Testing Feedback-regulated Star Formation in Gas-rich, Turbulent Disk Galaxies

The galaxies considered here are a subset of the DYNAMO survey galaxies Green et al. (2014). DYNAMO galaxies are selected from Sloan Digital Sky Survey (SDSS) DR4 based on high Hα line flux, while excluding active galactic nuclei (AGNs) from the sample. In Figure 1 we show that the galaxies in our sample have specific SFRs (SFR/M_*) that range from the low-z main-sequence values to those of z ≈ 1–2. The sample spans ${M}_{\mathrm{star}}=(0.9\mbox{--}9)\times {10}^{10}$ M_⊙, SFR ∼ 1–60 M_⊙ yr⁻¹, and extinction $A({\rm{H}}\alpha )\sim 0.2\mbox{--}1.7$ mag. Overall, galaxies similar to those in DYNAMO-HST are extremely rare in the local universe, with a space density of ∼10⁻⁸ to 10⁻⁷ Mpc⁻³.

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A number of observational results have been published showing that DYNAMO galaxies are more similar to z ≈ 1.5 main-sequence galaxies than local universe ultraluminous infrared galaxies (ULIRGs). DYNAMO galaxies have very high molecular gas fractions, ${f}_{\mathrm{gas}}\approx 0.2\mbox{--}0.8$ (Fisher et al. 2014; White et al. 2017), whereas local universe galaxies typically have molecular gas fractions of less than 10%. White et al. (2017) show that unlike local universe ULIRGs, DYNAMO galaxies have lower dust temperatures (T_dust ∼ 20–30 K).

An important similarity of DYNAMO galaxies to z ∼ 1–2 main-sequence galaxies is in the kinematics. DYNAMO galaxies are rotating systems with high gas velocity dispersions, $\sigma \sim 20\mbox{--}100$ km s⁻¹ (Green et al. 2010, 2014; Bassett et al. 2014; Bekiaris et al. 2016). Observations with ∼100 pc resolution confirm that these high dispersions are not caused by beam-smearing effects (Oliva-Altamirano et al. 2018). Moreover, Bassett et al. (2014) show that these kinematic signatures are also observed in the stellar kinematics of DYNAMO galaxies, thus indicating that it is not likely a gas disk inside a system of stars with different kinematics. Finally, Obreschkow et al. (2015) show that the angular momentum of DYNAMO galaxies is low for disks of their stellar mass but is very similar to what has since been observed in z ≈ 1.5 main-sequence galaxies (Swinbank et al. 2017).

Using Hubble Space Telescope (HST) Hα maps, Fisher et al. (2017b) show that DYNAMO galaxies are "clumpy," and when DYNAMO images are degraded to match z ≈ 2 observations, they meet quantified definitions of clumpy galaxies (e.g., Guo et al. 2015). Moreover, Fisher et al. (2017a) show that those DYNAMO galaxies identified as "disks" meet detailed predictions of Toomre (1964) instability theories, whereas control galaxies identified as mergers do not. Specifically, they find that the size of clumps correlates with the kinematics of disks, and clumps are only found in annuli that are unstable by the Toomre analysis.

Overall, the properties of DYNAMO galaxies most closely resemble galaxies at z ≈ 1–2. DYNAMO galaxies are therefore excellent laboratories for studying the processes in clumpy, turbulent disks with higher resolution and greater sensitivity.

In this analysis we only include targets that are identified as "disk galaxies." We use the same criteria as described in Fisher et al. (2017b) that disk galaxies both show rotating ionized gas in 2D velocity fields and are well fit by an exponentially decaying surface brightness profile. For surface photometry we use the FR647M continuum images for galaxies imaged with HST and the 1.9 μm continuum for galaxies observed with OSIRIS. For the galaxy G10-1 we use 500 nm continuum from GMOS for the stellar surface brightness profile, which has 1.2 kpc resolution. Galaxies G03-2, D00-2, and C14-2 only have SDSS imaging to measure the stellar surface brightness profile, which is considerably poorer resolution. However, all three of these galaxies have measured dust temperatures from Herschel data of 20–30 K (White et al. 2017), which strengthens the case that they are disk-like systems rather than major mergers. These three targets are plotted as different color points in all results figures. As discussed in Fisher et al. (2017b), galaxies H10-2 and G13-1 do not meet the criteria of disks, and from Oliva-Altamirano et al. (2018) galaxy E23-1 does not meet this definition of a disk galaxy.

For our analysis we make observations to measure molecular gas masses, ionized gas maps with ∼100 pc resolution, and kinematics of ionized gas. We present gas masses and kinematics on 17 DYNAMO galaxies, and we obtain ionized gas maps for 13 targets. All of these methods are well tested and have been used on the DYNAMO sample in previously published works. Here we summarize the basics of each.

We compile CO (1–0) observations from Fisher et al. (2014) and White et al. (2017) using the Plateau de Bure Interferometer (PdBI) and the Northern Extended Millimeter Array (NOEMA), respectively, with new NOEMA observations.

New NOEMA observations were made during the period 2016 May–December, with observing programs S16CK and W16BK (PI: Fisher). We use the same observational method and similar measurement techniques as were previously published in Fisher et al. (2014) and White et al. (2017). Similar to previous observations (Fisher et al. 2013; White et al. 2017), all observations were made with the WIDEX correlator. These observations comprise six of the targets listed in Table 1.

Table 1.  Sample Properties

Galaxy	z	Mstar	Mmol	Kinematic	SFR	R1/2	Emission	Gas
		${10}^{10}\,{M}_{\odot }$	${10}^{9}\,{M}_{\odot }$	Sourcea	M⊙ yr−1	kpc	Linea	Sourcea
C14-2	0.0562	0.56	1.83 ± 0.36	WiFeS3	1.12 ± 0.17	4.0	Hα3	IR SED3
D00-2	0.0813	2.43	5.08 ± 0.84	WiFeS3	5.14 ± 0.72	3.5	Hα3	IR SED3
G03-2	0.12946	0.65	5.16 ± 1.18	WiFeS3	4.6 ± 0.89	4.5	Hα3	IR SED3
G10-1	0.14372	2.75	13.45 ± 2.15	GMOS3	15.7 ± 1	1.2	Hα3	CO (1–0)3
D20-1	0.07049	2.95	5.93 ± 0.65	WiFeS3	4.7 ± 0.25	3.4	Hα3	CO (1–0)3
G04-1	0.12981	6.47	29.00 ± 2.10	GMOS2	21.32 ± 1	2.8	Hα2	CO (1–0)1
G20-2	0.14113	2.16	5.22 ± 0.59	GMOS2	18.24 ± 0.35	2.1	Hα2	CO (1–0)3
D13-5	0.07535	5.38	11.90 ± 0.36	GMOS2	17.48 ± 0.45	2.0	Hα2	CO (1–0)1
G08-5	0.13217	1.73	7.11 ± 0.79	GMOS2	10.04 ± 1	1.8	Hα2	CO (1–0)3
D15-3	0.06712	5.42	9.36 ± 0.18	WiFeS2	8.29 ± 0.35	2.2	Hα2	CO (1–0)3
G14-1	0.13233	2.23	4.94 ± 0.59	GMOS2	6.9 ± 0.5	1.1	Hα2	CO (1–0)3
C13-1	0.07876	3.58	5.91 ± 0.15	WiFeS2	5.06 ± 0.5	4.2	Hα2	CO (1–0)3
C22-2	0.07116	2.19	4.94 ± 0.35	OSIRIS4	3.2 ± 0.28	3.4	Pa α4	CO (1–0)5
SDSS 024921–0756	0.153	3.02	7.30 ± 0.56	OSIRIS4	10.54 ± 1.054	1.1	Pa α4	CO (1–0)5
SDSS 212912–0734	0.184	7.08	51.15 ± 3.77	OSIRIS4	53 ± 5.3	1.3	Pa α4	CO (1–0)5
SDSS 013527–1039	0.127	7.08	34.89 ± 1.61	OSIRIS4	25.27 ± 2.527	1.6	Pa α4	CO (1–0)5
SDSS 033244+0056	0.182	7.24	40.32 ± 2.39	OSIRIS4	60.5 ± 6.05	1.9	Pa α4	CO (1–0)5

