11institutetext:  National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA, USA 22903 11email<EMAIL_ADDRESS>22institutetext:
LERMA, Observatoire de Paris, PSL Research University CNRS Sorbonne Université
22email<EMAIL_ADDRESS>
# The Dark Neutral Medium is (Mostly) Molecular Hydrogen

H. Liszt and M. Gerin 1122<EMAIL_ADDRESS>
(received )

###### Abstract

Context. More gas is sometimes inferred in molecular cloud complexes than is
represented in HI or CO emission, and this is called dark neutral medium
(DNM).

Aims. Our aim is to extend a study of DNM along 13 directions in the outskirts
of Chamaeleon by determining the atomic or molecular character of the DNM
along a larger sample of sightlines.

Methods. We acquired ALMA ground rotational state absorption profiles of
$\mathrm{HCO^{+}}$ and other molecules toward 33 compact extragalactic
continuum background sources seen toward the Galactic anticenter, deriving
N(H2) = N($\mathrm{HCO^{+}}$)/$3\times 10^{-9}$ as before. We observed J=1-0
CO emission with the IRAM 30m telescope in directions where $\mathrm{HCO^{+}}$
was newly detected.

Results. $\mathrm{HCO^{+}}$ absorption was detected in 28 of 33 new directions
and CO emission along 19 of those 28. The five sightlines lacking detectable
$\mathrm{HCO^{+}}$ have three times lower $<$EB-V$>$ and $<$N(DNM)$>$. Binned
in EB-V, N(H2) and N(DNM) are strongly correlated and vary by factors of
50-100 over the observed range EB-V $\approx$ 0.05 - 1 mag, while N(HI) varies
by factors of only 2-3. On average N(DNM) and N(H2) are well matched, and
detecting $\mathrm{HCO^{+}}$ absorption adds little to no H2 in excess of the
previously inferred DNM. There are five cases where 2N(H2) $<$ N(DNM)/2
indicates saturation of the HI emission profile. For sightlines with ${\rm
W}_{\mathrm{CO}}$ $\geq$ 1 K-km s-1, the CO-H2 conversion factor N(H2)/${\rm
W}_{\mathrm{CO}}$ $=2-3\times 10^{20}\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm cm}^{-2}$/(1 K-km s-1) is higher than is derived from studies of resolved
clouds in $\gamma$-rays.

Conclusions. Our work sampled primarily atomic gas with a mean H2 fraction
$\approx$ 1/3, but the DNM is almost entirely molecular. CO fulfills its role
as an H2 tracer when its emission is strong, but large-scale CO surveys are
not sensitive to H2 columns associated with typical values N(DNM) = $2-6\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$. Lower $X_{\mathrm{CO}}$ values
from $\gamma$-ray studies arise in part from different definitions and usage.
Sightlines with ${\rm W}_{\mathrm{CO}}$ $\geq$ 1 K-km s-1 represent 2/3 of the
H2 detected in $\mathrm{HCO^{+}}$ and detecting 90% of the H2 would require
detecting CO at levels ${\rm W}_{\mathrm{CO}}$ $\approx$ 0.2-0.3 K-km s-1.

###### Key Words.:

interstellar medium – abundances

††offprints: H. S. Liszt

##  1 Introduction

Between the atomic and molecular interstellar gases there is a transition that
is imperfectly traced in $\lambda 21$cm HI and/or $\lambda 2.6$mm CO emission.
Gas that is not well represented in one or both tracers is described as dark
neutral medium (DNM) (Grenier et al., 2005; Planck Collaboration et al., 2015;
Remy et al., 2017, 2018), and the atomic or molecular character of the DNM can
only be decided by appealing to other information.

In the outskirts of Chamaeleon, we used millimeter-wave $\mathrm{HCO^{+}}$
absorption to show that the DNM was almost exclusively molecular, even while
the gas as a whole was mostly atomic (Liszt et al., 2018). $\mathrm{HCO^{+}}$
was detected along all 13 sampled sightlines, of which 12 lacked detectable CO
emission at a level of 1.5 K-km s-1. The amount of inferred H2 was comparable
to that of the DNM (confirming the amount of the detected DNM), and only for
three sightlines did it seem likely that saturation of the HI emission profile
could account for much of the DNM. Subsequent detections of CO absorption
along six directions by Liszt et al. (2019) showed that the lack of CO
emission was due to low CO column densities. The linewidth of CO absorption
was shown to be smaller than that of its parent molecule $\mathrm{HCO^{+}}$,
illustrating the complex chemical nature of the CO formation process in
diffuse gas.

In this work we extend the analysis of the composition of DNM to 33 sightlines
in directions toward the Galactic anticenter (Remy et al., 2017, 2018) and
consider the larger sample of 46 sightlines. The present study employs a
relatively small number of diffuse (AV $\leq 1$ mag) and translucent (AV $\leq
3$ mag) sightlines where background sources are serendipitously available, and
is not a revision of the earlier study, whose derived DNM column densities and
other results are assumed here. Rather, we use the newly derived H2
abundances, independent of the presence of CO emission, to consider such
topics as the atomic or molecular character of the DNM, the degree to which
the $\lambda$21cm HI profile represents N(HI) and the influence of the CO-H2
conversion factor on the DNM determination.

Section 2 summarizes the observational material considered here and Section 3
compares the prior results for DNM with atomic hydrogen measured in the
$\lambda$21cm emission and H2 as sampled by its proxy 89 GHz absorption of
$\mathrm{HCO^{+}}$. The role of CO emission data is explored in Section 4
where the CO-H2 conversion factor is derived. Section 5 presents a summary,
with conclusions and a discussion of later work.

##  2 Observations and data reduction

The methods of work were described in Liszt et al. (2018) and are not repeated
in full detail here. From the existing DNM analyses of Planck Collaboration et
al. (2015) and Remy et al. (2017, 2018) we take N(DNM) and $\lambda$21cm HI
column densities N(HI) and N(HI)cl representing respectively the line profile
integral integrated over all velocities and the column density that is
associated with the cloud and/or kinematic features hosting the neutral gas
and DNM, as derived by a decomposition that accounts for the compound nature
of HI emission profiles from individual clouds.

To the existing analyses we add ALMA observations of absorption from
$\mathrm{HCO^{+}}$ and profiles of $\lambda$2.6mm CO emission from the IRAM
30m telescope, as described below.

###  2.1 Millimeter-wave absorption

The new absorption data discussed here are spectra of $\mathrm{HCO^{+}}$ in 33
directions along Galactic anticenter sightlines toward ALMA phase calibrators
as given in the Tables here and projected on sky maps in Figure 1. As before,
the $\mathrm{HCO^{+}}$ spectra were acquired with a spectral resolution of
0.205 km s-1 sampled at 0.102 km s-1 intervals without the so-called spectral
averaging that bins data. We also acquired spectra of HCN, C2H, HCO, and
several isotopologs. Counts of detections are shown in Figure 2, but results
for species other than $\mathrm{HCO^{+}}$ will be discussed in detail in later
work. Statistics of the $\mathrm{HCO^{+}}$ detections are shown in Figure 9.
Spectra in the 28 directions with detectable $\mathrm{HCO^{+}}$ absorption are
shown in Figure 10.

###  2.2 IRAM 30m CO emission

We took frequency-switched CO emission profiles at the IRAM 30m telescope in
August 2019 toward the 28 anticenter sightlines with detected
$\mathrm{HCO^{+}}$ absorption. The CO spectra are shown in Figure 10. These CO
data were taken as five-point maps toward the target and displaced by 2 HPBW
(2x22′′) in the four cardinal directions, using a pointing pattern that was
designed to detect $\mathrm{HCO^{+}}$ and HCN emission in the presence of
spectral line absorption in the target direction. For the $\mathrm{HCO^{+}}$
emission that will be discussed in subsequent work, the $\mathrm{HCO^{+}}$
emission profile is formed by averaging spectra along the four outlying
directions. For CO the present results use the average of all five pointings
because contamination by absorption is not measurable given the strength of
the emission and/or the continuum. The results are presented on the main beam
antenna temperature scale that is native to the 30m telescope.

