PHYSICAL PROPERTIES OF MOLECULAR CLOUDS FOR THE ENTIRE MILKY WAY DISK

The usefulness of the Gaussian decomposition in the identification of molecular clouds is best seen in Figures 2 and 3. The top panel of Figure 2 presents a 2D cut through the CO data cube for a constant value of Galactic latitude (a so-called l–v diagram). Here we show a particularly complex region of the sky in the fourth quadrant. This representation of ${T}_{{\rm{B}}}(l,b,v)$ allows us to identify coherent structures that blend in an integrated intensity map. Even though individual clouds can be identified by eye, there is significant confusion and overlap between structures. Looking at this representation, one can appreciate why thresholding methods have been used in the past to isolate structures. But this isolation comes at the expense of leaving out the faint emission, below the lowest threshold level.

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The Gaussian decomposition is of great help in clearing some of the confusion. The bottom panel of Figure 2 shows the same l–v diagram but for a cube representing the integrated emission of the Gaussian components:

$$\begin{eqnarray}&&{W}_{\mathrm{CO}}(l,b,v)=\sqrt{2\pi }\,A(v)\,\sigma (v).\end{eqnarray} \tag{ 2 }$$

The cube ${W}_{\mathrm{CO}}(l,b,v)$ contains the integrated emission of every Gaussian component put at the central velocity position (v). In some cases, more than one Gaussian component will have the same central velocity v at a given (l, b) position. In such cases, ${W}_{\mathrm{CO}}(l,b,v)$ contains the sum of the column densities of all the components.

This cube contains all the emission of the original data cube described by the Gaussian decomposition, but without the spread of the velocity dispersion of each component. Figure 3 provides an example of a complex spectrum of the cube, where the black solid line is the observed spectrum, and the vertical blue solid line segments represent the integrated emission (${W}_{\mathrm{CO}}$—right axis) of each Gaussian component at its respective velocity. This spectral representation and the l–v cut of ${W}_{\mathrm{CO}}$ (Figure 2) illustrate how the Gaussian decomposition helps in separating blended emission and in identifying coherent structures.

The term "clustering analysis" refers to the task of identifying coherent groups in a data set. The appropriate clustering algorithm and parameter settings depend on the individual data set and on the intended use of the results. Here the cluster identification was done on the ${W}_{\mathrm{CO}}$ cube using a classical hierarchical algorithm with a threshold descent. The identification of coherent structures or clusters is done in position (l, b) and velocity (v) space at the same time. Because of this, and because it uses a threshold descent, the algorithm presented here has some similarity with clumpfind (Williams et al. 1994).

In detail, a first set of clusters was identified by grouping neighboring pixels with ${W}_{\mathrm{CO}}$ higher than a threshold value. This selection on ${W}_{\mathrm{CO}}$ creates islands in PPV space. The threshold value is then lowered and a new set of pixels in ${W}_{\mathrm{CO}}$ are selected. Lowering the threshold reveals the outskirts of already-defined clusters, as well as finding new islands with fainter peak emission. Each Gaussian component is either attached to a preexisting cluster, if one is nearby, or identified as the first component of a new cluster. This descent is continued for a number ${N}_{\mathrm{th}}$ of thresholds down to a minimum value.

In the descent we used ${N}_{\mathrm{th}}=15$ threshold values, spaced logarithmically; see Figure 3, where the horizontal gray lines depict the thresholds. The highest threshold value was set to ${W}_{\mathrm{CO}}=100$ K km s⁻¹, while the lowest threshold value was ${W}_{\mathrm{CO}}=0.8$ K km s⁻¹. The lowest threshold value corresponds to a Gaussian amplitude close to the noise level. For a narrow component with ${\sigma }_{i}=1.3$ km s⁻¹, the lowest threshold corresponds to A_i = 0.25 K; for 93% of the spectra the noise level is lower than 0.15 K (see Figure 22).

At each step of the process, we look for clusters that should be merged. We applied a strict criterion for cluster merging in order to avoid any runaway process that would lead to immense structures. Such overlinking is a common flaw of friends-of-friends algorithms. It is also known as the "chaining phenomenon" in single linkage clustering methods, where two clusters are linked together even if only a single element in each cluster is close to each other.

A cluster i is merged to a cluster j if its central coordinate is enclosed within the volume of j:

$$\begin{eqnarray}&&| {l}_{i}-{l}_{j}| \lt 2\,{\sigma }_{l,j}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&| {b}_{i}-{b}_{j}| \lt 2\,{\sigma }_{b,j}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&| \langle {v}_{i}\rangle -\langle {v}_{j}\rangle | \lt 2\,{\sigma }_{v,j},\end{eqnarray} \tag{ 5 }$$

where ${\sigma }_{l,j}$ is the ${W}_{\mathrm{CO}}$ weighted standard deviation of l for cluster j, and similarly for b, as defined in Equation (8) below. We also only allow the merging of clusters with less than 50 Gaussian components. The properties of the bigger clusters of the sample depend slightly on the merging procedure. If the merging criterion is too loose, big clusters merge into unrealistically large structures. The criteria selected here allow us to avoid this problem.

We explored several different values of ${N}_{\mathrm{th}}$ and different minimum and maximum threshold values. It appears that these details do not significantly affect the identification. The number of clusters identified depends mostly on the lowest threshold level. As we will describe later, the data revealed a large number of small, isolated, and faint clusters.

This method identified 8107 clusters that have at least 5 pixels on the sky. In the following we refer to these clusters as clouds. These clouds are composed of 89% of the Gaussian components ($5.8\times {10}^{4}$ Gaussian components are not associated with any cloud), which corresponds to 98% of the CO emission of the Dame et al. (2001) survey. The fact that the cluster identification includes almost the totality of the observed emission is illustrated in Figure 4, which shows the average T_B spectrum of the whole data set (black) and of the emission recovered in clouds (red).

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The histogram of the number of clouds per line of sight is shown in Figure 5. On 88% of the sky positions, there are four or fewer clouds on the line of sight; on 30% of the sky there is only one cloud on the line of sight. Roughly 11.5% of the surveyed sky was cloud-free using our criteria.

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For each cloud we define its central position as the intensity-weighted mean coordinate: the Galactic longitude, l, and latitude, b, are

$$\begin{eqnarray}&&l=\displaystyle \frac{{\displaystyle \sum }_{i}{W}_{\mathrm{CO}}^{i}\,{l}_{i}}{{W}_{\mathrm{CO}}^{\mathrm{tot}}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&b=\displaystyle \frac{{\displaystyle \sum }_{i}{W}_{\mathrm{CO}}^{i}\,{b}_{i}}{{W}_{\mathrm{CO}}^{\mathrm{tot}}}\end{eqnarray} \tag{ 7 }$$

where the sum is done over the ${N}_{\mathrm{comp}}$ Gaussian components, ${W}_{\mathrm{CO}}^{i}=\sqrt{2\pi }\,{A}_{i}\,{\sigma }_{i}\,{dv}$ is the integrated emission of a single Gaussian component, and ${W}_{\mathrm{CO}}^{\mathrm{tot}}={\sum }_{i}{W}_{\mathrm{CO}}^{i}$ the total integrated emission of all components.

