11institutetext: Max Planck Institut für Extraterrestrische Physik,
Giessenbachstrasse 1, 85748 Garching, Germany 22institutetext: Observatorio
Astronómico Nacional(OAN), 28014 Madrid, Spain 33institutetext: Ural Federal
University, 620002 Yekaterinburg, Russia 44institutetext: IRAP, Université de
Toulouse, CNRS, UPS, CNES, 31400 Toulouse, France 55institutetext: Laboratoire
d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy Saint-
Hilaire, 33615 Pessac, France

#  Gas phase Elemental abundances in Molecular cloudS (GEMS) V. Methanol in
Taurus††thanks: Based on observations carried out with the IRAM NOEMA
interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN
(Spain)

S. Spezzano 11 A. Fuente 22 P. Caselli 11 A. Vasyunin 33 D. Navarro-Almaida 22
M. Rodríguez-Baras 22 A. Punanova 33 C. Vastel 44 V. Wakelam 55

###### Abstract

Context. Methanol, one of the simplest complex organic molecules in the
Interstellar Medium (ISM), has been shown to be present and extended in cold
environments such as starless cores. Studying the physical conditions at which
CH3OH starts its efficient formation is important to understand the
development of molecular complexity in star-forming regions.

Aims. We aim at studying methanol emission across several starless cores and
investigate the physical conditions at which methanol starts to be efficiently
formed, as well as how the physical structure of the cores and their
surrounding environment affect its distribution.

Methods. Methanol and C18O emission lines at 3 mm have been observed with the
IRAM 30m telescope within the large program ”Gas phase Elemental abundances in
Molecular CloudS” (GEMS) towards 66 positions across 12 starless cores in the
Taurus Molecular Cloud. A non-LTE radiative transfer code was used to compute
the column densities in all positions. We then used state-of-the-art chemical
models to reproduce our observations.

Results. We have computed N(CH3OH)/N(C18O) column density ratios for all the
observed offsets, and two different behaviours can be recognised: the cores
where the ratio peaks at the dust peak, and the cores where the ratio peaks
with a slight offset with respect to the dust peak ($\sim$10000 AU). We
suggest that the cause of this behaviour is the irradiation on the cores due
to protostars nearby which accelerate energetic particles along their
outflows. The chemical models, which do not take into account irradiation
variations, can reproduce fairly well the overall observed column density of
methanol, but cannot reproduce the two different radial profiles observed.

Conclusions. We confirm the substantial effect of the environment onto the
distribution of methanol in starless cores. We suggest that the clumpy medium
generated by protostellar outflows might cause a more efficient penetration of
the interstellar radiation field in the molecular cloud and have an impact on
the distribution of methanol in starless cores. Additional experimental and
theoretical work is needed to reproduce the distribution of methanol across
starless cores.

###### Key Words.:

ISM: clouds - ISM: molecules - radio lines: ISM

##  1 Introduction

Methanol, CH3OH, and even more complex organic molecules (COMs) have been
widely observed in star-forming regions. COMs are defined as organic molecules
with $\geq$ 6 atoms (van Dishoeck, 2009), and their formation is considered to
be the first step in the chemical complexity that will eventually be inherited
by forming planets in the process of star and planetary system formation. In
the past decade several papers have reported on the detection of methanol and
other COMs towards starless cores showing that COMs are present and their
emission is extended in cold and shielded environments (Bacmann et al., 2012;
Bizzocchi et al., 2014; Vastel et al., 2014; Jiménez-Serra et al., 2016;
Scibelli & Shirley, 2020; Scibelli et al., 2021). Starless cores provide the
ingredients that will eventually build-up a planet, both the organic as well
as the refractory material. Therefore, it is valuable to understand the
chemistry that drives the build-up of chemical complexity in stellar nurseries
such as starless cores.
COMs have been widely observed towards hot cores and hot corinos, both
characterised by an active chemistry driven by ice sublimation at T$\geq$100 K
(e.g. Caselli et al. (1993); Cazaux et al. (2003); Caux et al. (2011);
Jørgensen et al. (2012); Belloche et al. (2014)). COMs are mostly formed on
the surface of dust grain via: i) hydrogenation of carbon monoxide (producing
small COMs like formaldehyde and methanol, Charnley et al. 1995; Watanabe &
Kouchi 2002); and ii) radical-radical (associative) reactions (producing
larger COMs like methyl formate and dimethyl ether, Garrod & Herbst 2006). In
order for radical-radical COM formation to happen, it is crucial to have a
central heating source like a young stellar object (YSO) because the mobility
of radicals on the surface appears at T$\geq$30 K. Because of the lack of a
central heating source, the detection of COMs toward starless cores challenged
our understanding of COMs in the interstellar medium. Large COMs like
acetaldehyde, dimethyl ether and methyl formate have been observed towards
dark cloud cores and starless cores (Bacmann et al., 2012; Vastel et al.,
2014; Jiménez-Serra et al., 2016; Nagy et al., 2019; Scibelli & Shirley, 2020;
Scibelli et al., 2021; Jimenez-Serra et al., 2021).

The high-level of molecular complexity revealed by these studies towards cold
cores has been a challenge for chemical models (Öberg et al., 2010). Recent
laboratory work has shown that COMs like acetaldehyde, vinyl alcohol, and also
glyicine is possible in interstellar ice analogs with non-energetic processes
(Chuang et al., 2020; Ioppolo et al., 2021). The formation of COMs on the
grains depends on the efficiency for the large radicals to diffuse on the
surfaces. Although it is an uncertain parameter, a larger diffusion efficiency
will produce more COMs on the surfaces (Ruaud et al., 2015; Walsh et al.,
2016). The formation in the ices also depends on the chemical network itself,
which is probably not complete for the larger molecules. Ruaud et al. (2015)
for instance have proposed new routes to form COMs at low temperature (and low
diffusion efficiency) via reactions induced by the formation of complexes with
the main ice components. New gas-phase pathways have also been proposed but
still require the efficient production and desorption of methanol (Vasyunin &
Herbst, 2013; Balucani et al., 2015). If the efficiency on the surface is
sufficient, remains the problem of desorbing these molecules strongly bound to
the surfaces. The most promising non-thermal desorption mechanisms seem to be
chemical reactive desorption from CO-rich surfaces (Minissale et al., 2016;
Vasyunin et al., 2017), chemical explosions (Rawlings et al., 2013; Ivlev et
al., 2015), and grain sputtering induced by cosmic-rays (Wakelam et al.,
2021).
Methanol is one of the simplest COMs and the precursor of many of the
O-bearing COMs, hence understanding the processes responsible for its release
in the gas phase is of pivotal importance to constrain chemical models and
reproduce the chemical complexity that we observe in starless cores.

In this paper we present our results on the emission of methanol towards 12
starless cores as well as towards the clouds where they are embedded. We study
the dependency, or the lack thereof, between the emission of methanol and
physical parameters such as visual extinction, volume density, as well as
environment. The paper is structured as follows: Section 2 describes the
observations, Section 3 presents the dataset as well as the TMC-1 molecular
cloud and the B213 filament, where the 12 observed starless cores are located.
The analysis of the observed data and its results are in Sections 4 and 5. The
results of the chemical modelling are presented in Section 6.2. Our
conclusions are discussed in Section 7. More detailed information on each
source is presented in the Appendice.

##  2 Observations

The molecular transitions used in this study are reported in Table 1. The 3 mm
methanol lines were observed within the ”Gas phase Elemental abundances in
Molecular CloudS” (GEMS) (PI: Asunción Fuente) IRAM 30m Large Program and the
observing procedure and receiver setups were described by Fuente et al.
(2019). The HPBW is varying with the frequency as HPBW($"$)=2460 /$\nu$ where
$\nu$ is in GHz. The observing mode was frequency switching with a frequency
throw of 6 MHz well adapted to remove standing waves between the secondary
mirror and the receivers. The Eight MIxer Receivers (EMIR) and the Fast
Fourier Transform Spectrometers (FTS) with a spectral resolution of 49 kHz
were used for these observations. The intensity scale is TMB, which is related
with T${}_{A}^{*}$ by $T_{MB}=(F_{eff}/B_{eff})T_{A}^{*}$ (see Table B.1 in
Fuente et al., 2019). GEMS observations in TMC 1 were completed with C18O
2$\rightarrow$1 observations carried out with the IRAM 30m telescope in a
previous project.

For TMC 1, we use observations of the CH3OH $J_{K_{a},K_{c}}$ = 10,1
$\rightarrow$00,0 line carried out with the Yebes 40m radiotelescope in Q-Band
(Tercero et al., 2020) during March-April 2018. The 40m telescope is equipped
with HEMT receivers for the 2.2-50 GHz range, and a SIS receiver for the
85-116 GHz range. Single-dish observations in K-band (21-25 GHz) and Q-band
(41-50 GHz) were performed simultaneously. The backends consisted of FFTS
covering a bandwidth of $\sim$2 GHz in band K and $\sim$9 GHz in band Q, with
a spectral resolution of $\sim$38 kHz. Central frequencies were 23000 MHz and
44750 MHz for the K and Q band receivers, respectively. The observing
procedure was position-switching, and the OFF-positions are RA(J2000) =
04h42m24s.24 Dec(2000):25∘41′27′′.6 for TMC 1-CP, RA(J2000) = 04h42m29s.52
Dec(2000):25∘48′07′′.2 for TMC 1-NH3, RA(J2000) = 04h42m32s.16
Dec(J2000):25∘59′42′′.0 for TMC 1-C. These positions were checked to be empty
of emission before the observations. The intensity scale is TMB with
conversion factors of 4.1 Jy/K in band K (TMB/T${}_{A}^{*}$=1.3) and in 5.7
Jy/K in band Q (TMB/T${}_{A}^{*}$=2.1). The HPBW of the telescope is 42$"$ at
7 mm and 84$"$ at 1.3 cm. As an example of our dataset, Figure 1 shows to the
left the H2 column density map of the TMC-1 molecular cloud (upper panel) and
the B213 filament (lower panel) computed from $Herschel$ and $Planck$ data
(Palmeirim et al., 2013; Rodríguez-Baras et al., 2021), and on the right a
zoom-in into the H2 column density map of B213-C1 where the observed offsets
are marked (upper panel) and the spectra of the 21,2-11,1 ($E_{2}$) transition
of methanol are overlaid with the 1-0 transition of C18O (lower panel). The
spectra observed towards the other sources in our sample are shown in Figures
7\- 17.

Figure 1: Left panels: H2 column density map of the TMC-1 molecular cloud
(lower panel) and the B213 filament (upper panel) in Taurus,computed from
$Herschel$ and $Planck$ data (Palmeirim et al., 2013; Rodríguez-Baras et al.,
2021). The offsets observed towards the starless cores in our dataset are
marked with a triangle.
Right panel: H2 column density of B213-C1 (upper panel). The triangles mark
the positions observed with the GEMS large project, and the red circles show
the beam size of the IRAM 30m telescope at 97 GHz ($\sim$26$"$). The spectra
of the 21,2-11,1 ($E_{2}$) transition of methanol (in red) overlaid with the
$J$ = 1-0 transition of C18O (in black) in all offsets towards B213-C1 are
shown in the lower panel.