Note.

^aReferences. (1) Fisher et al. 2014; (2) Fisher et al. 2017a; (3) White et al. 2017; (4) Oliva-Altamirano et al. 2018; (5) this work.

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Typical integration times were 1–2 hr in C or D configurations. The correlator was tuned to observe the redshifted CO (1–0) emission line in each target. This yielded an on-sky frequency of 97–107 GHz for targets ranging in redshifts from 0.18 to 0.07. These observations were then calibrated using standard GILDAS routines in CLIC and then cleaned with the MAPPING pipeline routine during an on-site visit to IRAM. Observations were reduced with the default channel width of 20 km s⁻¹ and then binned to 50 km s⁻¹, yielding cubes with typical rms ∼ 2–3 mJy. Final flux uncertainties for each target are given in Table 3.

Our targets are unresolved in these PdBI and NOEMA observations. Two of the sources (D20-1 and C22-2) have particularly elongated beams. However, these do not significantly affect the measurement of the flux. We have overlaid the beam shape and size on SDSS r-band images and checked each source for possible contamination from other sources. The only source with an additional source in the beam area is G20-2, which contains a point source representing less than 1% of the flux of G20-2 in the FR647M HST continuum image but is barely detectable in Hα. It is, therefore, not likely that this is changing the CO flux by a significant amount.

The spectrum for each target is measured in a polygon region containing the galaxy. These spectra are shown in Appendix A. Fluxes are obtained by binning the data into 50 km s⁻¹ channels and then integrating the resulting spectrum. The choice in range of channels to integrate over is made by starting at the redshifted frequency of the CO (1–0) line and summing all adjacent channels that are above the noise limit. The total line widths of our targets are typically 300–400 km s⁻¹. Noise levels for the observations were in the range of 0.5–1.0 mJy. We also consider that the choice in how to measure flux may affect the final value. We therefore also measure flux by fitting a single-component Gaussian function to the data, as well as measuring the flux in 20 km s⁻¹ channels. We take the standard deviation of the four different fluxes for each target and add this in quadrature with the noise in the spectrum to estimate the measurement uncertainty of flux. This value is shown in each spectrum in Appendix A. The range of S/N for new NOEMA observations was S/N = 10–22. Observational details of CO detections are listed in Table 3.

The CO (1–0) flux is converted to molecular gas mass (M_mol) in the usual fashion, in which M_mol = α_COL_CO, where L_CO is the luminosity of CO (1–0) and α_CO is the CO-to-H₂ conversion factor, including a 1.36× correction for heavier molecules. We adopted the standard value α_CO = 4.36. We have made multiple efforts to determine the most appropriate value of α_CO. First, based on SDSS spectra, DYNAMO galaxies have slightly subsolar metallicity. Recently, White et al. (2017) study the dust temperature for a set of DYNAMO galaxies, including several in this work. They find that DYNAMO galaxies have low dust temperatures, T_dust ∼ 20–30 K, implying Milky-Way-like conversion factors (as reviewed in Bolatto et al. 2013). Moreover, of the few galaxies that have both dust SED and CO (1–0) observations, the estimated gas masses agree to ∼25%. Finally, Fisher et al. (2014) used the formula from Bolatto et al. (2013), which estimates α_CO via the baryonic surface density, and found a result consistent with the Milky Way value.

We also include three targets from White et al. (2017) for which the gas mass is determined from Herschel observations. Here we estimate the dust mass by fitting blackbody models to the infrared spectral energy distribution (SED) and then convert to molecular gas mass by assuming a constant dust-to-gas ratio, similar to other works (e.g., Genzel et al. 2015).

Fisher et al. (2017b) present 10 HST Hα maps of DYNAMO galaxies, using ramp filters FR716N and FR782N with the Wide Field Camera on the Advanced Camera for Surveys (WFC) on HST (PID 12977; PI Damjanov). Integration times were 45 minutes in the narrowband filter and 15 minutes with the continuum filter. All images were reduced using the standard HST pipeline. We correct the fluxes measured in the image using an [N ii]/Hα ratio determined from the SDSS spectrum for each target. Seven of the DYNAMO-HST sample galaxies also have CO (1–0) fluxes and are included here. A detailed description of the observations, continuum subtraction, and clump measurement is given in Fisher et al. (2017b). The typical resolution of DYNAMO-HST observations is 60–150 pc.

We also observed five galaxies with Paα emission lines observed with Keck OSIRIS (Larkin et al. 2006). Our targets were observed with the laser guide star system with exposure times of 4 × 900 s, achieving resolution of 150–400 pc. We use the OSIRIS data reduction pipeline version 2.3. To flux-calibrate the OSIRIS spectra, we use the telluric stars observed each night (an average of three telluric stars per night). Our method corresponds to a first-order flux calibration consistent with the Two Micron All Sky Survey magnitudes within ∼20%. Detailed descriptions of both observation and reduction of OSIRIS data cubes are given in Bassett et al. (2017) and Oliva-Altamirano et al. (2018).

SFRs are calculated from the emission-line flux using the extinction-corrected Hα line luminosity by assuming SFR [M_⊙ yr⁻¹] = $5.53\times {10}^{-42}{L}_{{\rm{H}}\alpha }$ [ erg s⁻¹] (Hao et al. 2011). We calculate the intrinsic Hα extinction using the Hα and Hβ line ratios from SDSS spectra. For those targets with OSIRIS data we divide the Paα luminosity by the intrinsic luminosity ratio, L_Paα/L_Hα = 0.128 (Calzetti 2001). Using Hα fluxes from Green et al. (2017), we find that SFR_Hα – SFR_Paα ≈ ± 2.5 M_⊙ yr⁻¹.

