THE CO-TO-H₂ CONVERSION FACTOR AND DUST-TO-GAS RATIO ON KILOPARSEC SCALES IN NEARBY GALAXIES

We use dust mass surface density maps derived from pixel-by-pixel modeling of the infrared (IR) spectral energy distribution (SED) observed by Spitzer and Herschel with models developed by Draine & Li (2007). A detailed description of the modeling for NGC 0628 and NGC 6946 is presented in Aniano et al. (2012) and the full-sample results are presented in G. Aniano et al. (in preparation). The dust models include a description of the dust properties (size distribution, composition, and optical properties of the grains) with a variable fraction of dust in the form of polycyclic aromatic hydrocarbons (PAHs). The dust is illuminated by a radiation field distribution wherein a fraction of the dust is heated by a minimum radiation field U_min while the rest is heated by a power-law distribution of radiation fields extending up to U = 10⁷ U_MMP, where U_MMP is the solar neighborhood radiation field from Mathis et al. (1983). The fraction of the dust mass heated by radiation fields where U > 10² U_MMP is typically very small, so the dust mass surface density is not very sensitive to the exact value of the upper limit on U.

The resolution of the dust mass surface density map is equivalent to that of the lowest resolution IR map included in the modeling. In order to preserve spatial resolution while still covering the peak of the dust SED, we use the dust modeling at a resolution matched to the SPIRE 350 μm map (FWHM ∼ 25''). This limiting resolution allows us to include the following maps in the dust modeling: IRAC 3.6, 4.5, 5.8, and 8.0 μm; MIPS 24 and 70 μm; PACS 70, 100, and 160 μm; and SPIRE 250 and 350 μm. In theory, we could perform this analysis at even higher resolution using the SPIRE 250 resolution maps, which include all IRAC bands; MIPS 24 μm; all PACS bands; and SPIRE 250 μm. However, Aniano et al. (2012) found that maps where the limiting resolution exceeds MIPS 70 μm are less reliable due to the comparatively low surface brightness sensitivity of the PACS observations. At SPIRE 250 μm resolution, they find systematic errors of up to ∼30%–40% in the dust mass (compared to their best estimate, which includes all IRAC, MIPS, PACS, and SPIRE bands). However, when both SPIRE 250 μm and 350 μm observations are included, the systematic errors in the dust mass are ∼10% or less. We proceed by using the SPIRE 350 μm resolution dust modeling results.

Aside from the dust mass surface density, the Aniano et al. (2012) modeling also constrains a number of other quantities that we make use of later in interpreting the results. These quantities include $\overline{U}$, the average radiation field heating the dust; U_min, the minimum radiation field; f_PDR, the fraction of the dust luminosity that comes from dust heated by U > 100 U_MMP; and q_PAH, the fraction of the dust mass accounted for by PAHs with fewer than 10³ carbon atoms.

Statistical uncertainties on Σ_D and the other derived parameters were measured by Aniano et al. (2012) with a Monte Carlo approach. In most of the regions we consider, the statistical uncertainties on the dust mass surface densities are small, but the signal-to-noise ratio (S/N) of the dust mass maps is generally a function of radius and the outskirts can have S/N ∼ 5 in some cases. Our error estimates take these uncertainties into account and are described in Section 3.3. Possible systematic effects are discussed in Section 3.5.

To trace the atomic gas surface density in our targets, we use a combination of NRAO³⁵ Very Large Array (VLA) H i observations from THINGS (Walter et al. 2008) and supplementary H i, observations both new and archival, described in Leroy et al. (2013). The source and angular resolution of the H i maps for our targets are listed in Table 2.

The H i maps were converted from integrated intensities to column densities using Equation (5) of Walter et al. (2008). We convolved each map with an elliptical Gaussian kernel determined by its individual beam properties to produce a circular Gaussian point spread function (PSF). We then use kernels created following the procedures in Aniano et al. (2011) to convolve the circular Gaussian to match the SPIRE PSF at 350 μm. For the H i,  we assume the uncertainties to be the larger of either 0.5 M_☉ pc⁻² or 10% of the measured column density. Systematic uncertainties from H i opacity effects are discussed in Section 3.4.

To trace the molecular gas distribution in our targets, we use CO J = (2–1) mapping from the HERACLES survey (Leroy et al. 2009b). Integrated intensity maps were generated from the CO spectral cubes by integrating the spectra over a range in velocity around either (1) the detected CO line in that spectrum or (2) the expected CO velocity predicted from the H i velocity (since H i is detected at high S/N in almost all relevant pixels). We propagate uncertainties through these masking steps, creating in the end an integrated CO line map and an uncertainty map. The HERACLES maps have PSFs well approximated by a circular Gaussian with a FWHM of 134. We convolve the maps with kernels constructed using the techniques described by Aniano et al. (2011) to match the resolution of the SPIRE 350 μm maps. We have tested the effect of error beams (i.e., the extended wings of the PSF or stray light pick-up of the IRAM 30 m telescope) on the HERACLES maps and find that the effect is less than ∼5% for all galaxies and closer to 1% for most.

Our measurements of α_CO directly determine the conversion factor appropriate for CO J = (2–1). However, to compare with the standard CO J = (1–0) factor, which is more frequently used, we convert to (1–0) using a fixed value of the line ratio R₂₁ = (2–1)/(1–0) = 0.7. Due to revised telescope efficiencies, the R₂₁ we use differs slightly from that used in Leroy et al. (2009b). The R₂₁ we assume was found to be an appropriate average for the HERACLES sample (E. W. Rosolowsky et al., in preparation; note that we find good agreement with this R₂₁ by comparing the HERACLES measurements with published CO J = (1–0) measurements as described in the Appendix). We discuss the effects of assuming a fixed R₂₁ on our results for the (1–0) conversion factor in Section 3.4. In order to use the α_CO values with CO J = (2–1) observations, they should be divided by a factor of 0.7.

2.4.1. Metallicity

In the analysis presented in Section 4, we study the variations of α_CO and the DGR as a function of metallicity. Wherever possible, we make use of the metallicity measurements from Moustakas et al. (2010, hereafter M10), who derived characteristic metallicities as well as radial gradients in oxygen abundance for the SINGS sample. M10 present results using two different calibrations for the strong-line abundances—from Kobulnicky & Kewley (2004, KK04) and Pilyugin & Thuan (2005, PT05). Both calibrations are considered in the following analysis.

Our preferred metallicity measurement is a radial gradient from H ii region metallicities (M10, Table 8). For several galaxies, no radial gradient measurement is available, so we use a fixed metallicity for the entire galaxy equal to the "characteristic metallicity" (M10, Table 9). In the case of NGC 4236 and 4569, the only metallicity measurements available are from the B-band luminosity–metallicity (L–Z) relationship and we use those values from M10 with no gradient. Finally, two of our galaxies are not in the M10 sample. For NGC 3077, we use the metallicity 12 + log (O/H) = 8.9 given in KK04 with no gradient, from Calzetti et al. (1994). To obtain a PT05 measurement for NGC 3077, we subtract 0.6 dex, the average offset between the two calibrations found by M10. For NGC 5457 (a.k.a. M 101), a galaxy with a well-known radial metallicity gradient, we use the measurements from Bresolin (2007). These metallicities are from direct methods, so are not on the same scale as the measurements from either PT05 or KK04. The metallicities and gradients we use are listed in Table 3. These values are converted to match the r₂₅ values we adopt in this work, which can be slightly different from those adopted by M10. In order to compare with the MW, we use the metallicity of the Orion Nebula H ii region in the PT05 and KK04 calibrations, i.e., 12+log(O/H) = 8.5 for PT05 and 8.8 for KK04, which were obtained by applying the strong-line metallicity calibrations to the integrated spectrum of Orion from integral field spectroscopy³⁶ (Sánchez et al. 2007).

Table 3. Adopted Metallicities and Gradients

	PT05	PT05	KK04	KK04
Galaxy	Central Metallicity	Metallicity Gradient	Central Metallicity	Metallicity Gradient	Source
	(12 + log(O/H))	(dex $r_{25}^{-1}$)	(12 + log(O/H))	(dex $r_{25}^{-1}$)
NGC 0337	8.18 ± 0.07	⋅⋅⋅	8.84 ± 0.05	⋅⋅⋅	M10 Table 9
NGC 0628	8.43 ± 0.02	−0.25 ± 0.05	9.19 ± 0.02	−0.54 ± 0.04	M10 Table 8
NGC 0925	8.32 ± 0.01	−0.21 ± 0.03	8.91 ± 0.01	−0.43 ± 0.02	M10 Table 8
NGC 2841	8.72 ± 0.12	−0.54 ± 0.39	9.34 ± 0.07	−0.36 ± 0.24	M10 Table 8
NGC 2976	8.36 ± 0.06	⋅⋅⋅	8.98 ± 0.03	⋅⋅⋅	M10 Table 9
NGC 3077	8.30 ± 0.20	⋅⋅⋅	8.90 ± 0.20	⋅⋅⋅	K11
NGC 3184	8.65 ± 0.02	−0.46 ± 0.06	9.30 ± 0.02	−0.52 ± 0.05	M10 Table 8
NGC 3198	8.49 ± 0.04	−0.38 ± 0.11	9.10 ± 0.03	−0.50 ± 0.08	M10 Table 8
NGC 3351	8.69 ± 0.01	−0.27 ± 0.04	9.24 ± 0.01	−0.15 ± 0.03	M10 Table 8
NGC 3521	8.44 ± 0.05	−0.12 ± 0.25	9.20 ± 0.03	−0.52 ± 0.15	M10 Table 8
NGC 3627	8.34 ± 0.24	⋅⋅⋅	8.99 ± 0.10	⋅⋅⋅	M10 Table 9
NGC 3938	8.42 ± 0.20	⋅⋅⋅	9.06 ± 0.20	⋅⋅⋅	M10 L–Z
NGC 4236	8.17 ± 0.20	⋅⋅⋅	8.74 ± 0.20	⋅⋅⋅	M10 L–Z
NGC 4254	8.56 ± 0.02	−0.35 ± 0.08	9.26 ± 0.02	−0.39 ± 0.06	M10 Table 8
NGC 4321	8.61 ± 0.07	−0.31 ± 0.17	9.29 ± 0.04	−0.28 ± 0.11	M10 Table 8
NGC 4536	8.21 ± 0.08	⋅⋅⋅	9.00 ± 0.04	⋅⋅⋅	M10 Table 9
NGC 4569	8.58 ± 0.20	⋅⋅⋅	9.26 ± 0.20	⋅⋅⋅	M10 L–Z
NGC 4625	8.35 ± 0.17	⋅⋅⋅	9.05 ± 0.07	⋅⋅⋅	M10 Table 9
NGC 4631	8.12 ± 0.11	⋅⋅⋅	8.75 ± 0.09	⋅⋅⋅	M10 Table 9
NGC 4725	8.35 ± 0.13	⋅⋅⋅	9.10 ± 0.08	⋅⋅⋅	M10 Table 9
NGC 4736	8.40 ± 0.01	−0.23 ± 0.12	9.04 ± 0.01	−0.08 ± 0.10	M10 Table 8
NGC 5055	8.59 ± 0.07	−0.59 ± 0.27	9.30 ± 0.04	−0.51 ± 0.17	M10 Table 8
NGC 5457	8.75 ± 0.05	−0.75 ± 0.06	8.75 ± 0.05	−0.75 ± 0.07	B07
NGC 5713	8.48 ± 0.10	⋅⋅⋅	9.08 ± 0.03	⋅⋅⋅	M10 Table 9
NGC 6946	8.45 ± 0.06	−0.17 ± 0.15	9.13 ± 0.04	−0.28 ± 0.10	M10 Table 8
NGC 7331	8.41 ± 0.06	−0.21 ± 0.31	9.18 ± 0.05	−0.49 ± 0.25	M10 Table 8

References. Moustakas et al. (2010, M10), Kennicutt et al. (2011, K11), Bresolin (2007, B07).

