Observational Evidence for a Thick Disk of Dark Molecular Gas in the Outer Galaxy

We have observed the ground state, Λ-doubling OH emission lines at 18 cm with the 100 m Robert C. Byrd GBT, located in Green Bank, West Virginia, with multiple observational projects over the past five years in order to probe the characteristics of the OH emission from diffuse molecular gas in the outer Galaxy. The project IDs: AGBT14B_031A, AGBT14B_031B, AGBT15B_004, and AGBT19A_453 were all observed for various OH emission projects in the outer Galaxy, but later were reanalyzed to probe the robustness of the result presented. The first profile showing the broad emission feature was under Project ID AGBT15B_004 at l = 108°, b = 3°, and was thought to be a 'baseline feature' that was difficult to subtract.

A follow-up project, AGBT19A_484, was undertaken during the summer of 2019 with the GBT in order to directly observe the source geometry of the extremely broad OH emission as a function of Galactic longitude and latitude for the first time. These observations were completed by using frequency-switching by 2 MHz centered on 1617, 1665, and 1720 MHz (O'Neil 2002) using the VEGAS spectrometer in the L-band range (Prestage et al. 2015). The observations cover all four ground-state OH emission lines, although the 1612 band is commonly unusable due to radio-frequency interference (RFI). The beam size of the GBT at 18 cm is 76.

After an initial discovery in 2015 in a GBT spectrum toward l = 108000, b = 3000, we hypothesized that the broad signal was a faint baseline artifact caused by multipath reflections in the structure of the GBT itself and endeavored to prove that the OH signal was erroneous. This type of instrumental artifact was discovered during the commissioning of the GBT (Fisher et al. 2003), where ripples affect the Y linear polarization more than the X linear polarization due to the geometry of the telescope structure.

To test this hypothesis we observed the same position with the 20 m telescope at the GBO, and an additional position separated by a degree on the sky. The 20 m telescope has completely different structural reflections from the GBT due to its geometry. The detection of the broad, faint OH emission with two different telescopes eliminated the possibility that the broad OH signal was a systematic inherent in the 100 m dish of the GBT.

We observed the extended OH emission with the 20 m telescope at the Green Bank Observatory during the summer of 2018. Observing time on the 20 m telescope for this project was purchased using funds from the Director's Discretionary funds at Space Telescope Science Institute (STScI). The telescope operated in the L-band (1.3–1.8 GHz) range and provided two spectral windows—one centered on the 1667 and 1665 MHz lines and the other on the 1720 MHz satellite line. The beam size of the 20 m at 18 cm was 45'. While the observations with the GBT used frequency-switching ON/OFF observations, the stability of the 20 m receiver allowed for total-power observations.

To trace the total gas column density N_H, we use the all-sky dust reddening map from Schlegel et al. (1998) and Schlafly & Finkbeiner (2011)—the well-known "SFD" dust map. We integrate the entire sightline toward the outer Galaxy, hence the more updated 3D dust maps from Green et al. (2017) are not an appropriate estimation of the total gas column due to the lack of stellar measurements a few kiloparsecs outside the solar circle. Therefore, we opt to use the 2D SFD dust map. This map has an angular resolution of 61, which is comparable to the GBT beam (∼78). We query the 2D SFD dust map for the coordinates l = 108000, b = 3000, which returns a dust reddening value, E(B− V) of 1.86 mag.

We also use H i data to estimate the atomic hydrogen column density N(H). We use the H i profile toward l = 108000, b = 3000 from the Leiden–Argentine–Bonn (LAB) survey (Kalberla et al. 2005) and HI4PI database (Ben Bekhti et al. 2016). Additionally, for an estimate of the optical depth of H i toward this sightline, needed for an accurate estimate of N(H), we use two H i absorption–emission pairs from Strasser et al. (2007) to calculate the median H i optical depth along the entire line of sight.

The observations in AGBT15B_004 were experimented on in several stages of the frequency-switching process in order to test different parts of the intermediate-frequency (IF) system to make sure the signal was not introduced by some previously unknown systematic that was unaccounted for. Our observations are completed with in-band frequency-switching (see, e.g., O'Neil 2002); the reference measurement of ON–OFF is completed by shifting the central frequency 2 MHz and differencing the spectrum. When we perform in-band frequency-switching, the science signal is still present in the "OFF" measurement, and the final reduced signal is compiled by shifting and averaging the "OFF–ON" and "ON–OFF" measurements to decrease the noise in the final rms by $\sqrt{2}$. This process is demonstrated in Figure 1.

