DISTRIBUTION AND MASS OF DIFFUSE AND DENSE CO GAS IN THE MILKY WAY

In the inner Galaxy, the ¹²CO line was observed as part of the UMSB survey (Clemens et al. 1986; Sanders et al. 1986), a joint program between FCRAO and the State University of New York at Stony Brook performed between 1981 November and 1984 March. All of the observations were obtained using the FCRAO 14 m telescope. A grid sampled every 3' covering the range 18° < ℓ < 55°, and −1° < b < +1° was observed with a velocity resolution of 1 km s⁻¹ and an angular resolution of 45''. The UMSB survey covers the velocity range −10 km s⁻¹ < V_LSR < 140 km s⁻¹. The data were converted from radiation temperature scale (${T}_{R}^{*}$) to a main beam temperature scale (T_MB) via T_MB = ${T}_{R}^{*}$/0.7.

The GRS survey observed a ∼40° section of the inner galaxy (18° $\leqslant \;$ℓ $\leqslant \quad$ 557, −1° $\leqslant \;$b$\;\leqslant$ 1°) in ¹³CO J = $1\to 0$ , using the FCRAO. The observations were taken between 1998 and 2005 with the SEQUOIA multipixel array. The survey achieved an angular esolution of 47'', sampled on a 22'' grid, and a spectral resolution of 0.212 km s⁻¹ for a noise variance per voxel of σ(${T}_{A}^{*}$) = 0.13 K (${\sigma }_{{T}_{\mathrm{MB}}}=0.24$ K accounting for the main beam efficiency of 0.48). The survey covers the range of velocity −5 to 135 km s⁻¹ for Galactic longitudes ℓ$\;\leqslant$ 40° and −5 to 85 km s⁻¹ for Galactic longitudes ℓ$\;\geqslant$ 40°. The data were converted from the antenna temperature scale ${T}_{A}^{*}$ to a main beam temperature scale T_MB by correcting for the main beam efficiency of 0.48.

The GRS survey observed the CS $2\to 1$ line in a two-square-degree field located at Galactic longitudes ℓ = 443–463 and Galactic latitudes b = −05–05, with the same velocity coverage as the ¹³CO (−5 to 85 km s⁻¹). As for the ¹³CO survey, the CS survey is also half-beam-sampled (45'' resolution with 22'' pixels). It achieved a sensitivity of $\sigma ({{T}_{A}}^{*})=0.13$ K per voxel. The data were converted from the antenna temperature scale ${T}_{A}^{*}$ to a main beam temperature scale T_MB by correcting for the main beam efficiency of 0.48.

Data for the EXFC survey (C. M. Brunt et al. 2015, in preparation) were observed between 2003 and 2006 with the SEQUOIA beam array receiver. The survey spans two longitude ranges: ℓ = 55°–100° (hereafter EXFC 55–100), with the Galactic Latitude range −14 $\;\leqslant \quad$b$\;\leqslant$ +19, and ℓ = 135°–195° (hereafter EXFC 135–195), in the Galactic latitude range −36 $\leqslant \quad$b$\;\leqslant$ +56. The EXFC 135–195 survey covers the outer Galaxy only, while the EXFC 55–100 covers both the inner and outer Galaxy. ¹²CO and ¹³CO 1-0 were observed simultaneously with angular resolutions of 45'' and 48'', sampled on a 225 grid, and a spectral resolution of 0.127 km s⁻¹. The data were de-convolved to remove contributions by the antenna error beam and so are implicitly on a main beam temperature scale. We do not use the longitude range ℓ = 165°–195° because the radial velocity of CO emission in this range is close to zero, independent of distance (almost purely tangential motion).

The EXFC observations were split into 75 fields, sampling Galactic longitudes 55°–100° and 135°–195° every 3°. In this analysis, we do not use data with Galactic longitudes $\geqslant$165°, because at those longitudes the motion of the gas is almost purely transverse (no radial velocity component), and a kinematic distance can therefore not be estimated robustly. We resample and mosaic the full EXFC coverage below ℓ = 165° into 13 disjoint mosaic fields spanning 5° in longitude and the full latitude range of EXFC (−36–56 for EXFC 135–195, −14–19 for EXFC 55–100). The individual spectra composing the mosaics are weighted by their rms main beam temperature to produce the mosaics, and the original angular (225) and spectral (0.127 km s⁻¹) sampling are conserved. Due to i/o and memory limitations, the entire survey cannot be stored into a single mosaic file.

The GRS and UMSB surveys roughly cover the same area (ℓ ≃ 18°–55°, b ≃ −1°–1°). However, small differences exist in the mapping strategy between the ¹²CO and ¹³CO surveys, the GRS being half-beam-sampled on 225 pixels, while the UMSB is under-sampled on a 3' grid (both surveys being at 45'' resolution). To preserve some information about the spatial and spectral structure of the ¹³CO and CS observations, we resample the GRS and UMSB on a common, intermediate grid with voxels $1^{\prime} \times 1^{\prime} \times 0.3$ km s⁻¹. While interpolating the ¹²CO data does not improve its coarseness, it does allow us to more finely identify gas with ¹²CO emission and with ("dense") or without ("diffuse") ¹³CO emission.

An accurate estimation of the measurement errors is critical to this analysis. We compute a theoretical measurement error on the main beam temperature T_MB (for ¹²CO, ¹³CO, and CS) at each ℓ, b position during the mosaicking process, based on the rms of the data in its original form, and on the weights applied as part of the mosaicking process. However, residual (albeit small) baseline fluctuations between positions and within each spectrum can affect the noise rms. We therefore empirically determine the noise rms at each position on the sky using the following approach for the GRS and EXFC survey, in which the line emission is relatively sparse in the position–position–velocity (PPV) cubes. For each ℓ, b position, we determined the noise rms (on the T_MB scale) by fitting a Gaussian to the histogram of the spectrum (¹²CO, ¹³CO, or CS) in the range $\mathrm{min}({T}_{\mathrm{MB}})\leqslant {T}_{\mathrm{MB}}\leqslant -\mathrm{min}({T}_{\mathrm{MB}})$. The noise distribution is assumed to be Gaussian with zero mean, and therefore $\mathrm{min}({T}_{\mathrm{MB}})\;\lt$ 0. The resulting standard deviation of the Gaussian provides an accurate value of the noise rms of the observations. The fitted main beam temperature range ensures that most of the voxels included in the noise rms measurement do not include actual CO or CS emission, which would bias the noise estimation. This procedure resulted in two-dimensional maps of measurement errors for ¹²CO and ¹³CO emission for EXFC, and ¹³CO and CS emission for the GRS. Typical noise rms values (T_MB scale) in the GRS ¹³CO, ¹²CO (EXFC), and ¹³CO (EXFC) observations are 0.24 K per $1^{\prime} \times 1^{\prime} \times 0.3$ km s⁻¹ voxel, 2 K per $22\buildrel{\prime\prime}\over{.} 5\times 22\buildrel{\prime\prime}\over{.} 5\times 0.13$ km s⁻¹ voxel, and 0.7 K per $22\buildrel{\prime\prime}\over{.} 5\times 22\buildrel{\prime\prime}\over{.} 5\times 0.13$ km s⁻¹ voxel, respectively.

