Cloud-scale ISM Structure and Star Formation in M51

Schinnerer et al. (2013) and Pety et al. (2013) present PAWS, which mapped CO (1−0) emission from the central region of M51 at $1\buildrel{\prime\prime}\over{.} 16\times 0\buildrel{\prime\prime}\over{.} 97\sim 1\buildrel{\prime\prime}\over{.} 06\sim 40$ pc resolution with $\sim 5$ km s⁻¹ velocity resolution. PAWS includes short and zero-spacing information. Schinnerer et al. (2013) also summarize the multiwavelength data available for M51, with references (see their Table 2).

We also use broad-band maps of IR emission from Herschel and Spitzer. These were obtained as part of the Spitzer Infrared Nearby Galaxy Survey (Kennicutt et al. 2003) and the Herschel Very Nearby Galaxies Survey (Mentuch Cooper et al. 2012).

Integrated CO intensity: At $30^{\prime\prime}$ resolution, we use the PAWS single dish map (Pety et al. 2013) to measure the integrated CO intensity. At $10^{\prime\prime}$, we convolve the combined interferometer and single dish cube to a coarser $10^{\prime\prime}$ resolution and measure the integrated intensity from this degraded map. As discussed by Pety et al. (2013), the deconvolution of the hybrid 30m+PdBI map recovers 99% flux of the galaxy observed with the IRAM 30 m.

To collapse the $30^{\prime\prime}$ and $10^{\prime\prime}$ cubes to integrated intensity measurements, we sum over a broad velocity window from −70 to +70 km s⁻¹ about the local mean velocity. The signal-to-noise in CO (1−0) is very high, so some empty bandwidth is not a concern. We estimate the associated uncertainty by measuring the rms noise of the convolved line cube from the signal-free region. Then the statistical uncertainty in the integrated intensity is the sum in quadrature of the per-channel intensity noise across all channels in the velocity integration window multiplied by the velocity width of a channel.

Total infrared (TIR) surface brightness: We convolve the IR data to have Gaussian beams using the kernels of Aniano et al. (2011). Then, using the formulae of Galametz et al. (2013), we combine Spitzer 24 μm and 70 μm intensities with Herschel 160 μm and 250 μm intensities to estimate a TIR luminosity surface brightness, ${{\rm{\Sigma }}}_{\mathrm{TIR}}$, for each resolution element. This is our basic measure of star-formation activity throughout this paper.

At $10^{\prime\prime}$ resolution, we can only use the Herschel 70 μm data. We calculate the coefficient to translate I₇₀ to ${{\rm{\Sigma }}}_{\mathrm{TIR}}$ by comparing the two quantities at $30^{\prime\prime}$ resolution, where we know ${{\rm{\Sigma }}}_{\mathrm{TIR}}$ from the four-band calculation following Galametz et al. (2013). In the PAWS field, the ratio ${{\rm{\Sigma }}}_{\mathrm{TIR}}/{I}_{70}$ varies modestly as a function of radius, presumably reflecting a radial change in dust temperature. We find

$$\begin{eqnarray}\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{TIR}}}{{I}_{70}} & = & {10}^{6}\left\{\begin{array}{l}f({r}_{\mathrm{gal}})\,\mathrm{if}\,{r}_{\mathrm{gal}}\lt 2.5\,\mathrm{kpc}\\ 2.96\,\mathrm{if}\,{r}_{\mathrm{gal}}\gt 2.5\,\mathrm{kpc}\end{array}\right.\\ \mathrm{where}\quad f(r) & = & 1.93+0.01r+0.28{r}^{2}-0.048{r}^{3}.\end{eqnarray} \tag{ 1 }$$

Here ${r}_{\mathrm{gal}}$ refers to the deprojected galactocentric radius and f(r) is a polynomial fit to the ratio ${{\rm{\Sigma }}}_{\mathrm{TIR}}/{I}_{70}$ as a function of r inside ${r}_{\mathrm{gal}}=2.5\,\,\mathrm{kpc}$. ${{\rm{\Sigma }}}_{\mathrm{TIR}}$ has units of ${L}_{\odot }\,{\mathrm{kpc}}^{-2},{I}_{70}$ has units of MJy sr⁻¹, and ${r}_{\mathrm{gal}}$ has units of kpc. Outside ${r}_{\mathrm{gal}}\sim 2.5\,\,\mathrm{kpc}$, the ratio appears flat. The Appendix compares SFRs derived from I₇₀ using this approach to those using ${{\rm{\Sigma }}}_{\mathrm{TIR}}$ at $\theta =30^{\prime\prime}$. The two show a median ratio of 1, less than 10% scatter and no clear systematics across the PAWS field.

At $30^{\prime\prime}$, we measure the rms scatter in ${{\rm{\Sigma }}}_{\mathrm{TIR}}$ from the low intensity regions of the map to be $\sim 2.5\times {10}^{6}\,{L}_{\odot }$ kpc⁻². At $10^{\prime\prime}$, using only the $70\,\mu {\rm{m}}$ data, the rms scatter is higher, $\sim 8.5\times {10}^{6}\,{L}_{\odot }$ kpc⁻².