Similarly, the standard deviations along l and b are

$$\begin{eqnarray}&&{\sigma }_{l}=\sqrt{\displaystyle \frac{{\displaystyle \sum }_{i}{W}_{\mathrm{CO}}^{i}\,{({l}_{i}-l)}^{2}}{{W}_{\mathrm{CO}}^{\mathrm{tot}}}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\sigma }_{b}=\sqrt{\displaystyle \frac{{\displaystyle \sum }_{i}{W}_{\mathrm{CO}}^{i}\,{({b}_{i}-b)}^{2}}{{W}_{\mathrm{CO}}^{\mathrm{tot}}}}.\end{eqnarray} \tag{ 9 }$$

These quantities provide an estimate of the angular extent of each cloud along the Galactic longitude and latitude directions.

For each cloud we also compute the angle θ with respect to the Galactic plane ($b=0^\circ$). We first compute the slope p of the linear regression between b_i and l_i, weighted by ${W}_{\mathrm{CO}}^{i}$. Then the angle is simply $\theta ={\tan }^{-1}(p)$.

For a given cloud, one can define the total CO spectrum by adding up all the Gaussian components:

$$\begin{eqnarray}&&{T}_{{\rm{B}}}^{\mathrm{tot}}{(v)=\displaystyle \sum _{i=1}^{{N}_{\mathrm{comp}}}{A}_{i}\exp ((v-{v}_{i})}^{2}/{2\sigma }_{i}^{2}).\end{eqnarray} \tag{ 10 }$$

Using this total cloud CO spectrum, we compute the emission-weighted mean velocity $\langle v\rangle$ and the velocity dispersion ${\sigma }_{v}$ of each cloud:

$$\begin{eqnarray}&&\langle v\rangle =\displaystyle \frac{1}{{W}_{\mathrm{CO}}^{\mathrm{tot}}}\displaystyle \sum _{v}v\,{T}_{{\rm{B}}}^{\mathrm{tot}}(v)\,{dv}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\sigma }_{v}^{2}=\displaystyle \frac{1}{{W}_{\mathrm{CO}}^{\mathrm{tot}}}\displaystyle \sum _{v}{v}^{2}\,{T}_{{\rm{B}}}^{\mathrm{tot}}(v)\,{dv}-\langle v{\rangle }^{2}\end{eqnarray} \tag{ 12 }$$

where dv is the channel width.

The average H₂ column density, ${N}_{{{\rm{H}}}_{2}}$, and surface density, Σ, are defined as

$$\begin{eqnarray}&&{N}_{{{\rm{H}}}_{2}}=\displaystyle \frac{{W}_{\mathrm{CO}}^{\mathrm{tot}}\,{X}_{\mathrm{CO}}}{{N}_{\mathrm{pix}}}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\rm{\Sigma }}=\displaystyle \frac{{N}_{{{\rm{H}}}_{2}}\,2\,\mu \,{m}_{{\rm{H}}}\,{\mathrm{pc}}^{2}}{{M}_{\odot }}\end{eqnarray} \tag{ 14 }$$

where ${m}_{{\rm{H}}}$ is the mass of the hydrogen atom, $\mu =1.36$ is to take into account the contribution of helium and metals, and ${M}_{\odot }$ is the mass of the Sun. Throughout this paper we use ${X}_{\mathrm{CO}}=2\times {10}^{20}$ cm⁻² (K km s⁻¹)⁻¹ (Bolatto et al. 2013).

The identification of clouds and their description as the sum of Gaussian components provide some statistics on the global velocity dispersion of each cloud (${\sigma }_{v}$), as well as on the width of each of the Gaussian components. The latter provides information on the line-of-sight velocity dispersion.

The histogram of the velocity dispersion ${\sigma }_{i}$ of the individual Gaussian components (Figure 6, red line) shows a median value of 1.65 km s⁻¹ but a most probable value of only 0.85 km s⁻¹. The histogram is positively skewed, with values as large as 10 km s⁻¹. These large values are likely to be due to the velocity crowding effect in some areas of the inner Galaxy. The histogram is relatively narrow, with FWHM = 2.0 km s⁻¹.

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Figure 6 also shows the histogram of ${\sigma }_{v}$ for all the clouds in the catalog (black line). On the scale of clouds, the median velocity dispersion is 3.6 km s⁻¹, with a most probable value of 1.95 km s⁻¹. The histogram is broader (FWHM = 4.5 km s⁻¹) than that of the Gaussian components. For both ${\sigma }_{i}$ and ${\sigma }_{v}$, the contribution of the instrumental broadening was removed quadratically.

The histogram of ${\sigma }_{v}$ is broader and peaks at a higher value than the histogram of the individual Gaussian components. This is due to the fact that ${\sigma }_{v}$ includes both the average line-of-sight velocity dispersion and the variation of the centroid velocity over the cloud. In contrast, the statistics of the Gaussian width ${\sigma }_{i}$ sample only part of the line-of-sight component, because the spectrum of a cloud, at a given position on the sky, might be composed of more than one Gaussian component. The ratio of the most probable values is about 2.2, as is the ratio of the median values.

The velocity dispersion of a given cloud is the quadratic sum of the turbulent and thermal broadening. Gaussian decomposition is able to partly separate these two contributions to the line broadening. In that context, the ratio of the most probable values of these two histograms gives a lower limit on the Mach number of molecular clouds ($\mathrm{Mach}\gt 2.2$).

The probability distributions of Σ, M, R, and ${n}_{{{\rm{H}}}_{2}}$ are shown in Figure 7. In each panel, the solid black line shows the full sample, while the orange and blue lines show, respectively, the clouds located in the inner and outer Galaxy.

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The distribution of the mass surface density (Figure 7, top left) shows a range of $2\,{M}_{\odot }\lesssim {\rm{\Sigma }}\lesssim 300\,{M}_{\odot }$ pc⁻². The mean value of Σ is 28.6 M_⊙ pc⁻², with a large standard deviation of $29.3\,$M_⊙ pc⁻² and a skewness of 1.5. The mass surface density of clouds is higher in the inner Galaxy, with a mean value of 41.9 M_⊙ pc⁻², compared to 10.4 M_⊙ pc⁻² in the outer Galaxy. All those values are tabulated in Table 2.

