PROBABILITY DISTRIBUTION FUNCTIONS OF ¹²CO(J = 1 → 0) BRIGHTNESS AND INTEGRATED INTENSITY IN M51: THE PAWS VIEW

The CO data for M51 were obtained by PAWS (Schinnerer et al. 2013). PAWS observations mapped a total field of view of approximately 270'' × 170'' in the inner disk of M51 in the ABCD configurations of the PdBI between 2009 August and 2010 March. Since an interferometer filters out low spatial frequencies, the PdBI data were combined with observations of CO emission in M51 obtained using the IRAM 30 m single-dish telescope in 2010 May. In this section, we summarize the most important aspects of the observations and data reduction; the PAWS observing strategy, data reduction and combination procedures, and flux calibration are described in detail by Pety et al. (2013).

2.1.1. PdBI Observations

2.1.2. IRAM 30 m Observations

2.1.3. Combination of Single-dish and Interferometric Data

The final PAWS data cube is a joint deconvolution of the PdBI and IRAM 30 m data sets. The short-spacing visibilities not sampled by the PdBI were recreated from the single-dish map using the GILDAS/MAPPING software. For this, the map was deconvolved from the IRAM 30 m beam in the Fourier plane before multiplication by the PdBI primary beam in the image plane (as described by Rodriguez-Fernandez et al. 2008). After a further Fourier transform, pseudo-visibilities were sampled between 0 and 15 m (the diameter of a PdBI antenna) and these visibilities were then merged with the interferometric observations. For the joint deconvolution, we used an adaption of the Högbom CLEAN algorithm, as implemented in GILDAS/MAPPING. Supports defining the region to search for CLEAN components were defined for each velocity channel, based on where significant emission was detected in the 30 m cube. The convergence of the deconvolution was checked in three different ways. First, the cumulative flux as a function of the number of CLEAN components converged in each channel. Second, the residual channel images look like noise. Both criteria indicate a satisfying convergence of the deconvolution. Finally, we deconvolved the data a second time using exactly the same method except that we doubled the maximum number of CLEAN components from 320,000 to 640,000. The subtraction of both cubes again looks like noise.

The effective angular resolution of the final combined PAWS data cube is 116 × 097, corresponding to a spatial resolution of ∼40 pc at our assumed distance to M51 (7.6 Mpc; Ciardullo et al. 2002). The data cube covers the LSR velocity range 173–769 km s⁻¹ and the width of each velocity channel is 5 km s⁻¹. The mean rms of the noise fluctuations across the survey is σ_rms ∼ 0.4 K in a 5.0 km s⁻¹ channel. For a typical CO linewidth of 15 km s⁻¹, this corresponds to an average sensitivity of 3.5 K km s⁻¹ for the map of CO-integrated intensity. A map of the noise fluctuations across the PAWS field, overlaid with contours of CO-integrated intensity, is shown in Figure 1.

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For M33, we use the CO data presented by Rosolowsky et al. (2007), which combine observations by BIMA (Engargiola et al. 2003) and FCRAO (Heyer et al. 2004). The common field of view of the single-dish and interferometer surveys is 0.25 deg², covering most of M33's optical disk. The angular resolution of the combined cube is 132 × 129, corresponding to a spatial resolution of 53 pc for our assumed distance to M33 of 840 kpc (e.g., Galleti et al. 2004). The data cover the LSR velocity range [ − 400, 40] km s⁻¹and the velocity channel width is 2.0 km s⁻¹. The average rms noise per channel is 0.24 K.

The CO data for the LMC were obtained by the Magellanic Mopra Assessment (MAGMA). The MAGMA survey design, data acquisition, reduction procedures, and calibration are described in detail by Wong et al. (2011). MAGMA mapped CO cloud complexes that had been identified at lower resolution by NANTEN (Fukui et al. 2008), targeting 114 NANTEN GMCs with CO luminosities higher than 7000 K km s⁻¹ pc²and peak integrated intensities greater than 1 K km s⁻¹. The combined field of view of the MAGMA survey is ∼3.6 deg². Although the clouds targeted for mapping represent only ∼50% of the clouds in the NANTEN catalog, the region surveyed by MAGMA contributes ∼80% of the total CO flux measured by NANTEN. The MAGMA LMC data cube has an effective resolution of 45'', corresponding to a linear resolution of ∼11 pc at the distance of the LMC (50.1 kpc; Alves 2004). The velocity channel width is 0.53 km s⁻¹and the total LSR velocity range of the cube is 200–305 km s⁻¹. The average rms noise per channel across the MAGMA field is 0.3 K.

We construct PDFs of CO-integrated intensity I(CO) and CO brightness T_mb. The I(CO) PDF is simply a histogram of the (x, y) pixel values within a I(CO) map, while the T_mb PDF is a histogram of the (x, y, v) pixel values within a spectral line cube. The PDFs are constructed after applying a blanking mask that identifies genuine emission within the data cubes. For our analysis of the PAWS data cube, we identify significant emission using the masking method described by Pety et al. (2013). For our comparative analysis of M51, M33, and the LMC in Section 4.4, we construct an initial mask that contains pixels above a 5σ_rms threshold over two or more contiguous velocity channels. This mask defines a high significance core, which is then expanded to include all connected pixels above 1.2σ_rms over at least two velocity channels. We discuss the rationale for using these blanking masks and the influence of different masking techniques on the shape of the PDFs in Appendix A. The total number of independent data points within the I(CO) PDFs varies between ∼15, 000 and ∼250, 000 (for the T_mb PDFs, this increases by factor of ∼7). For the I(CO) PDFs, we normalize the histogram by the number of pixels within the survey field of view, not by the number of pixels where significant emission is detected. Likewise, we normalize the T_mb histogram by the number of independent (x, y, v) elements in the data cube.

As we discuss in Section 4.3, the PDFs of CO-integrated intensity and CO brightness T_mb within M51 exhibit diverse shapes that are often inconsistent with a simple functional form such as a LN or power-law distribution. We therefore parameterize the shape of each PDF using the brightness distribution index (BDI), a metric recently devised by Sawada et al. (2012b) to characterize the ratio between faint and bright ¹²CO(J = 1 → 0) emission within an 08 × 08 field in the Galactic plane. More precisely, we specify the BDI of CO brightness as

$$\begin{equation} {\rm BDI} = \log \left(\frac{\sum _{T_{2} < T_{i} < T_{3}} T_{i}}{\sum _{T_{0} <T_{i} < T_{1}} T_{i}} \right) , \end{equation} \tag{ 1 }$$

where T_i is the brightness of the ith pixel and (T₀, T₁, T₂, T₃) = (1.2, 2.5, 5, ∞) K are the thresholds that we use to define faint and bright emission. These are not the same thresholds adopted by Sawada et al. (2012b) for their analysis at ≲ 1 pc resolution, but are chosen such that variations in the shape of the PDF are captured (i.e., BDI is defined) for all the M51 environments that we analyze and that all the pixels included in the calculation contain significant CO emission (i.e., T_i > 3σ_rms). We define an equivalent parameter for the I(CO) PDFs, which we refer to as the integrated intensity distribution index (IDI). Analogous to Equation (1), we specify this as

$$\begin{equation} {\rm IDI} = \log \left(\frac{\sum _{I_{2} < I_{i} < I_{3}} I_{i}}{\sum _{I_{0} <I_{i} < I_{1}} I_{i}} \right), \end{equation} \tag{ 2 }$$

adopting (I₀, I₁, I₂, I₃) = (10.5, 25, 60, ∞) K km s⁻¹.