Cloud-scale properties: We measure the intensity-weighted cloud-scale properties of the gas in each beam following Leroy et al. (2016). In brief, we begin with the native 40 pc resolution PAWS cube. We recenter each spectrum about the local mean velocity. Next, we weight each spectrum by the integrated intensity along the line of sight and convolve from $1^{\prime\prime} \approx 40$ pc resolution (our "measurement scale") to $10^{\prime\prime} \approx 370$ pc or $30^{\prime\prime} \approx 1.1\,\,\mathrm{kpc}$ (our "averaging scales"). From these intensity-weighted, stacked spectra, we measure the integrated intensity and line width of the gas. This cross-scale weighted averaging also resembles that by Ossenkopf & Mac Low (2002).

Because of the intensity (∼mass for fixed ${\alpha }_{\mathrm{CO}}$) weighting, this approach captures the high-resolution structure of the emission within each larger averaging beam. Leroy et al. (2016) demonstrated that the results match those from mass-weighted averages of cloud catalogs well, but with far fewer assumptions. We write the resulting measurements as, e.g., $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$. This is read as "the mass-weighted average 40 pc surface density within a larger beam.²¹ " We focus on three such measurements.

• 1.  
  The cloud-scale molecular gas surface density, $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$. This is a linear translation of the integrated intensity, $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle ={\alpha }_{\mathrm{CO}}\,\langle {I}_{40\mathrm{pc}}\rangle$, where ${\alpha }_{\mathrm{CO}}$ is our adopted CO-to-H₂ conversion factor. If the line-of-sight length of the gas distribution, h, is known or assumed, then $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ can be used to estimate the volume density of the gas on $40\,\mathrm{pc}\times h$ scales, $\rho (\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$. From this, one can estimate the gravitational free-fall time, $\langle {\tau }_{\mathrm{ff},40\mathrm{pc}}\rangle$. We show in Section 3.2.2 that for published Milky Way and M51 cloud catalogs, ${{\rm{\Sigma }}}_{\mathrm{mol}}$ and ρ do correlate well.
• 2.  
  The rms line width of CO, $\langle {\sigma }_{40\mathrm{pc}}\rangle$, measured from the "equivalent width" and corrected for channelization and channel-to-channel correlation following Leroy et al. (2016). For a given temperature and when the line width is purely turbulent in nature, this corresponds to the turbulent Mach number, ${ \mathcal M }$. $\langle {\sigma }_{40\mathrm{pc}}\rangle$ may also contain a contribution from bulk motions unresolved at the 40 pc resolution of PAWS (Meidt et al. 2013; Colombo et al. 2014b, and S. Meidt et al., submitted). Thermal contributions to the line width are expected to be small.
• 3.  
  The dynamical state of the gas, as traced by the ratio $\langle {b}_{40\mathrm{pc}}\rangle$ with $b\equiv {{\rm{\Sigma }}}_{\mathrm{mol}}/{\sigma }^{2}$. This "boundedness parameter" also relies on an adopted CO-to-H₂ conversion factor. Within a length scale, b is proportional to UE/KE, the ratio of potential energy (UE) to kinetic energy (KE). This is the inverse of the virial parameter, ${b}^{-1}\,\propto {\alpha }_{\mathrm{vir}}\approx 2\mathrm{KE}/\mathrm{UE}$. When b is high, the gas should be more gravitationally bound.Also, within a length scale $\langle {b}_{40\mathrm{pc}}\rangle \propto {\tau }_{\mathrm{cross}}^{2}/{\tau }_{\mathrm{ff}}^{2}$, where ${\tau }_{\mathrm{ff}}\propto 1/\sqrt{M/{R}^{3}}$ is the free-fall time and ${\tau }_{\mathrm{cross}}\,\sim R/\sigma$ is the crossing time for the measured velocity dispersion. This ratio has been highlighted by Padoan et al. (2012) as a key driver for the star-formation efficiency per free-fall time.Note that in this paper we focus on $b\equiv {{\rm{\Sigma }}}_{\mathrm{mol}}/{\sigma }^{2}$. This differs from the $B\equiv {I}_{\mathrm{CO}}/{\sigma }^{2}$ discussed in Leroy et al. (2016) by a factor of ${\alpha }_{\mathrm{CO}}$, so that $b={\alpha }_{\mathrm{CO}}B$. While B has the advantage of being directly computed from observable quantities, $b\propto {\alpha }_{\mathrm{vir}}^{-1}$ is more closely linked to the physical state of the gas.

We estimate uncertainties in $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle ,\langle {\sigma }_{40\mathrm{pc}}\rangle$, and $\langle {b}_{40\mathrm{pc}}\rangle$ using a Monte Carlo approach. We measure the noise in the stacked, shuffled intensity-weighted spectra from the signal-free region. Then we realize 100 versions of each spectrum, adding random noise to the real spectrum. For each case, we remeasure $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle ,\langle {\sigma }_{40\mathrm{pc}}\rangle$, and $\langle {b}_{40\mathrm{pc}}\rangle$. We compare these to our measurements without added noise, which we take to be the true values for purpose of this exercise. The rms offset between the simulated noisy data and the true value yields our estimate of the noise. This approach is ad hoc but yields realistic statistical uncertainties and captures the covariance among the uncertainties on $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle ,\langle {\sigma }_{40\mathrm{pc}}\rangle$, and $\langle {b}_{40\mathrm{pc}}\rangle$.

We report our results in terms of simple transformations of observable quantities into physical parameters.