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2.4.2. Star Formation Rate and Stellar Mass Surface Density Maps

After all maps have been convolved to the SPIRE 350 μm resolution, we sample them with a hexagonal grid with approximately half-beam spacings (i.e., 125). Uncertainties in the dust mass surface density, CO integrated intensity, and H i column density are propagated through the necessary convolutions and samplings. Surface densities and other quantities have been deprojected using the orientation parameters listed in Table 1.

To perform the solution, we require multiple measurements of dust and gas tracers in the region. We also aim, however, to select the smallest possible regions, in order to ensure an approximately constant DGR. We proceed by dividing each target galaxy into hexagonal regions encompassing 37 of the half-beam spaced sampling points. We call these regions "solution pixels" (see panel (a) of Figure 1 for an example). The 37-point solution pixels are a compromise between small region sizes and a sufficient number of independent measurements needed to minimize statistical noise. The solution pixels tile the galaxy with center-to-center spacing of 375. Thus, neighboring solution pixels are not independent and share ∼40% of their sampling points. The overlap between solution pixels is illustrated in Figure 1. Such a tiling is optimal because it fully samples the data. To ensure that the final results are not dependent on the placement of the solution pixels, we performed a test where we distributed the solution pixels randomly throughout the galaxies and compared the resulting radial trends in α_CO to what we measured using the fixed grid described above. The radial trends were in agreement, demonstrating that our results are insensitive to the exact placement of the solution pixel grid. The solution pixels correspond to physical scales ranging from ∼0.6 kpc to 3.9 kpc. We have tiled each galaxy with solution pixels out to the maximum value of r₂₅ contained in the HERACLES maps.

For each sampling point i in the solution pixel, the measurements of Σ_{D, i}, $\Sigma _{{\rm H}\,{\scriptsize I}}$_{, i}, and I_{CO, i}, along with an assumed α_CO, determine the DGR_i. We step through a grid of α_CO values to find the α_CO that results in the most uniform DGR for all the DGR_i values in the solution pixel. In all solutions presented here, we use an α_CO grid with 0.05 dex spacing, spanning the range α_CO = 0.1–100 M_☉ pc⁻² (K km s⁻¹)⁻¹.

We determine the most "uniform" DGR in a solution pixel by minimizing the scatter in the DGR_i values as a function of α_CO. The scatter is measured with a robust estimator of the standard deviation³⁷ of the logarithm of the DGR_i values—this technique appears to work best because outliers have little effect on the measured scatter because of both the logarithmic units and the outlier suppression. After measuring the scatter in the DGR at every α_CO value, we find the α_CO at which the scatter (Δlog(DGR)) is minimized. This value is taken to be our best-fit α_CO value for the region. We consider a solution to be found when a minimum has been located in the DGR scatter within the range of our α_CO grid. This outcome does not occur in every solution pixel—some pixels do not have sufficient CO/H i contrast, others have too low S/N, and, for some, the minimum is at the edge of the allowable range and the solution is not well constrained. The failed solutions are not included in our further analysis.

An illustration of the technique is shown in Figure 1 for a region in NGC 6946. In panel (a), we show the HERACLES CO J = (2–1) map of the galaxy with our half-beam sampling grid; the hexagonal region shows the "solution pixel" in question, which includes 37 individual samples from the maps. Panels (b) and (c) show how varying α_CO affects the mean DGR and the scatter for the points in the region, illustrating how the scatter increases away from the best α_CO value. Panel (d) shows the scatter as a function of α_CO for the whole α_CO grid. A clear minimum exists for this solution pixel at α_CO ∼ 1.4 M_☉ pc⁻² (K km s⁻¹)⁻¹.

We judge the uncertainties on the "best-fit" α_CO and the DGR in several ways. First, to take into account statistical errors, we perform a Monte Carlo test on the solutions by adding random noise to our measured Σ_D, $\Sigma _{{\rm H}\,{\scriptsize I}}$, and I_CO values according to each point's measurement errors. We repeat the solution with the randomly perturbed data values 100 times and find the standard deviation of the results. We also perform a "bootstrapping" trial, which tests the sensitivity of each solution to individual measurements. In each bootstrap iteration for a given solution pixel, we randomly select 37 sampling points, with replacement, and derive the solution. The bootstrap procedure is repeated 100 times for each solution pixel and we measure the resulting standard deviation of the α_CO values. The standard deviations from the Monte Carlo and bootstrapping iterations are added in quadrature to produce the final quoted error for the α_CO values we determine. In addition, to check these uncertainties, we also estimate the scatter and bias in α_CO for the given technique from our simulated data trials described in the Appendix, based on the median CO S/N in the solution pixel and the measured minimum of Δlog(DGR). The uncertainties from Monte Carlo plus bootstrapping are comparable to what we expect given the simulated data trials.

3.4.1. Uncertainties on α_CO from R₂₁ Variations

3.4.2. Variations of Σ_D Linearity within Solution Pixels

3.4.3. Opaque H I

3.5.1. Absolute Calibration of Σ_D

We divided each of the 26 galaxies in our sample into solution pixels and performed the simultaneous solution for the DGR and α_CO in each pixel. As an example, we present the results for NGC 0628 in the following section. The results for all solution pixels in all galaxies can be found in the Appendix.

Figure 2 shows, from left to right, the H i, CO, and Σ_D maps used in our analysis. The circles overlaid on the maps represent the centers of the solution pixels we have defined. Figure 3 shows the same circles representing the solution pixel centers. The left panel shows the pixel centers now filled in with a color representing the best α_CO solution for that pixel. In the middle panel, a gray scale shows the uncertainty on that α_CO solution. The DGR values are shown in the panel on the right. Finally, in Figure 4, we show these measured α_CO values as a function of galactocentric radius (r₂₅). For comparison, Figure 4 also shows the local MW α_CO = 4.4 M_☉ pc⁻² (K km s⁻¹)⁻¹ value with a solid horizontal line (dotted lines show a factor of two above and below; see Section 5.1 for details on the measurement of the MW value). We note that the MW may show a gradient of α_CO with radius (also discussed in Section 5.1), but for purposes of comparison with the most widely used conversion factor, we use a constant α_CO on all plots.

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In general, NGC 0628 shows a α_CO value consistent with the MW value within a factor of two at all galactocentric radii. Outside a radius of r₂₅ ∼ 0.6, we find few good solutions. There is a weak trend for lower α_CO at smaller radii, with the central solution pixel having a conversion factor α_CO = 2.2 M_☉ pc⁻² (K km s⁻¹)⁻¹. Recent work by Blanc et al. (2013) found consistent results for α_CO by inverting the SFR surface density map using a fixed molecular gas depletion time.

The technique we have used to solve for the DGR and α_CO simultaneously only works if there is sufficient S/N in the CO map and a range of CO/H i ratios in each solution pixel. These two constraints impose limits on where in the galaxies we can recover solutions. In order to perform statistical tests on our sample of α_CO and DGR values, we need to understand what biases these limits introduce into our results. For example, the failure of the technique in regions with low CO S/N generally limits our good solutions to the inner parts of galaxies, where metallicities tend to be higher. In order to judge the existence of trends in α_CO versus metallicity, we therefore need to understand where we have achieved good solutions.

To investigate these effects, we have examined the fraction of solution pixels that have solutions and the uncertainty on the derived α_CO values as a function of the range of CO/H i ratios and mean I_CO in a pixel. In general, the H i in our target galaxies has a quite flat radial profile, while the CO drops off approximately exponentially (Schruba et al. 2011). Because of these trends, wherever there is sufficient signal in CO to achieve a good solution, the CO/H i range is adequate as well. Thus, we find that the mean I_CO of a solution pixel is the best predictor for the existence and quality of a solution. Figure 5 shows the fraction of pixels with solutions and the average uncertainty on the solutions as a function of mean I_CO. We identify a cut-off at I_CO > 1 K km s⁻¹ above which we obtain solutions >90% of the time and those solutions have an average uncertainty of <0.5 dex. For any statistical analysis that follows, we use only pixels above this cut-off.

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In Tables 4 and 5, we list the average α_CO derived galaxy by galaxy and for the full sample. These averages only include solution pixels where I_CO > 1 K km s⁻¹. The first two columns of Table 5 list the mean and standard deviation of the individual solution pixel measurements. This approach treats each solution pixel equally, regardless of how strongly it contributes to the total molecular gas mass. The third column instead lists the mean α_CO derived from the total $\Sigma _{{\rm H}_{2}}$ and the total I_CO for the galaxy (above I_CO = 1 K km s⁻¹), which is equivalent to a mean where the pixels are weighted by their I_CO, values. If a single α_CO value were to be applied to the data, this value would be the optimal quantity to use. For most galaxies, the CO-weighted mean is higher than the straight mean of the solution pixels. This result is due to the fact that the molecular gas is not evenly distributed across the galaxy and the area-weighted average is different then the molecular gas-weighted average.