Table 1: Spectroscopic parameters of the observed lines

Molecule | Transition | Rest frequencyaa$a$Frequencies and uncertainties from the CDMS (Müller et al., 2005) | Eup | A | n∗bb$b$n∗ is the critical density, calculated at 10 K.
---|---|---|---|---|---
|  | (MHz) | (K) | ($\times$10-5 s-1) | cm-3
CH3OH |  $J_{K_{a},K_{c}}$ = 10,1-00,0 ($A^{+}$) | 48372.460(2) | 2.3cc$c$Energy relative to the ground 00,0, $A$ rotational state. | 3.5$\times$10-7 | 7$\times$103
CH3OH |  $J_{K_{a},K_{c}}$ = 21,2-11,1 ($E_{2}$) | 96739.358(2) | 12.53cc$c$Energy relative to the ground 00,0, $A$ rotational state. | 2.6$\times$10-4 | 3$\times$104
CH3OH |  $J_{K_{a},K_{c}}$ = 20,2-10,1 ($A^{+}$) | 96741.375(2) | 6.96cc$c$Energy relative to the ground 00,0, $A$ rotational state. | 3.4$\times$10-4 | 3$\times$104
CH3OH |  $J_{K_{a},K_{c}}$ = 20,2-10,1 ($E_{1}$) | 96744.545(2) | 20.08cc$c$Energy relative to the ground 00,0, $A$ rotational state. | 3.4$\times$10-4 | 3$\times$104
CH3OH |  $J_{K_{a},K_{c}}$ = 20,2-10,1 ($E_{1}$-$E_{2}$) | 108893.945(2) | 13.10cc$c$Energy relative to the ground 00,0, $A$ rotational state. | 1.47$\times$10-5 | 5$\times$105dd$d$Calculated with the rate coefficient reported in Bizzocchi et al. (2014)
C18O |  $J$ = 1-0 | 109782.173(2) | 5.27 | 6.3$\times$10-8 | 2$\times$103
C18O |  $J$ = 2-1 | 219560.354(2) | 15.81 | 6.0$\times$10-7 | 2$\times$104

***

Table 2: Source sample

| Right Ascension | Declination
---|---|---
| (J2000) | (J2000)
TMC-1 C | 04:41:38.80 | +25:59:42.0
TMC-1 CP | 04:41:41.90 | +25:41:27.1
TMC-1 NH3 | 04:41:21.30 | +25:48:07.0
B213-1 | 04:17:41.80 | +28:08:47.0
B213-2 | 04:17:50.60 | +27:56:01.0
B213-5 | 04:18:03.80 | +28:23:06.0
B213-6 | 04:18:08.40 | +28:05:12.0
B213-7 | 04:18:11.50 | +27:35:15.0
B213-10 | 04:19:37.60 | +27:15:31.0
B213-12 | 04:19:51.70 | +27:11:33.0
B213-16 | 04:21:21.00 | +27:00:09.0
B213-17 | 04:27:54.00 | +26:17:50.0

##  3 Dataset

We have observed several lines of CH3OH and two lines of C18O towards the GEMS
sources in Taurus with the aim of understanding how these two chemically
related molecules behave towards several starless cores within the same
environment. The GEMS dataset allows us to study the abundance of both
molecules, as well as their ratios, towards different offsets within and
outside the cores, and hence tracing gas with different physical properties
such as volume density and temperature. Our source sample is listed in Table
2.
All transitions of methanol studied in this paper, except the
$J_{K_{a},K_{c}}$ = 10,1-00,0 ($A^{+}$) transition at 48 GHz, have been
observed towards every offset in our source sample (although they were not
always detected). Both C18O transitions have been observed towards every
offsets in the starless cores in TMC-1, while only the $J$ = 1-0 transition
has been observed towards the starless cores in the B213 filament. Figures 7\-
17 show the spectra of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid
with the 1-0 transition of C18O towards all offsets in our dataset, with the
exception of B213-C1, shown in Figure 1.

###  3.1 TMC-1 cloud

Taurus Molecular Cloud-1 (TMC-1) is a cold and dense cloud with a filamentary
structure of about 5$\arcmin\times$40$\arcmin$ in the centre of the Taurus
molecular cloud at a distance of 140.2${}^{+1.3}_{-1.3}$ pc (Galli et al.,
2019). The chemistry of TMC-1 has been the object of many studies in the
literature (e.g. Guelin et al. 1982; Pratap et al. 1997; Saito et al. 2002;
Schnee et al. 2007; Suutarinen et al. 2011). The most peculiar feature of
TMC-1 is the chemical differentiation between the southern part, the
cyanopolyyne peak (TMC-1 CP), where carbon-chain molecules are more abundant,
and the northern part, the ammonia peak (TMC-1 NH3), where NH3 and N2H+ are
more abundant. The origin of this differentiation has been extensively
discussed, and it can be explained by either a small variation in the chemical
timescale ($\sim$ 105 yr), and/or by the different density (with the ammonia
peak being denser) (Hirahara et al., 1992; Suzuki et al., 1992). Also a
variation in the C/O ratio can reproduce an abundance gradient similar to that
observed (Pratap et al., 1997). We present here the observations of methanol
and C18O towards the cyanopolyyne and ammonia peaks (TMC-1 CP and NH3). We
also present observations towards the pre-stellar core TMC1-C (Schnee et al.,
2007, 2010). Several observational evidences suggest that this core is a later
evolutionary stage than TMC1 (CP), presenting higher deuteration fractions,
more similar to those in L1544 (Crapsi et al., 2005; Navarro-Almaida et al.,
2021). The spectra of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid
with the 1-0 transition of C18O towards TMC-1C, TMC-1 CP and TMC-1 NH3 are
shown in Figures 7\- 9.

###  3.2 B213 filament

B213 (L1495) is one of the longest and most prominent filaments in the Taurus
molecular cloud, as it extends for over 80$\arcmin$, and it is located towards
the north-west of the cloud at a distance of 129.9${}^{+0.4}_{-0.3}$ pc (Galli
et al., 2019). The gas towards B213 presents a very complex velocity structure
that has been interpreted as the result of colliding filaments (Duvert et al.,
1986). B213 has been observed in CO isotopologues with large scale maps
(Onishi et al., 1996; Goldsmith et al., 2008). The denser gas has been studied
using ammonia, H13CO+, N2H+ and SO (Benson & Myers, 1989; Hacar et al., 2013).
Furthermore, some dense cores in B213 contain young stellar objects (Rebull et
al., 2010). We present here the observations of methanol and C18O towards nine
starless cores in B213. The spectra of the 21,2-11,1 ($E_{2}$) transition of
methanol overlaid with the 1-0 transition of C18O towards the starless cores
we observed in B213 are shown in Figure 1 and Figures 10\- 17.

##  4 Analysis

We have observed the three rotational transitions belonging to the 2-1
transitions of methanol at 96 GHz (symmetry A, E1 and E2) as well as the 1-0
transition at 108 GHz (symmetry E2-E1), in all cores of our sample. In
addition, we have observed the $J_{K_{a},K_{c}}$ = 10,1-00,0 ($A^{+}$)
transition of methanol at 48 GHz towards the three cores in TMC-1. The
intensity ratios among the four lines at 3 mm in our dataset cannot be
reproduced assuming LTE, implying that we have to use a non-LTE code if we
want to derive a reliable column density of methanol in all positions within
our dataset. Since methanol lines have a relatively high critical density
(between 0.7 and 3 $\times$104 cm-3, see Table 1), the correspondent column
density is very sensitive to variations in the H2 volume density. The H2
volume density towards all observed positions within the GEMS catalogue has
been calculated by modelling the emission of CS, C34S, and 13CS with the
radiative transfer code RADEX (van der Tak et al., 2007) in Rodríguez-Baras et
al. (2021). However, the lines of methanol and CS could come from different
velocity components and/or different layers within the same cloud. The line
profiles of C18O and CH3OH are different in many positions. Therefore, in
order to be consistent and derive accurate column densities we need to use the
volume density derived from CH3OH data. We hence decided to use the Markov
Chain Monte Carlo method (MCMC) together with the RADEX non-LTE radiative
transfer code within the CASSIS software (Vastel et al., 2015) in order to
compute the methanol column density, the excitation temperature, and the gas
volume density towards all positions in our dataset. To assess the goodness of
the fit, the reduced $\chi^{2}$ was computed for each model. The reduced
$\chi^{2}$ in our sample ranges from 1.5 to 4.5, with most of the models
having a reduced $\chi^{2}$ $\sim$ 2\. The results of the MCMC+RADEX analysis
with CASSIS on methanol are reported in Table LABEL:table:results. We used the
collisional rates for methanol reported in Rabli & Flower (2010) at 10 K.

We did not use the MCMC method for the C18O because for most of the cores we
observed only one transition (with the exception of the three cores in TMC-1),
and the observed lines of C18O are not very sensitive to variations in the
volume densities. Furthermore, C18O often has more velocity components than
CH3OH, and we are interested in deriving a column density for C18O from the
velocity component that corresponds to the one observed with CH3OH. We used
two approaches to derive the column densities of C18O in our dataset. In the
offsets with a good match between the rest velocity and line width of the
lines of methanol and C18O, we fit the lines of C18O and used RADEX to derive
the column density, assuming the Tkin and n${}_{H_{2}}$ derived for CH3OH. In
these cases, the errors derived on the vLSR and $\delta$v from the line fit
have been propagated to derive the error on the column density of C18O, see
for example the values for TMC-1C and B213-C6 reported in Table
LABEL:table:results. For the offsets with a bad match between the rest
velocity and line width of the lines of methanol and C18O (see for example the
spectra of TMC-1 NH3 in Figure 9), we have used the GUI of CASSIS and
performed a single-line fit with RADEX of the portion of the spectrum of C18O
that is comparable with methanol in velocity and line-width. In order to do
this we have fixed the vLSR and adjusted the line-width in order to be similar
to the line-width of methanol, and at the same time model well the line shape
of C18O. The error on the column densities derived in this manner, and
reported in Table LABEL:table:results, has been estimated to be 15$\%$. While
in the case of CH3OH our analysis shows that the molecules are not in LTE (as
the resulting Tex is lower than the kinetic temperature), this is not the case
for the C18O 1-0 transition, where Tex $\sim$ Tkin.

The visual extinction tabulated in Table LABEL:table:results has been
calculated from the H2 column density maps reported in Rodríguez-Baras et al.
(2021), using the formula AV [mag] = N(H2)[cm-2]/9.4$\times$1020[cm-2 mag-1]
(Bohlin et al., 1978).

We do not report on the upper limits for the column density of methanol
towards positions where the line was not detected because it is not possible
to derive them in the same fashion as we did for the methanol column densities
reported in Table LABEL:table:results, and we would have to assume a value for
the H2 volume density. The uncertainty induced by this assumption would make
the comparison of the upper limits on N(CH3OH) with the N(CH3OH) computed with
the MCMC+RADEX method not meaningful.