Galactocentric coordinates: Following Schinnerer et al. (2013), we assume an inclination $i=22^\circ$ (Colombo et al. 2014b) and a position angle ${\rm{P}}.{\rm{A}}.\,=\,172^\circ$ (Colombo et al. 2014b) with the galaxy center at ${\alpha }_{2000}={13}^{{\rm{h}}}{29}^{{\rm{m}}}{52.7}^{{\rm{s}}},{\delta }_{2000}\,=+47^\circ 11^{\prime} 43^{\prime\prime}$ (Hagiwara 2007). We adopt the 7.6 Mpc distance of Feldmeier et al. (1997) and Ciardullo et al. (2002).

CO (1−0)-to-H₂: We estimate H₂ mass from CO (1−0) emission using a CO-to-H₂ conversion factor ${\alpha }_{\mathrm{CO}}=4.35$ ${M}_{\odot }\,{\mathrm{pc}}^{-2}\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1})}^{-1}$, which includes helium. This is a standard value for the Galaxy (Bolatto et al. 2013b). In the Appendix, we show that a dust-based approach following Sandstrom et al. (2013) and Leroy et al. (2011) suggests approximately this value. Schinnerer et al. (2010) came to the same conclusion via a multi-line CO analysis of the spiral arms. B. Groves et al. (2017, in preparation) show that this value applies with only weak variations across the disk of M51a using several independent methods.

There have been other values suggested for M51, mostly lower than Galactic by a factor of $\approx 2$ based on dust observations (e.g., Nakai & Kuno 1995; Wall et al. 2016, though see our Appendix). We discuss the impact of a lower conversion factor in the text.

${{\rm{\Sigma }}}_{\mathrm{TIR}}$ to ${{\rm{\Sigma }}}_{\mathrm{SFR}}$: When relevant, we recast the TIR surface brightness as an SFR surface density using the conversion of Murphy et al. (2011), which assumes a Kroupa (2001) initial mass function and reduces to

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{SFR}}}{{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}}\approx 1.5\times {10}^{-10}\,\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{TIR}}}{{L}_{\odot }\,{\mathrm{kpc}}^{-2}}.\end{eqnarray} \tag{ 2 }$$

A large body of work explores the subtleties of SFR estimation, often in M51 (e.g., Calzetti et al. 2005; Leroy et al. 2008, 2012; Blanc et al. 2009; Liu et al. 2011). Our focus in this paper is on new diagnostics of the molecular medium. Given the overwhelming extinction in the inner region of M51, we adopt the simple, widely accepted SFR diagnostic of TIR surface brightness. As a check, the Appendix shows the impact of several alternative SFR prescriptions on our inferred molecular gas depletion time at $30^{\prime\prime}$ resolution. These matter mainly to the overall normalization. By using the TIR emission, it is likely that we somewhat overestimate ${{\rm{\Sigma }}}_{\mathrm{SFR}}$. One of our key findings is that ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ is low (Section 3.2); this result would be even stronger if we used a tracer that yields lower ${{\rm{\Sigma }}}_{\mathrm{SFR}}$. The systematic trends appear weak and, when present, are opposite the sense needed to yield a fixed ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$. We intend to revisit this assumption in more detail in future work, ideally using an extinction-robust SFR tracer with high angular resolution to measure SFR on the scale of individual clouds.

Galactic dynamics relate to molecular gas structure and star formation in M51 (e.g., see Koda et al. 2009; Hughes et al. 2013a; Colombo et al. 2014a; Meidt et al. 2015; Schinnerer et al. 2017). With this in mind, we separate our correlation analysis by dynamical regions (Section 3.4). To do this, we use the dynamical region masks created by Colombo et al. (2014a). We use their simplified region definition, which breaks the PAWS field into "arm," "interarm," and "central" regions. For each $10^{\prime\prime}$ or $30^{\prime\prime}$ sampling point, we convolve the PAWS integrated CO intensity map multiplied by the mask for each separate region to the working resolution. Then we note the fraction of the flux in each beam coming from each dynamical region. When most of the CO flux in a beam comes from one dynamical environment, we associate the results for that beam with that environment.

Note that the three-region version of the Colombo et al. (2014a) mask may still group together physically distinct environments. We treat the upstream and downstream interarm regions together (e.g., see Meidt et al. 2015), and the "center" groups together the star-forming central molecular ring and the nucleus, which is more quiescent and potentially contaminated by the active galactic nucleus (AGN, e.g., see Querejeta et al. 2016). We report all of our measurements in Table 1.

All other things being equal, high surface density gas should be denser, with a shorter collapse time, ${\tau }_{\mathrm{ff}}$. Do the variations in ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ in Figure 2 arise from changes in the cloud-scale gas density across the galaxy?

Figure 3 tests this expectation, plotting ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ as a function of $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$, the mass-weighted cloud-scale surface density in each beam. Table 2 quantifies what we see in the figure, reporting rank correlation coefficients between $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ and ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ for different ranges of $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$.

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We do find a weak anticorrelation between ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ and $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ over the range $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle \approx 100\mbox{--}350\,{M}_{\odot }$ pc⁻². Treating $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ as the independent variable yields ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}\propto {\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle }^{-\alpha }$ with $\alpha =0.25\mbox{--}0.35$ over this range. The rank correlation coefficient over this range is only −0.14, but still statistically significant.