In the inner Galaxy covered by the UMSB, which includes the molecular ring, the optically thick ¹²CO emission is ubiquitous and there are not enough voxels free of ¹²CO emission to estimate the rms in each spectrum. Therefore, instead of using the histogram of individual sight-lines, we estimate the noise rms from fitting a Gaussian to the histogram of the entire UMSB data set in the range $\mathrm{min}({T}_{\mathrm{MB}})\leqslant {T}_{\mathrm{MB}}\leqslant | \mathrm{min}({T}_{\mathrm{MB}})|$. The resulting noise rms (T_MB scale) for the UMSB is ∼0.47 K per $1^{\prime} \times 1^{\prime} \times 0.3$ km s⁻¹ voxel.

Since mass and luminosity of CO-emitting gas are proportional to distance squared, distances to each voxel (i.e., ℓ, b, v pixel location in the data) are required for our analysis. We compute kinematic distances to each voxel in the data, assuming that gas in the Galaxy rotates according to the rotation curve derived by Clemens (1985) for R_⊙ = 8.5 kpc and ${\theta }_{\odot }$ = 220 km s⁻¹.

In the outer Galaxy, there is a single solution for the distance for a given radial velocity and Galactic longitude. The luminosity (in ¹²CO or ¹³CO emission) of a voxel is therefore unambiguously determined.

In the inner Galaxy (R_gal $\leqslant$ 8.5 kpc, v_LSR $\geqslant$ 0), there are two distance solutions for a given velocity, a "near" and a "far" distance. This is the well-known problem of the kinematic distance ambiguity. The signal within a given voxel results from emission originating at either or both of those distances. Additional constraints are necessary to resolve this kinematic distance ambiguity. The H i self-absorption method (Knapp 1974; Burton et al. 1978; Jackson et al. 2002; Roman-Duval et al. 2009) cannot be used for individual voxels, and we therefore take the following approach, which uses a Monte Carlo simulation. For each of 10 statistically independent realizations, a near or far side distance is randomly assigned to each voxel based on the probability distribution of molecular gas height in the Galaxy. Specifically, we assume that the vertical density profile within the molecular disk is a Gaussian with FWHM thickness of 110 pc (see the review article by Heyer & Dame 2015, references therein, and Section 4.6). The probability of molecular gas to be preset at height z above or below the plane is also described by a Gaussian function with the same FWHM. For each of the near and far distance solutions, we compute the height above the Galactic plane of a voxel given its Galactic latitude b, ${z}_{\mathrm{near},\mathrm{far}}={d}_{\mathrm{near},\mathrm{far}}\times \mathrm{tan}(b)$, and the corresponding probabilities from the Gaussian vertical distribution ${p}_{g}({z}_{\mathrm{near}})$ and ${p}_{g}({z}_{\mathrm{far}})$. The relative probabilities of the emission in a voxel coming from the near and far distances are p(near) = ${p}_{g}({z}_{\mathrm{near}})$/$({p}_{g}({z}_{\mathrm{near}})+{p}_{g}({z}_{\mathrm{far}})$) and p(far) = ${p}_{g}({z}_{\mathrm{far}})$/$({p}_{g}({z}_{\mathrm{near}})+{p}_{g}({z}_{\mathrm{far}}))$. We then draw a random number from a uniform distribution between 0 and 1. If the random number is smaller than p(near), the voxel is assigned to the near distance. Otherwise, it is assigned to the far distance.

Once the distance is established, a CO luminosity and H₂ mass for each voxel is calculated. We save the four-dimensional data cubes (mass, luminosity, distance, galactocentric radius) in (ℓ, b, v, realization) space, and compute the spatial distribution of the luminosity and mass of CO gas for each Monte Carlo realization. The spatial distribution of CO gas are then averaged between the different realizations to produce the figures in this paper, and the standard deviation between different realizations is included in the error budget. The standard deviation between realizations is very small compared to other sources of errors, and 10 realizations are more than what is necessary to obtain an accurate error estimation.

In reality, the signal in a voxel can originate from emission at both the near and far distances. The advantage of the Monte Carlo method is that the final averaging between realizations distributes the signal in each voxel between the two distance solutions.

For each voxel and each realization, an error on the distance (near or far) is also calculated. The error computation assumes 10 km s⁻¹ non-circular motions, and computes the distance solutions ${d}_{\mathrm{near},\mathrm{far}}^{\pm }$ for $v\pm 10$ km s⁻¹, where v is the velocity of a voxel. The error on the distance is then $\delta {d}_{\mathrm{near},\mathrm{far}}$ = $| {d}_{\mathrm{near},\mathrm{far}}^{+}-{d}_{\mathrm{near},\mathrm{far}}^{-}|$/2. The distance error cubes (ℓ, b, v, realization) are also stored and used in this analysis. The median errors on the near and far distances in the inner Galaxy are 25% and 5% respectively. In the outer Galaxy, the median error on the distance is 70%. Generally, the fractional distance error increases with increasing longitude and with decreasing distance in the outer Galaxy.

Our primary goal is to determine the spatial distribution, both in luminosity and mass, of CO-emitting molecular gas. It is therefore crucial to capture the low-level extended emission. We are then faced with three difficulties. First, the sum of quantities (e.g., the main beam temperature or luminosity of a voxel) over a very large number of voxels (a single mosaic from EXFC contains approximately a billion voxels), which are potentially affected by small residual baseline offsets, can diverge or be dominated by those residual baseline effects. Second, capturing the low-level (low S/N) emission requires us to use a low threshold of detection (e.g., 1σ), which leads to positive biases in summed quantities, since positive noise peaks can be included and not their negative counterparts. Third, the relative contributions of those three components may depend on the S/N of the observations. For instance, if the S/N of the ¹³CO observations is lower than that of the ¹²CO data, then the fraction of diffuse gas could potentially be inflated because of the inability to robustly detect ¹³CO emission.

To circumvent those difficulties, we have developed a robust method to categorize a voxel into "noise" or "detection." First, the spectral cubes are smoothed spatially and spectrally. The size of the smoothing kernels is determined so that the ¹²CO, ¹³CO, and CS data have similar S/N, which ensures that the relative fraction of the diffuse extended, dense, and very dense CO components relative to the total detected CO emission does not depend on the sensitivity of the observations. We assumed the median rms measurement error in each survey to compute the kernels sizes, and there is therefore one kernel size per survey and per line. Additionally, we assumed main beam temperature ratios ${T}_{12}/{T}_{13}$ = 10 and ${T}_{12}/{T}_{\mathrm{CS}}$ = 15, based on the typical ratio observed in the line wings of individual spectra. These assumed ratios are only applied to determine the smoothing kernel widths and are not used for subsequent calculations of opacity.

The ${T}_{12}/{T}_{13}$ ratio exactly defines the optical depth of the ¹³CO line ${\tau }_{13}$ (see Section 3.4), under the assumption that the beam filling factors of ¹²CO and ¹³CO emission are the same, and that the excitation temperatures of the ¹²CO and ¹³CO lines are also the same. Under these assumptions, our goal of detecting ¹³CO in gas with ${T}_{12}/{T}_{13}$ < 10 corresponds to ${\tau }_{13}$ > 0.1. Variations in excitation temperature between the ¹²CO and ¹³CO lines, and differences (possibly of a factor two) in the beam filling factor of ¹²CO and ¹³CO emission, could increase ${\tau }_{13}$ by several.