###  2.3 Conversion from integrated HCO+ absorption to N(H2)

The suitability of $\mathrm{HCO^{+}}$ as a proxy for H2 in diffuse molecular
gas was explored extensively in Liszt & Gerin (2023) (hereafter Paper 1). As
in our earlier work, we use N(H2) = N($\mathrm{HCO^{+}}$)/$3\times 10^{-9}$
and N($\mathrm{HCO^{+}}$) = $1.10\times 10^{12}$$\Upsilon_{\mathrm{HCO^{+}}}$
where $\Upsilon_{\mathrm{HCO^{+}}}$ is the integrated $\mathrm{HCO^{+}}$
optical depth expressed in km s-1.

Figure 1: Sky maps of the observed sightlines. Left: Positions of the
background sources observed here are projected against a map of EB-V from the
work of Schlegel et al. (1998). The sightlines lacking detected
$\mathrm{HCO^{+}}$ absorption (largely due to weak continuum) are indicated
with smaller symbols. Right: Same as left, but shown against a map of N(DNM)
from Remy et al. (2017, 2018).

###  2.4 Reddening and dust optical depth

The 6′ resolution dust-emission maps scaled to optical reddening EB-V by
Schlegel et al. (1998) are cited here. Those reddening values can be converted
to Planck 353 GHz dust optical depth using the relationship established by
Planck Collaboration et al. (2014) between the 353 GHz dust optical depth and
the EB-V values of Schlegel et al. (1998), EB-V/$\tau_{353}=(1.49\pm
0.03)\times 10^{4}$ mag.

###  2.5 Conventions

Velocities presented with the spectra are taken with respect to the kinematic
definition of the Local Standard of Rest. N(H) is the column density of H
nuclei detected in neutral atomic and molecular form, N(H) = N(HI)+2N(H2), and
the molecular hydrogen and DNM fractions 2N(H2)/N(H) and N(DNM)/N(H) are
respectively represented by f${}_{{\rm H}_{2}}$ and fDNM. The integrated
absorption of the $\mathrm{HCO^{+}}$ profile in units of km s-1 is denoted by
$\Upsilon_{{\rm HCO}\mbox{${}^{+}$}}$ and similarly for other species. The
integrated emission of the J=1-0 CO line is denoted by ${\rm W}_{\mathrm{CO}}$
with units of K-km s-1 and the CO-H2 conversion factor N(H2)/${\rm
W}_{\mathrm{CO}}$ is denoted by $X_{\mathrm{CO}}$. Where reference is made to
a typical Galactic or standard CO-H2 conversion factor, the value N(H2)/${\rm
W}_{\mathrm{CO}}$ $=2\times 10^{20}\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm cm}^{-2}$/(1 K-km s-1) should be understood, as summarized in Table E.1
of Remy et al. (2017).

###  2.6 Overview

Observational results of importance to this work are given in Table 1 and
Table 2, and summaries of mean properties of various subsamples are presented
in Table 3. Some statistics of the continuum targets and noise in the
detections of $\mathrm{HCO^{+}}$ absorption are discussed in Appendix A and
the CO emission and $\mathrm{HCO^{+}}$ absorption line profiles are shown in
Figure 10 and discussed in Appendix B.

##  3 HI, DNM, and H2 sampled in HCO+ absorption

Figure 1 shows the new Galactic anticenter sightlines projected on maps of
EB-V and N(DNM). The region at $b<-30$o is described by Remy et al. (2017) as
Cetus, and that at $b>-22$o as Taurus-Main. California lies to the north
around $b=-10$o and Perseus is near (l,b) = 160o,-20o. Taurus-North overlays
most of the map region. The individual subregions are not discussed here owing
to the sparse sampling.

Counts of sources and detected molecular tracers binned in 0.05 mag intervals
of reddening are shown in Figure 2. $\mathrm{HCO^{+}}$ is the molecular tracer
detected most commonly (28 of 33 anticenter sightlines and 41 of 46 overall),
followed by C2H (26 total) and HCN (21 total) . The properties of molecular
absorption line tracers other than $\mathrm{HCO^{+}}$ will be discussed in
later work.

The Chamaeleon and Galactic anticenter subsamples have the same mode at EB-V =
0.25 - 0.3 mag in Figure 2, but the anticenter subsample has a high-reddening
tail at EB-V $>0.5$ mag that is absent in Chamaeleon. The Galactic anticenter
sample has higher mean extinction and molecular fraction. Overall (see Table
3) the anticenter sources have the following characteristics:

  * •

33% higher $<$EB-V$>$ = 0.36 vs 0.27 mag;

  * •

Same $<$N(HI)$>$ = $1.3\times 10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ and
fDNM = 0.12;

  * •

80% higher $<$N(HI)cl$>$ = 1.12 vs 0.62 $\times 10^{21}\leavevmode\nobreak\
{\rm cm}^{-2}$;

  * •

2 times higher $<$2N(H2)$>$ = 0.80 vs 0.39 $\times 10^{20}\leavevmode\nobreak\
{\rm cm}^{-2}$;

  * •

65% higher f${}_{{\rm H}_{2}}$ = 0.38 vs 0.23;

  * •

Higher fraction of detected CO, 9/33 (18/33 in the new work) vs 1/13;

  * •

Higher incidence of nondetection of $\mathrm{HCO^{+}}$, 5/33 vs 0/13, in some
cases due to low flux, see Figure 9.

The DNM fraction is nearly the same in the two samples (0.12-0.13) and much
smaller (0.06) in both subsamples when CO emission was detected at the level
of 1 K-km s-1. The DNM fraction is also noticeably smaller when molecular gas
sampled in $\mathrm{HCO^{+}}$ absorption is absent, and similarly at EB-V $<$
0.2 mag. The strongest variations in fDNM in Table 3 arise in selections based
on the strength of CO emission and are discussed in Section 4.

###  3.1 Quantitative summary results

Figure 3 shows an overview of trends in means of N(HI), N(DNM), and N(H2)
using data binned in 0.05 mag intervals of reddening as in Figure 2, plotted
horizontally at the mean EB-V in every bin. The counts in each bin are shown
and the bins at EB-V $>$ 0.5 mag are occupied by only one or two sightlines.
The association of DNM with molecular gas is unmistakable.

The mean HI column density varies by only a factor of 3 over a wider range
EB-V = 0.09 - 1 mag. By contrast, means of N(NDM) and N(H2) vary by factors of
50-100. The values of $<$N(DNM)$>$ and $<$N(H2)$>$ are comparable and increase
steadily for EB-V $\la$ 0.7 mag, beyond which $<$N(DNM)$>$ either fails to
increase or falls: the CO emission tracer used to derive N(DNM) was used
effectively to account for H nuclei in H2. The cloud-associated mean N(HI)cl
declines at EB-V $\geq 0.7$ mag as H2 sequesters H nuclei. The value of
$<$N(H2)$>$ increases up to the bins at highest EB-V where $<$N(H2)$>$
$\approx$ $<$N(HI$>$) $\approx 2-3$ $<$N(HI)cl$>$ and whole sightlines are
dominated by molecular gas.

Figure 2: Counts of sightlines and detected tracers binned in 0.05 intervals
of reddening, EB-V. The histogram of all sources is overlaid (dashed red) in
the other panels.

The correlation between DNM and H2 is remarkable considering the vastly
different scales on which EB-V, N(HI), and N(NDM) are measured (6-20′), as
contrasted with N(H2) $\propto$ $\Upsilon_{\mathrm{HCO^{+}}}$ that is derived
on submilliarcsecond scales from absorption against scatter-broadened compact
millimeter-wave ALMA calibrator continuum sources. The same kind of
correlation of tracers observed on different angular scales occurs in the
correlation of integrated $\lambda$21cm HI optical depth with EB-V (Liszt,
2021).