Our simplest expectation would be ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}\propto {\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle }^{-0.5}$. This would be expected if $\rho \propto \langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ (which appears reasonable, see Section 3.2.2), and stars formed from gas with a fixed efficiency per ${\tau }_{\mathrm{ff}}$. Gray lines in the figure illustrate this slope, which is steeper than the relation that we find. So over $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle \sim 100\mbox{--}350\,{M}_{\odot }$ pc⁻² denser (or at least higher $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$) gas does appear to form stars at a higher normalized rate, but the efficiency per free-fall time decreases (weakly) as $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ increases.

At higher $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle \gt 350\,{M}_{\odot }\,{\mathrm{pc}}^{-2},{\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ increases with increasing $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$, though with large scatter. This leads to the unexpected result pointed out by Meidt et al. (2013) that some of the least efficient star-forming regions in M51 have high cloud-scale molecular gas surface density. We show below that although these regions have high surface densities, they also appear to be less gravitationally bound (higher ${\alpha }_{\mathrm{vir}};$ Section 3.3.2).

Given a distribution of gas along the line of sight, $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ traces $\rho (\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$, the volume density of the gas averaged over the $\theta =40$ pc beam. In turn, $\rho (\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ determines the gravitational free-fall time, ${\tau }_{\mathrm{ff}}$. Contrasting ${\tau }_{\mathrm{ff}}$ with the measured ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ yields the efficiency per free-fall time, ${\epsilon }_{\mathrm{ff}}$. An approximately fixed ${\epsilon }_{\mathrm{ff}}$ is argued to hold across scale and system by, e.g., Krumholz et al. (2012), Krumholz & McKee (2005), and Krumholz & Tan (2007). More generally, ${\tau }_{\mathrm{ff}}$ is taken as the governing timescale for star formation, even when ${\epsilon }_{\mathrm{ff}}$ is low.

For gas with a depth h along the line-of-sight,

$$\begin{eqnarray}\rho (\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle ) & = & \langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle /h\\ {\tau }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle ) & = & \sqrt{3\pi /(32G\rho )}=81\,\mathrm{Myr}\,{\left(\displaystyle \frac{\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle }{{h}_{100\mathrm{pc}}}\right)}^{-0.5}\\ {\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle ) & = & {\tau }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )/{\tau }_{\mathrm{Dep}}^{\mathrm{mol}}\end{eqnarray} \tag{ 3 }$$

where h₁₀₀ is the depth of the molecular gas layer along the line-of-sight normalized to a fiducial value of 100 pc. ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ is the efficiency per free-fall time, obtained by contrasting ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ with ${\tau }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$.

The gray diagonal lines in Figure 3 show ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}\propto {{\rm{\Sigma }}}_{\mathrm{mol}}^{-0.5}$. If h₁₀₀ remains fixed, then each of these lines corresponds to a fixed ${\epsilon }_{\mathrm{ff}}$. In Figure 4, we show the distribution of ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ implied by our measurements. We plot results for both working resolutions and show values for a fixed h = 100 pc (left) and $h\propto {\langle {b}_{40\mathrm{pc}}\rangle }^{-1}$ (right, see explanation Section 3.2.1). We also illustrate the range of ${\epsilon }_{\mathrm{ff}}$ measured by several Milky Way studies.

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We find values of ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ that are low in both the absolute sense and relative to theoretical and Milky Way values. We also find ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ to vary as a function of environment and the local cloud population. Before discussing this in detail, we motivate our adopted h (Section 3.2.1) and demonstrate that $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ indeed should be a good predictor of $\rho (\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ (Section 3.2.2).

3.2.1. What Line-of-sight Depth to Use?

The depth of the gas layer along the line of sight h affects ${\tau }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ and so ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$. We do not observe h, but we can make a reasonable estimate. The most common approach is to measure the radius of a GMC on the sky and then assume spherical symmetry. In cloud catalogs for the Milky Way (Heyer et al. 2009; Miville-Deschênes et al. 2017) and M51 (Colombo et al. 2014b), most CO luminosity arises from clouds with radii $\sim 40\mbox{--}60$ pc. The left panel in Figure 5 shows the distribution of CO luminosity as a function of cloud radius for these three catalogs. The figure shows similar distributions for the Colombo et al. (2014b) M51 catalog and the inner (${r}_{\mathrm{gal}}\lt 8.5$ kpc) Milky Way portion of the recent Miville-Deschênes et al. (2017) catalog. In both cases, 68% of the luminosity comes from clouds with $\sim 30\,\mathrm{pc}\lt R\lt 95\,\mathrm{pc}$, with the mid-point for CO emission $R\sim 60$ pc. The Heyer et al. (2009) reanalysis of the Solomon et al. (1987) Milky Way clouds (their "A1") yields slightly smaller cloud sizes, $\sim 20\,\mathrm{pc}\lt R\lt 65\,\mathrm{pc}$, with median $R\sim 40$ pc.

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Estimates of the thickness of the molecular gas layer in both the Milky Way and M51 yield a similar value. Heyer & Dame (2015) compile estimates of the thickness of the molecular gas disk in the Milky Way (their Figure 6). They find $90\mbox{--}120$ pc (FWHM) within the Solar Circle. For M51, Pety et al. (2013) assumed the molecular gas to be in hydrostatic equilibrium. Following Koyama & Ostriker (2009), they calculated a mean FWHM thickness $\approx 94$ pc for the compact portion of the CO disk. If we consider the average density within FWHM $\approx \,90$ pc, then the corresponding h to use in Equation (3) is $h\approx 90/0.68=132$ pc.