When appropriate, we derive the best-fitting LN function to a PDF using a Levenberg–Marquardt fit to the function:

$$\begin{equation} P(s) = c_{0} \times \exp \left[\frac{-(\log s - \log s_{0})^{2}}{2x^{2}} \right]\,. \end{equation} \tag{ 3 }$$

We define the probability P(s) as the number of pixels in the bin divided by the total number of pixels in the map (or cube). Depending on context, s represents I(CO) or T_mb. Only bins to the right of the peak of the PDF, brighter than 4σ_rms and containing 10 or more pixels, are used to derive the fit. Several of the PDFs resemble power laws more than LN functions. In this case, we estimate the best-fitting slope of the power law using ordinary least squares linear regression.

In Section 5.2, we investigate whether there is a connection between the shape of the CO PDFs and the properties of GMCs identified within the PAWS field. For this, we use the GMC catalog presented by Colombo et al. (2013a). The catalog was constructed using the CPROPS package (Rosolowsky & Leroy 2006, hereafter RL06). CPROPS uses a dilated mask technique to isolate regions of significant emission within spectral line cubes and a modified watershed algorithm to assign the emission into individual clouds. Moments of the emission along the spatial and spectral axes are used to determine the size, linewidth, and flux of the clouds.

To generate the PAWS GMC catalog, CPROPS first identifies significant emission by finding pixels with CO brightness T_mb above a 4σ_rms threshold across two adjacent velocity channels, where the rms noise σ_rms is estimated from the median absolute deviation of each spectrum. This mask is then expanded to include all connected pixels with T_mb > 1.5σ_rms. Emission regions are then decomposed into GMCs by identifying emission that can be uniquely associated with local maxima. Full details of the decomposition procedure are presented in Colombo et al. (2013a).

We adopt the default CPROPS definitions of GMC properties. The cloud radius is defined as R = 1.91σ_R pc, where σ_R is the geometric mean of the second moments of the emission along the cloud's major and minor axes. The velocity dispersion σ_v is the second moment of the emission distribution along the velocity axis, which for a Gaussian line profile is related to the FWHM linewidth, Δv, by $\Delta v = \sqrt{8 \ln 2}\sigma _{\rm v}$. The CO luminosity of the cloud L_CO is the emission inside the cloud integrated over position and velocity, i.e.,

$$\begin{equation} L_{\rm CO} \; [{{\rm \, K\, km\, s}^{-1}\, {\rm pc}^{2}}] = D^{2} \left(\frac{\pi }{180 \times 3600} \right)^{2} \Sigma T \delta v \delta x \delta y\;, \end{equation} \tag{ 4 }$$

where D is the distance to the galaxy in pc, δx and δy are the spatial dimensions of a pixel in arcseconds, and δv is the width of one channel in km s⁻¹. The mass of molecular gas estimated from the GMC's CO luminosity M_CO is calculated as

$$\begin{equation} M_{\rm CO} \; [{{M}_{\odot }}] \equiv 4.4 \frac{{X_{\rm CO}}}{2 \times 10^{20} [{{\rm \, cm^{-2}\, (K\, km\, s^{-1})}^{-1}}]} L_{\rm CO}\;, \end{equation} \tag{ 5 }$$

where X_CO is the assumed CO-to-H₂ conversion factor and a factor of 1.36 is applied to account for the mass contribution of helium. The fiducial value of X_CO used by CPROPS is X_CO = 2.0 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. The virial mass is estimated as $M_{\rm vir} \; [{{M}_{\odot }}] = 1040 \sigma _{\rm v}^{2}R$, which assumes that molecular clouds are spherical with truncated ρ∝r⁻¹ density profiles (MacLaren et al. 1988). CPROPS estimates the error associated with a cloud property measurement using a bootstrapping method, which is described in Section 2.5 of RL06.

The final PAWS GMC catalog contains 1507 objects. The GMCs have peak brightness temperatures between ∼2 and 16 K, radii between 5 and 150 pc, and velocity dispersions between 1 and 30 km s⁻¹. The catalog and the properties of GMCs in different environments within the PAWS field are the subject of a companion paper (Colombo et al. 2013a). In general, the physical properties of the cataloged GMCs are similar to the GMCs identified by CO surveys of the inner Milky Way and other nearby galaxies, although GMCs in M51 tend to be larger, brighter, and have higher velocity dispersions and mass surface densities relative to their sizes than the GMCs in nearby low-mass systems such as the Magellanic Clouds and M33 (Hughes et al. 2013). The spatial resolution and sensitivity of PAWS is sufficient to resolve structures with size and mass comparable to a typical Galactic GMC (50 pc, 10⁵ M_☉; Blitz 1993). We note, however, that CO emission is almost ubiquitous across the PAWS field and that much of the emission resides in large (∼kpc-sized) regions of high brightness that bear little resemblance to Galactic GMCs. Overall, the cataloged GMCs account for approximately half of the total CO flux within the PAWS data cube, a fraction that varies from ∼40% in the interarm region to ∼60% in the spiral arms and central zone.

The PDF of CO-integrated intensity for the entire PAWS field is presented in Figure 2(a). The distribution is adequately described by an LN function, with a mean of 〈I(CO)〉 = 21.6 K km s⁻¹ and a logarithmic width of 0.44. The logarithmic dispersion in the fit residuals for bins above the 3σ sensitivity limit that contain more than five counts is 0.08. Relative to this LN function, there is some evidence for a truncation at high I(CO) values (≳ 200 K km s⁻¹). In principle, this could be due to the opacity of the ¹²CO(J = 1 → 0) emission line, an effect that is often observed on pc scales in regions of high extinction (A_V ≳ 5–10 mag; e.g., Lombardi et al. 2006; Pineda et al. 2008), but has rarely been considered for the scales probed by extragalactic observations (cf. Dickman et al. 1986). Against this interpretation, the 99th percentile of the CO peak brightness within the PAWS field is only ∼7 K, suggesting that there are negligible sightlines where the CO emission completely fills the telescope beam. Furthermore, recent numerical simulations that examine the ability of I(CO) to trace the H₂ column density on 20–60 pc scales show that saturation tends to produce a secondary peak in the I(CO) PDF rather than a smooth truncation (Shetty et al. 2011; Feldmann et al. 2012; see also Section 5.4). The finite resolution of observational data can also produce a truncation at high intensities (we explore this effect in Appendix B). In this case, however, we would expect the I(CO) PDFs for subregions within the PAWS field to exhibit similar thresholds, whereas several of them are consistent with pure LN functions (see Section 4.3).

