11institutetext: Laboratoire d’Astrophysique de Bordeaux, Univ. Bordeaux,
CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France
11email<EMAIL_ADDRESS>22institutetext: Purple Mountain
Observatory, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210023,
China 33institutetext: Kyushu Kyoritsu University, Jiyugaoka 1-8, Yahatanishi-
ku,Kitakyushu, Fukuoka, 807-8585, Japan 44institutetext: SRON Netherlands
Institute for Space Research, Landleven 12, 9747 AD Groningen, The Netherlands
55institutetext: Kapteyn Astronomical Institute, University of Groningen, NL
66institutetext: Laboratoire d’Astrophysique (AIM), CEA/DRF, CNRS, Université
Paris-Saclay, Université Paris Diderot, Sorbonne Paris-Cité, 91191 Gif-sur-
Yvette, France 77institutetext: Research Center for Intelligent Computing
Platforms, Zhejiang Laboratory, Hangzhou 311100, China 88institutetext:
Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China

#  Dense gas and star formation in the Outer Milky Way

J. Braine 11 Yan Sun 22 Y. Shimajiri 33 F.F.S. van der Tak 4455 Min Fang 22
Ph. André 66 Hao Chen 77 Yu Gao 8822

(Received xxxx; accepted xxxx)

We present maps and spectra of the HCN(1–0) and HCO+(1–0) lines in the extreme
outer Galaxy, at galactocentric radii between 14 and 22 kpc, with the 13.7
meter Delingha telescope. The 9 molecular clouds were selected from a CO/13CO
survey of the outer quadrants. The goal is to better understand the structure
of molecular clouds in these poorly studied subsolar metallicity regions and
the relation with star formation. The lines are all narrow, less than $2{\rm\
km\ s^{-1}}$ at half power, enabling detection of the HCN hyperfine structure
in the stronger sources and allowing us to observationally test hyperfine
collision rates. The hyperfine line ratios show that the HCN emission is
optically thin with column densities estimated at $N$(HCN) $\approx 3\times
10^{12}$cm-2. The HCO+ emission is approximately twice as strong as the HCN
(taken as the sum of all components), in contrast with the inner Galaxy and
nearby galaxies where they are similarly strong. For an abundance ratio
$\chi_{HCN}/\chi_{HCO^{+}}=3$, this requires a relatively low density solution
for the dense gas, with n(H2) $\sim 10^{3}-10^{4}$cm-3. The 12CO/13CO line
ratios are similar to solar neighborhood values, roughly 7.5, despite the low
13CO abundance expected at such large radii. The HCO+/CO and HCO+/13CO
integrated intensity ratios are also standard at about 1/35 and 1/5
respectively. The HCN is weak compared to the CO emission, with HCN/CO $\sim
1/70$ even after summing all hyperfine components. In low-metallicity
galaxies, the HCN deficit is attributed to a low [N/O] abundance ratio but in
the outer disk clouds it may also be due to a low volume density. At the
parsec scales observed here, the correlation between star formation, as traced
by 24 $\mu$m emission as is standard in extragalactic work, and dense gas via
the HCN or HCO+ emission, is poor, perhaps due to the lack of dynamic range.
We find that the lowest dense gas fractions are in the sources at high
galactic latitude ($b>2^{\circ}$, $h>300$pc above the plane), possibly due to
a lower pressure.

###### Key Words.:

Galaxies: Individual: Milky Way – Galaxies: ISM – ISM: clouds – ISM: Molecules
– Stars: Formation

##  1 Introduction

A general theory of star formation needs to be tested in multiple
environments. Most work (see e.g. Kennicutt & Evans, 2012) has studied either
galactic disk star formation, star formation in bright and perturbed
environments such as mergers or ULIRGs, or bright low metallicity objects such
as blue dwarf galaxies. Observations of the outer Galaxy provide a new
environment for the study of the star formation cycle (H i to $\rm H_{2}$ to
stars). The outer disk is cooler and has a lower gas content and in particular
less molecular gas than the brighter regions mentioned above. The star
formation rate is low (see e.g. Sodroski et al., 1997) and the metallicity is
subsolar but not extremely low in the outer disk (gradient of $\approx 0.04$
dex kpc-1, Pedicelli et al., 2009). The morphology and gravitational potential
are those of a rotating disk, unlike mergers and many dwarfs. Finally, the
outer disk, defined as beyond the R25 radius, is typically where the gas
surface density exceeds that of stars (see e.g. Figure 7 of Kennicutt & Evans
(2012) or Figure 1 of Hoekstra et al. (2001)). For a population of sun-like
stars, a brightness of 25 magnitudes per square arcsecond, defining the R25
radius, corresponds to a stellar surface density of 6.6M⊙pc-2. Hence, the
outer Galaxy represents a new environment (where $\Sigma_{\rm
gas}\ga\Sigma_{*}$) for the study of the star formation cycle.

A further motivation comes from the observations and simulations showing that
galaxies continue to be fueled by inflowing gas (see Linsky, 2003; Schmidt et
al., 2016, and references therein), particularly when a bar (i.e. a non-
axisymmetric potential) is present as is the case for the Galaxy. Thus the
outer disk represents a part of the future of our Galaxy.

The star formation rate (hereafter SFR) at kpc scales is well-known to be
linked to the presence of gas and particularly molecular gas (e.g. Kennicutt,
1989). Closer to cloud scales, the cloud life-cycle (pre-SF, embedded SF,
exposed SF, cloud dispersal) is such that much more scatter is present in the
SFR-$\rm H_{2}$ relation. More recently, the link between the dense gas mass,
generally as traced by the high-dipole moment molecules HCN or HCO+, has been
shown to be approximately linear in galactic disks at both kpc (e.g. Gao &
Solomon, 2004b, a; Kepley et al., 2014; Chen et al., 2017; Braine et al.,
2017) and pc scales (e.g. Wu et al., 2010; Shimajiri et al., 2017) with
approximately the same SFR/Mgas ratio (see Fig. 13 of Jiménez-Donaire et al.,
2019). In galactic centers, the SFR/Mgas ratio is clearly lower than in disks
(see lower panel of Fig. 7 in Chen et al., 2015; Usero et al., 2015). Table 5
and Figure 13 of Jiménez-Donaire et al. (2019) show that over about 10 orders
of magnitude in $L_{\rm HCN}$, there is a scatter of 0.37 dex in the $L_{\rm
IR}$/$L_{\rm HCN}$ ratio, where $L_{\rm IR}$ traces the SFR, in agreement with
Gallagher et al. (2018).

IRAS Far-IR observations and the follow-up in CO by Wouterloot & Brand (1989)
revealed molecular gas out to a galactocentric distance of about 18 kpc.
Perhaps surprisingly, Digel et al. (1994) found several molecular clouds
beyond 18 kpc but they are all much less CO-bright (although of comparable
mass) than Orion viewed at a similar distance and resolution. The atomic gas
extends considerably further (e.g. Kalberla & Kerp, 2009) but with little or
no evidence of star formation and young stars.

Recent increases in mapping speed and receiver bandwidth combined with post-
IRAS Far-IR surveys such as Herschel Hi-Gal (Molinari et al., 2016) and the
APEX Atlasgal (Schuller et al., 2009) has led to revived interest in the outer
Galaxy. Sun et al. (2015) surveyed a vast region covering $100<l<150^{\circ}$
and $-3<b<5.25^{\circ}$ in the $J=1-0$ transition of CO and isotopologues to
determine how much molecular gas is present at large galactocentric distances.
In addition to previously known clouds, they found 49 previously undetected
extreme outer Galaxy clouds ($R_{gal}>14$kpc). The CO emission is generally
quite weak, with integrated intensities below $10{\rm\ K}{\rm\ km\ s^{-1}}$ at
the CO peaks. 13CO(1–0) and C18O(1–0) were observed simultaneously and the
images and spectra are presented in Sun (2015). 13CO(1–0) emission is detected
in nearly half of the sources. The 12CO/13CO line ratio is typically about 5
in peak temperature and 7 in integrated intensity; the C18O lines were not
detected (see Sect. 4.1).

Sun (2015) started a comparison between star formation and the presence of
molecular gas in the outer Galaxy. One of the results of her thesis was that
while star formation was not always associated with molecular gas in these
extreme outer Galaxy clouds, the clouds hosting young stellar objects (YSOs)
were generally more massive than those without YSOs. Here we pursue the
comparison between tracers of star formation and molecular gas in the outer
Galaxy, focussing on the link between dense gas and star formation. At kpc
scales, there is a good correlation between the dense gas mass and the star
formation rate (Chen et al., 2015) as for whole galaxies (Gao & Solomon,
2004b). The low-$J$ transitions of the high dipole moment molecules HCN and
HCO+ are standard tracers of dense gas because high densities (close to
$10^{5}$ cm-3) are required for collisional excitation (Evans, 1999). However,
the lines are much (factor 10–100) weaker than the CO lines so the existing
12CO(1–0) and 13CO(1–0) observations were used to define regions to be
observed in HCN and HCO+ in the extreme outer Galaxy. The dataset obtained is
used to explore the physical conditions of the molecular medium far out in the
Galaxy and the link between dense gas and star formation in a new environment
(e.g. Glover & Clark, 2012). There is evidence that HCN is more sensitive to
metallicity variations than HCO+ (Rudolph et al., 2006; Nishimura et al.,
2016; Braine et al., 2017). The outer Galaxy has a subsolar metallicity but
also a generally weak UV field and thus provides complementary information to
observations of low metallicity galaxies, which often have high UV-fields.

Only a few observations of dense gas tracers in the outer Galaxy have been
made (Yuan et al., 2016) so we significantly increase this sample. Yuan et al.
(2016) observed a large sample of Planck Galactic Cold Clumps, detected in the
sub-mm range by the Planck satellite and generally close to the Sun. The
samples are quite different in that the Sun et al. (2015) survey was blind and
none of their 72 sources were detected in the C18O line. Very recently,
Fontani et al. (2022) observed molecular lines, including HCN and HCO+, in a
sample of actively star-forming IRAS-selected cores and Patra et al. (2022)
mapped the HCN and HCO+(1-0) lines around regions with massive stars, also in
the outer Galaxy. These two samples are quite different from ours, as we will
see (Sect. 4.2).

The paper is structured as follows: after presenting the observations and the
basic results, we estimate the excitation temperatures, optical depths, and
column densities of the CO lines and then for the HCN, where the satellite
line strengths are used to estimate the optical depth. In this part, we
initially follow Yuan et al. (2016). The following section presents radiative
transfer calculations of the HCN line ratios and also the HCN/HCO+ line ratio,
in order to constrain the volume density and determine whether the CO-emitting
gas can also be responsible for the HCN and HCO+ line emission. We then
discuss, with in particular the goal of understanding why the line ratios we
measure are not typical of galactic or most extragalactic observations, and
summarize the results for this rather unique data set.

