PHYSICAL PROPERTIES AND GALACTIC DISTRIBUTION OF MOLECULAR CLOUDS IDENTIFIED IN THE GALACTIC RING SURVEY

The sample of molecular clouds used in this analysis was identified by their ¹³CO J = 1 → 0 emission in the BU-FCRAO GRS (Jackson et al. 2006). The GRS was conducted using the FCRAO 14 m telescope in New Salem, Massachusetts between 1998 and 2005. The survey, which used the SEQUOIA multipixel array, covers the range of Galactic longitude 18° ⩽ ℓ ⩽ 557 and Galactic latitude −1° ⩽ b ⩽ 1°. The survey achieved a spatial resolution of 46'', sampled on a 22''grid, and a spectral resolution of 0.212 km s⁻¹ for a noise variance σ (T*_A) = 0.13 K ($\sigma _{T_{{\rm mb}}} = 0.26$ 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 ⩽ 40° and −5 to 85 km s⁻¹ for Galactic longitudes ℓ ⩾ 40°.

Because the GRS uses ¹³CO, which has an optical depth 50 times lower than that of ¹²CO, it allows for a better detection and separation of molecular clouds both spatially and spectrally than previous ¹²CO surveys. Using the algorithm CLUMPFIND (Williams et al. 1994) applied to the GRS data smoothed to 01 spatially and to 0.6 km s⁻¹ spectrally, 829 molecular clouds were identified by Rathborne et al. (2009). CLUMPFIND identifies as molecular clouds a set of contiguous voxels (i.e., (ℓ, b, v) positions) with intensity values higher than a given threshold, which has to be determined empirically so as to best identify molecular clouds at all levels of emission. We refer the reader to Rathborne et al. (2009) for the details of the identification procedure. Molecular cloud parameters such as Galactic longitude, latitude, and velocity of the ¹³CO emission peak were estimated by CLUMPFIND. Individual ¹³CO data cubes extracted from the GRS, and covering only the Galactic longitude, latitude, and velocity range of each individual molecular cloud were also created by Rathborne et al. (2009). We use kinematic distances to 750 out of the 829 GRS molecular clouds derived in Roman-Duval et al. (2009) using the Clemens (1985) rotation curve and H i self-absorption to resolve the kinematic distance ambiguity.

We make use of the ¹²CO UMSB survey (Sanders et al. 1986; Clemens 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 a spatial resolution of 44''. The sensitivity of the observations is 0.4 K per velocity channel. The UMSB survey only covers the velocity range −10 km s⁻¹ <V_LSR< 90 km s⁻¹, which excludes the GRS molecular clouds located at low longitude and at the tangent point.

The ¹²CO J = 1 → 0 excitation temperature of our sample of 580 molecular clouds was derived from the ¹²CO brightness temperature, with the assumption that the ¹²CO line is optically thick (i.e., τ (¹²CO) ≫ 1). In the optically thick regime, the observed ¹²CO brightness temperature T₁₂ and the excitation temperature T_ex are related by Rohlfs & Wilson (2004):

$$\begin{equation} T_{{\rm ex}} = 5.53 \frac{1}{{\rm ln} \left(1+\frac{5.53}{T_{12}+0.837} \right)}, \end{equation} \tag{ 1 }$$

where both temperatures are in K. This equation includes a background subtraction accounting for the cosmic microwave background at T = 2.73 K. The excitation temperature was calculated at each voxel (i.e., (ℓ, b, v) position) associated with each molecular cloud in the UMSB data resampled to the GRS grid. The ¹²CO excitation temperature was used as a proxy for the ¹³CO excitation temperature. The ¹³CO and ¹²CO excitation temperature should be identical in local thermal equilibrium since the energy levels of the two isotopomers are roughly the same. In reality, density gradients in molecular clouds and non-local thermal equilibrium situations can lead to differences in the excitation temperatures of ¹³CO and ¹²CO. These effects are described in Section 9.3. The mean excitation temperature of each molecular cloud was computed by averaging the excitation temperature over all the voxels that contain a ¹³CO brightness temperature greater than 4σ or, accounting for the main beam efficiency, a minimum brightness temperature T₁₃ > 1 K.

The ¹³CO optical depth was derived using (1) the ¹²CO excitation temperature as a proxy for the ¹³CO excitation temperature, and (2) the GRS ¹³CO data. Since the UMSB and GRS data have nearly the same beam width, corrections to account for different beam dilution between the two surveys can be neglected. The ¹³CO optical depth is thus given by (Rohlfs & Wilson 2004)

$$\begin{equation} \tau _{13} (\ell, b, v) = - {\rm ln} \left(1 - \frac{0.189 \: T_{13}(\ell, b, v)}{\big(e^{\frac{5.3}{T_{{\rm ex}}(\ell, b, v)}} -1 \big)^{-1}-0.16 } \right), \end{equation} \tag{ 2 }$$

where τ₁₃ is the ¹³CO optical depth and T₁₃ is the background-subtracted ¹³CO brightness temperature in K. The ¹³CO optical depth is evaluated at each GRS voxel (ℓ, b, v) associated with each molecular cloud (i.e., in the individual data cubes associated with each molecular cloud). The line center optical depth was computed at each pixel (i.e., position (ℓ, b) on the sky) where the ¹³CO integrated intensity is greater than 4σ = 0.23$\sqrt{N_v}$ K km s⁻¹, where N_v is the number of channel in the cube. We also computed the average of the line center optical depth over all GRS pixels associated with each cloud where the ¹³CO integrated intensity is greater than 4σ.

From the optical depths and excitation temperatures, ¹³CO column densities were derived at each GRS pixel (ℓ, b) associated with each molecular cloud (Rohlfs & Wilson 2004):

$$\begin{equation} \frac{N(^{13}\mbox{CO}) (\ell, b)}{\mathrm{cm}^{-2}} = 2.6 \times 10^{14}\: \int {\frac{T_{{\rm ex}} (\ell, b, v) \:\tau _{13}(v)}{1-e^{\frac{-5.3}{T_{{\rm ex}} (\ell, b, v)}}} \frac{dv}{\mathrm{km} \: \mathrm{s}^{-1}}}. \end{equation} \tag{ 3 }$$

¹³CO column densities are computed by integrating Equation (3) only over GRS voxels associated with each cloud where the ¹³CO brightness temperature is greater than 4σ = 1 K.

