Dust–Gas Scaling Relations and OH Abundance in the Galactic ISM

Observations of neutral hydrogen in the interstellar medium (ISM) have historically been dominated by two radio spectral lines: the 21 cm line of atomic hydrogen (H i) and the microwave emission from carbon monoxide (CO), particularly the CO(J = 1–0) line. The former provides direct measurements of the warm neutral medium (WNM), and the cold neutral medium (CNM), which is the precursor to molecular clouds. The latter is widely used as a proxy for molecular hydrogen (H₂), often via the use of an empirical "X-factor," (e.g., Bolatto et al. 2013). The processes by which CNM and molecular clouds form from warm atomic gas sows the seeds of structure into clouds, laying the foundations for star formation. Being able to observationally track the ISM through this transition is of key importance.

However, there is strong evidence for gas not seen in either H i or CO. This undetected material is often called "dark gas," following Grenier et al. (2005). These authors found an excess of diffuse gamma-ray emission from the Local ISM, with respect to the expected flux due to cosmic-ray interactions with the gas mass estimated from H i and CO. Similar conclusions have been reached using many different tracers, including γ-rays (e.g., Abdo et al. 2010; Ackermann et al. 2012, 2011), infrared emission from dust (e.g., Blitz et al. 1990; Reach et al. 1994; Douglas & Taylor 2007; Planck Collaboration et al. 2011, 2014a), dust extinction (e.g., Paradis et al. 2012; Lee et al. 2015), C ii emission (Pineda et al. 2013; Langer et al. 2014; Tang et al. 2016), and OH 18 cm emission and absorption (e.g., Wannier et al. 1993; Liszt & Lucas 1996; Barriault et al. 2010; Allen et al. 2012, 2015; Engelke & Allen 2018).

While a minority of studies have suggested that cold, optically thick H i could account for almost all the missing gas mass (Fukui et al. 2015), CO-dark H₂ is generally expected to be a major constituent, particularly in the envelopes of molecular clouds (e.g., Lee et al. 2015). In diffuse molecular regions, H₂ is effectively self-shielded, but CO is typically photodissociated (Tielens & Hollenbach 1985a, 1985b; van Dishoeck & Black 1988; Wolfire et al. 2010; Glover & Mac Low 2011; Lee et al. 2015; Glover & Smith 2016), meaning that CO lines are a poor tracer of H₂ in such environments. Indeed Herschel observations of C ii suggest that between 20%–75% of the H₂ in the Galactic plane may be CO-dark (Pineda et al. 2013).

For the atomic medium, the mass of warm H i can be computed directly from measured line intensities under the optically thin assumption. However, cold H i with spin temperature T_s ≲ 100 K suffers from significant optical depth effects, leading to an underestimation of the total column density. This difficulty is generally addressed by combining H i absorption and emission profiles observed toward (and immediately adjacent to) bright, compact continuum background sources. Such studies find that the optically thin assumption underestimates the true H i column by no more than a few tens of percent along most Milky Way sightlines (e.g., Dickey et al. 1983, 2000, 2003; Heiles & Troland 2003a, 2003b; Liszt 2014a; Lee et al. 2015); though, the fraction missed in some localized regions may be much higher (Bihr et al. 2015).

Since dust and gas are generally well mixed, absorption due to dust grains has been widely used as a proxy for total gas column density. Early work (e.g., Savage & Jenkins 1972, Bohlin et al. 1978) observed Lyα and H₂ absorption in stellar spectra to calibrate the relationship between total hydrogen column density N_H, and the color excess E(B − V). Similar work was carried out by comparing X-ray absorption with optical extinction, A_V (Reina & Tarenghi 1973, Gorenstein 1975). Bohlin et al's value of N_H/E(B − V) =5.8 × 10²¹ cm⁻² mag⁻¹ has become a widely accepted standard.

Dust emission is also a powerful tool and requires no background source population. The dust emission spectrum in the bulk of the ISM peaks in the FIR-to-millimeter range, and arises mostly from large grains in thermal equilibrium with the ambient local radiation field (Draine 2003, Draine & Li 2007). It has long been recognized that FIR dust emission could potentially be a better tracer of N_H than H i and CO (de Vries et al. 1987, Heiles et al. 1988, Blitz et al. 1990; Reach et al. 1994). An excess of dust intensity and/or optical depth above a linear correlation with N_H (as measured by H i and CO) is typically found in the range A_V = 0.3–2.7 mag (Planck Collaboration et al. 2011, 2014a, 2014b; Martin et al. 2012), consistent with the range where CO-dark H₂ can exist. Alternative explanations cannot be definitively ruled out, however. These include (1) the evolution of dust grains across the gas phases, (2) underestimation of the total gas column due to significant cold H i opacity, and (3) insufficient sensitivity for CO detection. It has also been impossible to rule out remaining systematic effects in the Planck data or bias in the estimate of τ₃₅₃ introduced by the choice of the modified blackbody model.

In this study, we examine the correlations between accurately derived H i column densities and dust-based proxies for N_H. We make use of opacity-corrected H i column densities derived from two surveys: the Arecibo Millennium Survey (MS, Heiles & Troland 2003b, hereafter HT03), and 21-SPONGE (Murray et al. 2015), both of which used on-/off-source measurements toward extragalactic radio continuum sources to derive accurate physical properties for the atomic ISM. We also make use of archival OH data from the Millennium Survey, recently published for the first time in a companion paper, Li et al. (2018). OH is an effective tracer of diffuse molecular regions (Wannier et al. 1993; Liszt & Lucas 1996; Barriault et al. 2010; Allen et al. 2012, 2015; Xu et al. 2016; Li et al. 2018), and has recently been surveyed at high sensitivity in parts of the Galactic plane (Dawson et al. 2014; Bihr et al. 2015). There exists both theoretical and observational evidence for the close coexistence of interstellar OH and H₂. Observationally, they appear to reside in the same environments, as evidenced by tight relations between their column densities (Weselak & Krełowski 2014). Theoretically, the synthesis of OH is driven by the ions O⁺ and ${{\rm{H}}}_{3}^{+}$ but requires H₂ as the precursor; once H₂ becomes available, OH can be formed efficiently through the charge-exchange chemical reactions initiated by cosmic-ray ionization (van Dishoeck & Black 1986). Here we combine H i, OH, and dust data sets to obtain new measurements of the abundance ratio, X_OH = N_OH/${N}_{{{\rm{H}}}_{2}}$—a key quantity for the interpretation of OH data sets.

The structure of this article is as follows. In Section 2, the observations, the data processing techniques, and corrections on H i are briefly summarized. In Section 3, the results from OH observations are discussed. Section 4 discusses the relationship between τ₃₅₃, E(B − V) and N_H in the atomic ISM. We finally estimate the OH/H₂ abundance ratio in Section 5 before concluding in Section 6.