Table 3: Results from the MCMC and RADEX analysis. |  |  |  CH3OH |  |  |  |  |  C18O |
---|---|---|---|---|---|---|---|---|---
|  AVaa$a$Derived from $Herschel$ and $Planck$ data (Palmeirim et al., 2013; Rodríguez-Baras et al., 2021). |  Ntot |  Tkin |  n${}_{H_{2}}$ |  vLSR |  $\delta$v |  Ntot |  vLSR |  $\delta$v
|  mag |  [1013cm-2] |  [K] |  [104cm-3] |  [km s-1] |  [km s-1] |  [1015cm-2] |  [km s-1] |  [km s-1]
TMC-1 C |  |  |  |  |  |  |  |  |
1 |  19.9 |  2.7${}^{+0.4}_{-0.30}$ |  10.4${}^{+0.2}_{-0.3}$ |  3.8${}^{+1.0}_{-0.7}$ |  5.18 |  0.34 |  3.62(1) |  5.21 |  0.46
2 |  18.5 |  2.7${}^{+0.3}_{-0.3}$ |  10.3${}^{+0.5}_{-0.12}$ |  2.5${}^{+0.5}_{-0.5}$ |  5.14 |  0.35 |  3.3(1) |  5.19 |  0.46
3 |  13.3 |  2.1${}^{+0.5}_{-0.2}$ |  12.5${}^{+0.3}_{-0.3}$ |  1.01${}^{+0.2}_{-0.3}$ |  5.12 |  0.32 |  2.67(2) |  5.16 |  0.46
4 |  4.8 |  1.4${}^{+0.3}_{-0.5}$ |  11.3${}^{+1.0}_{-0.6}$ |  0.15${}^{+0.08}_{-0.02}$ |  5.09 |  0.39 |  1.54(1) |  5.13 |  0.51
TMC-1 CP |  |  |  |  |  |  |  |  |
1 |  18.2 |  2.0${}^{+0.6}_{-0.3}$ |  10.5${}^{+0.9}_{-0.4}$ |  1.5${}^{+0.3}_{-0.5}$ |  5.63 |  0.33 |  2.3$\pm$0.34 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.28cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
1 |  18.2 |  1.2${}^{+0.1}_{-0.1}$ |  10.7${}^{+0.2}_{-0.4}$ |  3.87${}^{+0.30}_{-0.51}$ |  6.07 |  0.32 |  0.9$\pm$0.13 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.3cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  16.7 |  1.3${}^{+0.3}_{-0.2}$ |  12.1${}^{+0.3}_{-0.5}$ |  1.6${}^{+0.2}_{-0.5}$ |  5.66 |  0.38 |  2.2$\pm$0.33 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.28cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  16.7 |  1.1${}^{+0.2}_{-0.1}$ |  10.3${}^{+0.2}_{-0.2}$ |  3.5${}^{+0.3}_{-0.8}$ |  6.06 |  0.37 |  1.0$\pm$0.15 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.28cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  13.7 |  0.6${}^{+0.1}_{-0.1}$ |  10.7${}^{+0.2}_{-0.3}$ |  4.2${}^{+0.9}_{-0.6}$ |  5.68 |  0.36 |  1.8$\pm$0.27 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.28cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  13.7 |  0.7${}^{+0.1}_{-0.1}$ |  10.6${}^{+0.2}_{-0.5}$ |  4.2${}^{+0.3}_{-0.4}$ |  6.01 |  0.36 |  1.2$\pm$0.18 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.3cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
4 |  7.3 |  2.3${}^{+1.9}_{-1.0}$ |  10.3${}^{+0.4}_{-0.2}$ |  0.16${}^{+0.05}_{-0.04}$ |  5.67 |  0.33 |  1.2$\pm$0.18 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.3cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
4 |  7.3 |  0.3${}^{+0.1}_{-0.1}$ |  12.2${}^{+0.3}_{-0.7}$ |  0.34${}^{+0.07}_{-0.05}$ |  6.02 |  0.33 |  0.7$\pm$0.10 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.3cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
TMC-1 NH3 |  |  |  |  |  |  |  |  |
1 |  17.0 |  1.7${}^{+1.5}_{-0.3}$ |  11.3${}^{+0.4}_{-0.4}$ |  1.5${}^{+0.8}_{-1.0}$ |  5.59 |  0.43 |  0.7$\pm$0.10 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
1 |  17.0 |  1.0${}^{+0.2}_{-0.2}$ |  10.6${}^{+0.2}_{-0.4}$ |  2.8${}^{+0.7}_{-0.8}$ |  6.13 |  0.38 |  0.45$\pm$0.07 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  15.6 |  3.0${}^{+0.7}_{-0.6}$ |  11.0${}^{+0.4}_{-0.7}$ |  0.7${}^{+0.2}_{-0.1}$ |  5.60 |  0.37 |  0.8$\pm$0.12 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  15.6 |  3.8${}^{+1.4}_{-0.7}$ |  10.9${}^{+0.9}_{-0.5}$ |  0.4${}^{+0.1}_{-0.1}$ |  6.01 |  0.39 |  0.5$\pm$0.07 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  12.9 |  8.8${}^{+0.6}_{-0.7}$ |  13.8${}^{+0.5}_{-1.5}$ |  0.35${}^{+0.05}_{-0.04}$ |  5.75 |  0.44 |  1.1$\pm$0.16 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  12.9 |  1.7${}^{+0.5}_{-0.8}$ |  10.2${}^{+0.2}_{-0.1}$ |  0.29${}^{+0.19}_{-0.07}$ |  6.01 |  0.35 |  0.7$\pm$0.10 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
4 |  10.0 |  3.0${}^{+0.7}_{-0.4}$ |  10.6${}^{+1.5}_{-0.4}$ |  0.26${}^{+0.08}_{-0.06}$ |  5.84 |  0.44 |  0.9$\pm$0.13 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.45cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
B213-1 |  |  |  |  |  |  |  |  |
1 |  26.9 |  6.9${}^{+0.5}_{-0.5}$ |  10.1${}^{+0.1}_{-0.1}$ |  1.1${}^{+0.1}_{-0.1}$ |  5.97 |  0.50 |  1.3$\pm$0.19 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.6cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  13.6 |  5.8${}^{+0.4}_{-0.6}$ |  10.4${}^{+0.2}_{-0.2}$ |  1.1${}^{+0.2}_{-0.1}$ |  6.19 |  0.50 |  1.5$\pm$0.22 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.6cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  13.4 |  4.6${}^{+1.3}_{-1.1}$ |  10.1${}^{+0.1}_{-0.1}$ |  1.0${}^{+0.6}_{-0.3}$ |  6.21 |  0.48 |  1.2$\pm$0.18 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.6cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  13.4 |  3.2${}^{+1.7}_{-2.1}$ |  10.1${}^{+0.1}_{-0.1}$ |  1.7${}^{+0.7}_{-0.7}$ |  6.83 |  0.43 |  1.2$\pm$0.18 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.6cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
4 |  6.0 |  1.3${}^{+0.5}_{-0.3}$ |  10.2${}^{+0.2}_{-0.1}$ |  1.1${}^{+0.7}_{-0.4}$ |  6.62 |  0.66 |  1.3$\pm$0.19 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.6cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
5 |  4.2 |  0.4${}^{+0.3}_{-0.2}$ |  10.3${}^{+0.1}_{-0.1}$ |  1.0${}^{+0.8}_{-0.4}$ |  6.53 |  0.47 |  0.9$\pm$0.13 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.6cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
6 |  5.2 |  0.8${}^{+0.4}_{-0.2}$ |  10.3${}^{+0.4}_{-0.3}$ |  1.1${}^{+0.4}_{-0.4}$ |  6.65 |  0.43 |  0.71(1) |  6.63 |  0.47
7 |  10.5 |  0.7${}^{+0.2}_{-0.2}$ |  10.3${}^{+0.15}_{-0.16}$ |  4.4${}^{+1.5}_{-2.1}$ |  6.80 |  0.48 |  1.17(1) |  6.84 |  0.64
8 |  10.3 |  3.1${}^{+0.3}_{-0.2}$ |  10.1${}^{+0.12}_{-0.08}$ |  5.6${}^{+1.5}_{-1.2}$ |  5.97 |  0.49 |  1.24(3) |  6.06 |  0.61
9 |  5.5 |  2.8${}^{+0.2}_{-0.2}$ |  10.7${}^{+0.16}_{-0.19}$ |  6.5${}^{+0.8}_{-1.5}$ |  5.98 |  0.49 |  1.26(3) |  6.02 |  0.66
B213-2 |  |  |  |  |  |  |  |  |
1 |  20.9 |  3.5${}^{+2.2}_{-1.7}$ |  10.5${}^{+0.7}_{-0.4}$ |  2.4${}^{+1.0}_{-1.0}$ |  7.29 |  0.55 |  0.5$\pm$0.07 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  10.6 |  0.7${}^{+0.2}_{-0.2}$ |  10.3${}^{+0.1}_{-0.2}$ |  4.4${}^{+0.4}_{-1.3}$ |  7.02 |  0.46 |  0.6$\pm$0.09 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.6cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
8 |  16.5 |  1.5${}^{+1.1}_{-0.4}$ |  10.9${}^{+0.2}_{-0.3}$ |  2.4${}^{+1.1}_{-1.0}$ |  6.97 |  0.40 |  0.8$\pm$0.12 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
9 |  9.8 |  1.4${}^{+0.2}_{-0.2}$ |  10.2${}^{+0.2}_{-0.1}$ |  6.2${}^{+0.7}_{-2.6}$ |  6.85 |  0.33 |  0.9$\pm$0.05 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
B213-5 |  |  |  |  |  |  |  |  |
1 |  23.6 |  7.8${}^{+0.48}_{-0.9}$ |  10.2${}^{+0.1}_{-0.1}$ |  1.1${}^{+0.2}_{-0.1}$ |  6.26 |  0.50 |  1.5$\pm$0.22 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.7cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  13.0 |  3.0${}^{+0.5}_{-0.5}$ |  10.2${}^{+0.3}_{-0.1}$ |  1.2${}^{+0.3}_{-0.1}$ |  6.44 |  0.70 |  1.0$\pm$0.15 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.7cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  10.9 |  1.0${}^{+0.3}_{-0.1}$ |  10.1${}^{+0.06}_{-0.07}$ |  2.7${}^{+0.5}_{-1.5}$ |  6.73 |  0.68 |  1.0$\pm$0.15 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.7cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
4 |  13.3 |  0.3${}^{+0.1}_{-0.1}$ |  10.1${}^{+0.1}_{-0.1}$ |  1.0${}^{+0.1}_{-0.1}$ |  6.37 |  0.44 |  0.57(1) |  6.3 |  0.37
4 |  13.3 |  0.8${}^{+0.2}_{-0.2}$ |  10.1${}^{+0.1}_{-0.1}$ |  0.96${}^{+0.2}_{-0.1}$ |  7.07 |  0.37 |  4.72(1) |  7.11 |  0.55
5 |  14.0 |  0.2${}^{+0.1}_{-0.1}$ |  10.0${}^{+0.1}_{-0.1}$ |  0.6${}^{+0.6}_{-0.1}$ |  6.28 |  0.42 |  0.63(1) |  6.33 |  0.35
5 |  14.0 |  0.4${}^{+0.1}_{-0.1}$ |  10.4${}^{+0.1}_{-0.1}$ |  1.9${}^{+0.1}_{-0.2}$ |  7.09 |  0.37 |  4.88(2) |  7.13 |  0.55
6 |  14.5 |  1.2${}^{+0.7}_{-0.4}$ |  10.4${}^{+0.17}_{-0.15}$ |  4.8${}^{+2.0}_{-1.2}$ |  7.25 |  0.55 |  6.14(2) |  7.15 |  0.60
7 |  18.9 |  1.8${}^{+1.7}_{-1.1}$ |  10.7${}^{+0.7}_{-0.3}$ |  5.5${}^{+0.8}_{-0.9}$ |  7.3 |  0.51 |  7.75(2) |  7.21 |  0.59
8 |  12.9 |  2.9${}^{+0.3}_{-0.4}$ |  10.1${}^{+0.1}_{-0.1}$ |  1.2${}^{+0.2}_{-0.1}$ |  6.36 |  0.50 |  1.2$\pm$0.18 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.6cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
9 |  10.9 |  0.8${}^{+0.3}_{-0.1}$ |  10.3${}^{+0.1}_{-0.1}$ |  1.3${}^{+0.4}_{-0.6}$ |  6.06 |  0.48 |  0.77(1) |  6.12 |  0.52
9 |  10.9 |  0.3${}^{+0.2}_{-0.1}$ |  10.2${}^{+0.1}_{-0.1}$ |  1.0${}^{+0.6}_{-0.4}$ |  7.00 |  0.37 |  3.06(1) |  7.11 |  0.55
B213-6 |  |  |  |  |  |  |  |  |
1 |  22.2 |  3.5${}^{+0.5}_{-0.4}$ |  10.5${}^{+0.3}_{-0.2}$ |  2.5${}^{+1.3}_{-0.6}$ |  6.85 |  0.53 |  0.69(1) |  6.77 |  0.68
2 |  15 |  2.4${}^{+0.3}_{-0.3}$ |  10.6${}^{+0.1}_{-0.1}$ |  4.8${}^{+0.6}_{-1.0}$ |  6.77 |  0.40 |  0.83(1) |  6.79 |  0.46
3 |  5.6 |  0.7${}^{+0.5}_{-0.1}$ |  10.1${}^{+0.1}_{-0.1}$ |  3.3${}^{+1.4}_{-2.3}$ |  6.80 |  0.39 |  1.11(1) |  6.84 |  0.44
8 |  18.2 |  4.0${}^{+0.9}_{-0.5}$ |  10.1${}^{+0.1}_{-0.1}$ |  1.7${}^{+0.6}_{-0.6}$ |  6.94 |  0.37 |  0.73(1) |  6.92 |  0.49
9 |  11.3 |  2.5${}^{+2.5}_{-2.1}$ |  10.7${}^{+0.6}_{-0.6}$ |  5.2${}^{+3.3}_{-3.0}$ |  7.06 |  0.40 |  0.76(2) |  6.99 |  0.44
B213-7 |  |  |  |  |  |  |  |  |
1 |  20.2 |  9.5${}^{+2.7}_{-1.4}$ |  10.2${}^{+0.1}_{-0.1}$ |  0.48${}^{+0.15}_{-0.13}$ |  6.91 |  0.44 |  2.08(1) |  6.87 |  0.41
2 |  14.6 |  2.7${}^{+0.3}_{-0.3}$ |  10.4${}^{+0.1}_{-0.2}$ |  7.4${}^{+0.5}_{-3.0}$ |  6.92 |  0.41 |  2.38(2) |  6.91 |  0.47
3 |  7.2 |  6${}^{+1}_{-1}$ |  10.1${}^{+0.2}_{-0.1}$ |  4.5${}^{+1.7}_{-1.2}$ |  6.96 |  0.43 |  1.87(2) |  6.96 |  0.55
8 |  16.6 |  2.5${}^{+0.3}_{-0.2}$ |  10.3${}^{+0.3}_{-0.2}$ |  8.9${}^{+0.7}_{-1.5}$ |  6.91 |  0.47 |  1.82(2) |  6.84 |  0.43
9 |  8.7 |  7${}^{+1}_{-4}$ |  11.0${}^{+0.1}_{-0.8}$ |  0.31${}^{+0.67}_{-0.08}$ |  6.80 |  0.34 |  1.51(1) |  6.88 |  0.52
B213-10 |  |  |  |  |  |  |  |  |
1 |  20.7 |  1.3${}^{+0.2}_{-0.3}$ |  10.2${}^{+0.2}_{-0.1}$ |  1.2${}^{+0.7}_{-0.3}$ |  6.76 |  0.33 |  1.17(2) |  6.77 |  0.45
2 |  17.5 |  5.1${}^{+1.4}_{-1.2}$ |  10.5${}^{+0.2}_{-0.2}$ |  0.27${}^{+0.13}_{-0.08}$ |  6.69 |  0.27 |  1.34(2) |  6.68 |  0.47
3 |  11.8 |  4.9${}^{+0.8}_{-1.0}$ |  10.3${}^{+0.2}_{-0.1}$ |  0.27${}^{+0.11}_{-0.05}$ |  6.60 |  0.31 |  1.36(2) |  6.57 |  0.47
4 |  6.1 |  1.2${}^{+1.0}_{-0.4}$ |  10.1${}^{+0.1}_{-0.1}$ |  0.72${}^{+0.45}_{-0.39}$ |  6.58 |  0.32 |  0.90(1) |  6.59 |  0.45
7 |  17.6 |  2${}^{+1}_{-1}$ |  10.6${}^{+0.2}_{-0.4}$ |  0.58${}^{+0.62}_{-0.21}$ |  6.82 |  0.32 |  1.11(1) |  6.83 |  0.42
8 |  14.5 |  0.8${}^{+0.2}_{-0.1}$ |  10.1${}^{+0.1}_{-0.1}$ |  5.5${}^{+0.6}_{-2.0}$ |  6.89 |  0.31 |  1.31(1) |  6.86 |  0.40
9 |  9.1 |  0.6${}^{+0.1}_{-0.1}$ |  10.1${}^{+0.1}_{-0.8}$ |  4.9${}^{+0.4}_{-0.7}$ |  6.73 |  0.29 |  1.07(2) |  6.74 |  0.42
B213-12 |  |  |  |  |  |  |  |  |
1 |  22.1 |  1.4${}^{+0.4}_{-0.3}$ |  10.2${}^{+0.1}_{-0.1}$ |  0.83${}^{+0.3}_{-0.2}$ |  6.65 |  0.39 |  0.5$\pm$0.07 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  17.6 |  2${}^{+1}_{-1}$ |  10.8${}^{+0.2}_{-0.2}$ |  0.3${}^{+0.2}_{-0.1}$ |  6.69 |  0.39 |  0.45$\pm$0.07 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  11.5 |  6${}^{+1}_{-4}$ |  10.1${}^{+0.1}_{-0.1}$ |  0.13${}^{+0.78}_{-0.24}$ |  6.69 |  0.37 |  0.45$\pm$0.07 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
8 |  8.7 |  2.0${}^{+2.0}_{-0.8}$ |  10.5${}^{+0.21}_{-0.12}$ |  0.41${}^{+0.29}_{-0.19}$ |  6.51 |  0.32 |  0.7$\pm$0.10 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
B213-16 |  |  |  |  |  |  |  |  |
1 |  24.8 |  1.8${}^{+0.1}_{-0.11}$ |  10.1${}^{+0.09}_{-0.07}$ |  5.1${}^{+0.6}_{-1.0}$ |  6.68 |  0.34 |  1.3$\pm$0.19 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
2 |  13.3 |  1.2${}^{+0.1}_{-0.1}$ |  10.3${}^{+0.11}_{-0.10}$ |  4.3${}^{+0.7}_{-1.1}$ |  6.61 |  0.44 |  1.5$\pm$0.22 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
3 |  10.5 |  3.3${}^{+0.1}_{-0.1}$ |  10.2${}^{+0.14}_{-0.15}$ |  0.37${}^{+0.1}_{-0.1}$ |  6.48 |  0.43 |  1.5$\pm$0.22 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.5cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
4 |  6.9 |  3.3${}^{+0.2}_{-0.1}$ |  10.2${}^{+0.07}_{-0.07}$ |  0.16${}^{+0.1}_{-0.1}$ |  6.35 |  0.40 |  1.2$\pm$0.18 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
7 |  22.6 |  1.8${}^{+0.1}_{-0.1}$ |  10.34${}^{+0.12}_{-0.17}$ |  4.5${}^{+0.7}_{-0.9}$ |  6.71 |  0.33 |  1.2$\pm$0.18 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.35cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
8 |  9.6 |  1.3${}^{+0.3}_{-0.2}$ |  10.5${}^{+0.17}_{-0.20}$ |  2.4${}^{+0.8}_{-1.0}$ |  6.73 |  0.31 |  1.2$\pm$0.18 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.35cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
9 |  5.2 |  1.5${}^{+0.9}_{-0.6}$ |  10.3${}^{+0.46}_{-0.22}$ |  0.32${}^{+0.39}_{-0.15}$ |  6.56 |  0.41 |  1.0$\pm$0.15 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.35cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
B213-17 |  |  |  |  |  |  |  |  |
1 |  6.4 |  0.4${}^{+0.3}_{-0.2}$ |  10.4${}^{+0.3}_{-0.3}$ |  1.4${}^{+1.2}_{-0.7}$ |  7.24 |  0.42 |  0.8$\pm$0.12 |  bb$b$Fixed to the correspondent $v_{LRS}$ of CH3OH.  |  0.4cc$c$Fixed to a value that matches best the $\delta$v of CH3OH and the line shape of C18O.
7 |  12.0 |  4.1${}^{+2.8}_{-3.7}$ |  10.3${}^{+0.2}_{-0.2}$ |  7.4${}^{+0.9}_{-2.1}$ |  7.39 |  0.42 |  0.87(9) |  7.22 |  0.52
8 |  19.7 |  5.6${}^{+3.0}_{-2.2}$ |  10.5${}^{+0.4}_{-0.3}$ |  4.5${}^{+1.5}_{-2.4}$ |  7.31 |  0.42 |  0.83(1) |  7.21 |  0.48
9 |  10.2 |  0.8${}^{+0.6}_{-0.3}$ |  10.2${}^{+0.2}_{-0.1}$ |  3.7${}^{+1.1}_{-3.0}$ |  7.07 |  0.41 |  0.66(1) |  7.13 |  0.51
††† vLSR and $\delta$v are reported without uncertainty because the
statistical uncertainty is much lower than the spectral resolution.