The solid angle subtended by each GRS molecular cloud is computed by counting the number N_pix of GRS pixels (i.e., (ℓ, b) positions on the sky) associated with each molecular cloud that contain a ¹³CO integrated intensity greater than 4σ = 0.23$\sqrt{N_v}$ K km s⁻¹:

$$\begin{equation} \Omega = N_{\rm pix} \Delta l \Delta b, \end{equation} \tag{ 4 }$$

where Δℓ and Δb are the angular width of the pixels. Knowing the distance d to the molecular clouds, the area A and linear equivalent radius R of the molecular clouds are given by

$$\begin{eqnarray} A &=& \Omega d^2, \\ \ms R &=& \sqrt{\frac{A}{\pi }}. \end{eqnarray} \tag{ 5 }$$

The one-dimensional, intensity-weighted, velocity dispersion of a molecular cloud is defined as

$$\begin{equation} \sigma _{v_{{\rm 1D}}}^2 = \frac{\sum _{T_{13}(\ell, b, v) > 1{\rm K}}{ T_{13}(\ell, b, v) (v - \langle v\rangle)^2}}{\sum _{T_{13}(\ell, b, v) >1{\rm K}} {T_{13}(\ell, b, v)}}, \end{equation} \tag{ 7 }$$

where T₁₃ is the ¹³CO brightness temperature in K, and only positions and velocity channels where the ¹³CO main beam temperature is greater than 4σ = 1 K are taken into account. The average radial velocity 〈v〉 is given by

$$\begin{equation} \langle v\rangle = \frac{\sum _{T_{13}(\ell, b, v) > 1{\rm K}}{ T_{13}(\ell, b, v) v}}{\sum _{T_{13}(\ell, b, v) >1{\rm K}} {T_{13}(\ell, b, v)}}. \end{equation} \tag{ 8 }$$

Molecular hydrogen (H₂) and helium (He) are the main constituents of molecular clouds. In order to derive molecular cloud masses from CO observations, one must therefore make assumptions about the abundance of CO relative to H₂. First, we assume that the abundance of ¹³CO relative to H₂ and He is uniform. In reality, the CO abundance declines steeply with decreasing A_V due to photodissociation by the Galactic radiation field at A_V < 3 (Glover & Mac Low 2010). In contrast, self-shielded H₂ can exist at A_V as low as 0.2 (Wolfire et al. 2010). As a result, the ¹³CO/H₂ abundance is likely to be lower in the envelopes of molecular clouds than in their denser cores. The effects of abundance variations on the estimation of the molecular cloud masses are discussed in Section 9.2. Nevertheless, for simplicity, constant ratios n(¹²CO)/n(¹³CO) = 45 and n(¹²CO)/n(H₂) = 8 × 10⁻⁵ are assumed (Langer & Penzias 1990; Blake et al. 1987). A mean molecular weight of 2.72 accounts for the presence of both H₂ and He (Allen 1973; Simon et al. 2001). Under those assumptions, molecular cloud masses are derived from Equation (9) using kinematic distances from Roman-Duval et al. (2009):

$$\begin{equation} \frac{M}{M_{\odot }} = 0.27 \: \frac{d^2}{\mathrm{kpc}^2}\: \int _{\ell, b, v} { \frac{T_{{\rm ex}}(\ell, b, v) \: \tau _{13}(\ell, b, v)}{1-e^{\frac{-5.3}{T_{{\rm ex}} (\ell, b, v)}}}\frac{ dv}{\mathrm{km} \: \mathrm{s}^{-1}}\: \frac{d\ell }{^{\prime }} \frac{db}{^{\prime }} }. \end{equation} \tag{ 9 }$$

In this equation, the integrand is summed over all (ℓ, b) positions where the ¹³CO integrated intensity is greater than 4σ. Along those sightlines, only velocity channels where the brightness temperature is greater than 4σ = 1 K are included in the integration. The second criterion is necessary due to numerical issues with Equations (1), (2), and (9) when the brightness temperature value is negative. Hence, we do not include contributions from the noise in the integration described by Equation (9). However, some noise peaks in voxels isolated from molecular cloud emission, with values greater than 4σ, still remain even after filtering the integration with the first criterion. To remedy this problem, we filter the lines of sight (ℓ, b) along which the integration is performed by applying a threshold in ¹³CO integrated intensity (first criterion).

The mean number density of particles (H₂ and He) in the molecular clouds was estimated assuming spherical symmetry via

$$\begin{equation} \frac{n(\mathrm{H}_2 + \mathrm{He})}{\mathrm{cm}^{-3}} = 15.1 \times \frac{M}{M_{\odot }} \times \left(\frac{4}{3} \pi \:\frac{R^3}{\mathrm{pc}^3} \right)^{-1}. \end{equation} \tag{ 10 }$$

The surface mass density Σ of the molecular clouds is defined as

$$\begin{equation} \frac{\Sigma }{M_{\odot } \: \mathrm{pc}^{-2}} =\left(\frac{M}{M_{\odot }} \right) \left(\frac{A}{\mathrm{pc}^2}\right)^{-1}, \end{equation} \tag{ 11 }$$

where the area A of the molecular clouds is calculated as described in Section 3.4.

Molecular clouds are supported against gravitational collapse by various mechanisms, such as turbulence, thermal gas pressure, and magnetic fields (Heitsch et al. 2001; Klessen et al. 2000). Observations show that the line widths of molecular clouds are much wider than their thermal line widths. Transonic/supersonic turbulence must therefore be the main source of kinetic energy and support in molecular clouds (e.g., Larson 1981; Williams et al. 2000).

