The EDGE-CALIFA Survey: The Resolved Star Formation Efficiency and Local Physical Conditions

Figure 4 shows the relation between SFE_gas and galactocentric radius; the four different panels show the grouping of the 81 galaxies. Following modern studies, we use R_e to normalize galactocentric distances, except when we need to compare to published data that use r₂₅. Note that for the EDGE galaxies in this sample, r₂₅ ≈ 2.1R_e. In this figure for clarity we split the Sbc, Sc, and Scd galaxies into two groups by choosing the median of stellar masses of the EDGE-CALIFA sample log₁₀[M_⋆] = 10.7 (Bolatto et al. 2017). In general, there is a decreasing trend for SFE_gas with radius. It is important to note that SFE_gas is a fairly smooth function of radius for a given galaxy. In fact, variations between galaxies are frequently larger than variations between most annuli in a galaxy, indicating that the radial decrease in SFE_gas within a galaxy is often smooth and that galaxy-to-galaxy variations are significant.

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Figure 5 shows the radius at which our measured molecular surface density, averaged over an annulus, is the same as our assumed constant surface density in the atomic disk, Σ_mol = Σ_atom = 6 M_⊙ pc⁻². The typical radius at which this happens is r/R_e ∼ [1.1 ± 0.5], or r/r₂₅ ∼ [0.47 ± 0.28] (see inset panel), which agrees with the value of r/r₂₅ ∼ 0.43 ± 0.18 found by Leroy et al. (2008). Note that in Figure 4 the SFE_gas is generally smooth across that radius, suggesting that our assumption of a constant Σ_atom does not play a major role in determining the shape of the total gas SFE_gas.

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Figure 6 shows the average SFE_gas as a function of the normalized galactocentric radius for each of the four different groups of morphological classification used in Figure 4, with ±1σ variation indicated by the color bands. We note a systematic increase in the average SFE_gas from early-type (red shaded area) to late-type galaxies (blue shaded area). The SFE_gas tend to be lower for the early spirals (i.e., S0 and earlier; 10 galaxies), which have a steeper profile when compared with the rest of the morphological groups, and therefore show a significant anticorrelation between SFE_gas and r_gal (Pearson correlation coefficient of r = −0.6). This steepening may reflect the degree of central concentration seen in earlier-type galaxies. Sd−Ir galaxies show an SFE_gas flattening at r_gal < 0.45 r₂₅; however, their small amount (only two galaxies in our sample) does not allow us to conclude that this flattening is statistically significant. When looking at the average SFE_gas value, over r_gal for all the radial profiles (black circles), we find that the SFE_gas decreases exponentially even in regions where the gas is mostly molecular. In EDGE we see a continuous exponential profile for the SFE_gas averaged over all galaxies (black line in Figure 6). Although still within the error bars, this is in contrast to HERACLES, which sees a leveling of the SFE_gas in the inner regions. The greater range of SFE_gas values in our sample may be a reflection of the larger range of galaxy spiral types spanned by EDGE compared to HERACLES, which consisted mostly of late types. In fact, the Sbc, Sc, and Scd galaxies in EDGE-CALIFA (green band) are very consistent with the measurements of HERACLES. Where the gas is dominated by the atomic component, r ≳ 0.4r₂₅, the SFE_gas decreases rapidly to the galaxy edge. Because we assume a constant Σ_atom, this is fundamentally a reflection of the rapid decrease of SFR in the atomic disks.

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We can describe the behavior of the SFE_gas for our sample using an ordinary least-squares (OLS) linear bisector method to fit a simple exponential decay:

$$\begin{eqnarray}&&{\mathrm{SFE}}_{\mathrm{gas}}=[0.83\pm 0.07\,({\mathrm{Gyr}}^{-1})]\exp \left(\displaystyle \frac{-{r}_{\mathrm{gal}}}{[0.31\pm 0.02]\,{r}_{25}}\right).\end{eqnarray} \tag{ 15 }$$