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The truncation of the PDF is Figure 2(a) may therefore be physical. Elmegreen (2011) show that the density PDF should fall beneath a pure LN function at high gas densities if the Mach number decreases with increasing average gas density (as would be expected for a cloud that obeys the Larson (1981) size–linewidth relation). Alternatively, the truncation may reflect the efficacy of feedback processes that prevent the molecular gas from reaching very high mass surface densities ($\Sigma _{\rm H_{2}} \gtrsim 400\,{{M}_{\odot }\, {\rm pc}^{-2}}$, assuming X_CO = 2.0 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹). We discuss physical processes that could be influencing the shape of the I(CO) PDFs in M51 in more detail in Section 5.4.

The PDF of CO brightness for the PAWS cube is shown in Figure 2(b). The T_mb PDF is less like an LN function than the I(CO) PDF, with two roughly flat segments across 1 < T_mb < 3 K and T_mb > 5 K. Our LN fit to the PDF yields a mean 〈T_mb〉 ∼ 1.4 K and logarithmic width x ∼ 0.3. Assuming that the true distribution of T_mb values is LN, there are fewer high-brightness pixels than would be expected from this LN function. The truncation begins to occur at a CO brightness temperature of ∼5 K, which would seem too low to be due to opacity effects. Instead of an LN function, a broken power law with a slope of ∼ − 0.9 for 1 < T_mb < 5 K and a much steeper slope of ∼ − 4.1 for T_mb > 5 K—or, alternatively, a pure power law with a truncation at ∼5 K—may provide a better description of the PDF. The fit parameters and goodness of fit for the best-fitting LN and power-law functions to the T_mb PDF in Figure 2(b) are listed in Table 1.

Table 1. Fit Parameters for CO PDFs in M51, M33, the LMC, and Environments within M51

LN Fits
Figure	CO Property	Galaxy/Region	Mean	Logarithmic Width		Goodness of Fit
			s0	x
2(a)	I(CO)	PAWS field	21.6 K km s−1	0.44		0.08
2(b)	Tmb	PAWS field	1.4 K	0.31		0.18
4(a)	I(CO)	Bar	50.5 K km s−1	0.21		0.41
4(b)		Ring	40.8 K km s−1	0.47		0.18
4(c)		A1I	23.8 K km s−1	0.47		0.20
4(d)		A1O	36.6 K km s−1	0.35		0.08
4(e)		A1	27.4 K km s−1	0.44		0.28
4(f)		A2	25.4 K km s−1	0.35		0.10
4(g)		Up	18.8 K km s−1	0.25		0.09
4(h)		Down	24.0 K km s−1	0.22		0.13
5(a)	Tmb	Bar	1.5 K	0.30		0.11
5(b)		Ring	1.6 K	0.37		0.19
5(c)		A1I	1.0 K	0.38		0.17
5(d)		A1O	1.5 K	0.29		0.17
5(e)		A1	1.3 K	0.32		0.20
5(f)		A2	1.2 K	0.31		0.10
5(g)		Up	1.1 K	0.24		0.05
5(h)		Down	1.0 K	0.29		0.06
6(a)	I(CO)	M51	12.3 K km s−1	0.53		0.10
6(c)		LMC	1.8 K km s−1	0.26		0.13
6(d)	Tmb	M51	0.8 K	0.36		0.32
6(f)		LMC	0.1 K	0.28		0.14
Power-law fits
Figure	CO Property	Galaxy/Region	Slope	Slope 2	Domain	Goodness of Fit
			γ1	γ2
2(b)	Tmb	PAWS field	−0.88	−4.13	γ1: [1 < Tmb < 3] K	0.10
					γ2: [5 < Tmb < 8] K
4(c)	I(CO)	A1I	−0.85	...	γ1: I(CO) > 10 K km s−1	0.24
5(b)	Tmb	ring	−0.53	−6.26	γ1: [1 < Tmb < 5] K	0.09
					γ2: [8 < Tmb < 12.5] K
5(c)		A1I	−0.97	−4.68	γ1: [1 < Tmb < 4] K	0.07
					γ2: [5 < Tmb < 8] K
5(d)		A1O	−0.96	−4.97	γ1: [1 < Tmb < 4] K	0.07
					γ2: [5 < Tmb < 8] K
5(e)		A1	−0.97	−4.82	γ1: [1 < Tmb < 4] K	0.08
					γ2: [5 < Tmb < 8] K
6(d)	Tmb	M51	−1.30	−4.96	γ1: [1 < Tmb < 3] K	0.15
					γ2: [5 < Tmb < 8] K
6(e)		M33	−3.63	...	γ1: [0.8 < Tmb < 1.3] K	0.13
6(f)		LMC	−2.15	...	γ1: [0.2 < Tmb < 1] K	0.07

Notes. Parameters of best-fitting functions to PDFs in Figures 2, 4, 5, and 6. The parameters of the LN functions are determined from a Levenberg–Marquardt fit to Equation (3); the power-law and broken power-law fits are estimated using ordinary least squares regression. We use the logarithmic dispersion of the fit residuals to estimate the goodness of fit.

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An important question that we would like to address with the PAWS data is whether the organization and physical properties of molecular gas depend on galactic environment. Variations in the shape of the PDF could reflect differences in the relative importance of self-gravity, SF feedback, or gas flows in different parts of the galactic disk, which in turn might influence the ability of the molecular gas to form stars. Within the PAWS field, there are three main regions where the gas is likely to experience distinct physical conditions: within the strong spiral arms, the interarm region situated upstream and downstream of the spiral arms, and the central region, where the gas is influenced by the presence of a nuclear stellar bar (Zaritsky et al. 1993). These regions can be further classified according to their level of SF activity (as traced by, e.g., Hα) and/or gas flows, which we determine using the present-day torque profile (Meidt et al. 2013).

Here, we analyze seven regions within the PAWS field where we expect the molecular gas to experience different dynamical effects (see Figure 3). We define the different spiral arm regions according to the direction of gas flows driven in response to the underlying gravitational potential, which we derive from a map of M51's stellar mass distribution (Meidt et al. 2012). The widths of the spiral arms are defined with respect to the observed gas kinematics. We determine the zone of enhanced spiral streaming centered around the arm by measuring the (rotational) auto-correlation of azimuthal streaming velocities in the PAWS field (Colombo et al. 2013b). We construct azimuthal profiles of the auto correlation signal in a series of radial bins and take the width of the signal at 95% maximum as our measure of the kinematic arm width. The average kinematic width from along the two arms is centered on the spiral arm ridge line, defined by eye using the PAWS map of CO peak brightness. Both the location of the ridge and the width are assumed to be symmetric. The interarm region is divided into upstream and downstream by the midpoint of the spiral arm ridge lines. The definition of the spiral arm regions is based on the identification of distinct spiral patterns within the galactic disk (cf. Vogel et al. 1993; Shetty et al. 2007; Meidt et al. 2008; Dobbs et al. 2010) that we refined and describe in detail elsewhere (Meidt et al. 2013; Colombo et al. 2013b).

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The seven zones that we use to conduct our analysis are as follows.