Table 2.  Typical Values of Cloud Parameters

Parameter	Units	All Clouds		Inner Galaxy		Outer Galaxy		${\alpha }_{\mathrm{vir}}\leqslant 3$
		Avg	Median	Avg	Median	Avg	Median	Avg	Median
Σ	${M}_{\odot }$ pc−2	28.6	16.5	41.9	31.6	10.4	7.0	46.1	37.1
M	104 M⊙	15.1	3.8	22.6	7.8	5.4	1.6	40.7	12.0
R	pc	31.5	25.1	30.8	25.2	32.9	24.9	37.8	29.5
${n}_{{{\rm{H}}}_{2}}$	cm−3	24.1	9.6	33.7	16.9	11.0	3.3	30.5	19.2
${\sigma }_{v}$	km s−1	4.0	3.6	4.9	4.6	2.8	2.5	3.1	2.7
${\sigma }_{0}$	km s−1	0.8	0.8	1.0	0.9	0.6	0.5	0.5	0.5
${\alpha }_{\mathrm{vir}}$	⋯	22.4	8.5	20.7	7.3	25.0	11.3	2.0	2.1
${\tau }_{\mathrm{dyn}}$	106 yr	8.5	6.4	6.2	5.3	12.3	9.9	14.6	10.3
${\tau }_{\mathrm{ff}}$	106 yr	13.1	10.1	9.4	7.6	19.0	17.2	9.5	7.1
${P}_{\mathrm{int}}$	104 K cm−3	10.9	3.0	18.7	8.5	1.7	0.6	9.8	4.0
${\dot{E}}_{\mathrm{dis}}$	${L}_{\mathrm{Sun}}$	146.9	9.9	285.1	51.5	10.5	1.7	205.8	15.3
$2\epsilon$	10−27 erg cm−3 s−1	503.5	59.4	878.6	187.4	68.5	7.1	191.7	49.4

Note. Average and median values of physical quantities for different subsets of the catalog: all clouds, clouds located at ${R}_{\mathrm{gal}}\leqslant 8.5\,\mathrm{kpc}$, clouds located at ${R}_{\mathrm{gal}}\gt 8.5\,\mathrm{kpc}$, and clouds with ${\alpha }_{\mathrm{vir}}\leqslant 3$.

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The difference in Σ between the inner and outer Galaxy could be partly attributed to a variation of the ${X}_{\mathrm{CO}}$ factor that we assumed constant. Because the metallicity decreases with ${R}_{\mathrm{gal}}$ (Balser et al. 2011), ${X}_{\mathrm{CO}}$ is expected to increase toward the outer Galaxy. Therefore, assuming a constant ${X}_{\mathrm{CO}}$ systematically underestimates the surface density and mass of clouds in the outer Galaxy. Heyer & Dame (2015) estimated that this effect could be as high as ∼50 %. This effect is potentially large, but it is still smaller than the variation of a factor of four observed here.

In general, the values of Σ obtained here are similar to those found by Heyer et al. (2009). Both our mean values and those of Heyer et al. (2009) are much smaller than those obtained by Solomon et al. (1987) and Roman-Duval et al. (2010). For example, Solomon et al. (1987) mention that their typical GMCs have ${\rm{\Sigma }}\sim 170\,{M}_{\odot }$ pc⁻². This difference might be due to the fact that our method includes the emission down to the sensitivity limit of the data. Using a threshold in brightness, like Solomon et al. (1987) did (they identified clouds with ${T}_{{\rm{B}}}\gt 4$ K), one identifies clouds that are smaller, and their average mass surface density is systematically larger, as all the faint emission around the periphery of the cloud is missed. Possibly for the same reason, the large range in Σ reported here contrasts with the general idea that GMCs have a nearly constant value of Σ (Larson 1981; Solomon et al. 1987).

In addition, because our technique places almost all of the CO emission into clouds, we identified a large number of faint and smaller clouds that were missed by previous methods. Our analysis confirms the findings of Digel et al. (1990) that clouds in the outer Galaxy have a lower Σ. This is also similar to what is seen in M51, where diffuse and fainter CO emission is detected far away from the main spiral arms (Koda et al. 2009; Pety et al. 2013).

The total mass in the catalog is $1.6\times {10}^{9}\,{M}_{\odot }$, corresponding to a total H₂ mass of $1.2\times {10}^{9}\,{M}_{\odot }$ (assuming $\mu =1.36$). This is in good agreement with the review of Heyer & Dame (2015), who estimated the total H₂ mass to be ($1.0\pm 0.3)\times {10}^{9}\,{M}_{\odot }$ based on a compilation of previous studies.

The mass of clouds ranges from a few ${M}_{\odot }$ to $2.2\times {10}^{7}\,{M}_{\odot }$. The top right panel of Figure 7 shows the distribution ${dN}/d\mathrm{log}M$ for the entire catalog, as well as for the inner and outer Galaxy. The median mass of the sample is $3.8\times {10}^{4}\,{M}_{\odot }$. This is comparable to the molecular mass of several clouds of the Gould Belt: Taurus (Pineda et al. 2010), Perseus (Lee et al. 2015), Ophiuchus (Loren 1989), and the Coalsack (Cambrésy 1999) all have masses in the range $(1\mbox{--}3)\times {10}^{4}\,{M}_{\odot }$.

Most of the clouds are small, but they collectively contain little mass; the ∼5000 clouds with $M\leqslant {10}^{5}\,{M}_{\odot }$ contribute less than 10% of the total molecular gas mass. The cumulative histogram of the cloud mass (Figure 8) indicates that half of the mass is in clouds with $M\gt 8.4\times {10}^{5}$ ${M}_{\odot };$ there are ∼460 such clouds. The higher-mass end of the distribution follows a power law: ${dN}/d\mathrm{ln}M\sim {M}^{-2.0\pm 0.1}$.

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There is a significant difference between the cloud mass distributions of the inner and outer Galaxy. About 60% of the clouds are located in the inner Galaxy, but they make up 85% of the total molecular mass of the disk. Clouds are indeed significantly more massive in the inner Galaxy: there the median mass is $7.8\times {10}^{4}\,{M}_{\odot }$, while it is only $1.6\times {10}^{4}\,{M}_{\odot }$ outside the solar circle. These values are summarized in Table 2.

These statistical properties are similar to what has been estimated with smaller samples and over a much smaller range of scales than we use here (Solomon et al. 1987; Williams & McKee 1997; Heithausen et al. 1998; Kramer et al. 1998; Heyer et al. 2001; Rosolowsky 2005; Roman-Duval et al. 2010).

The clouds in the catalog have sizes ranging from less than 1 pc to about 150 pc (see Figure 7, bottom left panel). The most probable value of R is ∼30 pc, typical for GMCs. We note that the maximum size is comparable to the scale height of the molecular disk.

Interestingly, the distribution of R is very similar in the outer and inner Galaxy. Even though clouds in the inner Galaxy are more massive, have a larger Σ, and are in a much more crowded region in PPV space than in the outer Galaxy, their size distributions are surprisingly similar.

