A Search for Correlations between Turbulence and Star Formation in LITTLE THINGS Dwarf Irregular Galaxies

KED and FUV images were the inputs to the 2D cross-correlation. We geometrically transformed the FUV image to match the orientation and field of view (FOV) of the H i map using OHGEO in the Astronomical Image Processing System (AIPS) and then smoothed it to the H i beam using SMOTH in AIPS. We also blanked the pixels outside of the galaxy FUV emission, replacing the blanked pixels with zeros, so that pure noise would not add to the correlation coefficient C_coef. We constructed the KED maps as $0.5\times {N}_{{\rm{H}}{\rm\small{I}}}\times {v}_{\mathrm{disp}}^{2}$, where N_HI is the H i column density in hydrogen atoms per cm² and v_disp is the velocity dispersion in km s⁻¹. The conversion from counts in the KED maps to ergs pc⁻² is given for each galaxy in Table 2. Prior to executing the cross-correlations, we scaled both the FUV and KED images so that the pixel values were the same order of magnitude (roughly 100). These KED values determined from H i column density have not been multiplied by 1.36 to include helium and heavy elements. This factor will be used later when the efficiency of KED generation is calculated.

We decided not to remove the underlying exponential disks for the 2D cross-correlations. Although the SFR drops off with radius, the FUV image consists of knots of young stars, and there can be large FUV knots in the outer disks. For the H i moment 0 and 2 maps, the H i surface density and velocity dispersion do, on average, change with radius, too, but not in a regular and homogeneous fashion. Thus, in the 2D C_coef maps exponential structure could remain.

Here, a C_coef of 1 is perfectly correlated such that every bump and wiggle in one map is exactly reproduced in the other. A value of −1 is perfectly anticorrelated. The amplitude of the peak is a measure of the coincidence of features in each image. If the KED image correlates well with the local FUV flux, then the peak will be high and the breadth will be the average size of their rms summed feature sizes.

We used the commands correl_images and corrmat_analyze in IDL with a Python wrapper. We used this command to shift one image relative to the other over and over again to yield a map of C_coef. The peak pixel value in the C_coef map is the adopted C_coef. For example, for NGC 2366, we did a 150 × 150 array of offsets. That is, we calculated the C_coef for x, y offset of −150, −150 to x, y offset of +150, +150. This produces a matrix of 301 × 301 pixels. The peak, in this case, is at pixel 145, 147 and has a value of 0.3 compared to the center pixel 151, 151 value of 0.26. Thus, the maximum correlation is achieved when the FUV image is shifted relative to the KED image by the offset corresponding to x, y of 145, 147. We checked the C_coef of a piece of one of the galaxies "by hand" with a Fortran program we wrote, and we obtained the same peak C_coef. The peak C_coef and x, y shifts to that pixel are given in Table 2. The cross-correlation matrices are shown in Figure 1.

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The shift in x, y is also given in Table 2 relative to the disk scale length R_D, for better comparison to the size of the galaxy. The shifts vary between 0.02R_D (IC 1613) and 4.75R_D (Haro 36). Fifty percent of the galaxies (18) have shifts less than 0.5R_D, 33% (12) have shifts between 0.5R_D and 1R_D, and 17% (6) have shifts greater than 1R_D.

I. Bagetakos (private communication) examined the cross-correlation method on NGC 2403 as a function of image scale, focusing on scales of 0.23 to 3 kpc, and found correlations on various scales for different images such as star formation tracers, dust, and H i. Thus, we divided our images into square subregions of 16 × 16, 32 × 32, 64 × 64, and 128 × 128 pixels and computed the C_coef in each box. The coefficient images constructed from this just look like noise and show no particular connection to the FUV image. So, we do not consider them further.

