.93${}^{+0.04}_{-0.03}$ | 0.15${}^{+0.07}_{-0.04}$ | 1.88${}^{+0.04}_{-0.05}$ | 1.41${}^{+0.58}_{-0.30}$ | 57.36${}^{+31.46}_{-33.22}$
12.2 | 6.39 | 2.17${}^{+0.03}_{-0.03}$ | 0.24${}^{+0.05}_{-0.03}$ | 2.04${}^{+0.02}_{-0.02}$ | 1.85${}^{+0.53}_{-0.51}$ | 2.05${}^{+1.83}_{-0.79}$
19.9 | 5.53 | 2.04${}^{+0.63}_{-0.68}$ | 0.25${}^{+0.26}_{-0.10}$ | 2.06${}^{+0.41}_{-0.40}$ | 7.35${}^{+3.01}_{-2.17}$ | 39.41${}^{+36.41}_{-27.49}$
20.1 | 5.42 | 2.72${}^{+0.13}_{-0.13}$ | 0.11${}^{+0.01}_{-0.00}$ | 2.09${}^{+0.10}_{-0.10}$ | 3.82${}^{+0.31}_{-0.14}$ | 2.50${}^{+2.27}_{-1.12}$
23.1 | 5.03 | 2.68${}^{+0.09}_{-0.09}$ | 0.14${}^{+0.06}_{-0.03}$ | 1.95${}^{+0.08}_{-0.07}$ | 2.79${}^{+0.72}_{-0.77}$ | 9.88${}^{+11.43}_{-6.47}$
23.4 | 4.90 | 2.95${}^{+0.16}_{-0.16}$ | 0.11${}^{+0.02}_{-0.01}$ | 1.58${}^{+0.15}_{-0.15}$ | 2.60${}^{+0.35}_{-0.29}$ | 3.08${}^{+3.33}_{-1.56}$
25.541 | 4.62 | 1.07${}^{+0.23}_{-0.21}$ | 0.31${}^{+0.08}_{-0.05}$ | 1.93${}^{+0.14}_{-0.14}$ | 11.83${}^{+0.74}_{-1.29}$ | 88.35${}^{+8.52}_{-14.81}$
25.878 | 4.53 | 3.07${}^{+0.15}_{-0.18}$ | 0.39${}^{+0.99}_{-0.22}$ | 1.22${}^{+0.16}_{-0.15}$ | 6.28${}^{+3.68}_{-2.03}$ | 29.61${}^{+44.67}_{-24.72}$
37.7 | 4.52 | 2.73${}^{+0.05}_{-0.05}$ | 1.40${}^{+0.47}_{-0.43}$ | 1.91${}^{+0.09}_{-0.08}$ | 2.46${}^{+0.27}_{-0.30}$ | 4.12${}^{+4.73}_{-2.27}$
38.3 | 4.38 | 2.52${}^{+0.13}_{-0.13}$ | 0.39${}^{+1.13}_{-0.20}$ | 2.00${}^{+0.14}_{-0.23}$ | 5.16${}^{+2.21}_{-1.45}$ | 37.05${}^{+41.65}_{-29.37}$
38.5 | 4.35 | 2.60${}^{+0.17}_{-0.14}$ | 3.57${}^{+4.84}_{-3.40}$ | 1.53${}^{+0.53}_{-0.28}$ | 3.54${}^{+0.66}_{-0.15}$ | 2.84${}^{+31.72}_{-1.48}$
85.6 | 0.48 | 3.47${}^{+0.20}_{-0.19}$ | 2.21${}^{+4.75}_{-2.02}$ | 1.42${}^{+0.37}_{-0.29}$ | 3.27${}^{+1.82}_{-0.17}$ | 19.93${}^{+23.69}_{-1.81}$
86.6 | 0.48 | 1.63${}^{+1.24}_{-0.73}$ | 0.17${}^{+0.17}_{-0.06}$ | 2.03${}^{+0.49}_{-0.52}$ | 6.24${}^{+2.25}_{-1.90}$ | 51.23${}^{+25.50}_{-20.75}$
86.9 | 0.48 | 3.89${}^{+0.28}_{-0.31}$ | 0.25${}^{+1.22}_{-0.12}$ | 1.26${}^{+0.40}_{-0.32}$ | 4.32${}^{+1.62}_{-1.11}$ | 32.12${}^{+19.18}_{-13.20}$
W3(OH)
20.1 | 5.82 | 1.78${}^{+0.02}_{-0.02}$ | 0.13${}^{+0.03}_{-0.02}$ | 0.95${}^{+0.02}_{-0.02}$ | 2.90${}^{+0.42}_{-0.46}$ | 9.41${}^{+8.03}_{-5.79}$
23.4 | 5.32 | 1.75${}^{+0.04}_{-0.04}$ | 1.26${}^{+0.79}_{-0.72}$ | 0.65${}^{+0.05}_{-0.05}$ | 4.62${}^{+1.41}_{-0.32}$ | 7.05${}^{+20.61}_{-4.78}$
24.928 | 5.28 | 1.77${}^{+0.02}_{-0.02}$ | 3.18${}^{+0.74}_{-0.66}$ | 0.68${}^{+0.03}_{-0.03}$ | 5.23${}^{+0.19}_{-0.15}$ | 13.00${}^{+2.41}_{-1.86}$
24.933 | 5.27 | 1.74${}^{+0.02}_{-0.02}$ | 4.37${}^{+1.08}_{-0.86}$ | 0.62${}^{+0.03}_{-0.03}$ | 5.07${}^{+0.14}_{-0.12}$ | 11.15${}^{+1.65}_{-1.41}$
24.934 | 5.27 | 1.70${}^{+0.02}_{-0.02}$ | 1.51${}^{+0.56}_{-0.50}$ | 0.73${}^{+0.04}_{-0.04}$ | 4.54${}^{+0.49}_{-0.27}$ | 8.05${}^{+7.29}_{-3.93}$
24.959 | 5.26 | 1.74${}^{+0.02}_{-0.02}$ | 3.70${}^{+1.85}_{-1.50}$ | 0.65${}^{+0.07}_{-0.05}$ | 5.25${}^{+0.40}_{-0.22}$ | 12.24${}^{+4.94}_{-2.63}$
25.018 | 5.24 | 1.72${}^{+0.03}_{-0.03}$ | 4.53${}^{+1.67}_{-1.34}$ | 0.65${}^{+0.05}_{-0.04}$ | 5.28${}^{+0.20}_{-0.14}$ | 11.93${}^{+2.63}_{-1.61}$
25.124 | 5.20 | 1.62${}^{+0.05}_{-0.06}$ | 0.68${}^{+2.53}_{-0.37}$ | 0.83${}^{+0.09}_{-0.18}$ | 7.20${}^{+3.97}_{-2.38}$ | 36.31${}^{+49.27}_{-29.52}$
25.294 | 5.12 | 1.68${}^{+0.06}_{-0.06}$ | 2.14${}^{+2.40}_{-1.66}$ | 0.71${}^{+0.13}_{-0.10}$ | 4.65${}^{+2.80}_{-0.33}$ | 6.44${}^{+36.47}_{-4.19}$
25.541 | 5.03 | 1.60${}^{+0.05}_{-0.05}$ | 0.76${}^{+1.27}_{-0.52}$ | 0.81${}^{+0.07}_{-0.09}$ | 5.40${}^{+4.19}_{-1.14}$ | 18.47${}^{+57.08}_{-15.48}$
25.878 | 4.89 | 1.72${}^{+0.04}_{-0.04}$ | 1.90${}^{+1.32}_{-1.34}$ | 0.72${}^{+0.11}_{-0.07}$ | 4.69${}^{+2.14}_{-0.33}$ | 9.23${}^{+28.17}_{-4.23}$
151515$V_{\rm in}$ is the infall velocity, $\tau_{0}$ is the peak optical
depth of each layer, $\sigma$ is the velocity dispersion, $T_{\rm f}$ and
$T_{\rm r}$ are the excitation temperatures of the “front” and “rear” layers,
respectively. $\Phi$= 0.9 and $T_{\rm b}$=2.73 K are fixed for modelling the
lines in both W31C and W3(OH). $V_{\rm LSR}$ are fixed at $-$3.43 km s-1 and
$-$46.19 km s-1 for modelling the lines in W31C and W3(OH), respectively.