We also want to point out that, unlike Σ, the determination of the typical size of a cloud is less sensitive to the brightness threshold used for cloud identification. If R is determined with a brightness-weighted method (see Section 3.2), the low-brightness outskirts of clouds do not contribute significantly to the size. This explains why the values of R reported here are similar to previous studies (e.g., Solomon et al. 1987) while there are significant differences in Σ.

The mean H₂ number density of the whole sample is $\langle {n}_{{{\rm{H}}}_{2}}\rangle =24.1$ cm⁻³. The mean values in the inner and outer Galaxy are 33.7 and 11.0 cm⁻³, respectively. Such low values of ${n}_{{{\rm{H}}}_{2}}$, well below the CO critical excitation density of the order of 10³ cm⁻³, have been reported early on (Blitz & Thaddeus 1980). They indicate that the filling factor of dense gas in molecular clouds is significantly below unity (Perault et al. 1985). In other words, the low density values measured here indicate that emitting gas does not fill the CfA survey beam. Indeed, higher-resolution observations tend to find higher values of ${n}_{{{\rm{H}}}_{2}}$ (Roman-Duval et al. 2010).

It is generally believed that a substantial fraction of the molecular material in the Milky Way is located in a ring in the range $3\,\mathrm{kpc}\lesssim {R}_{\mathrm{gal}}\lesssim 7\,\mathrm{kpc}$. This was established by looking at the radial variation of the mass surface density of H₂ gas (i.e., ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ versus ${R}_{\mathrm{gal}}$) and modeling the CO emission as coming from a disk of varying scale height and midplane density with ${R}_{\mathrm{gal}}$ (Bronfman et al. 1988; Nakanishi & Sofue 2006).⁶ Because the catalog presented here contains almost all the CO emission, it is possible for the first time to estimate ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ using a cloud-based approach, i.e., by integrating the mass of all the clouds in an annulus of constant ${R}_{\mathrm{gal}}$. Our results are shown in the top left panel of Figure 9. They are compatible with most previous studies, especially with the analysis of Nakanishi & Sofue (2006). Close to the Galactic center, there is a departure from the estimates of Bronfman et al. (1988); this departure is explained by the fact that those authors did not include data from $| l| \lt 12$ in their analysis. With the same restriction in longitude, our estimate of ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ (not shown) is compatible with that of Bronfman et al. (1988).

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For ${R}_{\mathrm{gal}}\gt 3\,\mathrm{kpc}$, we note that the surface density falls off exponentially with a scale length of 2.0 kpc. This result is significantly different from the scale length of 3.5 kpc found by Williams & McKee (1997). These authors based their analysis on the outer Galaxy results of Wouterloot et al. (1990) (green filled circles in Figure 9, top left panel), which are not compatible with our estimates or with those of Nakanishi & Sofue (2006) or Pohl et al. (2008) (see also Figure 7 of Heyer & Dame 2015).

The molecular ring is best seen in the top right panel of Figure 9, which shows the total mass in rings of thickness 0.5 kpc as a function of ${R}_{\mathrm{gal}}$. We note that the increase of ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and total mass toward the inner Galaxy is produced in part by an increase of the number of clouds, but more significantly by an increase of the average mass surface density of individual clouds (Figure 9, bottom right panel).

Finally, it is interesting to compare the H₂ and H i surface density as a function of ${R}_{\mathrm{gal}}$. In the top left panel of Figure 9 we also show the ${{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$ data points of Nakanishi & Sofue (2016), which are similar to the ones obtained by Koda et al. (2016). In the inner Galaxy the H i surface density rises from a value of about 1 M_⊙ pc⁻² to 10 M_⊙ pc⁻² at ${R}_{\mathrm{gal}}\sim 10\,\mathrm{kpc}$. At larger ${R}_{\mathrm{gal}}$, the H i surface density falls exponentially with a scale length of ${R}_{{\rm{H}}{\rm{I}}}\approx 3.75\,\mathrm{kpc}$ (Kalberla & Kerp 2009). Both Nakanishi & Sofue (2016) and Koda et al. (2016) showed that the fraction of the gas in molecular form, ${f}_{{{\rm{H}}}_{2}}={{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/({{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}+{{\rm{\Sigma }}}_{{{\rm{H}}}_{2}})$, declines steadily with ${R}_{\mathrm{gal}}$. This is also shown in the bottom left panel of Figure 9 using our estimate of ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$. In the solar neighborhood, the molecular fraction is about 10% only.

Looking at external galaxies like M51 (Koda et al. 2009; Pety et al. 2013), it is likely that the molecular gas in the Milky Way is not located in a ring, but rather follows the spiral arms (Dobbs & Burkert 2012). Do GMCs follow the spiral structure of the Galaxy? Answering this question is challenging, due to the fact that we are located inside the disk. Velocity crowding, the kinematic distance ambiguity, noncircular motions, and the definition of cloud frontiers all add to the difficulty of the task. Many attempts were made in the past, on portions of the disk (Clemens et al. 1988; Solomon & Rivolo 1989). Recently Rice et al. (2016) have shown that the brightest molecular clouds seem to be spatially correlated with the spiral structures of the Galaxy. On the other hand, Koda et al. (2016) showed that there is rather modest variation of the fraction of molecular to atomic gas (about 20%) between the arm and the interarm regions of the Milky Way. These authors noticed that the main change in gas phase occurs at ${R}_{\mathrm{gal}}\gt 6\,\mathrm{kpc}$, where the molecular fraction drops below 50% (see Figure 9, bottom left panel).

In Figures 10 and 26–28 we present face-on views of the Milky Way representing the distribution of Σ, M, ${n}_{{{\rm{H}}}_{2}}$, as well as the distance z to the midplane. In every diagram, each dot represents a single cloud. The size of the dot is proportional to $\mathrm{log}(R)$. The color scale indicates the value of each parameter. Overlaid on these figures is the spiral arm model of Vallée (2008) adjusted to the Sun–Galactic center distance ${R}_{\mathrm{Sun}}=8.5\,\mathrm{kpc}$.

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The spatial distribution of Σ (Figure 10) is probably the most useful one to determine what the Milky Way might look like seen from the outside. Because Σ seems to be unrelated to the cloud physical size (see Section 4.3), it does not suffer as much as M and ${n}_{{{\rm{H}}}_{2}}$ do from a distance bias (see Appendix C). This map shows an increase in cloud surface density close to the four spiral arms, although the association is not clear everywhere. A significant fraction of the clouds are located between the arms. Like Rice et al. (2016), we notice a lack of clouds in the Perseus arm close to the Sun (for $l\gt 90^\circ$).