In this study, we use the all-sky optical depth (τ₃₅₃) map of the dust model data measured by Planck/IRAS (Planck Collaboration et al. 2014a—hereafter PLC2014a), the reddening E(B − V) all-sky map from Green et al. (2018), H i data from both the 21-SPONGE Survey (Murray et al. 2015) and the Millennium Survey (Heiles & Troland 2003a, HT03), OH data from the Millennium Survey (Li et al. 2018), and CO data from the Delingha 14 m Telescope, the Caltech Submillimeter Observatory (CSO), and the IRAM 30 m telescope (Li et al. 2018).

2.1. H i and OH

H i data from the Millennium Arecibo 21 cm Absorption-Line Survey (hereafter MS) was taken toward 79 strong radio sources (typically S ≳ 2 Jy) using the Arecibo L-wide receiver. The two main lines of ground state OH at 1665.402 and 1667.359 MHz were observed simultaneously toward 72 positions, and OH absorption was detected along 19 of these sightlines (see also Li et al. 2018). The observations are described in detail by HT03. Briefly, their so-called Z16 observation pattern consists of one on-source absorption spectrum toward the background radio source and 16 off-source emission spectra with the innermost positions at 1.0 HPBW and the outermost positions at $\sqrt{2}$ HPBW from the central source. The off-source "expected" emission spectrum, the emission profile we would observe in the absence of the continuum source, is then estimated by modeling the 17-point measurements. In this work, we use the published values of HT03 for the total H i column density, N_{H i} (scaled as described below), and use the off-source (expected) MS emission profiles to compute the H i column density under the optically thin assumption, ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$, where required. We compute OH column densities ourselves, as described in Section 3. All OH emission and absorption spectra are scaled to a main-beam temperature scale using a beam efficiency of η_b = 0.5 (Heiles et al. 2001), appropriate if the OH is not extended compared to the Arecibo beam size of 3'.

In order to increase the source sample, we also use H i data from the Very Large Array (VLA) 21-SPONGE Survey, which observed 30 continuum sources, including 16 in common with the Millennium Survey sample (Murray et al. 2015). 21-SPONGE used on-source absorption data from the VLA, combining them with off-source emission profiles observed with Arecibo. Murray et al. (2015) report an excellent agreement between the optical depths measured by the two surveys, demonstrating that the single dish Arecibo absorption profiles are not significantly contaminated with resolved 21 cm emission. Note that in this work we have used an updated scaling of the 21-SPONGE emission profiles, which applies a beam efficiency factor of 0.94 to the Arecibo spectra. The total number of unique sightlines presented in this work is therefore 93. The locations of all observed sources in Galactic coordinates are presented in Figure 1. Where sources were observed in both the MS and 21-SPONGE, we use the MS data.

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2.1.1. H i Intensity Scale Corrections

We check our ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$ against the Leiden–Argentine–Bonn survey (LAB, Hartmann & Burton 1997; Kalberla et al. 2005) and the HI4PI survey (HI4PI Collaboration et al. 2016). Both are widely regarded as a gold standard in the absolute calibration of Galactic H i. We find that the optically thin column densities derived from 21-SPONGE are consistent with LAB and HI4PI. However, the MS values are systematically lower than both LAB and HI4PI by a factor of ∼1.14. A possible explanation for this difference lies in the fact that (in contrast to 21-SPONGE) the MS did not apply a main-beam efficiency.

To bring the MS data set in-line with LAB, HI4PI, and 21-SPONGE, one might assume that both the on-source and off-source spectra must be rescaled, and the opacity-corrected column densities recomputed according to the method of HT03 (or equivalent). However, N_{H i} may in fact be obtained from the tabulated values of HT03, with no need to perform a full reanalysis of the data. For warm components, the tabulated values of N_{H i} are simply scaled by 1.14—appropriate since these were originally computed directly from the the integrated off-source (expected) profiles under the optically thin assumption. For cold components, we recall that the radiative transfer equations for the on-source and off-source (expected) spectra in the MS data set are given by:

$$\begin{eqnarray}&&{T}_{{\rm{B}}}^{\mathrm{ON}}(v)=({T}_{\mathrm{bg}}+{T}_{{\rm{c}}}){e}^{-{\tau }_{v}}+{T}_{{\rm{s}}}(1-{e}^{-{\tau }_{v}})+{T}_{\mathrm{rx}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{T}_{{\rm{B}}}^{\mathrm{OFF}}(v)={T}_{\mathrm{bg}}{e}^{-{\tau }_{v}}+{T}_{{\rm{s}}}(1-{e}^{-{\tau }_{v}})+{T}_{\mathrm{rx}},\end{eqnarray} \tag{ 2 }$$

where ${T}_{{\rm{B}}}^{\mathrm{OFF}}(v)$ and ${T}_{{\rm{B}}}^{\mathrm{ON}}(v)$ are the main-beam temperatures of the off-source spectrum and on-source spectrum, respectively. T_s is the spin temperature, τ_v is the optical depth, T_rx is the receiver temperature (∼25 K), and T_c is the main-beam temperature of the continuum source, obtained from the line-free portions of the on-source spectrum. T_bg is the continuum background brightness temperature including the 2.7 K isotropic radiation from CMB and the Galactic synchrotron background at the source position. Equations (1) and (2) may be solved for τ_v and T_s:

$$\begin{eqnarray}&&{e}^{-{\tau }_{v}}=\displaystyle \frac{{T}_{{\rm{B}}}^{\mathrm{ON}}(v)-{T}_{{\rm{B}}}^{\mathrm{OFF}}(v)}{{T}_{{\rm{c}}}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{T}_{{\rm{s}}}=\displaystyle \frac{{T}_{{\rm{B}}}^{\mathrm{OFF}}(v)-{T}_{\mathrm{bg}}{e}^{-{\tau }_{v}}-{T}_{\mathrm{rx}}}{1-{e}^{-{\tau }_{v}}}.\end{eqnarray} \tag{ 4 }$$

From Equation (3), it is clear that optical depth is unchanged by any rescaling, which will affect both the numerator and denominator of the expression identically. Only T_s must be recomputed. This is done on a component-by-component basis from the tabulated Gaussian fit parameters for peak optical depth, τ₀, peak brightness temperature (scaled by 1.14), and the linewidth Δv. The corrected N_{H i} is obtained from

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{{\rm{H}}{\rm{I}}}}{[{10}^{18}\,{\mathrm{cm}}^{-2}]}=1.94\cdot {\tau }_{0}\cdot \displaystyle \frac{{T}_{{\rm{s}}}}{[{\rm{K}}]}\cdot \displaystyle \frac{{\rm{\Delta }}v}{[\mathrm{km}\,{{\rm{s}}}^{-1}]},\end{eqnarray} \tag{ 5 }$$

where the factor 1.94 includes the usual constant 1.8224 and the 1.065 arising from the integration over the Gaussian line profile.