##  Appendix B Infall velocity as a function of the upper energy level,
critical density, and velocity dispersion

In this Appendix, we compare the derived infall velocity with the upper energy
level, critical density as well as velocity dispersion.

Figures 11 and 12 show the infall velocity in W31C and W3(OH) as a function of
the upper energy level and critical density, respectively. In W31C, apart from
the CH3OH transitions at 25.541, 86.6 and 86.9 GHz (which may have inaccurate
$V_{\rm LSR}$ or poor S/N ratio), the derived $V_{\rm in}$ shows an increasing
trend with increasing upper energy level and critical density. While in
W3(OH), $V_{\rm in}$ distributes within a narrow velocity range and shows no
clear trend with the upper energy level and critical density.

Figure 13 shows that there is a different behaviour between W31C and W3(OH)
when comparing $\sigma$ with $V_{\rm in}$ that are derived from the MCMC
fitting. We find that $V_{\rm in}$ is comparable to $\sigma$ in W31C for
$V_{\rm in}\lesssim$2.5 km s-1, indicating that the collapse can sustain the
turbulence in W31C at a large scale. For $V_{\rm in}\gtrsim$2.5 km s-1,
$\sigma$ becomes lower than $V_{\rm in}$. Because the higher $V_{\rm in}$ is
derived from lines with high energies and critical densities, the behaviour is
likely to occur at small scales. This is also the case for W3(OH). At the
small scales, the dissipation rate should be higher because of higher
densities (e.g. Elmegreen & Scalo 2004). The more efficient dissipation rate
may explain the observed behaviour that $\sigma$ is lower than $V_{\rm in}$.
If the observed $\sigma$ follows the typical line width-size scaling relation
(e.g. Larson 1981), the lower $\sigma$ corresponds to the smaller scale.
Hence, the trend that $V_{\rm in}$ increases with decreasing $\sigma$ might
indicate a multi-layer view of global collapse with hierarchical inflows in
W31C.

Figure 11: Infall velocity as a function of the upper energy level for both
W31C and W3(OH), respectively. The grey arrow roughly depicts a slight upward
trend in W31C.

Figure 12: Infall velocity as a function of the critical density for both W31C
and W3(OH), respectively. The grey arrow roughly depicts a slight upward trend
in W31C.  Figure 13: Velocity dispersion as a function of infall velocity for
both W31C and W3(OH) are marked with red triangles and blue squares,
respectively. The magenta dashed line represents where velocity dispersion
equals infall velocity.