A COMPARATIVE STUDY OF GIANT MOLECULAR CLOUDS IN M51, M33, AND THE LARGE MAGELLANIC CLOUD

In Section 4.1, we found that clouds in the arm+central region of M51 tend to have higher CO peak brightness and larger velocity dispersions than clouds in the two low-mass galaxies and M51's interarm region. One observational effect that could contribute to this difference is blending of the CO emission from discrete physical entities that overlap in (x, y, v) space and which cannot be decomposed at our resolution. This effect is unlikely to be dominant in regions where the CO emission is sparsely distributed and, for this reason, blending would seem to be an improbable explanation for the observed differences among the properties of clouds in M33, the LMC, and M51's interarm environment. On the other hand, clouds may become crowded in M51's spiral arms. This would tend to increase the size, brightness, and velocity dispersion of the structures that our decomposition algorithm identifies in the M51 spiral arm region.

Although higher-resolution observations—especially in the spectral domain—are required to unambiguously assess the prevalence of blending in M51's spiral arms, we suggest that blending is unable to fully explain the trends that we observe for several reasons. First, the scale height of the thin molecular disk in M51 is only ∼40 pc (Pety et al. 2013), which makes it unlikely that several ∼50 pc scale structures occurring along a single line of sight through the galaxy would be a common phenomenon. Furthermore, even though the typical linewidth of the matched-resolution clouds in the arm+central region of M51 is larger than the typical linewidth of clouds in M33, the LMC, and in M51's interarm region, it is still a factor of ∼3 smaller than the cloud-to-cloud velocity dispersion in M51 after we subtract a model of galactic rotation from the cloud radial velocities (∼16 km s⁻¹; see Section 5.2.2 for a description of how we subtract the galactic rotation model). This is consistent with the appearance of the CO line profiles in the M51 arm region, which typically exhibit a single peak or, more rarely, two peaks that are well separated along the velocity axis (which are then identified as discrete clouds by our decomposition algorithm). At our resolution, line profiles with multi-peaked velocity components are rare, even though we might expect a significant number of such profiles if emission in the spiral arms arose from distinct clouds with similar radial velocities that overlapped along the line of sight.

A further piece of evidence that the clouds identified in M51's spiral arms are discrete objects, as opposed to blended emission from multiple overlapping clouds, is that we do not detect any variation in the scaling between the virial mass estimate and CO luminosity for clouds of similar size in the M51 arm and interarm regions. Assuming that blending is not a significant problem in the interarm region—which is likely, since the observed interarm clouds are widely separated in (x, y) space—this agreement suggests that we are identifying discrete clouds in both environments, since mass estimates derived from applying the virial theorem to unbound associations of molecular clouds tend to be significantly larger than the masses inferred from the CO luminosity (e.g., Allen & Lequeux 1993; Rand 1995).

Finally, we note that if blending was solely responsible for raising the brightness temperature and velocity dispersion in M51's arm+central region, then we would expect the effect to be more pronounced in the spiral arms than in the central zone, where the CO emission is more sparsely distributed (see Figure 1(a)). Instead, we observe the opposite: the median peak brightness and velocity dispersion of clouds in the central zone is higher than in the arms by 1.1 K and 0.8 km s⁻¹, respectively. While this does not imply that blended emission from overlapping clouds is absent, it does suggest that there are physical processes besides—or in addition to—blending that determine the CO emission properties in these environments. In conclusion, although we cannot definitively exclude the possibility that blended emission from overlapping clouds contributes to the higher peak brightness and velocity dispersion of the clouds in the M51 arm+central region, it would seem insufficient to explain all the trends that we describe in Section 4.1.

A second potential explanation for the differences among the GMC populations of M51, M33, and the LMC is that there is a systematic difference in the way that ¹²CO(J = 1 → 0) emission traces the underlying H₂ distribution. If this were true, then the underlying physical properties of the molecular (i.e., H₂) clouds might be similar in all three galaxies, despite the variations that we infer from our CO observations. B08, for example, argued that the lower velocity dispersions and CO luminosities of molecular clouds in the Small Magellanic Cloud were best understood in terms of selective photodissociation of CO molecules. Feldmann et al. (2012), on the other hand, argued that X_CO also increases at high H₂ column densities once the CO-emitting clumps within molecular clouds shadow each other (i.e., when the CO filling factor within a velocity range corresponding to the channel width reaches unity) and CO emission from the cloud becomes globally optically thick.