Since there is a gradient in the ¹²CO/¹³CO abundance with galactocentric radius (Milam et al. 2005), this target ratio ${T}_{12}/{T}_{13}$ corresponds to H₂ surface densities between 5 and 10 M_⊙ pc⁻² (km s⁻¹)⁻¹ s in the Galactocentric radius range probed here (3–15 kpc), at an excitation temperature of 8 K (see Roman-Duval et al. 2010, and Section 3.4). Our goal of detecting CS emission with T₁₂/${T}_{\mathrm{CS}}$ < 15 corresponds to ${\tau }_{\mathrm{CS}}\gt 0.07$. Assuming an abundance ratio $n(\mathrm{CS})/n({{\rm{H}}}_{2})$ = $1\times {10}^{-9}$ (Neufeld et al. 2015), this implies the gas detected in CS emission has H₂ spectral surface densities > 20 M_⊙ pc⁻² (km s⁻¹)⁻¹.

Given these assumptions for the ${T}_{12}/{T}_{13}$ ratios, the sensitivity of the ¹³CO smoothed cubes must be 10 times better than the ¹²CO smoothed cubes in order for the fraction of diffuse and dense gas not to depend on the sensitivities of each spectral line data. Similarly, the CS smoothed cubes must be 15 times more sensitive than the ¹²CO smoothed observations. This constraint sets the number of elements in the smoothing kernels, via

$$\begin{eqnarray}&&\sqrt{\displaystyle \frac{{N}_{k13}}{{N}_{k12}}}=\left(\displaystyle \frac{{T}_{12}}{{T}_{13}}\right)\left(\displaystyle \frac{{\sigma }_{13}}{{\sigma }_{12}}\right),\end{eqnarray} \tag{ 1 }$$

where ${N}_{k12}$ and ${N}_{k13}$ are the number of voxels in the smoothing kernels for the ¹²CO and ¹³CO cubes, respectively, the line ratios are assumed as above, and (${\sigma }_{13}/{\sigma }_{12}$) is the ratio of sensitivities in the un-smoothed ¹²CO and ¹³CO cubes, taken to be the median rms in each survey and line. A similar equation applies to the CS cubes. Once the number of elements in the kernels are determined, the elements must be distributed in the spatial and spectral directions. Several constraints determine the size of the kernel in each direction. First, the size of the kernels must be the same in the Galactic longitude and latitude directions. Second, the size of the kernels in each direction must be an odd number. Third, because the UMSB is spatially under-sampled, we must minimize the size of the smoothing kernels in the spatial direction.

In the GRS+UMSB, where the native voxel is 1' × 1' × 0.3 km s⁻¹, we smooth the ¹²CO data with a (1, 1, 3) kernel, so ${N}_{k12}$ = 3. The sensitivity of the smoothed ¹²CO cubes is 0.25 K per voxel (0.47/$\sqrt{3.}$). We choose a kernel of size (3, 3, 7) for the ¹³CO, such that ${N}_{k13}$ = 63, which allows us to probe the ¹³CO line for ${T}_{12}/{T}_{13}$ ratios as high as 9 (close to the target value of 10). Similarly for the CS line, we can probe ${T}_{12}/{T}_{\mathrm{CS}}$ = 17 with a (5, 5, 9) kernel. In the EXFC survey, the native voxels are smaller (225 × 225 × 0.13 km s⁻¹), and the sensitivity per voxel is worse (2 K), and so the ¹²CO cubes are smoothed by a larger kernel of size (3, 3, 9) compared to the GRS+UMSB, corresponding to ${N}_{k12}$ = 81. This ensures a sensitivity of 0.22 K per voxel in the smoothed ¹²CO cubes, consistent with the GRS+UMSB. We smooth the ¹³CO cubes with a kernel of size (7, 7, 17) corresponding to ${N}_{k13}$ = 225, and ${T}_{12}/{T}_{13}$ = 9, also comparable to the GRS+UMSB. The sizes of the kernels are listed in Table 1.

In a second step, a detection mask is computed for each spectral cube (¹²CO, ¹³CO, CS). The mask is equal to 1 where the smoothed spectral cube has a main beam temperature T₁₂^sm (resp. T₁₃^sm) above $1{\sigma }_{12}^{\mathrm{sm}}$ (resp. ${\sigma }_{13}^{\mathrm{sm}}$), where the noise rms of the smoothed ¹²CO (resp. ¹³CO) cube ${\sigma }_{12}^{\mathrm{sm}}$ (resp. ${\sigma }_{13}^{\mathrm{sm}}$) is computed as the noise rms of the original cube divided by the square root of the number of voxels in the smoothing kernel: ${\sigma }_{12}^{\mathrm{sm}}$ = ${\sigma }_{12}/\sqrt{{N}_{k12}}$ (and similarly for ${\sigma }_{13}^{\mathrm{sm}}$). The mask is equal to zero everywhere else (non-detections).

Because we only use a threshold of 1σ, a significant amount of spurious noise peaks are still included in the detection mask at this stage. This is problematic because only positive noise peaks are included in the masked data. When summing masses or luminosities over a large number of voxels, as we do here, these remaining noise peaks can significantly and positively bias the summed or binned quantities. To remove those noise peaks, the mask is eroded and then dilated by a structure of size similar to the smoothing kernels. This effectively removes sharp features (such as noise peaks) smaller than structure used in the erosion/dilation procedure. The ERODE and DILATE functions in IDL are used for this purpose. Erosion and dilation are morphological operations commonly used in image processing, and are described in, e.g., Soille (1999). Finally, the eroded/dilated mask is applied to the un-smoothed data to separate the cubes' voxels into "detection" and "noise" categories. The resulting number of voxels in each category ("noise" or "detection") are listed in Tables 2 and 3 for the GRS+UMSB and EXFC surveys.

Table 2.  Number of Voxels in the "Noise" and "Detection" Categories for the ¹²CO, ¹³CO, and CS Lines in the GRS and UMSB (ℓ = 18°–55°)

	12CO	13CO	CS	Diffuse	Dense
Detection	$3.1\times {10}^{7}$	$2.2\times {10}^{7}$	$1.3\times {10}^{5}$	$1.2\times {10}^{7}$	$1.9\times {10}^{7}$
Noise	$9.6\times {10}^{7}$	$1.1\times {10}^{8}$	$2.0\times {10}^{6}$	...	...
Filling factor (PPV space)	24%	17%	6%	39%	61%

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Table 3.  Number of Voxels in the "Noise" and "Detection" Categories for the ¹²CO and ¹³CO Lines in the EXFC Survey

	EXFC 135–195				EXFC 55–100
	12CO	13CO	Diffuse	Dense	12CO	13CO	Diffuse	Dense
Detection	$1.0\times {10}^{8}$	$5.0\times {10}^{7}$	$5.1\times {10}^{7}$	$5.0\times {10}^{7}$	$1.1\times {10}^{8}$	$7.4\times {10}^{7}$	$3.9\times {10}^{7}$	$7.4\times {10}^{7}$
Noise	$6.3\times {10}^{9}$	$6.3\times {10}^{9}$	...	...	$3.1\times {10}^{9}$	$3.1\times {10}^{9}$	...	...
Filling factor (PPV space)	1.6%	0.8%	50%	50%	3%	2%	35%	65%

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Figure 1 shows examples of our detection procedure along one sight-line in each survey, with the total, detected, and noise spectra indicated by different colors. The velocity range in which the ¹²CO and ¹³CO lines are detected extends to very low main beam temperature levels, and is similar between the two lines. This constitutes an additional verification that the ¹²CO and ¹³CO lines are detected with similar S/N ratios.