Figure 6 shows several histograms illustrating the distribution of our measured α_CO values. The mean, derived with several different weighting schemes, is listed in Table 5. The top panels of Figure 6 show histograms of all solution pixels while the bottom panels show histograms of the galaxy averages from Table 4. On these histograms, we highlight galaxies with high inclinations (i > 65°) in green. It is clear that the high inclination galaxies tend to have higher α_CO values on average than the more face-on galaxies. In the highest inclination galaxies, the pixel will include contributions from gas at larger radii, this gas tends to have a lower DGR and be less molecular-gas-rich. This result is equivalent to the challenge faced by Leroy et al. (2011) in the SMC, where some of the H i along the line of sight originates in an essentially dust-free envelope, and in M31, where regions along the minor axis of the galaxy have contributions from gas and dust at a variety of radii. In addition, optical depth effects for H i may be accentuated for highly inclined galaxies. All galaxies with i > 65° show average α_CO values above the mean (except for NGC 0925, which has a somewhat uncertain inclination; de Blok et al. 2008). We thus eliminate all galaxies with i > 65°, leaving 782 total α_CO measurements.

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In Table 5, we list the average values of α_CO for the full galaxy sample. Excluding the high inclination galaxies, we find an average α_CO = 2.6 M_☉ pc⁻² (K km s⁻¹)⁻¹ for the individual solution pixel results. Weighted by I_CO, the average value is slightly higher, α_CO = 2.9 M_☉ pc⁻² (K km s⁻¹)⁻¹. To avoid highly resolved galaxies like NGC 5457 and 6946 contributing more points to the average, we also calculate averages where each galaxy contributes uniformly. These averages are listed in the last two columns of Table 5. The average value for our sample, α_CO = 3.1 M_☉ pc⁻² (K km s⁻¹)⁻¹, is only slightly lower than what is found in the MW disk.

The standard deviation in α_CO is 0.38 dex, when all lines of sight are treated equally (with our I_CO and inclination cut-offs). A key question we would like to answer is to what degree this scatter represents (1) the true scatter within each galaxy, (2) the galaxy-to-galaxy offsets, or (3) the variation of α_CO as a function of local parameters. If the scatter comes from variations with local environmental parameters, we may be able to generate a prescription for α_CO as a function of other observables. In the following sections, we explore the variations of α_CO within and among our galaxies to understand if and why α_CO varies.

In Figure 7, we present a summary of the α_CO values we find as a function of galactocentric radius. Plots of α_CO from each individual galaxy as a function of r₂₅ are presented in the Appendix. Each individual solution that makes our I_CO and inclination cut is shown in Figure 7 as a gray circle. We also display the mean and standard deviation of the α_CO values in 0.1 r₂₅ bins for each galaxy (this mean treats all solution pixels equally). The top panel of Figure 7 shows the measured α_CO values and the bottom panel shows those same values normalized by each galaxy's average α_CO from Table 4.

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The average radial profile of our galaxies is mostly flat as a function of r₂₅, but almost all galaxies show a decrease in α_CO in the inner ∼0.2 r₂₅ compared to their average value. The mean central decrease is ∼0.3–0.4 dex, but it can be as much as 0.8 dex. For several galaxies, this decrease leads to a central α_CO value that is an order to magnitude lower than the MW α_CO (e.g., NGC 4736, 5457, and 6946). The central depression persists even after correcting for differences in R₂₁ (see the following section for further discussion).

Normalizing each galaxy by its average α_CO and making each galaxy contribute equally to the average leads to a much smaller scatter at r₂₅ > 0.3; this result can be seen in the bottom panel of Figure 7 where the radial profiles show much less scatter than the individual measurements, which is not the case in the top, unnormalized, panel. Inside that radius, the scatter in the normalized α_CO profiles is reduced, but not by as much as outside that radius, indicating that individual galaxies show different central profiles of α_CO. The overall conclusions to be drawn from Figure 7 are: (1) outside r₂₅ ∼ 0.3, most of the scatter in our α_CO measurements can be explained by galaxy-to-galaxy differences, (2) inside that galactocentric radius, galaxy-specific trends in α_CO dominate the scatter, and (3) in general, the inner region of galaxies shows a lower α_CO value than the rest of the disk.

In Table 6, we list the central α_CO values and compare them to the MW value and the mean for the galaxy. Almost all of the central solution pixels have α_CO lower than the galaxy mean. The galaxies NGC 3351, 3627, 4321, 4736, 5457, and 6946 show α_CO values a factor of five or more lower than the MW value, at the 3σ confidence level. For NGC 4736, 5457, and 6946, in particular, the central solution pixel has a α_CO value an order of magnitude below the MW value. Only one galaxy, NGC 0337, shows a central α_CO higher than the MW value at a 3σ confidence level. In Table 6, we also list the CO (2–1)/(1–0) ratio measured as described in Section 3.4.1. While the central solution pixels do tend to have higher R₂₁ values than the value we adopted, the difference is less than a factor of two in all cases and does not alter our main conclusions about the low (1–0) conversion factor in these regions. In particular, for the 10 galaxies with measured R₂₁ and i < 65°, the central α_CO corrected for R₂₁ is still on average 0.3 dex lower (i.e., a factor of two) than the galaxy average.

The central solution pixels can be outliers from the rest of the pixels in our sample in the sense of having high CO/H i ratios. Since there are relatively few solution pixels with these conditions, they may not have been well represented in the simulated data we used to test the accuracy of the solution technique. High CO/H i ratios could bias the results: if the S/N of the H i maps is almost always higher than the S/N of the CO maps, the DGR scatter could be reduced in these conditions purely by decreasing the importance of CO in assessing the gas mass surface density. Since this bias would move the α_CO results toward lower values in the centers, we performed a test to judge whether our low central α_CO measurements could be due to this effect. The details of the test are described in the Appendix. In brief, we generated simulated datasets with known α_CO values where the $N_{{\rm H}\,{\scriptsize I}}$, I_CO, and Σ_D S/N is matched to the observations in each central solution pixel. We performed Monte Carlo trials to see how well the known input α_CO was recovered. In all cases, we found no evidence for a bias in the central solution pixel.

Taking into account the offset from R₂₁ variations, our results show that the central α_CO in NGC 3351, 3627, 4321, 4736, 5457, and 6946 is lower than the MW value by a factor of 4–10. Many of these galaxies show α_CO values closer to the MW value at larger radii, in line with the general trend seen in Figure 7. For NGC 3627, the mean α_CO for the galaxy is low as well, suggesting that the central region is not distinct from the rest of the disk (this galaxy is an interacting member of the Leo Triplet, so it is unique in our sample). Conversely, examination of the individual galaxy α_CO measurements as a function of radius (shown in the Appendix) for NGC 3351, 4321, 4736, 5457, and 6946 demonstrates that these galaxies show an unresolved region of lower-than-average α_CO values in their centers (note that our solution pixel grid oversamples the data, so an unresolved central depression affects the r₂₅ = 0 point and the six adjacent solution pixels). The closest galaxy that shows a central depression is NGC 4736 at D = 4.66 Mpc. At this distance, our solution pixel covers a region of radius 0.8 kpc. Even for this nearby example, we do not resolve the central depression. We tested whether the depressions were resolved using a grid of independent solution pixels (i.e., not overlapping) and found that the depression only affected the central pixel, consistent with it being unresolved. We note that our ability to detect any central depression may be a function of the galaxy's distance due to the increased size of the central solution pixel. Even at D = 14.3 Mpc, however, we detect a clear central depression in NGC 4321.

The type of nuclear activity in each galaxy is not a good predictor for whether or not it displays a central α_CO depression. Of the galaxies that show the clearest central depression, NGC 3351, 5457, and 6946 are classified as star-formation or H ii region dominated, while NGC 4321 and 4736 have signatures of active galactic nucleus (AGN) or low-ionization nuclear emission-line region activity (for details on the nuclear classifications; see Kennicutt et al. 2011). Several galaxies with flat radial α_CO profiles do show evidence for AGN activity: NGC 3627, 4254, and 4725, for example. Enhanced AGN activity could affect molecular gas properties in the nuclei (e.g., Krips et al. 2008), but the relatively weak AGNs present in the KINGFISH galaxies may not dominate on the kiloparsec scales we study here.

To summarize, we find that several galaxies in our sample show central α_CO values that are substantially lower than the MW value and also well below the galaxy average. This phenomenon appears to take the form of a depression in the central region that is unresolved by our solution pixel grid. We discuss these central regions and the physical conditions that may lead to low α_CO values further in Section 6.3.

Variations in α_CO may be related to variations in the local environmental conditions including metallicity, ISM pressure, interstellar radiation field strength, gas temperature, dust properties, or other variables. Correlations between α_CO and quantities that trace these environmental conditions may allow us to identify the drivers of α_CO variations. These correlations may also provide tools to predict the appropriate α_CO in a given environment. In the following, we examine the correlations of our measured α_CO with several observables that trace ISM conditions. For each tracer, we use the average value in each solution pixel. The physical interpretation of these correlations or lack thereof is discussed in Section 6.1.

In Figure 8, we plot our measured α_CO values as a function of the average radiation field intensity ($\overline{U}$), the PAH fraction (q_PAH), metallicity (12+log(O/H)), stellar mass surface density (Σ_*), SFR surface density (Σ_SFR), and the dust mass surface density (Σ_D). Each panel shows the individual measurements as gray symbols and the average for the galaxies as red symbols. The MW α_CO is highlighted with a horizontal gray line and the mean α_CO treating all solution pixels equally (α_CO = 2.6 M_☉ pc⁻² (K km s⁻¹)⁻¹) is shown as a dashed horizontal line. Overlaid on each panel is the binned mean and standard deviation and a linear fit to the gray points. In Table 7, we list the linear fit results as well as the rank correlation (RC) coefficient for each panel and its significance in standard deviations away from the null hypothesis. The RC coefficients suggest there are significant correlations between α_CO and all of the variables except metallicity. Due to possible inconsistencies between various metallicity measurements and incomplete knowledge of metallicity gradients in some galaxies, we examine the trends with metallicity separately in Section 4.6.1.

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Table 7. α_CO Correlation Properties

log(αCO) vs.	Correlationa	σ from	Linear Fitb
Variable	Coeff.	Null	Offset	Slope
r25	+0.19	5.2	0.29	+0.29
log($\overline{U}$)	−0.18	5.0	0.50	−0.31
qPAH (%)	+0.14	3.9	0.11	+0.10
12+log(O/H)	−0.05	1.4	2.39	−0.24
log(Σ*)	−0.26	7.1	1.05	−0.33
log(ΣD)	−0.15	4.1	0.26	−0.22
log(ΣSFR)	−0.18	5.1	0.00	−0.16

Notes. These correlations are shown in Figure 8. ^aSpearman rank correlation coefficient. ^bCorrelation with metallicity using a homogeneous sample of metallicity measurements is discussed in Section 4.6.1.

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Several other radiation field properties are measured from the dust SED modeling. These include U_min, the minimum radiation field heating the dust, and f_PDR, the fraction of the dust luminosity that arises in "photodissociation region (PDR) like" regions where U > 100 U_MMP. We have investigated the dependence of α_CO on these quantities and find weak trends with low significance.