Figure 1:  WISE 22 $\mu$m band images of sources 7, 14, 16, 18, 30, and 34
(identification in upper right corner). These are the sources for which only a
single pointing is available. The images are all 10 arcminutes in size,
oriented in $l$ and $b$. The central circle gives the HCN/HCO+ beamsize. The
transfer function (rainbow from black to white) of the WISE observations
differs from image to image. For cloud 7 it is from 91.5 to 98; for cloud 14
it is 100.5 to 107; for 16 it is 103 to 115; for 18 it is 102 to 114; for 30
it is from 108 to 112 and for cloud 34 it is from 108 to 135 counts. Following
Sun (2015), class I sources within our maps are shown as circles and class II
as squares. Figure 2:  WISE 22 $\mu$m band images of sources 21, 56, and 57
(identification in upper right corner). These are the sources which were
mapped. The images are all $600"\times 400"$ in size, oriented in $l$ and $b$.
The color scale (rainbow from black to white) of the WISE observations differs
from image to image. For cloud 21 it is from 95 to 116 and for clouds 56 and
57 the range is from 129 to 134 counts. The central circle gives the HCN/HCO+
beamsize and is the (0,0) position in Figures 4, 5, and 6 which show the
spectra.The box indicates the size of the region shown in Figures 4, 5, and 6.
Following Sun (2015), class I sources within our maps are shown as circles and
class II as squares. Figure 3: Spectra for the clouds that have not been
mapped, as in Fig. 1. The axes are the same for each cloud. 12CO is in black
and corresponds to the temperature scale. 13CO is in red and has been
multiplied by 5. HCO+, in green, has been multiplied by 35. The HCN spectra
have been multiplied by 35 for comparison with HCO+ but also multiplied by 9/5
to account for the satellite lines as explained in the text. The 12CO and 13CO
spectra have been convolved with a gaussian kernel to the angular resolution
of the HCN and HCO+ spectra. Figure 4: Spectra of Cloud 21. Positions are
indicated on the $x$ and $y$ axes as offsets in arcminutes and each box
represents 30$"$. 12CO is in black, 13CO in red, HCO+ in green and HCN in
blue. The velocity scale goes from $-106.5{\rm\ km\ s^{-1}}$ to $-97{\rm\ km\
s^{-1}}$. The temperatures scales are -.7 to 7K for 12CO, -.14 to 1.4K for
13CO, -.02 to .2 for HCO+ and for HCN. The HCN line (only central component
shown) has been multiplied by 9/5 so that the line intensity represents the
total line flux for comparison with other species. The 12CO and 13CO spectra
have been convolved to the same angular resolution as the HCN and HCO+. Figure
5: Spectra of Cloud 56. Positions are indicated on the $x$ and $y$ axes as
offsets in arcminutes and each box represents 30$"$. 12CO is in black, 13CO in
red, HCO+ in green and HCN in blue. The velocity scale goes from $-107.5{\rm\
km\ s^{-1}}$ to $-99{\rm\ km\ s^{-1}}$. The temperatures scales are -.7 to 7K
for 12CO, -.14 to 1.4K for 13CO, -.02 to .2 for HCO+ and for HCN. The HCN line
(only central component shown) has been multiplied by 9/5 so that the line
intensity represents the total line flux for comparison with other species.
The spectra in the corners show only CO as not all positions were observed in
HCN and HCO+. Figure 6: Spectra of Cloud 57. Positions are indicated on the
$x$ and $y$ axes as offsets in arcminutes and each box represents 30$"$. 12CO
is in black, 13CO in red, HCO+ in green and HCN in blue. The velocity scale
goes from $-106.5{\rm\ km\ s^{-1}}$ to $-98{\rm\ km\ s^{-1}}$. The
temperatures scales are -.7 to 7K for 12CO, -.14 to 1.4K for 13CO, -.02 to .2
for HCO+ and for HCN. The HCN line (only central component shown) has been
multiplied by 9/5 so that the line intensity represents the total line flux
for comparison with other species. The spectra in the corners show only CO as
not all positions were observed in HCN and HCO+. Figure 7:  Sum of spectra for
each of the clouds for which we were able to make maps. The lines are narrow
although each spectrum represents a sum of many positions. The clear
separation of the hyperfine components enables us to estimate the optical
depth. The parameters of the gaussian fits shown in green are given in Table
2. Figure 8:  Local Thermodynamic Equilibrium representation of hyperfine
component ratios as a function of main component (F=2-1) optical depth. The
observed ratios for clouds 21, 56, and 57 are indicated. The vertical dashed
lines indicate the uncertainties from Table 3. Figure 9:  Integrated
intensities of HCN and HCO+ lines for all positions observed. All data are on
the main beam temperature scale and the central HCN intensity has been
multiplied by 9/5 to account for the satellite hyperfine components. The line
indicates $I_{\rm HCO^{+}}=2I_{\rm HCN}$, close to the best-fit value of 1.92.
Class I and II sources are indicated by respectively red circles and squares.
Figure 10:  Integrated intensities of 12CO and 13CO lines for all positions
observed. The line indicates the best-fit: $I_{{}^{13}{\rm CO}}=\frac{I_{\rm
CO}}{8}$. Class I and II sources are indicated by respectively red circles and
squares.

##  2 Cloud sample and Observations

The sources were selected from the Sun et al. (2015) sample of outer Galaxy CO
clouds. All of the clouds are quite far from the galactic center, ranging from
14 to 22 kpc with the Reid et al. (2014) rotation curve. Although a different
rotation curve will yield slightly different results, these are clearly far
outer Galaxy objects, close to or beyond twice the solar circle distance.
Sources were chosen to have ($a$) an integrated CO(1–0) intensity $I_{\rm
CO}>7$ K km s-1 and ($b$) $I_{{}^{13}\rm CO}\geq 1.4$ K km s-1 and ($c$) a
peak 13CO temperature above 0.7 Kelvin. The 13CO data are from Sun (2015) and
the cloud characteristics, 12CO, and 13CO intensities are provided in Table 1.
This yielded a sample of nine clouds: 7, 14, 16, 18, 21, 30, 34, 56 and 57,
all of which have 13CO sizes between 2 and 5 square arcminutes and peak CO
temperatures above 4 Kelvins. The central positions of all clouds were
observed and clouds 21, 56, and 57, the brightest in HCN and HCO+, were
mapped. Fig. 1 presents WISE 22 $\mu$m band images, supposed to trace star
formation, with the positions observed and symbols indicating class I and
class II YSOs. Fig. 2 shows the same for the three clouds that were mapped in
HCN and HCO+ along with contours of CO intensity. The WISE 22 $\mu$m
band111https://irsa.ipac.caltech.edu/Missions/wise.html was chosen as it was
the only tracer of star formation available for all sources.

The YSOs within the clouds were identified with the criteria described in
Koenig et al. (2012) using the infrared data from the Two Micron All Sky
Survey (2MASS: Skrutskie et al., 2006) and the Wide-field Infrared Survey
Explorer (WISE: Wright et al., 2010). We restrain our sources to those with
photometric errors less than 0.1 mag for 2MASS data and signal-to-noise ratio
greater than 5 for WISE data. The criteria in Koenig et al. (2012) are
designed to search for young stars at Class I and Class II stages employing
two methods. The first is based on photometry in the WISE 3.4, 4.6 and 12 $\mu
m$ bands. The contamination from extragalactic sources (star-forming galaxies
and AGNs), shock emission blobs, resolved PAH emission objects can be removed
according to their locations in the [3.4]$-$[4.6] vs. [4.6]$-$[12] color-color
diagram and their WISE photometry. For the sources not detected in the WISE 12
$\mu m$ band, YSOs are identified from the dereddened $K_{\rm s}-$[3.4] vs.
[3.4]$-$[4.6] color-color diagram. In our work, the extinction used to
deredden the photometry is estimated from its location in the $J-H$ vs.
$H-K_{\rm s}$ color-color diagram as described in Fang et al. (2013).

The nominal central positions observed correspond to the CO maxima and not
necessarily a peak in star formation, as can be seen in Figures 1 and 2. Below
we give the assessment of the star formation for each observed region (Sun,
2015), starting with the single positions and followed by the maps.
Cloud 7: An IRAS point source is present as well as a class II source within
the DLH beam.
Cloud 14: No IRAS point source within the beam but a class I source at beam
center.
Cloud 16: No IRAS point source but there is a class II source within the DLH
beam.
Cloud 18: No IRAS point source but there is a class II source within the DLH
beam.
Cloud 30: No IRAS point source but there is a class II source within the DLH
beam.
Cloud 34: Class II source within DLH beam and IRAS source nearby (see Fig. 1).
Cloud 21: IRAS point source and class II source at edge of central beam but
within map, which also covers a class I source. Map shows no obvious
difference in line ratios at positions with star formation.
Cloud 56: No IRAS point source within but a class II source near nominal
center. Star formation (22 $\mu$m emission) extends to higher $b$ and HCN/HCO+
emission appears to follow.
Cloud 57: IRAS point source at center and two class I sources, one coincident
with the IRAS source. A class II source is also present further from the
center.
Class I sources are indicated as circles in Figures 1 and 2. Class II are
shown as squares, following Sun (2015).

The observations were carried out in May 2017 with the 13.7m telescope of the
Purple Mountain Observatory (PMO) in Delingha, Qinghai, China, hereafter DLH.
The beamsize at the HCN and HCO+ $J=1-0$ frequencies is 1′ which corresponds
to a linear scale of 3 pc at the typical distance of 10 kpc of our clouds
(except clouds 56 and 57 which are further such that the beam size is about
4.3 pc). The Superconducting Spectroscopic Array Receiver system, a nine-beam
(3X3beam, with separation of 3 arcmin), sideband- separating receiver (Shan et
al. 2012) was used as front end. The fast fourier transform spectrometer with
a bandwidth of 1 GHz provides 16384 channels and a spectral resolution of 61
kHz (see details in Shan et al., 2012), leading to a velocity resolution of
0.2 km s-1. The telescope is at an altitude of 3200 meters and the weather was
good during the observations, leading to system temperatures of 140 – 150K.

The standard chopper-wheel calibration was adopted to obtain the antenna
temperature, $T_{\rm A}^{*}$ (Kutner & Ulich, 1981). Spectral intensities are
further converted to a scale of main-beam brightness temperature using $T_{\rm
MB}=T_{\rm A}^{*}/\eta_{MB}$, where the main-beam efficiency $\eta_{\rm MB}$
is 0.57 at $\sim$90 GHz (see the status report of the 13.7m telescope
222http://english.dlh.pmo.cas.cn/fs/). Considering the small angular size of
our targets traced by CO, and the smaller region traced by dense gas, the
position-switching mode was used. Two beams with a separation of 6 arcmin were
used to observe on-source pointing and emission-free OFF position
simultaneously. The pointing accuracy is estimated to be better than 5 arcsec.
The mapping step is 30 arcsec.

##  3 Results

###  3.1 Maps and Spectra

Table 1 provides source numbers, positions, 12CO intensities, galactocentric
distance Rgal, heliocentric distance, all from Sun et al. (2015), and 13CO
intensities from Sun (2015). The CO data are on the main beam scale (i.e.
corrected for telescope efficiency) and come from observations with the
Delingha telescope which has a beam size of 48$"$ at the CO(1–0) frequency.

In Galactic observations, several clouds can be found along the line of sight
at different velocities. More common in the inner Galaxy, this is also the
case for these outer Galaxy observations. The outer Galaxy CO survey (Sun et
al., 2015; Sun, 2015) covered not only the peaks but a vast region in the
outer Galaxy and thus maps of each cloud. The spectra and maps of the clouds
in the appendix of Sun (2015) show that generally the CO and 13CO is strongest
in the distant cloud we are interested in and that even when this is not the
case (only cloud 30), the clouds can be visually separated in the maps.

Cloud # | $l$ | $b$ | ICO | Rgal | dist | I${}_{\rm{}^{13}CO}$
---|---|---|---|---|---|---
| $\circ$ | $\circ$ | K km s-1 | (kpc) | (kpc) | K km s-1
7 | 104.983 | 3.317 | 11.4 | 14.5 | 9.9 | 1.6
14 | 109.292 | 2.083 | 11.5 | 14.6 | 9.5 | 1.7
16 | 109.500 | 2.608 | 7.4 | 14.5 | 9.3 | 1.4
18 | 109.792 | 2.717 | 10.3 | 14.4 | 9.2 | 1.8
21 | 114.342 | 0.781 | 18.0 | 15.0 | 9.5 | 2.4
30 | 121.817 | 3.050 | 13.7 | 16.4 | 10.3 | 2.0
34 | 122.775 | 2.525 | 12.8 | 17.1 | 11.0 | 1.9
56 | 137.759 | -0.983 | 17.6 | 21.7 | 14.7 | 3.1
57 | 137.775 | -1.067 | 12.2 | 21.4 | 14.4 | 1.6
Table 1: Cloud sample: source numbers, positions, 12CO intensities,
galactocentric distance Rgal, heliocentric distance, all from Sun et al.
(2015), and 13CO intensities from Sun (2015).