The virial parameter α of a molecular cloud is the ratio of its virial mass M_vir to its mass. It describes the ratio of internal, supporting energy to its gravitational energy. The virial mass of a molecular cloud is defined as the mass for which a molecular cloud is in virial equilibrium, i.e., when the internal kinetic energy K equals half the gravitational energy U (2K + U = 0). It is given by

$$\begin{equation} M_{\rm vir} = 1.3 \frac{R\: \sigma _v ^2}{G} = 905 \frac{R}{\mathrm{pc}}\: \frac{\sigma _{v_{{\rm 1D}}}^2}{(\mathrm{km} \: \mathrm{s}^{-1})^2}, \end{equation} \tag{ 12 }$$

where R is the equivalent radius of a molecular cloud defined in Section 3.4 and $\sigma _{v_{{\rm 1D}}}$ is the one-dimensional velocity dispersion defined in Section 3.5. Note that the three-dimensional isotropic velocity dispersion, σ_v, is $\sqrt{3}$ times the one-dimensional velocity dispersion, which is measured in our spectroscopic data. For M > M_vir (α < 1), 2K + U < 0 and the molecular cloud is gravitationally bound. For M < M_vir (α > 1), 2K + U > 0 and the molecular cloud is not gravitationally bound.

The virial mass is proportional to the line width and to the radius of a molecular cloud, with the proportionality constant depending on the number density profile. We have computed the CO number density profile by assuming spherical symmetry and deprojecting the column density with an Abel transform. By fitting a power law to each density profile, we find that the average slope is −1.8. To be consistent with past literature (Solomon et al. 1987), we use the proportionality constant derived from a density profile of slope −2 to compute the virial mass.

The first 25 entries of the derived physical properties of the 580 GRS molecular clouds covered by the UMSB are listed in Table 1. The complete table (including molecular clouds not covered by the UMSB survey, see Section 5) can be found online. The first four columns indicate the molecular cloud name, Galactic longitude, latitude, and LSR velocity from the Rathborne et al. (2009) catalog. Column 5 indicates the FWHM velocity dispersion of the molecular clouds (derived from Section 3.5 via a conversion factor of $\sqrt{8 {\rm ln}(2)}$ between the 1σ and FWHM velocity dispersion). Column 6 gives the physical radii as defined in Section 3. Columns 7 and 8 indicate the masses of the molecular clouds and their uncertainty. The derivation of error bars on the mass estimates is discussed in detail in Section 9.4. Column 9 provides the mean number density of the molecular clouds. Columns 10 and 11 provide the mean excitation temperature and the mean ¹³CO center-of-line optical depth. Columns 12 and 13 show the mean surface density of each cloud and the virial parameter. Column 14 contains a flag "i" indicating that the cloud is covered by the UMSB survey.

The top panels of Figure 3 show the radius and mass spectra of the sample of 750 GRS molecular clouds, Ψ(R) = dR/dln(R) and Φ(M) = dN/dln(M). For M > 10⁵ M_☉, the mass spectrum follows a power law: Φ(M) ∝ M^{−1.64 ± 0.25}. The radius spectrum also follows a power law for R > 10 pc: Ψ(R) ∝ R^{−3.90 ± 0.65}. Within the error bars, the slope of the mass spectrum derived in this paper is consistent with the value of −1.5 obtained in previous work (e.g., Sanders et al. 1985; Solomon et al. 1987; Williams & McKee 1997). The slope of the radius spectrum is higher than the value obtained by Sanders et al. (1985), but is consistent, within the error bars, with the radius distribution obtained by Heyer et al. (2001).

Our sample of molecular clouds is complete above the turnover mass of M_to = 4 × 10⁴ M_☉, such that the slope of the mass spectrum of molecular clouds should not be affected by a lack of completeness. Our sample of molecular clouds was identified in a version of the GRS smoothed to 6' spatially and to 0.6 km s⁻¹ spectrally. GRS molecular clouds were detected by CLUMPFIND as contiguous voxels of brightness temperature greater than 0.2 K (Rathborne et al. 2009). Rathborne et al. (2009) furthermore applied the condition that molecular cloud candidates detected by CLUMPFIND must contain at least 16 smoothed voxels in order to be identified as a molecular cloud. Assuming that the ¹³CO line is optically thin, the minimum mass of a molecular cloud is given by

$$\begin{equation} \frac{M}{M_{\odot }} \ge 0.05 \: \frac{d^2}{\mathrm{kpc}^2}\: T_{{\rm ex}} \: e^{\frac{5.3}{T_{{\rm ex}}}} \times 16 T_{\rm min} \frac{\Delta \ell }{^{\prime }}\frac{\Delta b}{^{\prime }}\frac{\Delta V}{\mathrm{km} \mathrm{s}^{-1}}, \end{equation} \tag{ 14 }$$

where d is the distance in kpc, T_ex is the excitation temperature in K, and T_min = 0.2 K is the threshold brightness temperature (corrected for beam efficiency). Hence, with T_ex = 6.32 K (the average value observed in the GRS), Δℓ = Δb = 01, and ΔV = 0.6 km s⁻¹, the completeness limit is M_min = 50 d²_kpc, where d_kpc is the distance in kpc. Thus, M_min = 200 M_☉ at 2 kpc, 1250 M_☉ at 5 kpc, 5000 M_☉ at 10 kpc, and 11250 M_☉ at 15 kpc. Since the maximum distance probed by the GRS is 15 kpc (based on the GRS Galactic longitude and velocity coverage), the turnover mass is greater than the completeness limit of the GRS, and the slope of the mass spectrum of molecular clouds above the turnover mass should not be affected by completeness effects.