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Figure 2 shows the total ¹²CO, ¹³CO, and CS spectra (summed along all sight-lines) in the "detection" and "noise" categories, as well as their total in each survey. For the EXFC survey, the ℓ = 143° and ℓ = 86° are shown. Since noise voxels dominate in number (see Tablesapj522444t2 2 and 3), Figure 2 demonstrates that (1) there are no residual structures in the noise spectrum that resemble spectral lines, and our detection/masking algorithm has therefore successfully captured all the low-level extended emission, and (2) there is no thresholding-induced positive bias in the detected CO emission, which would appear as a systematically negative noise spectrum. We note that the dip in the "noise" ¹³CO spectrum in the GRS at about 12–15 km s⁻¹ is due to a contaminated "off" position in the GRS data, which causes an artificial negative feature in the baseline of the spectrum, with main beam temperature values around −1.5 to −1 at longitudes ℓ = 33°–36°. At some velocities, the "detected" spectrum is slightly larger than the "total" spectrum. This is due to small negative baseline fluctuations, and represents a very small effect, which is not seen in the combined fields.

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Figures 3, 4, 5, and 6 show integrated intensity images of the total, detected, and noise ¹²CO, ¹³CO, and CS emission in each survey. For the EXFC survey, the ℓ = 143° and ℓ = 86° fields are shown. The CS emission is only mapped in a two square degree GRS field at ℓ~45°. Our algorithm produces much smoother and cleaner maps than if a naive sum along the velocity axis were performed. There is no residual structure in the noise maps, indicating that all the low-level emission was captured in the detection mask.

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Once "noise" and "detection" masks are created for the ¹²CO, ¹³CO, and CS cubes, we define the "diffuse extended ¹²CO emission" as the ensemble of all voxels where ¹²CO is detected but ¹³CO is not detected. The "dense ¹²CO emission" corresponds to all voxels where both the ¹²CO and ¹³CO are detected. In the inner Galaxy field with CS observations, the "very dense ¹²CO emission" corresponds to voxels where ¹²CO, ¹³CO, and CS are detected. The definitions of the diffuse, dense, and very dense components are summarized in Table 4.

Table 4.  Definition of the Diffuse, Dense, and Very Dense Components

	12CO 1-0	13CO 1-0	CS 2-1	${{\rm{\Sigma }}}_{v}$(H2)
				(M⊙ pc−2 (km s−1)−1)
Diffuse	Detected	Undetected	Undetected	<10
Dense	Detected	Detected	Undetected	>10
Very dense	Detected	Detected	Detected	>20

Note. ${{\rm{\Sigma }}}_{v}$(H₂) is the approximate threshold spectral surface density between the different regimes "diffuse," "dense," and "very dense."

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The PPV cubes are smoothed to obtain the same S/N for the ¹²CO, ¹³CO, and CS observations. In Section 3.2, we determined that the ${T}_{12}/{T}_{13}$ ratio of 10 assumed to derive the sizes of the smoothing kernels corresponds to H₂ surface densities of approximately 5–10 M_⊙ pc⁻² (km s⁻¹)⁻¹ (under certain assumptions, see Section 3.2). Assuming a line width of 5 km s⁻¹ typical of GMCs, this corresponds to surface densities of 25–50 M_⊙ pc⁻². Thus, by construction, we can detect the ¹³CO line approximately down to "spectral" surface densities of 5–10 M_⊙ pc⁻² and the surface density threshold between "diffuse" and "dense" gas corresponds to H₂ surface densities of approximately 25–50 M_⊙ pc⁻². The density threshold between the diffuse and dense gas will vary depending on local conditions.

Similarly, we can detect CS emission with ${T}_{12}/{T}_{\mathrm{CS}}\lt 15$, which corresponds to H₂ spectral surface densities >20 M_⊙ pc⁻² (km s⁻¹)⁻¹, and surface densities >100 M_⊙ pc⁻².

Thus, the "diffuse," "dense," and "very dense" components correspond to different surface density regimes. The approximate threshold surface densities of the "diffuse," "dense," and "very dense" gas are reported in Table 4.

For each voxel with ¹²CO detection, the ¹²CO excitation temperature ${T}_{\mathrm{ex}}$ is computed following Equation (1) of Roman-Duval et al. (2010). For each voxel with both ¹²CO and ¹³CO detections, the ¹³CO optical depth ${\tau }_{13}$ is also computed using Equation (2) of Roman-Duval et al. (2010). Using the distances to each voxel (with a unique solution in the outer Galaxy, and 10 realizations of the near/far ambiguity in the inner Galaxy) and Equation (9) of Roman-Duval et al. (2010), we derive the ¹²CO and ¹³CO luminosities in all voxels with detections, as well as the H₂ mass M(H₂) in voxels with ¹²CO and ¹³CO detections. Roman-Duval et al. (2010) used a constant abundance ratio of 45 between ¹²CO and ¹³CO in order to convert the ¹³CO optical depth of a mass of H₂. In this work, which includes a much larger range in galactocentric radius, we adopt the abundance ratio derived in Milam et al. (2005), which is characterized by a radial gradient:

$$\begin{eqnarray}&&\displaystyle \frac{n({}^{12}\mathrm{CO})}{n({}^{13}\mathrm{CO})}=6.2\times {R}_{\mathrm{gal}}+18.7.\end{eqnarray} \tag{ 2 }$$

In voxels with ¹³CO main beam temperatures >2${\sigma }_{13}$, there is a tight linear relation between the ¹²CO luminosity of a voxel and its H₂ mass, as derived from ¹²CO and ¹³CO. The slope of this relation is the CO-to-H₂ conversion factor, ${X}_{\mathrm{CO}}$ (for column density) or ${\alpha }_{\mathrm{CO}}$ (for surface density), and increases with increasing Galactocentric radius, ${R}_{\mathrm{gal}}$. This relation between L(¹²CO) and M(H₂), derived from the combined data sets, is shown for ${R}_{\mathrm{gal}}$ = 5.6 kpc and 11 kpc in the top two panels of Figure 7. The bottom panel of Figure 7 displays the variations of the CO-to-H₂ conversion factor with Galactocentric radius. ${X}_{\mathrm{CO}}$ varies between $1.5\times {10}^{20}$ cm⁻² (K km s⁻¹)⁻¹ at ${R}_{\mathrm{gal}}\sim 3\;{\rm{kpc}}$ and $6\times {10}^{20}$ cm⁻² (K km s⁻¹)⁻¹ at ${R}_{\mathrm{gal}}\sim 15\;{\rm{kpc}}$. This is in agreement with the conclusions in Goldsmith et al. (2008), who found that the mass of both diffuse and dense CO-emitting gas in the Taurus molecular cloud is well traced by its luminosity, albeit with a slightly lower conversion factor of 4.1 M_⊙ pc⁻² (K km s⁻¹)⁻¹, corresponding to ${X}_{\mathrm{CO}}$ = $2\times {10}^{20}$cm⁻² (K km s⁻¹)⁻¹. Liszt et al. (2010) also reached similar conclusions in a study of diffuse Galactic sight-lines.