Three galaxies in our sample have neither HST nor OSIRIS data. For these we use the published Hα emission line fluxes measured from AAT/SPIRAL and AAT/WiFES (Green et al. 2017).

Data cubes containing intensity, velocity dispersion, and rotation velocity of each galaxy are extracted from the data by fitting Gaussian functions with point-spread function convolution to emission lines in individual spaxels. These are then fit with kinematic models by the method of least squares using the GPU-based software gbkfit (see Bekiaris et al. 2016). Inclination is taken from photometry. We model the rotation velocity, v_rot, with the function (Boissier et al. 2003)

$$\begin{eqnarray}&&{v}_{\mathrm{rot}}(r)={v}_{\mathrm{flat}}\left[1-\exp (-r/{r}_{\mathrm{flat}})\right].\end{eqnarray} \tag{ 1 }$$

The software then returns v_flat and r_flat.

The fit to the data cubes also includes an intrinsic component of velocity dispersion, σ. In the model the velocity dispersion is assumed to be constant across the disk. Fitting the velocity dispersion simultaneously in the model allows for accounting of the beam smearing in the data (Davies et al. 2011), as well as inclination. This flat velocity dispersion profile makes a necessary assumption that the galaxy is a disk and further requires our effort to exclude mergers as described above.

An alternative approach to modeling velocity dispersions is to make a weighted average of the velocity dispersion in the region of the galaxy that also shows a flat rotation curve. Oliva-Altamirano et al. (2018) investigate both methods with DYNAMO galaxies. They generate model galaxies designed to match clumpy DYNAMO disks. Then they fit these with both gbkfit and straightforward averages. They find that both methods recover similar values for velocity dispersion, with modeling being slightly more stable as it intrinsically accounts for rising dispersion in galaxy centers. On average the models and average methods have a difference of σ_ave–σ_model ≲ 5 km s⁻¹, which is consistent with our error bars.

The data sources for both σ and V include AAT/WiFES Hα observations (Green et al. 2014), Keck OSIRIS observations of Paα (Oliva-Altamirano et al. 2018), and Gemini GMOS observations of Hα and Hβ (Bassett et al. 2014; Fisher et al. 2017a). For a more detailed description of Gemini observations see Bassett et al. (2014). In those cases in which multiple observations are made on a single target we preference first the GMOS observations, as this data set offers both deep, high-S/N observations and subkiloparsec resolution, then OSIRIS observations as a result of the high resolution, and then AAT observations. The kinematic parameters we use here, derived from the three separate data sets, are found to generally agree on the order of the uncertainties, ∼±5–10 km s⁻¹ (Bassett et al. 2014; Bekiaris et al. 2016; Oliva-Altamirano et al. 2018).

Properties derived for analysis in this work (including kinematics, molecular gas depletion time, and pressure) are listed in Table 2.

In Figure 2 we compare t_dep to velocity dispersion of ionized gas (σ). For reference, Wisnioski et al. (2015) find that σ_gas increases from ∼10–20 km s⁻¹ in the local universe to σ > 30 km s⁻¹ at z > 1. In our sample, all galaxies with t_dep < 1 Gyr have σ > 40 km s⁻¹.

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Considering our entire DYNAMO data set, we find a strong, inverse relationship between the molecular gas depletion time and the gas velocity dispersion, with a Pearson's correlation coefficient of r = −0.8 and a p-value of 8.7 × 10⁻⁵. The best-fit relationship for these quantities is

$$\begin{eqnarray}&&\mathrm{log}({t}_{\mathrm{dep}})=-1.39\pm 0.23\times \mathrm{log}(\sigma )+2.27\pm 0.38,\end{eqnarray} \tag{ 2 }$$

where t_dep is in Gyr and σ is in km s⁻¹. The vertical scatter around this correlation is 0.12 Gyr. We note that in our data set this correlation has less scatter and a stronger correlation coefficient than that of t_dep and SFR/M_*.

To investigate how much the correlation depends on the galaxy with the largest velocity dispersion/shortest depletion timescale (G20-2), we remove it from the sample and recompute r and p. It is possible that this galaxy is at the least affecting the power law of the correlation to some degree. We therefore refit the data excluding galaxy G20-2. We still find a strong, inverse correlation for the data set that excludes this target, with a Pearson's correlation coefficient r = −0.78 and a p-value of 4.7 × 10⁻⁴. The best-fit quantity for this subset of the data that does not include G20-2 is

$$\begin{eqnarray}&&\mathrm{log}({t}_{\mathrm{dep}})=-1.04\pm 0.20\times \mathrm{log}(\sigma )+1.71\pm 0.33.\end{eqnarray} \tag{ 3 }$$

To be clear, the exclusion of G20-2 is a completely ad hoc treatment. We do not observe any special features of this target that lead us to believe that it is any more peculiar than the other DYNAMO galaxies. Fisher et al. (2017b) present both the Hα map and 600 nm continuum surface brightness profile of all DYNAMO galaxies, and G20-2 does not have significant asymmetries, aside from the presence of clumps. Moreover, there are no detectable companion galaxies in the HST images. We do note that G20-2 has a very prominent ring with a radius of ∼1 kpc. This may be driving a higher star formation efficiency or larger σ when compared to other DYNAMO galaxies. However, G04-1 and D13-5 both have rings as well, albeit less prominent than that of G20-2. We note that in Figure 4 of this work we will find that G20-2 is not an outlier.

As discussed in Robotham & Obreschkow (2015), the maximum likelihood fitting technique provides a robust treatment of uncertainties. Nonetheless, we also consider ordinary least-squares (OLS) bisector fits to the data. We do this to ensure that our fitting method is not somehow biasing the result. Also, OLS techniques have been in use for a much longer time (Isobe et al. 1990), allowing for standard comparison with previous work. We find that a fit to all our data recovers an OLS bisector of ${t}_{\mathrm{dep}}\propto {\sigma }^{-1.30\pm 0.18}$ and the fit to the data set in which G20-2 is omitted recovers an OLS bisector of ${t}_{\mathrm{dep}}\propto {\sigma }^{-1.03\pm 0.19}$. These different fitting methods, therefore, yield results that are within uncertainties of each other.