The new observations of the HCN(1–0) and HCO+(1–0) lines are shown along with
the pre-existing 12CO(1–0) and 13CO(1–0) lines in Figures 3, 4, 5, and 6. The
temperature scales are the same from one figure to another so that the figures
can be straightforwardly compared. We immediately see that the HCN(1–0) and
HCO+(1–0) lines were detected in all sources and that the HCO+ line is
systematically stronger than the HCN line. Central velocities are very similar
(generally $\pm$ one channel) and the optically thick 12CO line is generally
broader (by $\sim 40$%) than the dense gas tracers (compare Table 2 with Table
1 of Sun et al., 2015).

Table 2 presents the results of the HCN and HCO+observations. The integrated
intensity, velocity, and line width are given for the central spectra in the
HCO+(1-0) line and for the main line of HCN(1-0). Clouds 21, 56, and 57 were
partially mapped in HCN and HCO+and the lines indicated by sum are the fits to
the co-added (i.e. summed) spectra. The summed spectra have considerably
higher signal-to-noise ratios (see last column) so the individual hyperfine
components were fit and the results are presented in the same way.

###  3.2 HCN hyperfine structure

HCN and HCO+are considered to be tracers of dense gas due to their high
electric dipole moment. Their critical densities, where collisional de-
excitation equals spontaneous de-excitation (see Eq. 4 and Table 1 of Shirley,
2015), are $n_{crit,HCN(1-0)}\approx 4\times 10^{5}$ cm-3 and
$n_{crit,HCO^{+}(1-0)}\approx 6\times 10^{4}$ cm-3. These densities are
roughly 2 orders of magnitude higher than for CO so the typical molecular
cloud densities at which CO is collisionally excited do not suffice to
generate detectable HCN or HCO+ emission. The outer galaxy environment is
cool, with a weak radiation field, and as we shall see the HCN lines are
optically thin or nearly so.

Due to the nuclear quadrupole moment of Nitrogen (14N), there is hyperfine
structure which is observable in the ground-state rotational transition
($J=1\rightarrow 0$). Three lines are present and the line ratios can be used
to estimate optical depth. The main line is the $F=2\rightarrow 1$ transition
at 88.63185 GHz, with satellite lines $F=1\rightarrow 1$ and $F=0\rightarrow
1$ shifted by respectively 4.8 and $-7{\rm\ km\ s^{-1}}$. Figure 7 shows the
co-added HCN(1-0) spectra for clouds 21, 56, and 57. The individual spectra do
not have a sufficiently high signal-to-noise ratio to measure the HFS at
individual positions, particularly for the weakest component. The spectra are
presented for each cloud separately in Figures 4, 5, and 6, which show that
the co-added spectra are from regions with strong CO (and 13CO) emission. The
second-strongest component is clearly visible in the individual spectra of
cloud 56 (Fig. 5) which has the lowest noise level (see Table 2). Generally,
when spectra are stacked, the velocity is recentered because the stacking
concerns gas at different velocities – this shifting introduces additional
uncetainties because of the way the velocity difference is determined. This is
not the case here which is why we say “summed“ or “co-added“ rather than
stacked.

###  3.3 Relative line intensities

As can be seen in Figures 9 and 10, the CO and 13CO, and HCN and HCO+ line
fluxes are correlated. The relation is approximately $I_{\rm HCO^{+}}\approx
0.03I_{\rm CO}\approx 0.21I_{{}^{13}{\rm CO}}\approx 2I_{\rm HCN}$ for the set
of data. The correlation coefficients are $r=0.62$ between dense gas tracers
and $r=0.65$ between the 12CO and 13CO lines. Mixing the dense and total gas
tracers, for respectively CO-HCO+, CO-HCN, 13CO-HCO+, and 13CO-HCN, the
correlation coefficients are 0.53, 0.44, 0.44, and 0.43. Hence, the total gas
mass (clouds) and dense gas mass (cores) are linked but the dense gas tracers
(HCN, HCO+) are more strongly correlated between each other, as are the
general molecular gas tracers (CO, 13CO), than between clouds and cores.

##  4 Physical conditions from an LTE analysis

In this section we apply the standard formulae to estimate optical depths and
column densities of the species presented here. The formulae are presented in
Yuan et al. (2016), Mangum & Shirley (2015), and Shirley (2015) and the works
cited therein. We initially follow the reasoning in Yuan et al. (2016) for
straightforward comparison and then present another approach. Without making
ad hoc hypotheses, it is difficult to make non-LTE (local thermodynamic
equilibrium) calculations so the following section presents radiative transfer
calculations with RADEX (van der Tak et al., 2007).

The observations are of clouds between 14 and 22 kpc from the Galactic center
so we expect abundances to be subsolar (e.g. Pedicelli et al., 2009) and we
follow the Pineda et al. (2013) 12/13C gradient (see Eq. 4 in Yuan et al.,
2016) and the Fontani et al. (2012) 12CO abundance gradient (see Eq. 7 in Yuan
et al., 2016).

###  4.1 Molecular column densities: 12CO(1-0) and 13CO(1-0)

Following Yuan et al. (2016) Eqs. 3 and 4, we estimate the 13CO(1-0) optical
depth, $\tau_{13}$, and, multiplying by the 12/13C ratio, the optical depth of
the main line, $\tau_{12}$. This assumes that excitation temperatures and
surface filling factors (fraction of the beam occupied by emitting molecules)
are the same for both isotopologues. The 13CO(1-0) emission is optically thin,
with $0.15<\tau_{13}<0.32$ calculated in this way, yielding main isotope
optical depths of order 20. C18O(1-0) emission was not detected in these
sources, confirming the low optical depths of the 13CO(1-0) emission.

Assuming a filling factor of unity, the observed brightness temperature can be
used to calculate the excitation temperature of the 12CO(1-0) line. The
optical depths obtained previously mean that the main line is highly optically
thick so that it does not enter in the excitation temperature calculation (see
e.g. Eq. 88 of Mangum & Shirley, 2015) but also that the 13CO(1-0) column
density is almost independent of $\tau_{13}$; but we need to assume that the
excitation temperatures are the same (or known) for both isotopologues as in
Yuan et al. (2016). The excitation temperatures vary from 7.3 to 11.4 K for
the nine sources, apparently reasonable values for the outer Galaxy. Table 3
shows the optical depths, excitation temperatures, CO column densities, and H2
column density based on the 12CO and 13CO column density (see Sect. 4.5).
Table 4 provides the limits to the C18O(1-0) emission, obtained by averaging
C18O and 13CO spectra over the maps.

However, as explained in Shirley (2015), the effective critical density is
lower than the standard value for highly optically thick lines such as
12CO(1-0) but not the optically thin 13CO(1-0). As a result, we expect that
our hypothesis of equal excitation temperatures is inappropriate and we test
lower excitation temperatures for 13CO(1-0), halfway between the CMB
temperature and the 12CO(1-0) excitation temperature
$T_{ex,13}=T_{bg}+0.5\,(T_{ex,12}-T_{bg})$, so that the 13CO(1-0) alternative
excitation temperatures range from 5 to 7 K. While ad hoc, this preserves the
order among the sources and remains sufficiently far from $T_{bg}$ that the
column density is not close to diverging. The results with the modified
excitation temperature are also given in Table 3.

One may also wonder whether it is reasonable to assume the same filling factor
for the two lines, as we expect there to be a low-density region near the edge
of the cloud where 12CO is present (self-shielded) but not 13CO. This is
likely the case but the external radiation field in these extreme outer galaxy
sources is very low so the difference in filling factors should be small.
Filling factors are extremely structure-dependent and hence very difficult to
estimate, so we have not attempted to change the equal filling factor
assumption for 13CO(1-0).

###  4.2 Molecular column densities: HCN(1-0)

For LTE excitation, which is expected given that the energy levels are so
close, the optically thin hyperfine line ratios should be 5:3:1 (i.e. 0.6 and
0.2 when compared to the main line) due to the degeneracies of the levels
(2F$+$1), such that the total HCN line strength is 1.8 times that of the main
component. Figure 8 shows the HFS line ratios expected as a function of the
optical depth of the main line. Cases of ”anomalous” line ratios exist but
these all result in a high $\frac{(0-1)}{(2-1)}$ ratio (Kwan & Scoville, 1975;
Guilloteau & Baudry, 1981) and generally involve high optical depth and/or
double-peaked profiles. The ($F=1\rightarrow 1$)/($F=2\rightarrow 1$) and
($F=0\rightarrow 1$)/($F=2\rightarrow 1$) line ratios for clouds 21, 56, and
57 are respectively 0.57 and 0.21, 0.60 and 0.18, 0.69 and 0.325. These values
are indicated on Figure 8 and indicate that the HCN emission from clouds 21
and 56 is optically thin ($\tau\la 0.5$) but cloud 57 has an optical depth
$\tau\sim 1$. The agreement is excellent between the two line ratios for all
three clouds.

An independent estimate can be obtained by following Yuan et al. (2016) who
use the CLASS333https://www.iram.fr/IRAMFR/GILDAS/ hyperfine structure method
to fit the three lines of the summed spectra of clouds 21, 56, and 57. In
addition to velocity and velocity width, this method estimates the optical
depth of the main line. The optical depths obtained are $\tau_{21}=0.24\pm
0.15$, $\tau_{56}=0.66\pm 0.10$, $\tau_{57}=1.50\pm 1.47$ for clouds 21, 56,
and 57 respectively, in good agreement with the previous estimates.

Note that because these are line ratios of the same molecule at essentially
the same energy, the optical depths are those after taking into account any
filling factor, i.e. $not$ beam-averaged. The “naive“ and CLASS estimates are
consistent with optically thin ($\tau<1$) HCN emission from clouds 21 and 56
and $\tau\approx 1$ in cloud 57.

Optically thin means we can estimate the column density from the line
intensity. If we assume that the HCN excitation temperature is that of CO,
then the column densities range from $3\times 10^{10}$cm-2 to $2\times
10^{11}$cm-2 (cloud 57). If we assume that the HCN excitation temperature is
between the CMB temperature and that of CO (see previous section for 13CO),
then the column densities are a factor 3 higher, ranging from $10^{11}$cm-2to
$7\times 10^{11}$cm-2 (cloud 57). Yuan et al. (2016) assume a very low
excitation temperature, where T${}_{ex}=2hB/k$ (the first energy level, 4.25 K
for HCN), and this yields much higher HCN column densities. These column
densities are beam-averaged because they use the integrated intensities,
unlike the optical depths calculated earlier from the HFS ratios. The column
densities for the three excitation temperatures are provided in Table 3 but
our fiducial value is the intermediate one.

###  4.3 Molecular column densities: dense gas filling factor

Because we have an estimate of the optical depth, we can estimate the filling
factor by re-arranging Eq. 2 of Yuan et al. (2016) as follows

$T_{r}=\frac{h\nu}{k}[J(T_{ex})-J(T_{bg})]\,\tau\,f$ (1)

where $J(T)=[exp(h\nu/kT)-1]^{-1}$ which becomes

$f=\frac{k\,T_{r}}{h\nu\tau}\,[J(T_{ex})-J(T_{bg})]^{-1}$ (2)

We use the optically thin approximation $(1-exp(-\tau))\approx\tau$ but at the
moderate optical depths present here this has little effect. We can then use
the values of $\tau$ to deduce a limit for the filling factor $f$ for the two
$T_{ex}$ above.

In order to apply this reasoning to the other sources, we can average the 3
optical depths obtained from the CLASS fits. We obtain $<\tau>\approx 0.53$
when weighting either by the inverse of the uncertainty or by the square of
the inverse of the uncertainty. We apply this to the weaker sources which were
not mapped, hence probably over-estimating $\tau$. The filling factors
obtained range from 4-16% for our fiducial HCN(1-0) excitation temperature,
see Table 3. Wang et al. (2020) found dense gas filling factors of 6-28% in
the molecular filament GMF54 (Ragan et al., 2014) over the regions where 13CO
was detected (equally our case), so the values we find appear reasonable.
GMF54 is beyond the molecular ring and few appropriate comparison sources are
available. We then expect the true (not beam-averaged) column densities to be
a factor $1/f$ times higher (col. 10 divided by col 14 in Table 3). This
yields HCN column densities for the cores of $\approx 2\times 10^{12}$cm-2
except for clouds 56 and 57 which may reach $10^{13}$cm-2. Note that this is
for the total line width and not per ${\rm\ km\ s^{-1}}$ and assumes that the
dense gas occupies a relatively small fraction of the beam.