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In each separate ON–OFF and OFF–ON signal, we should expect two signals from the two OH lines, and the so-called "ghosts" of the subtracted signals from the shifted frequency. Whichever phase is "ON" or "OFF" is arbitrary. However, it is important that we check whether the broad feature is introduced in one part of the IF by appearing in one of the "phases" and not the other. If it were to appear in "ON" but not "OFF", this would raise the alarm of an unknown systematic. We show in Figure 1 that this is not the case, and the positive and negative signals of the OH emission are clearly visible in both reference spectra. The final product of the frequency-switched measurement is shown in the bottom panel of Figure 1, where the "OFF–ON" measurement is shifted and averaged together with the "ON–OFF" measurement.

In this final spectrum, we would expect the two signals as well as four "ghosts", or negative, subtracted signals. Higher-order polynomial fits are required for such wide, frequency-switched measurements, as opposed to the linear baseline fits typically used in position-switched measurements.

We were immediately suspicious that the broad OH signal was an erroneous signal introduced by high-order polynomial baseline fitting. We performed baseline fits for a wide range of polynomial orders to test the robustness of this method. The broad emission feature is detected in spectra with polynomial baseline fits from order 8 to 23. The precision of the measurement of the profile integral from the GBT spectra is then constrained by the fitting procedure, which introduces a systematic error of a few per cent. To test the robustness of the data pipelines; we created three different data pipelines utilizing different common signal reduction techniques: Fourier filtering (or bandstop filtering), ridge regression (or L2 regularization), and the standard general orthogonal polynomial regression used in GBTIDL. None of the data pipelines generated different results to the others.

In the case of our GBT observations, since the emission feature is very wide in velocity, and the data are frequency-switched, linear baselines cannot be removed. A new data fitting procedure in GBTIDL had to be created to deal with such wide profiles. Discontinuous regions of the bandpass that do not have "ghosts" (negative signal) introduced by the frequency-switching, or signal from the 1667 and 1665 MHz feature, were used (N_chans ∼ 6000) to fit a single high-order polynomial ranging from order 17 to 23 based on a few different criteria: (a) there should be no positive velocity emission due to Galactic structure, (b) the chosen polynomial fit does not preferentially subtract or undersubtract signal from the 1667 or 1665 emission near local gas velocities, resulting in a flat baseline, and (c) the rms in the highlighted baseline regions is minimized until no further improvement occurs. This procedure produced the most consistent results for the GBT spectrum and flat baselines in all regions where no signal or "ghost" was expected.

The pipelines were also used on the 20 m spectra to produce the broad OH signal with identical results to the GBT spectra. The only noticeable difference in the spectra was the "flattening" of "spiral arm" emission from discrete clouds due to beam dilution. The size of the beam at 18 cm for the GBT is 76, whereas the size of the beam for the 20 m is approximately 45'. While the surface brightness is larger in the larger beam, the similar surface brightness of the broad OH emission feature in both spectra indicates that the filling factor of the emitting gas is high, suggesting that the source geometry of the OH emission is extremely diffuse molecular gas that is volume-filling.

In addition to replicating the GBT result with the 20 m telescope, we also opted to forgo frequency-switching so that the baseline fitting of the 20 m spectra would be less complex. Baseline fits of order 2–3 were fit to the total power spectrum and clearly showed the broad OH feature. This result demonstrated that detection of the broad feature does not rely on the higher-order (i.e., 17–23) polynomial fits, which conceivably could have introduced spurious artificial signals into the GBT spectrum. In the 100 hour 20 m OH spectrum, the rms noise is ∼0.3 mK at a velocity resolution of 1 km s⁻¹.