The error bars in Figure 2 are representative of the measurement uncertainties propagated through to the physical quantities. In the case of depletion time, however, it is likely that the systematic uncertainty is somewhat larger. The systematic uncertainty on t_dep could be as high as a factor of 2 for any single object (e.g., Bolatto et al. 2013). To determine the impact of this on the robustness of the correlation in Figure 2, we carry out a simple bootstrap experiment. We randomly modify the log(t_dep) values to scatter around each point within a Gaussian distribution with σ_Gauss = 0.15 dex. We ran 1000 iterations, determining the correlation coefficient and p-value of each realization. We find that the median correlation coefficient is r = −0.63 with a standard deviation of 0.11 and the median p = 0.006 with a standard deviation of 0.04. We rerun this with the more pessimistic assumption of a flat distribution with width of ±0.15 dex and find a similar median correlation coefficient of r = −0.6 and a p-value still indicating a strong correlation, with median p = 0.01. Removing G20-2 from the fit reduces the robustness of the correlation, but with r = −0.53 and p = 0.03 the data still represent a high probability of correlation. Therefore, the correlation we observe in Figure 2 appears to be robust against the systematic uncertainties on the depletion time.

As we state above, exclusion of G20-2 in the fitted sample only marginally affects the robustness of the fit, measured by either Pearson's r or the p-value. Both sample choices satisfy definitions of "strong correlation" based on these statistical metrics. However, inclusion (or exclusion) of G20-2 does have a significant impact on the power law in the correlation between t_dep and σ. Future samples that contain more galaxies with t_dep ≲ 0.5 Gyr would be helpful to further constrain the exact value of the power law. With our current sample there appears to be enough evidence to support a statistically significant inverse correlation between t_dep and σ, with the power law ranging between σ⁻¹ and σ^−1.4. We will therefore consider predictions within this range consistent with our data set.

We note that the systematic observational effect of increased velocity dispersion on α_CO should result in the opposite trend. If the velocity dispersion of gas is increased by components other than internal cloud properties, then the effect is a systematic increase in observed CO luminosity (see discussion in Bolatto et al. 2013). This would increase the observed t_dep in galaxies with larger σ. Our observations of the opposite trend in Figure 2 then imply that this correlation is likely physical, and may even be steeper than our observations if we consider variations in α_CO.

Before we compare our observational results to predictions from theory, we point out an important assumption. Our measurement of velocity dispersion relies on ionized gas velocity dispersion maps, whereas theoretical predictions refer to total gas or molecular gas velocity dispersions. Ionized gas velocity dispersions include broadening due to thermal broadening. For a gas of 10⁴ K this broadening amounts to σ_br ≈ 10 km s⁻¹ and is added in quadrature with the line width due to the motion of gas, such that the observed velocity dispersion is ${\sigma }_{\mathrm{obs}}^{2}\approx {\sigma }_{\mathrm{gas}}^{2}+{\sigma }_{\mathrm{br}}^{2}$. For galaxy samples with lower measured velocity dispersions, such as in dwarf galaxies (as shown in Moiseev et al. 2015), this can be a significant contribution; however, our observed velocity dispersions range from 26 to 81 km s⁻¹. Thermal broadening will at most contribute 1–2 km s⁻¹.

Levy et al. (2018) find in nearby spiral galaxies that ionized gas gives systematically lower rotation velocities than molecular gas. They argue that in low-z spiral galaxies the molecular gas resides in a thin disk, where the ionized gas traces a thicker, more turbulent component. However, it is not clear whether the same result holds for galaxies, like DYNAMO and z = 1–2 main-sequence galaxies, that have significantly higher surface densities of gas and star formation. White et al. (2017) argue that for systems in dynamical equilibrium, which have large gas mass surface densities, the bulk of the gas will naturally have higher scale heights, which they show are well represented by the velocity dispersions measured with ionized gas in DYNAMO galaxies. Ultimately this field would significantly benefit from direct comparisons of ionized and molecular gas kinematics in gas-rich, turbulent disks; such studies are at present sparse. Recent work by Übler et al. (2018) compares ionized gas and molecular gas kinematics in a single star-forming disk galaxy at z = 1.4. They find that the kinematics, both rotation and velocity dispersion, for the two tracers agree well, within 1–5 km s⁻¹. More work on this comparison would certainly be welcome; nonetheless, the evidence thus far seems to suggest that at high SFR and high σ_ion the ionized gas is a good tracer of the gas kinematics. We therefore make the same assumption as other studies (e.g., Förster Schreiber et al. 2009; Lehnert et al. 2009; Green et al. 2010; Genzel et al. 2011; Wisnioski et al. 2015; Krumholz & Burkhart 2016) that for our sample σ_ion ≈ σ_gas.

Theories of the interstellar medium (ISM) that incorporate feedback from star formation predict coupling of t_dep and σ that is similar to our observations (Ostriker et al. 2010; Shetty & Ostriker 2012; Faucher-Giguère et al. 2013; Krumholz et al. 2018). In all of these theoretical derivations it is shown that in the limit of marginal stability the turbulent velocity has a linear relationship with the star formation efficiency. If we assume that in our targets the ionized gas velocity dispersion is mostly driven by turbulence, then these models predict ${t}_{\mathrm{dep}}\propto {\sigma }^{-1}$. This is represented as a dotted line in Figure 2 and is consistent with our data, assuming an arbitrary scale factor. In a subsequent section we will return to the topic of the scale factor in the σ−t_dep correlation in light of results presented there.

Salim et al. (2015) derive a star formation law using a multi-freefall prescription of the gas (see also Federrath & Klessen 2012). They predict that star formation efficiencies will depend on both the probability density distribution and the sonic Mach number of the turbulence. In the limit that the virial parameter is not significantly variable, the Mach number is directly proportional to the velocity dispersion. These models then predict ${t}_{\mathrm{dep}}\propto {\sigma }^{-4/3}$, which is shown as a dashed line in Figure 2. The prediction based on the multi-freefall model is almost exactly the same as our best-fit relation to our entire data set, ${t}_{\mathrm{dep}}\propto {\sigma }^{-1.39}$.

In the comparison of both the theories of feedback-regulated star formation and the multi-freefall model one must consider the variation of the freefall time, t_ff. A relationship between σ and t_dep in star formation theory can be taken from Equation (20) of Shetty & Ostriker (2012), which states that σ ∝ (t_ff/t_dep) (p_*/m_*). Therefore, our interpretation of Figure 2 assumes that t_ff varies less than t_dep and σ. Similarly, with the results of Salim et al. (2015), the simple relationship ${t}_{\mathrm{dep}}\propto {\sigma }^{-3/4}$ can only be derived by assuming that one can neglect variation in t_ff.