In general, however, we do not expect large variations in the value of X_CO among M51, M33, and the LMC. In a companion paper (Hughes et al. 2013), we argue that the absence of a truncation and the width of the probability distribution functions of CO-integrated intensity and brightness provide evidence that the velocity dispersion of the CO-emitting gas within the PAWS field is sufficiently high such that the saturation effect described by Feldmann et al. (2012) does not yet apply. Selective CO photodissociation should be an important effect at very low metallicities, but it is not expected to cause large variations in X_CO for systems with metal abundances greater than ∼0.3 Z_☉ (e.g., Bolatto et al. 2013). The metallicity of M51's inner disk is approximately solar (e.g., Moustakas et al. 2010; Bresolin et al. 2004) and several independent analyses of dust and molecular line emission in M51 indicate that the X_CO factor and dust-to-gas ratio are consistent with local Milky Way values (e.g., Schinnerer et al. 2010; Tan et al. 2011; Mentuch Cooper et al. 2012). M33 and the LMC have a lower metallicity than M51 by a factor of ∼2, but an empirical comparison between the H₂ masses inferred from CO and dust continuum emission also concludes that a Galactic value of the X_CO factor is applicable for GMCs in these two low-mass galaxies (Leroy et al. 2011). Based on these studies, we would not expect the X_CO factor to vary by more than a factor of a few among all three galaxies.

We can assess whether small variations in the CO-to-H₂ factor could nonetheless account for the differences in the mass surface density of GMCs that we infer using a virial analysis. The median values of the virial parameter in Table 3 are 1.6, 3.1, and 2.9 for resolved clouds in M51, the LMC, and M33, respectively. These values are obtained under the assumption that X_CO = 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. Alternatively, we can assume that the average dynamical state of GMCs is the same in all three galaxies and that the differences in the virial parameter instead reflect variations in the true value of X_CO. This is equivalent to attributing the small vertical offset between the GMC populations in Figure 8(b) to variations in X_CO, rather than inferring that the average dynamical state of GMCs varies among the different galaxies. If GMCs are typically just self-gravitating, then the median value of the virial parameter for a cloud population is 〈α〉 = 2 (e.g., Blitz et al. 2007; Leroy et al. 2011). Imposing this median value of α on all the cloud samples requires X_CO values of 1.6, 3.1, and 2.9 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ for M51, the LMC, and M33, respectively.

For resolved clouds identified in the matched-resolution cubes, the median mass surface densities that we infer for the LMC, M33, and for M51's interarm and arm+central environments (assuming X_CO = 2.0 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹) are 22, 86, 122, and 167 M_☉ pc⁻², respectively. The values that would be obtained using the galaxy-dependent X_CO values (i.e., derived under the assumption that 〈α〉 = 2) are 34, 124, 98, and 134 M_☉ pc⁻², respectively. We repeated the KS tests on the $\Sigma _{\rm H_{2}}$ distributions that we derive using the galaxy-dependent X_CO values and tabulate the results in Table 4. We find that there is still a statistically significant difference (p ⩽ 0.05) between the mass surface densities for the LMC clouds and all the other cloud populations and also between clouds in the arm and interarm regions of M51. On the other hand, there is no statistically significant difference between the mass surface density distributions of clouds in M33 and M51 using the galaxy-dependent X_CO values. Figure 7 shows that the clouds with CO surface brightnesses greater than 10 K km s⁻¹ in M33 are mostly small (R < 50 pc). If we restrict our virial analysis to clouds that are larger than this, then the median cloud mass surface density (obtained using the galaxy-dependent X_CO factor) for M33 reduces to 62 M_☉ pc⁻², but the difference between the mass surface densities of clouds with R ⩾ 50 pc in M33 and clouds in M51 remains statistically insignificant.

In summary, it is possible that the X_CO variations can account for some of the differences between the CO-derived properties of clouds in M33 and M51. The differences in CO-derived GMC properties that we find for the LMC clouds and among the M51 environments, on the other hand, cannot be fully explained by X_CO variations that remain consistent with either a virial analysis or independent analyses of dust emission and CO excitation. For these galactic environments, the differences in the CO-derived properties appear to reflect genuine variations in the physical properties of the molecular (i.e., H₂) clouds and not just the fidelity with which CO emission traces the underlying H₂ distribution.

A third possible explanation for the observed differences among the properties of GMCs in M33, the LMC, and the inner disk of M51 is that the typical density of GMCs is regulated by pressure variations in the ambient ISM. Traditionally, a large discrepancy between the high internal pressures of Milky Way GMCs and the much lower kinetic (thermal plus turbulent) ISM pressure has been thought to imply that GMCs are approximately in simple virial equilibrium (i.e., with internal kinetic energy equal to half their gravitational potential energy) and hence largely decoupled from the diffuse interstellar gas that surrounds them (e.g., Blitz 1993). This line of argument is problematic, however, based on renewed attention being paid to other potential sources of confining pressure for clouds, such as ram pressure from inflowing material (e.g., Heitsch et al. 2009) and the (static) weight of surrounding atomic gas (e.g., Heyer et al. 2001), as well as a long history of observations suggesting that pressure confinement is significant for molecular clouds in certain galactic environments, such as the outer Galaxy (e.g., Heyer et al. 2001) and at high latitude (e.g., Keto & Myers 1986). At the very least, we would expect the internal pressure of molecular clouds to be comparable to the external pressure, otherwise the clouds would be rapidly compressed and/or destroyed.