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In Figure 7, we fit a linear relation, thus forcing the slope in log–log space to be 1. However, to investigate potential deviations from a linear relation, we also performed a linear fit in log–log space (power-law fit), leaving the slope as a free parameter. The resulting slopes were between 0.96 and 1.001, indicating that the relation between CO luminosity and H₂ mass is well described by a linear function. Since we derive detection masks from smoothed data, the ¹³CO and ¹²CO main beam temperatures of voxels within the detection mask can be smaller than the uncertainties, or even negative. Instances of this effect are visible in the wings of the CO lines in Figure 1. While necessary to avoid thresholding-induced positive biases (as described in Section 3.2), this effect creates some numerical issues in the computation of ${\tau }_{13}$ and M(H₂). To circumvent this issue, the M(H₂) values in voxels with ¹³CO main beam temperatures lower than $2{\sigma }_{13}$ are replaced with estimates derived from the relation between the ¹²CO luminosity of a voxel and its mass. Since the CO–H₂ relation depends on Galactocentric radius, we bin the data in radial intervals of width 1 kpc, and derive a CO-to-H₂ conversion factor in each radial bin from the voxels with ¹³CO detections $\gt 2{\sigma }_{13}$. We then apply the same conversion factor between ¹²CO luminosity and H₂ mass to the voxels in that same radial bin, but with ¹³CO main beam temperatures $\lt 2{\sigma }_{13}$. This not only allows us to derive an H₂ mass for voxels in the dense mask, albeit with ¹³CO below the 2σ sensitivity, but also to derive an H₂ mass in the diffuse CO component, where ¹³CO is not detected and a mass estimate would otherwise not be possible.

Tablesapj522444t2 2 and 3 summarize the number of voxels in the "noise" and "detection" categories for each line (¹²CO, ¹³CO, CS), and the number of "diffuse," "dense," and "very dense" voxels in each survey. In Tablesapj522444t2 2 and 3, we also list the corresponding filling factors, computed as the number of detected voxels divided by the total number of voxels in the PPV cubes. CO and CS emission are in general relatively sparse in the PPV cubes, particularly in the outer Galaxy. The ¹²CO line has the highest filling factor in the PPV cubes, with 24% of voxels in the "detection" mask in the inner Galaxy covered by the GRS+UMSB. The filling factor in the PPV cubes of the ¹²CO line drops to <2% in the outer Galaxy. The filling factors in the PPV cubes of the ¹³CO line is about half that of the ¹²CO line. The CS-emitting gas is much more compact with a filling factor of <5%. This progression is qualitatively seen in Figures 3 and 4. Within the ¹²CO-emitting gas, the diffuse ¹²CO gas fills a slightly smaller volume (39%) than the dense CO gas (61%) in the GRS+UMSB coverage, but diffuse and dense CO gas occupy equal volumes in the outer Galaxy.

Figure 8 shows the distribution of the "spectral surface density of H₂" in each voxel, ${{\rm{\Sigma }}}_{v}$(H₂), expressed in M_⊙ pc⁻² (km s⁻¹)⁻¹, for the diffuse, dense, and very dense components, in each survey. Spectral surface densities correspond to the surface density of H₂ along the line of sight at the velocity of the voxel, per unit velocity. As expected, the distributions of the diffuse, dense, and very dense components peak at increasingly higher spectral surface densities (3, 8, and 12 M_⊙ pc⁻² (km s⁻¹)⁻¹, respectively). We note that, in order to obtain the surface density of a parcel of molecular gas, the spectral surface densities need to be multiplied by the velocity width of that parcel. Hence, the spectral surface densities shown in Figure 8 cannot be directly compared to the typical surface densities of GMCs. For comparison however, we also show in Figure 8 the distribution of the spectral surface densities in voxels within the GMCs identified in Roman-Duval et al. (2010), which closely resembles the spectral surface density distribution of the very dense gas, peaking at ${{\rm{\Sigma }}}_{v}$(H₂) = 15 M_⊙ pc⁻² and exhibiting a long tail to high spectral surface densities.

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The distribution of the H₂ spectral surface density in each voxel in the EXFC survey appears wider. However, the difference in the width of the distribution between the GRS+UMSB and EXFC is most likely due to the difference in measurement errors (${\sigma }_{12}$) at the original resolution of the data (typically ${\sigma }_{12}$ ∼ 2 K per native (225 × 225 × 0.13 km s⁻¹) voxel in EXFC, compared to ${\sigma }_{12}$ ∼ 0.24 to 0.5 K per native (1' × 1' × 0.3 km s⁻¹) voxel in the GRS+UMSB).

With the knowledge of the location (distance and coordinates), luminosity, and mass of each voxel, we have produced a face-on map of the Galactic distribution of ¹²CO-emitting molecular gas, separating the diffuse (¹²CO-bright, ¹³CO-dark) and dense (¹²CO-bright, ¹³CO-bright) CO components. The maps were obtained by summing in each 100 pc × 100 pc pixel the masses of all voxels located within a pixel, and dividing by the area of that pixel. The resulting face-on maps are shown in Figure 9. Strikingly, the surface density of molecular gas decreases by one to two orders of magnitude between Galactocentric radii of 3 and 15 kpc. In both sides of the solar circle, the diffuse CO component is smoother and more uniform than the dense component, which is consistent with the conclusions of Pety et al. (2013) in M51, who found that 50% of the CO luminosity in M51 originates from kiloparsec-scale diffuse emission.

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The large uncertainties on the distance will undoubtedly affect the detailed spatial distribution of molecular gas in the face-on maps. These maps should therefore not be used to derive the detailed structure of the Milky Way, but rather are meant to better conceptualize and visualize the transformations involved, between looking through the Galactic Plane and from above the Galactic Plane. In the next section, the radial and vertical distributions of H₂ and of the different CO components are computed by averaging those maps in bins of Galactocentric radius and vertical height above the plane.

We first examine the distribution of ¹²CO, ¹³CO, and CS average Galactic integrated intensities ${I}_{\mathrm{gal}}$ with Galactocentric radius ${R}_{\mathrm{gal}}$. Here, average Galactic integrated intensity corresponds to the total luminosity in a Galactocentric radius bin, divided by the surface area covered by the survey in that bin, projected onto the Galactic disk. ${I}_{\mathrm{gal}}$ is therefore the integrated intensity as seen from above the plane, averaged over radial bins of width 0.1 kpc, and is equivalent to the integrated intensity measurements for extragalactic surveys of face-on galaxies. In the inner Galaxy, the average Galactic integrated intensities are obtained for each Monte Carlo realization of distances separately. We then average the trends of ${I}_{\mathrm{gal}}({R}_{\mathrm{gal}})$ versus ${R}_{\mathrm{gal}}$ over all 10 realizations, and we include the standard deviation between realizations in the final error estimation. As an additional check that our detection algorithm picks up all the low-level extended emission, we also compute the average Galactic integrated intensities of the voxels with non-detections ("noise voxels") using a similar procedure. The resulting average Galactic integrated intensities for ¹²CO and ¹³CO, as well as their ratio, are plotted as a function of galactocentric radius in Figure 10 for each survey. The average Galactic integrated intensity in the CS 2-1 line is plotted versus R_gal in Figure 11. Of course, these trends are representative of the gas seen within the coverage of the surveys.