We note that we do not see clear breaks in this trend; instead, we find a continuous smooth exponential decline of SFE_gas as a function of r_gal. This is consistent with the rapid decline of star formation activity in the outer parts of galaxies (e.g., Leroy et al. 2008; Kennicutt 1989; Martin & Kennicutt 2001) and also is in agreement with previous results for low-redshift star-forming galaxies (e.g., Sánchez 2020; Sanchez et al. 2021). In particular, our results agree with the inside-out monotonic decrease of the SFE_gas shown by Sánchez (2020). Sánchez (2020) also finds that galaxies are segregated by morphology; for a given stellar mass, Sánchez shows that late-type galaxies present larger SFE_gas than earlier ones at any r_gal, which is consistent with the trend we observe in Figure 6. In the outer parts, our steeper profiles may be influenced by our assumption of constant H i surface density. However, this does not explain our steeper profiles we also observe in the inner galaxy. The top and bottom dashed lines in Figure 6 show how SFE_gas changes if instead of 6 M_⊙ pc⁻² we use Σ_atom = 3 and 12 M_⊙ pc⁻², which are the two extremes of Σ_atom values found in HERACLES (Leroy et al. 2008). A better match between EDGE and HERACLES would require using Σ_atom = 3 M_⊙ pc⁻², which appears extremely low. Note that these two studies use different SFR tracers: our extinction-corrected Hα may behave differently from the GALEX FUV that dominates the SFR estimate in the outer disks of HERACLES (e.g., Lee et al.2009).

How sensitive is the SFE_gas determination to the CO-to-H₂ conversion factor? To test this, we adopt a variable CO-to-H₂ conversion factor, α_CO, using Equation (4). This includes changes in the central regions caused by high stellar surface densities and changes due to metallicity. When comparing the effects of a constant and a variable prescription of α_CO (shaded area in Figure 6), we observe that the central regions present larger SFE_gas variations than the outer disks within the range of galactocentric distances we study, as the latter do not exhibit $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ significantly below 8.4 according to the O3N2 indicator, as shown in the top panel of Figure 7. Therefore, the variations of the CO-to-H₂ conversion factor are generally small and consistent with the assumption of a constant α_CO.

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So far, we have analyzed the SFE of the total gas, but it is also interesting to test whether the star formation efficiency responds to the phase of the ISM. The bottom panel of Figure 7 shows the star formation efficiency of the molecular gas, SFE_mol = Σ_SFR/Σ_mol (in yr⁻¹), as a function of the ratio between the molecular and the atomic surface densities, R_mol = Σ_mol/Σ_atom. Since we assume Σ_atom = 6 M_⊙ pc⁻², R_mol is a prescription for the Σ_mol normalized by a factor of 6. Although there is large scatter, the figure shows that the SFE_mol, averaged by R_mol bins (black filled circles), remains almost constant over the R_mol range, with an average log[SFE_mol] ∼−9.15 (blue dashed line in the bottom panel of Figure 7). The inset panel shows that the SFE_mol is also fairly constant over the range of galactocentric radii. These results are in agreement with Muraoka et al. (2019), who find a similar flattening in SFE_mol for annuli at r ≲ 0.6r₂₅ when analyzing 80 nearby spiral galaxies selected from the CO Multi-line Imaging of Nearby Galaxies survey (COMING; Sorai et al. 2019). Using CO, FUV+24 μm, and H_α +24 μm data for 33 nearby spiral galaxies selected from the IRAM HERACLES survey (Leroy et al. 2009), Schruba et al. (2011) found that H₂-dominated regions are well parameterized by a fixed SFE_mol equivalent to a molecular gas depletion time of ${\tau }_{\mathrm{dep},\mathrm{mol}}={\mathrm{SFE}}_{\mathrm{mol}}^{-1}\sim 1.4\,\mathrm{Gyr}$, which is consistent with our average τ_dep,mol ∼ 1.45 ± 0.23 Gyr. As for previous studies, these results support the idea that the vast majority of the star formation activity takes place in the molecular phase of the ISM instead of the atomic gas (e.g., Martin & Kennicutt 2001; Bigiel et al. 2008; Schruba et al. 2011).

We explore possible trends between SFE_gas, galactocentric radius, and nuclear activity. We adopt the nuclear activity classification performed by García-Lorenzo et al. (2015), who classify CALIFA galaxies (with signal-to-noise ratio larger than 3) into star-forming (SF), active galactic nuclei (AGN), and LINER-type galaxies, and we apply it, when available, for the 81 galaxies analyzed in this work (see column Nuclear in Table 1). We do not identify significant trends as a function of galactocentric radius for any of these three categories.