This calculation of the completeness limit does not take into account confusion, which in reality has an effect on our data. Confusion is more pronounced near the tangent point, where large physical separations correspond to small radial velocity differences. There is also the problem of molecular clouds that have similar radial velocities, but are located on either side of the tangent point, at the near and far kinematic distances. In these cases, if the velocity difference between the two clouds is small enough (typically less then the line width of the clouds), the clump-finding algorithm used to identify the GRS clouds, CLUMPFIND, will blend these two clouds into a single object. The extent of the calculation and modeling required to estimate how confusion affects the completeness limit by far exceeds the scope of this paper. The blending of two molecular clouds depends on many parameters such as the geometry (their location in the Galaxy), the structure of large-scale galactic features, the distance and radial velocity of the clouds (related by the rotation curve), their line widths, radii, angular separation, and the parameters used in CLUMPFIND. Nonetheless, confusion in the GRS, which uses ¹³CO as a tracer, is not nearly as severe as in surveys using the optically thick ¹²CO as a tracer for molecular gas.

It is worth mentioning that the GRS has a limited field of view (FOV), only covering the Galactic latitude range −1° < b < 1°. This in principle imposes a limit on a maximum cloud's radius, R_max, and mass, M^FOV_max, detectable by the GRS. We can use the relation between a cloud's mass and radius derived in Section 4 to estimate this maximum mass, with R_max = πd/180. Thus, we find M^FOV_max = 1.96 × 10⁵d^2.36_kpc. We will however show in Section 9.1 that this limit is never reached and thus does not affect our sample of molecular clouds and their physical properties.

The fourth panel of Figure 3 shows the histogram of the 750 molecular clouds' virial parameter, the median value of which is 0.46 ± 0.07. 70% of our molecular cloud sample (both in mass and number) have virial parameters <1. This analysis thus suggests that most of the molecular mass contained in identifiable molecular clouds is located in gravitationally bound structures.

The bottom left panel shows the mean density of H₂ in our sample of 750 molecular clouds, the median of which is 231 cm⁻³. This value is well below the critical density of the ¹³CO J = 1 → 0 transition, n_cr = 2.7 × 10³ cm⁻³, suggesting that the gas with density n > n_cr is not resolved by a 48'' beam (0.25 pc at d = 1 kpc), and that its filling factor is low.

The bottom right panel shows the surface mass density of the molecular clouds, with a median of 144 M_☉ pc⁻². Using the Galactic gas-to-dust ratio 〈N_H/A_V〉 = 1.9 × 10²¹ cm⁻² mag⁻¹ (Whittet 2003), this corresponds to a median visual extinction of 7 mag. This value is consistent with the prediction from photoionization dominated star formation theory (McKee 1989). A median surface mass density of 140 M_☉ pc⁻² is lower than the median value of 206 M_☉ pc⁻² derived by Solomon et al. (1987) based on the virial masses of a sample of molecular clouds identified in the ¹²CO UMSB survey. Note that Solomon et al. (1987) originally found a median surface mass density of 170 M_☉ pc⁻², assuming that the distance from the sun to the Galactic center is 10 kpc. Assuming a Galactocentric radius of 8.5 kpc for the sun, this value becomes 206 M_☉ pc⁻² (Heyer et al. 2009). The median surface density derived here is also higher than the value of 42 M_☉ pc⁻² derived by Heyer et al. (2009), who re-examined the masses and surface mass densities of the Solomon et al. (1987) sample using the GRS and a method similar method to ours. Similar to our analysis, Heyer et al. (2009) estimated the excitation temperature from the ¹²CO line emission and derived the mass and surface density from ¹³CO GRS measurements and the excitation temperature.

For the Solomon et al. (1987) molecular cloud sample, Heyer et al. (2009) found a median surface density of 42 M_☉ pc⁻² using the area A1 (the 1 K isophote of the ¹²CO line) defined by Solomon et al. (1987) to compute masses and surface mass densities. However, computing surface mass densities within the half power ¹²CO isophote (A2) yields a median surface mass density close to 200 M_☉ pc⁻² (see Figure 4 of Heyer et al. 2009). It is thus likely that the discrepancy between the surface densities derived here, in Heyer et al. (2009), and in Solomon et al. (1987) be explained by the different methods and thresholds used to compute the molecular clouds' properties. The ¹²CO line is about 4–5 times as bright as the ¹³CO line (Liszt 2006). As a result, it is likely that a significant fraction of the area A1 used by Solomon et al. (1987) to compute the molecular clouds' masses using the ¹²CO line only by assuming that molecular clouds are virialized does not exhibit ¹³CO line emission above the detection threshold of the GRS. Since the Heyer et al. (2009) derivation of molecular clouds' masses is based on ¹³CO as a tracer of molecular gas column density, this would result in a dilution of the surface mass density derived from ¹³CO, which would appear lower compared to the surface density derived from the brighter ¹²CO emission by Solomon et al. (1987). This effect likely contributes to the median surface density in Heyer et al. (2009) being lower than in Solomon et al. (1987) and in our analysis. Since our derivation of the molecular clouds' mass, radius, and surface mass density is only performed within the 4σ contour of the ¹³CO integrated intensity, it is likely biased toward the most opaque regions of molecular clouds, where the surface mass density is higher than in the envelopes, thus raising the median value of the surface mass density compared to Heyer et al. (2009).

Figure 4 shows the histograms of the excitation temperature and ¹³CO optical depth of the 580 GRS molecular clouds covered by the UMSB survey. The mean excitation temperature is 6.32 ± 0.04 K, and the mean optical depth of the ¹³CO line is 1.46 ± 0.02. Although the ¹³CO line is less opaque than its ¹²CO counterpart, optical depth effects should be taken into account in the derivation of ¹³CO column densities from ¹³CO spectral line mapping.