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To compute the total error on ${I}_{\mathrm{gal}}({R}_{\mathrm{gal}})$, indicated at 1σ by the thickness of the curves in Figures 10 and 11, we sum in quadrature the different sources of errors. These sources include errors on the near and far distance estimation due to non-circular motions, for the inner Galaxy, the standard deviation between Monte-Carlo distance realizations, and the residuals from the average Galactic integrated intensity of "noise" voxels. For a given voxel, the error on its luminosity incurred by the error on its distance is given by $\delta L=2\;L\delta d/d$, where d and $\delta d$ are the distance of the voxel and its error and L its luminosity. The error on the total luminosity in a Galactocentric radius bin is the quadratic sum of the errors on the luminosities of all the voxels included in that bin. We note that the standard deviation between near/far distance realizations in the inner Galaxy is negligible compared to the other sources of errors.

The average Galactic integrated intensity of ¹²CO and ¹³CO decreases by one to two orders of magnitude between ${R}_{\mathrm{gal}}$ ∼ 3 kpc and ${R}_{\mathrm{gal}}$ ∼ 15 kpc. ${I}_{\mathrm{gal}}({R}_{\mathrm{gal}})$ for ¹²CO and ¹³CO track each other closely throughout the Galactic plane, with an approximately constant ratio of 5 out to ${R}_{\mathrm{gal}}$ = 6.5 kpc. The ¹²CO/¹³CO integrated intensity ratio increases to 10–12 in the solar neighborhood, although the errors are larger in this case, and remains between 10 and 20 in the outer Galaxy, out to ${R}_{\mathrm{gal}}$ = 14 kpc. The ¹²CO/¹³CO integrated intensity ratio appears to be anti-correlated with ${I}_{\mathrm{gal}}({R}_{\mathrm{gal}})$, or in other words, with average Galactic surface density of CO-bright molecular gas. This factor-of-two increase in the ¹²CO/¹³CO luminosity ratio between 3 kpc and the solar neighborhood has previously been observed by Liszt et al. (1984). They interpret it as being a result of the volume density decreasing away from the Galactic center, which would be consistent with the decreasing star formation rate and resulting cloud temperatures seen in Roman-Duval et al. (2010). Additionally, the radial trend in the ¹²CO/¹³CO luminosity ratio could be explained by the decrease in the fraction of dense gas with decreasing surface density (and increasing Galactocentric radius), and the ¹³CO/¹²CO abundance gradient observed in Milam et al. (2005), which varies between 50 at R_gal = 5 kpc and 100 at R_gal = 15 kpc and could be consistent with the variations in luminosity ratio. Other possible effects that could explain the variations in the ¹²CO/¹³CO integrated intensity or luminosity ratio include more sub-thermally excited ¹³CO in the outer Galaxy.

Similarly, we derive the average Galactic H₂ surface density ${{\rm{\Sigma }}}_{\mathrm{gal}}({{\rm{H}}}_{2})({R}_{\mathrm{gal}})$ in the diffuse extended, dense, and very dense components, by summing the masses of all voxels in each CO component in Galactocentric radius bins of width 0.1 kpc, and dividing by the surface area of each survey projected on the Galactic Plane. The resulting radial distributions of the three CO gas components are shown in linear space separately for each survey in Figure 12 and in logarithmic space combining all data sets in Figure 13. As for the ¹²CO and ¹³CO average Galactic integrated intensity computation, all relevant sources of errors are included in the final error budget.

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In the inner Galaxy, the dense component dominates in mass. The mass fraction of dense gas decreases from 90% at ${R}_{\mathrm{gal}}$ = 4 kpc to 50% in the solar neighborhood. In the outer Galaxy, the mass fraction of dense gas varies between 40% and 80%, and is anti-correlated with surface density.

Assuming that the Galaxy is roughly axisymmetric and that the radial trends observed in our surveys are representative of the Galaxy as a whole, the total masses of the diffuse and dense CO components integrated between 3 and 15 kpc are $1.5\times {10}^{8}\;$ M_⊙ and $4.9\times {10}^{8}\;$ M_⊙, respectively, or 25% and 75% of the total H₂ mass traced by ¹²CO ($6.5\times {10}^{8}$ M_⊙) respectively. Statistical errors on integrated masses are ∼1%. Systematic uncertainties are ∼30% due to errors on the assumed abundances (¹³CO/¹²CO and ¹²CO/H₂) and the possibly inaccurate assumption of pure LTE at low column densities, which systematically underestimates masses in the diffuse regime. The total mass of H₂ derived here is compatible within errors to the number quoted in the review by Heyer & Dame (2015) or 9 × 10⁸ M_⊙. The total luminosities and masses of each component, in the inner, outer, and entire Galaxy, are listed in Table 5.

Table 5.  Total Luminosity and Molecular Mass in the Milky Way in the Diffuse and Dense Components Traced by ¹²CO

		Inner	Outer	Total
L(12CO)	Diffuse	$2.0\times {10}^{1}$	4.0	$2.4\times {10}^{1}$
	Dense	$1.1\times {10}^{2}$	3.8	$1.1\times {10}^{2}$
	Very dense	4.8	...	4.8
	Total	$1.3\times {10}^{2}$	7.7	$1.4\times {10}^{2}$
M(H2)	Diffuse	$9.3\times {10}^{7}$	$6.0\times {10}^{7}$	$1.5\times {10}^{8}$
	Dense	$4.6\times {10}^{8}$	$3.9\times {10}^{7}$	$4.9\times {10}^{8}$
	Very dense	$2.9\times {10}^{7}$	...	$2.9\times {10}^{7}$
	Total	$5.5\times {10}^{8}$	$9.9\times {10}^{7}$	$6.5\times {10}^{8}$

Note. Luminosities are given in units of K km s⁻¹ kpc². Masses are given in M_⊙. Statistical errors on integrated luminosities and masses are ∼1%. Systematic uncertainties are ∼30% due to uncertainties on abundances (¹³CO/¹²CO and ¹²CO/H₂) and the possibly non-applicable assumption of LTE in the diffuse regime, which systematically underestimates masses at low column densities.

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In the 2 deg² field where CS observations are available at ℓ = 45°, the very dense CO component (also bright in CS) is sparse. In the Galactocentric radius range probed by the observations (R_gal = 6–8.5 kpc), the fraction of the very dense component in the total surface density varies between zero and 50% locally in presumably massive star formation regions. Locally, the very dense component can thus comprise a significant fraction of the gas. However, the very dense gas has a relatively low filling factor. The very dense component traced by CS has a total mass of $2.9\times {10}^{8}\;$ M_⊙ in the observed range (R_gal = 6–8.5 kpc). The total molecular gas mass in this radial interval is $1.8\times {10}^{8}\;$M_⊙. Therefore, the very dense component represents only ∼14% of the total molecular gas mass traced by ¹²CO emission. However, it is possible that the fraction of very dense gas traced by CS be higher closer to the center of the Galaxy.

