PROPERTIES OF BULGELESS DISK GALAXIES. II. STAR FORMATION AS A FUNCTION OF CIRCULAR VELOCITY

Thirteen of our objects were observed in the CO(1–0) and CO(2–1) lines at 115 and 230 GHz in 2007 May with the IRAM 30 m telescope on Pico Veleta. Dual polarization receivers were used at both frequencies with the 512 × 1 MHz filterbanks on the CO(1–0) line and the 256 × 4 MHz filterbanks on the CO(2–1) line. The observations were carried out in wobbler switching mode with a wobbler throw of 200'' in the azimuthal direction. At the beginning of the observations, the frequency tuning was checked by observing a bright galaxy at a similar redshift. Observations of the same calibration source on different days allowed us to check the relative calibration, which was excellent (better than 10%) for CO(1–0). The calibration in CO(2–1) was equally good, except for one day when the calibration observation was different by ∼35%. Pointing was monitored on nearby quasars, Mars, or Jupiter every 60–90 minutes. During the observation period, the weather conditions were generally good, with pointing better than 3''. The typical system temperature was 300–500 K at 115 GHz and 500–1000 K at 230 GHz on the T*_A scale. At 115 GHz (230 GHz), the IRAM forward efficiency, F_eff, was 0.95 (0.91), the beam efficiency, B_eff, was 0.75 (0.54), and the half-power beam size is 21'' (11''). All CO spectra and line intensities are presented on the main beam temperature scale (T_mb), which is defined as T_mb = (F_eff/B_eff) × T*_A. For the data reduction, we selected the observations with good quality (taken during satisfactory weather conditions and showing a flat baseline), averaged the spectra from the individual scans of the source, and subtracted a constant continuum for the CO(1–0) spectra and a linear continuum for the CO(2–1) spectra.

Figure 1 shows the CO spectra for the objects observed in 2007 May. The black spectrum shows the CO(1–0) data and the red spectrum shows the CO(2–1) data. We did not use the CO(2–1) data in this paper, but show it for completeness and to corroborate some of the weaker CO(1–0) detections. The solid black horizontal line is the velocity range over which we integrated to derive the CO(1–0) line intensity. The dashed horizontal line is centered on the systemic velocity of the galaxy, derived from velocity field modeling of our H i emission line data. The width of the dashed line is the width of the H i emission line at 20% of the peak flux density (W₂₀; see Table 1 for these values). The integrated CO(1–0) line intensities are presented in Table 2.

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We did not detect ESO 544-G03, ESO 418-G008, ESO 501-G023, or UGC 6446. For these galaxies, we quoted the CO line intensities as upper limits, computed with I_CO < 3 σ_rms (W₂₀ Δv)^1/2, where σ_rms is the noise in the spectrum in K and Δv is the CO(1–0) spectrum channel width, which is 10 km s⁻¹ in the Hanning-smoothed spectra of Figure 1.

CO(1–0) and CO(2–1) spectra and line intensities for six of our objects—PGC 3853, NGC 2805, NGC 3906, NGC 4519, NGC 5964, and NGC 6509—were presented in Böker et al. (2003), also based on IRAM 30 m observations and reduced in the same manner. We did not obtain CO data for ESO 555-G027.

2.2.1. Integrated H i Line Intensity

The H i 21 cm data for our sample were obtained from the VLA/Expanded VLA, operated by the National Radio Astronomy Observatory,⁸ for projects AZ0133 (carried out in 2001 August), AL0575 (carried out in 2002 June and November), AM0873 (carried out in 2006 October and November), and AM0942 (carried out in 2008 May and 2009 July and August). The galaxies were observed in the C or CnB configurations, which provide a nominal angular resolution of 13''. The channel width is generally 5.2 km s⁻¹. The observations and reductions are described further in Paper I. We measured the integrated H i line intensity from velocity-integrated intensity maps (with units of Jy beam⁻¹ km s⁻¹) created from naturally weighted data cubes. The beam major axis FWHM (B_maj) and minor axis FWHM (B_min) for each data cube are listed in Table 3 of Paper I.

