THE CO-TO-H₂ CONVERSION FACTOR FROM INFRARED DUST EMISSION ACROSS THE LOCAL GROUP

The literature contains several working definitions of α_CO that apply to different scales or phases of the gas. In this paper and Equation (2), we define α_CO as the factor to convert from CO emission to total molecular gas mass on scales larger than individual clouds. Under this definition, α_CO includes any H₂ associated with C⁺ in the outer, poorly shielded parts of clouds as well as gas immediately mixed with CO. Indeed, our goal is to measure whether such envelopes become dominant below some metallicity. Because this definition integrates over cloud structure, it is possible to derive a single α_CO for a whole galaxy or part of a galaxy, and to view that α_CO as a function of large-scale environmental factors. We choose this definition of α_CO because it is directly applicable to CO measurements of distant galaxies on kiloparsec scales.

This is distinct from the ratio of CO to H₂ only for molecular gas mixed with CO. Dynamical measurements using CO emission at high spatial resolution may mainly probe this quantity with limited sensitivity to H₂ in an extended envelope not mixed with CO. The difference between this quantity and the quantity that we study has caused some confusion, leading to apparent contradictions between IR-based measurements and dynamical measurements. In fact, these may be largely attributed to the different regions being probed (Section 6).

Similarly, we do not define α_CO as the ratio of H₂ to CO along a pencil beam. We have no reason to expect that this quantity, or any ratio that places many elements across a cloud, remains reasonably constant across part of a galaxy. In contrast, comparisons of dust and CO emission imply dramatic variations in the CO-to-H₂ ratio within individual clouds (Pineda et al. 2008).

Our spatial resolution ranges from 45 to 180 pc (Table 1). For these resolutions, giant molecular clouds (GMCs) will mostly lie within one or two resolution elements (e.g., Heyer et al. 2009). This is ideal for our definition of α_CO. We wish to integrate over the structure of these clouds to make a single α_CO a more appropriate assumption.

This experiment leverages our knowledge of H i column to infer α_CO. It thus works best in regions where both H i and H₂ are important to the ISM mass budget. If we target areas where H i dominates then α_CO has little or no impact on δ_GDR while if we target areas with only H₂ then α_CO and δ_GDR are degenerate.

For our targets and resolution, being dominated by H₂ is not a concern for any reasonable α_CO. On the other hand, each target has large areas where the ISM is overwhelmingly H i. Especially in M 31 and M 33, much of the H i is at large radius and appears to have a different δ_GDR from the inner galaxy (Nieten et al. 2006). We mostly avoid these H i-only regions, instead targeting the part of each galaxy near where CO is detected. This gives us a range of total gas surface densities and relative contributions by H₂ and H i, allowing us to constrain α_CO and δ_GDR.

We define our target region by a ≈3σ intensity cut in the convolved CO map, I_CO ⩾ 1 K km s⁻¹ in M 31, M 33, and the LMC, I_CO ⩾ 0.25 K km s⁻¹ in the SMC, and I_CO ⩾ 0.03 K km s⁻¹ in NGC 6822. We reject small regions (area ≲ a resolution element) as likely noise spikes and then consider all area within about one resolution element of the remaining emission. Finally, we require a line of sight to have significant IR emission to be included; this is I₁₆₀ ⩾ 5 MJy sr⁻¹ in NGC 6822, I₁₆₀ ⩾ 10 MJy sr⁻¹ in all other targets. The result resembles loosely circling bright CO emission by hand. Figure 1 shows the target regions in contour on top of the 160 μm map.

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In M 31, M 33, and the SMC treating the whole galaxy at once causes problems with our model, which assumes a constant α_CO and δ_GDR across the area studied. Previous work arrived at similar conclusions. Nieten et al. (2006) observed δ_GDR to be higher in the inner part of M 31 than the 10 kpc ring containing most of the CO. In M 33, bright CO extends a fair distance out into the disk, but based on radial profiles of H i, CO, and 160 μm intensity it is immediately clear that this galaxy, too, has a strong radial gradient in δ_GDR. The SMC's CO emission is clustered into three distinct regions that show evidence for local variations in their δ_GDR, α_CO, and GMC properties (Leroy et al. 2007; Muller et al. 2010).