As a comparison, Battisti & Heyer (2014) found that the very dense component of molecular clouds, as traced by mm dust continuum emission, comprises ∼10% of the mass of molecular clouds identified with the CPROPS detection algorithm (Rosolowsky & Leroy 2006). In Section 5.1, we show that about 15% of the total molecular gas mass traced by ¹²CO in the Milky Way resides in such GMCs, and so the very dense gas fraction determined in Battisti & Heyer (2014) would represent about 1.5% of the total H₂ mass traced by ¹²CO, which is slightly lower than the very dense gas fraction derived here.

Figure 12 suggests that the fraction of dense CO gas is correlated with the Galactic surface density of molecular gas (traced by ¹²CO). We plot in the top panel of Figure 14 the relation between the mass fraction of dense CO gas and the Galactic molecular gas surface density, averaged in 100 pc wide pixels as seen from above the Galaxy (see Figure 9). At low molecular surface densities (${{\rm{\Sigma }}}_{\mathrm{gal}}^{100\;\mathrm{pc}}\lt {10}^{5}$ M_⊙ kpc⁻²), the fraction of dense gas is low (${f}_{\mathrm{DG}}\quad \lt$ 20%–30%). The dense gas fraction increases when the disk's molecular gas surface density increases, and reaches 80%–90% at high surface densities of 10⁷M_⊙ kpc⁻². A linear fit in log–log space to the ${{\rm{\Sigma }}}_{\mathrm{gal}}^{100\;\mathrm{pc}}$—f_DG relation yields f_DG = 0.02 ${{\rm{\Sigma }}}_{\mathrm{gal}}^{0.24\pm 0.01}$.

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The relation between Galactic surface density of H₂ and the fraction of very dense gas (traced by CS emission), f_VDG, is shown in the bottom panel of Figure 14. The mass fraction of very dense gas (presumably star-forming) also increases with increasing disk surface density, from f_VDG = 1% at ${{\rm{\Sigma }}}_{\mathrm{gal}}^{100\;\mathrm{pc}}$ = a few 10⁵M_⊙ kpc⁻², up to f_VDG = 30% at ${{\rm{\Sigma }}}_{\mathrm{gal}}^{100\;\mathrm{pc}}$ = 10⁷M_⊙ kpc⁻². A linear fit in log–log space yields f_VDG = $3.7\times {10}^{-7}$ ${{\rm{\Sigma }}}_{\mathrm{gal}}^{0.8\pm 0.1}$.

We derive the vertical distribution (i.e., perpendicular to the Galactic plane) of the total, diffuse, dense, and very dense CO components. Knowing the distance of each voxel, its height above the plane z was computed as $z=d\mathrm{tan}(b)$. We then summed the masses of all voxels in vertical height bins of width 5 pc and galactocentric radius bins of width 1 kpc, divided by the surface areas on the Galactic Plane covered by each survey in those radial bins, and divided by the bin width (5 pc) to obtain the average molecular gas density $\rho ({{\rm{H}}}_{2})$ (in M_⊙ pc⁻³) as a function of z and R_gal for the overall CO gas, as well as the diffuse, dense, and very dense CO gas components. In the inner Galaxy, the vertical profile of the molecular gas density was derived for each Monte Carlo realization and then averaged over all realizations. The resulting vertical profiles are shown in Figure 15. The profiles are fitted with Gaussians, and the resulting centroid and FWHM values are plotted as a function of Galactocentric radius in Figure 16.

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The total vertical profile of molecular gas in the inner Galaxy is well described by a Gaussian function with a FWHM of ∼110 pc. As seen in the radial distribution of diffuse and dense CO gas, the dense CO component dominates the inner Galaxy in mass. The profile of the diffuse component in the inner Galaxy is also Gaussian, but with a larger FWHM of 130–200 pc. In contrast, the very dense component is concentrated in the Galactic plane, with a non-Gaussian, double-peaked profile of FWHM ∼ 50 pc.

In the outer Galaxy, the molecular disk is more warped, with a centroid increasing from a few parsecs at the solar circle, up to 150 pc at R_gal = 14 kpc. The molecular disk is wider than in the inner Galaxy, with FWHM varying between 110 and 300 pc. The vertical profiles have multiple peaks and are thus not well fit by a Gaussian. The FWHM shown in Figure 16 thus represents a gross approximation of the profile width. In the outer Galaxy, the diffuse CO component has a similar mass as the dense CO gas, but their vertical profiles differ significantly. The vertical profile of the diffuse CO gas appears smoother and wider than the profile of the dense CO gas.

In both the inner and outer Galaxy, these results are consistent with previous estimates of the thickness and mid-plane displacement summarized by Heyer & Dame (2015). The larger vertical extent of the diffuse CO component suggests that it originates from a thick disk, which has already been suggested in the Milky Way by Dame & Thaddeus (1994), and in M51 by Pety et al. (2013).

Studies of the properties and distribution of molecular gas in galaxies commonly resort to a cloud identification algorithm, such as CLUMPFIND (Williams et al. 1994) or dendrogram (Rosolowsky et al. 2008) algorithms. These procedures allow catalogs of discrete objects and associated properties to be derived, including a distance derivation, which cannot be unambiguously determined in the inner Galaxy on a per voxel basis. However, it is not clear what fraction of the total CO emission this type of algorithm picks up. In the left panel of Figure 12, we show the radial distribution of H₂ within molecular clouds identified in Roman-Duval et al. (2010), within the same survey coverage. The molecular gas traced by clouds identified with CLUMPFIND in Roman-Duval et al. (2010) represents a small fraction of the total molecular gas in the inner Milky Way. The total mass of molecular gas in GMCs in the UMSB+GRS coverage is $4.6\times {10}^{7}$M_⊙, while in this analysis we derive a total molecular gas mass of $3.4\times {10}^{8}\;$ M_⊙ within the same coverage (not to be confused with the mass extrapolated to the entire galaxy in Section 4.4, or $6.5\times {10}^{8}$M_⊙). Thus, only ∼14% of the molecular gas mass in the Milky Way was identified within GMCs in the inner Milky Way based on their ¹³CO emission. This number is significantly smaller than the 40% quoted in Solomon & Rivolo (1989) and Williams & McKee (1997). However, these studies identified the GMCs in the ¹²CO cubes, whereas Roman-Duval et al. (2010) identified CO clouds in the GRS ¹³CO cubes. It is well known (e.g., Goldsmith et al. 2008; Heyer et al. 2009, and this work) that ¹²CO emission is more (approximately a factor of two) spatially extended than its ¹³CO counterpart, and so it is not surprising that the mass fraction of CO gas in ¹²CO-identified GMCs is larger than the mass fraction of CO gas in ¹³CO-identified GMCs. While the Milky Way is more confused than external galaxies, this suggests that studies of molecular gas relying on molecular cloud identification algorithms may be missing the majority of the molecular gas mass.