Since in the previous section we show a clear dependence of SFE_gas on galactocentric distance, it is expected that SFE_gas will also depend on the stellar surface density, Σ_⋆. Indeed, the top panel of Figure 8 shows an approximately power-law relationship between SFE_gas and Σ_⋆. We quantify this relation by using an OLS linear bisector method in logarithmic space to estimate the best linear fit to our data (excluding upper limits), obtaining

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}[{\mathrm{SFE}}_{\mathrm{gas}}({\mathrm{yr}}^{-1})] & = & [0.32\pm 0.27]\times \mathrm{log}[{{\rm{\Sigma }}}_{\star }\,({M}_{\odot }\,{{pc}}^{-2})]\\ & & -\ [10.13\pm 1.75].\end{array}\end{eqnarray} \tag{ 16 }$$

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When comparing the EDGE average SFE_gas over Σ_⋆ bins (black circles) with similar HERACLES bins (green squares), we find consistently slightly larger efficiencies at $\mathrm{log}[{{\rm{\Sigma }}}_{\star }({M}_{\odot }\,{{pc}}^{-2})]\lesssim 1.4$, although the HERACLES points are still within the error bars of our data. Since these points are in the outer regions of the EDGE galaxies, this result may be sensitive to the adoption of Σ_atom = 6 M_⊙ pc⁻². In the inner regions with $\mathrm{log}[{{\rm{\Sigma }}}_{\star }({M}_{\odot }\,{{pc}}^{-2})]\geqslant 2.6$, our average efficiencies are also higher, although we do not expect these regions to be sensitive to the choice of Σ_atom. Between these two extremes, however, there is good general agreement between the EDGE and HERACLES results.

The middle and bottom panels of Figure 8 show the relation between the H₂-to-H_I ratio (R_mol = Σ_mol/Σ_atom = Σ_mol/6 M_⊙ pc⁻²), Σ_*, and the gas surface density, Σ_gas = Σ_mol + Σ_atom, respectively. In the middle panel, we observe a tight correlation between R_mol and Σ_⋆. The relation is well described by a power law, and there is overall reasonable consistency between EDGE and HERACLES. Our measurements are also consistent with the resolved Molecular Gas Main Sequence relation (rMGMS, Σ_gas−Σ_⋆; Lin et al. 2019) found for EDGE-CALIFA galaxies by Barrera-Ballesteros et al. (2021). The bottom panel shows very good agreement between the EDGE and HERACLES results in the range $0.9\lesssim \mathrm{log}[{{\rm{\Sigma }}}_{\mathrm{gas}}]\lesssim 1.5;$ outside this range there are small differences, although there is still consistency within the error bars. Therefore, the discrepancies seen in the top panel are not the result of differences in efficiency at a given H₂-to-H_I ratio or gas surface density, but likely reflect small systematic differences in the relation between gas and stellar surface density in HERACLES and EDGE. Since we have both a broader morphological and a more numerous sample selection than HERACLES (particularly in the H_I-dominated regions), our results reflect on a more general power-law dependence of the SFE_gas on Σ_⋆. Observations have shown that the fraction of gas in the molecular phase in which star formation takes place depends on the pressure in the medium (Elmegreen 1993; Blitz & Rosolowsky 2006). These results suggest that high stellar densities in the inner regions of EDGE-CALIFA galaxies are helping self-gravity to compress the gas, resulting in H₂-dominated regions. Once the gas is predominantly molecular, our data suggest that a dependence of the SFE_gas on Σ_⋆ persists even in high-Σ_⋆, predominantly molecular regions.

Other studies have given different insights of the relation between star formation activity and the stellar surface density. For instance, analyzing 34 galaxies selected from the ALMA-MaNGA Quenching and STar formation (ALMaQUEST; Lin et al. 2019), Ellison et al. (2020) find that Σ_SFR is mainly regulated by Σ_mol, with a secondary dependence on Σ_⋆. Conversely, analyzing 39 galaxies selected from EDGE-CALIFA, Dey et al. (2019) find a strong correlation between Σ_SFR and Σ_mol; they show that the Σ_SFR − Σ_⋆ relation is statistically more significant. Sánchez et al. (2021), however, used the edge_pydb database to show that secondary correlations can be driven purely by errors in correlated parameters, and it is necessary to be particularly careful when studying these effects. Errors in Σ_gas, for example, will tend to flatten the relation between SFE_gas and Σ_gas because of the intrinsic correlations between the axes, and they will have the same effect on the relation between SFE_gas and Σ_* because of the positive correlation between Σ_* and Σ_gas.