The emission appears to be unpolarized. This is an important check because at centimeter wavelengths observers at the GBO will sometimes experience weak, transient, and usually polarized RFI that is time-dependent. We expect any real thermal OH emission to be unpolarized because it is due to particle motions and collisions in gas that result in dipole oscillations among the four ground-state lines. We checked that we could reproduce the characteristics of the broad and faint OH emission features in 1667 and 1665 MHz in the independent orthogonal linear polarizations (XX and YY). While the YY polarization experiences more baseline noise due to the aforementioned multipath scattering, we were able to replicate the spectrum and calculate a residual spectrum between the two polarizations that shows zero emission from the velocity range of the broad OH feature. If the emission were detected in only one polarization, that would indicate that it would be some type of systematic error that we were previously not accounting for.

In this experiment, we stacked five GBT OH spectra toward l = 108000, b = 300, keeping the linear polarizations separate throughout the reduction. Then, the spectra are smoothed to a velocity resolution of 1 km s⁻¹. The XX and YY spectra are displayed in the top two panels of Figure 2. The residual spectrum of XX – YYis also shown to demonstrate that the two polarizations behave differently at a very small level. The resulting integration time on the spectra in Figure 2 is approximately 40 hours with an rms noise of T_rms = 300 μK. The systematic from the YY spectrum is at the level of 0.5–0.6 mK, the brightness temperature of the 1667 MHz emission is higher at T_mb = 1–3 mK, and the 1665 MHz emission is about half of that, allowing a detection of this feature at high signal-to-noise ratio, especially if using only XX data. We rule out the possibility that the emission from this feature is a polarized systematic baseline ripple in the GBT.

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It became apparent that many sightlines could be stacked from multiple OH projects because of the extended nature of the broad OH emission, which appeared to be beam-filling. This allowed for artificially extremely long integration times (up to 300 hr). Depending on how many positions are averaged together, the rms noise in the spectra was ∼300–800 μK. The GBT spectra shown in the middle panel of Figure 3 come from an 80 hr stacked OH spectrum centered at 1666 MHz, which was then re-centered for the 1667 and 1665 MHz lines respectively. The positions used in this spectra are seven 10 hr exposures around the coordinates (l = 108°, b = 3°). The main beam efficiency of the 100 m in the L band is estimated to be η_mb = 0.95, which we used to convert antenna temperature to main beam temperature.

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After averaging 100 hr of observations at two separate positions (l = 108°, b = 3° and l = 109°, b = 25), separated by a degree on the sky, and smoothing to 1 km s⁻¹, the rms noise in the spectrum was ∼300 μK. We have aligned the local gas emission in the two spectra using an LSR (local standard of rest) velocity correction calculator kindly provided by F. Ghigo. ¹⁰ The resulting OH spectra at 1667 and 1665 MHz are shown in the bottom panel of Figure 3. The main beam efficiency of the 20 m is estimated to be η_mb = 0.77 (F. Ghigo, private communication), which we used to convert antenna temperature to main beam temperature.

On a GBT spectrum with 10 hr of integration time, the resulting rms at a velocity resolution of 1 km s⁻¹ is close to 1 mK. Typical brightness temperatures per channel are T_mb ∼ 1–3 mK, and thus the profile integral over the large velocity width (Δv_LSR ∼ 150 km s⁻¹) of the emission results in a signal-to-noise ratio of approximately 25. The emission also appears clearly in the two main lines of the OH ground state (ΔF = 0) at 1667 and 1665 MHz. The profile integrals of the observed emission regularly appear to be in the 9:5 LTE ratio set by the quantum mechanical transition strengths (Townes & Schawlow 1955; Turner & Heiles 1971). This ratio indicates that collisional excitation of OH is the dominant radiation mechanism in the gas, because the energy levels approach the typical Boltzmann distribution as described by an excitation temperature T_ex, yet the brightness temperature is not near the actual gas temperature.

The expression for the total molecular gas column density is (Mangum & Shirley 2015)

$$\begin{eqnarray}\begin{array}{rcl}{N}_{\mathrm{tot}} & = & \displaystyle \frac{3h}{8{\pi }^{3}| {\mu }_{\mathrm{lu}}{| }^{2}}\displaystyle \frac{{Q}_{\mathrm{rot}}}{{g}_{u}}\exp \left(\displaystyle \frac{{E}_{u}}{{{kT}}_{\mathrm{ex}}}\right)\\ & & \times {\left[\exp \left(\displaystyle \frac{h\nu }{{{kT}}_{\mathrm{ex}}}\right)-1\right]}^{-1}\displaystyle \int {\tau }_{\nu }d\nu \end{array}\end{eqnarray} \tag{ 1 }$$

where Q_rot is the rotational partition function, the statistical sum over all rotational energy levels in the molecule (for the OH hyperfine, Λ-doubling ground states, this sum is 16), and g_u is the degeneracy of the energy level u.