• 1.  
  Nuclear bar. The region at galactocentric radii R < 23''. The boundary is defined by the bar corotation resonance, inside of which the bar exerts negative torques and drives gas radially inward.
• 2.  
  Molecular ring. The region 23 < R < 35''. Here, the gas is influenced by both the bar and innermost spiral arms. Outside the bar corotation resonance, gas is driven radially outward, while the spiral drives gas radially inward inside its own corotation. These opposing torques accumulate gas in a ring-like structure. The region hosts some of the most active high-mass SF in M51.
• 3.  
  Inner density-wave spiral arm. The arm region 35 < R < 55''. The inner boundary is defined by the molecular ring and the outer boundary is the corotation radius of the density-wave spiral arms. Within this zone, gas is driven radially inward by negative spiral arm torquing. Despite the high gas surface densities in this region, there is little SF—as traced by Hα and 24 μm emission—that is directly associated with the brightest CO emission (Schinnerer et al. 2013).
• 4.  
  Outer density-wave spiral arm. The arm region 55 < R < 85''. This region extends from the density-wave corotation resonance to the start of the material spiral. Within this zone, gas is driven radially outward by positive spiral arm torquing.
• 5.  
  Material spiral arm. The arm region R > 85''. This region extends from the boundary of positive arm torques associated with the density wave spiral to the edge of the PAWS field. There is some indication that gas flows radially inward in this zone.
• 6.  
  Interarm region, downstream of the spiral arms.
• 7.  
  Interarm region, upstream of the spiral arms.

Finally, we note that the projected area of the seven regions is still quite large (between ∼2 and 17 kpc²) and each contains a statistically significant number of GMCs (≳ 100). The PDFs of CO emission in these regions are therefore more comparable with the PDFs of simulated galactic disks than the PDFs of individual clouds.

Figure 4 shows that the I(CO) PDFs for different M51 environments exhibit diverse shapes. The panels of Figure 4 are ordered such that the PDF amplitudes decrease from top left to bottom right, which reflects the fact that CO emission is more prevalent in the arms and central region of M51 than in the interarm region. The PDFs also tend to decrease in width, indicating that the fraction of pixels with bright CO emission declines with the overall frequency of CO detections. The PDFs of the spiral arms are notably wider than for the interarm environments; it is also evident that the PDF corresponding to the first spiral pattern (A1, i.e., the density wave spiral arm) is wider than the PDF for the second spiral (A2, the material arm). Since I(CO) is the integral of the CO brightness over the line profile, this variation in the PDF width would seem consistent with the results of previous studies (e.g., Garcia-Burillo et al. 1993; Kuno & Nakai 1997; Aalto et al. 1999; Schuster et al. 2007) that find that the average CO-integrated intensity and CO linewidth decreases with increasing distance along the arms and from the arm to the interarm region.

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The differences in the width of the PDFs among M51 environments are reflected in the IDI values that we derive, which become more positive as the number of pixels with I(CO) > 60 K km s⁻¹ increases (see Table 2). The development of more high-brightness emission appears to be accompanied by a change in the PDF shape: an LN function is a better description of the PDFs in the interarm region than in the spiral arm and ring regions. The I(CO) PDFs for the first spiral pattern, especially inside corotation (A1I), appear more like broken or truncated power laws than LN functions. The PDFs in the center of M51 also diverge from an LN shape: the I(CO) distribution in the molecular ring is essentially flat between ∼20 and 150 K km s⁻¹, while the PDF of the bar is the only region with an unambiguous decline at low intensities (I(CO) ≲ 50 K km s⁻¹). Several PDFs appear truncated near I(CO) ∼ 300 K km s⁻¹; this is seen most clearly in the molecular ring, but the distributions in the first spiral arm regions also decline steeply for I(CO) ≳ 300 K km s⁻¹.

In Figure 5, we present the PDFs of CO brightness for the different M51 environments. The distributions are more uniform than those of integrated intensity, but variations similar to those identified for the I(CO) PDFs are still evident. The PDF amplitude tends to decrease from panels (a) to (h) and only the interarm regions and second spiral arm (A2) yield PDFs that are approximately LN across the observed range of T_mb values (see Table 1). As noted for the I(CO) PDFs, regions with a relatively wide T_mb PDF (e.g., the ring, bar, and spiral arms) have more positive BDIs and also tend to diverge from an LN shape, in this case developing a pronounced change of slope near T_mb ≳ 6 K. This effect is most clearly seen for the PDFs of the molecular ring (panel (b)) and the first spiral pattern inside corotation (panel (c)), but generally it appears that an increase in the fraction of high-brightness CO emission is associated with a PDF that more resembles a truncated power law than an LN function. We discuss this result in relation to similar trends observed for PDFs of CO emission in the Galaxy in Section 5.1.

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Finally, we can compare the PDFs of CO-integrated intensity and CO brightness for the inner disk of M51 with the corresponding PDFs for other nearby galaxies. As we discuss in Appendix B, the shape of the PDF is sensitive to the resolution and sensitivity of the data. Prior to constructing the PDFs, we therefore degraded the M51 and LMC data cubes to the same spatial resolution as the M33 cube (∼53 pc) and folded the M33 and LMC data cubes along the velocity axis to the same channel width as the M51 cube (5  km s⁻¹). We interpolated all the cubes onto an (x, y) grid with the same pixel dimensions in physical space (15 × 15 pc). Significant emission was identified according to the method outlined in Section 3. The resulting masks were applied to the original data cubes and the integrated intensity images were constructed by summing unblanked pixels across the full velocity bandwidth of each survey.

The PDFs obtained from the I(CO) maps of M33, the LMC and M51 are shown in the left panels of Figure 6. The shape of the I(CO) PDF for M33 is highly uncertain due to the modest sensitivity of the BIMA+FCRAO data cube. Nonetheless, it is obvious that CO emission in the LMC and M33 are alike in the sense that the maximum observed I(CO) intensities are ∼10 K km s⁻¹ at the resolution of our analysis and not a few times 100 K km s⁻¹, as observed for M51. The I(CO) PDF for the LMC appears to be well represented by a narrow LN function with mean 〈I(CO)〉 = 2 K km s⁻¹and a logarithmic width x = 0.3. Since MAGMA is a targeted rather than a spatially complete survey of the LMC disk (see Section 2), the amplitude of its PDF is biased high compared with that of M51 and M33; normalizing by the full projected area of the LMC's H i disk, rather than the MAGMA field of view, would reduce the amplitude by more than an order of magnitude. The MAGMA survey strategy of targeting the brightest clouds in the NANTEN catalog also means that the shape of the PDF is biased toward high CO intensities; extending MAGMA to fainter clouds would recover a greater fraction of pixels with low CO brightness and narrow the PDFs in Figures 6(c) and (f). Relative to the observed I(CO) PDF of the LMC, the M51 I(CO) distribution peaks at higher CO intensity, 〈I(CO)〉 ∼ 12 K km s⁻¹, and is also wider by a factor of ∼2 in the logarithm (see Table 1). The difference between the best-fitting LN function derived for the M51 distribution in Figure 6(a) and that in Figure 2(a) is consistent with the differences that we observe for different masking techniques for identifying significant emission within the data cube (see Appendix A).