Rathborne et al. (2009) previously derived the excitation temperature and ¹³CO optical depth of the GRS catalog of molecular clouds and found a lower mean optical depth (0.13) and a higher mean excitation temperature (8.8 K). The difference between the results presented here and in Rathborne et al. (2009) can be explained by the use of different methods. Rathborne et al. (2009) first computed the mean ¹³CO and ¹²CO brightness temperatures each GRS molecular cloud, 〈T₁₃〉 and 〈T₁₂〉 before applying Equations (1) and (2) to obtain the "mean" excitation temperature and optical depth, T_ex(〈T₁₂〉) and τ₁₃(〈T₁₃〉, 〈T_ex〉). In contrast, we first computed the excitation temperature T_ex and the ¹³CO optical depth τ₁₃ from the ¹³CO and ¹²CO brightness temperatures using Equations (1) and (2) at each voxel associated with a particular molecular cloud. Only then did we average the excitation temperature over voxels with ¹³CO brightness temperature above the noise threshold (4σ), and the line center optical depth over pixels with ¹³CO integrated intensity above the 4σ noise level to obtain 〈T_ex(T₁₂)〉 and 〈τ₁₃(T₁₃, T_ex)〉. Because Equations (1) and (2) are nonlinear with T₁₂, T₁₃, and T_ex, the difference between performing the average before or after applying the equation can be quite large. For instance, for GRSMC G053.59+00.04, we find 〈τ₁₃(T₁₃, T_ex)〉 = 1.78, while τ₁₃(〈T₁₃〉, 〈T_ex〉) = 0.24, close to the typical optical depth found by Rathborne et al. (2009). In this respect, molecular cloud masses recently derived by Urquhart et al. (2010) likely underestimate the masses of GRS molecular clouds, since they use the (uniform) excitation temperatures and optical depths derived by Rathborne et al. (2009). We find that the masses derived by Urquhart et al. (2010) are lower than masses derived here by a factor 2–3.

8.1.1. Decline in CO Excitation Temperature with Galactocentric Radius

The equilibrium temperature of molecular clouds results from a balance between heating and cooling. The heating of gas in molecular clouds is due to: (1) electrons ejected from dust grains illuminated by the Interstellar Radiation Field (ISRF) due to the photoelectric effect (Bakes & Tielens 1994; Wolfire et al. 2003), (2) H₂ photodissociation (Black & Dalgarno 1977), (3) collisions between Galactic Cosmic Rays (GCRs) and molecules (Goldsmith & Langer 1978), (4) UV pumping of H₂ (Burton et al. 1990), and (5) formation of H₂ on dust grains (Hollenbach & McKee 1989). In the densest regions of molecular clouds that are completely shielded from the ISRF, heating is dominated by GCRs. In the regions closer to the CO boundary, the photoelectric effect likely plays an important role in the heating of the gas.

Fine structure lines of metals, most importantly [C ii] at 158 μm, and radiative CO rotational transitions, are responsible for the cooling of gas in molecular clouds. The decrease in the abundance of metals away from the Galactic center effectively decreases the cooling rate in molecular clouds as their Galactocentric radius increases (Quireza et al. 2006).

Because the gas temperature is the result of the equilibrium between heating and cooling, the effects of the variations in the heating and cooling rates with Galactocentric radius on the gas temperature are reflected in the variations of the gas temperature throughout the Galactic plane. The CO excitation temperature however depends on both the gas temperature and the local number density. T_ex can be significantly lower than the gas temperature if the density is lower than the CO critical density, in which case there are not enough particles to collisionally excite the CO molecule to the J = 1 level. This effect is known as sub-thermal excitation. Nonetheless, the CO excitation temperature is likely close to the gas temperature in the dense regions of molecular clouds, which dominate the emission. Hence, the variations of the CO excitation temperature with Galactocentric radius should constrain the balance between gas heating and cooling.

Figure 6 shows the CO excitation temperature versus R_gal. Only molecular clouds covered by the UMSB survey, for which we could derive a CO excitation temperature and optical depth, are taken into account. The top panel shows the maximum and mean excitation temperature in each molecular cloud. Since the mean excitation temperature is likely lower than the gas temperature due to sub-thermal excitation effects, the maximum excitation temperature in each cloud, corresponding to the densest regions, probably reflects the gas temperature more accurately. The bottom panel shows these temperatures averaged over 0.3 kpc Galactocentric radius bins. In both cases, the CO excitation temperature decreases smoothly with Galactocentric radius, from 14 K at R_gal ≃ 4 kpc down to 8 K at R_gal ≃ 8 kpc (for the maximum excitation temperature).

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This large-scale decline in the gas temperature away from the Galactic center indicates that the slight decline in metallicity (and hence cooling rate) away from the Galactic center is not large enough to overcome the decrease in the heating rate (due to a decrease in the strength of the ISRF and/or the GCR flux) with Galactocentric radius. We have modeled and quantified the contribution of variations in the GCR flux and the ISRF to the variations of the gas temperature with Galactocentric radius. The variations of the GCR flux have been investigated by Bloemen et al. (1986) and Strong et al. (1988). While the flux of cosmic ray nuclei ϕ_n appears constant with Galactocentric radius, the flux of cosmic ray electrons, ϕ_e, shows a gradient best described by ϕ_e(R_gal) = $\phi _e (R_{\odot })e^{-0.19(R-R_{\odot })}$. Note that Bloemen et al. (1986) originally cite a gradient of −0.16 kpc⁻¹, assuming R_☉ = 10 kpc. With the more recent value of R_☉ = 8.5 kpc (Kerr & Lynden-Bell 1986), this corresponds to a gradient of −0.19 kpc⁻¹. The heating rate by GCR is then Γ_GCR = ϕ_e(R_gal) Δ_cr n(H₂) (Goldsmith & Langer 1978), where Δ_cr = 17–26 eV is the energy deposited as heat as a result of the ionization by GCRs. The GCR flux at R_gal = R_☉ is in the range 1.5–3 × 10⁻¹⁷ s⁻¹ (Goldsmith & Langer 1978). Hence, the heating rate by GCRs is given by

$$\begin{equation} \Gamma _{{\rm GCR}} (R_{\rm gal}) = 5\times 10^{-28} n({\rm H_2}) e^{-0.19(R-R_{\odot })} \: \mathrm{[erg} \: \mathrm{cm}^{-3} {\rm s}^{-1} \mathrm{]}, \end{equation} \tag{ 15 }$$

where we chose a value of ϕ_eΔ_cr = 5 × 10⁻²⁸ erg s⁻¹, which best describes the excitation temperature trend in our data, and is in the range cited by Goldsmith & Langer (1978).