For the sample of galaxies in Figure 2 we expect that significant variation in the galaxy-averaged freefall time is unlikely. We expect that ${t}_{\mathrm{ff}}\propto 1/\sqrt{\rho }$ (Krumholz & McKee 2005), where ρ is the volume density. We can estimate the variation in ρ using the parameters Σ_gas/h_z, where h_z is the gas scale height. For galaxies with longer t_dep ∼ 1–2 Gyr the typical surface density in our sample is 10–50 M_⊙ pc⁻². We surmise that these lower-σ galaxies likely have a disk thickness that is similar to the Milky Way, of order ∼100 pc. For a clumpy galaxy with lower t_dep and higher σ the gas surface density is typically 100–1000 M_⊙ pc⁻². For these we assume a disk thickness that is similar to that of z ∼ 1 edge-on galaxies, 500–1000 pc (Elmegreen et al. 2005). Bassett et al. (2014) also present a discussion of two DYNAMO disk thicknesses based on kinematics of stars and gas; they conclude that h_z is in the range of 400–1000 pc. Even though the galaxies in the lower left portion of Figure 2 have higher surface densities, they are also thicker, and therefore the value of $1/\sqrt{{{\rm{\Sigma }}}_{\mathrm{gas}}/{h}_{z}}$ will not change much across the sample in Figure 2. This is consistent with the results of Krumholz et al. (2012), who find that the distributions of freefall times of "high-z disks" and "z = 0 spirals" overlap. Moreover, any variance in t_ff over this small dynamic range is likely to be significantly affected by stochastic scatter. However, if we extended our sample to either very lower surface density disks or perhaps very extreme starbursting disks at z ∼ 4, this assumption that t_ff does not significantly vary may not be as valid. We note that there is significant uncertainty on the scale height of the molecular disk in gas-rich disk galaxies, and more work needs to be undertaken to study this important quantity. We also note that these are galaxy-averaged quantities. It is very likely that measurements at the scale of individual molecular clouds have increased scatter and perhaps a different parameter dependency due to the more significant variation of the freefall time.

A number of theories and semianalytic models suggest that the gas depletion time should be directly connected to the dynamical time, t_dyn = 2πR_disk/V_flat, of the galaxy (e.g., Davé et al. 2011; Krumholz et al. 2012). These models are motivated by the well-known relationship Σ_SFR ∝ Σ_gas/t_dyn observed in nearby galaxies (Kennicutt 1998). Krumholz et al. (2012) argue that in the "Toomre regime," in which galaxies have high gas fractions and show marginal stability, the local star-forming region is not able to decouple from the ambient gas in the galaxy. From this concept they then derive a positive, linear correlation between t_dyn and t_dep.

In Figure 3 we show the relationship between depletion time and galaxy dynamical time for galaxies in our sample. The depletion time does have some dependency on t_dyn in that galaxies with t_dep ≲ 1 Gyr always have low t_dyn. An effort to fit a correlation returns a very weak, high-scatter relationship, with a Pearson's correlation coefficient of r = 0.3 and a p-value for this data set of 0.2. The best-fit relationship for our data set is

$$\begin{eqnarray}&&\mathrm{log}({t}_{\mathrm{dep}})=0.44\pm 0.24\mathrm{log}({t}_{\mathrm{dyn}})-0.44\pm 0.23.\end{eqnarray} \tag{ 4 }$$

This is significantly shallower than the prediction for gravity-driven turbulence (Krumholz et al. 2012). We find no correlation at all between t_dep and the product of Q × t_dyn, which is also predicted in the gravity-only theory. We find that using the same bootstrap method as used to analyze Figure 2, to account for systematic uncertainty in t_dep, results in a median correlation coefficient of r = 0.25 with a standard deviation of 0.16, and the median p-value is 0.3. These values indicate a very low probability of a correlation between t_dep and t_dyn in our data set.

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In this work we use the following formula from Elmegreen (1989) to estimate the midplane hydrostatic pressure within our galaxies:

$$\begin{eqnarray}&&P=\displaystyle \frac{\pi }{2}G{{\rm{\Sigma }}}_{g}\left[{{\rm{\Sigma }}}_{g}+\left(\displaystyle \frac{\sigma }{{\sigma }_{* }}\right){{\rm{\Sigma }}}_{* }\right].\end{eqnarray} \tag{ 5 }$$

Σ_gas and Σ_* represent the gas and stellar mass surface densities, and similarly σ and σ_* represent velocity dispersions of the gas and stars.

In Appendix B we outline our method to measure or estimate each of these parameters described in Equation (5), as well as discuss in detail the impact of the uncertainty in each parameter on the pressure. We briefly summarize our main sources of uncertainty here. We also note that Equation (5) was originally developed to describe subgalactic measurements of the ISM. Our use of it here requires the assumption that the ensemble averages of the galaxy are reflective of local values, though systematic biases may exist.

The largest source of uncertainty in pressure comes from our use of unresolved CO flux measurements. To calculate Σ_gas, we assume that the size of the ionized gas disk is equivalent to the size of the molecular gas disk. This assumption is consistent with observations of high-z galaxies (Tacconi et al. 2013; Bolatto et al. 2015; Dessauges-Zavadsky et al. 2015; Hodge et al. 2015) with an uncertainty of 20%–50%. However, we also consider the possibility that the molecular disk is as large as our unresolved CO measurement. This reduces the pressure by a factor of ∼4 in most targets and is reflected in the error bars of Figure 4. We also consider the uncertainty introduced from different assumptions of the atomic hydrogen surface density. We find that this only has a significant impact on the two lowest-pressure systems and is likewise reflected in the error bars. For more information on these uncertainties see Appendix B.

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To increase sampling at low pressures, we combine our sample set here with the observations of the THINGS sample (Walter et al. 2008), using derived measurements from Leroy et al. (2008) to calculate Σ_SFR and P. The combined data set spans nearly 6 orders of magnitude in midplane pressure and 4 orders of magnitude in Σ_SFR. We note that there are important differences between the THINGS and DYNAMO measurements of pressure. The THINGS sample has measured values of both molecular and atomic gas mass surface density, where the DYNAMO sample only has observations of molecular gas mass and atomic gas mass surface density is adopted. This is likely important at lower pressure, where the ratio of molecular to atomic gas mass surface density is lower. Alternatively, THINGS galaxies do not have measurements of ionized gas velocity dispersions. Leroy et al. (2008) adopt σ ≈ 11 km s⁻¹ as being consistent with their measurements of atomic gas. These considerations are discussed in more detail in Appendix B. We have five DYNAMO galaxies that overlap the range in derived pressure and Σ_SFR with those of the THINGS galaxies. We find the two samples to have similar values of Σ_SFR/P.

For DYNAMO galaxies we find very high values of P/k compared to local spirals like the Milky Way, reaching as much as 10⁵ times higher than the pressure in the Milky Way. This is similar to the pressure of the z ∼ 2 galaxy observed in Swinbank et al. (2011). DYNAMO galaxies have both significantly higher gas masses and smaller sizes compared to Milky-Way-like spirals. These differences lead to greater surface densities, which then create very high pressures that we observe. In almost all galaxies the "stellar term" of the pressure, (σ/σ_*)Σ_*, dominates over the "gas term," Σ_g. This is notable, as the gas fractions of DYNAMO galaxies are very high, f_gas ∼ 20%–60%. Even in this sample of gas-rich galaxies we find that on average P_star/P_gas ≈ 2.3 ± 1.1. We note that this is a prediction of feedback-regulated star formation, and that even in gas-rich systems the stars contribute a very important component to the pressure (Ostriker & Shetty 2011). We will return to this subject in our discussion of dynamical equilibrium pressure.