When $\mathrm{HCO^{+}}$ is not detected (5/46 directions) the following
characteristics are found compared to sightlines with detected
$\mathrm{HCO^{+}}$:

  * •

3 times lower $<$EB-V$>$ = 0.13 mag;

  * •

1.3 times higher $<$N(H)$>$/$<$EB-V$>$ = $8\times 10^{21}\leavevmode\nobreak\
{\rm cm}^{-2}$ mag-1;

  * •

3.5 times lower $<$N(DNM)$>$ $=8\times 10^{19}\leavevmode\nobreak\ {\rm
cm}^{-2}$;

  * •

2 times lower $<$fDNM$>$ = 0.07;

  * •

no DNM in 3/5 cases vs. 15/46 overall.

Instead, using 3$\sigma$ upper limits for N(H2) we find:

  * •

$>$5 times lower $<$2N(H2)$>\leq 1.5\times 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$ and

  * •

$>$2.5 times lower $<$f${}_{{\rm H}_{2}}$$>\leq 0.14$

The ensemble of sightlines with N(DNM) $<5\times 10^{19}\leavevmode\nobreak\
{\rm cm}^{-2}$ does not differ by more than 10-20% from other sightlines in
any mean property not involving N(DNM). The variations in N(DNM) and N(H2)
stand in great contrast with N(HI) that increases only by a factor of 3 across
the figure. The rise in N(H2) accounts for the failure of N(HI) to increase at
high EB-V, but this could in principle also be explained by increasing
saturation of the HI emission. As noted in Planck Collaboration et al. (2015)
and further refined by Remy et al. (2018), the global DNM analysis finds the
best fit with the optically thin estimate of N(HI), implicitly leaving a
bigger overall role for molecular gas than for saturation of the HI profile.

###  3.2 DNM and HI

Figure 3:  Trends in mean N(HI), N(H2), and N(DNM). The results are binned in
0.05 mag intervals of EB-V. Left: N(HI) is calculated over the whole line
profile. Right: N(HI) = N(HI)cl the HI column density associated with the
clouds studied here. The number of sightlines in each bin is shown in both
panels.

Shown in Figure 4 is the variation of $<$N(DNM)$>$ with $<$N(HI)$>$ and
$<$N(HI)cl$>$. Chamaeleon differs from the Galactic anticenter in having more
extraneous atomic gas along the line of sight and smaller values of cloud-
associated gas. The cloud-associated HI in Chamaeleon is clustered at the low
end of the range, even though the samples are evenly matched in the span of
their total HI column density and have the same mean N(HI) in Table 3.

Sightlines lacking DNM are present at all N(HI), but N(DNM) is uniformly small
along sightlines at the highest column densities N(HI) $\ga 2.3\times
10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ where CO emission is more commonly
observed. The sources with N(DNM) $>6\times 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$ and high N(DNM)/N(HI) ratios are perhaps the most obvious candidates
for hosting significant quantities of DNM in atomic form as the result of
saturation of the HI profile, but only two of these actually have small
contributions from N(H2). Assessing the DNM contribution arising from possible
saturation of the HI profile is discussed in Section 3.3 where it is seen that
the strongest cases of atomic DNM actually have more modest N(DNM).

###  3.3 DNM, EB-V, and N(H2)

As with HI, sightlines lacking DNM are present over the full range of EB-V in
Figure 5, top, but they are predominant at smaller N(H2). Sightlines with
2N(H2) $\la 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ lack DNM. Along
sightlines with column densities too low to host molecular gas, the atomic gas
is well represented by the optically thin estimates of N(HI) used in the DNM
analysis.

Care must be taken in interpreting the relationship of DNM and N(H2) in Figure
5 because the DNM in part represents material that is missing in the CO
emission tracer while some of the H2 traced by $\mathrm{HCO^{+}}$ is actually
visible in CO emission. To minimize this cross-contamination, Figure 6 shows a
plot of N(DNM) vs. N(H2) for sightlines lacking a detection of CO, using the
more sensitive IRAM data for the Galactic anticenter subsample. Included are
all but one (12/13) of the Chamaeleon sightlines and 60% of those toward the
anticenter.

Sightlines lacking DNM in the absence of CO emission in Figure 6 are almost
entirely confined at 2N(H2) $\la 2\times 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$, so the detections of $\mathrm{HCO^{+}}$ absorption do not imply
much additional gas along sightlines where DNM was not found. Missing from
Figure 6 are sightlines with 2N(H2) $\ga 10^{21}\leavevmode\nobreak\ {\rm
cm}^{-2}$ corresponding to ${\rm W}_{\mathrm{CO}}$= 2.5 K-km s-1 for the
Galactic CO-H2 conversion factor.

There are two or three cases in each subsample where N(DNM)/2N(H2) $\ga 2$ and
saturation of the $\lambda$21cm emission profile may be important. Most of
these sightlines have modest values N(DNM) $\approx 4-5\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ and are not the sightlines with
high N(DNM) that were singled out for discussion in Section 3.2 when comparing
N(DNM) and N(HI).

There are also two or three sightlines at N(H2) $>6\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ where N(DNM)/2N(H2) $\la 2$.
Overall $<$2N(H2)$>$ = 1.3$<$N(DNM)$>$ for sources lacking CO emission, so H2
accounts for DNM without adding extra “dark” gas, and the DNM is mostly
molecular. As before in Chamaeleon, the DNM is largely molecular, even if the
medium overall is predominantly atomic.

##  4 The role of CO emission

The CO emission that was detected in previous studies used to determine N(DNM)
along our sightlines was well above 1 K-km s-1 in every case (Table 2), with
$<$${\rm W}_{\mathrm{CO}}$$>$ = 4.9K-km s-1 (Table 3). This is very nearly the
same mean as for the sample of sightlines with ${\rm W}_{\mathrm{CO}}$ $\geq
1$K-km s-1 observed at the IRAM 30m, 5.5 K-km s-1 (see Table 2). Compared to
the overall average, the sightlines with $<$${\rm W}_{\mathrm{CO}}$$>$ $\geq$
1K-km s-1 have the following characteristics:

  * •

50-100% higher $<$EB-V$>$ = $0.5-0.7$ mag;

  * •

3-4 times higher $<$2N(H2)$>$ = $2-3\times 10^{21}\leavevmode\nobreak\ {\rm
cm}^{-2}$;

  * •

2 times higher $<{\rm f}_{\mbox{H${}_{2}$}}>$ = 0.6;

  * •

2 times lower $<$fDNM$>$ = 0.06.

Figure 4: N(DNM) plotted against total sightline N(HI) (bottom) and cloud-
associated atomic hydrogen N($\mathrm{H\textsc{i}}$)cl (top). Figure 5: N(DNM)
plotted against EB-V (top) and 2N(H2) (bottom). Larger symbols represent
sightlines where ${\rm W}_{\mathrm{CO}}$$\ga$ 1 K-km s-1.

The 50% smaller DNM fractions $<$fDNM$>$ = 0.06 along samples of sightlines
with $<$${\rm W}_{\mathrm{CO}}$$>$ $\geq$ 1K-km s-1 indicate that CO emission
is doing a good job of tracing H2 in gas where the molecular fraction is high
(Liszt, 2017), CO emission is strong, and the cloud-associated HI fraction
declines (Figure 2). However, CO emission in general represents a small
portion of the total molecular gas present along the sightlines in this work.
For instance, $<$2N(H2)$>$ = $7\times 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$ along 46 sightlines, while $<$${\rm W}_{\mathrm{CO}}$$>$ $\approx 5$
K-km s-1 along the 6-10 sightlines where $<$${\rm W}_{\mathrm{CO}}$$>$ $\geq$
1K-km s-1 (Table 3). For a typical Galactic CO-H2 conversion factor, the
summed molecular gas column represented in $\mathrm{HCO^{+}}$ is three to four
times that inferred from the summed CO emission. Most of the molecular gas in
our sample was hidden in the DNM prior to our work and is still only revealed
by the $\mathrm{HCO^{+}}$ absorption, even with more sensitve CO observations.
The IRAM 30m detections with $<$${\rm W}_{\mathrm{CO}}$$>$ below 1K-km s-1
comprise less than 20% of the total amount of CO emission.