The higher mass surface density of M51 clouds suggests that they should also have higher internal pressures than clouds in the low-mass galaxies (Section 4.1). We can estimate the internal pressure P_int of a molecular cloud according to

$$\begin{equation} \frac{P_{{\rm int}}}{k} = \rho _{g} \sigma _{\rm v}^{2} = 1176 \left(\frac{M}{M_{\odot }} \right)\left(\frac{R}{{\rm pc}} \right)^{-3}\left(\frac{\sigma _{\rm v}}{{{\rm \thinspace km\thinspace s}^{-1}}} \right)^{2}\,{{\rm \thinspace cm}^{-3} {\rm \thinspace K}}\,, \end{equation} \tag{ 4 }$$

where ρ_g is the H₂ volume density. For the resolved cloud populations identified in the matched-resolution cubes of M33, the LMC, and M51, we find median internal pressures of 〈P_int/k〉 ∼ 1.9 × 10⁵, 3.0 × 10⁴, and 4.3 × 10⁵ cm⁻³  K, respectively. We note that clouds in the spiral arms and central region of M51 tend to have higher internal pressures (〈P_int/k〉 ∼ 5.3 × 10⁵ cm⁻³  K) than clouds in the interarm region (〈P_int/k〉 ∼ 2.6 × 10⁵ cm⁻³  K).

The average kinetic pressure in the interstellar gas depends on the weight of the gas layer in the gravitational potential of the total mass (i.e., including gas, stars, and dark matter) that lies within the gas layer. Assuming the contribution from dark matter is negligible, we can approximate the external pressure at the boundary of a molecular cloud using the expression for the hydrostatic pressure at the disk midplane derived by Elmegreen (1989) for a two-component disk of gas and stars:

$$\begin{equation} P_{{\rm ext}} = \frac{\pi G}{2}\Sigma _{g} \left(\Sigma _{g} + \frac{\sigma _{g}}{\sigma _{*}}\Sigma _{*} \right). \end{equation} \tag{ 5 }$$

In this expression, Σ_g is the neutral (atomic + molecular) gas surface density, Σ_* is the stellar surface density, and σ_g and σ_* are the velocity dispersions of the gas and stars, respectively. This expression for the interstellar pressure accounts for the gravity forces due to the stars and gas, as well as the turbulent and thermal hydrodynamic pressure. It is obtained from the definition of the midplane pressure $P_{{\rm midplane}} = \rho _{g}\sigma _{g}^{2}$, after substituting ρ_g = Σ_g/2h_g and $h_{g} = ({\sigma _{g}^{2}}/{\pi G \Sigma _{{\rm total}}})$, where ρ_g is the gas density at the midplane, h_g is the scale height of the gas, and Σ_tot ≈ (Σ_g + (σ_g/σ_*)Σ_*) is an estimate of the total mass surface density within the gas layer.

Since our estimate for P_ext involves a combination of quantities, we plot the internal pressure of the GMCs as a function of Σ_g and (σ_g/σ_*)Σ_*, i.e., the gas and stellar components of the gravitational potential, in Figures 9(a) and (b), respectively. The origin and typical uncertainty associated with our observational estimates for Σ_g, Σ_*, and σ_* are discussed below. We adopt a constant gas velocity dispersion σ_g = 10 km s⁻¹ for all galaxies. The motivation for this choice is that a roughly constant gas velocity dispersion of 7–10 km s⁻¹ has been reported across the disks of several nearby galaxies (see van der Kruit & Freeman 2011 and references therein). In contrast, Tamburro et al. (2009) recently found that σ_g decreases linearly by 3–5 km s⁻¹ per ΔR₂₅, where R₂₅ is the optical radius of the galaxy beyond the optical radius for a subsample of THINGS galaxies, as well as typical values of σ_g ∼ 15–20 km s⁻¹ inside the optical radius. Assuming σ_g = 20 km s⁻¹ for all galaxies would shift the points in Figure 9(b) by ∼0.3 dex toward higher values along the x axis, i.e., increasing the contribution of stars to the gravitational potential acting on the gas layer. In addition to M51, M33, and the LMC, we include the GMC populations studied by B08 and the GMCs in M64 and NGC 6946 identified by Rosolowsky & Blitz (2005) and Donovan Meyer et al. (2012), respectively. For these additional data sets, our estimates for P_int/k are calculated using the published measurements of the GMC properties, i.e., we do not re-analyze the CO data cubes. The vertical error bars in each panel of Figure 9 reflect the median absolute dispersion of the P_int/k values, while the horizontal bars indicate the range of values that are observed across the field of view of each CO survey. For the B08 galaxies, the range on the x axis applies to R_gal < 0.4R₂₅. We adopt this region based on the GMC positions published by B08. For M51, M33, the LMC, M64, and NGC 6946, the range refers to the field of view of the original CO surveys. We note that our estimates of Σ_*, Σ_g, and σ_* are calculated using radial profiles of these quantities, rather than on a pixel-by-pixel basis.