We identify the "diffuse" gas reported here effectively based on its high ${T}_{12}/{T}_{13}$ ratio, which implies a low optical depth, and therefore a low surface density. In Section 3.2, we estimate that the spectral surface density transition between the gas components we classify as "dense" and "diffuse" is about 10 M_⊙ pc⁻² (km s⁻¹)⁻¹, corresponding to surface densities of 25–50 M_⊙ pc⁻² for typical line widths. In this context, we interpret the "diffuse" gas as being of low surface density, and likely gravitationally unbound and unable to form stars, while the "dense" gas corresponds to a gas component, the physical properties (density, surface density, viral parameter) of which are similar to those in the classical sense of molecular clouds. The "diffuse" gas is observed both in the form of isolated extended structures, but also in the envelopes of dense gas.

There are, however, other effects that can induce high ${T}_{12}/{T}_{13}$ ratios. In particular, the wings of optically thick ¹²CO emission from dense clouds can be broader than the corresponding ¹³CO line while emanating from the same dense gas. In this study, the emission corresponding to those optically thick ¹²CO line wings would be included in the "diffuse" component. Thus, we may be overestimating the emission and mass of truly diffuse gas, while underestimating the amount of truly dense gas. We cannot differentiate the emission from truly diffuse gas from the dense gas emission in the opacity-broadened wings of the ¹²CO line, because we do not segment the emission into clouds (there are no GMCs in our study). However, we observe that 40% of the gas mass classified here as "diffuse" in the outer Galaxy is located in sight-lines toward which no dense gas is detected. This implies that at least 40% of the "diffuse" gas mass fraction reported here in the outer Galaxy corresponds to truly diffuse gas. In the inner Milky Way, the "diffuse" gas mass fraction with no associated dense component is 15% in the EXFC 55–100 coverage, and 5% in the GRS+UMSB coverage. However, these numbers are not meaningful in the inner Milky Way because most line of sights exhibit more than one CO line detection.

To evaluate more quantitatively the fraction of gas that we report to be "diffuse," but actually corresponds to the opacity-broadened line wings of dense gas in sight-lines where both diffuse and dense gas are detected, we compute the centroid velocity maps of our "diffuse" and "dense" components. For sight-lines in which both "diffuse" and "dense" gas are detected, we then compute, for each survey, the cumulative mass distribution of "diffuse" gas as a function of the difference in centroid velocity between that "diffuse" and the "dense" gas along the same line of sight (Figure 17). If the high ${T}_{12}/{T}_{13}$ ratio gas that we classify as "diffuse" actually corresponds to the optically thick line wings of ¹²CO, then one would expect the centroid velocity of this component to be similar to the centroid velocity of the "dense" gas. In the GRS+UMSB surveys, 90% of the gas we report as "diffuse" has a centroid velocity farther than 5 km s⁻¹ from the centroid of the "dense" gas along the same sight-line. In the EXFC 55–100 and EXFC 135–195, 65% and 45% of the gas mass that we classify as "diffuse" has a centroid velocity farther than 5 km s⁻¹ from the centroid of the "dense" gas. The typical line width of CO clouds is 5 km s⁻¹, and so Figure 17 implies that most of the gas mass classified as "diffuse" in this study does correspond to truly diffuse gas, and not to emission from the opacity-broadened line wings of the ¹²CO line.

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Because of the high critical density of CS emission and because we detect CS 2-1 emission with ${T}_{12}/{T}_{\mathrm{CS}}$ < 15, corresponding to spectral surface densities > 20 M_⊙ pc⁻² (km s⁻¹)⁻¹ and surface densities >100 M_⊙ pc⁻² for typical line widths, we interpret the "very dense" gas as being relatively compact, gravitationally bound, and star-forming gas. Indeed, Lada et al. (2010) derive a threshold of 120 M_⊙ pc⁻² for star formation to occur, close to the "dense" to "very dense" threshold used here. It is however worth mentioning that Galactic Plane surveys of the CS line with higher sensitivity than the GRS (Liszt 1995; Helfer & Blitz 1997) have shown that every ¹³CO feature has emission from CS at a level ∼1%–2% of the ¹²CO emission. The densities derived from such weak emission are consistent with rather diffuse molecular gas (low hundreds cm⁻³). This is however not the gas we are probing here with CS emission at the level ${T}_{12}/{T}_{\mathrm{CS}}$ < 15.

In this work, we compute H₂ masses and surface densities in voxels with ¹²CO and ¹³CO emission detectable at >2σ under the assumption of LTE and a constant ¹²CO/¹³CO abundance. We derive the (constant) conversion factor between ¹²CO luminosity and H₂ mass in this sample of voxels. For voxels with ¹³CO emission below $2\sigma$ (and detected ¹²CO emission), we assume this same constant CO-to-H₂ conversion factor derived in the voxels detected in ¹³CO (see Section 3.4) to compute an H₂ mass. However, we note that the X_CO factor is likely to vary and increase for low surface density gas. This gas is less shielded against the interstellar radiation field, which affects CO more strongly than H₂. While H₂ is well protected against photodissociation above an extinction of ${A}_{V}\sim 1$, CO requires values of ${A}_{V}$ ≈ 2–3 under solar neighborhood conditions (see, e.g., Tielens & Hollenbach 1985; Wolfire et al. 1993; Röllig et al. 2007; Glover et al. 2010; Glover & Mac Low 2011; Shetty et al. 2011a, 2011b). Indeed observations of nearby clouds show strong spatial variations of X_CO (e.g., see Pineda et al. 2008; Lee et al. 2014, for a detailed analysis of the Perseus cloud). The fraction of diffuse gas scales linearly with the CO-to-H₂ conversion factor assumed. Since the CO-to-H₂ conversion factor may be significantly higher in diffuse gas compared to dense gas, the diffuse gas fraction derived here represents a lower limit. We note however that, when averaged over a large enough volume of the ISM or when focusing on the bulk of the molecular mass traced by CO lines, taking a roughly constant X_CO-factor gives acceptable results for solar-metallicity galaxies even when applied to the diffuse component (Solomon et al. 1987; Young & Scoville 1991; Liszt et al. 2010).

As alluded to in Section 1, there is a debate about the universality and slope of the K-S relation in nearby galaxies. Shetty et al. (2013, 2014b) argue in favor of galaxy-to-galaxy variation. Most, but not all, galaxies in their study portray a sub-linear relation between star formation rate surface density and molecular gas surface density (see also, Blanc et al. 2009; Ford et al. 2013). Shetty et al. (2014a) suggest that a non-linear K-S relation may result from the presence of CO not related to dense star-forming clouds, perhaps in a diffuse but pervasive molecular component. Our analysis confirms that this component of the ISM exists and contains about 25% of the total molecular ISM as traced by CO. If the SFR is linearly related to the amount of very high density gas, and if the radial trends we find in this work hold throughout the Galaxy, according to Shetty et al. (2014a), the underlying relationship between the star formation rate surface density and H₂ would be super-linear (see their Figure 3). We furthermore note that even in the inner Galaxy, which is clearly dominated by dense molecular gas, only 14% of molecular gas is associated with known molecular clouds as identified in the UMSB+GRS surveys (Roman-Duval et al. 2010). The bulk of this dense gas is found in a more distributed configuration. Our analysis suggests that the star formation process could simply be limited by the availability of such high-density gas at any given time.