###  4.1 The CO-H2 conversion factor

Comparison of the scaled $\mathrm{HCO^{+}}$ absorption and IRAM 30 emission
measurements provides the most direct determination of the actual CO-H2
conversion factor along the lines of sight studied in this work. Figure 7
shows the trends in $<$N(H2)$>$, $<$${\rm W}_{\mathrm{CO}}$$>$, and
$<$N(H2)$>$/$<$${\rm W}_{\mathrm{CO}}$$>$ binned in 0.05 mag increments of
EB-V as in Figure 3. Included are the 28 Galactic anticenter sightlines where
$\mathrm{HCO^{+}}$ was detected, with ${\rm W}_{\mathrm{CO}}$ taken at the
3$\sigma$ upper limit for sightlines where CO emission was not detected with
greater significance. Also included is the sightline toward J1733 in
Chamaeleon where CO emission was detected. The $3\sigma$ upper limits ${\rm
W}_{\mathrm{CO}}$ $\leq$ 1.5 K-km s-1 in Chamaeleon are not useful.

Figure 7 illustrates the behavior of the CO-H2 conversion factor that is
tabulated for different samples in Table 3. The values of $<$N(H2)$>$ and
$<$${\rm W}_{\mathrm{CO}}$$>$ both increase steadily with EB-V in Figure 7,
but at different rates so that their ratio declines by a factor $\approx 7$ to
$<$N(H2)$>$/$<$${\rm W}_{\mathrm{CO}}$$>$ $=2.5-3\times
10^{20}$H2$\leavevmode\nobreak\ {\rm cm}^{-2}$(K-km s-1)-1 for EB-V $\ga$ 0.5
mag. Variations of similar magnitude in individual diffuse and/or translucent
MBM clouds were reported by Magnani et al. (1998) and Cotten & Magnani (2013).

Mean values of N(H2)/${\rm W}_{\mathrm{CO}}$ are $2-2.5\times
10^{20}$H2$\leavevmode\nobreak\ {\rm cm}^{-2}$(K-km s-1)-1 for the old and new
samples with $<$${\rm W}_{\mathrm{CO}}$$>$ $\geq 1$ K-km s-1, increasing by
factors of 2-3 for the samples with IRAM 30m detections below 1K-km s-1 and
upper limits. Also see the inset in Figure 7 on this point..

Figure 6: N(DNM) plotted against N(H2) for sightlines lacking detected CO
emission in the analysis of Remy et al. (2018). Shown are loci at which the
number of H nuclei in H2 is 50, 100, and 200% of that in DNM. Figure 7:
Trends in mean N(H2) (black squares), ${\rm W}_{\mathrm{CO}}$ (pink diamonds),
and N(H2)/${\rm W}_{\mathrm{CO}}$ (blue triangles). The results are binned in
0.05 mag intervals of EB-V, as in Figure 3. Bins in which all sightlines have
only upper limits on ${\rm W}_{\mathrm{CO}}$ are indicated by upper and lower
limits. The scales for N(H2) $\leavevmode\nobreak\ {\rm cm}^{-2}$ and
N(H2)/${\rm W}_{\mathrm{CO}}$ ($\leavevmode\nobreak\ {\rm cm}^{-2}$ (K-km
s-1)${}^{-}1$) are read on the left vertical axis and that for ${\rm
W}_{\mathrm{CO}}$ (K-km s-1) on the right. The number of sightlines in each
bin is shown as in Figure 3 and the count is carried separately for ${\rm
W}_{\mathrm{CO}}$ at low EB-V. The light gray dashed line is a power-law fit
N(H2) $=10^{21.0973}$EB-V1.335.

The values of the CO-H2 conversion factor derived are comparable to those
derived in extant Galactic-scale $\gamma$-ray analyses (Table E.1 in Remy et
al. 2017) when ${\rm W}_{\mathrm{CO}}$ $\ga$ 1 K-km s-1, but are uniformly
larger than those derived in cloud-level studies like the DNM analysis whose
results are summarized in Table 2 of Remy et al. (2017). Those results ranged
from $1-1.6\times 10^{20}$H2$\leavevmode\nobreak\ {\rm cm}^{-2}$(K-km s-1)-1
for determinations based on dust and from $0.44-1.00\times
10^{20}$H2$\leavevmode\nobreak\ {\rm cm}^{-2}$ (K-km s-1)-1 for determinations
based on gamma rays.

There is no contradiction with the larger values found in this work whose
method of studying widely separated, semi-randomly placed sightlines is
similar to the larger scale studies. The strongly CO-emitting (${\rm
W}_{\mathrm{CO}}$ $>$ 10 K-km s-1) regions encountered in the cloud-level
studies that have generally small values of $X_{\mathrm{CO}}$ (see Figure 13
in Remy et al. 2018) were not sampled here.

There may also be a difference arising from the operational definition of the
conversion factor. The conversion factors in Table 2 in Remy et al. (2017) are
those that optimize the fit of the $\gamma$-ray or dust emissivity to a total
hydrogen column density that is represented schematically as N(H) =
N(HI)+N(DNM)+2N(H2) = N(HI)+N(DNM)+2$X_{\mathrm{CO}}$${\rm W}_{\mathrm{CO}}$.
After the analysis, there remains a DNM constituent whose molecular fraction
is undetermined and not explicitly considered in the definition of the
multiplier $X_{\mathrm{CO}}$; we showed that the DNM is largely molecular. By
contrast, the present analysis determines N(H2) independent of CO emission and
defines $X_{\mathrm{CO}}$ = N(H2)/${\rm W}_{\mathrm{CO}}$. This N(H2) includes
the molecular component of the DNM that we took pains to consider separately
in the discussion of Figure 6.

###  4.2 Achievable limits and detection thresholds for CO, DNM, and H2

Sightlines in our study often had rms CO emission noise $\Delta$${\rm
W}_{\mathrm{CO}}$ $\ga 1/3-1/2$ K-km s-1 in the prior DNM analysis (Table 2).
For a typical Galactic conversion 2N(H2)/${\rm W}_{\mathrm{CO}}$ $=4\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ (K-kms)-1, the equivalent $3\sigma$
threshold detection limits on the hydrogen column are 2N(H2) $=4-6\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$, above the actual values of N(DNM)
along most of the sightlines we observed. By contrast, the median $3\sigma$
detection threshold from $\mathrm{HCO^{+}}$ in our work is 2N(H2) $>1.1\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ and typical N(DNM) values are
N(DNM) $\ga 2\times 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ (Figure 6).

The effectiveness of reducing the detection threshold of CO emission below 1
K-km s-1 can be assessed by examining the distribution of CO detections and
upper limits in the IRAM 30m CO observations. The subsamples of IRAM CO
observations summarized in Table 3 show 6 detections with ${\rm
W}_{\mathrm{CO}}$ $>$ 1 K-km s-1, 12 detections with ${\rm W}_{\mathrm{CO}}$
$<$ 1 K-km s-1, and ten upper limits at levels ${\rm W}_{\mathrm{CO}}$
$<0.1-0.2$ K-km s-1 (see Table 1 for values of upper limits and Figure 3 for a
graphical representation of the data). $\mathrm{HCO^{+}}$ was detected along
all of these sightlines, and the three subsamples respectively represent
fractions 0.66, 0.26, and 0.08 of the total amount of H2.