In the analysis of OH emission line profiles, this equation can be converted to mostly observables. Consider a molecular cloud located in the ISM at a distance S along a line of sight from the Sun toward the outer Galaxy. The beam-averaged column density of OH molecules $\langle N({\rm{OH}}) \rangle$ along the line of sight through the cloud is (Liszt et al. 2010)

$$\begin{eqnarray} &&\langle N({\rm{OH}}) \rangle ={C}_{67}\left(\displaystyle \frac{{T}_{{\rm{ex}}}^{67}}{{T}_{{\rm{ex}}}^{67}-{T}_{c}}\right)\int {\rm{\Delta }}{T}_{b}^{67}(v)\,{dv},\end{eqnarray} \tag{ 2 }$$

where ${\rm{\Delta }}{T}_{b}^{67}(v)$ is the (main beam) brightness temperature of the OH emission profile from the cloud as observed in one of the 18 cm OH transitions (in this example the transition that gives rise to the 1667 MHz line), minus an estimate of the underlying radio continuum brightness at the same radial velocity (the "baseline"); T_ex ⁶⁷ is the excitation temperature for the 1667 line; T_c is the brightness of the Galactic continuum emission at 1667 MHz incident on the back surface of the cloud, and the integration over velocity includes all the molecular emission thought to arise in that particular cloud. This equation assumes that the Rayleigh–Jeans approximation applies, that the optical depth of the cloud in the OH line is small, that all the OH molecules are in the ground rotational–vibrational state, and that the four ground-state levels are populated according to LTE. Under these conditions, the constant C_ν consists of

$$\begin{eqnarray}&&{C}_{\nu }=\displaystyle \frac{8\pi k{\nu }_{\mathrm{ul}}{\rm{\Sigma }}{g}_{i}}{{{hc}}^{2}{A}_{\mathrm{ul}}{g}_{u}}\times ({\nu }_{\mathrm{ul}}/c)\end{eqnarray} \tag{ 3 }$$

where ν_ul is the transition line frequency, A_ul is the Einstein A coefficient for that transition, Σg_i is the sum of the statistical weights of the four levels giving rise to the 18 cm lines (= 16), g_u is the statistical weight of the upper energy level that gives rise to the specific line (e.g., 5 for the 1667 line), and k, h, and c designate the usual physical constants. This equation has been grouped into two terms; the first is from Goss (1968), and the second provides the conversion from line widths in hertz (used by Goss) to the currently more conventional units of Doppler velocity. Using the entries for the line frequencies and Einstein A coefficients found in the Splatalogue database ¹¹ appropriate for the 1667 MHz line, the value of C₆₇ is 2.257 × 10¹⁴ cm⁻² for ${\rm{\Delta }}{T}_{b}^{67}$ in kelvin and velocity in km s⁻¹.

In this analysis, we assume an excitation temperature of 5.1 K ± 1 K, because the excitation temperature is usually found to be within 1–2 K above the continuum temperature (Li et al. 2018; Engelke et al. 2020), although we stress caution because the real excitation temperature of the gas may be lower, closer to the Galactic background as found in Li et al. (2018). The possible range of excitation temperatures and continuum temperatures alone introduces systematic uncertainty in the resulting N(OH) values, which we explore below. To obtain an estimate of the continuum background brightness temperature, T_c , we estimate the synchrotron contribution from the 408 MHz continuum map of the Canadian Galactic Plane Survey (CGPS, Taylor et al. 2003) by adopting a temperature spectral index of 2.8 (Reich & Reich 1988):

$$\begin{eqnarray}&&{T}_{c}=2.7+{T}_{c,408}{(1667/408)}^{-2.8},\end{eqnarray} \tag{ 4 }$$

which results in T_c of approximately 4 K. An approved GBT program for this year will attempt to observe this dark disk in absorption and emission pairs, which should unambiguously provide an accurate measurement of T_ex in the future. The resulting column density is N(OH) = (7.41 ± 0.17) × 10¹⁴ cm⁻², which includes statistical error. For a sensitivity calculation on systematic errors, we calculate the possible ranges of OH column densities using 4.1 K < T_c < 4.5 K and 4.6 K < T_ex < 6.1 K. This range results in N(OH) = (1.0 ± 0.017 statistical ± 0.4 systematic) × 10¹⁵ cm⁻². Note that the systematic uncertainty due to the unknown values of T_c and T_ex dominates. We remind the reader here that the continuum temperature and excitation conditions are likely not constant through the entire outer Galaxy, and thus a more precise treatment in the future will be warranted when more data are available.