2.1.2. N_{H i} versus ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$

We show in Figure 2 the correlation between N_{H i} and ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$ toward all 93 positions. While optically thin H i column density is comparable with the true column density in diffuse/low-density regions with N_{H i} ≲ 5 × 10²⁰ cm⁻², opacity effects start to become apparent above ∼5 × 10²⁰ cm⁻².

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If a linear fit is performed to the data, the ratio f = N_{H i}/${N}_{{\rm{H}}\,{\rm{I}}}^{* }$ may be described as a function of log(${N}_{{\rm{H}}\,{\rm{I}}}^{* }$/10²⁰) with a slope of (0.19 ± 0.02) and an intercept of (0.89 ± 0.02) (see also Lee et al. 2015). Alternatively, a simple isothermal correction to the optically thin ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$ data with T_s ∼ 144 K also yields a good agreement with our data points, as illustrated in Figure 2 (see also Liszt 2014b). This approach also better fits the low N_{H i} plateau, N_{H i} < 5 × 10²⁰ cm⁻², below which ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$ ≈ N_{H i}. While a single component with a constant spin temperature is a poor physical description of interstellar H i, it can provide a reasonable (if crude) correction for opacity.

2.2. CO

2.3. Dust

The Millennium Survey OH data consists of on-source and off-source "expected" spectra for each of the OH lines. In our companion paper (Li et al. 2018), we use the method of HT03 to derive OH optical depths, excitation temperatures and column densities. Namely, we obtain solutions for the excitation temperature, T_ex, and τ via Gaussian fitting (to both the on-source and off-source spectra) that explicitly includes the appropriate treatment of the radiative transfer. In the present work, we use a simpler channel-by-channel method for the derivation of T_ex.

The radiative transfer equations for the on-source and off-source (expected) spectra are identical to those for H i, given above in Equations (1) and (2). All terms and their meanings are identical, with the exception that the spin temperature, T_s is replaced by T_ex. T_bg is the continuum background brightness temperature including the 2.7 K isotropic radiation from CMB and the Galactic synchrotron background at the source position. For consistency with HT03 and Li et al. (2018), we estimate the synchrotron contribution at 1665.402 and 1667.359 MHz from the 408 MHz continuum map of Haslam et al. (1982), by adopting a temperature spectral index of 2.8, such that

$$\begin{eqnarray}&&{T}_{\mathrm{bg}}=2.7+{T}_{\mathrm{bg},408}{({\nu }_{\mathrm{OH}}/408)}^{-2.8},\end{eqnarray} \tag{ 6 }$$

resulting in typical values of around 3.5 K. The background continuum contribution from Galactic H II regions may be safely ignored, since the continuum sources we observed are either at high Galactic latitudes or Galactic anti-center longitudes. Thus, in line-free portions of the off-source spectra:

$$\begin{eqnarray}&&{T}_{{\rm{B}}}^{\mathrm{OFF}}(v)={T}_{\mathrm{bg}}+{T}_{\mathrm{rx}}.\end{eqnarray} \tag{ 7 }$$

In the absence of information about the true gas distribution, we assume that OH clouds cover fully both the continuum source and the main beam of the telescope. We may therefore solve Equations (1) and (2) to derive T_ex and τ_v for each of the OH lines, as shown in Equations (3) and (4) for the case of H i.

We fit each OH opacity spectrum (cf. Equation (3)) with a set of Gaussian profiles to obtain the peak optical depth (τ_0,n), central velocity (v_0,n), and FWHM (Δv_n) of each component, n. Equation (4) is then used to calculate excitation temperature spectra. Examples of ${e}^{-{\tau }_{v}}$, ${T}_{{\rm{B}}}^{\mathrm{OFF}}$, and T_ex spectra are shown in Figure 3, together with their associated Gaussian fits. It can be seen that the T_ex spectra are approximately flat within the FWHM of each Gaussian component. We therefore compute an excitation temperature for each component from the mean T_ex in the range v_0,n ± Δv/2.

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Figure 4 compares the τ₀ and T_ex values obtained from our method with those of Li et al. (2018), demonstrating that the two methods generally return consistent results. Minor differences arise only for the most complex sightlines through the Galactic plane (G197.0+1.1, T0629+10), where the spectra are not simple to analyze; however, even these points are mostly consistent to within the errors.

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We compute total OH column densities, N_OH, independently from both the 1667 and 1665 MHz lines via:

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{\mathrm{OH},1667}}{[{10}^{14}\,{\mathrm{cm}}^{-2}]}=2.39\cdot {\tau }_{1667}\cdot \displaystyle \frac{{T}_{\mathrm{ex},1667}}{[{\rm{K}}]}\cdot \displaystyle \frac{{\rm{\Delta }}{v}_{1667}}{[\mathrm{km}\,{{\rm{s}}}^{-2}]},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{\mathrm{OH},1665}}{[{10}^{14}\,{\mathrm{cm}}^{-2}]}=4.30\cdot {\tau }_{1665}\cdot \displaystyle \frac{{T}_{\mathrm{ex},1665}}{[{\rm{K}}]}\cdot \displaystyle \frac{{\rm{\Delta }}{v}_{1665}}{[\mathrm{km}\,{{\rm{s}}}^{-2}]},\end{eqnarray} \tag{ 9 }$$

where the constants include Einstein A-coefficients of A₁₆₆₇ = 7.778 × 10⁻¹¹ s⁻¹ and A₁₆₆₅ = 7.177 × 10⁻¹¹ s⁻¹ for the OH main lines (Destombes et al. 1977). All values of τ₀, T_ex, and N_OH are tabulated in Table 1.

In this section, we will investigate the correlations between dust properties and the total gas column density N_H. Specifically, we consider dust optical depth at 353 GHz, τ₃₅₃, and reddening, E(B − V), with data sets sourced as described in Section 2.3. When these quantities are used as proxies for N_H, a single linear relationship between the measured quantity and N_H is typically assumed. In this work, our H i data set provides accurate (opacity-corrected) atomic column densities, while complementary OH and CO data allow us to identify and exclude sightlines with molecular gas (dark or not). We are therefore able to measure τ₃₅₃/N_H and E(B − V)/N_H along a sample of purely atomic sightlines for which N_H is very well constrained.