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Figures 9(a) and (b) show that the median internal pressure of the GMCs increases with both the gas and stellar components of the gravitational potential and that neither the gas nor the stellar term dominates our estimate for the external pressure. We note that a correlation with both quantities is to be expected if the internal pressure of molecular clouds is responding to the external pressure, since both stars and gas contribute to the total mass that determines gravitational force. In contrast, a good correlation with the stellar term would not be expected if the mass surface density of GMCs depends on a process such as the shielding of H₂ molecules against the interstellar radiation field, which depends on the local gas column density only. The robust correlations in both panels of Figure 9 further suggest that a relationship between our estimates for P_ext and P_int does not follow trivially from a correlation between Σ_g and the average mass surface density of the GMCs. Although both quantities are measures of the gas surface density, Leroy et al. (2013) have recently shown that due to the clumpiness of the molecular ISM, the CO surface brightness measured on ∼50 pc scales within galactic disks does not necessarily track measurements of the CO surface brightness on large (∼ kpc) scales or when using radial profiles, since the latter is a combination of both the intrinsic CO surface brightness of small-scale structures and the filling factor of such structures within the ∼ kpc scale region or annulus.

Our estimates for Σ_* have several sources. For M51, NGC 6946, and the B08 targets, we use the Σ_* radial profile published by Leroy et al. (2008), which uses an empirically calibrated conversion from the 3.6 μm intensity to the K-band flux and then a K-band mass-to-light ratio of 0.5 to convert from the K-band intensity to stellar surface density (Bell & de Jong 2001). The final calibration (Equation (C1) in Leroy et al. 2008) is

$$\begin{equation} \Sigma _{*} = 280 \, {\rm cos} i \, I_{3.6}, \end{equation} \tag{ 6 }$$

where I_3.6 is the 3.6 μm intensity in MJy sr⁻¹ and i is the galaxy inclination. We apply the same method to the Spitzer Local Volume Legacy Survey 3.6 μm map of M64 (Dale et al. 2009) to obtain a radial profile of Σ_* in that galaxy, assuming a central position of α₂₀₀₀ = 12:56:43.6, δ₂₀₀₀ = 21:40:59.3 and an inclination i = 60° (García-Burillo et al. 2003). The largest uncertainty in our estimates for Σ_* is the mass-to-light ratio, which depends on the metallicity, the initial stellar mass function, and the star formation history of galaxies. Bell et al. (2003) show that the K-band stellar mass-to-light ratios vary by 0.1 dex for redder galaxies and 0.2 dex for bluer galaxies like M51. We note that the stellar surface density profiles of NGC 6946 and M51 rise sharply at small galactocentric radii (R_gal ≲ 1 kpc), which may reflect the presence of a nuclear bulge. Since Equation (5) is invalid in such regions, we only include galactocentric radii where the stellar surface density follows a roughly exponential profile. For M33, we use the stellar surface density profile at a lookback time of 0.6 Gyr published by Williams et al. (2009; see their Figure 4). This profile, which agrees within ∼50% with the mass model of Corbelli (2003) inside our field of view (R_gal < 5.5 kpc), was constructed by modeling the star formation histories that best reproduce the color–magnitude diagrams obtained for four fields in M33, imaged with the Advanced Camera for Surveys on the Hubble Space Telescope. The uncertainties in Σ_* quoted by Williams et al. (2009) are ≲ 0.2 dex. To estimate Σ_* in the LMC, we use the stellar surface density map published by Yang et al. (2007), which was constructed using number counts of red giant branch and asymptotic giant branch stars in the Two Micron All Sky Survey Point Source Catalog (Skrutskie et al. 2006) and normalized to a total stellar mass of 2 × 10⁹ M_☉ in the LMC (Kim et al. 1998).