We explore the dependency of SFE_gas on the dynamical equilibrium pressure, P_DE. While the midplane gas pressure, P_h (Elmegreen 1989), is a well-studied pressure prescription in a range of previous works (e.g., Elmegreen 1993; Leroy et al. 2008), P_DE has been extensively discussed recently (e.g., Kim et al. 2013; Herrera-Camus et al. 2017; Sun et al. 2020; Barrera-Ballesteros et al. 2021). In both pressure prescriptions, it is assumed that the gas disk scale height is much smaller than the stellar disk scale height and the gravitational influence from dark matter is neglected. P_h and P_DE have an almost equivalent formulation, although they slightly differ in the term related to the gravitational influence from the stellar component (second term in Equations (6) and (8); see Section 3.3). We quantify this difference by computing the mean P_DE-to-P_h ratio averaged in annuli for our sample, obtaining P_DE/P_h ≈ 1.51 ±0.19. We use this value to convert the P_h from HERACLES into P_DE, since we perform our qualitative analysis using the dynamical equilibrium pressure.

The top panel of Figure 9 shows the SFE_gas as a function of P_DE (in units of K cm⁻³). The slope of the SFE_gas versus P_DE relation (averaged over P_DE bins; black circles) has a break at $\mathrm{log}[{P}_{\mathrm{DE}}]\sim 3.7$. Below $\mathrm{log}[{P}_{\mathrm{DE}}]\lesssim 3.7$ (i.e., where the ISM is HI dominated) we do not see a clear correlation between SFE_gas and P_DE. This is at the sensitivity limit existing data for EDGE, but it is also consistent with the overall behavior seen in HERACLES corresponding to a steepening of their mean relation. Above this pressure we find a clear linear trend in log–log space. For higher P_DE values (e.g., H₂-dominated regions) the EDGE average efficiencies are somewhat higher than those observed in HERACLES, which flatten out at high P_DE, although with a scatter that is within the respective 1σ error bars. For $\mathrm{log}[{P}_{\mathrm{DE}}]\gtrsim 3.7$ the EDGE average efficiencies are well described by the blue dashed line, which corresponds to 1% of the gas converted to stars per disk freefall time, τ_ff = (G ρ)^−1/2. To quantify this relation, we use an OLS linear bisector method to estimate the best linear fit to our data, obtaining

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}[{\mathrm{SFE}}_{\mathrm{gas}}({\mathrm{yr}}^{-1})] & = & [0.41\pm 0.29]\times \mathrm{log}[{P}_{\mathrm{DE}}/k({\rm{K}}\ {\mathrm{cm}}^{-3})]\\ & & -\ [11.32\pm 2.24].\end{array}\end{eqnarray} \tag{ 17 }$$

The bottom panel of Figure 9 shows the Σ_SFR versus P_DE, color-coded by galactocentric radius. When compared with other recent measurements (e.g., KINGFISH, Herrera-Camus et al. 2017; PHANGS, Sun et al. 2020), our annuli have the advantage of covering a somewhat wider dynamic range in both Σ_SFR and P_DE. We find a strong correlation between Σ_SFR and P_DE that is approximately linear for annuli at $\mathrm{log}[{P}_{\mathrm{DE}}/k]\gtrsim 3.7$, although below this limit we observe a break in the trend. As shown by the color-coding of the symbols, indicating r_gal in Figure 9, this limit is apparently related to the r_gal at which the transition from H₂-dominated to H i-dominated annuli happens. This transition may be due to the large range of physical properties covered by our sample, which span from molecular-dominated to atomic-dominated regimes. Where the ISM weight is higher (e.g., H₂-dominated regions), the SFR is stabilized by the increasing feedback from star formation to maintain the pressure that counteracts the P_DE (Sun et al. 2020). The lack of correlation we observe at $\mathrm{log}[{P}_{\mathrm{DE}}/k]\lesssim 3.7$ (r ≳ 0.7) is mainly because we are reaching our CO sensitivity in the H i-dominated regions. To quantify the correlation, we estimate the best linear fit by using an OLS linear bisector method in logarithmic space for annuli at r ≳ 0.7,