Table 2.  34 Atomic Sightlines

Sources	l/b	NH i	${N}_{{\rm{H}}\,{\rm{I}}}^{* }$	στ(OH1667)	${N}_{{{\rm{H}}}_{2}}$(upper limit)a	τ353	E(B − V)
(Name)	(°)	(1020 cm−2)	(1020 cm−2)	(10−4)	(1020 cm−2)	(10−6)	(10−2 mag)
3C33	129.4/−49.3	3.25 ± 0.0	3.2 ± 0.1	12.14	0.6	2.16 ± 0.07	3.54 ± 0.42
3C142.1	197.6/−14.5	25.11 ± 2.6	19.6 ± 0.8	10.55	0.52	21.53 ± 0.72	21.71 ± 0.81
3C138	187.4/−11.3	22.9 ± 1.1	19.9 ± 0.3	5.02	0.25	21.63 ± 0.81	17.47 ± 0.59
3C79	164.1/−34.5	10.86 ± 1.2	9.8 ± 0.8	37.03	1.84	9.23 ± 0.37	12.67 ± 0.78
3C78	174.9/−44.5	11.69 ± 0.5	10.3 ± 0.2	13.25	0.66	12.45 ± 0.63	14.64 ± 1.07
3C310	38.5/60.2	4.29 ± 0.1	4.0 ± 0.1	16.19	0.8	3.48 ± 0.15	2.75 ± 0.53
3C315	39.4/58.3	5.48 ± 0.4	4.7 ± 0.0	12.96	0.64	3.98 ± 0.09	5.63 ± 0.26
3C234	200.2/52.7	1.84 ± 0.0	1.9 ± 1.1	12.66	0.63	0.78 ± 0.03	1.64 ± 0.56
3C236	190.1/54.0	1.38 ± 0.0	1.3 ± 1.2	10.72	0.53	0.7 ± 0.03	2.04 ± 0.49
3C64	157.8/−48.2	7.29 ± 0.2	6.9 ± 0.8	33.12	1.65	6.55 ± 0.35	9.01 ± 0.36
P0531+19	186.8/−7.1	27.33 ± 0.7	25.4 ± 0.3	6.37	0.32	20.75 ± 0.62	20.54 ± 1.3
P0820+22	201.4/29.7	4.82 ± 0.2	4.8 ± 1.1	7.09	0.35	3.73 ± 0.11	2.56 ± 0.4
3C192	197.9/26.4	4.56 ± 0.1	4.5 ± 0.1	20.66	1.03	3.38 ± 0.08	4.05 ± 0.53
3C98	179.8/−31.0	12.7 ± 0.5	11.3 ± 1.3	12.26	0.61	13.72 ± 0.41	17.73 ± 1.02
3C273	289.9/64.4	2.35 ± 0.0	2.3 ± 0.1	21.0	1.04	1.3 ± 0.09	1.98 ± 0.41
DW0742+10	209.8/16.6	2.77 ± 0.0	2.8 ± 0.9	8.01	0.4	1.6 ± 0.03	1.99 ± 0.25
3C172.0	191.2/13.4	8.89 ± 0.2	8.6 ± 1.1	13.02	0.65	5.66 ± 0.08	4.7 ± 0.52
3C293	54.6/76.1	1.46 ± 0.1	1.5 ± 1.1	6.24	0.31	1.31 ± 0.09	2.83 ± 0.75
3C120	190.4/−27.4	18.17 ± 2.1	10.7 ± 0.1	28.29	1.41	29.26 ± 1.08	22.74 ± 1.03
CTA21	166.6/−33.6	10.97 ± 0.4	10.0 ± 0.8	27.35	1.36	10.39 ± 0.44	13.3 ± 0.96
P1117+14	240.4/65.8	1.79 ± 0.0	1.8 ± 0.3	15.0	0.75	1.5 ± 0.04	2.73 ± 0.54
3C264.0	237.0/73.6	1.97 ± 0.0	2.0 ± 0.4	6.29	0.31	1.64 ± 0.08	2.88 ± 0.34
3C208.1	213.6/33.6	3.15 ± 0.1	3.2 ± 0.2	18.67	0.93	3.12 ± 0.04	2.93 ± 0.43
3C208.0	213.7/33.2	3.41 ± 0.1	3.5 ± 0.2	19.69	0.98	3.38 ± 0.07	4.37 ± 0.47
4C32.44	67.2/81.0	1.23 ± 0.0	1.3 ± 0.6	9.88	0.49	0.81 ± 0.02	1.94 ± 0.29
3C272.1	280.6/74.7	2.82 ± 0.0	2.8 ± 0.3	10.24	0.51	1.73 ± 0.28	2.43 ± 0.31
4C07.32	322.2/68.8	2.43 ± 0.0	2.4 ± 0.3	30.7	1.53	2.32 ± 0.06	4.3 ± 0.41
3C245	233.1/56.3	2.39 ± 0.0	2.4 ± 0.2	11.36	0.56	2.22 ± 0.06	2.71 ± 0.36
3C348	23.0/29.2	6.56 ± 0.2	6.0 ± 0.1	16.51	0.82	5.7 ± 0.15	9.8 ± 0.33
3C286	56.5/80.7	2.33 ± 0.1	2.4 ± 2.7	7.13	0.35	0.81 ± 0.05	2.75 ± 0.74
4C13.65	39.3/17.7	10.56 ± 0.2	9.9 ± 0.1	20.01	0.99	11.8 ± 0.39	15.63 ± 0.6
3C190.0	207.6/21.8	3.21 ± 0.0	3.4 ± 0.9	17.57	0.87	2.41 ± 0.03	1.96 ± 0.42
3C274.1	269.9/83.2	2.74 ± 0.0	2.6 ± 0.1	13.13	0.65	2.34 ± 0.02	2.64 ± 0.42
3C298	352.2/60.7	2.39 ± 0.4	2.6 ± 0.1	10.69	0.53	1.3 ± 0.07	1.97 ± 0.39

Note.

^aEstimated from OH(1667) 3σ detection limits using T_ex = 3.5 K, FWHM = 1 km s⁻¹ and N_OH/${N}_{{{\rm{H}}}_{2}}$ = 10⁻⁷ (see Section 5).

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In the following, we consider 34/93 sightlines to be "purely atomic." These are defined as either (a) sightlines where CO and OH were observed and not detected in emission (16/93), or (b) sightlines where CO was not observed but OH was observed but not detected (18/93 sightlines). In both cases, we require that OH be undetected in the 1667 MHz line to a detection limit of N_OH < 1 × 10¹³ cm⁻² (see Li et al. 2018), which excludes some positions with weaker continuum background sources. We may confidently assume that these sightlines contain very little or no H₂ and note that all but one of them lie outside the Galactic plane ($| b| \gt 10^\circ$). Figure 5 shows maps of the immediate vicinity of these sightlines in τ₃₅₃ and E(B − V). Identical maps for the 19 sightlines with OH detections (see also Section 5), are shown in Figure 6.