We also applied alternate methods on one galaxy, NGC 2366 with a max C_coef of 0.3, to examine the robustness of our approach. This galaxy was chosen for initial and special tests because it has a giant H ii region and the H i velocity dispersion is high around this region, making it an interesting candidate for looking for a star formation–turbulence correlation. One problem with cross-correlations, in particular, can be caused by moderate signal-to-noise ratio (S/N) pixels dampening the value of C_coef. One simple diagnostic is a plot of the pixel values of the KED image against the pixel values of the FUV image, given that the FUV image has been geometrically transformed and smoothed to the same pixel grid and beam size as the KED image. We normalized the pixel values in each image to range from 0 to 1, and this plot is shown in Figure 2. If there were a notable correlation, we would expect a cluster of points in the top-right corner. If the images were anticorrelated, we would expect clusters of points around the top left and bottom right. We do not see either of these extremes. While there are some points in the top-left corner, it is not a distinct cluster; rather, it appears to be consistent with the typical tail end of a simple distribution of points from 0 to 1.

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We also tried variations of the weighted normalized cross-correlations (WNCCs) and a wavelet analysis to NGC 2366. The zero-mean normalized cross-correlation coefficient (ZNCC) is basically the standard Pearson correlation measure ρ for a 2D image. Applied to NGC 2366, ZNCC is 0.26. Like ρ or r coefficients, +1 is perfect correlation, −1 is perfect anticorrelation, and so 0.26, which is what we found for the max C_coef, implies a not very significant degree of correlation. One way to deal with pixels with low S/N is to use a WNCC. For this test, we weighted the pixels by the ratio of their signal to the standard deviation of values in the map, which is effective at down-weighting background noise pixels. Using this method, we obtain a WNCC value of −0.023—effectively zero, implying no significant correlation between the two images.

For our final test on NGC 2366, we used a wavelet analysis to see if the degree of correlation depends on scale/resolution. In this process, each image is convolved with progressively larger 2D kernels or wavelets, in this case, a Ricker or "Mexican hat" wavelet, and the cross-correlation is calculated at each of these scales or "lags." The result for NGC 2366 is shown in Figure 3. Strong correlations at a particular spatial scale would be evidenced by wavelet cross-correlation r_w values of ≳0.6 at that scale. Eventually, as the images are convolved to large-enough scales, they become less resolved and therefore naturally correlate. Figure 3 indicates that there is not much difference at any of the resolved scales. Using a slightly modified wavelet from the literature (e.g., Ossenkopf et al. 2008), there may be evidence of a slightly more prominent correlation between the images at 30 pixel scales (45''), but r_w is still not significant.

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Thus, we conclude that no matter how we look at the FUV and KED images of NGC 2366, the two images are mildly correlated at best, and this does not change much with scale.

The width of the peak signal in a cross-correlation matrix is expected to represent the scale of the correlation. However, in our matrices, the width is not well defined. The issue is demonstrated in Figure 4 where we show a radial plot and row and column cuts through the peak of the WLM C_coef matrix. The peak is, of course, obvious, but the radial plot is messy and the single row and column cuts show a complex background. The main feature in the cross-correlation maps is the exponential disk because both the KED and the SFR density peak in the center with the exponential disk. The width of the C_coef in Figure 4 is influenced more by the width of the disk than the scale of the 2D correlation. What to take as the baseline for a fit to the peak is also not clear. Therefore, we do not consider the widths of the peaks further here.

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We also calculated the C_coef in annuli from the center of the galaxy outward. The image was blanked outside of the target annulus, which was chosen to match those used by Hunter et al. (2012) to produce the H i radial profiles. We normalized the pixel values in the annulus with respect to the average in the annulus, so in effect large-scale variations, such as the exponential falloff with radius, are taken out. Then, we measured the C_coef for the annulus. Figure 5 shows the C_coef of the annuli as a function of the annulus distance from the center of the galaxy. The annuli used the galaxy center, ellipticity, and major axis position angle determined from V-band images and given by Hunter et al. (2012).

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We see a wide variety of profiles. The central points in NGC 4163 and in VIIZw403 reach a C_coef of nearly 0.95, and a few other galaxies have peaks as high as 0.9. By contrast, the peak in DDO 210 occurs in the outermost annulus and only reaches a value of 0.14. In many galaxies, the C_coef drops in value with radius, but in many others, it is relatively flat. In a few galaxies, the C_coef drops precipitously from a relatively high value for the innermost annulus to near zero beyond that radius (DDO 167, F564-V3, Haro 36).