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Thus both estimates of the thickness of the molecular disk and GMC catalogs support our adopted $h\sim 100$ pc. Because ${\tau }_{\mathrm{ff}}\propto {h}^{-0.5}$, modest variations in h do not have a large impact on ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$. Still, we test the impact of varying h by considering the case where the dynamical state of clouds (i.e., the virial parameter) is fixed. Then, $M\propto r{\sigma }^{2}$ and $r\propto {\sigma }^{2}/{\rm{\Sigma }}={b}^{-1}$. The same result applies for gas in a thin disk with only self-gravity. In this case,

$$\begin{eqnarray}&&{h}_{\mathrm{dyn}}\equiv 100\,\mathrm{pc}\,{\langle {b}_{40\mathrm{pc}}\rangle }^{-1}.\end{eqnarray} \tag{ 4 }$$

Note that in this situation, where b reflects a changing line-of-sight depth and not a changing dynamical state, ${\langle {b}_{40\mathrm{pc}}\rangle }^{-1}$ and $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ are both linearly proportional to $\langle {\rho }_{40\mathrm{pc}}\rangle$. Then, we expect a similar relation of ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ to both variables. Below, we show that this is not the case, and our best estimate is that b in fact does reflect a changing dynamical state, not a changing line-of-sight depth. Thus, we consider the case of fixed h = 100 pc to represent our basic result, and use Equation (4) to check the robustness of our conclusions.

3.2.2. Cloud Surface and Volume Density

3.2.3. Low Efficiency Per Free-fall Time

3.2.4. Possible Systematic Effects

At 370 pc resolution, we find 0.3 dex scatter in ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$, and Figure 3 shows a comparable scatter in ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$. Beyond only estimating ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$, we aim to understand how the mean gas properties in the beam and the region of the galaxy under consideration influence these two quantities. That is, how much of this scatter is random and how much results from changes in the local gas properties. Both ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ and ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ are of interest. ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ captures the SFR per unit gas and represents our most basic observation metric of whether gas in a part of a galaxy is good or bad at forming stars. ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ captures the efficiency of star formation relative to direct collapse, with ${\tau }_{\mathrm{ff}}$ representing the most common reference point for current theoretical models.

3.3.1. Surface Density

3.3.2. Velocity Dispersion

Surface density and volume density are not the only relevant properties of the gas. In a turbulence-regulated view of star formation, clouds with a high Mach number have a wider density distribution and include more dense gas (Padoan & Nordlund 2002). The Mach number also affects the critical density for the onset of star formation (e.g., Krumholz & McKee 2005), with a higher threshold density expected for higher Mach numbers.

Specific predictions differ from model to model (see Federrath & Klessen 2012), but most models predict an increase in ${\epsilon }_{\mathrm{ff}}$ for high ${ \mathcal M }$. If the temperature does not vary strongly across M51, and if the line widths that we observe are primarily turbulent in nature, then $\langle {\sigma }_{40\mathrm{pc}}\rangle$ should reflect the turbulent Mach number. In this case, if the turbulent models are right, then we would expect ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ to correlate with $\langle {\sigma }_{40\mathrm{pc}}\rangle$.

We test these expectations in the left panels of Figure 7. We plot ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ (top) and ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ (middle and bottom) as a function of $\langle {\sigma }_{40\mathrm{pc}}\rangle$. We do not observe a significant correlation between $\langle {\sigma }_{40\mathrm{pc}}\rangle$ and ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ at intermediate values of $\langle {\sigma }_{40\mathrm{pc}}\rangle \approx 6\mbox{--}12$ km s⁻¹. At high values of $\langle {\sigma }_{40\mathrm{pc}}\rangle$, we tend to find higher ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$. That is, where $\langle {\sigma }_{40\mathrm{pc}}\rangle$ appears high, gas appears inefficient at forming stars.

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Normalizing by the free-fall time, the middle and bottom left panels of Figure 7 show a steady decrease in ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ with increasing $\langle {\sigma }_{40\mathrm{pc}}\rangle$. The decline becoming steeper at high $\langle {\sigma }_{40\mathrm{pc}}\rangle$. The trend remains qualitatively the same for both treatments of line-of-sight depth. This anticorrelation is unexpected in turbulent theories. It suggests that the primary impact of the measured line width, whatever its origin, is to offer increased support against collapse rather than to increase the abundance of dense gas.

Based on modeling the velocity field, Meidt et al. (2013) and Colombo et al. (2014b) suggested that the line widths in M51 include substantial contributions from unresolved bulk motions. In this case, $\langle {\sigma }_{40\mathrm{pc}}\rangle$ may instead indicate the strength shearing or streaming motions, which can play a key role in suppressing star formation (Meidt et al. 2013). This seems very likely to explain the long depletion times at high $\langle {\sigma }_{40\mathrm{pc}}\rangle$ ($\gtrsim 12$ km s⁻¹).