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}[{{\rm{\Sigma }}}_{\mathrm{SFR}}({M}_{\odot }\,{{yr}}^{-1})] & = & [1.10\pm 0.11]\times \mathrm{log}[{P}_{\mathrm{DE}}/k\,({\rm{K}}\,{\mathrm{cm}}^{-3})]\\ & & -\ [7.28\pm 0.65].\end{array}\end{eqnarray} \tag{ 18 }$$

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Note that these results are potentially sensitive to the method we employ for the fitting. Nonetheless, using an orthogonal distance regression (ODR) to fit the same subsample, we obtain very comparable values $\mathrm{log}[{{\rm{\Sigma }}}_{\mathrm{SFR}}({M}_{\odot }\,{{yr}}^{-1})]$ $=[1.09\pm 0.05]\times \mathrm{log}[{P}_{\mathrm{DE}}/k$ (K cm⁻³)] − [7.25 ± 0.25]. Barrera-Ballesteros et al. (2021) analyze 4260 resolved star-forming regions of kiloparsec size located in 96 galaxies from the EDGE-CALIFA survey, using a similar sample selection (e.g., inclination, σ_gas, and Σ_atom constant values, among others), but they just consider equivalent widths for the Hα line emission EW(Hα) > 20 Å. Using an ODR fitting technique, they obtain $\mathrm{log}[{{\rm{\Sigma }}}_{\mathrm{SFR}}({M}_{\odot }\,{{yr}}^{-1})]=[0.97\pm 0.05]$ $\times \mathrm{log}[{P}_{\mathrm{DE}}/k\,({\rm{K}}\ {\mathrm{cm}}^{-3})]-[7.88\pm 0.48]$, which is in agreement with the distribution shown in the bottom panel of Figure 9. The figure also shows that the correlation agrees with hydrodynamical simulations performed by Kim et al. (2013) (green dashed line), in which they obtain a slope of 1.13. These results are also consistent with measurements obtained in other galaxy samples. Sun et al. (2020) obtain a slope of 0.84 ± 0.01 for 28 well-resolved CO galaxies (∼15, corresponding to ∼100 pc) selected from the ALMA-PHANGS sample by using a methodology very similar to ours. Smaller slopes have been referenced in local very actively star-forming galaxies (e.g., local ultraluminous infrared galaxies, ULIRGs), which at the same time may resemble some of the conditions in high-redshift submillimeter galaxies (e.g., Ostriker & Shetty 2011). Herrera-Camus et al. (2017) analyzed the [CII] emission in atomic-dominated regions of 31 KINGFISH galaxies to determine the thermal pressure of the neutral gas and related it to P_DE, obtaining a slope of 1.3 (dotted blue line). Our results bridge these two extremes; the strong correlation between Σ_SFR and P_DE and its linearity support the idea of a feedback-regulated scenario, in which star formation feedback acts to restore balance in the star-forming region of the disk (Sun et al.2020).

In the next two subsections, we exclude 21 galaxies (out of the 81) since their Hα rotation curves (taken from Levy et al. 2018) are either too noisy or not well fitted by the universal rotation curve parametric form. The top panel of Figure 10 shows SFE_gas versus τ_orb, the orbital timescale (in units of yr), color-coded by galactocentric radius. When analyzing our efficiencies averaged over orbital timescale bins (black symbols), we note that there is a slight flattening of the SFE_gas at $\mathrm{log}[{\tau }_{\mathrm{orb}}]\sim 7.9-8.1$. We also note that annuli at $\mathrm{log}[{\tau }_{\mathrm{orb}}]\lesssim 8.1$ are usually within the bulge radius in the SDSS i band (reddish star symbols). However, the error bars are consistent with SFE_gas decreasing as a function of τ_orb, including at $\mathrm{log}[{\tau }_{\mathrm{orb}}]\lt 8.1$. These results are in agreement with what is found in other spatially resolved galaxy samples (e.g., Wong & Blitz 2002; Leroy et al. 2008). The average gas depletion time for our subsample is ${\tau }_{\mathrm{dep}}={{\rm{\Sigma }}}_{\mathrm{gas}}/{{\rm{\Sigma }}}_{\mathrm{SFR}}\approx {2.8}_{-1.0}^{1.1}$ Gyr, which agrees fairly with the depletion time τ_dep = 2.2 Gyr found for HERACLES (not including early-type galaxies; Leroy et al. 2013). Utomo et al. (2017) computed the depletion times for 52 EDGE-CALIFA galaxies using annuli in the region within 0.7 r₂₅ (just considering the molecular gas); their average τ_dep ≈ 2.4 Gyr is in good agreement with our results.