Generally, the 2D C_coef indicates low levels of correlation between the FUV and KED images. In Figure 6, we plot the peak C_coef against the integrated SFR for each galaxy to see if a higher level of correlation is related to the overall SFR. There is no relationship between the two values. In annuli, C_coef can be as high as 0.9 in the center, indicating a correlation, but the values tend to be low overall, and the radial profiles exhibit a wide range of shapes. From the images, visually, most of the FUV is patchy and tends to be concentrated toward the central regions of the galaxies while the H i often extends quite far outside the optical/UV galaxy. So, the bird's eye view of a dIrr might expect a higher correlation in the central regions where there is ample H i and FUV, with little correlation as you go farther out where there are typically fewer FUV knots.

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By comparison, in the spiral NGC 2403, I. Bagetakos (private communication) found that the FUV and H i surface mass density are uncorrelated with a C_coef < 0.20. They did, however, find correlations between dust and star formation (C_coef > 0.55) and between PAHs and H i (C_coef ∼ 0.55). Bagetakos et al. chose NGC 2403 as their pilot galaxy because it is in the THINGS sample (Walter et al. 2008) with H i data, as well as images at 8 μm, 24 μm, Hα, and FUV, and is nearby with a H i beam of 136 pc × 119 pc. As an Scd spiral, it is significantly larger and more massive than the dIrr galaxies in this study.

Because star formation is usually lumpy, we ask whether the lack of correlation between FUV and KED images is because KED is smooth compared to FUV or because the lumps in the two images do not correlate. Figure 7 shows the KED maps and FUV images at full resolution. A contour of the FUV image is superposed on the KED map to facilitate comparison. We see that the FUV and KED maps are both generally lumpy, although the lumps are not necessarily located in the same place.

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View extended figure (13 panels)

To examine the degree of lumpiness, we looked at the fraction of pixels with raw values above a given percentage of the maximum pixel value in the image. Specifically, we counted the fraction of total pixels that have counts within 10%, 20%, 30%, 40%, and 50% of the maximum count value in each of the FUV and KED images. These data are shown in Figure 8 as a percentage of total pixels as a function of selected cutoff deviation from the maximum pixel value in the image. For example, in CVnIdwA, the percentage of pixels with values within 10% of the maximum value is 0.69% in the FUV image and 1.03% in the KED image, whereas the percentage of pixels with values within 50% of the maximum value is 5.65% in the FUV image and 15.98% in the KED image.

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To understand what these plots mean, we can compare the appearance of the galaxies in Figure 7 with the plots in Figure 8. We see in the images that galaxies like LGS 3, DDO 87, DDO 133, and SagDIG have a few small FUV knots but more or bigger KED knots. The KED knots fill more of the area and so a higher fraction of the pixels are close to the peak intensity. These galaxies have flat FUV profiles in Figure 8 because very few pixels are close to the peak intensity, i.e., the FUV is spotty, but they have KED profiles that increase with the percentage of the maximum pixel value because the KED is more uniform. DDO 43 is unique in this sample because it is the only one with an approximately flat KED profile and an FUV profile that increases with the percentage of the maximum pixel value. The reason is clear from Figure 7, which shows that the FUV image of DDO 43 is filled with bright spots, making most of the image close to the peak pixel value, while the KED image has weaker peaks that are more spread out. DDO 167, on the other hand, has FUV and KED knots that are comparable in size, and FUV and KED profiles that increase together with the percentage of maximum pixel value, as do DDO 47, DDO 101, F564-v3, and NGC 4163. Most of the galaxies have broader KED distributions than FUV emission, so their KED pixel percentages increase faster than their FUV pixel percentages as the top percentage of the maximum pixel value increases.