Unsurprisingly, the face-on view of M (Figure 26) indicates that more distant clouds are generally bigger and more massive than more nearby clouds. This is due to the limit imposed by the finite angular resolution and sensitivity of the observations. Naturally, the survey detects larger and more massive clouds with increasing distance (Heyer et al. 2001). The dependence on D of R and M and the detection limit of the catalog are detailed in Figure 24 in Appendix C. This effect does not imply that the catalog misses a large fraction of the emission from regions that are farther away, but rather that it does not have the ability to resolve faraway complexes into their substructures. Molecular clouds have a fractal structure and are not isolated in PPV space. This is clearly seen in CO observations of external galaxies (Koda et al. 2009; Pety et al. 2013). Therefore, in the Milky Way, closer molecular complexes are easily resolved into smaller pieces, while similar complexes that are far away are seen as a single entity. Unfortunately, this is inherent to the fact that we look at the Milky Way from the inside. This has to be kept in mind when looking at distance-dependent statistical properties of clouds.

Compared to the mass (∝D²), the density is less biased by distance (∝D⁻¹). It is true that closer clouds are generally smaller and tend to have a higher density (the distance bias), but the face-on view of ${n}_{{{\rm{H}}}_{2}}$ (Figure 27) also shows a general increase of the average density inside the solar circle. In the outer Galaxy, clouds are about 10 times less dense than in the spiral arms. We notice that the Scutum-Centaurus arm (at $X\approx 0\,\mathrm{kpc}$, $Y\approx 5\,\mathrm{kpc}$), where a significant fraction of the star-forming activity of the Milky Way takes place, shows a clear increase in gas density relative to other regions.

Finally, Figure 28 in Appendix E shows the distance to the midplane for each cloud. This diagram provides an interesting view of the Galactic warp seen in molecular emission. In the inner Galaxy, the molecular gas is concentrated in a layer close to z = 0, with a scale height of the order of 100 pc. On the other hand, in the outer Galaxy, the molecular layer is clearly flaring and is warped. Toward $l=90^\circ$ the molecular gas is above the midplane, while it is below toward $l=270^\circ$. This has been identified by many studies since the 1980s (see Figure 6 of Heyer & Dame 2015), but this figure is the first time that it has been shown over the whole plane with a single coherent sample of clouds.

Close to the Sun, most of the interstellar matter is seen at higher Galactic latitude. For instance, all the Gould Belt clouds are at intermediate latitudes ($| b| \sim 10^\circ \mbox{--}20^\circ$; see Perrot & Grenier 2003). Even though the current study concentrates only on CO emission in a very limited range in latitude ($-5^\circ \lt b\lt 5^\circ$), we notice that the global morphology of the solar neighborhood is reproduced. Figure 11 shows a $1\times 1\,\mathrm{kpc}$ face-on view of the clouds' surface density, centered on the Sun. Also shown are the isocontours of E(B − V)/pc of Lallement et al. (2014) obtained by an inversion of stellar color excess measurements obtained in the optical. This method uses parallax or photometric estimates of distance, providing a much more precise determination of the 3D structure of the ISM close to the Sun. The work of Lallement et al. (2014) clearly shows the position of nearby clouds with respect to the Sun. Several of these clouds are in the range $-30^\circ \lt l\lt 75^\circ$ and at a distance of 100–300 pc. They are part of the Gould Belt. These correspond to the Ophiuchus, Lupus, and Chamaeleon-Musca clouds, all located at $| b| \gt 5^\circ$. Therefore, they are not present in our catalog. On the other hand, complexes located at lower latitude seem to show a rather good correspondence with the clouds found here. This is the case for Vela, Cygnus, and Cassiopeia. Even part of the Acquila, Perseus, and Orion complexes are captured. One very striking feature of the solar neighborhood is the Local Bubble, a vast region with almost no dense gas that extends toward $l\sim 240^\circ$, corresponding to a diagonal to the upper left in Figure 11. This is very well captured by the current catalog.

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For every cloud in the sample, the internal (turbulent) pressure, ${P}_{\mathrm{int}}$ (in units of K cm⁻³), is estimated as

$$\begin{eqnarray}&&{P}_{\mathrm{int}}=\displaystyle \frac{2\,{m}_{p}}{k}\,{n}_{{{\rm{H}}}_{2}}\,{\sigma }_{v}^{2},\end{eqnarray} \tag{ 27 }$$

where k is the Boltzmann constant. The histogram of ${P}_{\mathrm{int}}$ (Figure 15, top panel) illustrates the large range of pressure found: over more than 3 orders of magnitude. The median value is ${P}_{\mathrm{int}}=3.0\times {10}^{4}$ K cm⁻³, but with a large standard deviation of $5.0\times {10}^{4}$ K cm⁻³.

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From Equation (27) we note that the total pressure scales like ${\rm{\Sigma }}/R\times {\sigma }_{v}^{2}$. As ${\sigma }_{v}^{2}\propto {({\rm{\Sigma }}R)}^{0.86}$, the total pressure scales almost like ${{\rm{\Sigma }}}^{2}$ and is thus very weakly dependent on distance. Therefore, as for Σ, the clear difference in ${P}_{\mathrm{int}}$ between the inner and outer Galaxy (see orange and blue histograms in Figure 15) is physical and not an observational bias. This is also clearly seen in the bottom panel of Figure 15, where the variation of ${P}_{\mathrm{int}}$ with ${R}_{\mathrm{gal}}$ appears similar to the variation of Σ with ${R}_{\mathrm{gal}}$ (Figure 9). The similarity between the spatial distribution of ${P}_{\mathrm{int}}$ and Σ is also seen in the face-on views (Figures 10 and 29).

As has been noticed before (e.g., Blitz 1993), the typical internal pressure of molecular clouds is much larger than the typical pressure of the diffuse H i, ${P}_{\mathrm{int}}=3.7\times {10}^{3}$ K cm⁻³ in the solar neighborhood according to Jenkins & Tripp (2011). These authors found excursions in H i pressure from 10³ to 10⁴ K cm⁻³, which is still less than the typical values found here.

As mentioned by Heyer et al. (2009), a ${\sigma }_{v}\mbox{--}({\rm{\Sigma }}R)$ relationship is expected if gravity is a dominant process in the dynamics. Specifically, one expects ${\sigma }_{v}\propto {({\rm{\Sigma }}R)}^{1/2}$ for clouds in virial equilibirum, i.e., ${\alpha }_{\mathrm{vir}}\approx 1\mbox{--}3$, where the virial parameter is defined as

$$\begin{eqnarray}&&{\alpha }_{\mathrm{vir}}=\displaystyle \frac{5\,{\sigma }_{v}^{2}\,R}{G\,M}=a\displaystyle \frac{2{ \mathcal T }}{| W| },\end{eqnarray} \tag{ 28 }$$

where ${ \mathcal T }$ is the total kinetic energy of the cloud and W is the gravitational energy associated with the cloud (ignoring tidal effects; Bertoldi & McKee 1992). We calculate the factor a in Appendix D, where we show that for fiducial values for the power-law index for the density $\rho \sim {r}^{-{k}_{\rho }}$ (${k}_{\rho }=1$) and for the index in Larson's size–line width relation $\sigma (r)\sim {r}^{p}$, with $p=1/2$, $a\approx 5/3$. In other words, the observed virial parameter ${\alpha }_{\mathrm{vir}}$ will be a factor of 5/3 larger than the ratio of twice the kinetic energy to the gravitational energy. If the latter is of order 2, meaning equal kinetic and gravitational energies, the observed virial parameter will be ${\alpha }_{\mathrm{vir}}\approx 3$.