##  5 Results

Figure 2 shows the column density ratio of methanol and C18O in all observed
positions in our sample plotted against the visual extinction and the dust
temperature derived from $Herschel$ and $Planck$ data (Palmeirim et al., 2013;
Rodríguez-Baras et al., 2021), as well as against the volume density derived
from the methanol MCMC+RADEX analysis, presented here in Section 4. Figure 2
shows no correlation among N(CH3OH)/N(C18O) and AV, Tdust, or n${}_{H_{2}}$.
This also holds if we consider B213 and TMC-1 separately or if we consider
solely the column density of methanol instead of the column density ratio with
respect to C18O. In Figure 2, the points belonging to B213 and TMC-1 are shown
as full circles and as crosses, respectively. In the case of multiple velocity
components reported for methanol and C18O in Table LABEL:table:results, we
assumed that most of the dust emission comes from the fiber with a larger
methanol column density (supposedly the densest). As a consequence, only the
velocity component with the largest methanol column density is plotted against
AV and Tdust in Figure 2. In particular, in the case of TMC-1 CP and NH3, we
plot the velocity component at 5.7 km/s. As we aim at studying methanol in
different positions across different cores, comparing solely the column
density of methanol would give us a biased picture because of the different
densities of the positions involved. On the other hand, comparing methanol
abundances with respect to molecular hydrogen is also not an optimal solution
given the presence of different velocity components in several spectra,
corresponding to different fibers on the line of sight. We decided to use the
column density ratio of methanol and carbon monoxide because they are
chemically related, as the depletion of carbon monoxide on the surface of dust
grains is necessary to the formation of methanol. Using the rare isotopologue
C18O we can limit, if not exclude, the risk of large optical depth.