The general increasing trend of the curves in Figure 8 is mostly the result of the exponential radial profile of the disk, with the peaks in KED and FUV standing for a nearly fixed fraction above the mean profile. Figure 9 shows the models for these curves assuming an exponential disk intensity profile I(r) = e^−r , so the radius as a function of intensity is $r(I)=-\mathrm{ln}(I)$. The radius at 10% of the peak is then $r(10 \% )=-\mathrm{ln}(1-0.1)$, and the number of pixels brighter than that is the area of the circle at this radius, or $\pi r{\left(10 \% \right)}^{2}$. In general, for an intensity that is a fraction x down from the peak intensity, the fraction of pixels in the total disk is

$$\begin{eqnarray}&&f(x)=\pi {\left(-\mathrm{ln}[1-x]\right)}^{2}/\left(\pi {r}_{\max }^{2}\right),\end{eqnarray} \tag{ 1 }$$

where ${r}_{\max }$ is the size of the disk measured in scale lengths. Figure 9 shows f(x) versus x in three cases. The top curve is for an exponential profile with a scale length 1.5 times larger than the middle curve and an overall galaxy size that is the same, ${r}_{\max }=2$ scale lengths. The lower curve also has a scale length 1.5 times larger than the middle curve, but the overall galaxy size for the lower curve is 1.5 times larger (${r}_{\max }=3$). Larger scale lengths for a given galaxy size make the percentage curves rise faster because more of the disk is close to the peak intensity at the center.

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The similarity of the model curves in Figure 9 to the observations in Figure 8 implies that the qualitative effect being captured by the fractional distribution is the result of the exponential disk. However, the percentage of pixels observed is much smaller than the model percentage, i.e., several percent or less for the observations compared to ∼10% at the 50% top percentage of the maximum pixel value. This difference implies that the peaks in the KED and FUV distributions stand above the exponential disk, so their areas are a small fraction, ∼10%, of the disk area, but the peak intensities have about the same radial dependence as the average disk, which means they are a fixed factor times the average disk brightness.

Another way of looking at the data is to compare individual pixels in pairs of images. We have done that, examining KED, the velocity dispersion of the gas v_disp, and H i surface density Σ_HI versus SFR surface density as determined from the FUV images, Σ_SFR. Recall that the FUV images were geometrically transformed and smoothed to match the pixel size and resolution of the H i images. For all galaxies but DDO 216 and Sag DIG, the pixel size is 1.5'', and for these two, it is 3.5''. To compensate for radial trends, we determined the azimuthally averaged Σ_SFR, KED, H i, and v_disp in annuli from the center of the galaxy and subtracted that from the observations. We used optically determined disk parameters of the center, minor-to-major axis ratio b/a, and position angle of the major axis from Hunter et al. (2012). The widths of the annuli, constant in a given galaxy, were chosen to be the same as those used to measure the H i surface density profiles of Hunter et al. (2012). The azimuthally averaged radial profiles of Σ_SFR, KED, v_disp, and Σ_HI are shown for each galaxy in Figure 10. The pixel–pixel plots of the excess KED, v_disp, and Σ_HI versus excess Σ_SFR are shown in Figures 11–13. All of these quantities except v_disp were corrected to a face-on orientation by multiplying the fluxes by the cosine of the inclination. The KED units are erg pc⁻², v_disp is in km s⁻¹, Σ_HI is in M_☉ pc⁻², and Σ_SFR is in units of M_⊙ yr⁻¹ pc⁻². KED values in Figures 10–13 have not been corrected for helium and heavy elements. Note that only the regions of relatively high Σ_SFR are plotted, i.e., with a positive excess above the annular average, and we plot the logarithm of this excess. For the quantities on the ordinate, we consider both positive and negative excess values over the average, so they are not plotted in the log. Some regions of locally high Σ_SFR have locally low KED, v_disp, or Σ_HI.

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View extended figure (7 panels)

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View extended figure (7 panels)

In the radial averages shown in Figure 10, we see that KED, Σ_SFR, and Σ_HI generally decrease with radius. Tamburro et al. (2009) found this also for spiral galaxies. They also found that v_disp decreases with radius in their sample, but in our sample of dIrr, the drop of v_disp with radius is very minor, if any. They also find a clear correlation of KED with Σ_SFR in pixel–pixel plots, whereas our Figure 11 does not show such a nice correlation.

Figures 11–13 typically show concentrations of points at a low excess Σ_SFR and a continuation of these points toward higher excess Σ_SFR. The low excess Σ_SFR is in the outer disks and the high excess Σ_SFR is in the inner disks. Some galaxies have two concentrations of points in these figures.