The histogram of ${\alpha }_{\mathrm{vir}}$ for the whole sample of clouds, as well as for the inner- and outer Galaxy clouds, is shown in Figure 16 (top panel). The distribution peaks at ${\alpha }_{\mathrm{vir}}$ of around 3–4, where the kinetic energy of the cloud is about equal to the gravitational energy and is strongly positively skewed. Clouds in the outer Galaxy are more likely to have ${\alpha }_{\mathrm{vir}}\gg 1$, something already noted by many studies (Sodroski 1991; Heyer et al. 2001). The Galactic profile (Figure 16, bottom panel) and the face-on view (Figure 31) of ${\alpha }_{\mathrm{vir}}$ indicate a slight decrease in the $3\,\mathrm{kpc}\lt {R}_{\mathrm{gal}}\lt 7\,\mathrm{kpc}$ region. We also note an increase of ${\alpha }_{\mathrm{vir}}$ in the inner part of the Galaxy (${R}_{\mathrm{gal}}\lt 3$ kpc), as well as an unexpected variation of ${\alpha }_{\mathrm{vir}}$ in the outer Galaxy (see Figure 31). The virial parameter is systematically larger in the third quadrant ($180^\circ \lt l\lt 270^\circ$) compared to the second quadrant ($90^\circ \lt l\lt 180^\circ$). Like ${P}_{\mathrm{int}}$, the virial parameter is very weakly dependent on distance.

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Figure 17 shows the two-dimensional histogram of ${\alpha }_{\mathrm{vir}}$ versus M. The correlation between these two quantities is modest (Pearson coefficient of −0.54). Overall, only 15% of the clouds have ${\alpha }_{\mathrm{vir}}\leqslant 3$, but they contribute 40% of the total mass in clouds. Assuming that ${\alpha }_{\mathrm{vir}}=2.3\times {10}^{3}\,{M}^{-0.53\pm 0.30}$ (see Figure 17), one could conclude that clouds with $M\gt 2.8\times {10}^{5}\,{M}_{\odot }$ are gravitionally bound. On the other hand, looking in detail at Figure 17, there are more unbound clouds with $M\gt 2.8\times {10}^{5}\,{M}_{\odot }$ than bound clouds. Therefore, even though more massive clouds tend to have a lower virial parameter, it seems difficult to conclude that mass is the main criterion that defines the gravitational stability state of a given cloud.

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These results are commensurate with previous estimates and with the idea that unbound (high ${\alpha }_{\mathrm{vir}}$) clouds are less efficient at forming stars: Liszt et al. (2010) argue that about 40% of the Galactic molecular gas is not forming stars, while Goldsmith et al. (2008) reached the same conclusion specifically for the Taurus molecular clouds.

The comparison of the free-fall time

$$\begin{eqnarray}&&{\tau }_{\mathrm{ff}}={\left(\displaystyle \frac{3\pi }{32G\rho }\right)}^{1/2}\end{eqnarray} \tag{ 29 }$$

where $\rho =\mu 2{m}_{p}{n}_{{{\rm{H}}}_{2}}$, and the dynamical time (or crossing time)

$$\begin{eqnarray}&&{\tau }_{\mathrm{dyn}}=\displaystyle \frac{R}{{\sigma }_{v}}\end{eqnarray} \tag{ 30 }$$

provides some insight into the physical processes involved in the efficiency with which molecular gas turns into stars.

The histograms of ${\tau }_{\mathrm{ff}}$ and ${\tau }_{\mathrm{dyn}}$ are shown in Figure 18. They both have a relatively narrow distribution. The median values are ${\tau }_{\mathrm{ff}}=1.0\times {10}^{7}\,\mathrm{yr}$ and ${\tau }_{\mathrm{dyn}}=6.4\times {10}^{6}\,\mathrm{yr}$. To first order, the free-fall time is about 1.5 times the value of the dynamical time. Both ${\tau }_{\mathrm{ff}}$ and ${\tau }_{\mathrm{dyn}}$ increase from the inner to the outer Galaxy, but their ratio, like ${\alpha }_{\mathrm{vir}}$, stays about the same.

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The ratio of the total molecular mass of the Milky Way to the typical free-fall time ($1.6\times {10}^{9}$ ${M}_{\odot }$/$1.0\times {10}^{7}\,\mathrm{yr}=160\,{M}_{\odot }$ yr⁻¹) provides an estimate of what the star formation rate would be if all molecular clouds were to form stars and if gravity alone were to drive the formation process. A finer estimate of this quantity can be obtained by summing $M/{\tau }_{\mathrm{ff}}$ for all the clouds in the sample. It gives a similar value: 215 M_⊙ yr⁻¹. This has to be compared with the actual star formation rate of the Milky Way, which is ∼2 M_⊙ yr⁻¹, indicating that the star formation efficiency is of the order of 1% on average.

In order to evaluate what processes drive turbulence in molecular clouds and regulate star formation in galaxies, it is essential to quantify the properties of turbulence. One important parameter is the rate at which energy is injected and dissipated in the turbulent cascade and the rate at which it transfers from large scales to small scales.

In a three-dimensional turbulent flow, the kinetic energy injected at large scales is transferred to smaller scales through nonlinear interactions, down to the dissipation scale, where it is converted into heat. In an incompressible turbulent flow, Kolmogorov (1941) showed that the energy tranfer rate per unit mass, ${v}^{3}/l$, is conserved. This means that, for a given turbulent flow with some energy injected at large scales, the energy transfer rate is constant whatever the scale at which it is estimated. Later, Lighthill (1955, p. 121) pointed out that, in a compressible turbulent flow, the quantity that is conserved is the mean volume energy transfer rate defined as

$$\begin{eqnarray}&&2\epsilon =\displaystyle \frac{\rho \,{\sigma }_{v}^{3}}{R}.\end{eqnarray} \tag{ 31 }$$

That led Kritsuk et al. (2007) to propose an extension of Kolmogorov's theory (Kolmogorov 1941) to compressible flow, which applies to molecular clouds. They showed that the quantity $u={\rho }^{1/3}v$ has the same statistical properties as the velocity field v in incompressible turbulence, whatever the Mach number of the flow.