In Figure 4 we compare the midplane hydrostatic pressure P, as described in Equation (5), with the SFR surface density. We find that a single correlation describes these data well, over 6 orders of magnitude in pressure. The best-fit relation is

$$\begin{eqnarray}&&\mathrm{log}({{\rm{\Sigma }}}_{\mathrm{SFR}})=(0.76\pm 0.06)\mathrm{log}(P/{k}_{B})-5.89\pm 0.35,\end{eqnarray} \tag{ 6 }$$

where Σ_SFR is in units of M_⊙ yr⁻¹ kpc⁻² and P/k_B is in units of cm⁻³ K. The 1σ vertical scatter around this relationship is represented as the shaded gray region in Figure 4.

It is predicted that ${{\rm{\Sigma }}}_{\mathrm{SFR}}=4{({p}_{* }/{m}_{* })}^{-1}\,{P}_{\mathrm{tot}}^{N}$, where the power-law index N is expected to be close to unity (e.g., Shetty & Ostriker 2012; Kim et al. 2013). Our correlation in Figure 4 is then qualitatively consistent with the prediction of a strong, positive relationship between these two quantities.

The power-law slope we observe in Figure 4 is significantly below unity, which is different from the predictions described above (shown as the dashed line in the figure). In simulations by Kim et al. (2013) the measured power law is slightly steeper than linear. In a subsequent section of this paper we discuss three possible options to explain this discrepancy: (1) that the relationship between pressure and Σ_SFR is truly nonlinear; (2) that the scale factor p_*/m_* is nonuniversal and increases for gas-rich, high-pressure disks; and/or (3) that alternate mechanisms, such as mass transport within a disk, also provide pressure support.

It is important to note that both systematic and observational uncertainties in the calculation of P/k and Σ_SFR can affect these results at the factor of a few level. For example, to calculate mass surface density, we estimate the size of the disk as R_disk = 2 R_1/2. However, since pressure scales as mass surface density squared, adopting a different characteristic radius of the galaxy would alter pressure more than Σ_SFR. The median value for these low-pressure systems is P/Σ_SFR ≈ 9 × 10³ km s⁻¹, which is only a factor of a few higher than the theoretical prediction, and as shown in Figure 4, the predicted value falls within the scatter of low-pressure systems. Considering this, we therefore conclude that for disks with lower values of pressure (P/k < 10⁶ cm⁻³ K) the data are within the uncertainty of the predictions from the models. This is to say that our data suggest that local universe spirals are consistent with theoretical predictions of Σ_SFR−P from feedback-regulated star formation models. This is similar to the results of Herrera-Camus et al. (2017), who find that KINGFISH galaxies, with P/k ∼ 10³.5–10⁴.5, are consistent with predictions from Kim et al. (2013).

High-pressure systems, P/k > 10⁶ cm⁻³ K, however, do not appear to be reconcilable with even generous estimations of the uncertainties. If we consider only those systems with P/k > 10⁶ cm⁻³ K, we find an average value of $\lt P/{{\rm{\Sigma }}}_{\mathrm{SFR}}\gt \approx 4.4\times {10}^{4}$ km s⁻¹. This is more than an order of magnitude larger than the theoretical prediction, as shown in Figure 4.

In light of these high values of P/Σ_SFR in DYNAMO galaxies, we now consider the zero-point in Figure 2. We can compare the scale factor in Figure 2 to that of Figure 4 by solving each theoretical prediction for p_*/m_*. Shetty & Ostriker (2012) predict that σ ≈ 0.366 (t_ff/t_dep) (p_*/m_*), where t_ff is the freefall time. Note that this is adopted from Equation (20) of Shetty & Ostriker (2012). The estimate using t_dep and σ does not require all of the assumptions that go into measurement of galaxy pressure (described in detail in Appendix B); most notably, galaxy sizes are not used in this case. However, this derivation depends on an assumption of the freefall time, which introduces a source of uncertainty. Nonetheless, we can think of this method of deriving the scale factor as a semi-independent check. The freefall time of a cloud will depend on the inverse square root of the local gas density (Krumholz et al. 2012). For a gas-rich, turbulent disk Krumholz et al. (2012) suggest values of t_ff ≈ 1–10 Myr. Using this range, we derive values of ${p}_{* }/{m}_{* }\approx {10}^{4}\mbox{--}{10}^{6}$ km s⁻¹ for DYNAMO galaxies, which is consistent with our estimate from P/Σ_SFR. We find that there is a high-scatter correlation, and on average, the values of p_*/m_* derived from the two methods are in agreement.

We caution that it is not necessarily the case that P/Σ_SFR is a unique, robust tracer of the true physical balance of these quantities over the entire range of galaxies. That is to say, systematic uncertainties could affect this correlation at some level. Most notably, the CO-to-H₂ conversion factor could be a function of pressure (Narayanan et al. 2011).

We also consider the "dynamical equilibrium pressure" as described in Kim et al. (2011),

$$\begin{eqnarray}&&{P}_{\mathrm{DE}}\approx \displaystyle \frac{\pi G{{\rm{\Sigma }}}_{g}^{2}}{2}+{{\rm{\Sigma }}}_{g}{\left(2G{\rho }_{\mathrm{SD}}\right)}^{1/2}\sigma ,\end{eqnarray} \tag{ 7 }$$

where ρ_SD is the total midplane density, including dark matter and stars, as defined in Ostriker & Shetty (2011). The total midplane density ${\rho }_{\mathrm{SD}}={{\rm{\Sigma }}}_{* }/{h}_{z}+{({V}_{\mathrm{flat}}/{R}_{\mathrm{disk}})}^{2}/(4\pi G)$. The quantity h_z is the disk thickness; we describe how we calculate it in Appendix B. As described before, we make the simplifying assumption that in galaxies with high gas velocity dispersion, like the DYNAMO sample, the velocity dispersion measured from ionized gas is a good representation of σ for the total gas. The dynamical equilibrium pressure is an alternate description of the pressure, under the assumption that the system has evolved to its equilibrium state, in which pressure balances the feedback mechanism.

An interesting feature of representing pressure as done in Equation (7) is that this directly ties the result in Figure 2 to the effect of increasing pressure. We note again that this is under the assumption that σ_z ≈ σ_los. If vertical dynamical equilibrium in a disk is satisfied, then Ostriker et al. (2010) predict that $P\propto {\rm{\Sigma }}\sqrt{{\rho }_{\mathrm{SD}}}$ and also, as discussed above, Σ_SFR ∝ P_DE. Therefore, in order for a linear (or nearly linear) relationship between pressure and also t_dep ∝ 1/σ to hold, the stellar gravity term ($\sigma \sqrt{2G{\rho }_{\mathrm{SD}}}$) from Equation (7) must dominate over the gas gravity term (πGΣ/2). Therefore, a direct prediction of the feedback-regulated theory of star formation in starbursting disk galaxies, which we can test with our data, is that $\pi G{\rm{\Sigma }}/2\lt \sigma \sqrt{2G{\rho }_{\mathrm{SD}}}$. We find that for all galaxies in the DYNAMO sample the stellar gravity term dominates over the gas term. We find that on average the ratio of star to gas terms is ∼7. The lowest values of star to gas terms are in systems with, as one would expect, larger gas fractions, M_mol/M_dyn ∼ 30%–80%; however, the ratio remains above 1, reaching down to as low as ∼3 for galaxies in our sample.