A possible reason for the lack of definite trends in Figure 2 might be that
methanol emits from the outer layers of starless cores (e.g. in L1544 as shown
in Bizzocchi et al. 2014; Vastel et al. 2014). Furthermore, the emission of
methanol is not homogeneous around pre-stellar and starless cores, and its
distribution has been shown to depend on the density structure around the
core, and hence the illumination onto the core due to the interstellar
radiation field (Spezzano et al., 2016, 2020). In Spezzano et al. (2016), for
example, the emission maps of methanol and cyclopropenylidene towards the
inner 2.5$"\times$2.5$"$ of the pre-stellar core L1544 show that both
molecules have asymmetric distribution around the core, with methanol peaking
towards the North-East and cyclopropenylidene towards the South-West. It is
important to note that the visual extinction at the methanol and the $c$-C3H2
emission peak in L1544 is the same, within error-bars. This shows that the H2
column density of the medium computed on the line-of-sight is not a good
enough indicator to discriminate among the regions where the illumination is
effective enough to keep more Carbon in its atomic form and hence available to
form hydrocarbons like $c$-C3H2, and the regions where there is enough
shielding to allow carbon to be locked in to carbon monoxide, and eventually
form methanol. In addition to the density structure around the core, also
larger scale environmental effects have shown to influence the distribution of
methanol around starless and pre-stellar cores. In Spezzano et al. (2020),
methanol was mapped towards 4 pre-stellar and 2 starless cores, and it was
shown that the methanol peak was influenced by the presence of nearby massive
stars in the two cores mapped in Ophiucus (OphD and HMM-1).
The GEMS dataset offers the opportunity to study the emission of methanol
towards 12 starless core, a larger sample with respect to previous studies.
Furthermore, the GEMS dataset is not limited to the cores, giving us
information on the transition from core to cloud. In order to spot trends that
might be related to the density structure surrounding the core, in Figures 18
to 29 we plot the column density ratios ordered by offset position across the
observed cut within the core for each source, while the offsets are shown on
top of the N(H2) column density maps of the cores in the upper panel of the
Figure to facilitate the comparison. The individual results are discussed in
detail in Appendix A. Generally, we can recognise two different behaviours,
shown in Figure 3: the cores where N(CH3OH)/N(C18O) peaks at the dust peak,
and the cores where the N(CH3OH)/N(C18O) ratios peak has a slight offset with
respect to the dust peak ($\sim$10000 AU). This trend is particularly clear in
the cores where more than 5 offsets have been observed. B213-C1 and C5 for
example show a clear peak of the column density ratio towards the dust peak
(see Figures 3, 21, and 23), while in B213-C10 and C16 the peak is slightly
shifted from the dust peak ($\sim$10000 au), (see Figures 3, 26 and 28). One
reason for this behaviour might be the external irradiation on the cores.
B213-C1 and C5 are in fact located towards the northern part of B213, that is
known to be actively forming stars with $\sim$40 Class 0/I/II sources present
in the vicinity of B213-C1, C5, C2, C6 and C7. B213-C10 and C16, on the other
side, are located in the central/southern part of B213, which only contains
$\sim$15 Class 0/I/II sources around B213-C10, C12 and C16, and 5 around
B213-C17. In Figure 31 the positions of the young stellar objects in B213 are
marked as red triangles on the H2 column density map of the filament. Low-mass
protostars can accelerate energetic particles along the outflow, and
consequently increase the local cosmic rays flux (Padovani et al., 2016; Fitz
Axen et al., 2021). This will in turn increase the destruction of CO, and
consequently decrease the production of methanol towards the outer layers of
starless cores in the North of B213.

The line-width of the methanol lines in the cores located in the North of B213
are $\sim$25$\%$ broader than the lines of methanol observed towards the
central and southern cores of B213, hinting at more prominent effects due to
stellar activity in the northern part for B213. This effect has already been
observed with ammonia in Seo et al. (2015). An additional indication of the
environmental effects is the fact that the dust temperature in the southern
cores of B213 is lower with respect to the northern cores, at comparable
visual extinctions, see Figure 30. In addition to the irradiation, there are
also differences in the velocity structure. In B213-C1, C2, and C7, the
profiles of C18O and CH3OH are very different. In B213-C6, C10 and C16 we have
however a better agreement between the profiles of C18O and CH3OH.

The N(CH3OH)/N(C18O) column density ratios towards the three cores in TMC-1
can be only calculated towards few offsets close to the dust peak, because
methanol was not detected at larger radii, in contrast with B213, likely
because the cores in B213 are denser. Towards TMC-1 NH3 the two different
velocities show peaks of the N(CH3OH)/N(C18O) ratio at different offsets,
maybe hinting at the fact that the gas they are tracing does not match in
chemical age. Towards TMC-1 CP, if we exclude the offset 4 for the component
at 5.6 km/s because of the large error bar, both velocity components show the
same trend, an increase of the N(CH3OH)/N(C18O) ratio towards the dust peak.
Towards TMC-1 C instead, the N(CH3OH)/N(C18O) ratio is constant within error-
bars.

Figure 2: Methanol and C18O column density ratios computed in all positions
within our sample plotted against the: visual extinctions derived from
$Herschel$ and $Planck$ data (Palmeirim et al., 2013; Rodríguez-Baras et al.,
2021) (upper panel); volume density derived from the MCMC+RADEX analysis of
the methanol lines (central panel); Tdust derived from $Herschel$ data in
Rodríguez-Baras et al. (2021) (lower panel).  Figure 3: Upper panel: methanol
and C18O column density ratio for two cores in the North of B213, C1 and C5.
Lower panel: methanol and C18O column density ratio for two cores in the South
of B213, C10 and C16. The column density ratios are plotted with respect to
the observed offsets, and offset 1 always refers to the dust peak (also shown
as a dashed green line). The positions of the offsets in the cores are shown
in Figures 21, 23, 26, and 28.

##  6 Chemical Modelling

###  6.1 Physical structure for spherical 1D static chemical model

For a proper chemical modeling of the region under study, it is necessary to
have a good knowledge of the density and temperature across the region.
However, sometimes our knowledge is limited to a few points, and geometrical
considerations and simplifications are mandatory. Temperature
parameterizations of prestellar cores can be found in, for example, Crapsi et
al. (2007), in which they choose the spherically symmetric profile

$T(r)=T_{\rm out}-\frac{T_{\rm out}-T_{\rm
in}}{1+\left(\frac{r}{r_{T0}}\right)^{\alpha}},$ (1)

where $T_{\rm out}$ and Tin are the temperatures at the outermost and
innermost observed positions, respectively. The dependence of the temperature
with the radius is controlled by $r_{T0}$, the flat radius of the temperature,
and $\alpha$, an asymptotic power index. A similar functional dependence with
the radius is found in Plummer-like density profiles. These profiles assume
spherical symmetry, and they have become popular density profiles to describe
dense cores (Tafalla et al., 2002; Priestley et al., 2018):

$n_{\rm H}(r)=\frac{n_{0}}{1+\left(\frac{r}{r_{0}}\right)^{\alpha}},$ (2)

where $n_{\rm H}$ is the total atomic hydrogen number density across the
profile, $n_{0}$ is the central density, $r_{0}$ is the flat radius, and
$\alpha$ is the asymptotic power index. These profiles allow us to obtain
density and temperature profiles across the B213 cuts in Table
LABEL:table:results. To have more data to derive physical profiles, we may fit
data of different cuts with similar temperatures simultaneously. Therefore, we
merge and average the data from the cut B213-C2 with B213-C6 (B213-C2-C6),
B213-C10 with B213-C12 (B213-C10-C12), and consider B213-C16 as independent
from the rest. The temperature profiles of B213-C2-C6, B213-C10-C12, and
B213-C16 can be determined by finding the set of parameters that best fit the
experimental data from Table LABEL:table:results. The results are shown in
Table 4.

Table 4: Parameter values for the temperature and density profiles presented in Section 6.1 Parameters | B213-C2-C6 | B213-C10-C12 | B213-C16
---|---|---|---
Tin (K) | $10.94\pm 0.05$ | $10.89\pm 0.04$ | $10.25\pm 0.07$
Tout (K) | $14.40\pm 0.05$ | $14.78\pm 0.24$ | $13.78\pm 0.41$
rT0 (au) | $(7.03\pm 0.20)\times 10^{3}$ | $(1.46\pm 0.14)\times 10^{4}$ | $(1.04\pm 0.27)\times 10^{4}$
$\alpha$ | $2.68\pm 0.16$ | $1.56\pm 0.13$ | $1.00\pm 0.16$
n0 (cm-3) | $(3.95\pm 7.22)\times 10^{5}$ | $(2.06\pm 3.85)\times 10^{5}$ | $(1.27\pm 8.47)\times 10^{6}$
r0 (au) | $(2.60\pm 4.37)\times 10^{3}$ | $(5.00\pm 9.08)\times 10^{3}$ | $(4.20\pm 37.28)\times 10^{2}$
$\alpha$ | $2.40\pm 1.58$ | $2.52\pm 2.88$ | $1.24\pm 1.39$

We may perform a similar computation to derive the parameters of the density
profiles. This time, however, we have information about the total hydrogen
column density in the measured visual extinction across B213-C2-C6,
B213-C10-C12, and B213-C16 in Table LABEL:table:results. Given the density
profile, we can estimate the total hydrogen column density, assuming spherical
symmetry, performing a projection along the line of sight:

${N_{{\rm H}}}(r)=2\times\sum_{i}\Delta l_{i}\frac{n_{\rm H}(s_{i})+n_{\rm
H}(s_{i+1})}{2},$

where $r$ is the impact parameter, $\Delta l_{i}=l_{i+1}-l_{i}$,
$s_{i}=\sqrt{r^{2}+l_{i}^{2}}$, $l_{i}$ is a discretization of the segment
along the line of sight $l_{\rm max}>\dots>l_{i+1}>l_{i}>\dots>0$, with
$l_{\rm max}=\sqrt{r_{\rm max}^{2}-r^{2}}$, and $r_{\rm max}$ the radius of
the density profile. The parameters that best describe the density of the B213
cuts would be those that best fit the visual extinction after the projection
along the line of sight. The total hydrogen column density $N_{H}$ is compared
to the visual extinction in Table LABEL:table:results using the relationship
Av = $N_{H}/1.88\times 10^{21}$ (Bohlin et al., 1978), where it is assumed all
hydrogen is in molecular form. To perform the fitting of the parameters to the
extinction values, we have subtracted the background extinction, that is, the
extinction at the outermost positions, to each cut. The results of the fitting
process are shown in Table 4.

###  6.2 Chemical modelling approaches

Two approaches were used to reproduce our observations with chemical models.
In the first, we used the physical structure of some of the B213 cores in our
sample derived in Section 6.1, and ran a spherical 1D static chemical model to
compute the radial abundance profiles of methanol and CO. In the second
approach, applied to the low-density parts of the filament beyond the starless
cores, we ran a 0D chemical model assuming a grid of temperature and AV to
compute the abundances of methanol.