Thus, a survey with a detection limit of 1 K-km s-1 would detect approximately
two-thirds of the molecular gas along these diffuse and/or translucent lines
of sight, and a much increased effort to reduce the detection limit to 0.2
K-km s-1 might find another one-fourth of the H2. Comparable reductions in the
fraction of undetected H2 with increasing CO sensitivity down to rms levels
$\Delta$${\rm W}_{\mathrm{CO}}$ $\approx 0.1$ K-km s-1 were achieved by Donate
& Magnani (2017). Missing one-third of the H2 at the 1 K-km s-1 threshold is
consistent with the dark gas fraction derived by Wolfire et al. (2010) and
Gong et al. (2018).

###  4.3 The wider view

In Section 4.2 we note that two-thirds of the H2 was found along the
sightlines with ${\rm W}_{\mathrm{CO}}$ $>$ 1 K-km s-1 and that a deeper
survey with a detection limit of 0.2 K-km s-1 would have found another 25% of
the H2. At this point we can ask how the results of this sparse sampling are
reflected in the region as a whole. We drew a hull around the observed
anticenter sightlines, as shown in Figure 11 and derived the pixel-by-pixel
statistics for the contained area, the amount of material (taken proportional
to EB-V), and the amount of H2. For H2 we used the N(H2)-EB-V relationship
N(H2) $=10^{21.0973}$EB-V1.335 111f${}_{{\rm H}_{2}}$ = 2N(H2)/N(H) = 0.4-0.5
at EB-V = 2 mag if N(H)/EB-V $=6-8\times 10^{21}{\rm mag}^{-1}$ and the fact
that ${\rm W}_{\mathrm{CO}}$ $\ga 1$ K-km s-1 at EB-V $\ga$ 0.4 mag or N(H2)
$\ga 4\times 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ in Figure 7 (see also
Figure 5 of Paper 1).

Figure 8 shows the probability densities (at left) and cumulative
distributions for the derived quantities. Reading values off the cumulative
probability distributions at right in Figure 8, conditions with EB-V $\geq$
0.4 mag, N(H2) $\ga 4\times 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ occur
over one-fourth of the total area containing one-half of the total projected
gas column and two-thirds of the H2. Apparently, the sampled sightlines were
representative of the region as a whole regarding the H2 distribution.

Sampling 90% of the H2 would require detecting CO emission at EB-V $\ga 0.2$
mag where N(H2) $\approx 2\times 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$.
The sampling here is too sparse to derive an equivalent value of ${\rm
W}_{\mathrm{CO}}$ in Figure 7, but the broader sample in Paper 1 suggests
${\rm W}_{\mathrm{CO}}$ $\ga$ 0.3 K-km s-1 would be appropriate. Sightlines
with EB-V $<0.32$ mag or AV $<1$ mag comprise two-thirds of the area, one-
third of the mass, and one-fourth of the H2.

Figure 8:  Probability distributions of area, amount of material, and
molecular hydrogen with respect to EB-V. Left: Counts (probability density)
scaled to the same area. Right: Cumulative distribution function. N(H2) is
calculated using the mean power-law fit N(H2) $=10^{21.0973}$EB-V1.335 derived
as shown in Figure 7.

###  4.4 Comparison with predictions and models

For models, the critical factors in predicting the amount of dark gas are the
threshold where ${\rm W}_{\mathrm{CO}}$ reaches the 1 K-km s-1 brightness
level and the amount of material above the threshold. In the regime of diffuse
molecular gas the integrated brightness of the J=1-0 line is determined almost
exclusively by the CO column density, with ${\rm W}_{\mathrm{CO}}$ (K-km s-1)
= N(CO)$/10^{15}\leavevmode\nobreak\ {\rm cm}^{-2}$ for hydrogen number
densities n(H) $=64-500\leavevmode\nobreak\ {\rm cm}^{-3}$ (Liszt, 2007, 2017;
Hu et al., 2021) and kinetic temperatures appropriate to the local thermal
pressure. This simple proportionality between column density and brightness
persists well beyond the CO column density where the J=1-0 transition becomes
optically thick, owing to the strongly subthermal excitation (Goldreich &
Kwan, 1974).

Thus, the CO chemistry dominates the CO brightness, and the sky map of CO
emission is a map of the chemistry. The observations can be reproduced by an
ad hoc CO formation chemistry in which a fixed relative abundance of
$\mathrm{HCO^{+}}$ N($\mathrm{HCO^{+}}$)/N(H2) $=3\times 10^{-9}$ thermally
recombines to CO if the H2 formation and shielding of H2 and CO are treated
self-consistently along with the heating and cooling Liszt (2017). However,
networks of chemical reactions in quiescent gas may fail to form the observed
quantity of $\mathrm{HCO^{+}}$ and produce too little CO. This causes the CO
brightness to reach 1 K-km s-1 at overly large values of EB-V and N(H2) where
shielding of CO by dust and H2 make up for the deficit in the CO formation
rate (e.g., Gong et al. 2018 Figure 7; Hu et al. 2021 Figure 9).

Models with a weak CO formation chemistry have an innate tendency to
overestimate the amount of CO-dark gas, but the actual amount of CO-dark gas
depends on the distribution of material. In practice the DNM fraction seen by
Remy et al. (2018) varied from 0 to 0.3 (their Figure 8) and the sightlines
sampled here had $<$fDNM$>$ = 0.13.