The existence of a correlation between α_CO and environmental parameters does not directly identify the cause of the variations in α_CO, particularly since all of the parameters change radially to first order. In Figure 9, we illustrate the radial variations of the same variables by plotting them normalized by their average value as a function of r₂₅. All of the variables show significant correlations with radius. Table 8 lists the correlation coefficients and linear fits to the normalized radial profiles of the parameters. In all cases, the correlation of the variables with r₂₅ is more significant than the correlation of α_CO with those variables.

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Table 8. Correlation Properties versus r₂₅

Variable vs.	Correlationa	σ from	Linear Fit
r25	Coeff.	Null	Offset	Slope
log(αCO/〈αCO〉gal)	+0.18	5.0	−0.10	+0.23
log($\overline{U}/\langle \overline{U}\rangle _{\rm gal}$)	−0.57	15.9	0.17	−0.39
qPAH/〈qPAH〉gal	+0.19	5.3	−0.17	+0.40
log((O/H)/〈O/H〉gal)	−0.65	18.2	0.07	−0.17
log(Σ*/〈Σ*〉gal)	−0.73	20.5	0.43	−0.97
log(ΣD/〈ΣD〉gal)	−0.68	19.1	0.25	−0.56
log(ΣSFR/〈ΣSFR〉gal)	−0.31	8.8	0.37	−0.86

Notes. These correlations are shown in Figure 9. ^aSpearman rank correlation coefficient.

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If all of the variations of these parameters were primarily related to radius, we would not expect to find stronger correlations between α_CO and any parameter than between α_CO and radius. Our results suggest, however, that the correlation between α_CO and Σ_* is stronger (r_s = −0.25) and more significant (7.1σ) than the correlation with radius r₂₅ (r_s = +0.19, 5.2σ). One possible explanation is that our Σ_* measurement may trace the stellar profile better than r₂₅ itself. Other explanations for the correlation of α_CO with Σ_* will be discussed in Section 6.1.

Figure 7 illustrates that the normalized radial profile of α_CO shows a factor of ∼2 standard deviation in a typical radial bin. This uncertainty can be explained primarily by the uncertainty on our α_CO solutions themselves, which is typically close to a factor of ∼2. It is therefore unlikely that we could predict α_CO as a function of other variables to better precision than a factor of ∼2 unless the galaxy average α_CO is highly correlated with that variable. In general, the galaxy averages shown in Figure 8 do not appear to be more tightly correlated with the environmental parameters. Figure 8 thus illustrates that aside from outliers at the extremes of these plots, the minimum standard deviation of α_CO in these bins is a factor of two or more and correlations with environmental parameters do not allow us to predict the behavior of α_CO within the galaxies better than our normalized radial profile.

As previously discussed, the average profile of α_CO versus radius is mostly flat with a central depression in some galaxies. Many of the environmental parameters we plot in Figures 8 and 9 show radial trends extending over the entire range we cover, although the normalized trends generally span less than an order of magnitude. This range does not provide the leverage necessary to separate the dominantly radial correlations of multiple variables. Future studies attempting to associate changes in α_CO with environmental parameters will need either higher precision measurements of α_CO or observations spanning a greater range of environments.

One possible way to overcome the limitations of separating various radial trends is to normalize all of the variables by their mean in radial bins and then search for residual trends between them. We have investigated such non-radial variations in 0.1 r₂₅ bins and, in general, find only very weak trends. NGC 4254 is one of few galaxies that shows a marginally significant correlation—the normalized α_CO measurements correlate with normalized $\overline{U}$ and Σ_* at the level of 4σ–5σ from the Spearman RC coefficient. In both cases, the correlation is positive, such that α_CO increases in regions with higher $\overline{U}$ and Σ_* compared to the average in that radial bin. NGC 6946 also shows a correlation at the 5σ level between normalized α_CO and f_PDR. In this case, the correlation is negative, meaning that α_CO decreases in regions with high f_PDR.

NGC 6946 is unique in that the non-radial structure is clearly visible in the maps of α_CO shown in the Appendix. The low α_CO values in that galaxy appear to track the spiral arm structure seen in the I_CO and Σ_D maps. To quantify this observation, we have investigated the correlation between the radially normalized α_CO, I_CO, Σ_D, and $\Sigma _{{\rm H}\,{\scriptsize I}}$ values. These correlations are weak (all are <5σ significance in the Spearman RC) but more widespread. NGC 3627, 4254, 4321, 5457, and 6946 all show some degree of correlation at >3σ with radially normalized I_CO, $\Sigma _{{\rm H}\,{\scriptsize I}}$, or Σ_D values. For all galaxies aside from NGC 6946, the correlations are positive, in the sense that the conversion factor is higher where there is more H i and dust or more CO emission. NGC 6946 shows negative correlations with all of these parameters.

4.6.1. α_CO versus Metallicity

Metallicity has been suggested by several theoretical and observational studies to be an important driver for α_CO variations. Unfortunately, metallicity measurements in nearby galaxies are often very uncertain and subject to systematic errors from different calibrations and techniques. In Figure 8, we show our α_CO measurements as a function of metallicity from the PT05 calibration (middle left panel). Our measurements span ∼0.5 dex in metallicity with no statistically significant trend in α_CO. One possible reason for the lack of a clear trend in this figure is that we have combined metallicity measurements obtained with different techniques. In addition, many of the galaxies in our sample lack constraints on possible gradients.

To explore any metallicity trends that may be washed out due to systematic effects when combining metallicities from different sources, we isolate a sample of galaxies with uniformly determined metallicities from M10 and plot those separately in Figure 10. We have eliminated galaxies whose metallicities have been determined from integrated spectrophotometry (those listed as "M10 Table 9" in Table 3) because the drift scan observations were not always along angles that allow a robust gradient measurement. We also isolate our highest confidence α_CO measurements by showing only those with uncertainties less than 0.3 dex (a factor of ∼2). We show metallicities with both the PT05 and KK04 calibrations for comparison.

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In general, there is a slight negative correlation of α_CO with metallicity for this subsample. With the PT05 calibration, we find a RC coefficient of r = −0.2, which is 3.8σ from the null result given the number of measurements. Using the KK04 metallicities, we find r = −0.1 at 2.0σ from the null result. If we remove the S/N cut on the α_CO measurements, the correlations in both PT05 and KK04 essentially disappear, yielding results that are less than 1σ from the null result.

The weakness of the correlation between α_CO and metallicity suggests that in the regions of the galaxies we are studying, metallicity may not be the primary driver of α_CO variations. Along these lines, it is interesting to note the contrast between a galaxy like NGC 6946, which has a relatively shallow metallicity gradient, and galaxies like NGC 0628 and 3184, which are thought to have much steeper gradients. NGC 6946 has α_CO values spanning a range of an order of magnitude, while NGC 0628 and 3184 have much smaller ranges—the opposite of what we would expect if their metallicity gradients were the dominant factor controlling α_CO.

In the Appendix, we show the measured DGR for all galaxies and solution pixels in figures similar to Figure 3. The DGR we report is the mean value for all of the individual sampling points in a solution pixel. Generally, the scatter in the DGR_i values within a solution pixel is small (<0.1 dex). This result means that the uncertainty on the α_CO value generally dominates the uncertainty on the DGR as well. Therefore, it is important to note that the errors in the DGR and α_CO values are highly correlated. We assign a representative uncertainty on the DGR using the ±1σ bounds on α_CO. In general, when good solutions are obtained, the DGR measurements vary smoothly across the galaxy. This result supports our key assumption that the DGR varies on scales larger than our solution pixels. The smoothness of the DGR maps can be seen by inspecting the figures in the Appendix and the results for NGC 0628 shown in Figure 3.

In Figure 11, we show a histogram of all of the measured DGR values for our galaxies that comply with our cuts in inclination and I_CO. The average of our measurements treating all lines of sight equally is log(DGR) = −1.86 with a standard deviation of 0.22 dex. This finding is slightly higher than the value typically adopted for the solar neighborhood of log(DGR_MW) = −2.0. Forcing all galaxies to contribute equally to the average despite differing numbers of solution pixels gives log(DGR) = −1.96. The lower scatter in the DGR measurements for all solution pixels compared to the α_CO measurements, despite the uncertainty on α_CO dominating the uncertainty on the DGR, is due to the contribution of H i to the total gas mass surface density. Because H i makes up some fraction of the gas mass, the DGR has a smaller possible range over which to vary compared with α_CO.

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The DGR may also vary as a function of environmental parameters. To explore any such trends, we show our measured DGR values as a function of the same tracers we have previously studied in Figure 12. We list the correlation coefficients and linear fit parameters in Table 9.

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Table 9. DGR Correlation Properties

log(DGR) vs.	Correlationa	σ from	Linear Fitb
Variable	Coeff.	Null	Offset	Slope
r25	−0.33	9.2	−1.71	−0.35
log($\overline{U}$)	+0.01	0.3	−1.87	+0.02
qPAH (%)	+0.00	0.1	−1.85	+0.00
12+log(O/H)	+0.26	7.3	−6.50	+0.55
log(Σ*)	+0.28	7.9	−2.27	+0.21
log(ΣSFR)	+0.03	0.8	−1.79	−0.03

Notes. These correlations are shown in Figure 8. ^aSpearman rank correlation coefficient. ^bCorrelation with metallicity using a homogeneous sample of metallicity measurements is discussed in Section 4.8.

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The DGR shows significant correlations with galactocentric radius, metallicity, and Σ_*. Even with heterogeneously determined metallicities, it shows a much clearer trend with 12+log(O/H) than α_CO does over the same range. A correlation of the DGR with metallicity is expected—given the high depletions of elements such as Mg, Si, Ca, and Ti in the local area of the MW, to first order the mass of dust should be proportional to the amount of heavy elements.

In Figure 13, we show the DGR as a function of metallicity for the same sample of galaxies in Figure 10. In addition to the best linear fit, shown with a dotted black line, we also plot a prediction for the linear scaling of the local MW DGR with metallicity (solid black line). The dashed lines above and below show a factor of two higher and lower DGRs. The scaling of the plot is such that it covers three orders of magnitude, the same range covered in Figure 10. It is clear that the DGR values have much less scatter than the α_CO values over the same range of metallicity—a product of the limited allowable range of DGRs set by the fact that some dust is associated with H i.