An accurate measurement of the column density of H i is needed to calculate the presumed amount of H₂ in the sightline from the gas content as traced by dust reddening. We estimate the optical depth of H i, τ, as a constant value across the sightline by examining two absorption–emission pairs from Strasser et al. (2007) that we read from their online data: l = 108446, b = 40498 and l = 108753, b = 25772. Although τ changes along the line of sight, we take the median value from these sightlines as a constant for the calculations of the column density of H i at l = 108°, b = 3° because they are close to each other and representative of the diffuse ISM in this direction. The median across the line of sight in these two directions results in τ = 0.103 ± 0.03, which we use for τ in the following H i column density calculations.

We use Equation (5) in Dickey & Benson (1982) to calculate the column density of H i using the LAB survey H i spectrum at l = 108°, b = 3°:

$$\begin{eqnarray} &&\langle N({\rm{H}}\,{\rm\small{I}}) \rangle ={C}_{0}\int {T}_{{\rm{mb}}}(v)\displaystyle \frac{\tau }{1-{e}^{-\tau }}{dv}\end{eqnarray} \tag{ 5 }$$

where C₀ = 1.82 × 10¹⁸ ${{\rm{cm}}}^{-2}/({\rm{K}}\,{\rm{km}}\,{{\rm{s}}}^{-1})$. This calculation assumes an isothermal medium along the line of sight, with constant H i spin temperatures, which is likely somewhat incorrect because there are different environmental regions along the entire line of sight toward the outer Galaxy. However, because our estimate of the total gas content, N_H, is from dust infrared emission, and hence does not contain radial velocity information, we cannot reliably decompose the H i and OH spectra into their different components if the H₂ content is of pertinent interest. The resulting H i column density is N(H) = (1 ± 0.3) × 10²² cm⁻². The resulting N(OH)/N(H) ratio is a few × 10⁻⁷.

The HI4PI server ¹² returns an H i column of N(H) = 9.58 × 10²¹ cm⁻². Our column density differs (albeit within error) because we restrain our column density calculation only to the Milky Way disk (−150 km s⁻¹ < V_LSR < 100 km s⁻¹) and correct for τ, whereas the HI4PI N(H) maps calculate their optically thin column densities over their entire radial velocity range, −600 km s⁻¹ < V_LSR < 600 km s⁻¹. This may include not only the Milky Way disk material, but also H i gas in the Milky Way halo, intermediate- and high-velocity gas complexes, and extragalactic H i objects.

While several estimates of the N(OH)/N(H₂) abundance ratio exist for several Galactic environments (Liszt & Lucas 1996; Nguyen et al. 2018; Rugel et al. 2018; Jacob et al. 2019), it is unclear which value is appropriate to use for this sightline, because most of these estimates were calculated toward discrete clouds with absorption sources, and typically near the solar neighborhood. We estimate our own abundance ratio in this sightline by using dust reddening as a tracer for the total gas content, to see if it is consistent with the latest abundance ratio calculations from the THOR survey (Rugel et al. 2018) and using CH as a calibration tracer (Jacob et al. 2019). We closely followed the prescription of this method as laid out in Nguyen et al. (2018). Reddening is caused by dust grains through the absorption and scattering of light, defined as

$$\begin{eqnarray}&&E(B-V)=\displaystyle \frac{{A}_{V}}{{R}_{V}}=1.086\displaystyle \frac{\kappa }{{R}_{V}}r\mu {m}_{{\rm{H}}}{N}_{{\rm{H}}}\end{eqnarray} \tag{ 6 }$$

where A_V is the dust extinction, R_V is an empirical coefficient, κ is the emissivity per cross section, r is the dust-to-gas mass ratio, μ is the molecular weight (= 2), m_H is the weight of a hydrogen atom, and N_H is the total gas content, defined as N_H = N(H) + 2N(H₂). R_V is typically assumed to be 3.1 (Cardelli et al. 1989). While it has been shown that R_V varies on Galactic scales, it does not vary significantly (Schlafly et al. 2016). The SFD dust map assumes a value of 3.1, which we use in this paper.