###  4.4 Molecular column densities: HCO+(1-0)

HCO+ is believed to be less abundant than HCN (Godard et al., 2010; Watanabe
et al., 2017) and the turbulent linewidths and dipole moments are similar such
that if HCN is optically thin, then HCO+ should be as well. Taking HCO+ to be
optically thin, we can do the same calculations, again assuming our fiducial
excitation temperature midway between $T_{\rm bg}$ and Tex,CO (Sect 4.1). The
column densities are given in Table 3. We can see that overall the HCN and
HCO+ column densities are similar: $0.4<\frac{N_{HCN}}{N_{HCO^{+}}}<2$ (see
columns 10 and 13 of Table 3).

Having established that the HCN(1-0) line, and likely the HCO+line, are
optically thin or nearly so, we can ask ourselves whether the apparently
similar HCN and HCO+ column densities necessarily reflect the true abundance
ratio. The critical density of HCN is about 7 times higher than that of HCO+
(see Shirley, 2015, Table 1 or, for HCO+, Fig. 3 extrapolating to our column
densities), potentially resulting in more severely subthermal excitation for
the HCN line. This could reduce the HCN line intensities, from which we
estimate the column densities, more than the HCO+, such that the true HCN/HCO+
abundance ratio could be higher. As it is difficult analytically, the
following section will explore this more quantitatively via non-LTE
calculations.

###  4.5 Molecular column densities: H2

Observations of 12CO(1-0), 13CO(1-0), HCN(1-0), and HCO+(1-0) have been
presented. The latter two molecules are considered dense gas tracers as their
critical densities are about 100 times higher than for CO where
$n_{crit}\approx 1000$cm-3and less when highly optically thick. We can
estimate the average H2 column density from the CO observations by simply
dividing by the fractional abundance (Pineda et al., 2013; Fontani et al.,
2012), such that $N(H_{2})=N(^{12}CO)/\chi_{{}^{12}CO}$ or
$N(H_{2})=N(^{13}CO)/\chi_{{}^{13}CO}$ where the column densities are given in
columns 5 and 8 of Table 3. The $N(H_{2})$ values are presented in the last
two columns of Table 3. The H2 column densities derived in this way are
significantly higher than when using the standard large-scale $N({\rm
H}_{2})/I_{\rm CO}$ factor used in extragalactic observations (e.g. Bolatto et
al., 2013). This is expected as the CO emission from the outer galaxy is weak
compared to the inner galaxy (Sodroski et al., 1995; Digel et al., 1994),
presumably due to lower abundances and lower temperatures. Extragalactic
observations by Sandstrom et al. (2013) found that the $N({\rm H}_{2})/I_{\rm
CO}$ ratio did not change significantly within the inner disk (out to 0.7
$R_{25}$). However, the positions observed here are far beyond
$R_{25}$444Estimates of the disk mass scale length of the Galaxy range between
2 and 3 kpc (Binney & Vasiliev, 2023; Bovy & Rix, 2013) and following Gusev et
al. (2012) we estimate the blue-band scale length to be about 3.5kpc, yielding
$R_{25}\approx 11$kpc assuming a central disk surface brightness of 21.65
mag/″2 from Freeman (1970). and the ratios we find by dividing the N(H2) from
Table 3 by ICO from Table 1 are still within the scatter of their Figure 4,
such that there is no conflict.

Rather standard values of N(H2) are found, of order $10^{22}$H2 cm-2,
suggesting that the procedure is reasonable and that outer galaxy, even
extreme outer galaxy, molecular clouds have similar masses and column
densities as local clouds. Taking the maps in Figures 4, 5, and 6 as
representative of the size of the clouds, we obtain sizes of 10–15 pc. If the
depth is the same, the average density (i.e. for a volume filling factor of
unity) is $<n>\approx 10^{22}/3\,10^{19}\approx 300$cm-3. This is sufficient
to thermalize the highly optically thick 12CO(1-0) line but not the HCN or
HCO+. In order to reach densities where the dense gas tracers are excited,
either the cloud would be a thin sheet with a line-of-sight depth about 1% of
the extent perpendicular to the line of sight, or the HCN and HCO+ emission is
produced by dense clumps within the cloud which do not contribute
significantly ($>10$%) to the cloud mass or the 12CO(1-0) emission. We will
assume the latter.

It is difficult to estimate the H2 column density of the dense clumps as the
HCN and HCO+ abundances are not well known. HCN sticks to dust and is thus
less abundant in cool dense environments. Lahuis & van Dishoeck (2000) suggest
that the HCN abundance with respect to H2 is $\chi_{\rm HCN}\approx 10^{-8}$
and higher in hot environments. In the outer Galaxy clouds, we are clearly not
in the ”hot environment” case. A study of Planck cold cores by Yuan et al.
(2016) found very low abundances, roughly $1.5\times 10^{-10}$ for both HCN
and HCO+. Although these clouds were generally much closer, with an average
distance of 1.3kpc, this work has the physical conditions closest to the outer
disk observations presented here. The Fontani et al. (2022) and Patra et al.
(2022) sources (massive star-forming regions) are very different (although in
the outer disk) with broader lines, outflows, strong HCN and HCO+ emission and
even H13CN emission; they do not estimate abundances. Pirogov et al. (1995)
find an HCN abundance $\chi_{\rm HCN}>10^{-10}$ towards a high-latitude cloud.
Turner et al. (1997) estimate $\chi_{\rm HCN}>10^{-9}$ from observations of
translucent molecular clouds. Watanabe et al. (2012) and Watanabe et al.
(2017) find $\chi_{\rm HCN}\approx 10^{-9}$ towards a low-mass class 0
protostar and $\chi_{\rm HCN}\approx 2\times 10^{-9}$ towards W 51
respectively. In their extragalactic work, Gao & Solomon (2004a) assume
$\chi_{\rm HCN}=2\times 10^{-8}$. Watanabe et al. (2014) observed two
positions in M 51 and find fractional abundances $\chi_{\rm HCN}\approx
10^{-9}$. Martín et al. (2006) and Aladro et al. (2011) find values within a
factor two of $10^{-9}$ for respectively NGC 253 and M 82 such that the column
density of dense gas is:

$N(H_{2})_{\rm dense}=10^{21}\times\frac{N_{\rm
HCN}}{10^{12}}\times\frac{10^{-9}}{\chi_{\rm HCN}}\times\frac{1}{f}{\rm
cm}^{-2}$ (3)

where $N_{\rm HCN}$ is the beam averaged value but, because we divide by the
filling factor $f$, $N(H_{2})_{\rm dense}$ is the estimated N(H2) of the dense
clumps. Note that this expression keeps the HCN abundance as a free parameter
but introduces reasonable values in order to provide an illustrative value of
the H2 column density. The range of $\chi_{HCN}$ that we regard as likely in
the extreme outer disk environment is $10^{-9}\ga\chi_{HCN}\ga 10^{-10}$.

A beam averaged HCN or HCO+ column density of a few $10^{11}$cm-2(Table 3
cols. 10 and 13) corresponds to beam-averaged dense gas column of
$N(H_{2})_{dense}\approx 10^{21}\,{\rm cm^{-2}}$, to be compared with
$N(H_{2})\approx 10^{22}\,{\rm cm^{-2}}$ from cols. 15 and 16 of Table 3.
Thus, depending on the abundance, the mass in the clumps dense enough to
excite the HCN and HCO+ molecules represents of order 10% of the mass.