Table 1: Line-of-sight properties and new results for $\mathrm{HCO^{+}}$ and
CO

Source | RA(J2000) | Dec(J2000) | $l$ | $b$ |  EB-V1 | 89 GHz flux |  $\Upsilon_{{\rm HCO}\mbox{${}^{+}$}}$2 |  $\sigma$$\Upsilon_{{\rm HCO}\mbox{${}^{+}$}}$3 |  ${\rm W}_{\mathrm{CO}}$4 |  $\sigma$${\rm W}_{\mathrm{CO}}$5
---|---|---|---|---|---|---|---|---|---|---
| hh.mmsss | dd.mmsss | degrees | degrees | mag | Jy | km s-1 | km s-1 |  K-km s-1 |  K-km s-1
J0203+1134 | 2.03464 | 11.34492 | 149.6826 | -47.4992 | 0.144 | 0.126 |  $\leq$0.263 | 0.088 |  |
J0209+1352 | 2.09357 | 13.52045 | 150.1800 | -44.8290 | 0.094 | 0.223 | 0.20 | 0.050 |  $\leq$0.071 | 0.024
J0211+1051 | 2.11132 | 10.51348 | 152.5781 | -47.3674 | 0.145 | 0.462 | 0.76 | 0.029 | 0.36 | 0.025
J0213+1820 | 2.13105 | 18.20255 | 148.7289 | -40.4014 | 0.130 | 0.093 |  $\leq$0.345 | 0.115 |  |
J0231+1322 | 2.31459 | 13.22547 | 157.0917 | -42.7380 | 0.121 | 0.430 | 0.14 | 0.025 |  $\leq$0.064 | 0.021
J0242+1742 | 2.42243 | 17.42589 | 157.0180 | -37.7033 | 0.077 | 0.227 |  $\leq$0.151 | 0.050 |  |
J0252+1718 | 2.52077 | 17.18427 | 159.7420 | -36.7885 | 0.220 | 0.172 | 0.25 | 0.077 |  $\leq$0.096 | 0.032
J0325+2224 | 3.25368 | 22.24004 | 163.6700 | -28.0213 | 0.213 | 1.162 | 1.01 | 0.017 | 0.94 | 0.051
J0329+3510 | 3.29154 | 35.10060 | 155.9152 | -17.4042 | 0.267 | 0.570 | 0.51 | 0.032 |  $\leq$0.143 | 0.048
J0329+2756 | 3.29577 | 27.56155 | 160.7030 | -23.0743 | 0.198 | 0.158 |  $\leq$0.193 | 0.064 |  |
J0334+0800 | 3.34533 | 8.00144 | 177.2396 | -37.0871 | 0.391 | 0.150 | 0.44 | 0.088 |  $\leq$0.162 | 0.054
J0336+3218 | 3.36301 | 32.18293 | 158.9998 | -18.7650 | 0.733 | 1.689 | 0.16 | 0.009 |  $\leq$0.219 | 0.073
J0356+2903 | 3.56085 | 29.03423 | 164.6120 | -18.4927 | 0.212 | 0.139 | 1.50 | 0.090 | 1.62 | 0.042
J0357+2319 | 3.57216 | 23.19538 | 169.0302 | -22.4661 | 0.185 | 0.224 | 0.28 | 0.027 |  $\leq$0.192 | 0.064
J0400+0550 | 4.00117 | 5.50431 | 184.2710 | -33.7266 | 0.266 | 0.159 | 0.25 | 0.063 |  $\leq$0.172 | 0.057
J0401+0413 | 4.01199 | 4.13344 | 186.0261 | -34.4947 | 0.341 | 0.405 | 0.49 | 0.021 | 0.19 | 0.048
J0403+2600 | 4.03056 | 26.00015 | 168.0260 | -19.6483 | 0.201 | 0.331 | 0.60 | 0.029 | 0.62 | 0.067
J0406+0637 | 4.06343 | 6.37150 | 184.7075 | -32.0009 | 0.283 | 0.264 | 0.54 | 0.051 | 0.63 | 0.040
J0407+0742 | 4.07291 | 7.42075 | 183.8723 | -31.1558 | 0.265 | 0.387 | 0.53 | 0.031 | 0.19 | 0.042
J0426+0518 | 4.26366 | 5.18199 | 189.3631 | -28.7705 | 0.291 | 0.516 | 0.12 | 0.020 |  $\leq$0.170 | 0.057
J0426+2327 | 4.26557 | 23.27396 | 173.8881 | -17.4457 | 0.539 | 0.304 | 2.57 | 0.057 | 4.84 | 0.045
J0427+0457 | 4.27476 | 4.57083 | 189.8857 | -28.7306 | 0.335 | 0.414 | 0.62 | 0.024 | 0.39 | 0.045
J0437+2037 | 4.31038 | 20.37343 | 176.8096 | -18.5565 | 0.532 | 0.217 | 1.54 | 0.073 | 0.67 | 0.026
J0431+1731 | 4.31574 | 17.31358 | 179.4942 | -20.3579 | 0.464 | 0.104 | 1.01 | 0.110 | 0.70 | 0.049
J0433+0521 | 4.33111 | 5.21156 | 190.3730 | -27.3967 | 0.298 | 4.911 | 0.35 | 0.003 |  $\leq$0.109 | 0.036
J0437+2940 | 4.37044 | 29.40138 | 170.5818 | -11.6609 | 0.979 | 0.059 | 5.92 | 1.138 | 10.42 | 0.027
J0438+3004 | 4.38049 | 30.04455 | 170.4116 | -11.2283 | 0.952 | 0.689 | 6.25 | 0.038 | 7.11 | 0.026
J0439+3045 | 4.39178 | 30.45076 | 170.0655 | -10.5913 | 0.867 | 0.195 | 5.05 | 0.082 | 6.69 | 0.027
J0440+1437 | 4.40211 | 14.37570 | 183.2538 | -20.5438 | 0.681 | 0.337 | 1.21 | 0.031 | 0.83 | 0.029
J0445+0715 | 4.45014 | 7.15539 | 190.4535 | -23.8898 | 0.121 | 0.275 |  $\leq$0.083 | 0.028 |  |
J0449+1121 | 4.49077 | 11.21286 | 187.4274 | -20.7365 | 0.504 | 0.521 | 0.65 | 0.022 | 0.23 | 0.033
J0502+1338 | 5.02332 | 13.38110 | 187.4143 | -16.7456 | 0.564 | 0.271 | 1.81 | 0.059 | 2.46 | 0.031
J0510+1800 | 5.10024 | 18.00416 | 184.7304 | -12.7895 | 0.328 | 2.411 | 0.13 | 0.005 | 0.17 | 0.036

1From Schlegel et al. (1998)
2Integrated optical depth or $3\sigma$ upper limit over a 3 km s-1 interval
3Integrated emission from our results or $3\sigma$ upper limit over a 3 km s-1
interval
4 rms of detected emission or rms over a 3km s-1 interval