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The DGR values show a stronger correlation with metallicity than the α_CO values do. With the PT05 calibration, we find a RC coefficient of 0.35, which is significant at the 6.4σ level. For KK04, the correlation coefficient is 0.39 at 6.9σ. In a given metallicity bin, the standard deviation of the DGR values is ∼0.15 dex for PT05 and ∼0.18 for KK04. This result suggests that, on average, we should be able to predict DGRs to better than a factor of two given the metallicity in one of these calibrations. In addition, we do not see evidence for major galaxy-to-galaxy offsets in the DGR versus metallicity plots, which distinguishes the DGR and α_CO behaviors; most galaxies appear to have similar slopes in the plots.

Neither the KK04 nor PT05 calibration values fall directly along the line of the scaled MW DGR. On the PT05 scale, almost all of our sample has higher DGRs for a given metallicity than the MW scaling, while the opposite is true on the KK04 scale. For purposes of scaling the MW metallicity, we have used strong-line abundance measurements calculated from the integrated spectrum of the Orion Nebula.

The best linear fit to the data is shown with a dotted line in both the PT05 and KK04 panels. This fit is parameterized by the following equation, where uncertainties have been determined using bootstrapping:

$$\begin{equation} {\rm log}({\rm DGR}) = a + b(12+{\rm log}({\rm O/H})-c). \end{equation} \tag{ 4 }$$

For the PT05 measurements, we find the following constants:

$$\begin{eqnarray*} &&a = -1.86\pm 0.01\\ &&b = 0.85\pm 0.11\\ &&c = 8.39. \end{eqnarray*}$$

For the KK04 measurements, the constants are:

$$\begin{eqnarray*} &&a = -1.86\pm 0.01\\ &&b = 0.87\pm 0.11\\ &&c = 9.05. \end{eqnarray*}$$

In these equations, the constant c is the mean metallicity of the sample of points. We use this parameterization in order to avoid covariance in the offset (a) and slope (b). In both cases, the slope is below unity. It can be seen in Figure 13 that the flatter-than-unity slope is mainly due to NGC 3184 (blue stars). Leaving this galaxy out of the fits produces slopes of 1.13 ± 0.10 and 1.01 ± 0.11 for the PT05 and KK04 calibrations, respectively. Omitting this galaxy also increases the significance of both correlations by 1σ–2σ.

We note that it is necessary to use the same strong-line metallicity calibration that we have (i.e., KK04 or PT05) in order to predict the DGR (or α_CO) using our fits. To the extent that applying our DGR(Z) relation is merely an interpolation over a given metallicity range, there is relatively little uncertainty introduced in the process. Any systematic errors in the metallicity scale itself are minimized if the same scale is used in the calibration and the application.

The local region of the MW is seen to have a conversion factor close to α_CO = 4.4 M_☉ pc⁻² (K km s⁻¹)⁻¹. This standard value has been recovered to within a factor of two by various techniques: γ-ray measurements (Digel et al. 1996; Abdo et al. 2010); virial masses (Solomon et al. 1987); and dust (Dame et al. 2001; Pineda et al. 2008; Planck Collaboration et al. 2011b). A number of studies of the Galactic center, however, have found very different answers. Using γ-ray observations, Strong et al. (2004) found evidence that α_CO in the Galactic center was a factor of 5–10 lower than in the disk. Similar results using dust as a tracer for total gas mass have been found by Sodroski et al. (1995), who suggest that α_CO is lower by a factor of 3–10. Dahmen et al. (1998) modeled multiple ¹²CO, ¹³CO, and C¹⁸O lines and found that the standard conversion factor from the disk overestimated the molecular gas mass in the Galactic center by an order of magnitude. Thus, several independent measurements suggest that the MW has a α_CO value near its center that is between 3 and 10 times lower than that in the solar neighborhood. The properties of the transition from the standard disk value to this lower number are not well constrained, although γ-rays provide some hint of a gradient (Strong et al. 2004).

The agreement among the various techniques in the Galactic center provides confidence in the resulting low α_CO. On their own, each technique is subject to a number of important systematic uncertainties. For γ-ray modeling, a key limitation is our knowledge of the cosmic ray distribution. In the case of multi-line modeling approaches, it is not clear that the approximations involved adequately represent the variety of excitation conditions in the molecular gas (Mao et al. 2000; Bayet et al. 2006). Virial mass-based techniques rely on the assumption that the clouds are in virial equilibrium, with gravity opposed primarily by turbulent motions. If magnetic stresses and thermal pressures are significant, or if GMCs are actually not self-gravitating, the virial technique may fail to reflect the true conversion factor, particularly in regions like the Galactic center (e.g., Dahmen et al. 1998).

Measurements of α_CO using various techniques in nearby galaxies have provided somewhat contradictory results. Virial mass studies of GMCs in nearby galaxies tend to find α_CO values very similar to the MW disk (Wilson 1995; Bolatto et al. 2008), even in galaxy centers (Donovan Meyer et al. 2012) and in low-metallicity galaxies like the SMC (Bolatto et al. 2008). In contrast, a number of studies using dust-based techniques have found very high α_CO values in low-metallicity galaxies, suggesting substantial amounts of "dark molecular gas" or "CO-free H₂" around GMCs (Israel 1997; Dobashi et al. 2008; Leroy et al. 2009a, 2011). Leroy et al. (2011) used dust to trace the total amount of gas and determine α_CO for the major galaxies of the Local Group. Their results suggested that (1) at low metallicity, α_CO can be well above the standard MW value, (2) in the central few kiloparsecs of M31, the conversion factor was slightly lower than the standard value, and (3) the 10 kpc ring of M31, M33, and the LMC (which collectively span ∼0.5 dex in metallicity) all showed α_CO values in agreement with the MW value. Studies of the centers of several nearby galaxies with "large velocity gradient" (LVG) or related modeling have found evidence for α_CO up to an order of magnitude lower than the MW value (e.g., Weiß et al. 2001; Israel 2009a, 2009b).

The study presented here is the first to create maps of α_CO in galaxies outside the Local Group. Therefore, we are able to study α_CO across a range of environments and connect the various trends that have been found in previous works. We find that for I_CO > 1 K km s⁻¹, α_CO is relatively constant for a range of radii and metallicities in galaxies. In the central ∼kiloparsec, we observe that galaxies often show a lower-than-average α_CO value, sometimes by up to factors of 10. These low α_CO values we find in several galaxy centers are not unprecedented, given the results from multi-line modeling studies and the evidence that the Galactic center shows a similar depression. Our lowest metallicities are still above the transition abundance seen by Leroy et al. (2011), where galaxies in the Local Group began to display much higher α_CO (i.e., 12+log(O/H) ≲ 8.2). The picture that emerges from the synthesis of these literature results is one where α_CO is generally around the standard MW value with a factor of ∼2 variability in the disks of galaxies, in the regime where CO is bright and metallicity is greater than ∼1/2 Z_☉, with some galaxies showing lower α_CO in their central kiloparsec.

In several cases, our α_CO measurements overlap with previous ¹²CO J = (1–0) virial mass based studies. Bolatto et al. (2008) derived GMC properties and virial masses for NGC 2976 and 3077. Their results for NGC 3077 are in good agreement with previous studies by Meier et al. (2001) and Walter et al. (2002). Recently, Donovan Meyer et al. (2012) and Donovan Meyer et al. (2013) determined α_CO in the central regions of NGC 6946 and 4736, respectively, using virial masses. In Table 10, we provide a summary of the galaxies in our sample with existing literature measurements. Figure 14 shows a comparison between these literature virial mass-based α_CO measurements with the values derived in this study. In almost all cases, we find a lower α_CO using the dust-based technique than the virial mass technique.

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For NGC 2976, Bolatto et al. (2008) found conversion factors ∼4 times larger than the MW value, on average. Almost all of our measurements for NGC 2976 show α_CO being consistent with the MW value, within their uncertainties. The central solution pixel has α_CO a factor of 4–5 below the MW value. In contrast, the virial mass and dust-based techniques are in agreement for NGC 3077, both returning a conversion factor consistent with that of the MW. The center of NGC 6946 stands out in our study as a place with a particularly low α_CO. However, virial mass measurements find α_CO more consistent with the MW value. Figure 14 illustrates that the discrepancy decreases with radius for NGC 6946—many of the points with galactocentric radii closer to r₂₅ ∼ 0.2 agree with our dust-based measurements. We will discuss the discrepancy between the virial and dust-based α_CO values in galaxy centers further in Section 6.3. It is interesting to note that the normalcy of NGC 3077, currently undergoing a starburst, compared to the also highly star-forming centers of 4736 and 6946, suggests that conditions other than just high star-formation surface density must contribute to determining α_CO.

The center of NGC 6946 has been observed extensively in molecular gas lines and modeled in a variety of ways. Israel & Baas (2001) modeled ¹²CO and ¹³CO lines plus [C i] (492 GHz) observations and found a conversion factor ∼10 times lower than the MW value in the center. Using LVG modeling of multiple CO isotopomers including ¹²C¹⁸O, Walsh et al. (2002) found a conversion factor 4–5 times below the MW value. Similarly, Meier & Turner (2004) used an LVG analysis of C¹⁸O observations, in addition to multiple CO lines, to argue that the central region had a conversion factor four times below the MW value. All of these studies suggest a central α_CO between 4–10 times lower than the MW value, in good agreement with what we observe with dust, but in contrast with the virial mass results.

Several other galaxies in our sample have been modeled with multi-line techniques, however they are at high inclination and therefore do not have reliable α_CO measurements from our work. Israel (2009b) used multiple ¹²CO and ¹³CO lines plus [C i] (492 GHz) observations toward the central region of NGC 4631 to constrain α_CO and found a value six times lower than the standard MW value. Similarly, Israel & Baas (1999) argued for α_CO ∼ 5 times lower than the MW value in the center of NGC 7331. In both of these highly inclined galaxies, we find α_CO a factor of two or more higher than the MW α_CO. This result is due to the failure of our technique at high inclinations.

Regan et al. (2001) found that approximately half of their sample of galaxies observed in the BIMA SONG survey showed excess CO emission in their centers compared to the extrapolation of their best-fit exponential disk profile. They argued that this result could be due to enhanced reservoirs of molecular gas in the centers or a change in the conversion factor. Table 6 lists the central excess classification for the galaxies in our sample studied by Regan et al. (2001). Indeed, we find that in every case where these authors found excess CO above the exponential profile, we find a conversion factor significantly below the MW value. Conversely, for the three galaxies we have observed and that they found to not have an excess, we find conversion factors consistent with the MW value or slightly higher. Similar results are found when comparing with the radial profiles determined from the HERACLES CO maps we utilize in this study (see radial profiles in Schruba et al. 2011). In general, the offset between the central α_CO and galaxy average α_CO from our measurements is similar in magnitude to the excess observed at the centers compared to the extrapolated exponential disk profile.