In Figure 2 we show a strong correlation (r ≈ −0.8 and p-value = 10⁻⁴) between the molecular gas depletion time and velocity dispersion of the ionized gas, such that ${t}_{\mathrm{dep}}\propto {\sigma }^{-1.39\pm 0.23}$. We have tested this correlation against the systematic uncertainties introduced from the scaling of ionized gas emission lines to SFRs, as well as converting CO (1–0) line flux to molecular gas mass. We find the correlation in our sample to be robust. Indeed, we find that t_dep correlates more strongly with σ than any other parameter we compared it to (e.g., SFR/M, Σ_SFR, t_dyn, Σ_star).

The inverse correlation between t_dep and σ seems inconsistent with predictions in which the turbulence is driven only by gravity, which do not incorporate more complex treatments of turbulence (e.g., Federrath & Klessen 2012). Krumholz et al. (2012), in their Equation (18), determine that for a system in which the turbulence is driven exclusively by gravity t_dep ∝ Qt_dyn, where Q is Toomre's stability parameter and t_dyn is the dynamical time of the galaxy. The depletion time of marginally stable disks, such as our sample, is therefore predicted to mostly be driven by the dynamical time; however, our data set does not support a strong correlation between t_dyn and t_dep. We note the caveat that although there is very little evidence to support a galaxy-averaged relationship between t_dep and t_dyn, Krumholz et al. (2012) suggest that a local relationship may be stronger (where t_dep and t_dyn are measured in radial bins). This is not possible to measure with our current data set; resolved observations of CO in turbulent disks would be very helpful to this end.

Theories in which turbulence is driven by feedback (e.g., Ostriker & Shetty 2011; Shetty & Ostriker 2012; Faucher-Giguère et al. 2013), however, predict an inverse relationship t_dep ∝ σ⁻¹. As we show in Figure 2, this power-law slope is consistent with our observations.

Using a multi-freefall timescale prescription for the gas also agrees with our results (Federrath & Klessen 2012; Salim et al. 2015). The critical parameters in these models include the probability density function and sonic Mach number of the gas. These models are not necessarily inconsistent with feedback-driven models, in that feedback could still drive the compressive form of turbulence, which then produces a different distribution of freefall times.

5.1.1. Possible Implications for Cosmic Evolution of ${t}_{\mathrm{dep}}$

The results we find in this paper may shed some light on the observed shallow evolution of depletion time with redshift (Genzel et al. 2015; Schinnerer et al. 2016; Scoville et al. 2016; Tacconi et al. 2017). Theories that drive gas depletion time via the cosmic evolution of the dynamical time predict an order-of-magnitude decrease in t_dep. However, recent observations of high-redshift galaxies find that depletion times drop by a factor of 2–5 from z = 0 to z = 4. As we have discussed above in the theory of feedback-regulated star formation, a lower gas depletion time is natural in systems that have both high pressure and high internal velocity dispersion.

In Figure 5 we compare the observed cosmic evolution of molecular gas depletion time to the prediction based on a simple model using a t_dep ∝ σ⁻¹ dependency set to match our observations in Figure 2. Wisnioski et al. (2015) present the gas velocity dispersion, from emission lines, over a range in redshift z ∼ 0–3.5. We use the average cosmic evolution of σ(z), from Wisnioski et al. (2015), to determine t_dep(σ) as a function of redshift. The observed cosmic evolution of t_dep is from the empirical fit to the main-sequence evolution of the composite data set of Tacconi et al. (2017). The predicted values of t_dep agree with observations for z > 1. This implies first that our results likely hold on high-z galaxies, at least in a bulk sense. Moreover, this may imply that star formation efficiencies at high redshift are regulated not only by the availability of gas but also by the feedback within the disk.

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At z ∼ 0 our simple model overpredicts the data by a factor of 2–3. This is not at all surprising. Many of the assumptions that go into our analysis are not valid for low-z spirals. (We remind the reader that DYNAMO galaxies are atypical galaxies for the low-z universe.) First, unlike our galaxies, the velocity dispersion for typical low-z spirals is quite low (σ < 20 km s⁻¹). As we discuss above, ionized gas measurements are more significantly affected by thermal broadening at low dispersion, and ionized gas in more typical, low-SFR, low-z spirals may overestimate the true gas velocity dispersion. Also, at z ∼ 0 the gas almost certainty becomes far more dominated by the atomic component than in z > 1 galaxies (Obreschkow & Rawlings 2009).

In Figure 5 we also show the expected evolution of the depletion time if it were exclusively a function of the dynamical time, ${t}_{\mathrm{dep}}\propto {t}_{\mathrm{dyn}}\propto {(1+z)}^{-3/2}$ (see arguments in Davé et al. 2011; Krumholz & Burkhart 2016). This predicts significantly more evolution than in observations. Similarly, Lagos et al. (2015) predict a very steep evolution to the depletion time. We interpret this to imply that internal processes, such as the regulation of star formation via feedback, are therefore a dominant factor in determining the evolution of the depletion time over the past ∼10 billion years.

We find that in our data set Σ_SFR and the midplane pressure of galaxies have a very tight correlation across almost 6 orders of magnitude in pressure using two different formulations of midplane pressure (Figure 4). These correlations are found to be sublinear with ${{\rm{\Sigma }}}_{\mathrm{SFR}}\propto {P}^{0.77}$. Our measurements are therefore not as steep as theories that predict Σ_SFR ∝ P (Ostriker et al. 2010; Kim et al. 2011).

There appear to be three possibilities to reconcile our observations with the theoretical predictions: (1) the relationship between SFR surface density and pressure is sublinear, (2) the feedback momentum injected into the ISM per unit new stars formed (p_*/m_*) changes as a function of the local ISM properties, and/or (3) alternate mechanisms could drive turbulence and provide support against gravitational pressure.

From our data alone we cannot uniquely distinguish between these scenarios. We will consider these options below. We note to the reader that there is also the possibility that each of these options is contributing to the offsets we observe.