Cloud | I${}_{\rm HCO^{+}}$ | V${}_{\rm HCO^{+}}$ | fwhm${}_{\rm HCO^{+}}$ | IHCN | VHCN | fwhmHCN | rms
---|---|---|---|---|---|---|---
| K km s-1 | km s-1 | km s-1 | K km s-1 | km s-1 | km s-1 | mK
7 | 0.148$\pm$ 0.015 | -102.89 $\pm$ 0.07 | 1.484 $\pm$0.181 | 0.022 $\pm$ 0.018 | -102.92 $\pm$0.18 | 0.494 $\pm$0.779 | 17
14 | 0.133 $\pm$ 0.015 | -101.11$\pm$ 0.08 | 1.407$\pm$ 0.178 | 0.023 $\pm$ 0.008 | -101.28 $\pm$0.06 | 0.373$\pm$ 0.135 | 19
16 | 0.072 $\pm$0.012 | -99.69 $\pm$ 0.09 | 1.161 $\pm$0.250 | 0.031 $\pm$0.010 | -99.36 $\pm$0.09 | 0.615 $\pm$0.249 | 18
18 | 0.117 $\pm$ 0.013 | -99.40 $\pm$0.09 | 1.616 $\pm$0.192 | 0.046 $\pm$0.010 | -99.87 $\pm$0.21 | 1.628 $\pm$0.365 | 12
30 | 0.100 $\pm$ 0.013 | -104.02 $\pm$0.09 | 1.297 $\pm$0.181 | 0.058 $\pm$0.014 | -104.48 $\pm$0.17 | 1.246 $\pm$0.245 | 22
34 | 0.201 $\pm$0.018 | -107.25 $\pm$0.09 | 2.221 $\pm$0.255 | 0.038 $\pm$0.012 | -107.17 $\pm$0.18 | 1.143 $\pm$0.386 | 17
21 | 0.553 $\pm$ 0.018 | -101.10 $\pm$0.03 | 1.801 $\pm$0.073 | 0.086 $\pm$0.012 | -101.20 $\pm$0.08 | 1.020 $\pm$0.166 | 19
sum | $2-1$ |  |  | 0.067 $\pm$0.006 | -101.35 $\pm$0.05 | 1.276 $\pm$0.136 | 3.4
sum | $1-1$ |  |  | 0.038 $\pm$0.009 | -96.53 $\pm$0.16 | 1.321 $\pm$0.402 | 3.4
sum | $0-1$ |  |  | 0.014 $\pm$0.008 | -108.32 $\pm$0.24 | 0.827 $\pm$0.624 | 3.4
56 | 0.319 $\pm$0.013 | -103.55 $\pm$0.05 | 2.345 $\pm$0.107 | 0.126 $\pm$0.013 | -103.45 $\pm$0.09 | 1.828 $\pm$0.239 | 12
sum | $2-1$ |  |  | 0.096 $\pm$0.009 | -103.57 $\pm$0.08 | 1.794 $\pm$0.189 | 4.3
sum | $1-1$ |  |  | 0.058 $\pm$0.013 | -98.73 $\pm$0.19 | 1.744 $\pm$0.448 | 4.3
sum | $0-1$ |  |  | 0.017 $\pm$0.011 | -110.70 $\pm$0.37 | 1.126 $\pm$0.761 | 4.3
57 | 0.474 $\pm$0.016 | -102.23 $\pm$0.03 | 2.013 $\pm$0.082 | 0.133 $\pm$0.016 | -102.17 $\pm$0.09 | 1.463 $\pm$0.200 | 16
sum | $2-1$ |  |  | 0.083 $\pm$0.009 | -102.37 $\pm$0.10 | 1.799 $\pm$0.236 | 5.9
sum | $1-1$ |  |  | 0.057 $\pm$0.012 | -97.38 $\pm$0.19 | 1.821 $\pm$0.446 | 5.9
sum | $0-1$ |  |  | 0.027 $\pm$0.014 | -109.59 $\pm$0.51 | 1.980 $\pm$1.116 | 5.9
Table 2: Results of observations of central positions, deduced from gaussian fits. Intensities are in main beam scale but for HCN are only of the main $F=2\rightarrow 1$ transition. The spectra for clouds 21, 56, and 57 have been summed (see Fig. 7 and the lines marked ”sum” provide the fits to the individual hyperfine components. The expected velocity differences are 4.842 and $-7.064$ km s-1 with respect to the main (2-1) component. The satellite lines were fit using as initial guess (for the fit algorithm of CLASS) the main line velocity $+$ 4.842 or -7.064 for the $1-1$ and $0-1$ transitions respectively. That the velocities do not show exactly these differences provides an independent estimate of the uncertainty in the velocity determination. Rms noise levels are in mK converted to main beam scale and per 0.2 km s-1 channel. Cl | TCO | $\chi_{\rm 12/13}$ | Tex,CO | N${}_{\rm CO}^{a}$ | N${}_{\rm{}^{13}CO}^{a}$ | T${}_{\rm ex,^{13}CO}^{b}$ | N${}_{\rm{}^{13}CO}^{b}$ | N${}_{\rm HCN}^{a}$ | N${}_{\rm HCN}^{b}$ | N${}_{\rm HCN}^{c}$ | N${}_{\rm HCO^{+}}^{a}$ | N${}_{\rm HCO^{+}}^{b}$ | fb | N(H2)a | N(H2)b
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | $10^{17}$ | $10^{15}$ |  | $10^{15}$ | $10^{11}$ | $10^{11}$ | $10^{11}$ | $10^{11}$ | $10^{11}$ | % | log | log
| K |  | K | cm-2 | cm-2 | K | cm-2 | cm-2 | cm-2 | cm-2 | cm-2 |  |  |  |
7 | 5.4 | 93.20 | 8.75 | 1.82 | 1.17 | 5.74 | 4.23 | 0.29 | 0.91 | 2.91 | 0.63 | 2.01 | 5.8 | 21.67 | 22.00
14 | 4.5 | 93.67 | 7.80 | 2.83 | 1.71 | 5.27 | 6.33 | 0.40 | 1.28 | 3.04 | 0.75 | 2.43 | 7.2 | 21.87 | 22.19
16 | 4.0 | 93.20 | 7.30 | 2.52 | 1.74 | 5.02 | 6.48 | 0.63 | 2.07 | 4.10 | 0.48 | 1.57 | 10.9 | 21.81 | 22.19
18 | 5.4 | 92.73 | 8.75 | 1.96 | 1.35 | 5.74 | 4.87 | 0.60 | 1.91 | 6.09 | 0.50 | 1.59 | 12.1 | 21.69 | 22.06
30 | 5.0 | 102.13 | 8.35 | 0.58 | 1.61 | 5.54 | 5.88 | 0.84 | 2.71 | 7.68 | 0.47 | 1.53 | 16.4 | 21.28 | 22.29
34 | 5.8 | 105.42 | 9.15 | 2.08 | 1.24 | 5.94 | 4.45 | 0.45 | 1.41 | 5.03 | 0.77 | 2.44 | 9.3 | 21.87 | 22.23
21 | 8.0 | 95.55 | 11.40 | 0.96 | 0.89 | 7.06 | 3.04 | 0.55 | 1.66 | 9.84 | 1.32 | 4.03 | 12.3 | 21.42 | 21.90
56 | 6.0 | 127.04 | 9.35 | 3.14 | 1.91 | 6.04 | 6.81 | 1.45 | 4.58 | 17.18 | 1.16 | 3.68 | 8.6 | 22.31 | 22.75
57 | 5.5 | 125.63 | 8.85 | 2.98 | 1.16 | 5.79 | 4.18 | 2.27 | 7.23 | 23.72 | 1.96 | 6.26 | 4.0 | 22.27 | 22.52
Table 3: Temperatures, column densities, and filling factors as described in Section 4. Col. 2 is the observed CO line temperature, col. 3 the radially varying 12/13CO abundance ratio, col. 4 is the CO excitation temperature, and columns 5 and 6 are the deduced CO and 13CO column densities, all following Sect. 3.2 of Yuan et al. (2016). Col. 7 is our fiducial excitation temperature for the optically thin or close to thin species (see Sect. 4.1). Columns 8 - 13 are the column densities derived using the excitation temperatures from Sect. 4.1 and Yuan et al. (2016), see footnotes below. Col. 14 is the estimated filling factor from Sect. 4.3. The H2 column densities in the last two columns are derived as in the first paragraph of Sect. 4.5, where col 16. is the H2 column deduced from the 13CO line assuming an excitation temperature as in ($b$) below. ($a$) Assumes all excitation temperatures are those derived for CO (col. 4). ($b$) Our fiducial case which attempts to account for subthermal excitation temperatures such that T${}_{\rm ex}=T_{bg}+0.5\times(T_{\rm ex,CO}-T_{bg})$. ($c$) Follows Yuan et al. (2016) hypothesis where HCN and HCO+ excitation temperatures are equal to $2hB/k$. Cloud |  13CO | C18O | ratio | $\chi_{13/18}$
---|---|---|---|---
| K km s-1 | K km s-1 |  |
21 | $0.60\pm 0.06$ | $0.03\pm 0.06$ | $\ga 20\pm 10$ | 9.6
56 | $1.39\pm 0.07$ | $0.20\pm 0.07$ | $\ga 7.0\pm 2$ | 10.3
57 | $0.85\pm 0.06$ | $0.11\pm 0.06$ | $\ga 7.7\pm 3$ | 10.3
Table 4: 13CO(1-0) and C18O(1-0) intensities averaged over the maps.
$\chi_{13/18}$ is the 13CO to C18O abundance ratio at the galactocentric
distance of the cloud, which for optically thin emission and equal excitation
temperatures would be the intensity ratio. The last two columns can
legitimately be compared.

##  5 Physical conditions via radiative transfer calculations

The analytical calculations in the previous section are necessarily based on
the hypothesis that the molecules are in Local Thermal Equilibrium. LTE is a
strong assumption which is probably not valid, particularly for high-dipole
moment molecules like HCN and HCO+. We made a first attempt to account for
this by allowing for a lower excitation temperature for the optically thin (or
nearly so) molecular emission. In this section we make non-LTE calculations in
order to go a step further and in particular to treat the HCN and HCO+
molecules separately because although they both have high dipole moments,
their critical densities are different and the non-LTE transfer calculations
enable us to interpret the factor $\sim 2$ difference in intensity.

###  5.1 Model setup

To constrain the volume and column densities of our observed outer Galaxy
clouds, we have run a grid of non-LTE models of the HCN and HCO+ line emission
using the radiative transfer program
RADEX555https://personal.sron.nl/$\sim$vdtak/radex/index.shtml (van der Tak et
al., 2007). This program solves for the relative populations of the rotational
energy levels of the molecules, taking collisional and radiative
(de-)excitation into account, and treats the line radiative transfer with an
escape probability formalism. The free parameters in the calculation are the
kinetic temperature, the gas density, and the molecular column density. In the
calculations, we assume a background temperature of 2.73 K, a turbulent line
width of $1.0\leavevmode\nobreak\ {\rm\ km\ s^{-1}}$, and a uniform medium
without velocity gradients. The linewidths of these clouds are slightly above
$1.0\leavevmode\nobreak\ {\rm\ km\ s^{-1}}$ and any column density per
turbulent linewidth should be multiplied by the linewidth (Table 2) to obtain
the total column density.

Watanabe et al. (2017) indicate that $\chi_{\rm HCO^{+}}\approx\chi_{\rm
HCN}/$3 on average in the sources they consider (their Figures 10, 11), with
$\chi_{\rm HCO^{+}}\la 10^{-9}$. Although somewhat uncertain, we take this
abundance ratio for our calculations. The decreasing [N/O] ratio would favor
HCO+ in the outer disk but HCO+ requires a source of ionization. The
competition between these effects is difficult to evaluate and, to our
knowledge, model calculations have not provided insight to this. Based on the
estimates in Yuan et al. (2016), we see no evidence for lower HCO+/HCN
abundance ratios.

Spectroscopic data for the HCN and HCO+ rotational and hyperfine structure
were taken from the Cologne Database for Molecular Spectroscopy
(CDMS)666http://cdms.de (Müller et al., 2005). The HCO+ calculations use
collision data with H2 by Flower (1999), which cover rotational levels up to
$J$=20 and temperatures of 40–200 K. For the collisional (de)excitation of the
lowest rotation-hyperfine levels of HCN with H2, two calculations exist, which
both cover our relevant energy levels (up to $J$=8–10) and temperatures (down
to 5 K and up to 30–100 K), but which differ in the ratio between the rates
for radiative ($\Delta$J=1) and non-radiative ($\Delta$J=0,
$\Delta$F=0,$\mp$1) transitions. The calculation by Ben Abdallah et al. (2012)
used a potential energy surface for the interaction with a reduced
dimensionality, averaging over H2 orientations, and found that collisional
deexcitation through non-radiative (cross-$F$) channels is $\approx$10$\times$
faster than through $\Delta$J=1 channels. However, after upgrading the close
coupling calculations to full dimensionality, Hernández Vera et al. (2017)
found that the collisional deexcitation through both types of channels
proceeds at approximately equal rates. For comparison, the quasi-classical
calculation by Green & Thaddeus (1974) did not resolve the hyperfine structure
of HCN and used He as collision partner. The corresponding datafile on the
Leiden Atomic and Molecular Database (LAMDA) (Schöier et al.,
2005)777http://home.strw.leidenuniv.nl/$\sim$moldata/ assumes that the
radiative transitions have equal collision rates, while the cross-$F$ rates
are zero. For lack of experimental data on the relative propensities of
cross-$F$ rates, we have explored all 3 sets of collision rates in our models
for HCN.

Figure 11: Intensity ratios of the hyperfine components of the HCN $J$=1–0
line, calculated with RADEX as a function of $N$(HCN). Grey areas indicate
observed ranges for the $F$=1–1/2–1 (top) and 0–1/2–1 (bottom) ratios,
respectively 0.6, 0.6, 0.7 and 0.2, 0.2, 0.3 for clouds 21, 56, and 57. The
numbers indicating the rates refer to the years of publication of the rates.

###  5.2 Model results

To constrain the column densities of the clouds, we model the intensity ratios
of the hyperfine components of the HCN $J$=1–0 transition. Figure 11 shows the
result for an assumed $T_{\rm kin}$ of 10 K and $n$(H2) = 104 cm-3, and
calculations for $T_{\rm kin}$ in the 10–30 K range and $n$(H2) in the
103–3$\times$104 cm-3 range indicate that our results depend only weakly on
the assumed gas density and kinetic temperature. Column densities of $N$(HCN)
$\approx$3$\times$1012 cm-2 are seen to be consistent with both observed
ratios (indicated by the grey areas), but only if the Hernández Vera et al.
(2017) collision data with the similar cross-$F$ and cross-$J$ rates are used.
Using the data with the large cross-$F$ rates, there is no column density of
HCN which matches both observed ratios simultaneously. The collision data with
the cross-$F$ rates equal to zero are also able to match the observed
hyperfine ratios, but the inferred $N$(HCN) is less reliable since these
hyperfine collision rates were assumed rather than calculated. These rates
also lead to unrealistically high hyperfine ratios toward high HCN column
densities. The models also reproduce the observed absolute intensities of the
HCN lines, for similar column densities as indicated by the hyperfine ratios.
We conclude that the HCN line emission indicates $N$(HCN)
$\approx$3$\times$1012 cm-2, using the collision data by Hernández Vera et al.
(2017).

Figure 12: Intensity ratio of the HCN and HCO+ $J$=1–0 lines, calculated with
RADEX as a function of $n$(H2).

To constrain the volume densities of the clouds, we model the ratios of the
HCN and HCO+ $J$=1–0 lines. These calculations assume $T_{\rm kin}$ = 20 K and
an abundance ratio of [HCN]/[HCO+] = 3 (Watanabe et al., 2017). Figure 12
shows the result as calculated with RADEX using the HCN collision rates by
Hernández Vera et al. (2017). Our observed line ratio of $\approx$1/2 (summed
over the HCN hyperfine components) is seen to correspond to H2 densities in
the $10^{3}$–$10^{4}$ cm-3 range. The analytical calculations in the previous
section yielded similar HCN and HCO+ columns but the degree of subthermal
excitation could not be assessed. Nonetheless, if $\chi_{\rm
HCO^{+}}>\chi_{\rm HCN}/3$, then a higher density would be required to yield
the observed HCN/HCO+ line ratio. For example, if $\chi_{\rm
HCO^{+}}/\chi_{\rm HCN}=0.5$ (and not 1/3), then we would deduce a density of
$n$(H2) $\approx 10^{4.5}\,{\rm cm^{-3}}$.