To estimate Σ_g, we use radial profiles of H i-and CO-integrated intensity, assuming X_CO = 2.0 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹, optically thin H i emission, and a helium contribution of 1.36 by mass to convert between measurements of integrated intensity and gas mass surface density. For M51, NGC 6946, and the B08 galaxies, we use the gas radial profiles published by Leroy et al. (2008). For the LMC, we use the radial profiles published by Wong et al. (2009). For M33, we use the radial profiles published by Gratier et al. (2010), re-calculating the molecular gas surface density using X_CO = 2.0 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. For M64, we measured radial profiles of the atomic and molecular gas surface densities using the THINGS and BIMA-SONG integrated intensity maps of H I and CO emission (Walter et al. 2008; Helfer et al. 2003), adopting the same central position and inclination as for Σ_*.

With the exception of the LMC, our estimates for σ_* are based on measurements of the central stellar velocity dispersion obtained by Ho et al. (2009) using the Palomar spectroscopic survey of nearby galaxies (Ho et al. 1995, 1997). For the LMC, we use the velocity dispersion of carbon stars measured by van der Marel et al. (2002). Many studies have indicated that the exponential scale height of stellar disks is roughly constant with galactocentric radius (e.g., van der Kruit & Searle 1981; de Grijs & Peletier 1997; Kregel et al. 2002), while σ_* declines (van der Kruit & Freeman 2011 and references therein). To estimate σ_* across each CO survey's field of view, we assume that σ_* decreases exponentially according to σ_* = σ_{*, 0}exp (− R_gal/2R_*) (e.g., Bottema 1993; Boissier et al. 2003), where σ_{*, 0} is the central stellar velocity dispersion and R_* is the exponential scale length of the stellar disk R_*. We rely on literature values for R_* (see Table 7 for exact references).

Finally, in Figure 10, we plot the median internal pressure of the GMC populations as a function of the external pressure. As in Figure 9, the vertical error bars correspond to the median absolute dispersion of the P_int/k measurements of the GMCs in each galaxy, while the horizontal error bars indicate a range of P_ext/k values that characterize the region of the galactic disk where the GMCs are located. It is clear from Figure 10 that there is a good correlation between the internal and external pressures of GMCs, suggesting that the variation in GMC mass surface densities that we observe among M51, M33, and the LMC may arise because the external ISM pressure plays a role in regulating the internal pressure (and hence velocity dispersion and density) of molecular clouds. Figure 10 further suggests that GMCs are not greatly overpressured with respect to their environment (i.e., 〈P_int/k〉 ∼ P_ext/k). Rather than simple virial equilibrium between their gravitational and internal kinetic energies, the implication is that GMCs may instead tend toward a pressure-bounded equilibrium configuration. This result is satisfying insofar as it suggests that the traditional dichotomy between strongly gravitationally bound GMCs in the inner disk of the Milky Way and the pressure-confined, low-mass clouds in the outer Galaxy and at high galactic latitude may be more apparent than real. If molecular structures are bound by a combination of self-gravity and external pressure, then self-gravity may appear dominant for samples that preferentially include objects with high masses and densities, while pressure confinement should appear more important for samples of low-mass, low-density objects. Moreover, the trend in Figure 10 would seem to confirm that the higher GMC mass surface densities and linewidths reported by studies of M64 and M82 (Rosolowsky & Blitz 2005; Keto et al. 2005)—i.e., nearby systems where the disk surface density is intermediate between conditions in Local Group galaxies and true starbursts—arise because GMCs exhibit a continuum of properties (as previously suggested by Rosolowsky 2007), rather than an intrinsic bimodality between molecular gas properties in "normal" and "starburst" environments.

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An important observable consequence of external pressure regulating the properties of GMCs is that the scaling between a GMC's size and linewidth—i.e., the coefficient of the size–linewidth relation—should depend on the external pressure. We discuss whether there is evidence for such variations in the GMC populations of M51, M33, and the LMC in Section 5.2.2. Furthermore, if the internal pressure of GMCs is comparable with the interstellar pressure, then this shallow pressure gradient across GMC boundaries means that the clouds' evolution should be more susceptible to pressure fluctuations in the surrounding ISM than classical GMC models have tended to assume. A detailed investigation of the importance of dynamical pressure for the stability of gas and global patterns of star formation in M51 is the subject of a companion paper (Meidt et al. 2013; see also Jog 2013).

Larson's third "law" describes an inverse relationship between the density of a molecular cloud and its size, implying that molecular clouds have roughly constant molecular gas column density. Several studies of extragalactic GMC populations via their CO emission (e.g., B08) have reported that the average H₂ surface density of extragalactic GMCs is roughly constant within galaxies and, moreover, that it is in good agreement with the value that is observed for GMCs in the inner Milky Way, $\langle \Sigma _{\rm H_{2}} \rangle \sim 100$ M_☉ pc⁻². Our results in Section 4.2, in contrast, suggest that there are subtle but genuine variations in the characteristic H₂ surface densities of the GMC populations of M51, M33, and the LMC.