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In all of the following subsections, N_H is taken to be equal to N_{H i}, the opacity-corrected H i column density, as derived along sightlines with no molecular gas detected in emission.

4.1. N_H from Dust Optical Depth τ₃₅₃

We adopt the all-sky map of dust optical depth τ₃₅₃ computed by PLC2014a. This was derived from an MBB empirical fit to IRAS and Planck maps at 3000, 857, 545, and 353 GHz, described by the expression:

$$\begin{eqnarray}&&{I}_{\nu }={\tau }_{353}{B}_{\nu }({T}_{\mathrm{dust}}){\left(\displaystyle \frac{\nu }{353}\right)}^{{\beta }_{\mathrm{dust}}}.\end{eqnarray} \tag{ 10 }$$

Here, τ₃₅₃, dust temperature, T_dust, and spectral index, β_dust, are the three free parameters, and B_ν(T_dust) is the Planck function for dust at temperature T_dust which is, in this model, considered to be uniform along each sightline (see PLC2014a for more details). The relation between dust optical depth and total gas column density can then be written as:

$$\begin{eqnarray}&&{\tau }_{353}=\displaystyle \frac{{I}_{353}}{{B}_{353}({T}_{\mathrm{dust}})}={\kappa }_{353}r\mu {m}_{{\rm{H}}}{N}_{{\rm{H}}}={\sigma }_{353}{N}_{{\rm{H}}},\end{eqnarray} \tag{ 11 }$$

where σ₃₅₃ is the dust opacity, κ₃₅₃ is the dust emissivity cross-section per unit mass (cm² g⁻¹), r is the dust-to-gas mass ratio, μ is the mean molecular weight, and m_H is the mass of a hydrogen atom.

Figure 7 shows the correlation between N_H and τ₃₅₃. A tight linear trend can be seen with a Pearson coefficient of 0.95. The value of σ₃₅₃ from the orthogonal distance regression (Boggs & Rogers 1990) linear fit is (7.9 ± 0.6) × 10⁻²⁷ cm² H⁻¹ (the intercept is set to 0), where the quoted uncertainties are the 95% confidence limits estimated from pair bootstrap resampling. This is consistent to within the uncertainties with that obtained by PLC2014a based on all-sky H i data from LAB, (6.6 ± 1.7) × 10⁻²⁷ cm² H⁻¹. Note that here we have quoted the PLC2014a measurement made toward low N_{H i} positions, because the lack of any H i opacity correction in that work makes this value the most reliable. However, our fit is consistent with all of the σ₃₅₃ values presented in that work (which was based on the Planck R1.20 data release), to within the quoted uncertainties.

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Small systematic deviations from the linear fit, evident at the high and low column density ends of the plot, are discussed further in Section 4.3.

In order to examine the possible contribution of molecular gas to N_H along the 34 atomic sightlines, we estimate upper limits on ${N}_{{{\rm{H}}}_{2}}$ from the 3σ OH detection limits using an abundance ratio of X_OH = 10⁻⁷ (see Section 5). These values are tabulated in Table 2, and the resulting upper limits on N_H are shown as gray triangles in Figure 7. As expected, the σ₃₅₃ obtained from the fit to these upper limits is lower, at (6.4 ± 0.3) × 10⁻²⁷ cm² H⁻¹. However, while some molecular gas may indeed be present at low levels, these limits should be considered as extreme upper bounds on the true molecular column density. This is particularly true for the most diffuse sightlines with the lowest column density (N_{H i} < 5 × 10²⁰ cm⁻²), where the observational upper limits may appear to raise N_H by up to ∼50%. Molecules are not expected to be well-shielded at such low columns (and indeed even CNM is largely absent along these sightlines in our data). Even for higher column density data points, it can be readily seen from Figures 5 that all sightlines considered in this analysis lie well away from even the faintest outskirts of CO-bright molecular gas complexes. We also note that the deviations from the linear fit that will be discussed in more detail below could not be removed by any selective addition of molecular gas at levels up to these limits.

We next compare our results with the dust opacity σ₃₅₃ derived by Fukui et al. (2015) (plotted on Figure 7 as a dashed line). These authors derived a smaller value than in the present work (by a factor of ∼1.5), by restricting their fit to only the warmest dust temperatures, under the assumption that these most reliably select for genuinely optically thin H i. They then applied this factor to the Planck τ₃₅₃ map (excluding $| b| \lt 15^\circ$ and CO-bright sightlines) to estimate N_{H i}, assuming that the contribution from CO-dark H₂ was negligible. This resulted in N_{H i} values ∼2–2.5 times higher than under the optically thin assumption, and motivated their hypothesis that significantly more optically thick H i exists than is usually assumed. However, we find that while the σ₃₅₃ of Fukui et al. (2015) may be a good fit to some sightlines in the very low N_{H i} regime (≲3 × 10²⁰ cm⁻²), it overestimates N_{H i} at larger column densities by ∼50%. Indeed, as will be discussed below, σ₃₅₃ is not expected to remain constant as dust evolves. This (combined with some contribution from CO-dark H₂) may reconcile the apparent discrepancy between their findings and absorption/emission-based measurements of the opacity-corrected H i column.

4.2. N_H from Dust Reddening E(B − V)

Reddening caused by the absorption and scattering of light by dust grains is defined as:

$$\begin{eqnarray}&&E(B-V)=\displaystyle \frac{{A}_{V}}{{R}_{V}}=1.086\displaystyle \frac{{\kappa }_{V}}{{R}_{V}}r\mu {m}_{{\rm{H}}}{N}_{{\rm{H}}},\end{eqnarray} \tag{ 12 }$$

where A_V is the dust extinction, R_V is an empirical coefficient correlated with the average grain size, and all other symbols are defined as before. In the Milky Way, R_V is typically assumed to be 3.1 (Schultz & Wiemer 1975), but it may vary between 2.5 and 6.0 along different sightlines (Goodman et al. 1995; Draine 2003).

The ratio $\langle {N}_{{\rm{H}}}/E(B-V)\rangle =5.8\times {10}^{21}\,{\mathrm{cm}}^{-2}\,{\mathrm{mag}}^{-1}$ (Bohlin et al. 1978) is a widely accepted standard, used in many fields of astrophysics to connect reddening measurements to gas column density. This value was derived from Lyα and H₂ line absorption measurements toward 100 stars (see also Savage et al. 1977), and has been replicated over the years via similar methodology (e.g., Shull & van Steenberg 1985; Diplas & Savage 1994; Rachford et al. 2009). However, a number of recent works using H i 21 cm data have found significantly higher values (PLC2014a; Liszt 2014a; Lenz et al. 2017).