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The PDFs of CO brightness for the three galaxies are presented in the right panels of Figure 6. For M51, we fit the distribution of CO brightness with an LN function with mean 〈T_mb〉 = 0.8 K and logarithmic width x = 0.4. This is consistent with the best-fitting LN function derived for the PAWS data in Figure 2(b), which used a more sophisticated masking technique to identify significant emission. Alternatively, a broken power law with a shallow slope of ∼ − 1.3 between T_mb = 1 and 4 K and a much steeper slope (∼ − 5.0) above 4 K also fits the T_mb distribution for M51 reasonably well. For the LMC, the best-fitting LN function has mean 〈T_mb〉 = 0.1 K and logarithmic width x = 0.3; there is no sign of a truncation. A simple power law with a slope of ∼ − 2.2 also adequately represents the distribution. There are insufficient pixels with significant emission in the M33 data cube to attempt to fit the T_mb PDF with an LN function. A simple power law with a slope of ∼ − 3.6 provides a reasonable fit to the distribution for pixel values T_mb ≳ 0.3 K. We note that the slopes of the power laws that fit the T_mb PDFs reproduce the trends observed for the GMC mass distribution in the three galaxies, i.e., a shallow slope for M51 (below T_mb ∼ 4 K) and a much steeper brightness distribution for the low-mass galaxies (A. Hughes et al., in preparation). We discuss the connection between the shape of the CO PDFs and the GMC mass distribution further in Section 5.2.

Both high resolution and wide-field coverage are necessary to characterize the CO emission in galaxies on spatial scales that are relevant for SF, hence few extragalactic studies have produced PDFs of CO brightness and/or integrated intensity that represent a significant fraction of a galactic disk. One exception is an analysis of the LMC by Wong et al. (2011), which found that the I(CO) PDF was roughly consistent on ∼10 pc scales, with a narrow LN function ($\langle \Sigma _{{\rm H}_{2}} \rangle = 16$ M_☉ pc⁻², x ∼ 0.3 dex) at high column densities. The authors noted some evidence for a truncation around $\Sigma _{\rm H_{2}} = 200$ M_☉ pc⁻², which they tentatively attributed to opacity effects in the ¹²CO(J = 1 → 0) line. Notably, however, the MAGMA data show no evidence for a power-law excess at high column densities, as has been observed on pc scales within star-forming Galactic clouds (e.g., Kainulainen et al. 2009), suggesting that the structure of LMC molecular clouds is still dominated by turbulence on ∼10 pc scales.

The CO emission in M51 itself has been analyzed by numerous authors (e.g., Vogel et al. 1988; Rand 1993; Aalto et al. 1999; Helfer et al. 2003; Shetty et al. 2007). With the exception of the recent survey by Koda et al. (2009), however, observations with high spatial resolution have mostly focused on a spiral arm segment (e.g., Schinnerer et al. 2010; Egusa et al. 2011), while studies covering a significant fraction of the disk (e.g., Garcia-Burillo et al. 1993; Schuster et al. 2007) have had a resolution of a few hundred pc or greater, i.e., insufficient resolution to resolve individual GMCs. Rather than examining PDFs, these lower-resolution studies have typically examined radial trends in properties such as the gas velocity dispersion, Toomre's Q parameter, and the molecular gas depletion time $\tau _{\rm H_{2}}$ (e.g., Schuster et al. 2007; Hitschfeld et al. 2009). From the PDFs in Section 4.3, we would expect to observe a radial decline in the average value of $\Sigma _{\rm H_{2}}$ constructed from azimuthal averages, since the fraction of bright CO emission (as parameterized by the IDI values) decreases along the spiral arms and also because the interarm region occupies an increasing fraction of the disk area with increasing galactocentric radius. Since the mass surface density is an important input for the determination of Toomre's Q and $\tau _{\rm H_{2}}$, radial trends in these quantities may likewise reflect a combination of differences between the arm and interarm zones and variations along the spiral arms (an interpretation that would seem to be supported by the map of Toomre's Q presented in Figure 15 of Hitschfeld et al. 2009, for example).

More generally, we note that differences in the basic properties of the CO emission (i.e., peak brightness, velocity dispersion) between the arm and interarm regions of M51 have been reported by several previous studies and often interpreted as evidence for changes in the physical state of the molecular gas as it passes through the spiral arms. While the precise identification of M51's arm and interarm zones varies (usually because M51's gaseous spiral arms appear wider at lower spatial resolution), CO emission in the interarm has been shown to have lower velocity dispersion (e.g., Garcia-Burillo et al. 1993; Aalto et al. 1999; Hitschfeld et al. 2009), lower peak brightness (e.g., Garcia-Burillo et al. 1993; Tosaki et al. 2002), lower SF efficiency (as inferred from the ratio of H α to ¹²CO(J = 1 → 0) emission, e.g., Rand 1993; Tosaki et al. 2002), higher ¹²CO(J = 1 → 0)/¹³CO(J = 1 → 0) isotopic ratios (e.g., Tosaki et al. 2002), and lower ¹²CO(J = 2 → 1)/¹²CO(J = 1 → 0) transitional ratios (e.g., Koda et al. 2012) than emission in the spiral arms. Most of these results suggest that molecular gas in the interarm region has a lower characteristic density than gas within the spiral arms.

Spatial resolution is rarely a limitation for studies of molecular gas in the Milky Way, although previous analyses of Galactic CO emission have tended to focus on the physical properties of GMCs (e.g., Solomon et al. 1987; Roman-Duval et al. 2010), which were quickly recognized to be the preferred—perhaps only—site of high-mass SF in the Galaxy. Recent work has emphasized, however, that faint, spatially extended CO emission contributes significantly to a region's total CO flux (e.g., Goldsmith et al. 2008; Heyer et al. 2009; Liszt et al. 2010). Very recently, Sawada et al. (2012b) presented PDFs of ¹²CO(J = 1 → 0) and ¹³CO(J = 1 → 0) brightness for an 08 × 08 field toward the Galactic plane at l ≈ 38°, observed using the Nobeyama Radio Observatory (NRO) 45 m telescope. These authors find clear differences between the PDFs constructed from the emission at radial velocities corresponding to the Sagittarius arm and those corresponding to the interarm regions, showing that the structural properties of the molecular gas vary in response to Galactic structure. They conclude that compact, high-brightness CO structures develop downstream of the molecular spiral arms, where they are spatially coincident with signatures of active SF (e.g., H ii regions). Sawada et al. (2012b) point out that their result is a rediscovery of a conclusion that had already been drawn by earlier studies. Sanders et al. (1985), for example, observed a connection between the location of Galactic molecular clouds in longitude–velocity space and their peak brightness temperature: "hot" (i.e., high-brightness) clouds were preferentially located in the spiral arms traced by H ii regions (Georgelin & Georgelin 1976). Egusa et al. (2011) present a qualitatively similar scenario for a ∼2 kpc segment of M51's inner spiral arm, showing that both high-mass (∼10⁶ M_☉) CO clumps and H ii regions are preferentially located downstream of the spiral arm ridge line (note, however, that the high-brightness CO structures described by Sawada et al. (2012b) occur on much smaller spatial scales than the structures observed by Egusa et al. (2011) in M51).