In this analysis we assume that the total gas content is a linear combination of atomic and molecular hydrogen, N_H = N(H) + 2N(H₂). We subtract the column density of atomic hydrogen and assume that the residual is molecular hydrogen. The uncertainties in this method include the uncertainty in the dust reddening, the uncertainty in the N_H/E(B − V) ratio, and the uncertainty in the optical depth, τ, of the H i.

Using the opacity-corrected H i column density from Section 4.5, we calculate a column density of protons, N_H = 1 × 10²² cm⁻², and that of H₂, N(H₂) = 3 × 10²⁰ cm⁻². We use these to calculate an abundance ratio, N(OH)/N(H₂) = (1.5–4) × 10⁻⁶, which is consistent with the range of abundance ratios from the literature (Nguyen et al. 2018; Rugel et al. 2018; Jacob et al. 2019). Our calculations and the predictions from theory are also consistent with expectations for diffuse regions, A_v < 0.1 mag (Neufeld & Wolfire 2016).

In 2019 we undertook a GBT survey in longitude and latitude to probe the structure of the broad OH emission under project ID AGBT19A_484. It was quickly discovered that the emission experiences the same expected velocity structure as the large-scale H i in the outer Galaxy. That is, the velocity distribution of the emission approaches zero with increasing l, as shown in Figure 4. This unambiguously implies that the broad OH profile is indeed a large-scale structure in the Galaxy that obeys the geometry of the velocity field (Burton 1985). The H i and OH emission morphologies appear to behave the same in velocity space, implying a cospatial nature of this phase of molecular gas with atomic gas in the outer Galaxy. The H i associated with the Galactic disk is known to extend outward from the center farther than optically observed portions of the Galaxy such as stars, as well as known molecular components detected with CO. As such, this result extends the distance outward in the Galaxy at which molecular gas is found. At the same longitude and latitude, we demonstrate that this gas is by definition dark, because it lacks a similar detection in a sensitive CO spectrum (Heyer et al. 1998); see Figure 5.

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While the longitude–velocity dependence is suggestive of a large-scale structure, we also checked for structure in the range −5° < b < 5° toward l = 105° by stacking in latitude the data from AGBT15B_004 in longitude. We were able to detect the broad OH emission to some level in every spectrum. It was apparent, however, that the OH emission is very weak (T_mb = 1 mK) at high latitudes, b = −5° and b = 5°. This suggests that the feature has some scale height of ∼200 pc. In a future paper we will investigate this further. Additionally, it appears that the gas has a bimodal distribution around b = 1°. This may be because of the Galactic warp in this direction (Yuan & Wallace 1973), which warps the stellar and gas disk above the Galactic plane.

Owing to Galactic rotation, the distance s along the line of sight and the radial velocity v at that point are approximately linearly related in the outer Galaxy. This suggests that we can estimate the volume density n_OH at any point s along the line of sight by differentiating Equation (2) with respect to distance s along the line of sight to obtain

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{n}_{s}\left({\rm{OH}}\right) & \approx & {C}_{67}\left[\displaystyle \frac{{T}_{{\rm{ex}}}^{67}}{{T}_{{\rm{ex}}}^{67}-{T}_{c}(l\geqslant L)}\right]\displaystyle \frac{d}{{ds}}\displaystyle \int {\rm{\Delta }}{T}_{b}^{67}\left(v\right){dv}\\ & \approx & {C}_{67}\left[\displaystyle \frac{{T}_{{\rm{ex}}}^{67}}{{T}_{{\rm{ex}}}^{67}-{T}_{c}(l\geqslant L)}\right]{\rm{\Delta }}{T}_{b}^{67}{\left.(v)\displaystyle \frac{{dv}}{{ds}}\right|}_{s=S}\end{array}\end{array}\end{eqnarray} \tag{ 7 }$$