###  5.3 HCN and HCO+ as dense gas tracers

Depending on the HCN and HCO+ abundances, the H2 column density deduced from
HCN could be comparable to that deduced from the CO observations (Table 3).
Can the gas be the same? We can test this using the intensities of the lines.
Assuming that the majority of the CO emission comes from gas with a density
$n\la 3000\,{\rm cm^{-3}}$ (see discussion in Sect. 4.5), RADEX calculations
show that for $n\la 3000\,{\rm cm^{-3}}$, $I_{HCN}/I_{CO}\la 0.001$ (H2 column
assumed is $10^{21}\,{\rm cm^{-2}}$ with abundances $\chi_{CO}=10^{-5}$ and
$\chi_{HCN}=10^{-9}$). Since $I_{HCN}/I_{CO}\ga 0.01$ from Tables 1 and 2, the
HCN emission does not come from the gas emitting the majority of the CO
emission but rather from clumps or cores within the less dense ambient
molecular gas. n principle, the CO could come from gas with $n>3000\,{\rm
cm^{-3}}$. However, not only would the clouds necessarily be much thinner
along the line of sight but also the 13CO emission would be quite unusual,
much stronger in the higher$-J$ transitions than in 13CO(1–0). The few data
available for the 13CO(2–1) line, or even CO(2–1), at large scales in the
outer parts of galaxies suggest that the higher-$J$ transitions are not
stronger (Sakamoto et al., 1997; Braine et al., 1993). Hence, in these outer
disk clouds, HCN and HCO+ trace dense gas ($n>10^{3}\,{\rm cm^{-3}}$).

Figure 9 shows that the HCO+ line is on average twice as strong as the HCN
line. To make this figure, the HCN(1–0) main line intensities were multiplied
by 9/5 to account for the flux in the weaker satellite lines which have low
signal-to-noise ratios and thus can add considerable noise to the sum of the
intensities. As seen in Sect. 4, the optical depth is low so this procedure
should be appropriate. Since both lines are optically thin, we have used RADEX
to estimate the typical density at which the HCO+(1–0) is twice the total
HCN(1–0) line intensity. For a column density ratio of 3, this occurs at $n\la
10^{4}\,{\rm cm^{-3}}$ (T${}_{\rm kin}=20$K). If in fact the HCO+ abundance is
higher (than 1/3 of the HCN abundance), then the density implied by the
observed flux ratio is higher. A larger fraction of the HCO+ emission,
compared to HCN, comes from the lower density CO-emitting gas and this is why
the above calculation sets a lower limit to the typical density of the dense
component. This is also seen in Fig. 11 of the observations of W3(OH) by
Nishimura et al. (2017) which shows that the HCN(1–0)/HCO+(1–0) ratio
decreases towards the more extended and more diffuse material.

We obtain a typical clump or core size of $L\approx N/n$ where $N$ is the
dense gas column density defined in Eq. 3 such that

$M_{cl}=\rho V\approx n2m_{p}f_{h}^{-1}(N/n)^{3}$ (4)

where $2m_{p}f_{h}^{-1}$ is the average molecular mass. Adopting reference
values of $N_{HCN}=$ 3$\times$1011 cm-2, $n=10^{4}\,{\rm cm^{-3}}$,
$\chi_{HCN}=10^{-9}$, and $f=0.1$, we can express the clump or core mass as
follows by substituting Eq. 3 into Eq. 4

$\begin{split}M_{cl}&=\frac{2m_{p}}{f_{h}}10^{-8}\left(\frac{10^{4}}{n}\right)^{2}(3\times
10^{21})^{3}\left(\frac{N_{HCN}}{3\times
10^{11}}\right)^{3}\left(\frac{10^{-9}}{\chi_{HCN}}\right)^{3}\left(\frac{0.1}{f}\right)^{3}\\\
&=0.62\left(\frac{10^{4}}{n}\right)^{2}\left(\frac{N_{HCN}}{3\times
10^{11}}\right)^{3}\left(\frac{10^{-9}}{\chi_{HCN}}\right)^{3}\left(\frac{0.1}{f}\right)^{3}M_{\odot}\end{split}$
(5)

with a hydrogen fraction of $f_{h}=0.73$. The mass in such a clump is of order
a solar mass but highly dependent on the abundance, such that if
$\chi_{HCN}=10^{-10}$ then the clump mass becomes $620M_{\odot}$. The other
parameters are constrained by observations: $N_{HCN}$ by the HCN flux,
$N_{HCN}/f$ by the HCN optical depth limit, and $n$ by the HCN/HCO+ flux
ratio.

##  6 Link between star formation and dense gas fraction

Figure 13: Comparison of molecular line intensities and tracers of star
formation. The bottom panel compares the SFR tracers: background-subtracted 24
$\mu$m emission and 8 $\mu$m emission, also background-subtracted (the
background emission is considerably stronger than any stellar contribution at
8 $\mu$m). The middle panel shows how the 12CO intensity varies with SFR. The
top panel shows how the dense gas fraction, as traced by the HCO+/CO ratio,
varies with SFR. The red triangles indicate the sources which are at high
galactic latitude. It should be immediately apparent that these sources are
not distinguishable from the others in the lower two frames but occupy a
specific region with low dense gas fraction in the top panel. Neither 8 nor 24
$\mu$m data are available for cloud 30. Figure 14: Link between dense gas mass
(HCN or HCO+ luminosity on the x-axis) and star formation rate (IR luminosity
on the y-axis). The line shows a linear relation with $L_{FIR}/L_{HCN}$ or
$L_{FIR}/L_{HCO^{+}}=1000L_{\odot}({\rm\ K}{\rm\ km\
s^{-1}}{\mathrm{p}c}^{2})^{-1}$. HCN data are shown as filled triangles and
HCO+ as open squares. The outer disk data presented in this paper are shown in
blue. The sources for the data are: Chin et al. (1997), Chin et al. (1998),
Gao & Solomon (2004b), Gao et al. (2007), Brouillet et al. (2005), Wu et al.
(2005), Ma et al. (2013), Buchbender (2014), Krips et al. (2008), Juneau et
al. (2009), Privon et al. (2015), Graciá-Carpio et al. (2008), García-Burillo
et al. (2012), Chen et al. (2015), Chen et al. (2017), Usero et al. (2015).

Gao & Solomon (2004b) suggested that the star formation rate is proportional
to the mass of dense gas such that starburst galaxies have high dense gas
masses. In ultraluminous IR Galaxies (ULIRGs), there is generally a huge
amount of gas concentrated near the center and hence the dense gas mass (HCN
flux) and fraction (typically the HCN/CO flux ratio) are high. Here we examine
smaller scales so, to compare dense gas with star formation, we have compared
these line ratios for positions with and without a class I or II source. The
HCN/CO and HCO+/CO flux ratios are marginally higher for the positions with a
class I or class II source but not significantly.

Aside from the good correlation between 8 $\mu$m and 24 $\mu$m emission, the
main result is that the HCN/CO and HCO+/CO ratios are a factor two lower for
the sources at high galactic latitude ($b>2^{\circ}$, sources 7, 14, 16, 18,
30, and 34, see top panel of Fig. 13). These sources are identified by red
triangles in Fig. 13.

In a spiral galaxy, the molecular-to-atomic gas ratio at kpc scales is largely
controlled by the local hydrostatic pressure (see Blitz & Rosolowsky, 2004),
itself a function of the stellar and gaseous surface densities. As the stellar
surface density decreases with galactocentric distance (spiral disks are
approximately exponential), so does the molecular fraction. The same logic
applies to the dense gas fraction which is presumably why the HCN/CO and
HCO+/CO flux ratios we find are below the inner disk ratios (e.g. Usero et
al., 2015). It likely also explains why the high-latitude clouds (200pc or
more above the galactic mid-plane) of our sample or from Yuan et al. (2016)
have lower HCN/CO and HCO+/CO flux ratios.

Far-IR emission comes mainly from dust heated by young stars and as such is a
standard tracer of star formation. There is a clear correlation between the
presence of star formation, as traced by Far-IR emission, and molecular line
emission at large scales. We will use the Spitzer 24 $\mu$m and 8 $\mu$m
emission to trace the SFR because our sources are unobscured and the Spitzer
data have high angular resolution, respectively 6″ and 2″. The 24 $\mu$m
emission is a standard tracer of star formation (Calzetti et al., 2007) but
the 8 $\mu$m emission varies strongly as a function of environment and
particularly metallicity (Draine et al., 2007). However, for a given
environment (outer disk in the present case), it can be used to trace the SFR
(Boquien et al., 2015).

The 24 $\mu$m and 8 $\mu$m
emission888https://sha.ipac.caltech.edu/applications/Spitzer/SHA were
background subtracted by examining a histogram of the values in the maps. The
value considered to be ”background” is where the number of pixels reaches
about 1% of the maximum number of pixels in the histogram. Thus we are not
sensitive to variations in a small number of pixels yet we use low values in
the map. The 8 and 24 micron measurements are continuum, meaning that they see
all matter along the line of sight and, unlike for the spectra, outer galaxy
and more local material cannot be distinguished. As we are looking in the
plane of the galaxy, there is emission virtually everywhere such that the
background value is not well defined. The 8 and 24$\mu$m fluxes provided can
only be considered best estimates. Towards our clouds, with the exception of
cloud 30, the outer galaxy cloud is the main source (this is deduced from the
relative CO line strength at each velocity). However, along other lines of
sight within the 8 and 24 $\mu$m maps, this is not necessarily the case so we
subtract a low value rather than the most common value (which reflects the
emission and not the background). We make sure not to take extremely low
values which could be influenced by noise. Figure 13 shows that the 8 and
24$\mu$m fluxes are highly correlated, which would not be the case if the
estimated background fluxes were highly incorrect. The fluxes were measured
with two different instruments (IRAC and MIPS on the Spitzer satellite),
making a coincidence even less likely.

Figure 13 shows how the molecular line intensity varies as a function of the
SFR. Not all sources have both 24 $\mu$m and 8 $\mu$m available so, after
showing that the 24 $\mu$m and 8 $\mu$m fluxes are highly correlated
($r=0.82$, bottom panel), we trace the SFR with the 24 $\mu$m fluxes but use 8
$\mu$m fluxes when no 24 $\mu$m data are available. The CO intensities are
weakly correlated with SFR ($r\approx 0.2$, middle panel) and the HCN and
HCO+intensities (not shown in Fig. 13) are not significantly correlated with
SFR. The HCO+/CO ratio (top panel), tracing the dense gas fraction, is not
significantly correlated with SFR but the sources further than 300 pc above
the galactic plane stand out as having very low HCO+ (and HCN, not shown)
emission. These sources do not stand out in other ways.

Comparing to larger scales, the dense gas tracers (or dense gas mass following
Gao & Solomon, 2004b), and star formation rate, as calculated from the 8 or 24
$\mu$m flux (Boquien et al., 2010; Galametz et al., 2013), follow the
extrapolation of the trends found in Chen et al. (2015) and Shimajiri et al.
(2017), as shown in Fig. 14. The reason for the absence of a correlation
between LHCN and SFR is likely the similarity of the regions we observe, all
occupying the same area of Fig. 14.

##  7 Comparison with other environments

In the extreme outer Galaxy clouds observed here, the line intensity ratios
are approximately: CO/HCN$\sim 70$, CO/HCO${}^{+}\sim 35$, and CO/13CO$\sim
7.5$. Thus, the CO emission is stronger relative to the other lines than in
the majority of sources observed thus far (see e.g. Fig. 8 of Chen et al.,
2015).