Here we use the all-sky map of E(B − V) from Green et al. (2018) to estimate the ratio N_H/E(B − V) for our sample of purely atomic sightlines, at $| b| \gt 5^\circ$. The results are shown in Figure 8. It can be seen that E(B − V) and N_H are strongly linearly correlated, with a Pearson coefficient of 0.93. The ratio obtained from the linear fit is N_H/E(B − V) = (9.4 ± 1.6) × 10²¹ cm⁻² mag⁻¹ (the intercept is also set to be 0), where the quoted uncertainties are the 95% confidence limits estimated from pair bootstrap resampling. This value is a factor of 1.6 higher than that in Bohlin et al. (1978).

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The value obtained here is consistent with the estimate of Lenz et al. (2017): N_H/E(B − V) = 8.8 × 10²¹ cm⁻² mag⁻¹ (no uncertainty is given in that work). These authors compared optically thin H i column density from HI4PI (Collaboration et al. 2016) with various estimates of E(B − V) from Schlegel et al. (1998), Peek & Graves (2010), Schlafly et al. (2014), PLC2014a, and Meisner & Finkbeiner (2015). We note that the estimate of Lenz et al. (2017) is only valid for N_H < 4 × 10²⁰ cm⁻², where it seems safe to assume that the 21 cm emission is optically thin. Our value is also close to that of Liszt (2014a), who find N_{H i}/E(B − V) = 8.3 × 10²¹ cm⁻² mag⁻¹ (also given without uncertainty) for $| b| \geqslant 20^\circ$ and 0.015 ≲ E(B − V) ≲ 0.075, by comparing H i data from LAB and E(B − V) from Schlegel et al. (1998). The methodology used by these two studies differs in a number of details. For instance, Liszt (2014a) did not apply a gain correction to the Schlegel et al. (1998) map (whereas Lenz et al. 2017 scaled it down by 12%), and did not smooth it to the LAB angular resolution (30'). However, Liszt (2014a) did apply an empirical correction factor to account for H i opacity (albeit one whose effects on high-latitude sightlines was small). These details may account for the difference between the values obtained by these two otherwise similar studies.

We also note that, like the present work, these studies did not take into account the potential contribution of dust associated with the diffuse warm ionized gas (WIM). This would tend to produce a flattening of the E(B − V) versus N_{H i} relation at low N_{H i} and therefore increase the value of N_{H i}/E(B − V) artificially. Because we are able to accurately probe a large column density range (up to 3 × 10²¹ cm⁻²), we would naively expect our estimate of N_H/E(B − V) to be less affected by WIM bias than either Liszt (2014a) or Lenz et al. (2017; which would tend to have a greater effect on lower column data points). While more work is needed to quantify the contribution of the WIM on dust emission/absorption measurements at low E(B − V), we consider it unlikely to account for the difference between our work and historically lower measurements of the N_H/E(B − V) ratio.

Despite minor differences between these three studies, it is clear that they point to a N_{H i}/E(B − V) value of (∼8–9) × 10²¹ cm⁻² mag⁻¹. This is 40%–60% higher than the traditional value of Bohlin et al. (1978), which has been used by most models of interstellar dust as a reference point to set the dust-to-gas ratio (e.g., Draine & Fraisse 2009; Jones et al. 2013). We note that if N_H is replaced with upper limits (as discussed in Section 4.1), N_H/E(B − V) climbs yet higher, leaving this key conclusion unaffected.

4.3. Disentangling the Effects of Grain Evolution and Dark Gas on σ₃₅₃

A number of studies have used the correlation between τ₃₅₃ and N_H, particularly with regards to the search for dark gas (e.g., Planck Collaboration et al. 2011; Fukui et al. 2014, 2015; Reach et al. 2015). It is clear that τ₃₅₃ and N_H are in general linearly correlated only if σ₃₅₃ is a constant. However, it is recognized that σ₃₅₃ is sensitive to grain evolution, and significant variations in the ratio N_H/τ₃₅₃ have been observed, particularly when transitioning to the high-density, molecular regime (e.g., Planck Collaboration et al. 2014a, 2015; Okamoto et al. 2017; Remy et al. 2017). The origin of observed variations in σ₃₅₃ may relate to a change in dust properties via κ₃₅₃, and/or a variation in the dust-to-gas ratio r, but may also include a contribution due to the presence of dark gas, if this is unaccounted for in the estimated N_H.

PLC2014a presented the variation in σ₃₅₃ with N_H at 30' resolution over the entire sky. In that work, N_H was derived from (${N}_{{\rm{H}}\,{\rm{I}}}^{* }$ + X_CO W_CO), thus dark gas (both optically thick H i and CO-dark H₂) was unaccounted for. We reproduce their data in Figure 9. It can be seen that σ₃₅₃ is roughly flat and at a minimum in a narrow, low column density range N_H =(1−3) × 10²⁰ cm⁻², then increases linearly until N_H = 15 ×10²⁰ cm⁻², by which point it is almost a factor of 2 higher. It then remains approximately constant for the canonical value of X_CO = 2.0 × 10²⁰ cm⁻² K⁻¹ km⁻¹ s. A key issue for dark gas studies is disentangling how much of the initial rise in σ₃₅₃ is due to changing grain properties and how much is due to the contribution of unseen material, whether it be opaque H i or diffuse H₂. (Note also the upturn in σ₃₅₃ seen at the lowest N_H, which may be due to the presence of unaccounted-for protons in the warm ionized medium.)

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The column density range probed by our purely atomic sightlines, N_H = (1 ∼ 30) × 10²⁰ cm⁻² well samples the range where σ₃₅₃ undergoes its first linear increase. Dark gas is also fully accounted for in our data, since H i is opacity-corrected, and no molecular gas is detected in emission along these sightlines. To quantify the effect of ignoring H i opacity on σ₃₅₃, we compare σ₃₅₃ deduced from the true, opacity-corrected N_{H i} with that deduced under the optically thin assumption. The results are shown in Figure 10. In low column density regions (N_H < 5 × 10²⁰ cm⁻²), each σ₃₅₃ pair from N_{H i} and ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$ are comparable. However, at higher column densities (N_H > 5 × 10²⁰ cm⁻²) σ₃₅₃ from true N_{H i} is systematically lower than that measured from ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$. On average, σ₃₅₃ obtained from optically thin H i column density increases by ∼1.6 when going from low to high column density regions; whereas σ₃₅₃ from true N_{H i} increases by ∼1.4. This suggests that if H i opacity is not explicitly corrected for, it can account for around one-third (1/3) of the increase of σ₃₅₃ observed during the transition from diffuse to dense atomic regimes. The remaining of two-thirds (2/3) must arise due to changes in dust properties.