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The orbital timescale has a strong correlation with radius, and theoretical arguments expect SFE_gas to be closely related to orbital timescale in typical disks (Silk 1997; Elmegreen 1997; Kennicutt 1998). A correlation between SFE_gas and τ_orb is based on the "Silk−Elmegreen" relation, which states that Σ_SFR = _orb Σ_gas/τ_orb, where _orb is the fraction of the gas converted into stars per orbital time (also called "orbital efficiency"). Therefore, because SFE_gas = Σ_SFR/Σ_gas, SFE_gas and τ_orb are related by

$$\begin{eqnarray}&&{\mathrm{SFE}}_{\mathrm{gas}}=\displaystyle \frac{{\epsilon }_{\mathrm{orb}}}{{\tau }_{\mathrm{orb}}}.\end{eqnarray} \tag{ 19 }$$

It is interesting to analyze the relations between the different timescales since they can give intuition about the physical processes underlying the star formation activity (e.g., Semenov et al. 2017; Colombo et al. 2018). Equation (19) shows that the timescale to deplete the gas reservoir and the orbital timescale are related through _orb. Although there is large scatter, the median values of τ_orb and τ_dep for our sample are $({2.0}_{-0.7}^{+0.9})\times {10}^{8}$ yr and $({2.8}_{-1.0}^{+1.1})\times {10}^{9}$ yr, respectively. These values are in good agreement with previous EDGE-CALIFA sample results found by Colombo et al. (2018), who analyze a more limited subsample of 39 galaxies without the benefit of CO line stacking and more constrained to inclination below 65°, with ${\tau }_{\mathrm{orb}}=({3.2}_{-1.2}^{+2.0})\times {10}^{8}$ yr and ${\tau }_{\mathrm{dep}}=({2.8}_{-1.2}^{+2.3})\times {10}^{9}$ yr. The black dashed line in the top panel of Figure 10 corresponds to the best fit to our binned data (black symbols); our fit excludes lower limits (shown as triangles in the figure), and it shows that _orb ≈ 5% of the total gas mass is converted to stars per τ_orb. This average efficiency is lower than but similar to the _orb ≈ 7% of efficiency found by Wong & Blitz (2002) and Kennicutt (1998) and the _orb ≈ 6% efficiency for HERACLES (Leroy et al. 2008). Also, this efficiency is the same as the average molecular gas orbital efficiency found by Colombo et al. (2018) for their subsample of EDGE-CALIFA galaxies (_orb ≈ 5%). Similar to our results, all of these studies did not find a clear correlation between SFE_gas and τ_orb in the inner regions of disks, where the ISM is mostly molecular.

Like Colombo et al. (2018), however, we find that a constant _orb is not a good approximation for the data. The efficiency per orbital time depends on the Hubble morphological type, with _orb increasing from early to late types. This is shown in the bottom panel of Figure 10, which shows the data grouped according to the same four morphological classes used in Figure 6. Our results show that annuli from Sbc, Sc, and Scd galaxies, which are the most numerous in our sample, seem to group around _orb ∼ 5%. This value is also representative of the typical _orb seen for the morphological bins composed by Sa—Sd and Sdm—Ir types in the range $8.0\lt \mathrm{log}[{\tau }_{\mathrm{orb}}]\lt 8.4$. However, these groups also show _orb ≲ 5% in the ranges $\mathrm{log}[{\tau }_{\mathrm{orb}}]\lt 8.0$ and $\mathrm{log}[{\tau }_{\mathrm{orb}}]\gt 8.4$. However, early-type galaxies (with admittedly limited statistics, 21 annuli in total) show substantially lower _orb, with a median of _orb = 1.2%. These values are in agreement with previous results for EDGE-CALIFA galaxies by Colombo et al. (2018), even though sample selection and processing were different. They observe an _orb ∼ 10% for Sbc galaxies (most numerous in their subsample) and a systematic decrease in orbital efficiencies from late- to early-type galaxies.