Ten of our objects have B_maj and B_min < 21''. To adjust the H i beam to match the CO(1–0) beam, we used the AIPS task CONVL to convolve the integrated intensity map so the output Gaussian beam has B_maj = B_min = 21''. To approximately match the H i beam to the CO(1–0) beam for the remaining 10 objects with B_maj > 21'' (on average, B_maj = 29'') and B_min < 21'', we either used the original integrated intensity map or convolved the map to have B_min = 21'' and the beam major axis approximately equal to the original B_maj.

Using the resulting maps, we measured the average H i line intensity in Jy beam⁻¹ km s⁻¹ within the 21'' diameter circular aperture ($\langle {I_{\rm H\,\mathsc{i}}} \rangle$) with the AIPS task IMEAN. These values are listed in Table 2. The integrated H i line intensity in K km s⁻¹ is given by

$$\begin{equation} {\langle {I_{\rm H\,\mathsc{i}}} \rangle \, [{\rm K \, km \, s^{-1}}] = \frac{6.07 \times 10^{5} \ \langle I_{\rm H\,\mathsc{i}} \rangle \, [{\rm Jy \, beam^{-1} \, km \, s^{-1}}]}{B_{\rm maj} \, B_{\rm min}}}, \end{equation} \tag{ 1 }$$

where B_maj and B_min are in arcseconds and now refer to the beam of the map on which we made the measurements. These values are listed in Table 2.

For the objects with B_maj > 21'', we assumed that the average line intensity within a 21'' beam is equal to the measured line intensity from our image with a larger beam. We estimated the uncertainty introduced by this assumption by convolving the integrated intensity maps of three objects with B_maj < 21'' such that the convolved beams match those of the 10 objects with B_maj > 21''. We measured the average line intensity within the 21'' diameter circular aperture and compared this to the true value measured from the map with a 21'' beam. We found that the line intensities in our test cases differ from the true values by up to 11% (in the case of the NGC 4519 beam, which has a B_maj = 5191) and by 4% on average. The test measurements both over and under estimate the true value depending on the emission distribution. Therefore, we included this in our uncertainty estimate, but make no correction.

The main contributors to the final uncertainty in our H i line intensities are flux calibration (5%), aliasing (up to 11% and described in Paper I), and using an image with a beam larger than 21'' to estimate the line intensity within 21'' (on average 4%). Not all objects are subject to the latter two uncertainties. Nonetheless, we conservatively assigned the quadrature sum of these uncertainties (13%) as the generic uncertainty associated with our H i line intensities within the 21'' diameter aperture.

NGC 6509 shows H i in absorption on the east side of the galaxy because it is in the foreground of the radio source 4C +06.63. The average line intensity within 21'' is unaffected because the eastern edge of the aperture and the western edge of the radio lobe are separated by about 20''.

2.2.2. Epicyclic Frequency

Hα and continuum images of the galaxies were obtained at the MDM 2.4 m Hiltner telescope over the course of four observing runs in 2007 January, 2007 November, 2007 May/June, and 2008 January. Each galaxy was observed for between 30 minutes and 2.5 hr through a pair of matched, custom-made 15 nm wide narrowband filters centered at 663 nm and 693 nm, hereafter the 663bp15 and 693bp15 filters, respectively. The Hα emission line falls within the 663bp15 bandpass for all of the galaxies in this sample. Observations were obtained with the Direct CCD camera "Echelle," which has 2048×2048 pixels. The CCD was binned over 2×2 pixels to produce a plate scale of 055 pixel⁻¹, which was well matched to the 1''–15 image quality measured on most nights. The field of view was 94. Conditions were photometric for most of the observations and a series of spectrophotometric standards were observed for flux calibration, as were a series of twilight flats. The exposure times and observation dates are listed in Table 3.

Overscan subtraction, flat fielding, cosmic-ray rejection, and bad pixel removal were performed with IRAF.⁹ All of the 663bp15 and 693bp15 images of each galaxy were registered to a common coordinate system with SCAMP (Bertin 2006) and a combined image for each filter was constructed with SWARP (Bertin et al. 2002).

Observations of spectrophotometric standard stars were used to determine the absolute flux calibration for the 663bp15 filter (including an airmass correction) and the relative throughput of the two filters. We calculated the expected ratio of stellar flux in these filters for late-type galaxies by using the SYNPHOT package in IRAF to convolve the filter transmission functions with a series of stellar population synthesis models (e.g., Bruzual & Charlot 2003) with both continuous and exponentially declining (τ = 1 Gyr) star formation histories. The range of flux ratios from these models and the relative throughput of the two filters were used to scale the 693bp15 image and subtract it from the 663bp15 image to create an Hα image for each galaxy.