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From Equation (11), we see that σ₃₅₃ is a function of the dust-to-gas mass ratio, r, and the dust emissivity cross-section, κ₃₅₃, which depends on the composition and structure of dust grains. Given the uncertainties on the efficiency of the physical processes involved in the evolution of interstellar dust grains, it is difficult at this point to conclude if the variations of σ₃₅₃ observed here are due to an increase of the dust mass (i.e., r) or to a change in the dust emission properties (i.e., κ₃₅₃). Using the dust model of Jones et al. (2013), Ysard et al. (2015) suggest that most of the variations in the dust emission observed by Planck in the diffuse ISM could be explained by relatively small variations in the dust properties. That interpretation would favor a scenario in which the increase of σ₃₅₃ from diffuse to denser gas is caused by the growth of thin mantles via the accretion of atoms and molecules from the gas phase. Even though this process would increase the mass of grains (and therefore increase r), the change of the structure of the grain surface would lead to a larger increase in κ₃₅₃. Alternatively, it is possible that this systematic variation of τ₃₅₃/N_H could be due to residual large-scale systematic effects in the Planck data, or to the fact that the modified blackbody model introduces a bias in the estimate of τ₃₅₃. Neither of these explanations can be ruled out.

Figure 9 shows σ₃₅₃ as a function of N_H superimposed on the results from PLC2014a. It can be seen that we observe a similar rise in σ₃₅₃ in the column density range (∼5–30) × 10²⁰ cm⁻², but less extreme. In particular, most of our data points in the higher column density range (N_H > 5 × 10²⁰ cm⁻²) are found below the PLC2014a trend, which is derived from the mean values of σ₃₅₃ over the whole sky in N_H bins. This is true even if we use ${N}_{{\rm{H}}\,{\rm{I}}}^{* }$ rather than N_{H i} to derive σ₃₅₃, indicating that optically thick H i alone cannot shift our data points high enough for a perfect match. This is consistent with the fact that we are examining purely atomic sightlines, and likely happens because we are sampling comparatively low number densities (n_H ≲ 10–100 cm⁻³; a mixture of WNM and CNM), whereas the sample in PLC2014a includes molecular gas in the N_H bins, presumably with a higher κ₃₅₃. However, in diffuse regions with N_H < 5 ×10²⁰ cm⁻², the mean value of σ₃₅₃ from our sample is comparable with that from PLC2014a.

4.4. E(B − V) as the More Reliable Proxy for N_H?

We have seen that along 34 atomic sightlines E(B − V) shows a tight linear correlation with N_H in the column density range N_H = (1 ∼ 30) × 10²⁰ cm⁻². τ₃₅₃ also shows a good linear relation with N_H but with systematic deviations as described above.

Figure 11 replicates Figure 10 but for E(B − V) rather than τ₃₅₃. Although the sample used here is small, these figures demonstrate clearly that the ratio E(B − V)/N_H is more stable than τ₃₅₃/N_H over the range of column densities and sightlines covered by our analysis. In fact, with N_H corrected for optical depth effects, our data are compatible with a constant value for E(B − V)/N_H, up to N_H = 30 × 10²⁰ cm⁻². On the other hand, we have observed an increase of τ₃₅₃/N_H with N_H, which we suggest may be due to an increase of the dust emissivity (an increase of r and/or κ₃₅₃ without significantly affecting the dust absorption cross-section). While we are unfortunately unable to follow how these relations evolve at higher A_V and in molecular gas, our results nevertheless suggest that the E(B − V) maps of Green et al. (2018) are a more reliable proxy for N_H than the current release of Planck τ₃₅₃ in low-to-moderate column density regimes.

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The rotational lines of CO are widely used to probe the physical properties of H₂ clouds, but in diffuse molecular regimes where CO is not detectable in emission other species and transitions must be considered as alternative tracers of H₂. Among these, the ground-state main lines of OH are a promising dark gas tracer; they are readily detectable in translucent/diffuse molecular clouds (e.g., Magnani & Siskind 1990; Barriault et al. 2010), and since OH is considered to be a precursor molecule necessary for the formation of CO in diffuse regions (Black & Dalgarno 1977; Barriault et al. 2010), it is expected to be abundant in low-CO density/abundance regimes.

The utility of OH as a tracer of CO-dark H₂ depends on our ability to constrain the OH/H₂ abundance ratio, X_OH =N_OH/${N}_{{{\rm{H}}}_{2}}$. From an observational perspective, this requires good estimates of both the OH and H₂ column densities, the latter of which often cannot be observed directly. Many efforts (both modeling and observational) have been devoted to deriving X_OH in different environmental conditions, which we summarize below:

• 1.  
  Astrochemical models by Black & Dalgarno (1977) found X_OH ∼ 10⁻⁷ for the case of ζ Ophiuchi cloud.
• 2.  
  Nineteen comprehensive models of diffuse interstellar clouds with n_H from 250 to 1000 cm⁻³, T_k from 20 to 100 K and A_V from 0.62 to 2.12 mag (van Dishoeck & Black 1986) found OH/H₂ abundances from 1.6 × 10⁻⁸ to 2.9 × 10⁻⁷.
• 3.  
  The OH abundance with respect to H₂ from chemical models of diffuse clouds was found to vary from 7.8 × 10⁻⁹ to 8.3 × 10⁻⁸ with A_V = (0.1–1) mag, T_K = (50–100) K and n = (50–1000) cm⁻³ (Viala 1986).
• 4.  
  Six model calculations (that differ in depletion factors of heavy elements and cosmic-ray ionization rate) by Nercessian et al. (1988) toward molecular gas in front of the star HD 29647 in Taurus found OH/H₂ ratios between 5.3 × 10⁻⁸ and 2.5 × 10⁻⁶.
• 5.  
  From OH observations toward high-latitude clouds using the 43 m NRAO telescope, Magnani et al. (1988) derived X_OH values between 4.8 × 10⁻⁷ to 4 × 10⁻⁶ in the range of A_V = (0.4–1.1) mag, assuming that ${N}_{{{\rm{H}}}_{2}}$ = 9.4 ×10²⁰ A_V. However, we note that the excitation temperatures of the OH main lines were assumed to be equal, T_ex,1665 = T_ex,1667, likely resulting in overestimation of N_OH (see Crutcher 1979; Dawson et al. 2014).
• 6.  
  Andersson & Wannier (1993) obtained an OH abundance of ∼10⁻⁷ from models of halos around dark molecular clouds.
• 7.  
  Combining N_OH data from Roueff (1996) and Felenbok & Roueff (1996) with measurements of ${N}_{{{\rm{H}}}_{2}}$ from Savage et al. (1977, using UV absorption), Rachford et al. (2002, using UV absorption) and Joseph et al. (1986, using CO emission), Liszt & Lucas (2002) find X_OH = (1.0 ± 0.2) × 10⁻⁷ toward diffuse clouds.
• 8.  
  Weselak et al. (2010) derived OH abundances of (1.05 ± 0.24) × 10⁻⁷ from absorption-line observations of five translucent sightlines, with molecular hydrogen column densities ${N}_{{{\rm{H}}}_{2}}$ measured through UV absorption by (Rachford et al. 2002, 2009).
• 9.  
  Xu et al. (2016) report that X_OH decreases from 8 × 10⁻⁷ to 1 × 10⁻⁷ across a boundary region of the Taurus molecular cloud, over the range A_V = 0.4–2.7 mag. ${N}_{{{\rm{H}}}_{2}}$ was obtained from an integration of A_V-based estimates of the H₂ volume density (assuming ${N}_{{{\rm{H}}}_{2}}$ = 9.4 × 10²⁰ A_V).
• 10.  
  Recently, Rugel et al. (2018) report a median X_OH ∼ 1.3 × 10⁻⁷ from THOR Survey observations of OH absorption in the first Milky Way quadrant, with ${N}_{{{\rm{H}}}_{2}}$ estimated from ¹³CO(1-0).