It has been seen in several numerical simulations that the turbulent energy decays in one turnover time if it is not maintained (see the review of Hennebelle & Falgarone 2012). Following Mac Low (1999), the energy dissipation rate can be estimated as the total kinetic energy divided by the dynamical time:

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{dis}}=-\displaystyle \frac{1}{2}\,\displaystyle \frac{M\,{\sigma }_{v}^{3}}{R},\end{eqnarray} \tag{ 32 }$$

which is simply the kinetic energy transfer rate integrated over the whole volume of a cloud.

We computed and ${\dot{E}}_{\mathrm{dis}}$ for each cloud in the catalog.

The histogram of ${\dot{E}}_{\mathrm{dis}}$ and its variation with ${R}_{\mathrm{gal}}$ are shown in Figure 19. The face-on view of ${\dot{E}}_{\mathrm{dis}}$ is also shown in Figure 32. The variation of the median value of ${\dot{E}}_{\mathrm{dis}}$ as a function of ${R}_{\mathrm{gal}}$ is quasi-bimodal, with ${\dot{E}}_{\mathrm{dis}}\sim 2\,{L}_{\odot }$ for ${R}_{\mathrm{gal}}\gt 7\,\mathrm{kpc}$ and ${\dot{E}}_{\mathrm{dis}}\sim 100\,{L}_{\odot }$ at smaller ${R}_{\mathrm{gal}}$.

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The histogram of $2\epsilon$ (Figure 20) shows a significant spread in values, of about 4 orders of magnitude, similar to the one of ${\dot{E}}_{\mathrm{dis}}$. There is a clear difference in the value of between the inner and outer Galaxy; it drops steadily for ${R}_{\mathrm{gal}}\gt 5\,\mathrm{kpc}$, with an exponential scale length similar to what we found for the mass surface density Σ (Figure 9). For ${R}_{\mathrm{gal}}\lt 9\,\mathrm{kpc}$, the value of $2\epsilon \sim {10}^{-25}$ erg cm⁻³ s⁻¹ is similar to previous estimates obtained for molecular clouds and H i clouds in the solar neighborhood (Hennebelle & Falgarone 2012).

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We observed no variation of with GMC radius R, which is not inconsistent with the presence of a turbulent cascade. However, we are unable to evaluate whether is constant through scales in individual clouds, as was done in Hennebelle & Falgarone (2012). As mentioned in Section 5.2, we are only sampling the large-scale properties of compressible turbulence for isolated clouds, not their multiscale nature.

Finally, we note that ${\dot{E}}_{\mathrm{dis}}$ and depend significantly on distance with ${\dot{E}}_{\mathrm{dis}}\propto {D}^{2.3}$ and $\epsilon \propto {D}^{-0.7}$. One could worry that this would affect the radial distributions shown in Figures 19 and 20. The fact that these two quantities depend in an opposite manner on D (${\dot{E}}_{\mathrm{dis}}$ tends to increase with D, while decreases) provides some guidelines to evaluate the impact of the distance bias on the trends seen here. The 2 orders of magnitude difference between the inner and outer Galaxy, seen for both quantities, cannot be the result of an observational bias. On the other hand, the exact shape of the variations in the range $5\,\mathrm{kpc}\lt {R}_{\mathrm{gal}}\lt 15\,\mathrm{kpc}$ could be partly due to the different distance bias of ${\dot{E}}_{\mathrm{dis}}$ and .

One might look more globally at the problem of turbulence driving. Throughout the course of its evolution, a galaxy loses energy, by radiation, and gas, by the star formation process and through galactic scale outflows. Without mass accretion from the circumgalactic medium, the intergalactic medium, or both, the star formation rate and the level of turbulent energy would rapidly die off.

Klessen & Hennebelle (2010) suggested that gas accretion is the main process that maintains turbulence at the scale of a galaxy, of a molecular cloud, and of a protoplanetary disk. They argue that what maintains the star formation activity and the level of turbulence in galaxies is the accretion of matter from the intergalactic medium. At the scale of clouds, they proposed that turbulence could be driven by gas accretion due to large-scale motions in the disk.

According to Klessen & Hennebelle (2010), the turbulent motions in molecular clouds are inherited from the cloud formation process itself. Dense clouds in galaxies are formed in regions of higher pressure where the gas can cool efficiently (Saury et al. 2014). These regions will then become molecular and gravitationally unstable. Converging flows, possibly caused by perturbation of the gravitational potential (e.g., spiral density waves) or supernova explosions, are one likely source of pressure increase. In this scenario the turbulent energy is provided by the continuous accretion onto GMCs induced by the large-scale converging flow.

Following Klessen & Hennebelle (2010), the energy injection by mass accretion is

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{inj}}=\displaystyle \frac{1}{2}\,{\dot{M}}_{\mathrm{acc}}\,{v}_{\inf }^{2}\end{eqnarray} \tag{ 33 }$$

where ${v}_{\inf }$ is the infall velocity and

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{acc}}=4\pi \,{R}^{2}\,{v}_{\inf }\,{\rho }_{\mathrm{ISM}}\end{eqnarray} \tag{ 34 }$$

is the mass accretion rate of diffuse matter of mass density ${\rho }_{\mathrm{ISM}}$ onto a cloud of radius R. Assuming a mean ISM volume number density of ${n}_{\mathrm{ISM}}=1$ cm⁻³ and a typical molecular cloud radius of R = 30 pc and scaling to an infall (or converging flow) velocity ${v}_{\inf }=10$ km s⁻¹ produces an energy injection rate of ${\dot{E}}_{\mathrm{inj}}\approx 32\,{L}_{\odot }$. We have chosen an inflow velocity corresponding to the sound speed of ionized gas, for reasons that will become apparent.

To see whether this process provides enough energy to maintain turbulence, it has to be compared with the turbulent energy dissipation rate. In the outer Galaxy, our results are compatible with a constant value of ${\dot{E}}_{\mathrm{dis}}\approx 2\,$L_⊙ (see Figure 19). This is a factor of 15 lower than the input energy by converging flows of 10 km s⁻¹. The energy conversion efficiency from converging flow to turbulent motion of dense clouds has been estimated to be in the range of 0.01–0.1 (Klessen & Hennebelle 2010). We conclude that the turbulent motions of non-star-forming molecular clouds in the outer disk of the Milky Way can be explained by the general process of cloud formation and mass accretion by converging flows of H i gas.