At lower $\langle {\sigma }_{40\mathrm{pc}}\rangle$ the picture is less clear. M51 obeys the standard GMC scaling relations Colombo et al. (2014a), including when analyzed beam-by-beam Leroy et al. (2016), so we do expect that over most of the galaxy $\langle {\sigma }_{40\mathrm{pc}}\rangle$ reflects the turbulent line width to a reasonable degree (though see S. Meidt et al. 2017, submitted). In this case, Figure 7 presents a result not expected in turbulent theory: that high line width implies a low efficiency per free-fall time. Making similar measurements in other galaxies will help illuminate whether this effect is general or indeed driven by the large-scale dynamics of M51.

3.3.3. Dynamical State

M51 exhibits strong spiral and radial structure. Large-scale gas flows have been linked to the ability of M51's gas to form stars (e.g., Koda et al. 2009) and to the suppression of star formation by streaming motions (e.g., Meidt et al. 2013). Figure 8 shows how ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ and ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ vary with $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ and $\langle {b}_{40\mathrm{pc}}\rangle$ region-by-region. Here, we color the points according to the dynamical region from which most of the CO emission in the beam originates. We show the two-dimensional distributions of $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle ,\langle {\sigma }_{40\mathrm{pc}}\rangle ,\langle {b}_{40\mathrm{pc}}\rangle ,{{\rm{\Sigma }}}_{\mathrm{mol}},{{\rm{\Sigma }}}_{\mathrm{SFR}}$, and ${({\tau }_{\mathrm{Dep}}^{\mathrm{mol}})}^{-1}$ in Figure 9.

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Figure 8 shows ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ as a function of $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ for arm (green), interarm (purple), and central (blue) parts of the galaxy. As previously shown by Koda et al. (2009), Hughes et al. (2013b), and (Colombo et al. 2014a), the cloud-scale surface density increases dramatically, moving from the interarm to the arm region. The center of the galaxy exhibits high gas surface densities.

Although the arms concentrate molecular gas, we do not observe a decrease in ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ moving from the interarm to arm regions. Combining the arm and interarm regions, ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ remains approximately constant as a function of surface density until it rises at the highest values of $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$. This is the apparent suppression of star formation—despite high surface densities—observed in the arms by Meidt et al. (2013). These observations are also consistent with the observation by Foyle et al. (2010) of a weak contrast in ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ between arm and interarm regions in M51.

The inner part of M51 has high $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$, similar to that found in the spiral arms. Here, however, the high surface densities are accompanied by low ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$. As a result, in the top left panel of Figure 8, the points at high $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ separate in ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ according to the region from which they arise. As Figure 9 shows, many of the lowest ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ arise from the star-forming ring of the galaxy. These correspond to the high ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ points in the scaling relations in Figure 2. The few points at the galaxy center, in which AGN contamination (Querejeta et al. 2016) and beam smearing (e.g., S. Meidt et al. 2017, in preparation) contribute most, has little effect on the overall trend.

The top right panel of Figure 8 shows that, although the parts of the galaxy separate in ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ versus $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ space, they overlap much better when ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ is plotted as a function of $\langle {b}_{40\mathrm{pc}}\rangle$. That is, the long depletion times observed at high $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ in the arms appear to be there because that gas has low $\langle {b}_{40\mathrm{pc}}\rangle$, i.e., it appears weakly gravitationally bound. We observe an anticorrelation between ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ and b in both the arm and interarm regions. The central region, which has the lower ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$, also has the strongest self-gravity, traced by b.

We do observe an offset between the median ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ in the arm and interarm region at fixed b. At the same $\langle {b}_{40\mathrm{pc}}\rangle$, points in the arms have typically 0.13 dex ($\sim 35 \%$) longer ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$. This could reflect evolutionary effects on scales larger than our averaging beam. For example, Schinnerer et al. (2017) show the formation of stars along spurs displaced downstream from the arms. Or it could be driven additional suppression of star formation in the arms by dynamical effects not captured by $\langle {b}_{40\mathrm{pc}}\rangle$ (Meidt et al. 2013). Alternatively, it could reflect a lower filling fraction in the interarm region, so that beam dilution affects the interarm points more, lowering b relative to its true value. It could also reflect a low level bias in our SFR tracers, which affects the lower-magnitude ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ in the interarm more than in the arm.

When recast from ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ to ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ as a function of $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ (bottom row), the galaxy again separates. Here the arms appear as outliers. They show low ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$, significantly lower than the interarm region or the center. That is, given the high surface densities in the arms, we would expect collapse to proceed quickly. But the observed ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ does not support this expectation. The contrast between these low ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ in the arms and the higher values in the interarm regions drive the anticorrelation between ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ and $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ observed across the whole galaxy.

The bottom right panel of Figure 8 shows ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ as a function of $\langle {b}_{40\mathrm{pc}}\rangle$ region-by-region. When we considered the whole galaxy (Figure 7), only a weak correlation related ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ to $\langle {b}_{40\mathrm{pc}}\rangle$. Here the individual regions show a stronger positive correlation between ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ and $\langle {b}_{40\mathrm{pc}}\rangle$. There is some indication that at least the interarm regions match the sense of the Padoan et al. (2012) prediction (the black line). The picture for the arm and center regions is less clear. Together they may show a weak positive correlation between ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ and $\langle {b}_{40\mathrm{pc}}\rangle$, but it is not clear that they should be grouped together. The offset between the interarm and arm regions at fixed $\langle {b}_{40\mathrm{pc}}\rangle$ appears even stronger in ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ than for ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$. At fixed $\langle {b}_{40\mathrm{pc}}\rangle$ interarm regions have typically ∼0.24 dex, almost a factor of two, higher ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ than arm regions with the same $\langle {b}_{40\mathrm{pc}}\rangle$.