We measured the Hα flux for each galaxy within the 21'' diameter circular aperture and within a circle of diameter D₂₅, the B-band major isophotal diameter at 25 mag arcsec⁻², with aperture photometry. All Hα flux measurements were multiplied by a factor of 0.75 to account for emission from [N ii] λλ6548, 6584 in the 663np15 bandpass (Kennicutt 1983). The Hα flux measurements were also corrected for Galactic extinction using the extinction law of O'Donnell (1994) assuming R_V = 3.1 and reddening values from Schlegel et al. (1998) and tabulated on NED.¹⁰ The final values within the 21'' aperture are listed in Table 2.

A number of our galaxies show substantial variation in the Hα flux within the 21'' aperture if we vary the center of the aperture by the ∼2''–3'' pointing accuracy of the IRAM 30 m. Including this uncertainty, as well as uncertainty in the [N ii] correction and absolute calibration uncertainties, we estimate the uncertainty in the Hα flux within the 21'' diameter aperture to be about 30%.

Figure 2 compares our Hα fluxes measured within D₂₅ relative to published values from Koopmann et al. (2001), James et al. (2004), Moustakas & Kennicutt (2006), and Epinat et al. (2008), where we corrected the published values for [N ii] emission and Galactic extinction if necessary. Our values are on average 20% smaller than the published values. This systematic offset is comparable to our calibration uncertainty and may be due to differences in the filter profiles, the angular extent of the apertures, and the treatment of bright stars with potentially large subtraction residuals. We note that this offset corresponds to an uncertainty in the SFR surface density of less than 0.1 dex, which is smaller than the 0.2 dex uncertainty that we assign due to the variable contribution to dust heating from non-star-forming populations.

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Fourteen of the twenty galaxies were observed as part of our Spitzer Cycle 5 Program 50102; the remaining six (NGC 337, UGC 1862, ESO 418-G008, NGC 2805, NGC 4519, and NGC 4561) were observed for various other programs. Observations for Program 50102 consisted of five, dithered observations with a frame time of 30 s in all four IRAC channels (3.6 μm, 4.5 μm, 5.8 μm, and 8 μm). Data for the other six archival data sets were comparable to our observations, with the exception that the data for UGC 1862 only include the galaxy in channels 2 and 4. In the analysis described below, we use the channel 1, 2, and 4 data.

The basic calibrated data for all 20 galaxies were processed with Sean Carey's artifact mitigation software,¹¹ which corrects for a variety of effects such as muxbleed, column pulldown/pullup, electronic banding, and first frame effect. We then created mosaics for each channel with the Spitzer Science Center's MOPEX (MOasicker and Point source EXtractor) package. This package corrects individual images for background variations and optical distortions, and then projects them onto an output mosaic image for each channel. These mosaic images were used for five measurements for each galaxy: the inclination (i), position angle of the major axis (P.A.), the PAH flux density, the 4.5 μm flux density, and the exponential disk scale length.

2.4.1. Inclination and Position Angle

2.4.2. PAH Flux Density

2.4.3. 4.5 μm Flux Density

2.4.4. Exponential Disk Scale Length

We estimated the exponential scale length of each galaxy using the IRAF ellipse task on the 4.5 μm data, allowing the center, P.A., and ellipticity to vary as a function of SMA. We fit an exponential to the mean isophotal intensity profile to derive the central surface brightness and scale length. We excluded PSF and/or bar components from the profile fit based on visual examination of the images and a provisional GALFIT (Peng et al. 2002) analysis. We created a two-dimensional image representing the ellipse profile with bmodel, subtracted the model from the original image, and found that the standard deviation in the region of the galaxy within the residual image is typically less than about 10 times the standard deviation in a galaxy-free region of the original image. Given the small scale structures present in most of the galaxies, we accepted these values and fits. The scale lengths are listed in Table 4 and are between 0.69 and 3.4 kpc.