As noted by several previous authors (e.g., Kegel 1989; Scalo 1990; Ballesteros-Paredes & Mac Low 2002), the limited surface brightness sensitivity of extragalactic CO observations is an evident source of bias for CO-based estimates of GMC mass surface density since sightlines with low-to-intermediate H₂ column densities will fall beneath the CO detection limit and hence be excluded from the regions that are identified as molecular. At high brightness, on the other hand, CO observations may underestimate the true H₂ column density if the CO-emitting regions within shadow each other in velocity space (e.g., Feldmann et al. 2012). Coupled with the fact that wide-field extragalactic ¹²CO(J = 1 → 0) observations are rarely designed to achieve surface brightness sensitivities much deeper than 5σ for a typical ∼10⁵ M_☉ cloud, these effects suggest that the range of H₂ column densities inferred from CO observations will inevitably be quite restricted. Indeed, even though the minimum CO-derived estimate of $\Sigma _{\rm H_{2}}$ for the PAWS GMCs still likely reflects the survey's limiting CO surface brightness sensitivity (see Figure 7), the large velocity dispersion of the CO-emitting gas within the PAWS field may explain why the dispersion of the size–CO luminosity relation (and hence the width of the inferred $\Sigma _{\rm H_{2}}$ distribution) is larger for M51 GMCs than for the GMC populations of M33, the LMC, and other Local Group systems (see Figure 3 of B08).

Some further insight is provided by comparing the size–luminosity relations obtained using different decompositions of the PAWS data cube in Figure 7. In particular, it is evident that the CO surface brightness values obtained using a method that preferentially identifies structures with a characteristic size scale (i.e., the "cloud-based" decompositions in Figures 7(a) and (b)) cover a wider range than the values obtained when the boundaries of the identified structures are defined using a fixed intensity threshold (i.e., the "island-based" decompositions in Figures 7(c) and (d)). Quantitatively, we find that the scatter in the logarithm of the residuals about the best-fitting size–luminosity relationships increases from ∼0.2 dex for islands in both the intrinsic and matched-resolution M51 data cubes to ∼0.5 dex for the cloud structures identified in the same cubes.

These decomposition-dependent results for the scatter in the CO surface brightness (and hence $\Sigma _{{\rm H}_{2}}$) values derived from the PAWS data are qualitatively similar to the two cases considered by Lombardi et al. (2010) in their analysis of nearby Galactic clouds using dust extinction to trace H₂ column density: the average CO surface brightness of molecular structures above a fixed brightness threshold is approximately constant, while equivalent measurements over a fixed size scale yield much larger variations in $\Sigma _{{\rm H}_{2}}$, both among and within the GMC populations of the three galaxies that we investigate. Lombardi et al. (2010) argue that the former result arises because molecular clouds have an approximately universal log-normal column density distribution. While this hypothesis can be empirically verified using extinction data for local clouds, the resolution and dynamic range of the extragalactic CO data is insufficient to recover the detailed shape of the I(CO) distribution for individual extragalactic GMCs. In our case, the narrow range of CO surface brightness measurements in Figures 7(c) and (d) arises because a large fraction of the pixels within an island structure sample emission that is close to the observational sensitivity limit. For the intrinsic-resolution M51 cube, we find that for over half the islands (51%), pixels with integrated intensity values less than 5σ make up more than half the total number of pixels within the structure. This fraction (i.e., where pixels with values <5σ constitute the majority of pixels within the structure) is similar in the LMC (46% of islands) and even greater in M33 (78% of islands).

Our conclusion is that the appearance of constant CO surface brightness among extragalactic GMCs is mostly an artifact due to the combination of several conspiring effects: first, the algebraically imposed covariance of L_CO and R, which yields a robust yet trivial correlation between these quantities; second, the strategy of designing extragalactic CO surveys to detect a "typical" Milky Way GMC at the ∼5σ level, which limits a survey's CO surface brightness sensitivity and may only reveal the high-mass, CO-bright GMCs in low-mass galaxies (rather than the bulk of the molecular cloud population); and third, the limited range of environmental conditions that had been probed by extragalactic CO observations with cloud-scale resolution prior to PAWS.

The possibility that the size–linewidth relation is an observational artifact has received less attention in the literature, with most debates focusing on whether it is a sign that GMCs attain approximate virial balance between their gravitational and internal kinetic energies or a manifestation of interstellar turbulence (see, e.g., Ballesteros-Paredes 2006 and references therein). Nevertheless, our analysis (see Figure 6) suggests that some caution in interpreting the extragalactic GMC results is required. In particular, we find little evidence for a correlation between the size and linewidth of cloud structures within M51, M33, or the LMC. A combined sample of GMCs from all three galaxies, on the other hand, yields a size–linewidth relation similar to σ_v∝R^0.5 simply due to the variations in spectral and spatial resolution of the input data sets. For reasons noted in Section 4.1, decomposition algorithms preferentially identify structures in position and velocity space close to the resolution of a data cube. Based on our analysis of the M51, M33, and LMC data, we would therefore recommend that if measurements from high-resolution data sets yield the clouds with small radii and narrow linewidths and low-resolution data sets populate the large R and high σ_v end of a size–linewidth relation, then this bias should be explicitly excluded before a physical explanation for the correlation is invoked.