Together, Figures 8 and 9 paint a picture of M51 that qualitatively resembles that seen in many barred galaxies: despite the high surface densities in the inner dynamical features (here the arms), gas in this region appears stabilized against collapse. But flows along the arms feed gas condensations (the star-forming ring) in the inner regions (see Querejeta et al. 2016), where star-formation activity does proceed at a high level in both an absolute and normalized sense. Despite our averaging over moderately large (370 pc) areas, timescale effects may also be at play. The ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ map in Figure 9 shows significant azimuthal structure, and as shown by Schinnerer et al. (2017) star formation tends to occur in spur-like structures downstream of the arms. We refer the reader to extensive discussions in Meidt et al. (2013), Colombo et al. (2014a), and Querejeta et al. (2016), Schinnerer et al. (2017), and references therein, for more discussion.

Within the next year, it should be possible to conduct a similar analysis as we present here for M51 for a diverse sample of local galaxies. These include the other five galaxies treated by Leroy et al. (2016) and targets of new ALMA mapping surveys that achieve cloud-scale resolution across $\sim 10$ star-forming galaxies. Such tests will establish: (1) if our observed very low ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ is universal, (2) if the apparent role of self-gravity traced by b is unique to M51 or a general feature, and (3) whether the gravitational free-fall time estimated from high-resolution imaging indeed appears to be a controlling parameter. The combination of these cloud-scale measurements with density-sensitive spectroscopy (e.g., Usero et al. 2015; Bigiel et al. 2016; Leroy et al. 2017) will also help connect structural analysis at the GMC-scale to the internal density structure of clouds, which plays a key role in their ability to form stars.

Our structural analysis follows the "beamwise" approach described in Leroy et al. (2016), but more literature exists estimating GMC properties for nearby galaxies (e.g., Fukui & Kawamura 2010; Colombo et al. 2014a, the latter for M51). Following similar studies in the Milky Way (e.g., Murray 2011; Evans et al. 2014), these measurements can be compared to ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ in a similar way to what we do here. A. Schruba et al. (2017, in preparation) present such an analysis for a large collection of galaxies with GMC property measurements.

Finally, two major observational steps could address the tension between our measurements and those of the Milky Way. First, by observing CO from a large part of a star-forming galaxy at very high spatial resolution, one could attempt to mimic the Milky Way observations with full knowledge of the surrounding medium. Second, pairing extinction-robust SFR tracers with high-resolution gas mapping would allow for the kind of population studies carried out by Lee et al. (2016). The need to leverage low-resolution IR maps to estimate the SFR limits current efforts to consider population averages at a few hundred parsec scales.

We use TIR intensity as our tracer of SFR. At $\theta =30^{\prime\prime}$, we calculate ${{\rm{\Sigma }}}_{\mathrm{TIR}}$ using four bands and the SED-fitting based prescription of Galametz et al. (2013). At $\theta =10^{\prime\prime}$, we use a linear translation of I₇₀ into ${{\rm{\Sigma }}}_{\mathrm{TIR}}$, with the coefficient derived from comparing 70 μm intensity to TIR intensity at $30^{\prime\prime}$ resolution. We then translate ${{\rm{\Sigma }}}_{\mathrm{TIR}}$ to ${{\rm{\Sigma }}}_{\mathrm{SFR}}$, following Murphy et al. (2011).

The main impact of ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ in this paper is on the estimate of ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$. To assess the impact of our choice of estimator, Figure 10 shows the effect on ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ of replacing our adopted TIR-based SFR with estimates using a different approach. We only have access to all of the required data at $\theta =30^{\prime\prime}$, so this plot shows only results for that resolution over the PAWS field, our area of interest.

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First, we show results using only 70 μm emission and the formulae quoted in Sections 2.2 and 2.3. This is our approach at $\theta =10^{\prime\prime}$, where Herschel's 70 μm map is our only available IR band. We also show results using only 24 μm, using Hα assuming one magnitude of extinction, hybridizing Hα and 24 μm emission, and combining Hα with 24 μm after subtracting a "cirrus" (non-star-forming) component from the 24 μm emission. Except for the 70 μm emission, the prescriptions used for the other tracers are taken from Leroy et al. (2012), which builds heavily on Calzetti et al. (2007) and Murphy et al. (2011). We use the gas-based cirrus prediction, which assumes a typical dust-to-gas ratio and that all of the gas is illuminated by a radiation field 0.6 times that found in the Solar Neighborhood. The final panel shows the result for quadrupling the radiation field used in the cirrus estimate. Including an FUV-based hybrid (as in Leroy et al. 2008) would not add much to the analysis given the heavily extinguished nature of the region in question (see Leroy et al. 2012). Each panel quotes the median and scatter in the logarithm of the ratio between ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ estimated using this other tracer to that used in the main body of the paper.