However, in the nearly square degree region of Orion B mapped by Gratier et
al. (2017), they obtain very similar ratios for their median values of CO,
13CO, HCO+, and HCN, although the HCN emission is stronger after correction
for flux in the satellite lines. The median values tend to sample the more
extended lower column density regions, as also found recently by Evans et al.
(2020).

The Shimajiri et al. (2017) work on nearby molecular clouds showed that HCN
and HCO+ emission often traced star formation while the H13CN and H13CO+ lines
were sensitive to the total H2 column. These regions are much denser than the
outer Galaxy clouds studied here and the HCN and HCO+ emission is highly
optically thick. The net effect is that the intensity of HCN and HCO+ is
highly dependent on gas temperature (thus star formation) rather than density.
In the highest column density regions, the main isotope lines are strongly
self-absorbed. HCN emission in outer Milky Way clouds is optically thin and
does not trace star formation and so it appears akin to the H13CN in Shimajiri
et al. (2017), which traces column density in high volume density regions.
While Shimajiri et al. (2017) do not calculate H13CN abundances, the H13CO+
abundances are within a rather narrow range:$1.5-5.8\times 10^{-11}$ with
respect to H2. These values are consistent with $\chi_{HCO+}\sim 10^{-9}$ and
the column densities ($N_{H^{13}CO^{+}}\sim 10^{12}\,{\rm cm^{-2}}$) are
similar to our estimates for the main isotope.

Yuan et al. (2016) present HCN and HCO+ observations, also with the PMO
telescope, of cool clumps identified by the Planck satellite. They are at
approximately the solar circle (mean distance of 1.3kpc), neither outer nor
inner Milky Way. Their estimated column densities are variable but have median
NHCN and N${}_{\rm HCO^{+}}\approx 10^{12}\,{\rm cm^{-2}}$, although the ratio
is quite variable. Their abundance estimates differ from source to source but
are mostly $\chi\sim 10^{-10}$ for both HCN and HCO+. Other Galactic works
include Park et al. (1999) and Barnes et al. (2011) who assume higher
abundances, typically $\chi\sim 10^{-9}$. In the cool clouds, the HCO+ line is
stronger than HCN.

So far we have compared with other Galactic observations but extragalactic
sources allow for a much greater range in properties. A relatively distant
region in M 51 was observed with the IRAM interferometer by Chen et al. (2017)
who found HCN/CO ratios similar to ours but generally higher HCN/HCO+ ratios.
M 31 was observed by Brouillet et al. (2005) and, in the outer parts, a
similarly low HCN/CO ratio was found but again a higher HCN/HCO+ ratio. The
Large Magellanic Cloud (LMC) provides an environment with a lower metallicity
and a higher radiation field than our observations of the outer Milky Way but
at similar scales. Chin et al. (1997) observed several lines in the LMC and
found again that the HCO+ emission was stronger than the HCN (or HNC), unlike
large Milky Way molecular clouds or star-forming galaxies (their Table 10).
Subsequently, Nishimura et al. (2016), Anderson et al. (2014), Galametz et al.
(2020), and Seale et al. (2012) observed CO-strong molecular clouds in the LMC
and consistently confirm the weak emission from Nitrogen-bearing molecules, as
well as strong CCH emission. Braine et al. (2017) observed the low-metallicity
galaxies M 33, IC 10, and NGC 6822 in the local group and found ratios quite
similar to those presented here. These authors concluded that the low HCN/HCO+
ratio was due to weak HCN emission rather than particularly strong HCO+
emission. This was attributed to the decrease in N/O with metal deficiency.

The low HCN/HCO+ ratios we observe in the extreme outer Galaxy are probably
due to a combination of low volume density and subsolar metallicity. A
gradient in N/O exists in the Galaxy (Magrini et al., 2018). However, while
Arellano-Córdova et al. (2021) find a N/O gradient of $-$0.015 dex kpc-1 from
optical observations, Rudolph et al. (2006) find a N/O gradient of $-$0.03 dex
kpc-1 from Far-IR spectral line data. As a result, the N/O ratio for the
distant clouds should be approximately half of the solar circle value. If the
new values underestimate the gradient, then the low HCN fluxes could be
entirely due to a decrease in abundance. Two features lead us to believe that
volume density is also an issue. First of all, the new gradient uses high-
quality GAIA DR2 distances. Secondly, the low molecular fraction in and the
diffuse nature of the outer parts of spiral galaxies argues for a lower
hydrostatic pressure as compared to the inner disk (see e.g. Blitz &
Rosolowsky, 2004). While the molecular line ratios are not clearly related to
the level of star formation, there is a clear difference between the clouds in
the plane and those at $b>2^{\circ}$ in that the HCO+/CO ratio is lower for
the latter (Fig. 13, but also HCO+/13CO and the same ratios with HCN). Our
clouds at higher latitude are at $h\ga 300$pc from the galactic plane such
that the local pressure should be lower.

We find a similar effect in the Yuan et al. (2016) sample. Clouds with
$h>180$pc above the plane have significantly lower average and median
I${}_{HCO^{+}}$/N(13CO) and IHCN/N(13CO) values than the clouds at $h<180$pc.
The mean values are 50% higher in the plane and the median is twice as high
for both I${}_{HCO^{+}}$/N(13CO) and IHCN/N(13CO). N(13CO) has been used as
the reference column density but the higher dense fraction (stronger HCN and
HCO+ emission) close to the Galactic plane is equally found using the 12CO
column density (see Table 3 of Yuan et al., 2016).

##  8 Conclusions

The dense gas tracers HCN(1-0) and HCO+(1-0) were observed in a sample of
extreme outer Galaxy clouds ($15\la R_{gal}\la 21$kpc) from the Sun et al.
(2015) unbiased CO survey. The clouds were chosen for their reasonably strong
CO ($I_{CO}>7{\rm\ km\ s^{-1}}$) and 13CO ($I_{{}^{13}CO}>1.4{\rm\ km\
s^{-1}}$) emission. None were detected in C18O(1-0) but the limits are close
to what would be expected for optically thin emission so far out in the disk.

1\. The HCN and HCO+ emission is weak, both in an absolute sense and relative
to the CO lines (Tables 1 and 2). $I_{\rm HCO^{+}}\approx 0.03I_{\rm
CO}\approx 0.21I_{{}^{13}{\rm CO}}\approx 2I_{\rm HCN}$ after accounting for
the satellite HCN lines. These ratios are typical of a subsolar metallicity
environment, where N is more affected than O.

2\. The 12CO(1-0) emission is highly optically thick but the 13CO(1-0) is
optically thin, $\tau_{13}\la 1$. The estimated H2 column densities are $\sim
10^{22}$cm-2, depending on the hypotheses (Table 3), but in all cases several
times higher than would be deduced from a standard extragalactic $N({\rm
H}_{2})/I_{\rm CO}\sim 2\times 10^{20}\,{\rm cm^{-2}/(K{\rm\ km\ s^{-1}})}$
conversion factor (e.g. Bolatto et al., 2013).

3\. The lines are narrow and the hyperfine ratios of HCN indicate that the
emission is optically thin, $\tau_{HCN(1-0)}\la 1$. We obtain beam-averaged
HCN column densities, for our fiducial excitation temperature, of $1-7\times
10^{11}$cm-2 (Table 3, col 10). With the optical depths, we can estimate the
dense gas filling factor, again dependent on the assumed $T_{ex}$, and we
obtain values of $10\pm 6$% (Table 3), in reasonable agreement with other
estimates. Other molecules and transitions can trace gas substantially denser,
which would then have a lower filling factor. Similar results are obtained for
the HCO+ line (see Table 3).

4\. The H2 volume densities required to collisionally excite the optically
thin HCN and HCO+ lines are about 100 times those required to excite the CO
lines. Hence, see Sect. 4.5, we conclude that the HCN and HCO+ emission does
not trace the bulk of the gas but the denser clumps. Combined with the derived
filling factors, and dependent on the assumed but poorly known abundances of
HCN and HCO+, we estimate (Sect. 4.5) that the gas emitting the HCN(1-0) and
HCO+(1-0) flux represents some 10% of the total gas mass.

5\. The critical densities of HCN and HCO+ are not the same and we use RADEX
non-LTE models to obtain estimates of the HCN column density and use the
HCN/HCO+intensity ratio to constrain the volume density of the ”dense” gas.
Because these lines are optically thin, the intensity ratio depends on the
abundance ratio. For $\chi_{\rm HCO^{+}}=\chi_{\rm HCN}/3$ (Watanabe et al.,
2017), the ”dense” gas is likely $n(H_{2})\la 10^{4}$cm-3 but if the relative
HCO+abundance is higher, as obtained in Section 4, then denser gas is required
for the same line ratio.

6\. As part of the modeling, it was realized that the high cross-F rates
initially found by Ben Abdallah et al. (2012) could not fit our data but that
the revised rates by Hernández Vera et al. (2017) provided a good fit. It was
thus possible to observationally test hyperfine collision rates.

7\. The original motivation was to explore the link between star formation and
dense gas in the poorly known outer disk environment. Our data fall on the
general $L_{HCN}-L_{IR}$ plot (Fig. 14) but, within our sample, there is no
correlation between the SFR, as traced by the 8 and/or 24$\mu$m flux, and the
flux in the HCN and HCO+ lines or the dense gas fraction as traced by the
HCN/CO or HCO+/CO line ratios. Instead, we find that the dense gas fraction
decreases away from the galactic plane, presumably due to lower pressure. An
analysis of the Yuan et al. (2016) sources reveals the same trend.

###### Acknowledgements.

We would like to dedicate this work to our dear friend and colleague Prof. Yu
Gao, who passed away during the revision of the article. We would like to
thank the DeLingHa telescope staff and the Purple Mountain Observatory for
making these observations possible. JB would like to thank NanJing University
for hosting. We would also like to ackowledge grants from the french PNGC and
the ANR program for financing through the ANR-11-BS56-010 project STARFICH.
Y.S. acknowledges support by the Youth Innovation Promotion Association, CAS
(Y2022085), Light of West China Program, CAS (No. xbzg-zdsys-202212), and the
NSFC through grant 11773077. HC is supported by Key Research Project of
Zhejiang Lab?No. 2021PE0AC0. Y.G. was supported by the National Key Basic
Research and Development Program of China (2017YFA0402700), the National
Natural Science Foundation of China (11861131007, U1731237, 12033004), and
Chinese Academy of Sciences Key Research Program of Frontier Sciences
(QYZDJSSW-SLH008). This publication makes use of data products from the Wide-
field Infrared Survey Explorer, which is a joint project of the University of
California, Los Angeles, and the Jet Propulsion Laboratory/California
Institute of Technology, funded by the National Aeronautics and Space
Administration.