Although the spatial resolution of the PAWS data is considerably worse than that of Galactic surveys, the PDFs in Figure 5 show similar trends as those reported by Sawada et al. (2012b). As we noted in Section 4.3, the PDFs of the interarm region resemble narrow LN functions, while the PDFs in the central and spiral arm regions reach higher maximum intensities and tend to be better represented by broken power laws. These variations in shape are reflected by the BDI and IDI values: low-brightness emission dominates the total flux in both the arm and interarm environments, but the relative contribution from bright emission increases in the spiral arms. Bright emission is most dominant in the center of M51, where ∼25% of the total CO flux arises from pixels with T_mb > 4 K (in comparison, less than 5% of the emission in the interarm region is brighter than 4 K). Similar to Sawada et al. (2012b), we find that the BDI and IDI values are higher on the downstream side of the spiral arms than on the upstream side. Tracers of high-mass SF, e.g., Hα, 24 μm, and far-ultraviolet emission, also appear to be preferentially located downstream of arms (Schinnerer et al. 2013), again consistent with the Galactic results. We discuss the connection between SF and the shape of the CO PDFs for different M51 environments in more detail in Section 5.3.

The fact that a similar relationship between CO emission properties and spiral arm structure is observed in both M51 and the Milky Way would seem to support the argument by Sawada et al. (2012b) and Sawada et al. (2012a) that the arm–interarm variations they observe reflect genuine changes in the density distribution of the Galactic CO-emitting gas. One important caveat, however, is that the PAWS data have much lower spatial resolution (∼40 pc) than the Galactic NRO data (≲ 1 pc). In particular, the lower CO brightness temperatures that we observe in M51 (T_mb ≲ 16 K) indicate that our PAWS measurements reflect a combination of the average kinetic temperature and the filling factor of the CO-emitting gas within a resolution element. Variations in CO brightness on ≲ 1 pc scales, in contrast, should mostly track variations in gas temperature and/or density since beam dilution should be minimal on these scales. Some of the variation between high and low BDI values in M51 will reflect changes in the covering fraction of the CO emission for the different M51 environments, as well as differences in the intrinsic brightness temperature of the CO-emitting structures that are more directly comparable with the variations described by Sawada et al. (2012b).

In Section 4.3, we described variations in the characteristic shape of the CO PDFs for different M51 environments. To what extent are these differences manifested in variations of the properties of GMCs within each environment or of the ensemble properties of a GMC population (e.g., its mass distribution)? Intuitively, we would expect some connection between GMCs and the presence of high-brightness CO emission, since most methods for identifying GMCs from CO data cubes invoke either a brightness threshold or local maximum in the CO brightness distribution in order to define cloud structure. The connection may be rather indirect, however, since the fraction of CO emission above the PAWS sensitivity limit that is associated with the observationally defined GMCs varies between 40% and 65%, depending on galactic environment (Colombo et al. 2013a).

We examined the relationship between GMC properties and the shape of the CO PDFs using the cloud catalog presented by Colombo et al. (2013a) and the BDI and IDI values calculated in Section 4.3. Since the BDI and IDI values themselves exhibit a tight one-to-one correlation, for simplicity we refer only to the IDI values in the following sections. We illustrate some of these correlations in the left column of Figure 7. Environments where bright CO emission is more dominant (i.e., with more positive IDI values) are associated with a higher maximum GMC mass M_{gmc, 95}, which we estimate using the 95th percentile of the GMC virial mass distribution (panel (a)), a greater number surface density of GMCs $\mathcal {N}_{{\rm gmc}}$ (panel (b)), and a higher average surface density for individual GMCs $\langle \Sigma _{\rm H_{2}} \rangle$ (panel (c)). The IDI is also strongly correlated with the slope of the GMC mass spectrum γ_gmc: in environments with more bright emission, the mass spectrum is shallower (panel (d)). For observations with low resolution (i.e., where a single resolution element is much larger than the characteristic size of a GMC), a good correlation between the prevalence of bright CO emission and the mass and mass surface density of identified cloud structures might arise simply due to higher filling factors of CO emission, i.e., increases in the measured CO-integrated intensity reflect a greater number of CO-emitting clouds within the telescope beam, rather than changes in the intrinsic properties of GMCs. We do not consider this to be the cause of the good correlations in Figure 7, however, since the PAWS resolution (∼40 pc) is well matched with the characteristic size of an individual Galactic GMC (50 pc, e.g., Blitz 1993) and considerably less than the typical spacing between the identified GMCs (a few times hundred pc or greater; Colombo et al. 2013a). The peak CO brightness temperatures of the GMCs range from ∼2 to 16 K, which is comparable with the values observed for Galactic GMCs (5–10 K; Solomon et al. 1987). Since the molecular gas in M51 GMCs appears to have a similar kinetic temperature as in Galactic GMCs (∼10 K; Schinnerer et al. 2010), this again suggests that the filling factor of the CO emission in M51 GMCs is close to unity.

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It is remarkable that the GMC properties are often more strongly correlated with the shape of the CO PDFs than other quantities with which they might also be expected to correlate. In particular, we note that M_{gmc, 95} and γ_gmc are more tightly correlated with the IDI than with the total CO luminosity or the total number of GMCs in each region (the latter plots are not shown). This would seem to confirm that an increase in the maximum GMC mass is not simply due to an increase in the available gas reservoir and more adequate sampling of the top end of the GMC mass function (i.e., a sample size effect) and that the good correlation among the IDI, M_{gmc, 95}, and γ_gmc arises because the density distribution of the molecular ISM plays a role in regulating the GMC mass distribution. It is also noteworthy that the IDI increases with both the number density of GMCs $\mathcal {N}_{{\rm gmc}}$ and the average surface density of the individual clouds $\langle \Sigma _{\rm H_{2}} \rangle$. This suggests that a distinction that is sometimes drawn by empirical studies of extragalactic SF between an increase in the number of GMCs per resolution element and variations in the H₂ surface density on the scale of individual clouds (e.g., Bigiel et al. 2008) is somewhat artificial: at least in the inner disk of M51, clouds in environments with more GMCs per unit area also tend to have higher average surface densities.

The good correlation between the IDI and γ_gmc in panel (d) of Figure 7 is especially noteworthy. Considerable theoretical and observational effort has been devoted to showing how the shape of the stellar initial mass function might be inherited from the density structure of interstellar gas (e.g., Hopkins 2012; Chabrier & Hennebelle 2010 and references therein), with many studies adopting the shape of the GMC mass function as a description of the latter. A major problem with using GMC mass spectra for this purpose, however, is that the decomposition algorithm has a major impact on the identification and parameterization of cloud structures and hence the shape of the resulting mass distribution (e.g., Wong et al. 2011; Reid et al. 2010). Moreover, many widely used decomposition methods are not flux conservative, discarding a considerable fraction of the CO emission that is unambiguously detected within a spectral line data cube. As a description of how dense gas is distributed within galaxies, PDFs avoid these ambiguities even though, as we show in Appendix B, the resolution of the observational data must be well matched to the physical scales of interest in order to accurately capture the shape of the PDF. The PDF also conveys no information about the characteristic size of dense gas structures, moreover, so a more complete description of the organization of the dense ISM strictly requires an analysis of the CO PDF in conjunction with a metric such as the spatial power spectrum (as suggested by, e.g., Bournaud et al. 2010). Nonetheless, the plots in the left column of Figure 7 indicate that there is a strong relationship between the shape of the PDF and the mass distribution and properties of GMCs within M51 environments, suggesting that in real galactic disks the presence of bright emission and the development of massive molecular structures are physically linked. Testing whether a similar connection between the GMC mass function and shape of the PDF holds across a range of galaxy types is a project that should become feasible once ALMA acquires cloud-scale imaging of CO emission across the full galactic disk for a large sample of nearby galaxies.