To isolate regions with fixed α_CO and δ_GDR, we separate each galaxy into several zones. We treat the "inner" part of M 31 separately from the 10 kpc ring and further divide the ring into a "north" and "south" part. We exclude 60° around the minor axis (in the plane of the galaxy) of M 31 (see Section 4.3). We break M 33 into an "inner" zone where r_gal < 2 kpc and an "outer" zone where 2 kpc <r_gal < 4 kpc. We divide the SMC into three parts: a "west" region, a "north" region, and an "east" region. Our regions in the SMC deliberately exclude two clouds near the center of the galaxy identified in Leroy et al. (2007) to have high CO-to-IR ratios; including these clouds leads to even worse solutions for this part of the SMC. Black lines and labels in Figure 1 indicate the divisions for each galaxy.

After the target regions are defined, we sample each map using a hexagonal grid spaced by 1/2 the resolution (i.e., 225 in M 31, 20'' in M 33, 2' in the LMC, 225 in NGC 6822, and 13 in the SMC). This yields a matched, approximately Nyquist-sampled set of measurements of I₇₀, I₁₆₀, I_CO, and $\Sigma _{\rm H\,\mathsc{i}}$.

We follow the approach of Draine & Li (2007) and Draine et al. (2007) to estimate the amount of dust along the line of sight. These papers present models that can be used to estimate the dust mass and incident radiation field from IR intensities. Following Draine et al. (2007) and Muñoz-Mateos et al. (2009), we search a grid of models illuminated by different radiation fields to find the best-fit dust mass surface density, Σ_dust, for each set of IR intensities. A slight difference between our fits and those papers is that we quote the geometric mean of the dust mass across the region of parameter space where χ² < χ²_min + 1 (i.e., within 1 of the χ² for the best-fit model). We present the results of our own direct grid-search fits, but during analysis made extensive use of the work of Muñoz-Mateos et al. (2009), who parameterized the results of model fitting over a wide range of parameter space have been as functions of the IR intensity at 24, 70, and 160 μm.

We only need a linear tracer of dust mass to solve for α_CO, the normalization affects δ_GDR but not α_CO. While values of δ_GDR that we find are interesting on their own, the overall normalization of the Draine & Li (2007) models do not affect our results for α_CO. Before settling on the Draine & Li (2007) models, we also used modified blackbody fits with a range of emissivities to estimate the dust opacity, τ₁₆₀. We obtained very similar results for α_CO using that approach, in the end preferring the Draine & Li (2007) models mainly for their more straightforward handling of the 70 μm band, which can include significant out-of-equilibrium emission. When we derive the uncertainties associated with our assumptions (Section 4.4 and Appendix B), we adopt either a modified blackbody or the Draine & Li (2007) models with equal probability. When using a modified blackbody fit, we include a variable fraction of 70 μm emission from an out-of-equilibrium population and emissivity power-law index in the calculation.

M 31. M 31 is quiescent compared to our other targets. It has a very low I₇₀/I₁₆₀, implying very low radiation fields or colder dust temperatures (Gordon et al. 2006). Montalto et al. (2009) showed that with only Spitzer data the radiation field illuminating the dust is unconstrained. In addition to poor constraints on the model, these low I₇₀/I₁₆₀ make ratio maps extremely sensitive to artifacts or contamination by point sources. M 31, of all our targets, will benefit most from the additional spectral energy distribution (SED) coverage offered by Herschel. Our approach in the meantime is to treat all points in M 31 with as though they had the median IR color across the whole galaxy. We implement the recommendation by Draine et al. (2007) that in the absence of submillimeter data, the radiation field be limited to 0.7 times the local value. This treatment effectively assumes that value everywhere in M 31 and uses the I₁₆₀ μm to estimate the dust mass. We note this uncertainty in Table 2.