Although cloud structures identified in the matched-resolution data cubes of all three galaxies fail to exhibit a strong correlation between their size and linewidth, the vertical offset between the cloud populations in M51 and the low-mass galaxies in Figure 6(b) suggests there may be a genuine difference in the physical state of their cloud populations. More precisely, clouds in M51—and especially in the spiral arms and central regions—have larger linewidths compared with clouds of an equivalent size in the LMC or M33. Since regions with high velocity dispersions in M51's inner disk are often associated with low levels of star formation activity (Meidt et al. 2013) and the star formation rates of M33 and the LMC are high relative to their global CO luminosities (assuming a universal molecular gas depletion time of ∼2 Gyr; e.g., Leroy et al. 2008), it seems unlikely that this segregation is due to higher levels of internal turbulence generated by star formation feedback. On the other hand, such an offset would be expected if the dynamical state of the clouds is influenced by the external pressure, as we suggested in Section 5.1. Following Elmegreen (1989; e.g., see also Chieze 1987 and Field et al. 2011 for alternative derivations), clouds that achieve equilibrium among self-gravity, the external pressure, and their internal kinetic energy should follow:

$$\begin{equation} \sigma _{\rm v} \propto \left(\frac{P_{{\rm ext}}/k_{B}}{10^{4} {{\rm \thinspace cm}^{-3} {\rm \thinspace K}}} \right)^{1/4} \left(\frac{R}{\rm pc} \right)^{1/2}\,. \end{equation} \tag{ 7 }$$

According to our estimates in Section 5.1, the external pressure experienced by clouds in M51 is approximately an order of magnitude higher than that felt by clouds in M33 and the LMC. From Equation (7), we would then expect M51 clouds to exhibit linewidths ∼1.8 times larger than clouds of similar size in the low-mass galaxies, in good agreement with the vertical offset in the size–linewidth plot that we observe (a factor of ∼2 at a fixed size scale).

A second important result from our analysis in Section 4.2 is that a tighter correlation between the size and linewidth of molecular structures becomes apparent when we identify GMCs as regions of connected CO emission (i.e., islands). Arguably the most convincing example is for islands identified in the LMC data cube with its intrinsic resolution (Figure 6(c)), which follow σ_v = (0.16 ± 0.03)R^{0.84 ± 0.05} over approximately two orders of magnitude in size. In the intrinsic resolution cube, island structures in the spiral arms and central region of M51 exhibit a weak correlation that is shallower than for LMC islands: σ_v = (1.1 ± 0.4)R^{0.5 ± 0.1}. On one hand, this could indicate a genuine difference in the density structure of the molecular ISM between the two environments. Numerical simulations by Dobbs & Bonnell (2007) show that if the size–linewidth relation arises due to gas clumps being brought together at the location of a shock (e.g., a spiral shock or the interface of two colliding flows), then a steeper size–linewidth relation is expected if the molecular gas is more clumpy. The higher peak brightness of the CO emission in M51 suggests that CO-emitting regions may fill the beam more uniformly in M51 than in the LMC, which is at least qualitatively consistent with the LMC hosting more clumpy molecular material than M51, but higher-resolution observations that probe the internal density structure of GMCs in the LMC and M51 would be required to validate this model. On the other hand, we caution that the shallower correlation for islands in M51's spiral arms may be partly driven by the ∼10 objects with 100 < R < 500 pc. An inspection of Figure 6(c) suggests that these objects appear to follow a shallower relationship than the scattered trend exhibited by the smaller islands. Excluding islands with R > 150 pc from the fit yields σ_v = (0.6 ± 0.3)R^{0.6 ± 0.1} for the M51 arm+central region, which has a slope that is more similar to the LMC relation.