Table 4. Derived Properties

Source	$R_{21\mbox{$^{\prime \prime }$}}$	${\rm log} \, \Sigma _{\rm H\,\mathsc{i}}$	log ΣH2	log ΣSFR	log Σ*	log M*	12 + log(O/H)	l*	κ	Qgas	Q*	Qgas + stars	log Ph
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
NGC 0337	2.11	1.33	1.18	−1.67	2.49	9.9	8.93	1.7	0.17	2.7	2.9	2.0	5.32
PGC 3853	1.16	0.78	0.67	−2.68	1.88	9.1	8.68	1.6	0.22	12	8	5	4.42
PGC 6667	2.51	1.09	0.42	−2.44	1.73	9.3	8.75	2.4	0.12	4.8	6	3.2	4.50
ESO 544-G030	1.42	1.02	<0.09	−2.83	1.51	8.6	8.46	1.1	0.12	6.0	5	2.9	4.39
UGC 1862	2.27	0.62	0.29	−2.70	1.67	9.2	8.70	1.5	0.07	7	3.0	2.3	4.04
ESO 418-G008	2.40	0.84	<0.34	−2.72	1.40	8.7	8.50	1.1	0.07	4.5	3	2.0	4.21
ESO 555-G027	2.47	0.78	...	−2.44	1.77	9.3	8.74	1.9	0.12	...	5	...	...
NGC 2805	2.85	0.74	0.92	−2.28	1.93	9.8	8.91	2.3	0.041	1.8	1.6	1.1	4.52
ESO 501-G023	0.71	0.40	<0.32	−3.16	1.01	7.8	8.00	0.72	0.12	16	8	5	3.77
UGC 6446	1.83	0.73	< − 0.008	−3.21	1.01	8.6	8.47	1.1	0.07	6.5	5	2.9	3.88
NGC 3794	1.96	0.75	0.27	−2.64	1.93	9.2	8.72	1.5	0.15	12	5	3.9	4.26
NGC 3906	1.86	0.69	0.20	−2.42	1.94	9.1	8.67	0.93	0.09	8	2.1	1.8	4.27
UGC 6930	1.73	0.66	0.72	−2.63	1.62	9.1	8.66	2.1	0.18	11	10	6	4.24
NGC 4519	2.00	0.82	0.99	−1.93	2.46	9.8	8.90	2.0	0.22	8	4	3.7	4.83
NGC 4561	1.25	1.23	0.40	−2.32	2.15	8.9	8.58	0.69	0.14	4.2	2.3	1.7	4.98
NGC 4713	1.52	0.99	1.15	−1.98	1.99	9.2	8.70	1.4	0.25	6.3	7	4.0	4.94
NGC 4942	2.90	0.88	0.69	−2.35	2.52	10.0	8.96	1.8	0.12	5.8	2.1	1.9	4.72
NGC 5964	2.52	0.71	0.53	−2.44	1.70	9.5	8.82	3.4	0.055	3.9	3	2.2	4.11
NGC 6509	2.87	0.89	1.31	−1.92	2.18	9.7	8.88	1.7	0.22	4.7	6	3.3	5.06
IC 1291	3.21	1.10	0.17	−2.19	1.35	8.9	8.60	2.1	0.18	8	13	5	4.37

Notes. Column 1: object name. Column 2: physical size of 21'' (kpc). All surface density measurements are within a 21'' diameter aperture. Column 3: H i mass surface density (M_☉ pc⁻²). The typical uncertainty is 0.06 dex. Column 4: H₂ mass surface density (M_☉ pc⁻²), computed with X_CO = 2.8 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. The typical uncertainty is 0.07 dex. Column 5: SFR surface density, computed assuming the Kroupa-type IMF from Calzetti et al. (2007; M_☉ yr⁻¹ kpc⁻²). We assume a typical uncertainty of 0.2 dex. Column 6: stellar mass surface density (M_☉ pc⁻²). We assign a typical uncertainty of 0.2 dex. Column 7: total stellar mass (M_☉). We assign typical uncertainty of 0.3 dex. Column 8: oxygen abundance derived from the total stellar mass and the mass–metallicity relation of Tremonti et al. (2004). We derive a typical statistical uncertainty of 0.11 dex, but note that strong-line metallicity methods return 12 + log(O/H) values as disparate as 0.5 dex. Column 9: stellar scale length (kpc). We estimate the uncertainty to be 20%. Column 10: epicyclic frequency (km s⁻¹ pc⁻¹), evaluated at 525. We estimate the uncertainty to be 30%. Column 11: gas stability parameter. The typical uncertainty is 30%. Column 12: stellar stability parameter. The typical uncertainty is 60%. Column 13: combined gas and stellar stability parameter. The typical uncertainty is 40%. Column 14: mid-plane pressure (K cm⁻³). The typical uncertainty is 0.17 dex.