Another motivation for constructing the CO PDFs across a range of M51 environments is to assess whether there are connections among empirical tracers of SF activity and the density distribution of molecular gas. Several empirical calibrations for the SF rate exist in the literature (for a detailed comparison of the limitations and assumptions of different methods, see Leroy et al. 2012), but here we restrict our analysis to comparing the shape of the CO PDFs with the properties of young stellar clusters identified by Chandar et al. (2011) using multi-color images of M51 obtained by the Advanced Camera for Surveys onboard the Hubble Space Telescope (Mutchler et al. 2005). The interested reader is referred to Chandar et al. (2011) for a description of the methods used to select clusters and to derive physical quantities such as their age and mass. The relationship between CO emission and other SF tracers within the PAWS field is discussed in several companion papers (Meidt et al. 2013; Schinnerer et al. 2013).

As for GMCs, we find evidence for a strong connection between the prevalence of bright CO emission and M51's young (τ ⩽ 10⁷ Myr) cluster population. In particular, more positive IDI values are associated with a higher maximum young cluster mass M_{yc, 95} (defined analogously to M_{gmc, 95}) and with a higher number surface density $\mathcal {N}_{{\rm yc}}$ and combined mass surface density $\mathcal {M}_{{\rm yc}}$ of young clusters (Figures 7(e)–(g)). The origin of these trends would seem to lie in a physical—as opposed to statistical—connection between young clusters and GMCs: M_{yc, 95}, $\mathcal {N}_{{\rm yc}}$, and $\mathcal {M}_{{\rm yc}}$ are better correlated with the average GMC mass surface density ($\langle \Sigma _{\rm H_{2}} \rangle$) and maximum GMC mass (M_{gmc, 95}) than with the total number of young clusters or GMCs within each M51 environment.

An exception to the good correspondence between the shape of the CO PDFs and the properties of GMCs and young stellar clusters is the slope of the cluster mass distribution: while there is a clear trend for the GMC mass spectrum to become shallower in regions where bright CO emission is more prevalent, a connection between the slope of the young cluster mass function and the shape of the CO PDFs is less obvious (see panels (d) and (h) of Figure 7). There is some indication that the mass distribution of the young cluster populations in the arm and interarm regions follows the same trend with IDI as GMCs, but within the central kpc of M51 (i.e., the molecular ring and nuclear bar regions) the young cluster mass distributions are steep (≲ − 2.5) even though bright CO emission is relatively dominant there.

In Figure 8, we plot the slope of the young cluster mass spectrum directly against the slope of the GMC mass spectrum for the different M51 environments. Various techniques for estimating the slope of the mass spectrum have been used by empirical studies of young clusters and GMCs (e.g., differential versus cumulative mass distributions, bins of equal width versus bins containing an equal number of objects) and the derived slope is known to be sensitive to factors such as cloud decomposition algorithm, the adopted low-mass completeness limit, and undersampling and/or the existence of a physical truncation to the distribution at high masses. We constructed both differential and cumulative mass distributions for the GMC and young cluster populations and, for each variant, we estimated the slope multiple times using different binning strategies (in the case of the differential mass distributions) and mass ranges to calculate the fit. We describe these tests more fully in Appendix C. The two panels of Figure 8 show the results using a cumulative representation (panel (a)) and a differential representation (panel (b)) for the mass distributions. In both panels, the error bars reflect the dispersion in the estimated slopes of the GMC and young cluster mass spectra in each environment. Despite the systematic uncertainties that limit the accuracy of any individual measurement of the mass distribution slope, we can therefore confidently draw two conclusions from Figure 8. The first is that while there is reasonable agreement between γ_gmc and γ_yc for the arm and interarm environments of M51, the slope of the mass distribution is not universal, i.e., the same values of γ_gmc and γ_yc do not hold everywhere within M51. Second, agreement between the slopes of the GMC and young cluster mass distributions is not ubiquitous. When averaged across the entire PAWS field and for some of the arm and interarm regions, γ_gmc and γ_yc agree to within ∼0.3 dex, but for the molecular ring and upstream environments $\gamma _{{\rm gmc}} \not\approx \gamma _{{\rm yc}}$, regardless of the method used to represent the mass functions.

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The trends in Figure 8 are remarkable since the observed similarity between γ_gmc ≈ −1.7 and γ_yc ≈ −2.0 is frequently cited as evidence for the weak mass dependence of both the efficiency of SF in GMCs and probability of cluster disruption. In their investigation of stellar feedback and disruption of GMCs, for example, Fall et al. (2010) derive relations between γ_gmc and γ_yc, which for isolated, bound systems in the absence of magnetic support are linked via the slope of the mass versus size relationship of GMCs. Fall et al. (2010) examine the limiting regimes of energy- and momentum-driven feedback, arguing that the constant surface density of GMCs (M∝R²) ensures that γ_gmc ∼ γ_yc regardless of the type of feedback that dominates GMC disruption.

In a globally averaged sense, the young cluster and GMC populations of M51 would seem to conform to the model outlined by Fall et al. (2010). In this case, γ_gmc = −1.7, γ_yc = −2.1, and, for momentum-driven feedback, the predicted slope of the GMC mass-size relation is α = 1.8, in acceptable agreement with the observed value (α_obs = 2.1; Colombo et al. 2013a). However, our results for individual environments within M51 suggest a more nuanced interplay among molecular gas, young clusters, and galactic structure. Downstream of the spiral arms, for example, γ_gmc ∼ γ_yc ≈ −2.0 and α_obs = 1.8, in good agreement with the model prediction for energy-driven feedback. Yet upstream of the spiral arms (where γ_gmc ∼ γ_yc ≈ −3.0), in the molecular ring (γ_gmc ≈ −1.4, γ_yc ≈ −2.7) and in the first spiral arm pattern of M51 (γ_gmc ≈ −1.5, γ_yc ≈ −2.1), the exponent of the mass–size relationship predicted by Fall et al. (2010) is too shallow compared with the observed value of 0.4–1.0 dex, regardless of the assumed feedback mechanism. This suggests that there may be regions within galactic disks where physical processes that would seem to be excluded by Fall et al. (2010), e.g., cluster coalescence, mass-dependent cluster/GMC disruption, or a dominant role for energy-driven feedback, are in fact important. More generally, however, it highlights how valuable information can be lost in the calculation of galaxy-wide averages. With data sets that yield statistically significant samples of GMCs and other star-forming phenomena within ∼kpc-scale regions, it is timely that physical quantities (for example, based on a consideration of galaxy dynamics and/or ISM properties) determine the environments where GMC properties and extragalactic SF are investigated, rather than relying solely on radial profiles and/or apertures that are "blind" to their location with respect to galactic structure.