## References

  * Aladro et al. (2011) Aladro, R., Martín, S., Martín-Pintado, J., et al. 2011, A&A, 535, A84
  * Anderson et al. (2014) Anderson, C. N., Meier, D. S., Ott, J., et al. 2014, ApJ, 793, 37
  * Arellano-Córdova et al. (2021) Arellano-Córdova, K. Z., Esteban, C., García-Rojas, J., & Méndez-Delgado, J. E. 2021, MNRAS, 502, 225
  * Barnes et al. (2011) Barnes, P. J., Yonekura, Y., Fukui, Y., et al. 2011, ApJS, 196, 12
  * Ben Abdallah et al. (2012) Ben Abdallah, D., Najar, F., Jaidane, N., Dumouchel, F., & Lique, F. 2012, MNRAS, 419, 2441
  * Binney & Vasiliev (2023) Binney, J. & Vasiliev, E. 2023, MNRAS
  * Blitz & Rosolowsky (2004) Blitz, L. & Rosolowsky, E. 2004, ApJL, 612, L29
  * Bolatto et al. (2013) Bolatto, A. D., Wolfire, M., & Leroy, A. K. 2013, ARA&A, 51, 207
  * Boquien et al. (2015) Boquien, M., Calzetti, D., Aalto, S., et al. 2015, A&A, 578, A8
  * Boquien et al. (2010) Boquien, M., Calzetti, D., Kramer, C., et al. 2010, A&A, 518, L70+
  * Bovy & Rix (2013) Bovy, J. & Rix, H.-W. 2013, ApJ, 779, 115
  * Braine et al. (1993) Braine, J., Combes, F., & van Driel, W. 1993, A&A, 280, 451
  * Braine et al. (2017) Braine, J., Shimajiri, Y., André, P., et al. 2017, A&A, 597, A44
  * Brouillet et al. (2005) Brouillet, N., Muller, S., Herpin, F., Braine, J., & Jacq, T. 2005, A&A, 429, 153
  * Buchbender (2014) Buchbender, C. 2014, PhD thesis, Universidad de Granada, http://hdl.handle.net/10481/31707
  * Calzetti et al. (2007) Calzetti, D., Kennicutt, R. C., Engelbracht, C. W., et al. 2007, ApJ, 666, 870
  * Chen et al. (2017) Chen, H., Braine, J., Gao, Y., Koda, J., & Gu, Q. 2017, ApJ, 836, 101
  * Chen et al. (2015) Chen, H., Gao, Y., Braine, J., & Gu, Q. 2015, ApJ, 810, 140
  * Chin et al. (1998) Chin, Y.-N., Henkel, C., Millar, T. J., Whiteoak, J. B., & Marx-Zimmer, M. 1998, A&A, 330, 901
  * Chin et al. (1997) Chin, Y.-N., Henkel, C., Whiteoak, J. B., et al. 1997, A&A, 317, 548
  * Digel et al. (1994) Digel, S., de Geus, E., & Thaddeus, P. 1994, ApJ, 422, 92
  * Draine et al. (2007) Draine, B. T., Dale, D. A., Bendo, G., et al. 2007, ApJ, 663, 866
  * Evans (1999) Evans, Neal J., I. 1999, ARA&A, 37, 311
  * Evans et al. (2020) Evans, Neal J., I., Kim, K.-T., Wu, J., et al. 2020, ApJ, 894, 103
  * Fang et al. (2013) Fang, M., Kim, J. S., van Boekel, R., et al. 2013, ApJS, 207, 5
  * Flower (1999) Flower, D. R. 1999, MNRAS, 305, 651
  * Fontani et al. (2022) Fontani, F., Colzi, L., Bizzocchi, L., et al. 2022, A&A, 660, A76
  * Fontani et al. (2012) Fontani, F., Giannetti, A., Beltrán, M. T., et al. 2012, MNRAS, 423, 2342
  * Freeman (1970) Freeman, K. C. 1970, ApJ, 160, 811
  * Galametz et al. (2013) Galametz, M., Kennicutt, R. C., Calzetti, D., et al. 2013, MNRAS, 431, 1956
  * Galametz et al. (2020) Galametz, M., Schruba, A., De Breuck, C., et al. 2020, A&A, 643, A63
  * Gallagher et al. (2018) Gallagher, M. J., Leroy, A. K., Bigiel, F., et al. 2018, ApJ, 858, 90
  * Gao et al. (2007) Gao, Y., Carilli, C. L., Solomon, P. M., & Vanden Bout, P. A. 2007, ApJL, 660, L93
  * Gao & Solomon (2004a) Gao, Y. & Solomon, P. M. 2004a, ApJS, 152, 63
  * Gao & Solomon (2004b) Gao, Y. & Solomon, P. M. 2004b, ApJ, 606, 271
  * García-Burillo et al. (2012) García-Burillo, S., Usero, A., Alonso-Herrero, A., et al. 2012, A&A, 539, A8
  * Glover & Clark (2012) Glover, S. C. O. & Clark, P. C. 2012, MNRAS, 426, 377
  * Godard et al. (2010) Godard, B., Falgarone, E., Gerin, M., Hily-Blant, P., & de Luca, M. 2010, A&A, 520, A20
  * Graciá-Carpio et al. (2008) Graciá-Carpio, J., García-Burillo, S., Planesas, P., Fuente, A., & Usero, A. 2008, A&A, 479, 703
  * Gratier et al. (2017) Gratier, P., Bron, E., Gerin, M., et al. 2017, A&A, 599, A100
  * Green & Thaddeus (1974) Green, S. & Thaddeus, P. 1974, ApJ, 191, 653
  * Guilloteau & Baudry (1981) Guilloteau, S. & Baudry, A. 1981, A&A, 97, 213
  * Gusev et al. (2012) Gusev, A. S., Guslyakova, S. A., & Khramtsova, M. S. 2012, Astronomical and Astrophysical Transactions, 27, 233
  * Hernández Vera et al. (2017) Hernández Vera, M., Lique, F., Dumouchel, F., Hily-Blant, P., & Faure, A. 2017, MNRAS, 468, 1084
  * Hoekstra et al. (2001) Hoekstra, H., van Albada, T. S., & Sancisi, R. 2001, MNRAS, 323, 453
  * Jiménez-Donaire et al. (2019) Jiménez-Donaire, M. J., Bigiel, F., Leroy, A. K., et al. 2019, ApJ, 880, 127
  * Juneau et al. (2009) Juneau, S., Narayanan, D. T., Moustakas, J., et al. 2009, ApJ, 707, 1217
  * Kalberla & Kerp (2009) Kalberla, P. M. W. & Kerp, J. 2009, ARA&A, 47, 27
  * Kennicutt (1989) Kennicutt, R. C., J. 1989, ApJ, 344, 685
  * Kennicutt & Evans (2012) Kennicutt, R. C. & Evans, N. J. 2012, ARA&A, 50, 531
  * Kepley et al. (2014) Kepley, A. A., Leroy, A. K., Frayer, D., et al. 2014, ApJL, 780, L13
  * Koenig et al. (2012) Koenig, X. P., Leisawitz, D. T., Benford, D. J., et al. 2012, ApJ, 744, 130
  * Krips et al. (2008) Krips, M., Neri, R., García-Burillo, S., et al. 2008, ApJ, 677, 262
  * Kutner & Ulich (1981) Kutner, M. L. & Ulich, B. L. 1981, ApJ, 250, 341
  * Kwan & Scoville (1975) Kwan, J. & Scoville, N. 1975, ApJL, 195, L85
  * Lahuis & van Dishoeck (2000) Lahuis, F. & van Dishoeck, E. F. 2000, A&A, 355, 699
  * Linsky (2003) Linsky, J. L. 2003, Sp.Sci.Rev., 106, 49
  * Ma et al. (2013) Ma, B., Tan, J. C., & Barnes, P. J. 2013, ApJ, 779, 79
  * Magrini et al. (2018) Magrini, L., Vincenzo, F., Randich, S., et al. 2018, A&A, 618, A102
  * Mangum & Shirley (2015) Mangum, J. G. & Shirley, Y. L. 2015, PASP, 127, 266
  * Martín et al. (2006) Martín, S., Mauersberger, R., Martín-Pintado, J., Henkel, C., & García-Burillo, S. 2006, ApJS, 164, 450
  * Molinari et al. (2016) Molinari, S., Schisano, E., Elia, D., et al. 2016, A&A, 591, A149
  * Müller et al. (2005) Müller, H. S. P., Schlöder, F., Stutzki, J., & Winnewisser, G. 2005, Journal of Molecular Structure, 742, 215
  * Nishimura et al. (2016) Nishimura, Y., Shimonishi, T., Watanabe, Y., et al. 2016, ApJ, 818, 161
  * Nishimura et al. (2017) Nishimura, Y., Watanabe, Y., Harada, N., et al. 2017, ApJ, 848, 17
  * Park et al. (1999) Park, Y. S., Kim, J., & Minh, Y. C. 1999, ApJ, 520, 223
  * Patra et al. (2022) Patra, S., Evans, Neal J., I., Kim, K.-T., et al. 2022, AJ, 164, 129
  * Pedicelli et al. (2009) Pedicelli, S., Bono, G., Lemasle, B., et al. 2009, A&A, 504, 81
  * Pineda et al. (2013) Pineda, J. L., Langer, W. D., Velusamy, T., & Goldsmith, P. F. 2013, A&A, 554, A103
  * Pirogov et al. (1995) Pirogov, L., Zinchenko, I., Lapinov, A., Myshenko, V., & Shul’Ga, V. 1995, A&AS, 109, 333
  * Privon et al. (2015) Privon, G. C., Herrero-Illana, R., Evans, A. S., et al. 2015, ApJ, 814, 39
  * Ragan et al. (2014) Ragan, S. E., Henning, T., Tackenberg, J., et al. 2014, A&A, 568, A73
  * Reid et al. (2014) Reid, M. J., Menten, K. M., Brunthaler, A., et al. 2014, ApJ, 783, 130
  * Rudolph et al. (2006) Rudolph, A. L., Fich, M., Bell, G. R., et al. 2006, ApJS, 162, 346
  * Sakamoto et al. (1997) Sakamoto, S., Hasegawa, T., Handa, T., Hayashi, M., & Oka, T. 1997, ApJ, 486, 276
  * Sandstrom et al. (2013) Sandstrom, K. M., Leroy, A. K., Walter, F., et al. 2013, ApJ, 777, 5
  * Schmidt et al. (2016) Schmidt, T. M., Bigiel, F., Klessen, R. S., & de Blok, W. J. G. 2016, MNRAS, 457, 2642
  * Schöier et al. (2005) Schöier, F. L., van der Tak, F. F. S., van Dishoeck, E. F., & Black, J. H. 2005, A&A, 432, 369
  * Schuller et al. (2009) Schuller, F., Menten, K. M., Contreras, Y., et al. 2009, A&A, 504, 415
  * Seale et al. (2012) Seale, J. P., Looney, L. W., Wong, T., et al. 2012, ApJ, 751, 42
  * Shan et al. (2012) Shan, W. L., Yang, J., & Shi, S. C., e. a. 2012, ITTST, 2, 593
  * Shimajiri et al. (2017) Shimajiri, Y., André, P., Braine, J., et al. 2017, A&A, 604, A74
  * Shirley (2015) Shirley, Y. L. 2015, PASP, 127, 299
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163
  * Sodroski et al. (1997) Sodroski, T. J., Odegard, N., Arendt, R. G., et al. 1997, ApJ, 480, 173+
  * Sodroski et al. (1995) Sodroski, T. J., Odegard, N., Dwek, E., et al. 1995, ApJ, 452, 262
  * Sun (2015) Sun, Y. 2015, PhD thesis, Purple Mountain Observatory, Chinese Academy of Sciences
  * Sun et al. (2015) Sun, Y., Xu, Y., Yang, J., et al. 2015, ApJL, 798, L27
  * Turner et al. (1997) Turner, B. E., Pirogov, L., & Minh, Y. C. 1997, ApJ, 483, 235
  * Usero et al. (2015) Usero, A., Leroy, A. K., Walter, F., et al. 2015, AJ, 150, 115
  * van der Tak et al. (2007) van der Tak, F. F. S., Black, J. H., Schöier, F. L., Jansen, D. J., & van Dishoeck, E. F. 2007, A&A, 468, 627
  * Wang et al. (2020) Wang, Y., Beuther, H., Schneider, N., et al. 2020, A&A, 641, A53
  * Watanabe et al. (2017) Watanabe, Y., Nishimura, Y., Harada, N., et al. 2017, ApJ, 845, 116
  * Watanabe et al. (2012) Watanabe, Y., Sakai, N., Lindberg, J. E., et al. 2012, ApJ, 745, 126
  * Watanabe et al. (2014) Watanabe, Y., Sakai, N., Sorai, K., & Yamamoto, S. 2014, ApJ, 788, 4
  * Wouterloot & Brand (1989) Wouterloot, J. G. A. & Brand, J. 1989, A&AS, 80, 149
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868
  * Wu et al. (2005) Wu, J., Evans, II, N. J., Gao, Y., et al. 2005, ApJ, 635, L173
  * Wu et al. (2010) Wu, J., Evans, II, N. J., Shirley, Y. L., & Knez, C. 2010, ApJS, 188, 313
  * Yuan et al. (2016) Yuan, J., Wu, Y., Liu, T., et al. 2016, ApJ, 820, 37