The figure shows that the IR-based estimates agree well with one another and yield higher ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ than estimates using Hα. Indeed, the main result of changing the SFR tracer is usually to lower ${{\rm{\Sigma }}}_{\mathrm{SFR}}$, thereby increasing ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$. The magnitude of the shift is a factor of $\sim 2$ if only Hα with 1 mag of extinction is used or a very large cirrus component is adopted (which also amounts to only weakly correcting Hα for extinction). A main result of our analysis is a low ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$. Lower ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and higher ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ would drive ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ to even lower values. In detail, given the gas-rich, dusty nature of the inner few kiloparsecs of M51, we do not necessarily expect these lower ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ estimates to be more correct, but if they are then it would not change our qualitative conclusions.

Note that data at higher $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ tend to show a larger discrepancy between IR-based SFR estimates and Hα with little or no correction. The sense of this trend is that most alternatives to the IR-based ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ would yield longer ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ at higher $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$. The result would be an even lower ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ at high $\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle$ than we already observe. That is, none of the alternatives in Figure 10 push the data toward a more nearly fixed ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$.

Furthermore, recall that Figure 2 shows that our IR-based approach yields measurements that overlap the Paα+24 μm based estimates from Kennicutt et al. (2007). They studied selected apertures, while we sample the whole inner disk, so there are methodological differences. But the overall magnitude of both the gas and SFR estimates agrees well.

Finally, note from the first panel that ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$ estimated using only 70 μm emission and our adopted scaling agrees very well with that estimated using the four-band Galametz et al. (2013) fit. That is, the approach that we use at $\theta =30^{\prime\prime}$ agrees well with that which we are forced to use $\theta =10^{\prime\prime}$. The median ratio agrees by construction, but the small scatter gives us confidence in our use of 70 μm emission and our application of Equation (1) (though see Boquien et al. 2016, for a more in depth consideration of IR emission as a function of scale).

We translate CO (1−0) emission into molecular mass assuming a fixed ${\alpha }_{\mathrm{CO}}=4.35$ ${M}_{\odot }\,{\mathrm{pc}}^{-2}\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1})}^{-1}$. This value is supported by multi-line (Schinnerer et al. 2010) and cloud virial mass (Colombo et al. 2014a) studies. Schinnerer et al. (2010) provide a thorough summary of the literature on ${\alpha }_{\mathrm{CO}}$ in M51, which has so far yielded results that break down into either an approximately Galactic conversion factor or values ∼0.5 times Galactic. If the lower ${\alpha }_{\mathrm{CO}}$ holds, there would be less molecular gas mass than we infer in the main paper, and a shorter ${\tau }_{\mathrm{Dep}}^{\mathrm{mol}}$. This would increase ${\epsilon }_{\mathrm{ff}}(\langle {{\rm{\Sigma }}}_{40\mathrm{pc}}\rangle )$ by ${({\alpha }_{\mathrm{CO}}/{\alpha }_{\mathrm{MW}})}^{-1.5}$, because the conversion factor also affects the density and so ${\tau }_{\mathrm{ff}}\propto 1/\sqrt{\rho }$.

Figure 11 shows that an approximately Galactic conversion factor is also supported by the dust-based approach of Sandstrom et al. (2013) and Leroy et al. (2011). We compare ${{\rm{\Sigma }}}_{\mathrm{dust}}$, the dust mass surface density estimated from Herschel multiband data, to the measured CO intensity and the Hi column density from VLA imaging. The CO map is the PAWS single dish map, the Hi map comes from THINGS (Walter et al. 2008). The dust map is the result of fitting using the Draine & Li (2007), Draine et al. (2007) models to the Herschel and Spitzer photometry, following Aniano et al. (2012) and modified by the correction to dust mass suggested in Planck Collaboration et al. (2016).

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For this application, we assume that the dust-to-gas ratio is constant over the range ${r}_{\mathrm{gal}}=1\mbox{--}8\,\,\mathrm{kpc}$. The approximately constant metallicity of the galaxy supports this assumption (e.g., Croxall et al. 2015). The figure shows that ${\alpha }_{\mathrm{CO}}\approx {\alpha }_{\mathrm{MW}}=4.35$ ${M}_{\odot }\,{\mathrm{pc}}^{-2}\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1})}^{-1}$ yields an approximately flat dust-to-gas ratio as a function of radius. A lower conversion factor, as suggested by Nakai & Kuno (1995) and Wall et al. (2016) yields a strong gradient in dust-to-gas ratio as a function of radius. The right panel shows the formal results of minimizing scatter in DGR, while varying ${\alpha }_{\mathrm{CO}}$ treating either each ring (black) or each $\theta =30^{\prime\prime}$ line of sight (gray) as independent measurements. Both approaches yield a best fit ${\alpha }_{\mathrm{CO}}\approx 4.5\mbox{--}5.0$ ${M}_{\odot }\,{\mathrm{pc}}^{-2}\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1})}^{-1}$.

Uncertainties apply to this dust-based approach, including phase- or density-dependent depletion (Jenkins 2009), emissivity variations (e.g., Ossenkopf & Henning 1994), and the presence of sufficient dynamic range in the H₂/Hi ratio to achieve a good fit (Sandstrom et al. 2013). The interplay of these uncertainties with ${\alpha }_{\mathrm{CO}}$ variation are discussed at length in Israel (1997), Leroy et al. (2007, 2011), Sandstrom et al. (2013), and Roman-Duval et al. (2014), and are beyond the scope of this paper. The key point, for us, is that the best current available dust and gas maps suggest that our adopted ${\alpha }_{\mathrm{CO}}\approx {\alpha }_{\mathrm{MW}}$ represents a reasonable choice.