Our analysis in this paper was prompted, in part, by numerical simulations showing that the gas density distribution in galactic disks is well represented by a single LN function spanning several orders of magnitude (e.g., Wada & Norman 2007). If this is an accurate description of real galactic disks, then the result offers insight into the origin of the KS law, which can be reproduced from a LN density PDF with a limited number of plausible assumptions, such as a critical density threshold for SF (Elmegreen 2002) or a direct proportionality between the local gas and SF rate densities (e.g., Kravtsov 2003). While the overall I(CO) and T_mb PDFs for the PAWS field are roughly LN, our results in Section 4.3 show that this average PDF shape obscures considerable diversity among the PDFs observed for different ∼kpc-sized regions within M51. Since the regions that we use to investigate the PDFs are defined according to dynamical criteria, our basic result is that large-scale dynamical processes in M51's inner disk have an observable effect on the density (and column density) distribution of M51's molecular ISM.

There are several further characteristics of our observed CO PDFs that are noteworthy in relation to the simulation results. For example, Wada & Norman (2007) find an increase in the logarithmic width of the PDF of ∼0.3 dex for an order of magnitude increase in the mean gas density. This is roughly consistent with the variation in the width of the I(CO) PDFs of M51 and the LMC, between which the average H₂ column density also varies by a factor of ∼10 (see Section 4.4). However, we caution that several effects limit the extent to which we can compare our observed CO PDFs with the density PDFs from simulations. The first is that the simulations describe gas density across six orders of magnitude, corresponding not only to the molecular ISM but also to the atomic and warm ionized phases. In practical terms, this makes the numerical result difficult to verify, since different observational tracers must be used to probe different phases of the interstellar gas and each of them are sensitive to a much narrower range of densities (and column densities) than the full dynamic range of the simulated PDFs.

The few simulations that include explicit treatment of molecular chemistry tend to find a range of densities between ∼1 and 10³ cm⁻³ for the H₂ gas in galactic disks and that the distribution exhibits a sharp cut-off below the H₂ self-shielding limit (n ≲ 5  cm⁻³; see, e.g., Figure 11 of Dobbs et al. 2008). The use of ¹²CO(J = 1 → 0) emission to trace the H₂ column density should narrow the observed PDF even further, since at moderately low extinction (A_V ∼ 1 mag, e.g., Wolfire et al. 2010) H₂ can self-shield while CO molecules are photodissociated. At high H₂ column densities (N(H₂) ≳ 10²² cm⁻²), on the other hand, the CO-emitting structures within a GMC will start to overlap and shadow each other, leading to a saturation of I(CO) intensities (e.g., Shetty et al. 2011). Feldmann et al. (2012) show that this saturation should occur at I(CO) intensities near a few 100 K km s⁻¹. In general, however, the I(CO) PDFs in Figure 4 show no evidence for a peak at these intensities caused by a "pile-up" of saturated pixels, but instead more closely resemble their pure N(H₂) PDFs after they exclude pixels with low CO-integrated intensities (<0.2 K km s⁻¹; see Figure 8 of Feldmann et al. 2012). We suggest that this is because the typical CO linewidths in M51 are much larger than the two possibilities considered by Feldmann et al. (2012; i.e., a constant linewidth of 3 km s⁻¹or a virial scaling of the linewidth with mass surface density). As a consequence, the majority of the CO-emitting molecular structures in M51 do not shadow each other in velocity space and hence I(CO) remains a relatively good tracer of the H₂ column density. This is consistent with recent studies of gas and dust in M51, which suggest that the X_CO factor has a roughly Galactic value throughout the disk (e.g., Schinnerer et al. 2010; Tan et al. 2011; Mentuch Cooper et al. 2012).

What insights can models provide regarding the diversity of distribution shapes that we observe? As noted in the Introduction, the PDF for supersonically turbulent isothermal gas is LN when the influence of gravity is negligible. In this case, the logarithmic width of the PDF x varies with the Mach number $\mathcal {M}$ according to $x^{2} \approx \ln (1 + 0.25 \mathcal {M}^{2})$ (Padoan et al. 1997). Across a galactic disk, however, the temperature and average density of the molecular gas will vary with location. Thus, a more realistic expectation for the observed density PDF on global to kpc scales may be the convolution of the local LN PDF (reflecting the distribution of densities within a region over which the average gas density ρ_ave and Mach number are relatively constant) with the PDFs of ρ_ave and $\mathcal {M}$ within the galactic disk. Elmegreen (2011) recently considered such a model, presenting convolution PDFs for several idealized cases of gas clouds with different radial density profiles and variable Mach numbers (see his Figure 1). The resulting PDFs clearly diverge from a pure LN shape. As clouds become more centrally condensed (and hence more dominated by self-gravity), the convolution PDFs develop a power-law tail at high densities, with a slope that varies inversely with the slope of the density profile. If the Mach number decreases at higher average densities, on the other hand, the convolution PDF appears truncated relative to a pure LN since the local PDFs get narrower with increasing ρ_ave.

In broad terms, the analysis by Elmegreen (2011) suggests that the diverse shapes of the I(CO) PDFs in Figure 4 reflect large-scale variations in the average density, temperature, and/or velocity fluctuations for the molecular gas within different M51 environments. In reality, these properties are likely to vary simultaneously, so attributing a specific PDF morphology to a variation in one physical quantity and/or process is not straightforward. Nevertheless, it is suggestive that the I(CO) PDFs in the interarm region—where we might expect the temperature, density, and velocity structure of the molecular gas to be determined by cloud-scale processes—resemble the pure LN shapes expected for isothermal supersonically turbulent molecular gas, whereas the I(CO) PDFs in M51's spiral arms—where the molecular gas not only reaches higher densities, but its velocity structure can be influenced by large-scale dynamical effects such as streaming motions and the spiral shock—more obviously diverge from a LN shape. The molecular ring region, where the I(CO) PDF is very broad and almost flat-topped, is arguably the extreme case both in terms of dynamical effects and SF activity. Although shear and large-scale non-circular motions should be low in the ring, the molecular gas accumulates here due to opposing bar and spiral torques and the average gas velocity dispersion is relatively high (Colombo et al. 2013a). The level of SF activity in the ring is also high (Schinnerer et al. 2013), so feedback from nascent stars may also have a strong effect on the distribution of gas densities in this region.

Finally, we note that Hopkins et al. (2012) have recently shown that the dominant mode of stellar feedback (and not just the total amount of SF) has an observable effect on the shape of the density PDF for the cold gas component in their simulated galaxies. Like Wada & Norman (2007), they find that the width of the density PDF decreases for systems with lower average gas densities. The density PDFs of their simulated gas disks show striking departures from lognormality (see their Figures 10 and 11), however, which they attribute to the inclusion of cooling, self-gravity, and a physically motivated implementation of different feedback mechanisms in their simulations. In particular, they find that radiation pressure is crucial for suppressing a pile-up of gas with high densities (n ≳ 10⁴ cm⁻³), since pure gas heating (e.g., by supernovae, stellar winds, and H ii photoionization) is ineffective in disrupting dense gas clumps where the cooling time is much shorter than the dynamical time. While the overall shapes of the I(CO) and T_mb PDFs for different M51 environments almost certainly reflect the combined action of several distinct physical processes, the absence of a secondary peak in the PDFs at high CO intensities would seem to confirm that the dominant feedback mechanism in M51 must be effective at preventing the build-up of high-density material.