To model the emission of methanol in cold starless cores, it is essential to
utilize a so-called three-phase astrochemical model. By three phases, we mean
gas phase, surface of icy mantles of interstellar grains, and bulk ice of
grain mantles. The usage of a three-phase model, a more advanced approach in
comparison to two-phase (gas phase + ice without distinction between surface
and bulk), is justified by the importance of the details of chemical processes
on interstellar grains for the formation of methanol and its abundance in the
gas phase. Since Geppert et al. 2006, the consensus is that observed
abundances of interstellar methanol cannot be formed in gas-phase chemical
reactions. In contrast, formation of methanol during hydrogenation of CO
molecules on interstellar grains is shown to be very efficient (e.g. Watanabe
& Kouchi 2002). The delivery of methanol formed on cold ($\sim$10 K) grains to
the gas phase is most likely due to the process of reactive desorption (Garrod
et al., 2007; Minissale et al., 2016). The efficiency of reactive desorption
depends on the details of a particular chemically reacting system and the
composition of the environment, i.e., the underlying ice surface (Minissale et
al., 2016; Chuang et al., 2018). Thus, in order to model properly a
complicated relation between CO and CH3OH in prestellar clouds, we need to
utilize the reasonably detailed approach to grain chemistry and gas-grain
interaction. To model the emission of methanol, we utilized the MONACO model
previously applied to studies of chemistry in a number of prestellar cores
(Vasyunin & Herbst, 2013; Vasyunin et al., 2017; Nagy et al., 2019; Lattanzi
et al., 2020; Harju et al., 2020; Scibelli et al., 2021; Jimenez-Serra et al.,
2021). MONACO is a rate equations-based numerical code capable of simulation
chemistry in interstellar medium under a three-phase approach. As described in
details in Vasyunin et al. (2017), the code includes treatment of chemical
reactions in the gas phase, as well as diffusive chemistry on surfaces of icy
mantles on interstellar grains, and in the bulk of icy mantles. The code
includes treatment of thermal desorption, cosmic ray-induced desorption
(Hasegawa & Herbst, 1993), photodesorption with photodesorption yield equal to
10-3 particles per incident photon (Öberg et al., 2008) and reactive
desorption following parametrization by Minissale et al. (2016). The
efficiency of reactive desorption is further adjusted to the fraction of ice
surface covered with non-water species (Vasyunin et al., 2017). We assume low
diffusion-to-desorption energy ratio Ediff/Edes = 0.3 for thermal diffusion of
species on ice surface. In addition, quantum tunneling as a source of mobility
for atomic and molecular hydrogen is assumed through a rectangular barrier for
diffusion with thickness of 1.2 angstroms (Vasyunin et al., 2017). For
diffusion of bulk species, Ediff/Edes is assumed twice that of for surface,
i.e., 0.6 following Garrod (2013). The results of the spherical 1D static
chemical models are presented in Figures 4 and 5. More details on the physical
profiles used for the cores can be found in Section 6.1.
To convert abundances of species per unit volume to directly observed values
of column densities, we assumed that modeled starless cores are spherically
symmetric with radial density profiles derived as described in Section 6.1.
Convolution with IRAM 30m beam has also been taken into account. The procedure
is described in more details in Jiménez-Serra et al. (2016).

To explore the abundances of CO and CH3OH in the molecular gas that surrounds
starless cores in the filament, we utilized the same chemical model as for the
cores. However in contrast to starless cores, the surrounding gas does not
exhibit a well-shaped spatial structure. Thus, we utilized simple 0D chemical
models with constant values of temperature and density. Since both parameters
can vary in the surrounding gas, we calculated grid of 0D models on a
parameter space that we believe reasonable covers the ranges of temperature
and density in the filament gas. The results of such gridded modeling allows
us to explore the behavior of CO and CH3OH abundances beyond the starless
cores in B213 filament. The results of the 0D models are presented in Figure
6.

###  6.3 Modelling results

Figure 4 compares the N(CH3OH) derived from the spherical 1D static chemical
model used with the physical structures derived for B213 C2-C6, C10-12 and C16
at 105, 106 and 107 years, with the column densities derived in this paper for
B213-C6, C10 and C16 and reported in Table LABEL:table:results. The models at
105 yr reproduces fairly well (within a factor of a few) the column densities
of methanol. The model using the physical structure derived for the cores C2
and C5 (left panel in Figure 4) reproduces very well also the radial profile
of observed column densities of methanol towards B213-C5. We decided to plot
only the N(CH3OH) computed for C5 and not for C2 because the latter has larger
error bars. The radial profiles observed for C10 and C16 (central and right
panels in Figure 4) are instead not very well reproduced by the models. While
it is important to keep in mind the intrinsic large uncertainties of the
column densities computed by chemical models (Vasyunin et al., 2004), we
observe that the changes in N(CH3OH) as a function of radii cannot be
reproduced by the chemical models for C10 and C16, while it can reproduced
fairly well for C6. This is shown even if we look at the radial profiles of
methanol and C18O in Figures 18 to 29, with respect to the one predicted by
the models in Figure 5. It is important to note that the physical structures
used to run the spherical 1D static chemical model have been derived from
$Herschel$ observations, which are not sensitive to the central and densest
part of the cores because of the large beam ($\sim$40$"$). The volume density
in starless cores could in fact have a steep increase towards the inner 40$"$,
especially in the more evolved pre-stellar cores (see for example Figure 5 in
Crapsi et al. 2007). Figure 5 shows the radial distribution of the
N(CH3OH)/N(C18O) column density ratio, as well as the column densities of
CH3OH and C18O (calculated from the main isotopologue assuming a 16O/18O ratio
of 500, Solar value in Penzias 1981) in order to facilitate the comparison
with our observational results shown in Figures 18 to 29. Independently from
the physical structure used, the model predicts a drop of CH3OH column density
of about one order of magnitude from 105 to 106 years, and an additional,
although less significant, drop from 106 to 107 years. The column density of
C18O is underpredicted by the model. However, the model is only considering
the core, and not also the cloud where the core is embedded. To check if the
difference in C18O column densities between model and observation is only due
to the lack of the contribution from the cloud, we have calculated the C18O
column density at offsets outside of the cores, for example offset 7 in
B213-C2, and then added it to the column densities predicted by the model, and
we saw that the contribution of the cloud can account for the missing CO.

In some of our cores the observed cuts cover over 40000 AU, hence merging into
the cloud, while the physical models of the cores are derived assuming a
Bonnor-Ebert sphere with a radius of 10000 AU. We therefore computed a grid of
models in the temperature range of 8-14 K and AV range of 2-8 mag. Please note
in this case the AV is the local AV, so it needs to be multiplied by a factor
of two to be compared with the AV tabulated in Table LABEL:table:results. The
column density of methanol resulting from the grid models at 105, 106 and 107
yr are shown in Figure 6. Also in this case, the models cannot reproduce the
variety of profiles that we observe. They predict a substantial drop of the
methanol in gas-phase between 105 and 106 yr because of freeze-out, as well as
a defined peak of the methanol column density, independent from the dust
temperature, at AV $\sim$ 4 mag (corresponding to a total visual extinction of
$\sim$8 mag along the line of sight). This is in contrast with what we
observe, i.e. that the cores where the dust temperature is higher, tend to
have the methanol peaking at larger AV. It is important however to note that
protostellar feedback (e.g. higher cosmic-ray ionization rate and/or shocks)
is not included in the models.

Figure 4: Methanol column densities calculated with the spherical 1D static
chemical model using the physical structures derived for the cores B213-C2 and
C6 (left panel), C10 (central panel), and C16 (right panel). The yellow, green
and grey squares represent the results of the models extracted at 105, 106,
and 107 yr. The column densities observed towards B213-C6, C10 and C16 are
shown as dark blue, red and light blue in the left, central and right panels,
respectively.

Figure 5: Results of the spherical 1D static chemical model using the physical
structure derived for B213 C2 and C6 (left panel), C10 and C12 (central
vertical panel) and C16 (right panel). The column density profiles of CH3OH
and C18O (red and blue points) as well as their ratio (black points) have been
computed at 105 years (upper panel), 106 (central horizontal panel) and 107
years (lower panel).

Figure 6: Methanol column densities calculated with the grid 0D chemical model
at 105 years (left panel), at 106 years (central panel), and at 107 years
(right panel). The volume density used is a function of AV defined by the
density profiles of C2-C6 cores used for the spherical 1D static chemical
model.

##  7 Conclusions

Our observations of methanol with the GEMS dataset show that methanol is
present and its emission is extended in cold starless cores and in their
surrounding clouds. In our dataset in fact, methanol has been detected at
visual extinction ranging from 4 to 27 mag. Furthermore, the different
behaviours in our sample of starless cores in Taurus show that local
environmental differences are significative and have an impact on the
distribution of methanol.

Previous work on emission maps of methanol towards 4 starless and 2 pre-
stellar cores suggested that the asymmetric distribution of methanol is linked
to the large scale density structure around the cores as well as different
amount of illumination from the external radiation field (Spezzano et al.,
2016, 2020). With this work we can confirm that environmental effects have a
strong impact on the methanol distribution. However, we show that the column
density and temperature of the medium computed on the line-of-sight are not
good enough indicators to identify the effects of the environment on the
methanol, and that the level of star formation activity needs to be taken into
account. It is clear from our data that low-mass protostars in the surrounding
environment have an impact on the distribution of methanol towards starless
cores. Low-mass protostars in fact can accelerate energetic particles and
increase the local cosmic-ray flux and ionisation rate. This can locally
increase the abundance of He+, the main destruction partner of CO in dark
clouds, thus reducing its abundance in the gas and solid form (the latter
being the first step toward CH3OH production). Increased fluxes of cosmic rays
can also increase the CO desorption rate (e.g. Hasegawa & Herbst 1993), again
reducing the amount of available solid CO for its transformation into methanol
via successive hydrogenation. The higher CO desorption rate could compensate
the gas phase destruction of CO via He+; indeed we do not see any local
reduction of the C18O column density around the cores in the North. Dedicated
observations targeting molecular ions are needed to map the variation of
ionization rate across low-mass star forming regions in order to test the
effect on the chemistry with the chemical models. Another consequence of the
presence of low-mass stars nearby is that they contribute to the clumpiness of
the medium, escavated by the cones of the outflows, which could lead to a more
efficient penetration of the interstellar radiation field in the molecular
cloud. In these conditions, only the densest regions (i.e. the centers of the
starless cores) will be screened enough from UV photons to allow CO to
accumulate on the surface of dust grains. The higher dust temperature measured
by $Herschel$ in the northern region of B213 could indeed hint to this
scenario, but higher resolution mapping of the dust continuum emission and gas
tracers are needed to make quantitative conclusions.

The comparison with state-of-the-art chemical models shows that overall the
agreement of the modelled column densities with the observed ones is quite
good, and that reactive desorption is a very efficient method to release the
methanol formed on the surface of dust grains into the gas-phase, reproducing
the observations. The discrepancies among models and observations in the
reproduction of the radial profiles might be due to incomplete
formation/destruction/desorption mechanisms in the chemical models, to the
fact that the protostellar feedback is not included in the models, as well as
to the intrinsic limit of using spherical models to reproduce an asymmetric
emission. Additional experimental and theoretical work on, for example, the
chemical desorption and cosmic ray sputtering on different ice mixtures would
be helpful for a better match between observations and models. Collisional
rates for methanol with molecular hydrogen at temperatures lower than 10 K
would also be beneficial.