Table 2. Dust-based α_CO for Local Group Galaxies

System	αCOa	Uncertaintyb				Correlationc	Scatterd	Notes
		Stat.	Rob.	Assump.	Tot.		(log10δGDR)
M 31
Inner	2.1	0.14	0.03	0.26	0.29	0.68 ± 0.02	0.21	U unconstrained
South	10.0	0.05	0.01	0.12	0.13	0.85 ± 0.02	0.11	U unconstrained
North	4.2	0.05	0.02	0.19	0.20	0.83 ± 0.02	0.15	U unconstrained
M 33
Inner	8.4	0.07	0.03	0.15	0.17	0.76 ± 0.03	0.08
Outer	5.0	0.09	0.05	0.17	0.20	0.76 ± 0.03	0.10
LMC	6.6	0.04	0.02	0.14	0.14	0.84 ± 0.01	0.14
NGC 6822	24	0.13	0.13	0.26	0.31	0.56 ± 0.07	0.18	Averaged colors
SMC
West	67	0.06	0.02	0.08	0.10	0.91 ± 0.04	0.09	Subtracted $\Sigma _{\rm H\,\mathsc{i}}=40$ M☉ pc−2e
East	53	0.08	0.04	0.44	0.45	0.95 ± 0.11	0.07	Subtracted $\Sigma _{\rm H\,\mathsc{i}}=60$ M☉ pc−2e
North	85	0.16	0.05	0.11	0.20	0.47 ± 0.06	0.15	Subtracted $\Sigma _{\rm H\,\mathsc{i}}=20$ M☉ pc−2e

Notes. ^aBest fit. Units of M_☉ pc⁻² (K km s⁻¹)⁻¹ including helium. ^bEstimated 1σ uncertainty in log₁₀α_CO for each source of uncertainty. "Stat."—statistical (from Monte Carlo); "Rob."—robustness to removal of data (from bootstrapping); "Assump."—variation of assumptions. ^cLinear correlation coefficient relating Σ_gas and Σ_dust for best-fit α_CO. Uncertainty gives 1σ scatter for random re-pairings of data. ^d1σ scatter in log₁₀δ_GDR in dex for best-fit α_CO. ^eBased on the fit of $\Sigma _{\rm H\,\mathsc{i}}$ to Σ_dust at low Σ_dust.

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NGC 6822. NGC 6822 shows the faintest IR emission of any of our targets. As a result, many of the variations in I₇₀/I₁₆₀ appear driven by noise and artifacts rather than T_dust variations. Gratier et al. (2010) noted the difficulty of deriving dust temperatures for each line of sight. Based on plots of I₇₀ versus I₁₆₀, we identify two main "colors" in our data. We assign each line of sight the median color for its group, with the cut at I₇₀/I₁₆₀ = 0.6. We note this uncertainty in Table 2.

Gas with different δ_GDR may be superposed along the line of sight, in which case our assumption of a single δ_GDR no longer applies. In some cases, we expect there to be a diffuse H i component with little or no associated dust along the line of sight. Such a component requires that we introduce an additional term into Equation (2) so that Σ_gas remains finite once Σ_dust ∼ 0.

The simplest example of this is an edge-on spiral galaxy with a strong radial gradient in δ_GDR. M 31 is inclined and does show a gradient in δ_GDR (Nieten et al. 2006). The pile-up of many different radii along a single line of sight will be most severe along the minor axis and Equation (2) does not appear to describe these data as well as those along the major axis. Because we have plenty of data, we simply exclude 60° around the minor axes from our analysis. This improves the stability of our solution. No other correction appears necessary.

The SMC has an extended distribution of H i that may also be very elongated along the line of sight. In this H i envelope and away from the main star-forming regions (in the "Wing" and "Tail"), δ_GDR is observed to be high (Bot et al. 2004; Leroy et al. 2007; Gordon et al. 2009). Similar results are found for other dwarf irregulars (Walter et al. 2007; Draine et al. 2007). Given the very high column densities found in the SMC, it is plausible that low δ_GDR H i lies along the line of sight.