Of course, there must be a source of free energy to drive the gas inflow onto the cloud. We note that in the outer Galaxy the virial parameter of the clouds is ∼10, so that the potential energy of the GMC is small compared to the kinetic energy of the turbulence, i.e., the gravitational potential energy of the GMC is not a significant source of energy. This is not true in the inner Galaxy, a point we will return to. Since there is very little star formation in the outer Galaxy, we are left with two sources of free energy, accretion onto the disk from the circumgalactic or intergalactic medium, or accretion through the disk, whether driven by the magnetorotational instability or gravity-driven instability, e.g., via spiral arms (which are seen in the outer disk). The gas depletion time of the Milky way is of order 1 Gyr, much shorter than the age of the Galaxy, suggesting that accretion either onto or through the disk, or both, is ongoing; assuming that the situation is at least near an equilibrium, we scale the accretion to the star formation rate, which is of order $1-2\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. The potential energy released per second by the accretion is roughly

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{acc}}\approx {\dot{M}}_{\mathrm{acc}}{v}_{c}^{2}\\ &&\,\approx 2.9\times {10}^{40}\left(\displaystyle \frac{{\dot{M}}_{\mathrm{acc}}}{1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right){\left(\displaystyle \frac{{v}_{c}}{220\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 35 }$$

This should be compared to the turbulent luminosity of the disk, roughly

$$\begin{eqnarray}{\dot{E}}_{\mathrm{dis},\mathrm{disk}} & = & \displaystyle \frac{1}{2}{M}_{g}{\sigma }_{v}^{3}/H\approx {10}^{39}\left(\displaystyle \frac{{M}_{g}}{2\times {10}^{9}\,{M}_{\odot }}\right)\\ & & \times {\left(\displaystyle \frac{{\sigma }_{v}}{5\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{3}{\left(\displaystyle \frac{H}{100\mathrm{pc}}\right)}^{-1}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 36 }$$

We conclude that accretion energy is sufficient to drive the turbulence in the disk and that accretion of gas onto GMCs in the outer disk is sufficient to drive the turbulence in individual clouds outside the solar radius.

The situation in the inner Galaxy is very different, however. The observed GMC turbulent energy dissipation rate in the inner Galaxy is a factor of 100 larger than that in the outer Galaxy. The mean density of the atomic/ionized ISM is similar in both locations, as are the mean GMC radii, and we expect that the typical infall velocities at large distances from the GMC are similar as well. Thus, it would appear that a new source of turbulent energy is called for.

The virial parameters of clouds inside the solar circle are noticeably smaller than those of clouds in the outer Galaxy; see the bottom panel of Figure 16. In other words, the potential energy in inner Galaxy GMCs is comparable to the kinetic energy, in contrast to the GMCs in the outer Galaxy. This suggests that contraction of these GMCs may power the higher levels of turbulence we see.

This does not explain why GMCs in the inner Galaxy are more tightly bound than those in the outer Galaxy.

More significantly, the fact that the GMCs have low virial parameters and are contracting raises the question, what halts the contraction? It is clearly not the turbulence.

Turbulence in molecular clouds that are not forming stars is comparable to that in more actively star-forming clouds. Williams et al. (1994) noticed that two molecular clouds, one actively forming stars while the other is not, have very similar size–line width relations and clump mass spectra. This is also found in the LMC by Kawamura et al. (2009). In addition, statistical studies of the velocity field of individual molecular clouds indicate no characteristic scale. Both points support the idea that the turbulent energy injection occurs at scales larger than the size of clouds (Ossenkopf & Mac Low 2002; Brunt et al. 2009), and in particular that feedback does not strongly and directly affect the turbulence at scales below the GMC scale.

To take a specific example, jets from protostars can drive turbulence in molecular gas; Matzner (2007) estimates the total impulse produced per stellar mass formed of ${v}_{* }\approx 40\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Using this value, one finds that $\sim {10}^{4}\,{M}_{\odot }$ in young (jet-emitting) stellar objects are needed to power the turbulence we find in an inner Galaxy GMC. In Lee et al. (2016) we show that only about 50 of the ∼500 GMCs with $M\gt {10}^{6}\,{M}_{\odot }$ have star clusters this massive, assuming a normal initial mass function. In other words, neither the energetics nor (the lack of) a feature in the turbulent spectrum on the scale of jets is compatible with protostellar jets being the source of turbulence in inner Galaxy GMCs.

This does not mean that other forms of stellar feedback have no role in the generation of turbulence. For example, O star winds, H ii regions, and radiation pressure produce momentum outputs on much larger scales than do protostellar jets, as do supernovae. Estimates for the first three forms of feedback give ${v}_{* }\approx 300\,\mathrm{km}\,{{\rm{s}}}^{-1}$, while supernovae provide ${v}_{* }\approx 3000\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (e.g., Ostriker & Shetty 2011). Thus, the problem with the energetics of jet-driven turbulence might not apply with these other forms of feedback. But while each form of feedback may produce large-scale turbulence, they all act initially on small scales, so that we might expect to see many GMCs with excess turbulence on small scales; such features are not common in any data set we are aware of. Thus, it appears unlikely that stellar feedback is directly responsible for the high GMC dissipation rates we find in the inner Galaxy.

However, the normalization of the line width, ${\sigma }_{0}={\sigma }_{v}/{R}^{1/2}$, increases by a factor of two from the outer Galaxy to the inner Galaxy (Figure 13). The variation in the GMC turbulent dissipation rate is far more dramatic; it is a factor of 100 larger in the inner Galaxy, where star formation is active (Figure 19). This implies that the turbulent energy injection rate is also 100 times higher than the injection rate per cloud in the outer Galaxy.

This increase can be explained by contraction of inner Galaxy GMCs (e.g., Ballesteros-Paredes et al. 2011; Ibanez-Mejia et al. 2016); this is the solution we advocate here. However, contraction of the GMC cannot be the entire solution, since continued contraction would lead to very dense massive GMCs, which do not exist. Something must halt, or rather completely reverse, the contraction.

We suggest that some combination of the four feedback processes described above reverses the contraction of GMCs in the inner Galaxy. This reversal must occur over about a GMC dynamical time, since we do not see many GMCs that are clearly in the act of dispersing. This input of turbulent energy on the GMC size scale would necessarily have to equal, or more likely exceed, the amount of turbulent energy dissipated during the earlier collapse.

In this scenario, the contraction of GMCs is the proximate or direct driver of the large turbulent dissipation rates we see, but the ultimate source of the turbulent energy is the burning of nuclear fuel in massive stars. Some fraction of the nuclear energy released unbinds the GMC hosting the massive stars, restoring the gravitation potential energy lost in the contraction, and more. This potential energy is then available to power the turbulence in the next generation of GMCs.

If the increase of turbulent energy injection were due to stellar feedback, we could expect to see an associated increase of the velocity dispersion of the gas, as indeed we do. We note, however, that it is likely that only a fraction of the energy released by stellar feedback goes into random motions. For example, shocks created by supernovae would inject energy more efficiently into compressive than solenoidal modes; the interstellar gas gets more compressed than agitated. Indeed, like the turbulent energy dissipation rate, the internal GMC pressure in the inner Galaxy is about 2 orders of magnitude larger than in the outer Galaxy.