Table 2: Line-of-sight properties and derived quantities

Source | E(B-V) |  N(HI)1 |  N(HI)cl1 |  2N(H2) 2 |  ${\rm W}_{\mathrm{CO}}$3 |  N(DNM) 1 |  2N(H2)/N(HI) | N(DNM)/N(HI) |  $X_{\mathrm{CO}}$4
---|---|---|---|---|---|---|---|---|---
| mag | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ |  K-km s-1 | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ |  |  |
J0203+1134 | 0.144 | 0.664 | 0.646 |  $\leq$0.193 |  | 0.201 |  $\leq$0.290 | 0.303 |
J0209+1352 | 0.094 | 0.696 | 0.686 | 0.144 |  | 0.00 | 0.206 | 0.00 |
J0211+1051 | 0.145 | 0.618 | 0.617 | 0.561 |  | 0.240 | 0.908 | 0.388 |
J0213+1820 | 0.130 | 0.933 | 0.911 |  $\leq$0.253 |  | 0.177 |  $\leq$0.271 | 0.190 |
J0231+1322 | 0.121 | 0.926 | 0.852 | 0.103 |  | 0.00 | 0.111 | 0.00 |
J0242+1742 | 0.077 | 0.820 | 0.819 |  $\leq$0.111 |  | 0.00 |  $\leq$0.135 | 0.00 |
J0252+1718 | 0.220 | 1.142 | 1.139 | 0.184 |  | 0.299 | 0.161 | 0.262 |
J0325+2224 | 0.213 | 0.999 | 0.926 | 0.742 | 2.82 | 0.180 | 0.743 | 0.180 | 1.314
J0329+3510 | 0.267 | 1.418 | 1.131 | 0.372 |  | 0.150 | 0.263 | 0.106 |
J0329+2756 | 0.198 | 1.129 | 1.002 |  $\leq$0.142 | $\leq 1.97$ | 0.00 |  $\leq$0.126 | 0.00 |
J0334+0800 | 0.391 | 1.893 | 1.855 | 0.322 |  | 0.223 | 0.170 | 0.118 |
J0336+3218 | 0.733 | 1.435 | 1.165 | 0.119 |  $\leq$2.60 | 0.606 | 0.082 | 0.423 |
J0356+2903 | 0.212 | 0.666 | 0.520 | 1.099 | 1.40 | 0.178 | 1.650 | 0.267 | 3.931
J0357+2319 | 0.185 | 1.022 | 0.944 | 0.205 |  $\leq$0.79 | 0.00 | 0.200 | 0.00 |
J0400+0550 | 0.266 | 1.266 | 1.210 | 0.182 |  | 0.238 | 0.144 | 0.188 |
J0401+0413 | 0.341 | 1.209 | 1.202 | 0.357 |  | 0.426 | 0.296 | 0.352 |
J0403+2600 | 0.201 | 0.830 | 0.667 | 0.438 | 1.99 | 0.00 | 0.528 | 0.00 | 1.101
J0406+0637 | 0.283 | 1.182 | 1.143 | 0.397 |  | 0.266 | 0.335 | 0.225 |
J0407+0742 | 0.265 | 1.033 | 1.006 | 0.387 |  | 0.258 | 0.374 | 0.250 |
J0426+0518 | 0.291 | 1.305 | 1.141 | 0.091 |  $\leq$1.40 | 0.414 | 0.070 | 0.317 |
J0426+2327 | 0.539 | 1.586 | 1.331 | 1.887 | 5.78 | 0.00 | 1.190 | 0.00 | 1.632
J0427+0457 | 0.335 | 1.361 | 1.192 | 0.457 | 2.49 | 0.590 | 0.336 | 0.433 | 0.918
J0437+2037 | 0.532 | 2.324 | 1.992 | 1.130 |  $\leq$1.61 | 0.00 | 0.486 | 0.00 |
J0431+1731 | 0.464 | 1.714 | 1.457 | 0.739 | 3.09 | 0.00 | 0.431 | 0.00 | 1.198
J0433+0521 | 0.298 | 1.159 | 0.950 | 0.255 |  $\leq$1.48 | 0.795 | 0.220 | 0.686 |
J0437+2940 | 0.979 | 1.320 | 0.936 | 4.345 | 11.98 | 0.126 | 3.292 | 0.095 | 1.814
J0438+3004 | 0.952 | 2.019 | 1.002 | 4.586 | 7.70 | 0.771 | 2.271 | 0.382 | 2.979
J0439+3045 | 0.867 | 2.361 | 0.745 | 3.701 | 9.42 | 0.00 | 1.567 | 0.00 | 1.964
J0440+1437 | 0.681 | 1.659 | 1.524 | 0.888 |  $\leq$1.97 | 1.230 | 0.535 | 0.741 |
J0445+0715 | 0.121 | 1.044 | 0.915 |  $\leq$0.061 |  | 0.00 |  $\leq$0.058 | 0.00 |
J0449+1121 | 0.504 | 1.416 | 1.347 | 0.476 |  | 0.791 | 0.336 | 0.558 |
J0502+1338 | 0.564 | 2.188 | 1.957 | 1.324 |  $\leq$1.90 | 0.672 | 0.605 | 0.307 |
J0510+1800 | 0.328 | 2.347 | 1.986 | 0.092 |  $\leq$2.00 | 0.00 | 0.039 | 0.00 |
J0942-7731 | 0.336 | 1.000 | 0.617 | 0.837 |  $\leq$1.50 | 0.382 | 0.838 | 0.382 |
J1058-8003 | 0.152 | 0.651 | 0.470 | 0.147 | “ | 0.371 | 0.226 | 0.570 |
J1136-6827 | 0.460 | 2.407 | 0.915 | 0.910 | “ | 0.172 | 0.378 | 0.071 |
J1145-6954 | 0.387 | 1.925 | 0.998 | 0.638 | “ | 0.223 | 0.331 | 0.116 |
J1147-7935 | 0.300 | 2.506 | 0.773 | 0.040 | “ | 0.00 | 0.016 | 0.00 |
J1152-8344 | 0.283 | 1.088 | 0.505 | 0.176 | “ | 0.515 | 0.162 | 0.473 |
J1224-8313 | 0.257 | 0.962 | 0.563 | 0.693 | “ | 0.335 | 0.720 | 0.348 |
J1254-7138 | 0.282 | 1.902 | 0.660 | 0.102 | “ | 0.00 | 0.053 | 0.00 |
J1312-7724 | 0.476 | 1.257 | 0.671 | 0.199 | “ | 0.396 | 0.158 | 0.315 |
J1550-8258 | 0.107 | 0.679 | 0.416 | 0.177 | “ | 0.027 | 0.261 | 0.039 |
J1617-7717 | 0.091 | 0.659 | 0.431 | 0.043 | “ | 0.045 | 0.065 | 0.068 |
J1723-7713 | 0.255 | 0.834 | 0.500 | 1.105 | 2.40 | 0.195 | 1.325 | 0.234 | 2.302
J1733-7935 | 0.139 | 0.792 | 0.525 | 0.041 |  $\leq$ 1.50 | 0.00 | 0.052 | 0.00 |

1 HI and DNM column densities are from Planck Collaboration et al. (2015) and
Remy et al. (2017, 2018)
2 N(H2) = N($\mathrm{HCO^{+}}$)/$3\times 10^{-9}$
3 Integrated J=1-0 CO emission used by Planck Collaboration et al. (2015) and
Remy et al. (2017, 2018)
4 $X_{\mathrm{CO}}$ = N(H2)/${\rm W}_{\mathrm{CO}}$

Table 3: Mean line-of-sight properties and derived quantities

Sample | E(B-V) | N(HI) | N(HI)cl |  2N(H2) | ${\rm W}_{\mathrm{CO}}$ | N(DNM) | f${}_{{\rm H}_{2}}$ | fDNM | $X_{\mathrm{CO}}$
---|---|---|---|---|---|---|---|---|---
| mag | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ |  K-km s-1 | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ |  |  |
Galactic anticenter | 0.362 | 1.324 | 1.119 | 0.798 | 5.185 | 0.268 | 0.38 | 0.13 | 1.93
# | 33 | 33 | 33 | 33 | 9 | 33 | 33 | 33 | 9
Chamaeleon | 0.271 | 1.282 | 0.619 | 0.393 | 2.400 | 0.205 | 0.23 | 0.12 | 2.30
# | 13 | 13 | 13 | 13 | 1 | 13 | 13 | 13 | 1
All LOS | 0.336 | 1.312 | 0.977 | 0.684 | 4.906 | 0.250 | 0.34 | 0.13 | 1.95
# | 46 | 46 | 46 | 46 | 10 | 46 | 46 | 46 | 10
EB-V $\leq 0.2$mag | 0.131 | 0.818 | 0.710 | 0.168 |  | 0.082 | 0.18 | 0.08 |
# | 13 | 13 | 13 | 13 |  | 13 | 13 | 13 |
Lacking $\mathrm{HCO^{+}}$1 | 0.134 | 0.918 | 0.859 |  $\leq$0.152 |  | 0.076 |  $\leq$0.14 |  $\geq$0.08 |
# | 5 | 5 | 5 | 5 |  | 5 | 5 | 5 |
DNM $\leq 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ | 0.273 | 1.373 | 0.953 | 0.539 | 5.070 | 0.004 | 0.28 | 0.00 | 1.67
# | 17 | 17 | 17 | 17 | 4 | 17 | 17 | 17 | 4
CO detected earlier2 | 0.502 | 1.369 | 0.928 | 1.910 | 4.906 | 0.204 | 0.58 | 0.06 | 1.95
# | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10
30m CO detected | 0.467 | 1.491 | 1.197 | 1.312 | 2.170 | 0.318 | 0.47 | 0.11 | 3.02
# | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18
30m ${\rm W}_{\mathrm{CO}}$ $>$1K-km s-1 | 0.685 | 1.690 | 1.082 | 2.824 | 5.522 | 0.291 | 0.63 | 0.06 | 2.56
# | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6
30m ${\rm W}_{\mathrm{CO}}$ $\leq$1K-km s-1 | 0.358 | 1.391 | 1.255 | 0.555 | 0.494 | 0.332 | 0.29 | 0.17 | 5.62
# | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12
30m UL $\leq$1K-km s-1 | 0.287 | 1.226 | 1.107 | 0.198 |  $<$0.140 | 0.273 | 0.14 | 0.19 |  $>$7.07
# | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10
Sample | E(B-V) | N(HI) | N(HI)cl |  2N(H2) | ${\rm W}_{\mathrm{CO}}$ | N(DNM) | f${}_{{\rm H}_{2}}$ | fDNM |  XCO
| mag | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ |  | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ |  |  |

1 Limits in this table use $3\sigma$ upper limits on undetected quantities
2 All sightlines detected earlier in CO had ${\rm W}_{\mathrm{CO}}$ $>$1K-km
s-1

##  5 Summary

We took 89.2 GHz ALMA $\mathrm{HCO^{+}}$ absorption spectra toward 33 compact
millimeter-wave extragalactic continuum sources seen against the Galactic
anticenter (Figure 1 and the tables above). We also observed J=1-0 CO emission
at the IRAM 30m telescope in the 28 anticenter directions where
$\mathrm{HCO^{+}}$ was detected. Inferring N(H2) from N($\mathrm{HCO^{+}}$)
using the ratio N($\mathrm{HCO^{+}}$)/N(H2) $=3\times 10^{-9}$, we combined
these results with those from our earlier study of 13 directions where
$\mathrm{HCO^{+}}$ absorption was detected in the outskirts of Chamaeleon. We
compared the inferred N(H2) with prior determinations of the column densities
of the dark neutral medium, the neutral gas of uncertain (atomic or molecular)
character that had been found to be missing in HI and/or CO emission when
compared with the submillimeter dust and $\gamma$-ray emissivities of large-
scale molecular cloud complexes.