In the following, we explore the correlations or lack thereof between α_CO and parameters that may influence it. Since our resolution elements are large compared to an individual GMC, our α_CO value is an average over a population of GMCs, including any molecular gas that may be in a more diffuse phase and not in self-gravitating clouds.

For an individual, self-gravitating, turbulence-supported molecular cloud, there are several key parameters that influence α_CO: the density and temperature of the gas and the fraction of the mass that exists in the "CO-dark" phase (where H₂ self-shields and CO is photodissociated). The general dependence of α_CO on density and temperature can be illustrated using a simple model—a spherical, homogeneous cloud where self-gravity is opposed primarily by turbulence and ¹²CO J = (1–0) is optically thick. Following the derivation in Draine (2011),

$$\begin{equation} \alpha _{\rm CO} = 3.4 n_3^{0.5} (e^{5.5/T_{{\rm ex}}}-1)\ \hbox{$M_\odot $}\ \mathrm{pc}^{-2} (\mathrm{K\ km\ s}^{-1})^{-1}. \end{equation} \tag{ 5 }$$

Here, n₃ is the H nucleon density n_H = 10³n₃ cm⁻³ and T_ex is the excitation temperature. Real molecular clouds are certainly not spherical and homogenous, but this model allows us to understand the basic dependence of α_CO on density and temperature (note that simulated molecular clouds with a non-spherical, non-homogenous structure show similar basic scalings; Shetty et al. 2011). Molecular clouds with increased density will have higher α_CO while clouds with increased temperature will have lower α_CO.

Because CO is optically thick at solar metallicity, the abundance of C or O does not directly influence α_CO. However, metallicity may indirectly influence α_CO by altering the density or temperature structure of the gas, since the heating and cooling rates may be affected by metallicity or correlate with metallicity. At lower metallicities, the transition between ionized and neutral carbon to CO may shift relative to the H i/H₂ transition due to a lack of dust shielding, leading to layers of "CO-dark" H₂ (Tielens & Hollenbach 1985; van Dishoeck & Black 1988; Wolfire et al. 1993; Kaufman et al. 1999; Bell et al. 2006). α_CO will increase when there are significant amounts of gas in these layers. Recent work by Wolfire et al. (2010) used PDR modeling and a spherical model for a turbulent molecular cloud to study variations in the conversion factor. These authors found that the "dark molecular gas" fraction (i.e., the fraction of the molecular cloud mass where H₂ exists but CO is photodissociated) depends primarily on the extinction through the cloud or the DGR. Studies of simulated molecular clouds have also found a dependence of α_CO on extinction through the cloud (Glover & Mac Low 2011).

The simple model we have described above assumes that the cloud's self-gravity is balanced by turbulence. If there are other forces that contribute to the virial equation, α_CO may be altered. For a self-gravitating cloud, an increase in the pressure on the surface of the cloud will lead to a larger velocity dispersion. Because CO is optically thick, the amount of emission that "escapes" in the CO line is directly tied to the velocity dispersion of the cloud and increasing the velocity dispersion increases the CO emission for the same mass of H₂. Therefore, if a self-gravitating cloud were subject to significant external pressure, we would expect a lower α_CO. Conversely, if magnetic fields play an important role in supporting the cloud, this effect would lower the velocity dispersion and raise α_CO. In ultra-luminous infrared galaxies (ULIRGs), the molecular gas may not be in individual self-gravitating clouds, but instead in a large-scale molecular medium where stellar mass contributes to the gravitational potential (e.g., Downes & Solomon 1998). In this case, the velocity dispersion is larger than what would be expected from the gas mass alone and the CO emission is enhanced for a given gas mass (resulting in a lower α_CO).

In a ∼kiloparsec region of a galaxy, many individual molecular clouds will be averaged together in our measurement. Therefore, another factor that could contribute to the variation of our measured α_CO is changes in the cloud populations. In addition, any significant component of diffuse CO emission would be included as well. Several studies have suggested that CO emission from diffuse gas may not be negligible, even in the local area of the MW (Liszt & Lucas 1998; Goldsmith et al. 2008). Interestingly, it appears that locally, α_CO is similar in diffuse molecular gas and self-gravitating GMCs (Liszt et al. 2010). Recent work by Liszt & Pety (2012) has shown that under differing environmental conditions, the conversion factor appropriate for diffuse gas can vary substantially. Therefore, depending on the amount of diffuse molecular gas and the local conditions in a galaxy, α_CO may be very different from what is observed in the local area of the MW. The contribution of diffuse molecular gas with a different α_CO has been cited previously as a cause for discrepancies between different techniques for measuring α_CO in M51 (Schinnerer et al. 2010).

To summarize our theoretical expectations, α_CO measured in ∼kiloparsec regions of nearby galaxies can vary as a function of environment for many reasons: changes in the excitation temperature and density of the gas or contributions from pressure or magnetic support in self-gravitating clouds; envelopes of "CO-dark" gas; changes in the molecular cloud population; or contributions from diffuse CO emission. Unfortunately, directly measuring temperature, density, and velocity dispersion in large samples of extragalactic GMCs is challenging due to the need to resolve individual clouds in multiple molecular gas emission lines. Therefore, we are left with more indirect tracers of changes to the GMC properties. In Section 4.6, we examined the correlation of α_CO with $\overline{U}$, q_PAH, metallicity, Σ_*, Σ_SFR, Σ_D, and galactocentric radius.

The average radiation field $\overline{U}$, measured from the dust SED, could influence α_CO through the gas excitation temperature. If photoelectric heating dominates over other heat sources (e.g., cosmic rays) at the τ_CO ∼ 1 surface, then the intensity of the radiation field may play a role in determining T_ex (Wolfire et al. 1993). Likewise, q_PAH and metallicity could both influence the efficiency of the photoelectric effect (Bakes & Tielens 1994; Röllig et al. 2006), thereby changing the heating rate. We expect that Σ_SFR should be responsible for higher $\overline{U}$ in many regions. Enhanced SFRs leading to higher radiation field intensities has been suggested as the explanation for the observed high α_CO in several outer-disk molecular clouds in M33 (Bigiel et al. 2010). If the gas excitation temperature were affected by the radiation field or photoelectric heating, we would expect negative correlations (i.e., lower α_CO at higher $\overline{U}$, Σ_SFR, q_PAH, etc.). The correlations we observe between α_CO and $\overline{U}$, q_PAH, and Σ_SFR are generally weak. $\overline{U}$ and α_CO show the strongest association and the slope of the trend is negative, with lower α_CO at higher $\overline{U}$. This result is consistent with the expectation of higher radiation field intensities leading to warmer molecular gas temperatures, but the correlation is weak and other variables may play a more important role.

We see a weak trend of α_CO with metallicity. Since our observations are limited to regions with metallicities similar to or higher than that of the MW, we may not expect to see a strong correlation between α_CO and metallicity. The fraction of gas mass in the layer where CO is photodissociated is predicted to be ∼30% at the MW metallicity/DGR (Wolfire et al. 2010). Thus, increasing the DGR should only have a minimal effect on the conversion factor at MW metallicity and above. To illustrate this point, in Figure 15 we plot our α_CO measurements as a function of metallicity and overlay the Leroy et al. (2011) Local Group measurements and the models of Wolfire et al. (2010) and Glover & Mac Low (2011). The model predictions are normalized to α_CO = 4.4 M_☉ pc⁻² (K km s⁻¹)⁻¹ at the MW metallicity (i.e., 12+log(O/H) = 8.5 in PT05 and 8.8 in KK04; from the Orion Nebula). Both models assume a linear scaling of the DGR with metallicity and a fixed gas mass surface density for molecular clouds. The model predictions are discussed in detail in Leroy et al. (2013). In both calibrations, our measurements do not extend into the regime where "CO-dark" H₂ dominates α_CO. We note that an investigation of the α_CO–DGR relationship using our results would require taking into account the strong correlation of the uncertainties in these parameters.

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The strongest correlation we observe for α_CO is that with Σ_*. This correlation is the only one that is stronger than the correlation of α_CO with galactocentric radius r₂₅. In essence, our definition of r₂₅ is based on a scale length of the stellar disk, defined by a B-band surface brightness, so it is possible the stronger correlation of α_CO with Σ_* is due to it being a better proxy for the disk scale length than r₂₅. In addition, of the environmental parameters we have available for our analysis, Σ_* is the most straightforward to measure and may show the smallest systematic uncertainties. α_CO/Σ_* may be the strongest correlation simply because the other parameters are more uncertain. The question remains, however, as to why α_CO would correlate with the stellar mass surface density. There are a number of possibilities: pressure contributions to GMC virial balance, enhanced fractions of diffuse CO emission correlated with ISM pressure, changes to the population of GMCs, or other effects. We do not speculate here about what causes this correlation, but rather note that observations of multiple molecular gas lines at GMC resolution will help understand these trends and are possible with ALMA.

Although we see some evidence for weak correlations between α_CO and environmental variables, it is not possible with this dataset to distinguish the cause of these variations. Our primary result is that in regions with I_CO > 1 km s⁻¹ in these galaxies, α_CO is mostly insensitive to environment except in the central regions. This result may be due to our limited S/N for each individual α_CO measurement plus the relatively small range of environmental conditions we probe. Another possibility is that in the disks of most galaxies, the properties of molecular gas are insensitive to environment because most of the gas resides in molecular clouds that are not under significant external pressure or subject to enhanced turbulence or magnetic fields. The conversion factor appropriate for such clouds should be similar to the standard MW value.

Because of the resolution of our observations, our measurements average over a population of molecular clouds. This result means that environmental variations in the cloud population can also be responsible for changes in α_CO. Therefore, it is interesting to compare our results to what has been found with galaxy simulations, despite the fact that these simulations are forced to adopt a sub-grid model that prescribes α_CO for unresolved clouds. In Figure 16, we show our measured α_CO values as a function of the average I_CO in each solution pixel. Due to our completeness cut, we have no points below 1 K km s⁻¹. The mean and standard deviation of the α_CO measurements in bins of 0.25 K km s⁻¹ are shown with black circles and error bars.

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Overlaid on Figure 16 we show the predictions of two recent simulations. Feldmann et al. (2012, F12) couples full galaxy simulations with the Glover & Mac Low (2011) cloud-based conversion factor predictions assuming a fixed gas temperature and either a constant line width of 3 km s⁻¹ or a virial scaling. The Narayanan et al. (2012, N12) predictions include sub-grid semi-analytic models for the chemical and thermal state of the gas and dust. In the N12 models, the temperature of the molecular gas is not fixed and depends on heating by cosmic rays and, in dense regions, on energy transfer with dust. For both models, it is necessary to adopt a filling factor or clumping correction due to the difference in resolution between the simulations and our observations. The simulations of F12 and N12 both have ∼60–70 pc resolution, so we apply the same correction to both. For purposes of comparing with the trend of our observed α_CO as a function of CO integrated intensity, we have adopted a filling factor correction of three. This correction factor is larger than purely the ratio of the resolutions because our beam will contain more than one molecular cloud. Verifying that this absolute scaling is correct would require a more detailed investigation that is beyond the scope of this paper.