The heating rate by photoelectric effect is given in Bakes & Tielens (1994). For neutral grains, the heating rate is given by Γ_pe = 4.86 × 10⁻²⁶G₀(R_gal)n_H [erg cm⁻³ s⁻¹], in an unattenuated medium, where G₀ is the strength of the ISRF in Habing units (G₀ varies with Galactocentric radius), and n_H is the hydrogen density. In a molecular cloud of visual extinction A_V, the heating rate due to photoelectric effect is therefore

$$\begin{equation} \Gamma _{{\rm pe}} = 4.86\times 10^{-26} G_0 (R_{\rm gal}) e^{-5.4 A_V} n_{\rm H} \: \mathrm{[erg} \: \mathrm{cm}^{-3} {\rm s}^{-1}\mathrm{].} \end{equation} \tag{ 16 }$$

In this equation, we assumed that CO was present for A_V > 1 (Wolfire et al. 2010), and used the standard Milky Way extinction curve to relate A₁₀₀₀, the FUV extinction at 1000 Å, and A_V (A₁₀₀₀ = 4–5 A_V, see Gordon et al. 2003). The strength of the ISRF, G₀, can be observed via the dust temperature (Bernard et al. 2008): G₀ ∝ T^4+β_d, where β is the emissivity index of the dust (β = 1.5–2; see Boulanger et al. 1996; Gordon et al. 2010), and T_d is the equilibrium temperature of large dust grains. In addition, Sodroski et al. (1997) derived the dust equilibrium temperature versus Galactocentric radius. They found T_d(R_gal) = 28 K–6.8 K × (R_gal/R_☉). Hence, with the local IRSF having strength G₀(R_☉) = 1.7, the variations of G₀ throughout the Galactic plane are given by

$$\begin{equation} G_0(R_{\rm gal}) = G_0(R_{\odot }) \left(1.32 - 0.32 \frac{R_{\rm gal}}{R_{\odot }}\right)^{(4+\beta)}. \end{equation} \tag{ 17 }$$

This expression for G₀(R_gal) can then be used in Equation (16) to calculate the heating due to photoelectric effect as a function of R_gal.

The cooling rate by a variety of fine structure atomic and molecular lines is given in Goldsmith & Langer (1978). For n(H₂) = 300 cm⁻³, which is close to the median density ($n_{\rm H_2}^{{\rm med}}$ = 230 cm⁻³) in our sample of molecular clouds, the cooling rate is Λ = 4.7 × 10⁻²⁷T^1.6 [erg cm⁻³ s⁻¹].

In thermal equilibrium, the gas temperature can then be determined by equating the cooling and heating rates: Λ = Γ. The gas temperature resulting from the balance between heating by GCR and line emission cooling is shown as a blue line in Figure 6. For the H₂ number density, n(H₂), we use the median value of our sample ($n_{\rm H_2}^{{\rm med}}$ = 230 cm⁻³) in Equation (15). Our simple calculation reproduces the variations of the CO excitation temperature with Galactocentric radius within the error bars. In Figure 6, we also show the gas temperature obtained from the equilibrium between cooling and heating by photoelectric effect (dashed green line). We used Equations (16) and (17), with the median number density found in our molecular cloud sample, and β = 1.5 (Gordon et al. 2010). In this case, our prediction also reproduces the temperature trend within the error bars.

Thus, both a decline in the GCR flux and in the strength of the ISRF can explain the decrease in CO excitation temperature derived in this analysis based on the GRS. It is likely that the GCR flux and the strength of the ISRF are correlated, because they both depend on the star formation rate. In this case, the decline in gas temperature with R_gal would reflect a decrease in the star formation rate with Galactocentric radius. Indeed, young massive stars produce most of the ionizing and dissociating radiation responsible for gas heating in the envelopes of molecular clouds. GCRs are thought to be produced in supernova remnants via Fermi acceleration. Since supernovae remnants are the products of massive star formation, the GCR flux is therefore tied to massive star formation. Because of the large mean free path of GCRs, however, it is not clear how the heating rate due to GCRs in molecular clouds relates to the local star formation rate. Nonetheless, a gradient of −30%/R_☉ in the star formation rate (see Equation (17)) would explain the decline of the gas temperature with Galactocentric radius.

8.1.2. Enhancement at R_gal = 6.4 kpc

The maximum CO excitation temperature in each cloud is enhanced by about 2 K at a Galactocentric radius of 6.4 kpc. This enhancement is significant at the 3σ level with respect to the uncertainty plotted in Figure 6. Over the longitude coverage of the GRS, a Galactocentric radius of 6.4 kpc corresponds to the Sagittarius arm. An elevated gas temperature at 6.4 kpc may be related to massive star formation occurring in the Sagittarius arm (both through the strength of the radiation field and the GCR flux). In Figure 6, an increase of 50% in the GCR flux (blue line) and in the strength of the ISRF (green line) at R_gal = 6.4 kpc reproduces the trend observed in the GRS.

The idea that star formation locally increases the gas temperature inside molecular clouds can be further tested by comparing different data sets. Anderson et al. (2009) found an enhancement of H ii regions at a Galactocentric radius of ≃ 6 kpc. In addition, the 15 hottest molecular clouds located at Galactocentric radii 6–7 kpc (which stand out in Figure 6) all have confirmed star formation activity—polycyclic aromatic hydrocarbon emission in the 8 μm band, and 24 μm emission tracing warm dust—in the GLIMPSE and MIPSGAL infrared surveys. Eight of these 15 molecular clouds are associated with the star formation region W51 located at a Galactic longitude of 49°–50° and at a distance of 5.5 kpc, which places it near the Sagittarius arm. The seven others belong to different star formation regions and contain known masers and H ii regions. To test the hypothesis that star formation activity increases the gas temperature of molecular clouds, we compare the excitation temperatures of "active" molecular clouds, containing H ii regions, to the temperature of "quiescent" molecular clouds. In Roman-Duval et al. (2009), a catalog of such "active" and "quiescent" molecular clouds was established based on the coincidence between the morphology of the 21 cm continuum and ¹³CO emission, and the velocity of ¹³CO and recombination lines from Anderson et al. (2009). This catalog can readily be used to compare the excitation temperatures of active and quiescent molecular clouds. The histograms of the CO excitation temperature of both samples are shown in Figure 7. On average, "active" molecular clouds have a slightly higher temperature, 〈T_ex〉 = 6.96 ± 0.22, compared to "quiescent" molecular clouds (〈T_ex〉 = 6.28 ± 0.04). A Kolmogorov–Smirnov test shows that the active and quiescent molecular cloud populations have a less than 1% chance of being from the same distribution. This difference of temperature between active and quiescent molecular clouds, significant at the 3σ level, supports the idea that star formation increases the gas temperature (and hence CO excitation temperature) of molecular clouds.