To account for an envelope of dust-poor H i, we subtract a diffuse component from the SMC H i map before fitting for α_CO. We estimate this component via an ordinary least squares bisector fit relating Σ_dust and $\Sigma _{\rm H\,\mathsc{i}}$ at low Σ_dust, where H i is likely to represent most of the gas. This line implies a value of $\Sigma _{\rm H\,\mathsc{i}}$ where Σ_dust = 0, which we take as our estimate of the diffuse H i. We estimate a diffuse component of $\Sigma _{\rm H\,\mathsc{i}} = 40$, 20, and 60 M_☉ pc⁻² for the western, northern, and eastern parts of the SMC. Similar fits to the other targets suggest a diffuse H i component with magnitude below 10 M_☉ pc⁻², usually consistent with zero. Our uncertainty estimates include a 20% (1σ) uncertainty on this diffuse component in the SMC and ±5 M_☉ pc⁻² in the case of galaxies other than the SMC.

This subtraction of diffuse H i differs from Leroy et al. (2007, 2009, though the latter applied this approach in some of the analysis). The effect on α_CO is largest in the "east" section of the SMC, which is embedded in the mostly diffuse Wing.

Appendices A and B lay out our method of solution and uncertainty estimates in detail. We identify the best-fit α_CO for each data set as the value that minimizes the point-to-point (rms) scatter in log₁₀δ_GDR. We estimate the uncertainty from three sources: (1) statistical noise, (2) robustness to removal of individual data, and (3) assumptions. These are reported in Table 2 and we take the overall uncertainty to be the sum of all three terms in quadrature.

Our model has several important limitations. It cannot recover a pervasive, CO-free H₂ component like that suggested for the LMC by Bernard et al. (2008). To derive such a component, we would need to make strong assumptions about δ_GDR. We derive α_CO for H₂ associated with CO emission on ∼100 pc scales, essentially α_CO for GMCs and their envelopes. Based on UV spectroscopy (e.g., Tumlinson et al. 2002), we consider a pervasive H₂ phase unlikely, but our experiment makes no test of this idea one way or the other.

We assume that δ_GDR does not vary between the atomic and molecular ISM. In fact, observations and theory suggest the δ_GDR does correlate with density. The depletion of heavy elements from the gas phase increases with increasing density (Jenkins 2009), though absorption measurements cannot probe to very high densities. Meanwhile, accounting for the observed dust abundance appears to require buildup of dust in molecular clouds (Dwek 1998; Zhukovska et al. 2008; Draine 2009). A lower δ_GDR in dense, molecular gas directly, linearly scales our derived α_CO. Dust already accounts for ∼50% of the relevant heavy elements, so this effect cannot exceed a factor of ∼2 in magnitude. In fact, the effect should be even less severe because we already study mainly the peaks of the gas distribution, where we expect H i to also have high densities.

Our dust modeling also implicitly assumes that the dust emissivity does not vary between the atomic and molecular gas. Observations of suggest that dust properties do change between the diffuse and dense ISM, with an enhanced emissivity in dense gas. The best evidence for this comes from a ∼30%–50% increase in far-IR optical depth relative to optical extinction at high columns (∼1 mag; e.g., Arce & Goodman 1999; Dutra et al. 2003; Cambrésy et al. 2005). Direct comparison of virial masses to submillimeter emission suggests a similar enhancement (Bot et al. 2007). The origin of this enhanced emissivity is thought to be a change in the size distribution of dust, with "fluffy," low-albedo grains created in dense environments (Dwek 1997; Stepnik et al. 2003; Cambrésy et al. 2005). As with δ_GDR variations, emissivity variations directly scale the derived α_CO.