We find that the predicted trends of α_CO with CO integrated intensity from N12 and F12 match the average observed behavior well, although both underpredict α_CO in the central regions of some galaxies. There is considerable scatter around this trend, however. F12 argue that on ∼kiloparsec scales metallicity is the primary driver of the conversion factor variations. This metallicity dependence reflects, to first order, the Glover & Mac Low (2011) dependence between α_CO and extinction through the cloud with the addition that F12 assume a metallicity-dependent DGR. Our α_CO results do not show a strong dependence on metallicity, but may not extend to low enough metallicities to clearly distinguish such variations. N12 do see a decrease in α_CO in regions with high SFR surface densities due to enhanced gas temperatures and velocity dispersions, similar to what we have found in some galaxy centers.

For some galaxy centers in our sample, we measure α_CO values lower than the typical MW value by factors of 5–10. Interestingly, the virial mass-based α_CO measurements for several of these regions do not agree with the values found here, instead recovering α_CO values close to that of the MW. From Figure 14, we found that the discrepancy between virial mass and dust-based α_CO decreases with radius for NGC 6946—suggesting that this discrepancy is a property of the galaxy centers rather than an issue with the resolution of the virial mass techniques (which tends to be comparable to the size of the GMCs). For NGC 6946, results of multi-line modeling studies are generally in much better agreement with our results. In the following, we discuss processes that could lower α_CO measured from dust and multi-line modeling techniques, while leaving virial mass-based α_CO values similar to those of the MW.

For self-gravitating GMCs, all three techniques should derive a lower α_CO if the gas excitation temperature were changing independently of the other variables. In contrast, if the velocity dispersion of the cloud changed due to external pressure or additional sources of turbulence, the virial and dust techniques could arrive at different results. In such a situation, the measured line width would suggest a larger virial mass and the cloud's CO emissivity would also increase. Thus, one could derive a conversion factor similar to the MW value from the comparison of CO to virial mass since both have increased. Oka et al. (1998a) have suggested that external pressure in the Galactic center can account for up to an order of magnitude change in the conversion factor. However, Donovan Meyer et al. (2012) found that GMCs in NGC 6946 did not generally show enhanced velocity dispersions at a given size or luminosity, in contrast to the Galactic center clouds identified by Oka et al. (1998b). At the resolution of our CO maps, we cannot distinguish between changes in GMC internal velocity dispersion (which would affect α_CO) and changes in the velocity dispersion within the population of clouds, so we cannot test whether there is enhanced velocity dispersion in GMCs with our measurements.

It is possible that molecular gas in the centers of some galaxies is not in self-gravitating GMCs. If that is the case, the velocity dispersion may reflect other hydrodynamical processes and the virial mass estimate will be incorrect. For instance, if the molecular gas is not in GMCs but rather in a larger-scale molecular medium that is gravitationally bound to the stars and gas, the velocity dispersion will be larger and more CO emission will be produced for a given amount of gas. This explanation has been suggested by Downes et al. (1993) and Downes & Solomon (1998) to explain the properties of molecular gas in the nuclear disks of ULIRGs (see also recent work by Papadopoulos et al. 2012). Another option is a contribution to the CO emission from diffuse molecular gas, which could potentially have a lower α_CO than GMCs (Liszt et al. 2010).

The dust-based and multi-line modeling α_CO results may agree better than the virial results in galaxy centers because they do not assume a relationship between the velocity dispersion and the mass of molecular gas, which would cause issues in the cases outlined above. At the moment, only NGC 6946 (out of our i < 65° galaxies) has both virial and multi-line modeling results. In the other galaxies, we may be able to find some indication of how optical depth, excitation, and velocity dispersion affect α_CO by examining the ratios of ¹²CO and ¹³CO lines. A. Usero et al. (in preparation) present measurements of ¹³CO and ¹²CO lines toward regions of several HERACLES galaxies. Selecting all of their pointings toward the disks of galaxies in our sample (in regions above our I_CO and inclination cuts), the uncertainty weighted mean of R_12/13 = ¹²CO J = (1–0)/¹³CO J = (1–0) is 8.24  ±  0.11. The only galaxy centers with ¹³CO measurements are NGC 0628, 3184, 5055, and 6946. NGC 0628, 3184, and 5055 show R_12/13 values consistent with or slightly lower than the average disk value. NGC 6946, on the other hand, has R_12/13 = 18.53 ± 0.81. The high R_12/13 for the center of NGC 6946 has been noted in several previous studies (Paglione et al. 2001; Israel & Baas 2001; Meier & Turner 2004).

As shown in Figures 12 and 13, we observe a good correlation of the DGR with metallicity. Unlike most previous studies of the DGR resolved within galaxies, we made no assumption about α_CO in determining these values. Although we do not probe a wide range of metallicity due to the limitations on CO S/N, it is clear in Figure 13 in particular that the DGR is correlated with metallicity with less than a factor of two scatter over 0.5 dex in metallicity. A linear dependence of the DGR on metallicity suggests a constant fraction of heavy elements are locked up in dust grains. Chemical evolution and dust life-cycle models have varying predictions for the dependence of the DGR on metallicity (Dwek 1998; Lisenfeld & Ferrara 1998; Hirashita et al. 2002). Most of the dramatic differences between the models, however, occur at lower metallicities and observations of dwarf galaxies may provide more leverage on distinguishing between models (e.g., Herrera-Camus et al. 2012).

Due to the requirements on CO S/N to apply the technique we have used, our measurements only tell us about α_CO above I_CO = 1 K km s⁻¹ in the galaxies we have targeted. Generally, this requirement limits us to the inner parts of galaxies (<r₂₅), where metallicities are comparable to those in the MW disk. However, these are the regions where H₂ contributes most significantly to the total gas mass (Schruba et al. 2011), so applying our conversion factor results to unresolved galaxies should work reasonably well for recovering the total H₂ mass. A forthcoming paper extending our study to the H i dominated regions of the KINGFISH galaxies will test this assertion. Note that our recommendations are only applicable to spiral galaxies with metallicity similar to that of the MW.

For an unresolved galaxy, we recommend adopting our galaxy-based average α_CO of 3.1 M_☉ pc⁻² (K km s⁻¹)⁻¹ with uncertainty of 0.3 dex (a factor of ∼2). These values are the mean and standard deviation of the average value for all galaxies with inclinations less than 65°.

In dealing with resolved galaxies, we recommend adopting a flat radial α_CO profile for regions with I_CO > 1 K km s⁻¹, except in the central ∼0.1r₂₅, where the average α_CO is a factor of two lower. When dealing with a single galaxy, our average radial profile from Figure 7 suggest that the α_CO values have ∼0.2 dex of scatter in each radial bin (0.3 dex for the central bin). Therefore, for studies of molecular gas within a single galaxy, the relative values of α_CO adopted from this profile have a factor of ∼1.5 uncertainty. When comparing resolved galaxies, it is necessary to further recognize that the absolute normalization of the radial profiles, i.e., the galaxy average value, has an additional 0.3 dex uncertainty.

In the case where additional information about the galaxy is available, such as profiles of stellar mass surface density, SFR surface density, or line ratios of ¹²CO (2–1)/(1–0) or ¹²CO/¹³CO, we suggest using these maps to pick out regions that may have α_CO values very different from the mean. These regions have Σ_* ≳ 1000 M_☉ pc⁻², Σ_SFR ≳ 0.1 M_☉ yr⁻¹ kpc⁻², or R₂₁ ≳ 1 at our working resolution of ∼1 kpc. In these regions, α_CO can be 5–10 times lower than the MW α_CO, and are often systematically lower than the average radial profile.

The general procedure for finding the most "uniform" DGR in the solution pixel given spatially resolved measurements of $\Sigma _{{\rm H}\,{\scriptsize I}}$, Σ_D, and I_CO is to step through a grid of α_CO values, calculate the DGR at each sampling point in the solution pixel using that α_CO, measure some statistic that can be used to judge how uniform the DGR values are across the region, and, finally, either minimize or maximize that statistic as a function of α_CO. Below, we describe the various statistics we have tested to judge the uniformity of the DGR values in a given region. The reason we investigate all of these possible statistics is because each weights outliers differently, and it is not obvious a priori which will work best to recover the underlying α_CO. For techniques that involve minimizing scatter in the DGR, we use three different statistics to measure scatter: the standard deviation (RMS), the median absolute deviation (MAD), and a robust estimator of the standard deviation from Tukey's biweight (BIW; Press et al. 2002).

• 1.  
  Minimum fractional scatter in the DGR (FS). At each value of α_CO, we calculate the DGR for all points in the region in question. We then divide each DGR value by the mean DGR in the region and measure the scatter of the resulting values (with RMS, MAD, and BIW statistics).
• 2.  
  Minimum logarithmic scatter in the DGR (LS). At each α_CO, we calculate the scatter of the logarithm of the DGR values (with RMS, MAD, and BIW). This technique is similar to the minimization of fractional scatter, but with a different weighting for the individual points. This method also throws away data points with negative values, since their logarithm is undefined.
• 3.  
  Maximum correlation coefficient of  Σ_D and Σ_Gas (LC or RC). At each value of α_CO, we compute the linear correlation (LC) and rank correlation (RC) coefficients between Σ_D and Σ_gas = $\Sigma _{{\rm H}\,{\scriptsize I}}$ + α_COI_CO. The best α_CO value will maximize the correlation between these two quantities.
• 4.  
  Minimum χ² of best-fit plane to I_CO, ${N}_{{\rm H}\,{\scriptsize I}}$, and Σ_D (PF). The Σ_D, $\Sigma _{{\rm H}\,{\scriptsize I}}$, and $\Sigma _{\rm H_2}$ values describe a plane with two free parameters. At each value of α_CO, we fit a plane to the measurements to determine the DGR. The best value of α_CO will be the one that minimizes the χ² of the best-fit plane.
• 5.  
  Minimum correlation coefficient of the DGR versus I_CO/${N}_{{\rm H}\,{\scriptsize I}}$ (CHC). In this technique, we search for the α_CO value that minimizes the correlation of the DGR with I_CO/$N_{{\rm H}\,{\scriptsize I}}$. This technique weights the points differently depending on how much local Σ_gas depends on α_CO.