where dv/ds is in units of km s⁻¹/cm for the value of C₆₇ adopted here. To determine an approximate value for dv/ds appropriate for any given Galactic longitude, we use the values of S for a given value of v returned by the "Revised Kinematic Distance Calculator" for very long baseline interferometry developed by the "BeSSeL" collaboration ¹³ ; for additional background information see the review paper by Reid & Honma (2014). This approach allows us to estimate values for s(v) and ds/dv that are less prone to local perturbations in the velocity field of the outer Galaxy. For instance, at l = 1080, b = +30, and at, e.g., v = −80 km s⁻¹ (the approximate radial velocity of the outer arm of the Galaxy at this value of l) we obtain the linear approximation s = a − bv with a ≈ 0 and b ≈ 0.0875 kpc/(km s⁻¹) at a distance of ≈7.0 kpc from the Sun. As an illustration, the volume density at this location is 1.8 × 10⁻⁹ OH molecules cm⁻³ per mK of 1667 MHz main beam brightness temperature, assuming we are far enough in the outskirts of the Galaxy to set T_c (s ≥ S) = 3 K (i.e., slightly above the cosmic microwave background alone) and choosing ${T}_{{\rm{ex}}}^{67}=4.1\,{\rm{K}}$ (a reasonable value between the ones found in Engelke & Allen 2018 and Li et al. 2018). Using the abundance ratio N(OH)/N(H₂) = 2 × 10⁻⁶, the corresponding H₂ volume density is ∼7 × 10⁻³ H₂ molecules cm⁻³ at this location. By changing the excitation and continuum temperatures within reasonable assumptions (±2 K, for either), the volume density could be higher or lower by a factor of 10. This is consistent with the volume density of the diffuse molecular gas phase suggested by Papadopoulos et al. (2002), and consistent with the average molecular volume density of the ISM found by Bohlin et al. (1978).

An estimate of the H₂ mass implied by the broad OH emission from the diffuse OH emission can be calculated using several reasonable assumptions. It is, however, important that we treat this number with caution because the uncertainties involved in this method amount to roughly an order of magnitude. Since we calibrate our OH/H₂ abundance ratio for this sightline in an integrated fashion with dust reddening, we have no knowledge of the evolution of the OH/H₂ abundance ratio in radial velocity (and hence distance toward the outer Galaxy). With the assumption of excitation and continuum temperature above, we move forward by transforming our column density of OH to H₂ using the abundance ratio calculated in Section 4.6. The uncertainty in the abundance ratio is the dominant error in the following calculation. While we calculate N(OH)/N(H₂), it is subject to all of the assumptions implicit in the dust reddening method, and we should also not be using one number for the abundance ratio for the sightline, because there are multiple environments in the spectrum. Nevertheless, if we use the derived abundance ratio and the X(CO) number we can transform the T_mb measurements of the H i, OH, and CO spectra into proton units. This exercise, displayed in Figure 6, shows that diffuse molecular gas (as traced by OH) may be a component of mass equal to that of the atomic gas traced by H i in the outer Galaxy.

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We use a simple geometric model of the outer Galaxy: a cylinder with a hole in the middle corresponding to the solar circle (R= 8 kpc). We calculate the volume of this cylinder as $V=\pi h({r}_{1}^{2}-{r}_{2}^{2})$, where we take the height of the dark molecular disk h as 200 pc, which is motivated by observations but will be further investigated in a future paper. In broad strokes, the z-distribution of OH was investigated at the gas velocity of the "Perseus Arm" (V_LSR = −60 km s⁻¹); we were able to measure the scale height of the gas features by fitting a Gaussian. We take this as the scale height because it appears that the disk of emission peaks in surface brightness at this feature. We assume a line-of-sight disk length of R = 25 kpc, which is approximately the extent of the H i disk. The resulting volume of this hollow cylinder is 1 × 10⁶⁷ cm³. Using T_mb =4 mK, the number density is 7 H₂ molecules per liter. The mass in the outer Galaxy can be calculated as $M=2{m}_{p}{{Vn}}_{{{\rm{H}}}_{2}}$ ∼ 1 × 10⁸ M_⊙. The main uncertainties here are the excitation temperature of OH and the abundance ratio of OH to H₂, the latter being dominant.