Our best estimate is that emissivity and δ_GDR variations bias our measured α_CO high by a factor of ∼1.5–2.0. Because this estimate distills a variety literature results, none of them definitive, applying such a correction after the fact would simply confuse our results. More importantly, we have no handle on how these factors vary with environment, though we expect weaker effects at low metallicity, where shielding is weaker. We highlight a quantitative understanding of δ_GDR and dust emissivity vary with environment as the most important systematics that must be addressed to improve dust-based derivations of α_CO.

Finally, our data cannot rule out a pervasive population of very cold dust. We consider this unlikely and neglect it throughout the paper (e.g., see Draine et al. 2007). If such a population exists, the Herschel Space Observatory will identify it and our picture of dust will change dramatically over the next few years. However, we note that preliminary results suggest that dust masses are not dramatically affected by the inclusion of Herschel data (Gordon et al. 2010). Other systematics in using dust to trace H₂ have been discussed at length elsewhere (Israel 1997a, 1997b; Schnee et al. 2005; Leroy et al. 2007, with references to many other works therein) and we do not repeat them here. Our best estimate is that none of these represent a major concern. Almost all will be addressed by the inclusion of long-wavelength Herschel data (and are already improved relative to IRAS by using Spitzer's 160 μm band).

Before arriving at these results, we varied our methodology significantly. We tried different dust treatments, methods of solution, sampling regions, and treatments of diffuse H i. Most variations yield similar results to what we present here but a few cases are worth comment.

First, the northern part of the SMC is not well described by our model even after we excluded the high CO-to-IR clouds. Several distinct groups of data appear in the scatter plots for this region, suggesting local variations in δ_GDR or α_CO. The data for this region are already good (Muller et al. 2010), so improved modeling—multiple dust and H i components, local variations in α_CO and δ_GDR—seems like the most acute need. In the meantime, our solution minimizes δ_GDR and yields a result consistent with the other SMC regions.

NGC 6822 yields reasonable solutions but is somewhat unstable to the choice of methodology. Adjusting the dust treatment, background subtraction, or weighting can change the best-fit α_CO by ∼50%. This instability results mostly from the low signal-to-noise ratio and dynamic range in the data. These, in turn, are low because of the large distance of NGC 6822 compared to the SMC or the LMC, which makes observations challenging. The resulting low intensities also make confusion with the Milky Way an issue (Cannon et al. 2006). Improved resolution from Herschel or ALMA should allow a greater range of Σ_dust as individual clouds are resolved. This will enable a stronger constraints on α_CO.

The inner part of M 31 is also somewhat unstable to the choice of methodology—especially region definition and fitting method. Best-fit solutions can range from α_CO ∼ 1 to 5. The most likely cause is variations in δ_GDR and α_CO within the region studied—the data do not appear to be a plane in CO–H i–dust space. Both improved modeling and better IR SED coverage will help this case.

We emphasize that our approach to the SMC is conservative. We subtract a diffuse H i foreground, correct for high optical depth H i, and solve for δ_GDR in the star-forming regions only—rendering us insensitive to any pervasive CO-free H₂ component. All of these choices lower α_CO, bringing it closer to the other galaxies, but the SMC still exhibits notably high α_CO. The more straightforward approaches employed in Israel (1997a), Leroy et al. (2007), and Leroy et al. (2009) yield even higher α_CO.

Figure 6 shows α_CO (left) and δ_GDR (right) as a function of metallicity. Table 2 gives the adopted metallicity for each target, which unfortunately represent a major source of uncertainty. Literature values span several tenths of a dex for most of our systems and internal variations in 12 + log (O/H) often exist even where strong gradients are absent. We show a 0.2 dex uncertainty in 12 + log (O/H) in our plots to reflect this uncertainty.

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Figure 6 shows that δ_GDR is a clear function of metallicity, increasing with decreasing metallicity. This clear, one-to-one dependence of δ_GDR on metallicity provides a consistency check on our solutions for α_CO. The data are roughly consistent with a linear relation (i.e., a fixed fraction of metals in dust), though a fit yields a slightly sublinear slope, log₁₀δ_GDR = (9.4 ± 1.1) − (0.85 ± 0.13)[12 + log (O/H)]. Our focus on star-forming regions likely biases us toward high δ_GDR; studies of all material find a steeper relation (Lisenfeld & Ferrara 1998; Muñoz-Mateos et al. 2009).