The possibility that the GMC linewidths that we measure include gas motions unrelated to the cloud's intrinsic velocity dispersion is underscored by the fact that structures with R > 500 pc (the magenta squares in Figures 6(c) and (d)) also seem to follow a relation that is roughly consistent with the canonical size–linewidth relationship derived for Galactic GMCs by S87. This is somewhat surprising if the origin of the size–linewidth relation is due to GMCs achieving dynamical equilibrium or turbulence in the molecular ISM, since we would expect the size–linewidth relationship to break down on scales corresponding to the largest virialized structures or the spatial scale on which the turbulence is driven, usually thought to be comparable with the scale height of the gas disk (∼200 pc in M51). For structures on scales much larger than a typical GMC, the linewidths that we measure are likely to be broadened by systematic motions within the galactic disk, such as galactic rotation or spiral arm streaming motions. To assess the importance of the former effect on the correlations in Figures 6(c) and (d), we subtracted a model of the contribution of galactic rotation to the GMC linewidths from the M51 data cube. In practice, we did this by generating a map of the line-of-sight velocity that would be expected at each spatial position within the PAWS and MAGMA fields from our preferred model of M51's and LMC's rotation (Wong et al. 2011; Meidt et al. 2013). After identifying islands of significant emission within the original data cubes, we used the MIRIAD task specshift to offset the individual CO spectra belonging to an island along the spectral axis such that the radial velocity corresponding to the model velocity field at each (x, y) position was shifted to the central channel of a new "velocity-shifted" data cube. This is equivalent to subtracting the modeled rotation component from the observed velocity field map, although we manipulate the data in (x, y, v) space so that the result is a data cube that can be analyzed using CPROPS. Next, we estimated the physical properties of each "velocity-shifted" island using the same method that we applied to the original data cubes (see Section 3). The aim of this shifting procedure was to suppress the contribution of galactic rotation to the global velocity dispersion measurement that we obtain for each island.

In Figure 11, we plot the velocity dispersion versus the radius for the "velocity-shifted" islands (red diamonds) identified in M51 (panel (a)) and the LMC (panel (b)). In both galaxies, there are fewer islands where CPROPS is able to measure the velocity dispersion after the shifting procedure. In M51, many of these objects correspond to islands in the original sample with large velocity dispersions compared with their size, so the outliers and upper envelope of the main distribution of red diamonds in Figure 6 are effectively removed. The result is to bring the bulk of the "velocity-shifted" M51 data points slightly closer to the measurements for the low-mass galaxies and to the size–linewidth relation for inner Milky Way GMCs (the dashed line in Figure 11). Indeed, a BCES bisector fit to the "velocity-shifted" M51 islands yields σ_v = (0.85 ± 0.24)R^{0.51 ± 0.06}, which is indistinguishable from the canonical S87 result. In the LMC, the data points appear less scattered after the galactic rotation model is subtracted, but the overall distribution is not shifted toward significantly lower velocity dispersions, an effect that is seen for large (R ≳ 300 pc) islands in M51. The best-fitting size–linewidth relation for the velocity-shifted LMC islands is identical to the relation obtained from islands identified in the original data cube. In summary, we find that contamination of GMC linewidths by systematic motions associated with galactic rotation is significant for the largest CO-emitting structures in M51, but contamination does not appear to strongly determine the correlation between size and linewidth for structures with spatial scales corresponding to the characteristic size of GMCs (i.e., 10–100 pc).

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Once again, our conclusion is that further investigation is required to establish whether extragalactic GMC populations follow the same size–linewidth relation as GMCs in the inner Milky Way and that particular care must be taken to eliminate the effects of resolution, survey design—since the observing configuration of most extragalactic CO surveys is selected to optimize sensitivity to structures with sizes and linewidths similar to Galactic GMCs—and analysis methods. Alternative explanations for the physical origin of the size–linewidth relation—besides simple virial equilibrium and interstellar turbulence—also merit further consideration. Here, we have suggested that the larger velocity dispersion of M51 clouds relative to clouds with similar size in the LMC and M33 may be due to the higher pressure at the cloud surface in M51's inner disk. In contrast, the best explanation for the more robust size–linewidth relations that we recover when we identify GMCs as "islands" of CO emission may be that an islands decomposition yields regions where external processes—such as a converging flow or a spiral shock—are bringing pre-existing smaller molecular structures into the same spatial location. These "cloud associations" are likely to be globally unbound, with CO linewidths that reflect the macroscopic motions of their constituent gas clumps, rather than interstellar turbulence per se (e.g., Dobbs & Bonnell 2007). In this case, a scaling between the size and linewidth might simply reflect the inhomogeneous density distribution and velocity fluctuations in the neutral ISM: over small scales, the constituent clumps encounter relatively homogenous material with similar density and local velocity, resulting in a low clump–clump velocity dispersion. Over larger scales, however, the clumps interact with material with a larger range of densities and peculiar velocities and hence experience different decelerations. This produces a higher velocity dispersion between widely separated clumps. In other words, the tighter correlation between size and linewidth when we identify GMCs as regions of connected CO emission may simply reflect a more adequate sampling of the density and velocity structure of the ISM by the dense clumps that constitute the CO islands (e.g., Bonnell et al. 2006).