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We assigned an error of 20% to the scale lengths based on the comparison of the scale lengths computed as described above to scale lengths computed from ellipse runs where we held the P.A. and ellipticity fixed as a function of SMA at the values from Table 1. We confirmed that scale lengths derived from the 3.6 μm data are generally consistent with the values derived from the 4.5 μm data within the uncertainty (they differ by 9% on average).

The H i surface density is given by

$$\begin{equation} \Sigma _{\rm H\,\mathsc{i}} \, [M_{\odot } \, {\rm pc^{-2}}] = 0.015 \, \langle I_{\rm H\,\mathsc{i}} \rangle \, {\rm cos}(i), \end{equation} \tag{ 2 }$$

where $\langle I_{\rm H\,\mathsc{i}} \rangle$ is the integrated H i line intensity within the 21'' diameter circular aperture in K km s⁻¹ from Section 2.2.1. We did not include a correction for He. The $\Sigma _{\rm H\,\mathsc{i}}$ uncertainty is dominated by the contribution from the line intensity uncertainty and the typical uncertainty in ${\rm log} \, (\Sigma _{\rm H\,\mathsc{i}})$ is 0.06 dex. The H₂ surface density is given by

$$\begin{equation} \Sigma _{\rm H_{2}} \, [M_{\odot } \, {\rm pc^{-2}}]\! =\! 3.2 \, \frac{X_{\rm CO}}{2.0 \times 10^{20} \, {\rm cm^{-2} \, (K \, km \, s^{-1})^{-1}}} \, I_{\rm CO} \, {\rm cos}(i), \end{equation} \tag{ 3 }$$

where X_CO is the CO-to-H₂ conversion factor and I_CO is the CO line intensity in K km s⁻¹ from Section 2.1. We used a constant X_CO of 2.8 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. We used this value rather than the Milky Way value of 2.0 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ so we can plot the Kennicutt (1998) total hydrogen star formation law relative to our data. Again, we did not include a correction for He.

The typical uncertainty in ${\rm log} \, (\Sigma _{\rm H_{2}})$ is 0.07 dex, due mainly to the CO line intensity uncertainty. We did not include uncertainty due to X_CO. Leroy et al. (2011) studied five local group galaxies and concluded that X_CO is relatively constant at the solar value for 12 + log (O/H) ≳ 8.4 and increases with decreasing oxygen abundance below 12 + log (O/H) ∼ 8.2–8.4. We have only one galaxy where we estimated that the oxygen abundance is below 12 + log (O/H) = 8.4 (Section 3.3), so we do not expect much X_CO variation in our sample.

We also use the total hydrogen surface density, $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}} = \Sigma _{\rm H\,\mathsc{i}} + \Sigma _{\rm H_{2}}$. The typical uncertainty in ${\rm log} \, (\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}})$ is 0.05 dex.

We used Hα emission to trace the unobscured SFR and PAH emission to trace the obscured SFR. The Hα emission is due to recombination in H ii regions, which are ionized by O and early B stars. PAHs are small dust grains (or large molecules) that are primarily excited by single UV photons (Sellgren 1984; Leger & Puget 1984). Because ionizing radiation is not required to heat PAHs (and in fact there is evidence that ionizing radiation destroys PAHs; Helou et al. 2004; Peeters et al. 2004), PAH emission traces lower-mass, longer-lived stars than those ultimately responsible for the Hα emission. PAH emission has been used to trace the total SFR (e.g., Zhu et al. 2008), but there is variation in the 8 μm luminosity at a given SFR due to environment, especially metallicity (e.g., Calzetti et al. 2007). Kennicutt et al. (2009) used 75 galaxies to calibrate SFR estimates based on Hα and PAH emission by comparing the combined Hα and PAH luminosity to the Hα luminosity corrected for extinction using the Balmer decrement. The authors found that SFRs calculated with Hα and PAH emission agreed with their reference SFRs with as little scatter as SFRs calculated with Hα and 24 μm emission. We used the SFR calibration of Kennicutt et al. (2009) to calculate the SFR surface density within the 21'' diameter circular aperture:

$$\begin{eqnarray} \Sigma _{\rm SFR} [M_{\odot } \, {\rm yr^{-1} \, kpc^{-2}}]&=& 7.30 \times 10^{10}(F_{\rm H\alpha } + 1.1\nonumber\\ && \times 10^{-28} \, \nu \, F_{\rm PAH}) \, {\rm cos}(i), \end{eqnarray} \tag{ 4 }$$

where F_Hα and F_PAH are the Hα flux in erg s⁻¹ cm⁻² and the PAH flux density in mJy within the 21'' diameter circular aperture, ν is the central frequency of the 8 μm IRAC band in Hz, and the constant includes the aperture area. This assumes the initial mass function (IMF) from Calzetti et al. (2007), which is similar to that presented in Kroupa (2001). The Kennicutt et al. (2009) SFR was calibrated for galaxy-averaged data, but should be appropriate for our data because we generally probe regions that are a couple of square kpc in area and should therefore contain a number of star-forming regions. Even using both Hα and PAHs to trace the SFR, galaxies with low metallicity could have underestimated SFRs because the fraction of dust mass in PAHs decreases at low metallicity, particularly below Z_☉/4 (12 + log (O/H) ≲ 8.3; Draine et al. 2007; Smith et al. 2007). In the following section, we estimate the oxygen abundance of each galaxy from the stellar mass and find that only one galaxy (ESO 501-G023) has 12 + log (O/H) < 8.3. Therefore, we do not expect significant SFR underestimates in our sample.

Kennicutt et al. (2009) discussed that the dominant source of uncertainty in their SFRs is due to varying contributions to dust heating from older stellar populations (≳ 100 Myr). Based on their suggestions, and including the fact that our sample has a limited range in morphology and therefore star formation history, we assigned a general uncertainty of 0.2 dex to our SFR surface densities. This dominates over the contribution due to Hα flux and PAH flux density measurement uncertainties.

In upcoming sections, we use the SFE defined as $\Sigma _{\rm SFR}/\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ in yr⁻¹. The typical uncertainty in the SFE is 0.2 dex.

We derived the stellar mass surface density within the 21'' diameter circular aperture and the total stellar mass from the 4.5 μm flux densities. We chose to use the 4.5 μm data over the 3.6 μm data because there are no PAH emission features in the 4.5 μm band.

To estimate the stellar mass, we used a relationship between K-band mass-to-light ratio and color from Bell & de Jong (2001), derived from stellar population synthesis modeling:

$$\begin{equation} {\rm log} \, (\Upsilon _{\ast }^{\rm K}) = 1.43(J-K_{\rm s})-1.53, \end{equation} \tag{ 5 }$$

where ϒ^K_* is the mass-to-light ratio in the K band in M_☉/L_{K, ☉}, L_{K, ☉} is the solar luminosity in the K band, and the J and K_s magnitudes are from the Two Micron All Sky Survey (2MASS; mainly the Extended Source Catalog, but three galaxies are in the Large Galaxy Atlas of Jarrett et al. 2003) and are listed in Table 2. The original relationship uses Johnson K magnitudes, but use of K_s magnitudes does not introduce significant error to the mass. This relation is a linear combination of the ϒ^K_* − (V − J) and ϒ^K_* − (V − K) relations presented in Table 1 of Bell & de Jong (2001) and we subtracted 0.15 dex to convert from the scaled Salpeter IMF that the authors use to a Kroupa (2001) IMF. For three galaxies that do not have 2MASS data, we used

$$\begin{equation} {\rm log} \, \Upsilon _{\ast }^{\rm K} = 0.21(B-K_{\rm s})-1.11 = 0.21(B-[4.5]) -1.23, \end{equation} \tag{ 6 }$$

where we used B-band magnitudes from the Third Reference Catalogue of Bright Galaxies (RC3; de Vaucouleurs et al. 1991, also listed in Table 1). This equation is similar to that used in Lee et al. (2006), but we used the Bell & de Jong (2001) Table 1 model results, we have converted to a Kroupa (2001) IMF, and we assumed (K_s − [4.5]) = 0.58. This color is the average value from the 11 galaxies in our sample where the 2MASS aperture radius ("r_m_ext") and our aperture SMA (D₂₅/2) differ by less than 40% of our SMA. Six galaxies have spuriously large/red (K_s − [4.5]) colors because the 2MASS aperture radius is much smaller than our aperture radius (as small as 20% of our aperture radius; for a similar effect, see Dale et al. 2009).