In the left panel of Figure 6, we plot the best-fit α_CO as a function metallicity. A gray area shows the commonly accepted range of values for the Milky Way (Bloemen et al. 1986; Solomon et al. 1987; Strong & Mattox 1996; Dame et al. 2001; Heyer et al. 2009). A dashed line shows the α_CO argued by Draine et al. (2007) to offer the best consistency between dust and gas measurements in SINGS galaxies. The molecular ring in M 31, both parts of M 33, and the LMC span a factor of ∼2–3 in metallicity but show little clear gradient in α_CO. All three appear broadly consistent with the Milky Way. We find higher α_CO in NGC 6822 and the SMC, but the rise relative to the higher metallicity systems much steeper than linear. Figure 6 therefore suggests a jump from "normal" to "high" α_CO at metallicity ∼1/4 the solar value rather than a steady dependence of α_CO on metallicity. At the high-metallicity end, the inner part of M 31 does suggest a low α_CO. It is unclear how much weight to ascribe to this point and whether the low α_CO is primarily a product of metallicity: the solution is somewhat unstable and a number of other conditions differ between the bulge of M 31 and our other targets. However, the point does highlight that lower-than-Galactic α_CO has often been observed, especially in extreme, dust-rich systems (e.g., Downes & Solomon 1998).

Does a rapid increase in α_CO over a narrow range in metallicity make physical sense? The metallicity dependence of α_CO is likely driven by "CO-dark" H₂ at low extinctions. In regions with low metallicity, a significant mass of H₂ is found in the outer parts of clouds where the carbon is mostly C⁺ rather than CO (e.g., Maloney & Black 1988; Israel 1997b; Bolatto et al. 1999). Both Glover & Mac Low (2010) and Wolfire et al. (2010) recently studied this problem by modeling molecular clouds. They find that the dust extinction is the key parameter determining the fraction of CO-dark gas or α_CO. For clouds with mean extinction A_V ∼ 8 mag, such as those in the Milky Way, Wolfire et al. (2010) find a fraction of CO-dark gas f_DG ∼ 30%, in rough agreement with several observations of nearby clouds (Grenier et al. 2005; Abdo et al. 2010). At intermediate metallicities, the CO-dark H₂ is not dominant, so that even a significant fractional change does not impact α_CO very much. For example, in the calculations by Wolfire et al. (2010) going from the metallicity of M 31 to that of the LMC causes a doubling of the CO-dark gas fraction, from f_DG ≈ 20% to f_DG ≈ 40%, but the corresponding effect on α_CO is only a factor of ∼1.3, easily within the uncertainties of our study. As the mean extinction through the cloud decreases due to the effects of metallicity on the gas-to-dust ratio the increase in the fraction of CO-dark H₂ makes it the dominant molecular component, at which point α_CO is very rapidly driven upward.

Our measurements suggest that CO-dark gas becomes an important component in the metallicity range 12 + log (O/H) ∼ 8.2–8.4. With such a small sample, we cannot be sure that this is a general result, but it is in good agreement with the expectation from Wolfire et al. (2010). Above these metallicities, variations in the fraction of CO-dark gas will still exist, but their influence on the value of α_CO will be small. Excitation of the molecular gas (particularly due to temperature variations) will be likely the dominant factor setting α_CO in molecule-rich systems. This is probably the cause of the low α_CO observed in LIRGs and ULIRGs (e.g., Downes & Solomon 1998), which have solar or slightly subsolar metallicities (e.g., Rupke et al. 2008). Within galaxies, the radiation field incident on the molecular gas may also play a role, a fact emphasized by Israel (1997a). Unfortunately, the size scales involved make this extragalactic measurement difficult. While Israel (1997a) found strong quantitative support for the influence of the radiation field on α_CO, Pineda et al. (2009) and Hughes et al. (2010) recently failed to detect this effect in carefully controlled experiments in the LMC.