###### Acknowledgements.

The authors are grateful to the anonymous referee for insightful comments.
A large part of the data analysis described in this paper was performed during
the spring of 2020, in the beginning of the COVID pandemic and during a hard
lockdown. S.S. wishes to thank the Max Planck Society for the flexibility that
was allowed during the pandemic, because it contributed to maintaining a clear
and focus mind during the hours that she could dedicate to her work, and
overall to keep calm, while waiting for the ”storm” to pass.
Based on analysis carried out with the CASSIS software
(http://cassis.irap.omp.eu;) and CDMS and JPL spectroscopic databases and
LAMDA molecular databases. CASSIS has been developed by IRAP-UPS/CNRS. S.S.
wishes to thank the Max Planck Society for the Independent Max Planck Research
Group funding. AF, DNA and MRB are funded by Spanish MICINN through
PID2010-106235GB-I00 national research project. V.W. acknowledges the CNRS
program ”Physique et Chimie du Milieu Interstellaire” (PCMI) co-funded by the
Centre National d’Etudes Spatiales (CNES). A.V. and A.P. are the members of
the Max Planck Partner Group at the Ural Federal University. A.V. and A.P.
acknowledge the support of the Russian Ministry of Science and Education via
the State Assignment Contract no. FEUZ-2020-0038.

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##  Appendix A Additional information on the single cores

###  A.1 TMC-1 C

We have observed CH3OH and C18O (both lines) towards four positions in TMC-1
C. The $J_{K_{a},K_{c}}$ = 21,2-11,1 ($E_{2}$), 20,2-10,1 ($A^{+}$), 20,2-10,1
($E_{1}$), and 20,2-10,1 ($E_{1}$-$E_{2}$) transitions of methanol have been
observed towards all four positions, while the 10,1-00,0 ($A^{+}$) transition
has been observed only towards the offsets 3 and 4. All lines have been
detected with a signal to noise larger than 3 towards all positions. The
spectra of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid with the
$J$ = 1-0 transition of C18O are shown in Figure 7, and they show a very good
match both in $v_{LSR}$ and in line-width.
Figure 18 shows in the upper panel the observed offsets within TMC-1 C on the
H2 column density map derived from $Herschel$ and $Planck$ data (Rodríguez-
Baras et al., 2021) data, and the variation of the CH3OH and C18O column
density ratio, as well as the single column densities N(CH3OH) and N(C18O) in
the observed cut across TMC-1 C in the lower panel. Both N(CH3OH) and N(C18O)
increase when going towards dust peak of TMC-1 C. The column density ratio is
constant, within error-bars, across the observed cut.

###  A.2 TMC-1 CP

We have observed CH3OH and C18O (both lines) towards four positions in TMC-1
CP. The $J_{K_{a},K_{c}}$ = 21,2-11,1 ($E_{2}$), 20,2-10,1 ($A^{+}$),
20,2-10,1 ($E_{1}$), and 20,2-10,1 ($E_{1}$-$E_{2}$) transitions of methanol
have been observed towards all four positions, while the 10,1-00,0 ($A^{+}$)
transition has been observed only towards the offset 4. All lines have been
detected with a signal to noise larger than 3 towards all positions. The
spectra of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid with the
$J$ = 1-0 transition of C18O are shown in Figure 8. TMC-1 CP is known to have
multiple velocity components (Fuente 2019 and references therein). From the
spectra shown in Figure 8 we see that methanol is tracing two of the many
velocity components seen in C18O.
Two velocity components are present towards TMC-1 CP, one at 5.6 km/s and one
at 6.0 km/s, with the latter having column densities of both CH3OH and C18O
larger by about a factor of 2 with respect to the lower velocity component.
Figure 19 shows the observed offsets on the H2 column density map in the upper
panel, and the variation of the CH3OH and C18O column density ratio, as well
as the single column densities N(CH3OH) and N(C18O) in the observed cut across
the core for both velocity components, in the lower panel. In the lower
velocity component the column density of both methanol and C18O increase
towards the dust peak. Excluding the offset 4, that has a larger error in
comparison with the other positions, also the column density ratio increases
towards the dust peak.

###  A.3 TMC-1 NH3

We have observed CH3OH and C18O (both lines) towards four positions in TMC-1
NH3. The $J_{K_{a},K_{c}}$ = 21,2-11,1 ($E_{2}$), 20,2-10,1 ($A^{+}$),
20,2-10,1 ($E_{1}$), and 20,2-10,1 ($E_{1}$-$E_{2}$) transitions of methanol
have been observed towards all four positions, while the 10,1-00,0 ($A^{+}$)
transition has been observed only towards the offset 4. All lines have been
detected with a signal to noise larger than 3 towards all positions. The
spectra of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid with the
$J$ = 1-0 transition of C18O are shown in Figure 9. The velocity structure
that we can see in the C18O lines is quite complex, and the mismatch with the
peaks of the methanol lines suggests the presence of more velocity components
that we can actually resolve, at least in C18O.
TMC-1 NH3 is the ammonia peak in TMC-1 and supposed to be the most evolved
core, with respect to TMC-1 CP for example (Hirahara et al., 1992). Two
velocity components are present for methanol, while a more complex structure
is present in C18O, see Figure 9. Figure 20 shows in the upper panel the
observed offsets within TMC-1 NH3 on the H2 column density map derived from
$Herschel$ and $Planck$ data (Rodríguez-Baras et al., 2021), and the variation
of the CH3OH and C18O column density ratio, as well as the single column
densities N(CH3OH) and N(C18O) in the observed cut across the core for both
velocity components, in the lower panel. In the component at 5.7 km/s, the
column densities of both molecules have a peak at the offset 3, and then
decreases moving towards the dust peak. The column density ratio shows the
same behaviour. In the component at 6.0 km/s instead, while the column density
of C18O decreases towards the dust peak, the methanol column density has a
peak at the offset 2, and so does the column density ratio.

###  A.4 B213 C1

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C1. The observed lines have been detected with a signal
to noise larger than 3 towards all positions. The spectra of the 21,2-11,1
($E_{2}$) transition of methanol overlaid with the $J$ = 1-0 transition of
C18O are shown in Figure 1. The velocity structure is quite complex, with both
molecules showing multiple velocity components. A shift in the vLSR of the
main velocity component across the observed cut, is also clearly seen.
The upper panel in Figure 21 shows the H2 column density map of B213-C1 as
well as the offsets of our observations. The observed stripe is centred at the
denser peak, labeled as offset 1, but also passes through a less dense tail
towards the East (offsets 6 and 7). In the lower panel of Figure 21 are shown
both the column density ratio of CH3OH/C18O and the single column densities
across the observed stripe, which all decrease rather sharply towards the East
after the offset 3 and peak towards the dust peak.

###  A.5 B213 C2

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C2. The observed lines have been detected with a signal
to noise larger than 3 towards positions 1, 2, 8 and 9. The spectra of the
21,2-11,1 ($E_{2}$) transition of methanol overlaid with the $J$ = 1-0
transition of C18O are shown in Figure 10. The C18O transition shows two
velocity components in all position with the exception of positions 6 and 7.
The methanol transition instead shows a brighter velocity component at $\sim$7
km/s.
The H2 column density map of B213-C2 is shown in the upper panel of Figure 22
with all the offsets observed within B213-C2 in the GEMS large project,
however methanol was only detected towards the densest offsets (1,2 8 and 9).
Both the variation of the CH3OH and C18O column density ratio, as well as the
single column densities N(CH3OH) and N(C18O) are shown in the lower panel. The
column densities of CH3OH and C18O show different behaviours. While C18O is at
its lowest at the dust peak and increases towards the West (offsets 8 and 9),
methanol peaks at the dust peak and decreases towards offsets 8 and 9,
although the error bars on the methanol column density at the dust peak are
rather large so it might be only a slight increase. The column density ratio
also peaks towards the dust peak of B213-C2.

###  A.6 B213 C5

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C5. The observed lines have been detected with a signal
to noise larger than 3 towards all positions. The spectra of the 21,2-11,1
($E_{2}$) transition of methanol overlaid with the $J$ = 1-0 transition of
C18O are shown in Figure 11. The velocity structure is rather complex, with
the methanol line tracing a portion of a lower velocity wing in the C18O line
towards the positions 1, 2 3, and 8. Two main velocity components are visible
in both lines towards the positions 4, 5, and 9. While a single velocity
component in both lines is present at the positions 6 and 7.
The upper panel in Figure 23 shows the H2 column density map of B213-C5 as
well as the offsets of our observations. The observed stripe is centred at a
dense peak towards the West of the map, labeled as offset 1, and also passes
through an another dense clump towards the East (offsets 6 and 7). The column
densities and column density ratio plots in the lower panel of Figure 23 show
a very clear behaviour: the C18O column density decreases when moving from
East to the West in the observed offsets, and both the methanol column density
and the column density ratio have a very well defined peak at the dust peak of
B213-C5 (offset 1).

###  A.7 B213 C6

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C6. The observed lines have been detected with a signal
to noise larger than 3 towards positions 1, 2, 3, 8 and 9. The spectra of the
21,2-11,1 ($E_{2}$) transition of methanol overlaid with the $J$ = 1-0
transition of C18O are shown in Figure 12. In this case, the velocity
structure is quite simple compared to the others cores in B213, with only one
prominent velocity component. Both vLSR and line widths match very well for
C18O and CH3OH.
The H2 column density map of B213-C6 is shown in the upper panel of Figure 24
with all the offsets observed within B213-C2 in the GEMS large project,
however methanol was only detected towards the densest offsets (1,2, 3, 8 and
9). The column densities and column density ratio plots are shown in the lower
panel: the C18O column density decreases towards the denser offsets, while
both the methanol column density and the column density ratio increase towards
the dust peak of B213-C6.

###  A.8 B213 C7

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C7. The observed lines have been detected with a signal
to noise larger than 3 towards positions 1, 2, 3, 4, 5, 8 and 9. Only C18O has
been observed towards the position 6, and neither C18O nor CH3OH have been
observed towards position 7. The spectra of the 21,2-11,1 ($E_{2}$) transition
of methanol overlaid with the $J$ = 1-0 transition of C18O are shown in Figure
13. Different velocity components are present in the different positions. Two
velocity components (at $\sim$6 and 7 km/s) are visible in positions 1, 8 and
9 for C18O, with only the brightest being observed also in methanol. Only the
component at 7 km/s is visible in both molecules towards the positions 2 and
3. Positions 5 and 6 show only one velocity component at $\sim$6 km/s. In
positions 4 two velocity components are also present, but with a smaller
separation in velocity with respect to positions 1, 8 and 9.
The H2 column density map of B213-C7 is shown in the upper panel of Figure 25
with all the offsets observed within B213-C2 in the GEMS large project,
however methanol was only detected towards the densest offsets (1,2, 3, 8 and
9). The column densities and column density ratio plots are shown in the lower
panel of Figure 25. The column density of C18O has a peak at the offset 2 and
then decreases towards the West. The methanol column density, as well as the
column density ratio have an oscillating pattern.