Binning the HI, H2, and DNM column densities in reddening, we showed in Figure
3 that the mean DNM and molecular gas column densities were comparable and
varied compatibly by factors of 50-100 over the observed range EB-V = 0.09 - 1
mag, while N(H(I) varied by only factors of 2-3. The means of N(H2) and N(DNM)
are small at low mean reddening, and increase in similar fashion up to EB-V=
0.5 mag where molecular gas begins to dominate and CO emission is strong.
N(H2) continues to increase with higher reddening, but N(DNM) and the column
density of cloud-associated HI fall where H2 dominates (sequestering hydrogen)
and CO emission is stronger and more closely representative of N(H2).

We made detailed individual sightline-level comparisons of N(DNM) with EB-V,
N(HI), and N(H2) in Figures 4-6. Sightlines with appreciable DNM appear at all
N(HI) (Figure 4); the overall mean DNM fraction fDNM = N(DNM)/(N(HI)+2N(H2)) =
0.12 is modest. Sightlines with appreciable DNM are lacking when EB-V $\la$
0.15 mag (Figure 5) and when 2N(H2) $\la 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$ or 2N(H2) $\ga 10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ (Figure
6). In Figure 6 we compared N(DNM) and N(H2) in directions lacking detected CO
emission in order to eliminate the case that H2 was already represented by CO
emission in the DNM analysis. This figure showed that there were 2-3
sightlines in each subsample (Chamaeleon and anticenter) or 5-6/46 overall
where 2N(H2) $<\leq$ N(DNM)/2 and H2 accounted for the minority of the DNM.
There are also a few directions at 2N(H2) $\approx 10^{21}\leavevmode\nobreak\
{\rm cm}^{-2}$ where 2N(H2) $>$ 2N(DNM). Overall, the amounts of DNM and H2
are similar $<$2N(H2)$>$ = 1.3 $<$N(DNM)$>$ for the unambiguous cases lacking
CO emission. The form of the DNM is overwhelmingly molecular hydrogen.

Directions with $<$${\rm W}_{\mathrm{CO}}$$>$ $>$1 K-km s-1 have two times
smaller mean DNM fractions $<$fDNM$>$ = 0.06, while sightlines with $<$${\rm
W}_{\mathrm{CO}}$$>$ $<$ 1 K-km s-1 have three times higher fDNM $\ga$
0.17-0.19 (Table 3). The relatively few sightlines with ${\rm
W}_{\mathrm{CO}}$ $\geq$ 1 K-km s-1 contain two-thirds of the H2 detected in
$\mathrm{HCO^{+}}$, and detecting 90% of the H2 would require detecting CO at
levels ${\rm W}_{\mathrm{CO}}$ $\approx$ 0.2-0.3 K-km s-1.

The CO-H2 conversion factor falls steadily with increasing EB-V or ${\rm
W}_{\mathrm{CO}}$ in Figure 7. Because the H2 is concentrated in the
sightlines with ${\rm W}_{\mathrm{CO}}$ $\geq$ 1 K-km s-1, the overall mean
CO-H2 conversion factors in our work are $<$N(H2)$>$/$<$${\rm
W}_{\mathrm{CO}}$$>$ = $2-3\times 10^{20}$H2$\leavevmode\nobreak\ {\rm
cm}^{-2}$ for samples of sightlines with detectable CO emission. These values
are comparable to previously determined large-scale Galactic averages, and are
substantially higher than the global values determined by the cloud-level
analyses quoted here to derive N(DNM). We ascribed these differences in part
to the present sampling of widely scattered sightlines (i.e., that we did not
do a cloud-level analysis) and perhaps to differences in the operational
definition of the conversion factor, as discussed in Section 4.1.

Subsequent papers derived from the observations discussed here will focus on
the physics and chemistry of the molecules observed in the course of this
work, chiefly $\mathrm{HCO^{+}}$, C2H, and HCN.

###### Acknowledgements.

The National Radio Astronomy Observatory (NRAO) is a facility of the National
Science Foundation, operated by Associated Universities, Inc. This work was
supported by the French program “Physique et Chimie du Milieu Interstellaire”
(PCMI) funded by the Conseil National de la Recherche Scientifique (CNRS) and
Centre National d’Etudes Spatiales (CNES). This work is based in part on
observations carried out under project number 003-19 with the IRAM 30m
telescope]. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN
(Spain). This paper makes use of the following ALMA data –
ADS/JAO.ALMA#2016.1.00714.S
– ADS/JAO.ALMA#2018.1.00115.S ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. We thank Isabelle Grenier for providing results of the DNM analysis and
we thank the anonymous referee for many helpful remarks. HSL is grateful for
the hospitality of the ITU-R and the Hotel de la Cigogne in Geneva during the
completion of this manuscript.

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##  Appendix A Statistics of HCO+ detections

Figure 9:  Integrated $\mathrm{HCO^{+}}$ optical depth
$\Upsilon_{\mathrm{HCO^{+}}}$ plotted against its rms error. The upper limits
along five sightlines not detected in $\mathrm{HCO^{+}}$ are $3\sigma$.

$\mathrm{HCO^{+}}$ absorption was not detected in five directions, all in the
Galactic anticenter sample. Figure 9 shows $\Upsilon_{\mathrm{HCO^{+}}}$
plotted against its rms error $\Delta$$\Upsilon_{\mathrm{HCO^{+}}}$, in
essence against the strength of the continuum background since all sightlines
were observed for the same length of time. The plot shows that even the
noisiest sightlines miss relatively small amounts of molecular gas. The bias
in the plot, whereby $<$$\Upsilon_{\mathrm{HCO^{+}}}$$>$ increases with
$\Delta$$\Upsilon_{\mathrm{HCO^{+}}}$, was to some extent built into the
observing to increase the yield. That is, the source selection began with
flux-limited samples that would achieve a minimum signal-to-noise ratio on the
weakest source, and added 50% additional targets at lower flux Sν and higher
EB-V/Sν that could be serendipitously observed without a proportionate
increase in the required observing time.

##  Appendix B Spectra of HCO+ and CO

Figure 10: Spectra of $\mathrm{HCO^{+}}$ absorption (red) and CO emission
(black) from the Galactic anticenter sample. For an explanation of the
appearance of the frequency-switched IRAM 30m emission spectra, see Appendix
B.

Spectra of $\mathrm{HCO^{+}}$ absorption and CO emission in the 28 directions
with detected $\mathrm{HCO^{+}}$ absorption are shown in Figure 10. The
$\mathrm{HCO^{+}}$ absorption is the negative-going signal shown in red. The
black histogram shows the CO emission. The frequency-switched IRAM 30m CO
spectra were delivered only after folding in frequency, preventing a
separation of the phases of the frequency-switching cycle using the methods of
Liszt (1997). Negative-going features in the CO spectrum are artifacts; only
the positive-going CO signal coincident with $\mathrm{HCO^{+}}$ absorption is
interstellar.

##  Appendix C Area used to derive large-scale statistics

Figure 11: As in Figure 1, left, but showing the outlines of an area
containing the observed sightlines used to derive large-scale properties of
the anticenter region (see Section 4.3).

To derive the statistical distributions shown in Figure 8 and discussed in
Section 4.4, we drew a hull around the observed sightlines as illustrated in
Figure 11 and summed over the interior pixels.