As concluded in Colombo et al. (2018), our results support the idea of a nonuniversal efficiency per orbit for the "Silk−Elmegreen" law. Figure 10 shows that not only does _orb depend on morphological type, but the behavior also varies with galactocentric radius: at short orbital timescales ($\mathrm{log}[{\tau }_{\mathrm{orb}}]\lesssim 8.3$), or small radii ($\mathrm{log}[r/{R}_{{\rm{e}}}]\lesssim 1.1-1.3$), the efficiency per unit time SFE_gas tends to be constant, and as a consequence, the observed _orb tends to systematically decrease as τ_orb decreases. This is best seen in the top panel in the departure of the binned data (black symbols) from the dashed line of constant _orb. Note that this is also the approximate radius of the molecular disk, the region where molecular gas dominates the gaseous disk (Figure 5).

Other studies have also reported SFE_gas deviations as a function of morphology. Koyama et al. (2019) analyze CO observations of 28 nearby galaxies to compute the C-index = R90_petro,r /R50_petro,r as an indicator of the bulge dominance in galaxies (where R90_petro,r and R50_petro,r are the radius containing 90% and 50% of the Petrosian flux for SDSS r-band photometric data, respectively). Although they do not detect a significant difference in the SFE_gas for bulge- and disk-dominated galaxies, they identify some CO-undetected bulge-dominated galaxies with unusually high SFE_gas values. Their results may reflect the galaxy population during the star formation quenching processes caused by the presence of a bulge component, and they could explain the flattening shown in the top panel (mostly dominated by annuli within bulges) and bottom panel (mainly due to early-type and Sb−Scd galaxies' annuli) of Figure 10.

The formulation of the Toomre Q gravitational stability parameter (Toomre 1964, see Section 3.3 for more details) provided a useful tool to quantify the stability of a thin disk disturbed by axisymmetric perturbations. Some studies have shown that the star formation activity is widespread where the gas disk is Q-unstable against large-scale collapse (e.g., Kennicutt 1989; Martin & Kennicutt 2001).

First, we examine the case where only gas gravity is considered; the top left panel of Figure 11 considers this case, showing the SFE_gas as a function of both the Toomre instability parameter for a thin disk of gas (x-bottom axis), Q_gas, and galactocentric radius (indicated by circle color). The vertical black dashed line marks the limit where the gas becomes unstable to axisymmetric collapse. The vast majority of our points are in stable (or marginally stable) annuli with an average Q_gas = 3.2. There is no apparent correlation of SFE_gas with Q_gas (Pearson correlation coefficient of 0.17), and that is independent of galaxy mass (middle left panel) or type (bottom left panel). In other words, SFE_gas does not decrease as stability increases (i.e., as Q_gas increases). This is in agreement with similar results reported in previous studies. For example, using H_I observation for 20 dwarf irregular galaxies selected from the Local Irregulars That Trace Luminosity Extremes, The H_I Nearby Galaxy Survey (LITTLE THINGS; Hunter et al. 2012), Elmegreen & Hunter (2015) find that dIrr galaxies are Q_gas-stable, with a mean Q_gas ∼ 4. They also find that their galaxies have relatively thick disks, with typical (atomic) gas scale heights of h_gas ∼ 0.3 − 1.5 kpc. Consequently, they are more stable than the infinitely thin disks for which the Q_gas = 1 criterion is derived.

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Stars represent the dominant fraction of mass in disks at galactocentric radii with active star formation. Thus, it makes sense to account for their gravity when determining the stability of the ISM in these regions. The top right panel of Figure 11 shows the SFE_gas as a function of Toomre's instability parameter modified by Rafikov (2001) to include the effects of both gas and stars, Q_stars+gas; again galactocentric radius is indicated by color. As expected, we find that disks become more unstable when stellar gravity is included in addition to gas, with a few points appearing in the nominally unstable region for thin disks. The bulk of the annuli, however, are found at around Q_stars+gas ≈ 1.6. This is roughly consistent with calculations of Q in other samples (Romeo 2020). There is, however, no correlation of SFE_gas with Q.

The middle panels of Figure 11 show the SFE_gas values versus Q_gas and Q_stars+gas, but this time splitting the points into two groups of different galaxy stellar mass; as in Section 4.2.1, we choose log₁₀[M_⋆] = 10.7 to split the groups. Although the two groups separate in Q_gas, with annuli from galaxies with log₁₀[M_⋆] < 10.7 tending to be in general more stable, the separation disappears once the stars are taken into account in the Q calculation.