5.2.1. Is ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ versus P Truly Nonlinear?

5.2.2. Does ${p}_{* }/{m}_{* }$ Vary from Local Spirals to High-pressure Turbulent Disks?

5.2.3. Are Alternate Sources of Pressure Support Important in Gas-rich Disks?

The size of the starlight is measured from the continuum observations corresponding to each emission-line measurement. Similarly, the sizes used to calculate the SFR surface density are measured from the resolved emission-line maps for each target. For THINGS galaxies Leroy et al. (2008) measure the sizes in stars, SFR, and molecular gas, and for those targets we use the corresponding measurement to directly measure the associated surface brightness. Our observations of CO (1–0) in most DYNAMO targets are unresolved source detections, with a handful that are marginally resolved, in which R_disk is only slightly larger than the beam size. To calculate the total midplane pressure for DYNAMO galaxies, we therefore must make an assumption on the sizes of the molecular gas.

A straightforward method to estimate the molecular gas disk is to assume that the half-light radius of the ionized gas is roughly equivalent to that of the molecular gas. In the local universe the surface brightness profiles of gas in disk galaxies have been shown to be well behaved with a regular, exponential decay (Bigiel & Blitz 2012). In the THINGS sample Leroy et al. (2008) find that the average ratio of CO to SFR scale lengths is <l_CO/l_SFR = 1.0 ± 0.2. Assuming that the distribution of ionized gas is a good proxy for the distribution of the star formation, the assumption that R_1/2(CO) ≈ R_1/2(Hα) would be well justified in similar galaxies.

Due to the difficulties of such observations, there is considerably less work comparing the distribution of molecular gas to the distribution of stars, or star formation in turbulent disks. Hodge et al. (2015) measure the size of gas, CO (2–1), and star formation, 880 μm, in a rare double-lensed system at z ∼ 4 with ∼1 kpc resolution, finding that the respective physical size of the disk is 14 kpc in CO and ∼10 kpc in star formation. Bolatto et al. (2015) study high-resolution maps of CO (1–0) and CO (3–2) in two z ∼ 2 targets, finding that the CO gas in these targets R_1/2(CO) − R_1/2(optical) ≈ 1 kpc. Similarly, the sample of Tacconi et al. (2013), which has considerably lower spatial resolution, nonetheless shows that on average R_1/2(CO)/R_1/2(optical) ≈ 1 with a standard deviation of 0.5. Dessauges-Zavadsky et al. (2015) map CO (2–1) in a sample of strongly lensed galaxies at z ∼ 2 and find R_1/2(CO) ∼ 1–4 kpc, which is similar to our estimates from Hα. We note that more extreme differences have been observed; however, those targets are typically found to have multiple velocity components, indicating that they are likely ongoing mergers (e.g., Spilker et al. 2016). Overall, the observations of main-sequence galaxies at z > 1 seem to suggest that the uncertainty on our estimation of R_1/2(CO) ≈ R_1/2(Hα) would be at the ∼20%–50% level. Increasing the size of the molecular gas disk by 50% (i.e., R = 1.5 R_1/2(Hα)) would decrease our measured pressures by a factor of 2–3 in DYNAMO galaxies.

For our unresolved observations we can use the measured beam size as an estimate of the "largest possible size" for the molecular gas disk in DYNAMO galaxies, which will then correspond to a lowest possible pressure. For the five galaxies in which the beam is slightly smaller than R_disk we use the radius at which the flux from the galaxy is equivalent to the noise. The median largest circularized radius of CO disk in the DYNAMO is ∼1.6 times larger than the corresponding R_disk measured from the ionized gas. This corresponds to a decrease in the measured midplane pressure by a factor of 3–4.

The THINGS observations suggest that for normal spirals, low-pressure disks, assuming that the molecular gas and star formation have similar disk sizes, are safe approximations. Observations of higher-redshift sources suggest that this is roughly a safe assumption, though molecular disks in these higher-pressure systems may be slightly larger. We opt for the simple assumption that R_disk(CO) ≈ R_disk(ion), as this is consistent with the data and allows us to make a single correction for low- and high-pressure systems. We then use the maximum observed CO (1–0) size for each target as the lower bound error on pressure that is introduced from the CO observations. This uncertainty will be added in quadrature with other uncertainties to determine the lower limit of pressure for each target.

Based on results from Bassett et al. (2014), which compare the velocity dispersion of gas and stars in DYNAMO galaxies, we find that the standard approximation for stellar velocity dispersion, ${\sigma }_{* }\approx 1/2\sqrt{\pi {{Gl}}_{* }{{\rm{\Sigma }}}_{* }}$, where the disk scale length l_* ≈ R_half,*/1.76, reproduced measured velocities within ±10 km s⁻¹. We also consider a simpler formulation where σ_* ≈ σ +15 km s⁻¹. We find that these result in similar overall values of P when inserted into Equation (5). For the sake of consistency with previous studies (e.g., Leroy et al. 2008) we use ${\sigma }_{* }\approx 1/2\sqrt{\pi {{Gl}}_{* }{{\rm{\Sigma }}}_{* }}$. Note that, as done in Bassett et al. (2014), we assume that for DYNAMO galaxies σ ≈ σ_z. More work on the stellar kinematics of turbulent disks would certainly be informative for these assumptions.

A significant source uncertainty in the calculation of the midplane pressure is the estimation of the total gas mass surface density. Our observations of CO (1–0) only allow for observation of the molecular gas. However, estimations of midplane pressure refer to the entire gas mass surface density, atomic and molecular. Observing atomic gas masses on z ∼ 0.1 galaxies is difficult with present facilities (Catinella & Cortese 2015) and is a primary goal of future SKA pathfinders. We caution that since the midplane pressure depends on ${{\rm{\Sigma }}}_{\mathrm{gas}}^{2}$, even small differences in Σ_gas may significantly affect the slope of correlations.

We use a multistep method to estimate the total gas density. Observations of local spirals find that Σ_atm ∼ 5–10 M_⊙ pc⁻² (e.g., Bigiel et al. 2008). We first use a conservative estimate assuming the constant Σ_atm ∼ 5 M_⊙ pc⁻² to estimate the midplane pressure. Blitz & Rosolowsky (2006) give a correlation between the ratio of molecular to atomic gas, R_mol, and the total pressure, P/k. Using our initial estimate of pressure, which assumed Σ_atm ∼ 5 M_⊙ pc⁻², we then calculate the expected R_mol for DYNAMO galaxies. Note that in DYNAMO galaxies the stellar and molecular surface densities are more likely to drive the value of the pressure. We then recalculate the midplane pressure with the new estimate of Σ_atm = Σ_mol/R_mol. We find that for galaxies with high surface densities of gas (Σ_mol ≳ 30 M_⊙ pc⁻²) the difference in calculated midplane pressure (constant versus variable Σ_atm) is small, less than 0.01 dex. However, for the two lowest surface density galaxies in our sample the difference in pressures reaches 0.12 dex. This difference will be reflected in error bars in the associated figures. We rerun this estimation assuming the larger value of Σ_atm = 15 M_⊙ pc⁻² as an initial guess. We find that this has at most a difference of 0.04 dex in determined pressure, and only on the two targets with the lowest Σ_gas.