A key aspect of the technique we use to constrain the DGR and α_CO is defining the region over which we assume there to be one value each of the DGR. Under ideal circumstances, the regions would be as small as possible given the resolution of the maps. Realistically, however, there are several considerations that must be taken into account when defining the regions, including covering an adequate range of CO/H i ratios, having enough points to measure scatter accurately, and keeping the region to ∼kiloparsec scales over which it is reasonable to assume that the DGR is indeed constant.

To complement the hexagonal, half-beam spaced grid with which we sampled the original data (i.e., our "sampling points"), we again use hexagonal spacings to define the "solution pixels" as the individual samples can be divided naturally into concentric hexagons. We tested a variety of solution pixel sizes ranging from 19-point to 271-point hexagons.

We found that 37-point solution pixels were a good compromise between size and solution quality. Because of the underlying half-beam spaced sampling grid, the 37 point solution pixel contains ∼9 independent measurements. For the next smallest possible hexagon (19-points), there are not sufficient numbers of independent measurements to reliably judge the scatter in the DGR.

In order to fully sample the map, the center of each hexagonal pixel is offset by 1/2 of the spacing from its neighbor. This technique results in the "solution pixels" not being independent—each neighboring pixel shares ∼40% of the same data (the oversampling can be exactly described as n²/(3n² − 3n + 1) for concentric hexagons).

We have investigated the efficacy of the various techniques using simulated data modeled on our observations. Our goal is to identify the technique or techniques that return the most accurate values of α_CO and the DGR over the range of conditions and S/N levels in our dataset. We also aim to characterize any biases in the recovered α_CO as a function of input α_CO and S/N. In performing these tests, it is important to model the simulated data on our real observations because the ability of the techniques to judge the DGR uniformity depends not only on the S/N of the measurements but also on the range of CO/H i ratios. Since CO and H i can be correlated, anti-correlated, or independent of each other on kiloparsec scales, depending on the state of the ISM, it is difficult to generate a reasonable set of simulated data from scratch that encompasses the range of CO/H i behaviors in the observations. Therefore, we expect the best test of the technique will come from simulated data that are closely modeled on the observations. In the following, we describe how the simulated data are generated and discuss the results of the test.

The simulated data are generated via a Monte Carlo procedure in which we randomly choose a galaxy from our sample (listed in Table 1) at SPIRE 350 μm resolution, choose a solution pixel, and create simulated data based on the selected CO, H i, and Σ_D samples. For each trial, we randomly sample a range of α_CO values between 0.5 and 50 M_☉ pc⁻² (K km s⁻¹)⁻¹ and intrinsic scatter per axis (evenly divided among the I_CO, $\Sigma _{{\rm H}\,{\scriptsize I}}$, and Σ_D axes, for lack of better knowledge of the sources of intrinsic scatter) between 0.0 and 0.4 dex. We note that because of the properties of lognormal distributions, the scatter in the DGR is not simply $\sqrt{3}\times$ the input intrinsic scatter per axis, so we approximate it after the fact using the input scatter per axis and the mean H₂/H i ratio for the simulated measurements.

Our procedure is the following: (1) select the observations with the sampling described above, (2) use the I_CO and $\Sigma _{{\rm H}\,{\scriptsize I}}$ measurements along with the random α_CO value to create a simulated Σ_gas vector, (3) calculate the DGR value that preserves the average S/N level of the Σ_D measurements and create a simulated, noise-free Σ_D vector by multiplying the Σ_gas from step 2 by this DGR, (4) apply the randomly selected intrinsic scatter to each axis, and (5) add simulated measurement errors according to the observed uncertainties in the I_CO, $\Sigma _{{\rm H}\,{\scriptsize I}}$, and Σ_D maps. These simulated measurements are then used as inputs to solve for α_CO and the DGR using the techniques described above.

From this Monte Carlo simulation, we can then choose the technique that is the most robust (i.e., most frequently achieves a solution) and accurate (i.e., gets closest to the known input α_CO). In Figure 17, we show the results of this simulation for the techniques we have outlined. The panels show (left) the mean difference between the input and recovered α_CO value, (middle) the standard deviation of the difference of the input and recovered α_CO, and (right) the percent of the Monte Carlo trials that achieve a solution. These values are binned in a two-dimensional space defined by the intrinsic scatter added to the simulated data and the median S/N of the simulated CO measurements.

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It is clear that the FS and LS techniques provide the least bias in the recovered α_CO over the relevant range of intrinsic scatter and S/N (except for the FS_RMS, which is significantly biased at low CO S/N). The correlation coefficient (LC and RC), planefit (PF), and CO/H i correlation (CHC) techniques all provide much more limited regions with low bias in the recovered α_CO, as well as having a lower fraction of solutions. In general, the MAD based techniques have slightly less bias toward the low CO S/N region, but show a larger scatter between the input and recovered α_CO. Overall, we find that the LS_BIW (robust BIW mean of the logarithmic scatter) is the technique that shows the best performance in accuracy, precision, and robustness over the range of characteristics of our dataset.

In addition to helping us choose the best technique, the tests with simulated data also allow us to investigate how the measured minimum in the logarithmic scatter corresponds to the known intrinsic scatter added to the simulated data. In addition to being an interesting quantity scientifically, the measured scatter can also help us to identify which region of the parameter space a given solution falls in, thereby allowing us to judge how biased we expect it to be given our knowledge from these simulated data tests. Figure 18 shows a comparison of the known input intrinsic scatter with the measured minimum in the logarithmic scatter for the LS_BIW technique. Most of the outliers come from regions with low CO S/N. At very low levels of intrinsic scatter, the measured scatter is larger due to the influence of observational uncertainties. In general, however, the measured minimum log scatter is a very good proxy for the intrinsic scatter in the DGR.

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Based on these results with simulated data, we can use the measured median CO S/N and the measured minimum of the scatter in the DGR values to predict how biased and how uncertain a given determination of α_CO is for our dataset. To this end, we use the measured minimum in the DGR scatter as an estimate of the intrinsic DGR scatter (it is biased slightly higher, but this effect is minor and represents a conservative estimate of the intrinsic scatter). Then, using this value and the median CO S/N in the solution pixel, we interpolate in the grid presented in Figure 17 for "LS_BIW" to predict the bias inherent in the technique.

One of the key results of our work is the measurement of low α_CO values in the centers of some galaxies (see Section 6.3). These locations are outliers from the average solution pixel in the sense of having high CO S/N and often a relative deficiency in $\Sigma _{{\rm H}\,{\scriptsize I}}$. Since these locations make up a small subset of all solution pixels, our Monte Carlo simulations from the previous section may not adequately judge the bias of the technique in these regions since they are not sampled frequently enough (only 26 out of ∼900 solution pixels). We therefore perform a separate Monte Carlo simulation for the galaxy centers in order to verify that these low α_CO values are not due to biases in the technique (we have demonstrated that the average pixel in the sample is not biased using the experiment in the previous section).

The results of the technique are biased if the simulation returns an α_CO systematically different from the true α_CO. For this test, we will define several different α_CO values. First, α_{CO, meas} is the measured value for a given solution pixel; we do not know if this measured value is biased as it is only what we have measured for that pixel given the true data. Second, α_{CO, in} is a known α_CO value we have used to generate the simulated data. Finally, α_{CO, out} is the α_CO value that we recover for α_{CO, in} using our solution technique.

The essence of this test is that we take a range of α_{CO, in} values and map them to their α_{CO, out} values by generating simulated data given α_{CO, in} and running it through our solution technique. This simulated data generation proceeds as described in the previous section and is designed to preserve the CO/H i ratio and S/N of the measurements for a given pixel. In addition to observational noise, we also add intrinsic scatter between 0.01 and 0.1 dex to the simulated data, which reflects the level of intrinsic scatter we have found in the real solutions. For each solution pixel, we generate 10⁴ random α_{CO, in} values that span our range of α_CO = 0.1–100.

With this simulation, we can determine for any given value of α_{CO, out} which values of α_{CO, in} ended up there. If a solution is biased, the correspondence between α_{CO, in} and α_{CO, out} will not be one-to-one. For example, a bias may make us consistently recover α_{CO, out} = 1 M_☉ pc⁻² (K km s⁻¹)⁻¹ when α_{CO, in} = 0.1 M_☉ pc⁻² (K km s⁻¹)⁻¹.

In judging the biases of the actual solutions for our central solution pixels, we then select all of the α_{CO, out} measurements equal to α_{CO, meas} and find the mean and standard deviation of the α_{CO, in} values that gave those results. In Figure 19, we show a plot of the mean and standard deviation of the relation between the α_{CO, in} values and the α_{CO, meas} values. This figure shows that within the uncertainties of α_COmeas, none of the central pixel measurements show a bias.

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In addition, a second check on these low central α_CO, at least for NGC 6946, is shown in Figure 20 where we show the minimization curve for several techniques toward its central solution pixel. All techniques agree within their errors, highlighting the minimum at α_CO ∼ −0.4. This result emphasizes that the low α_CO found in the center is not an artifact of the particular technique we are using, but can be recovered via minimization of scatter in the DGR, minimization of the correlation coefficient between CO/H i and the DGR, and minimization of the χ² of the best-fit plane to CO, H i, and Σ_D.

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To summarize, we have tested a number of variations on the basic technique of finding the most "uniform" DGR in a given region by stepping through a grid of α_CO values. We created simulated data with known α_CO and DGR intrinsic scatter matched to the range of S/N in our dataset. Via a Monte Carlo simulation, we found the robust mean of the logarithmic scatter to be the technique that spans the range of CO S/Ns and possible intrinsic DGR scatters with the highest accuracy and rate of success. Given knowledge of the CO S/N for a region and the measured minimum of the logarithmic scatter, we can use the simulated data tests to constrain how biased our α_CO measurement can be. We have also specifically checked that the central regions of galaxies, which only comprise 26 pixels out of the sample, are not biased due to their sometimes unusual CO/H i ratios. We found no significant bias for these solutions and show in addition that a variety of techniques can reproduce the low central α_CO solutions, not just the minimization of the DGR scatter.

In Figure 21, we show the measured CO (2–1)/(1–0) line ratio in the solution pixels we have defined for each galaxy (the black points) and in radial profiles (colored lines). The assumed R₂₁ is shown with a horizontal solid black line and factors of two above and below that value are marked with black dashed lines. There are variations in R₂₁, but in regions with good CO S/N (the only regions in which we will achieve good solutions) these variations are generally less than a factor of two away from the R₂₁ we have assumed. This deviation is within the uncertainties on the α_CO solutions, which typically have 0.2 dex uncertainty, at minimum.

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