In one of the ideas on how stars relate to SFE_gas, Dib et al. (2017) show that star formation may be associated with the fastest-growing mode of instabilities. In that case, the relation between SFR and gas in spiral galaxies may be modulated by the stellar mass, which will contribute to the gravitational instability and regulation of star formation (like in the case of NGC 628; Dib et al. 2017). Also, the ${{\rm{\Sigma }}}_{{\mathrm{SFE}}_{\mathrm{gas}}}$−Σ_⋆ relation, known as the "extended Schmidt law," suggests a critical role for existing stellar populations in ongoing star formation activity, and it may be a manifestation of more complex physics where Σ_⋆ is a proxy for other variables or processes (Shi et al. 2011). Our results may reflect the importance of instabilities in enhancing the SFE_gas due to the strong gravitational influence from stars, particularly in galaxies with log₁₀[M_⋆] > 10.7. But in the aggregate there is no apparent evidence for a trend showing that annuli with more unstable Q have higher star formation efficiencies.

The bottom panels of Figure 11 show the same relations as the top panels, but this time the data are grouped in four bins by morphological type. In both panels crosses correspond to the "center of mass" for each morphological group. Although annuli in early-type galaxies are more "Toomre stable," the statistics are very sparse and the Toomre calculation may not apply (since these are not thin disks). Otherwise, we do not find a clear trend between morphology and stability based on the Toomre parameter for stars and gas. Previous studies have reported that Q_star+gas increases toward the central parts of spirals. For example, Leroy et al. (2008) found that although molecular gas is the dominant component of the ISM in the central regions, HERACLES galaxies seem to be more stable there than near the H₂-to-H_I transition. If the type of gravitational instability that Q is sensitive to plays a role in star formation in galaxies, we would expect to see some links between Q and molecular gas abundance. It is therefore interesting to test whether there is dependence of the H₂-to-H_I ratio, R_mol = Σ_mol/Σ_atom, on the degree of gravitational instability in EDGE galaxies. Since we assume a constant Σ_atom, however, for us R_mol is simply a normalized molecular gas surface density, Σ_mol. We use the typical H₂-to-H_I transition radius found in Section 4.2.1 to split the annuli into three groups: (i) annuli at r < 0.3r₂₅ (r < 0.6R_e; red points), which should be strongly molecular; (ii) annuli between 0.3r₂₅ < r and r < 0.5r₂₅ (0.6R_e < r < 1.4R_e; yellow points), which should be around the molecular to atomic transition region; and (iii) annuli at r > 0.5r₂₅ (r > 1.4R_e; blue points), which should be dominated by atomic gas. The top panel of Figure 12 shows that Σ_mol has a large scatter and does not seem to depend strongly on Q_star+gas. Within each range, however, we find that annuli with smaller galactocentric radii tend to be slightly more stable.

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A suggestive trend emerges when we limit the range of galactocentric radii. We compute a principal component analysis (PCA; Pearson 1901) to find the main axis along which the three populations vary most. The top panel of Figure 12 shows the PCA major and minor axis for annuli in the three defined zones. The axes have been normalized to fit the minor or major axes of the elliptical contours that enclose 50% of the annuli over a given range. The figure suggests that, within a given range, we tend to find more plentiful molecular gas in regions where annuli are more Toomre unstable. A concern, however, is that the axes in this plot have a degree of intrinsic correlation since the computation of Q_star+gas includes Σ_mol. Therefore, to assert that the correlation we observe is physically meaningful, we need to show that it is stronger than that imposed by the mathematics of the computation. We quantify the strength of the correlations using the Spearman rank correlation coefficient, which is a nonparametric measure of the monotonicity of the observed correlations. To investigate the degree to which the axes are internally correlated, we randomize the Σ_mol data (within each range) and recompute Q_star+gas in 200 realizations, to obtain the distributions of the Spearman rank correlation coefficient for each randomized group. Clearly, in the randomized data we would expect only the degree of correlation caused by the mathematical definition of the quantities. The bottom panel of Figure 12 shows that the Spearman rank correlation coefficients for the actual data (dashed red, dashed yellow, and dashed blue vertical lines) are consistent with the distributions seen in the randomized histograms. These results suggest that the correlation between Σ_mol and Q_star+gas seen in the top panel of Figure 12 is purely driven by the implementation of Equation (14), in which Q_star+gas depends on Σ_mol.