###  A.9 B213 C10

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C10. The observed lines have been detected with a
signal to noise larger than 3 towards all positions except positions 5 and 6.
Only C18O has been observed towards the position 5. The spectra of the
21,2-11,1 ($E_{2}$) transition of methanol overlaid with the $J$ = 1-0
transition of C18O are shown in Figure 14. Two velocity components at
$\sim$5.5 and $\sim$7 km/s are present for C18O towards the positions 1, 2, 3,
7, 8, and 9. Only the brightest velocity component is observed in methanol.
The upper panel in Figure 26 shows the H2 column density map of B213-C10 as
well as the offsets of our observations. Towards B213-C10, methanol was
observed in all offset with the exception of offsets 5 and 6. The column
densities and column density ratio plots are shown in the lower panel of
Figure 26. The C18O column density shows two peaks, one around offsets 2 and
3, and another at offset 8. The methanol column density, as well as the column
density ratio instead have a clear peak only at offsets 2 and 3, towards a
tail of in the H2 column density map.

###  A.10 B213 C12

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C12. The observed lines have been detected with a
signal to noise larger than 3 towards positions 1, 2, 3, 4, 8 and 9. The
spectra of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid with the
$J$ = 1-0 transition of C18O are shown in Figure 15. The C18O lines show a
complex velocity structure, with multiple velocity components unresolved
towards the positions 1, 2, 3, and 4, and two main velocity components towards
positions 8 and 9. The methanol line shows only one velocity components at
$\sim$7 km/s.
The H2 column density map of B213-C12 is shown in the upper panel of Figure 27
with all the offsets observed within B213-C12 in the GEMS large project,
however methanol was only detected towards the densest offsets (1,2, 3 and 8).
Both the variation of the CH3OH and C18O column density ratio, as well as the
single column densities N(CH3OH) and N(C18O) are shown in the lower panel. The
column densities of CH3OH and C18O show different behaviours: C18O increases
from offsets 3 to offset 8, passing through the dust peak at offset 1, while
methanol shows the opposite behaviour. The column density shows the same trend
as the methanol column density.

###  A.11 B213 C16

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C16. The observed lines have been detected with a
signal to noise larger than 3 towards all positions except positions 5 and 6,
where only the line of C18O have been observed. The spectra of the 21,2-11,1
($E_{2}$) transition of methanol overlaid with the $J$ = 1-0 transition of
C18O are shown in Figure 16. The C18O lines show wings in most positions,
hinting at the presence of more than one velocity component on the line of
sight. The methanol lines do not show wings, with the exception of the
position 4.
The upper panel in Figure 28 shows the H2 column density map of B213-C16 as
well as the offsets of our observations. Towards B213-C16, methanol was
observed in all offset with the exception of offsets 5 and 6. The column
densities and column density ratio plots are shown in the lower panel of
Figure 28. The C18O column density shows a very shallow peak around offsets 2
and 3. Both the methanol column density, as well as the column density ratio
show a steep increase moving from offset 2 towards offset 4 (opposite with
respect to the direction of the dust peak), and have a shallower at offsets 1
and 7.

###  A.12 B213 C17

We have observed all lines of CH3OH reported in Table 1, with the exception of
the 10,1-00,0 ($A^{+}$) transition, and the 1-0 transition of C18O towards
nine positions in B213-C17. The observed lines have been detected with a
signal to noise larger than 3 towards positions 1, 2, 3, 7, 8, and 9 for C18O,
and positions 1, 7, 8, and 9 for methanol. The spectra of the 21,2-11,1
($E_{2}$) transition of methanol overlaid with the $J$ = 1-0 transition of
C18O are shown in Figure 17. The lines show a single velocity component at the
positions 1, 7, 8 and 9; while the C18O line shows a double peak structure at
the positions 2 and 3.
The H2 column density map of B213-C17 is shown in the upper panel of Figure 29
with all the offsets observed within B213-C17 in the GEMS large project,
however methanol was only towards few offsets (1, 7, 8 and 9). Both the
variation of the CH3OH and C18O column density ratio, as well as the single
column densities N(CH3OH) and N(C18O) are shown in the lower panel and show a
shallow peak towards the denser offsets observed (7 and 8).

Figure 7: Spectra of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid
with the $J$ = 1-0 transition of C18O in TMC-1 C.  Figure 8: Spectra of the
21,2-11,1 ($E_{2}$) transition of methanol overlaid with the $J$ = 1-0
transition of C18O in TMC-1 CP.  Figure 9: Spectra of the 21,2-11,1 ($E_{2}$)
transition of methanol overlaid with the $J$ = 1-0 transition of C18O in TMC-1
NH3.  Figure 10: Spectra of the 21,2-11,1 ($E_{2}$) transition of methanol
overlaid with the $J$ = 1-0 transition of C18O in B213-C2  Figure 11: Spectra
of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid with the $J$ = 1-0
transition of C18O in B213-C5  Figure 12: Spectra of the 21,2-11,1 ($E_{2}$)
transition of methanol overlaid with the $J$ = 1-0 transition of C18O in
B213-C6  Figure 13: Spectra of the 21,2-11,1 ($E_{2}$) transition of methanol
overlaid with the $J$ = 1-0 transition of C18O in B213-C7  Figure 14: Spectra
of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid with the $J$ = 1-0
transition of C18O in B213-C10  Figure 15: Spectra of the 21,2-11,1 ($E_{2}$)
transition of methanol overlaid with the $J$ = 1-0 transition of C18O in
B213-C12  Figure 16: Spectra of the 21,2-11,1 ($E_{2}$) transition of methanol
overlaid with the $J$ = 1-0 transition of C18O in B213-C16  Figure 17: Spectra
of the 21,2-11,1 ($E_{2}$) transition of methanol overlaid with the $J$ = 1-0
transition of C18O in B213-C17  Figure 18: Upper panel: H2 column density of
TMC-1 C derived from $Herschel$ and $Planck$ data (Rodríguez-Baras et al.,
2021). The triangles mark the positions observed with the GEMS large project.
Lower panel: CH3OH and C18O column densities, and column density ratios
computed in the different offsets in TMC-1 C. In the plot of the CH3OH and
C18O column density ratio, the AV in each offset is marked in red.  Figure 19:
Upper panel: H2 column density of TMC-1 CP derived from $Herschel$ and
$Planck$ data (Rodríguez-Baras et al., 2021). The triangles mark the positions
observed with the GEMS large project. Lower panel: CH3OH and C18O column
densities, and column density ratios computed in the different offsets in
TMC-1 CP in the two velocity components observed for methanol. In the plot of
the CH3OH and C18O column density ratio, the AV in each offset is marked in
red.  Figure 20: Upper panel: H2 column density of TMC-1 NH3 derived from
$Herschel$ and $Planck$ data (Rodríguez-Baras et al., 2021). The triangles
mark the positions observed with the GEMS large project. Lower panel: CH3OH
and C18O column densities, and column density ratios computed in the different
offsets in TMC-1 NH3 in the two velocity components observed for methanol. In
the plot of the CH3OH and C18O column density ratio, the AV in each offset is
marked in red.  Figure 21: Upper panel: H2 column density of B213-C1 derived
from $Herschel$ and $Planck$ data (Palmeirim et al., 2013). The triangles mark
the positions observed with the GEMS large project. The red triangle shows the
position of a Class II/III protostellar core (Rebull et al., 2010).Lower
panel: CH3OH and C18O column densities, and column density ratios computed in
the different offsets in B213-C1. In the plot of the CH3OH and C18O column
density ratio, the AV in each offset is marked in red.  Figure 22: Upper
panel: H2 column density of B213-C2 derived from $Herschel$ and $Planck$ data
(Palmeirim et al., 2013). The triangles mark the positions observed with the
GEMS large project. Lower panel: CH3OH and C18O column densities, and column
density ratios computed in the different offsets in B213-C2. In the plot of
the CH3OH and C18O column density ratio, the AV in each offset is marked in
red.  Figure 23: Upper panel: H2 column density of B213-C5 derived from
$Herschel$ and $Planck$ data (Palmeirim et al., 2013). The triangles mark the
positions observed with the GEMS large project. The red triangles show the
positions of Class I/flat (full triangle) or Class II/III (empty triangle)
protostellar cores (Rebull et al., 2010). Lower panel: CH3OH and C18O column
densities, and column density ratios computed in the different offsets in
B213-C5. In the plot of the CH3OH and C18O column density ratio, the AV in
each offset is marked in red.  Figure 24: Upper panel: H2 column density of
B213-C6 derived from $Herschel$ and $Planck$ data (Palmeirim et al., 2013).
The triangles mark the positions observed with the GEMS large project. Lower
panel: CH3OH and C18O column densities, and column density ratios computed in
the different offsets in B213-C6. In the plot of the CH3OH and C18O column
density ratio, the AV in each offset is marked in red.  Figure 25: Upper
panel: H2 column density of B213-C7 derived from $Herschel$ and $Planck$ data
(Palmeirim et al., 2013). The triangles mark the positions observed with the
GEMS large project. Lower panel: CH3OH and C18O column densities, and column
density ratios computed in the different offsets in B213-C7. In the plot of
the CH3OH and C18O column density ratio, the AV in each offset is marked in
red.  Figure 26: Upper panel: H2 column density of B213-C10 derived from
$Herschel$ and $Planck$ data (Palmeirim et al., 2013). The triangles mark the
positions observed with the GEMS large project. The red triangle shows the
position of a Class I/flat protostellar core (Rebull et al., 2010). Lower
panel: CH3OH and C18O column densities, and column density ratios computed in
the different offsets in B213-C10. In the plot of the CH3OH and C18O column
density ratio, the AV in each offset is marked in red.  Figure 27: Upper
panel: H2 column density of B213-C12 derived from $Herschel$ and $Planck$ data
(Palmeirim et al., 2013). The triangles mark the positions observed with the
GEMS large project. The red triangles show the positions of Class I/flat
protostellar cores (Rebull et al., 2010). Lower panel: CH3OH and C18O column
densities, and column density ratios computed in the different offsets in
B213-C12. In the plot of the CH3OH and C18O column density ratio, the AV in
each offset is marked in red.  Figure 28: Upper panel: H2 column density of
B213-C16 derived from $Herschel$ and $Planck$ data (Palmeirim et al., 2013).
The triangles mark the positions observed with the GEMS large project. The red
triangles show the positions of Class I/flat (full triangle) or Class II/III
(empty triangle) protostellar cores (Rebull et al., 2010). Lower panel: CH3OH
and C18O column densities, and column density ratios computed in the different
offsets in B213-C16. In the plot of the CH3OH and C18O column density ratio,
the AV in each offset is marked in red.  Figure 29: Upper panel: H2 column
density of B213-C17 derived from $Herschel$ and $Planck$ data (Palmeirim et
al., 2013). The triangles mark the positions observed with the GEMS large
project. The red triangle shows the position of a Class I/flat protostellar
core (Rebull et al., 2010). Lower panel: CH3OH and C18O column densities, and
column density ratios computed in the different offsets in B213-C17. In the
plot of the CH3OH and C18O column density ratio, the AV in each offset is
marked in red.  Figure 30: AV and Tdust in the B213 cores within our sample
are plotted against each other to show the effect of the larger stellar
activity in the northern part of B213. In this Figure, the cores in the North
of B213 are plotted in different shades of red that go from darker to lighter
when moving from North to South. The cores in the central and southern part of
B213 are plotted in different shades of blue, with the darkest being the core
located more to the South.  Figure 31: H2 column density map of the B213
filament derived from $Herschel$ and $Planck$ data (Palmeirim et al., 2013).
The young stellar objects reported in Rebull et al. (2010) are shown as
triangles.