To quantify the pixel distributions, we determined the excess Σ_SFR and other quantities at the plotted concentrations. For each galaxy, we made a histogram of the log of the excess Σ_SFR (the abscissa value) and found the peak at the low-density concentration. The excess log Σ_SFR in that peak was determined from the average value in the three bins of the histogram centered there. The bin width was 0.2 in the log of the excess Σ_SFR. Then, for these three bins around the histogram peak for log excess Σ_SFR, we determined the mean value of the quantity plotted on the ordinate in the figures, i.e., the excess KED, v_disp, and Σ_HI. For the higher-excess Σ_SFR, we took the mean value of the excess Σ_SFR and the other quantities for all regions where the log of the excess Σ_SFR was larger than the high-star-formation edge of the concentration of points, typically at −9.8 but ranging from −9.5 to −10.2 depending on the plotted galaxy. When there was only one prominent concentration of points in the figure, we determined the values there.

Figure 14 shows the mean excess KED corrected for helium and heavy elements, v_disp, and Σ_HI versus the mean of the log of the (positive) excess Σ_SFR for all galaxies, with dots corresponding to the low-Σ_SFR concentrations in the outer disks and crosses corresponding to the high Σ_SFR in the inner disks. The curves in the KED plot show fitted relationships between the KED generated by supernovae and the Σ_SFR for the indicated efficiencies of converting supernova energy into turbulence and for galaxy scale heights of 850 pc and 540 pc. These theoretical KEDs come from Equation (3.7) in Bacchini et al. (2020), which is

$$\begin{eqnarray}&&{{KED}}_{\mathrm{SN}}=\eta {{\rm{\Sigma }}}_{\mathrm{SFR}}{f}_{\mathrm{cc}}{E}_{\mathrm{SN}}(2H/{v}_{\mathrm{turb}}),\end{eqnarray} \tag{ 2 }$$

where η is the efficiency of energy conversion from supernova to turbulence, ${f}_{\mathrm{cc}}=1.3\times {10}^{-2}\,{M}_{\odot }^{-1}$ is the number of core-collapse supernovae per solar mass of stars, ${E}_{\mathrm{SN}}={10}^{51}$ erg is the supernova energy, H is the disk thickness, and v_turb is the turbulent gas velocity dispersion (the ratio of these latter two quantities gives the turbulent dissipation time). Bacchini et al. (2020) compare the radial profiles of turbulent energies in 10 nearby galaxies with the SFRs and derive an average efficiency of ${1.5}_{0.8}^{1.8}$% if all of the turbulence comes from star formation. Because the required efficiency is relatively low, they concluded that supernovae related to star formation can drive most of the interstellar turbulence.

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For the dwarf galaxies studied here, we evaluate Equation (2) using scale heights and velocity dispersions from the average values for 20 dIrrs in Elmegreen & Hunter (2015), in Table 2 of that paper. For the concentrations of pixel values corresponding to the outer regions of the galaxies, we take the average scale height and v_disp at 2 scale lengths, which are H = 850 pc and v_disp = 9.7 km s⁻¹. For the inner regions, we take the values at 1 scale length, which are 540 pc and 10.7 km s⁻¹. We also include helium and heavy elements in the KED by multiplying the H i mass surface density by 1.36. Then, with f_cc and ${E}_{\mathrm{SN}}$ given above, Equation (2) is fitted for the efficiency in the two cases, using for the moment v_disp instead of v_turb. The results are drawn as curves in the left panel of Figure 14. The average local efficiencies for the conversion of supernova energy to KED are η = 0.0080 ± 0.045 and 0.0050 ± 0.0075 for the outer- and inner-disk regions, with these assumptions.

The total dispersions used to evaluate η include thermal and turbulent motions, which were distinguished in several limiting cases by Bacchini et al. (2020) to get the desired v_turb. If we assume Mach ∼ 1 turbulence in the general H i ISM, then v_turb = v_disp/2^0.5 and the derived values of η decrease by a factor of 0.7, preserving the ratio η/v used to match the KED. Alternatively, we could use thermal dispersions of 4.9 km s⁻¹ and 6.1 km s⁻¹ modeled for NGC 4736 and NGC 2403, respectively, by Bacchini et al. (2020) to estimate that v_turb/v_disp ∼ 0.8, given that v_disp ∼ 10 km s⁻¹ here. Then, our derived η should decrease by ∼0.8. The Bacchini et al. (2020) galaxies were more massive than our dIrr galaxies, but the thermal contributions to v_disp are not likely to be much different. These corrections change the average value of η = 0.0065 for the two regions in Figure 14 to η ∼ 0.0048, using a mean correction factor of 0.75.