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The variations of the masses and surface mass densities of the GRS molecular clouds as a function of Galactocentric radius should help understand and predict the variations in the star formation activity resulting from this reservoir of molecular gas throughout the Galactic disk. The azimuthally averaged Galactic surface mass density of molecular gas, Σ_gal, was obtained by summing the molecular clouds' masses over 0.5 kpc Galactocentric radius bins. Since the range of longitudes covered by the GRS data is limited, each radial bin covers only a limited range in azimuth, which varies with Galacotcentric radius. To convert the mass measurements to a more uniform measure of gas content, the total mass contained in a radial bin was therefore divided by the surface area covered by the GRS data within the radial bin. As a result, we obtained an azimuthally averaged surface mass density of molecular gas encompassed by GRS molecular clouds at each Galactocentric radius. The left panel of Figure 8 shows Σ_gal as a function of Galactocentric radius R_gal. Σ_gal peaks at R_gal = 4.5 kpc (Σ_gal = 2.5 × 10⁶ M_☉kpc⁻²) and decreases steeply with Galactocentric radius down to 10⁵ M_☉kpc⁻² at R_gal = 7.5 kpc.

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The middle panel of Figure 8 shows the molecular clouds' masses averaged over 0.5 kpc Galactocentric radius bins. Since each Galactocentric radius bin covers a range of distances, and since the completeness limit varies with distance, only molecular clouds with masses greater than the completeness limit at 15 kpc, or 1.1 × 10⁴ M_☉ pc⁻², were taken into account. The mean molecular cloud mass in each radial bin appears rather constant with R_gal out to 5.5 kpc, at which point the mean molecular cloud mass starts to drop from 10⁵ M_☉ at R_gal = 5.5 kpc down to 5 × 10⁴ M_☉ at R_gal = 7.5 kpc.

The right panel of Figure 8 shows the molecular clouds' surface mass density, Σ_c, averaged over 0.5 kpc Galactocentric radius bins. Σ_c increases from 3 kpc to 4 kpc, remains constant between 4 kpc and 6 kpc, and decreases beyond this point, from 170 M_☉ pc⁻² down to 120 M_☉ pc⁻² at R_gal = 7.5 kpc. It has been suggested by (McKee 1989) that the star formation rate in a molecular cloud is governed by ambipolar diffusion, and subsequently by the surface mass density. In a molecular cloud with low surface mass density, the ionized fraction sustained by the ISRF penetrating the molecular cloud is high enough to prevent collapse and subsequent star formation due to magnetic support. In contrast, star formation proceeds more rapidly in molecular clouds that exhibit a high surface mass density. The decline in the average molecular cloud surface mass density with Galactocentric radius beyond 6 kpc may therefore suggest a decrease in the star formation rate with Galactocentric radius.

The top panel of Figure 9 shows the mass versus the distance to each molecular cloud. The mass (and radius) of the GRS molecular clouds is observed to on average increase with distance. This is due to (1) a Malmquist bias resulting from the increasing completeness limit and probability of observing a massive cloud with distance, and to a minor extent (2) the effects of finite resolution and the method used to identify molecular clouds. Indeed, it is hard to define one set of parameters for CLUMPFIND (spatial and spectral smoothing, brightness threshold and brightness increment) that identify a molecular cloud at all emission levels.

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9.1.1. Modeling Biases

In order to analytically describe the effects of increasing completeness limit and probability of observing rarer, more massive clouds as a function of distance, we assume a power-law mass spectrum, ϕ(M) = ϕ₀M^−1.64 (see Section 6). The average mass of a molecular cloud detected by the GRS at a distance d in the GRS field is

$$\begin{equation} \langle M\rangle (d) = \frac{\int _{M_{\rm min}(d)}^{M_{\rm max}(d)} {M\phi (M) dM}}{\int _{M_{\rm min}(d)}^{M_{\rm max}(d)} {\phi (M) dM}} \end{equation} \tag{ 18 }$$

or

$$\begin{equation} \langle M\rangle (d) =\frac{M_{\rm max}(d)^{0.36}-M_{\rm min}(d)^{0.36}}{M_{\rm min}(d)^{-0.36} - M_{\rm max}(d)^{-0.36}}, \end{equation} \tag{ 19 }$$

where M_min(d) is the minimum mass that can be detected by the GRS at distance d, i.e., the completeness limit. M_min(d) is given by M_min(d) = 50d²_kpc (see Section 6), where d_kpc is the distance in kpc. M_max is a physical upper limit on the masses of molecular clouds used to integrate Equation (18), which would diverge for M_max = ∞. Indeed, molecular clouds cannot grow infinitely due to disruptive mechanisms (e.g., turbulence, Galactic shear), the limited molecular scale height, and limited molecular content of the Milky Way. In addition, the FOV of the GRS is limited to Galactic latitudes −1° < b < 1°, such that the maximum mass detectable in the survey cannot exceed the limit derived in Section 6: M^FOV_max = 1.96 × 10⁵d²_kpc. However, Figure 10, showing the maximum mass detected in each distance bin and the theoretical upper limit due to the limited FOV, demonstrates that the maximum mass allowed by the limited FOV of the GRS, M^FOV_max, is never reached. Thus, we use a constant upper bound, M_max, in the integration of Equation (18). We use M_max = 10⁶M_☉, the largest molecular cloud mass observed in the GRS, and also a typical high end on the mass of massive giant molecular clouds (Solomon et al. 1987).