Overall, while model calculations tend to produce some variation in the OH abundance ratio over different parts of parameter space (8 × 10⁻⁹–4 × 10⁻⁶), observationally determined measurements of X_OH cluster fairly tightly around 10⁻⁷, with some suggestion that this may decrease for denser sightlines.

In this paper, we determine our own OH abundances, using the MS data set to provide N_OH and N_{H i}; then employing τ₃₅₃ and E(B − V) (along with our own conversion factors) to compute molecular hydrogen column densities as ${N}_{{{\rm{H}}}_{2}}=\tfrac{1}{2}({N}_{{\rm{H}}}-{N}_{{\rm{H}}{\rm{I}}})$. We note that since this dust-based estimate of ${N}_{{{\rm{H}}}_{2}}$ cannot be decomposed in velocity space, the OH abundances are determined in an integrated fashion for each sightline, and not on a component-by-component basis. While CO was detected along all but one sightline, it was not detected toward all velocity components, meaning that our abundances are generally computed for a mixture of CO-dark and CO-bright H₂ (for further details, see Li et al. 2018).

The OH column densities derived in Section 3 are derived from direct measurements of T_ex and τ. This means that they should be accurate compared to methods that rely on assumptions about these variables (see, e.g., Crutcher 1979; Dawson et al. 2014). In computing N_H, we assume that the linear correlations (deduced from τ₃₅₃, E(B − V) and N_{H i} toward 34 atomic sightlines) still hold in molecular regions. In this manner, estimates of the OH/H₂ abundance ratio can be obtained within a range of visual extinction A_V = (0.25–4.8) mag. We note that, of our 19 OH-bright sightlines, 5 produce ${N}_{{{\rm{H}}}_{2}}$ that is either negative or consistent with zero to within the measurement uncertainties; these are excluded from the analysis.

Figure 12 shows N_OH and X_OH as functions of ${N}_{{{\rm{H}}}_{2}}$. We find that N_OH increases approximately linearly with ${N}_{{{\rm{H}}}_{2}}$, and the OH/H₂ abundance ratio is approximately consistent for the two methods, with no evidence of and systematic trends with increasing column density. Differences arise due to the overestimation of N_H derived from τ₃₅₃ along dense sightlines compared to N_H from E(B − V). As discussed in Section 4, σ₃₅₃ varies by up to a factor of 2 in the range of N_H = (1 ∼ 30) × 10²⁰ cm⁻², whereas the ratio $\langle {N}_{{\rm{H}}}/E(B-V)\rangle$ is quite constant. The mean and standard deviation of the X_OH distribution deduced from E(B − V) is (0.9 ± 0.6) × 10⁻⁷, which is close to the canonical value of ∼1 × 10⁻⁷, and double the X_OH from τ₃₅₃, (0.5 ± 0.3) ×10 ⁻⁷. We regard the higher value as more reliable.

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We have combined accurate, opacity-corrected H i column densities from the Arecibo Millennium Survey and 21-SPONGE with thermal dust data from the Planck satellite and the new E(B − V) maps of Green et al. (2018). We have also made use of newly published Millennium Survey OH data and information on CO detections from Li et al. (2018). In combination, these data sets allow us to select reliable subsamples of purely atomic (or partially molecular) sightlines, and hence assess the impact of H i opacity on the scaling relations commonly used to convert dust data to total proton column density N_H. They also allow us to make new measurements of the OH/H₂ abundance ratio, which is essential in interpreting the next generation of OH data sets. Our key conclusions are as follows:

• 1.  
  H i opacity effects become important above N_{H i} > 5 ×10²⁰ cm⁻²; below this value the optically thin assumption may usually be considered reliable.
• 2.  
  Along purely atomic sightlines with N_H = N_{H i} = (1–30) × 10²⁰ cm⁻², the dust opacity, σ₃₅₃ = τ₃₅₃/N_H, is ∼40% higher for moderate-to-high column densities than low (defined as above and below N_H = 5 × 10²⁰ cm⁻²). We have argued that this rise is likely due to the evolution of dust grains in the atomic ISM, although large-scale systematics in the Planck data cannot be definitively ruled out. Failure to account for H i opacity can cause an additional apparent rise of the order of ∼20%.
• 3.  
  For purely atomic sightlines, we measure a N_H/E(B − V) ratio of (9.4 ± 1.6) × 10²¹ cm⁻² mag⁻¹. This is consistent with Lenz et al. (2017) and Liszt (2014a), but 60% higher than the canonical value from Bohlin et al. (1978).
• 4.  
  Our results suggest that N_H derived from the E(B − V) map of Green et al. (2018) is more reliable than that obtained from the τ₃₅₃ map of PLC2014a in low-to-moderate column density regimes.
• 5.  
  We measure the OH/H₂ abundance ratio, X_OH, along a sample of 16 molecular sightlines. We find X_OH ∼ 1 ×10⁻⁷, with no evidence of a systematic trend with column density. Since our sightlines include both CO-dark and CO-bright molecular gas components, this suggests that OH may be used as a reliable proxy for H₂ over a broad range of molecular regimes.

J.R.D. is the recipient of an Australian Research Council (ARC) DECRA Fellowship (project number DE170101086). D.L. thanks the supports from the National Key R&D Program of China (2017YFA0402600) and the CAS International Partnership Program (No.114A11KYSB20160008). N.M.-G. acknowledges the support of the ARC through Future Fellowship FT150100024. L.B. acknowledges the support from CONICYT grant PFB06. We are indebted to Professor Mark Wardle for providing us with valuable advice and support. We gratefully acknowledge discussions with Dr. Cormac Purcell and Anita Petzler. Finally, we thank the anonymous referee for comments and criticisms that allowed us to improve the paper.

This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.