Many authors have measured α_CO using a variety of techniques. We will not attempt to summarize the literature, but focus on comparisons to two sets of measurements: (1) previous applications of IR-based techniques and (2) high spatial resolution measurements of dynamical masses using CO emission.

We measure high α_CO in the SMC and NGC 6822. This agrees with a larger trend in which IR photometry and spectroscopy suggest high α_CO in the range 12 + log (O/H) ≲ 8.0–8.2. Israel (1997a) saw this in a number of irregulars. Rubio et al. (2004), Leroy et al. (2007), and Leroy et al. (2009) found similar results in the SMC, though as we note in Section 5.1 we actually solve for a somewhat lower α_CO than these studies—a fact we attribute to our focus on deriving δ_GDR from the H i immediate associated with the star-forming region. Gratier et al. (2010) found the same high α_CO for NGC 6822 using several approaches. Using far-infrared spectroscopy, Madden et al. (1997) found indications of high α_CO in IC 10, a Local Group galaxy with metallicity similar to NGC 6822 (we do not include IC 10 in this study because it lies near the Galactic plane and has considerable IR foregrounds). Pak et al. (1998) reached similar conclusions in the SMC. Dust continuum modeling by Bernard et al. (2008) and Roman-Duval et al. (2010) found a mixed picture in the LMC, also consistent with our findings. Note that in contrast with this broad agreement, recent work on diffuse lines of sight in the Milky Way (Liszt et al. 2010) suggests that the ratio of CO brightness to H₂ column density is not a strong function of column density.

Figure 7 compares α_CO as a function of metallicity between this study and the literature. The points in red indicate IR-based measurements. In detail, our measurements (circles) yield lower α_CO than previous IR-based studies. We suspect that this is mainly because we solve for α_CO without assuming δ_GDR or measuring it far away from the region of interest. One likely sense of systematic variations in δ_GDR is that δ_GDR is likely to be higher in the dense gas close to molecular complexes, which tend to reside mainly in the stellar disk, than in a diffuse, extended H i disk (e.g., Stanimirovic et al. 1999; Draine et al. 2007; Muñoz-Mateos et al. 2009). If δ_GDR is taken to be too high, Equation (2) yields a corresponding overestimate of α_CO. In the SMC, our attempt to remove a diffuse H i component along the line of sight also leads to lower α_CO (Section 5.1), though it is less clear that our approach is correct in that case. Regardless of the cause, by simultaneously solving for α_CO and δ_GDR in the regions of interest in a uniform way across a heterogeneous sample we improve on literature studies of individual galaxies.

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A long standing discrepancy exists between IR-based results and high-resolution virial mass measurements based on CO observations. Using virial masses, Wilson (1995), Rosolowsky et al. (2003), and Bolatto et al. (2008) all found weak or absent trends in X_CO as a function of metallicity. The blue points in Figure 7 show virial mass results from CO observations with resolution better than 30 pc. The two approaches agree up to about the metallicity of M 33 or the LMC, and then strongly diverge in the SMC. This divergence is most easily understood if the additional H₂ traced by IR lies in an extended envelope outside the main CO-emitting region (Bolatto et al. 2008). Such an envelope can reconcile the virial mass and dust measurements and naturally explains the scale dependence of α_CO observed by Rubio et al. (1993) in the SMC. These envelopes could perhaps still have an effect on the velocity dispersion of the material inside it (and consequently the measured virial mass) via surface pressure. Structures with virial parameters α ⩽ 1, however, are often observed inside local molecular clouds, suggesting that at least in some instances the velocity dispersion does not appreciably show the impact of the surrounding material. An alternative view is argued by Bot et al. (2007, 2010), who observed discrepancies between dust-based masses and virial masses even at fairly small scales. They suggest that magnetic support becomes very strong at low metallicities, perhaps because of higher ionization fractions inside clouds.