This η value is the average for the peak regions of star formation. It measures how efficiently star formation energy gets into H i turbulent motions locally in units of the supernova energy per unit mass of young stars. When normalized this way, other types of energy related to star formation such as expanding H ii regions and stellar winds are included in η, too. What is not included as a source of turbulent motion is energy unrelated to star formation, such as gravitational energy from gas collapse on the scale of the ISM Jeans length, or collapse energy from transient spiral arms driven by combined gas and stellar masses, or shock energy from the relative motions of gas and stellar spiral density waves. If η ∼ 0.0048 measured locally gives the actual efficiency for star formation to pump turbulence in the H i gas, then the global turbulent energy pumped by all of the star formation in a galaxy should equal our local η multiplied by the global SFR (along with the other factors in Equation (2)). Because Bacchini et al. (2020) found that the global turbulent energy is ${1.5}_{0.8}^{1.8}$% of the energy derived from the SFR, there would seem to be more energy required than what star formation alone can provide. The excess energy needed is a factor of ∼0.015/0.0048 − 1 ∼ 2 times the star formation energy.

This factor has many uncertainties, both from the range in global values derived by Bacchini et al. (2020) and from the galaxy-to-galaxy or inner-disk to outer-disk variations derived here. For example, our η ∼ 0.0048 is closer to that of the dwarf galaxy DDO 154 in Bacchini et al. (2020), which had η = 0.009 assuming a pure warm phase H i. Also, our average η for the inner-disk regions in Figure 14 was higher than the average for the inner and outer disks combined (which gave the value 0.0048) by a factor of 1.2. But even within this range, the global energy from turbulence seems to be larger than what can be pumped from star formation alone, if we use local SFRs as the basic means of calibrating η.

Figure 14 for the KED excess has two high points for the inner disk which were not included in the efficiency fit. These are the galaxies Haro 29 with excess KED = 16 × 10⁴⁵ erg pc⁻² and NGC 1569 with excess KED = 141 × 10⁴⁵ erg pc⁻². Correspondingly, Figure 11 shows a scatter of individual pixel points to very high values of KED for these galaxies.

The middle panel of Figure 14 shows the average v_disp excess at each concentration of excess Σ_SFR in Figure 12. The excess velocity dispersion is rarely larger than 1 km s⁻¹ and averages 0.45 km s⁻¹ in the outer disk, −0.34 km s⁻¹ in the inner disk, and 0.37 km s⁻¹ overall. Some local star formation regions have lower H i velocity dispersions than the average at that galactocentric radius, giving negative excesses in Figure 14. These typically small excesses in the local H i velocity dispersion are consistent with the small feedback efficiencies found above. There is relatively little generation of turbulence at the positions of star-forming regions.

The right-hand panel of Figure 14 shows the average Σ_HI excess at each concentration of excess Σ_SFR in Figure 13. There is a clear trend toward excess H i at local star formation regions, although in a few cases the H i is less than the azimuthal average. This general excess corresponds to a ratio of Σ_HI to Σ_SFR that equals 6.5 Gyr in the outer disk, 1.2 Gyr in the inner disk and 1.6 Gyr overall, where this latter fit is shown by the curve in the figure. For this fit, the high point that is plotted in Figure 14 is excluded, that is, for NGC 1569, where the ratio is 31 Gyr. This average ratio of ∼1.6 Gyr is comparable to the consumption time for molecules, which is about 2 Gyr in Bigiel et al. (2008) and Leroy et al. (2008). If only molecular clouds form stars, then this similarity implies that the molecular fraction is about 50% in the inner disk, as suggested using other properties of H i and star formation in recent papers (Hunter et al. 2019, 2021; Madden et al. 2020).