The current literature values of OH/H₂ cluster around ∼1 × 10⁻⁷ (Liszt & Lucas 1996; Nguyen et al. 2018; Rugel et al. 2018; Jacob et al. 2019); a possible value of the mass using this ratio (which results in 150 H₂ molecules per liter) is ∼2 × 10⁹ M_⊙, which is about equal to current estimates of CO-traced H₂ mass in the Galaxy at ∼1 × 10⁹ M_⊙ (Heyer & Dame 2015), and within an order of magnitude of the H i mass, at ∼8 × 10⁹ M_⊙ (Kalberla & Kerp 2009).

If we use our calculated value for the abundance ratio for the lower limit of mass, and the literature value of the abundance ratio for the upper limit, the resulting range of molecular disk mass values is ∼10^8–9 M_⊙. The existence of a mass of dark H₂ in our Galaxy equal to that of the CO-bright-traced H₂ has been inferred from the indirect observations of γ-rays (Grenier et al. 2005) and dust (Planck Collaboration et al. 2011). With this measurement we provide concrete evidence of the existence of these diffuse molecules and, with the radial velocity distribution, the geometry of the diffuse molecular gas as a thick disk beyond the solar circle. Note that the newly discovered diffuse molecular disk contains CO-dark molecular gas in addition to the previously reported clouds of CO-dark molecular gas found during our OH observing program in the arms of the Galaxy (Allen et al. 2015; Busch et al. 2019). Moreover, it is interesting to examine to what extent the diffuse molecular disk might affect the total baryonic mass of the Galaxy. Using an estimate of stellar mass of ∼6 × 10¹⁰ M_⊙ (Licquia & Newman 2015; McMillan 2017) and an estimate of the known mass of the interstellar medium of ∼1 ×10¹⁰ M_⊙ (Kalberla & Kerp 2009), we roughly estimate that this newly discovered diffuse molecular disk could increase the baryonic mass estimate of the Galaxy by ∼0.1%–2%. We anticipate that further sensitive surveys of this broad OH emission throughout the Galaxy will refine the measurement of the mass of the diffuse molecular disk in the future.

The scale height of the diffuse OH emission corresponds extremely well with the thick diffuse molecular gas disk found in M51 in extended CO emission (Pety et al. 2013). This comparison raises the question as to why the CO sightlines in the Milky Way are CO-dark when the diffuse molecular gas in certain other galaxies is not. One possible explanation for the detectable CO emission in M51 is the geometric effect of M51 being face-on. This orientation could result in enough radiative trapping in the z-direction for the emission from the diffuse component to be detectable at a sensitivity of ∼1 K km s⁻¹. We show in Busch et al. (2019) that CO is subthermally excited below its critical density of 1000 cm⁻³, and below this threshold the surface brightness of CO is heavily dependent on gas volume density. While the CO emission in Pety et al. (2013) appears to come from diffuse molecular gas of n ∼ 300–500 cm⁻³, the OH emission may yet still be tracing an even more diffuse component in addition to the CO-traced diffuse molecular gas because its critical density is n ∼ 1 cm⁻³. A future study intended to search for OH emission from diffuse disks in external galaxies known to contain CO-bright disks, as well as several that do not contain detected CO disks, could provide more information about the differences between conditions in the disks of these galaxies and the properties that lead to a diffuse disk being CO-bright or CO-dark. Such a study could also demonstrate whether OH traces a larger total mass of diffuse molecular gas in these disks even for galaxies that already contain a detectable disk in CO.

Papadopoulos et al. (2002) lay out a theoretical basis for both H₂ formation and shielding at large Galactic distances. They argue that both the ingredients (H i and dust grains) and environmental conditions exist for H₂ formation well inside an H i disk. The existence of abundant dark H₂ may also shed light on the puzzling star formation behaviors in H i-dominated regions (Krumholz 2013). Star formation correlates with H₂ down to the lowest column densities that can be detected by CO, but then also begins to correlate with H i in the outer parts of the Galaxy. If there were a thick disk of diffuse molecular gas in the outer Galaxy, it might help explain the star formation in the H i-dominated portions of the outer Galaxy, as opposed to alternative explanations such as high-velocity runaway stars (Andersson et al. 2021).

Modeling and observations in future papers will have to study the distinction between the OH- and CO-traced diffuse molecular gas disks in order to fully understand the transition zone between CO-bright and CO-dark molecular gas.