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9.1.2. Biases Observed in the GRS

9.1.3. Effects of Biases on Molecular Clouds' Physical Properties

The derivation of the GRS molecular clouds' masses is based on the assumption that the abundance of ¹³CO relative to H₂ is uniform within a molecular cloud. Owing to photodissociation and fractionation, the abundance ratio n(¹³CO)/n(H₂) may however decrease significantly between the shielded dense interiors of molecular clouds and their diffuse, UV-exposed envelopes (Liszt 2007; Glover & Mac Low 2010). Molecular hydrogen (H₂) being self-shielded due to numerous optically thick absorption lines in the UV, it can resist the photodissociating ISRF at lower extinctions (A_V of a few tenth) than its CO counterpart, which can only exist for A_V > 1–3 (Glover & Mac Low 2010; Wolfire et al. 2010). A significant amount of molecular gas is therefore likely hidden in CO dark envelopes around molecular clouds. As a result, our method probably underestimates the masses of the GRS molecular clouds. Wolfire et al. (2010) predicts that 30% of a molecular cloud's mass is in the form of CO-dark molecular envelope, while Goldsmith et al. (2008) predict 50%. This is nonetheless the order of magnitude of the molecular mass not traced by CO in the Milky Way.

In addition, the CO molecule tends to leave the gas phase and freeze out onto dust grains (thus forming a mantle) in cold, dense molecular cores. This effect could also potentially render some of the molecular mass inside a molecular cloud invisible to CO observations. Indeed, using other molecules such as N₂H⁺ and the dust continuum as a tracer for the densest molecular phase, Bacmann et al. (2002) have measured a CO under-abundance of 5–15 in molecular cores of density 10⁵–10⁶ cm⁻³. The under-abundance depends on the density (Bacmann et al. 2002), such that CO mantling on dust grain does not occur for densities lower than 10⁴ cm⁻³. The typical size of such cores is 0.05–0.2 pc (Bacmann et al. 2000; Rathborne et al. 2010), which at a distance of 5 kpc represents an angular size of 15''. Thus, a core's projected area represents 1% of the GRS beam area. Therefore, we do not expect CO mantling on dust grain to have a significant effect on the estimation of a molecular cloud's mass compared to photodissociation effects.

In order to derive ¹³CO optical depths and molecular cloud masses, the CO excitation temperature was assumed to be identical for ¹³CO and ¹²CO. Since CO is usually thermalized within molecular clouds due to its low dipole moment, this is a reasonable assumption. This assumption might however break down in the more diffuse envelopes of molecular clouds, where the optically thick ¹²CO can remain thermalized due to radiative trapping, while the optically thin ¹³CO is sub-thermally excited. In this case, the ¹³CO excitation temperature would be lower than the ¹²CO excitation temperature. Nonetheless, emission from sub-thermally excited ¹³CO in diffuse molecular cloud envelopes is probably under the GRS detection threshold (Heyer et al. 2009).

In addition to the effects of sub-thermal excitation and abundance variations, noise intrinsic to the data and errors on kinematic distances affect the accuracy of a molecular cloud's mass estimate. The estimation of the masses of the molecular clouds is based on the knowledge, at each voxel, of the excitation temperature (derived from the ¹²CO brightness temperature), of the optical depth (derived from the excitation temperature and the ¹³CO brightness temperature), and the distance of each molecular cloud. The ¹³CO and ¹²CO brightness temperatures are affected by Gaussian noise, with a standard deviation of $\sigma _{T_{{\rm mb}}} = 0.26$ K for both the UMSB (¹²CO) and the GRS (¹³CO) surveys (Jackson et al. 2006). The accuracy of kinematic distances was discussed in Roman-Duval et al. (2009). The error on the distance stems from the error in the velocity of a molecular cloud with respect to its LSR velocity, the difference being caused by cloud-to-cloud dispersion and non-circular motions associated with spiral arms. The cloud-to-cloud dispersion amounts to 3–5 km s⁻¹ (Clemens 1985) and local velocity perturbations associated with spiral arms are of order 15 km s⁻¹ (Clemens 1985).

The error on the molecular clouds' masses due to each of those factors was estimated using Monte-Carlo simulations. For every molecular cloud in the GRS, we computed the velocity dispersion and peak brightness temperature of the ¹³CO line toward each GRS pixel. The ¹³CO line toward each pixel of the molecular cloud was then modeled by a Gaussian line with the same peak brightness temperature and velocity dispersion. Gaussian noise of standard deviation $\sigma _{T_{{\rm mb}}} = 0.26$ K was added to the model molecular cloud, and its ¹³CO optical depth estimated assuming an excitation temperature of 6.3 K (the average value in Figure 4). We reproduced the contribution of the distance uncertainty on the mass error using the following method. First, a random Gaussian velocity error e_v of dispersion $\sigma _{e_v}$ = 5 km s⁻¹ was added to the LSR velocity of the molecular cloud, V₀, to obtain a flawed estimate of the molecular cloud LSR velocity V₁ = V₀ + e_v. The kinematic distance d₁ corresponding to V₁ was computed using the Clemens (1985) rotation curve and the Galactic longitude of the molecular cloud, conserving the near/far kinematic distance assignment of the molecular clouds from Roman-Duval et al. (2009). The distance d₁ was finally used, along with the ¹³CO optical depth of the model molecular cloud, to compute its mass. This process was repeated 20 times for the standard deviation of the mass of the model molecular cloud to converge. The error on the mass estimation of the molecular cloud was then computed as σ_M = $\sqrt{\langle (M_i-M_{\mathrm{cloud}})^2\rangle _{i=0,20} }$, where M_cloud is the mass of the GRS molecular cloud and M_i is the mass of the ith realization of the model molecular cloud. Error bars on the mass estimate are specified in Table 1 and in the online complete version of the table.