We then calculated the stellar mass within the 21'' diameter circular aperture and the total stellar mass with

$$\begin{equation} {\rm log} \, (M_{\ast }/M_{\odot }) = {\rm log} \, \Upsilon _{\ast }^{\rm K} + {\rm log} \, (L_{4.5}/L_{4.5,\odot }) - 0.4(K_{\rm s}-[4.5]), \end{equation} \tag{ 7 }$$

as in Lee et al. (2006). To calculate the luminosity of the galaxy, L_4.5, we used either the 4.5 μm flux density within the 21'' diameter circular aperture or the total 4.5 μm flux density from Table 2, the distances presented in Table 1, the absolute 4.5 μm magnitude of the Sun, M_{4.5, ☉} = 3.3 (Lee et al. 2006; Oh et al. 2008), and the zero-magnitude flux density of the 4.5 μm band, F⁰_4.5 = 179.7 Jy (Reach et al. 2005). This equation uses the fact that (K − [4.5]) ∼ 0 for the Sun. We again assumed (K_s − [4.5]) = 0.58, for all galaxies in this case. For the stellar mass surface density (Σ_*), we then divided by the deprojected area of the 21'' diameter circular aperture.

The average offset between masses computed with Equations (5) and (6) is 0.2 dex, with masses computed with Equation (6) generally smaller. This offset is smaller than our estimate of the uncertainty in the mass, described further below. The masses computed with Equation (7) above are in good agreement with masses computed using the Oh et al. (2008) relationship between K-band and 4.5 μm band mass-to-light ratios, derived from stellar population synthesis modeling.

Bell & de Jong (2001) cite uncertainties of 0.1–0.2 dex in their mass-to-light ratios, which include model and dust uncertainties and allow for small star formation bursts. The uncertainty introduced to the stellar mass by uncertainties in L_4.5 is of similar order, mainly due to distance uncertainties. We assigned a general uncertainty of 0.3 dex to the stellar masses, which is the quadrature sum of the above uncertainties. This is in agreement with the Conroy et al. (2009) estimate of typical stellar mass uncertainties from stellar population synthesis modeling. The uncertainty in the stellar mass surface density is dominated by the contribution from the mass-to-light ratio. We therefore assigned a typical uncertainty of 0.2 dex.

We estimated a representative oxygen abundance for each galaxy to help assess the accuracy and precision of our molecular hydrogen and SFR surface densities (Sections 3.1 and 3.2) and also so we can compare our data to the Krumholz et al. (2009) model (Section 4.4). We estimated the oxygen abundance with the total stellar masses calculated from the 4.5 μm flux density and the mass–metallicity relation of Tremonti et al. (2004, their Equation (3)), which assumes a Kroupa (2001) IMF, consistent with the rest of our analysis. We derive an average uncertainty in 12 + log(O/H) of 0.11 dex.

In Section 4.3, we study trends between SFE and stability, parameterized by a generalized Toomre Q parameter (Toomre 1964; Rafikov 2001). We calculated the stability parameters within the 21'' diameter circular aperture. The stability parameter for the gas and stellar components of the disk are

$$\begin{equation} Q_{\rm gas} = \frac{\kappa \sigma _{\rm gas}}{\pi {\rm G} \Sigma _{\rm gas}}, {\rm and} \end{equation} \tag{ 8 }$$

$$\begin{equation} Q_{\rm stars} = \frac{\kappa \sigma _{\ast, \rm r}}{\pi {\rm G} \Sigma _{\ast }}, \end{equation} \tag{ 9 }$$

respectively, where κ is the epicyclic frequency and σ_gas and σ_{*, r} are the gas and radial stellar velocity dispersions. Rafikov (2001) calculated the stability parameter that includes both gas and stars in a thin rotating disk:

$$\begin{equation} \frac{1}{Q_{\rm gas+stars}} = \frac{2}{Q_{\rm stars}} \, \frac{q}{1+q^{2}} + \frac{2}{Q_{\rm gas}} \, R \, \frac{q}{1+q^{2}R^{2}}, \end{equation} \tag{ 10 }$$

where q = kσ_{*, r}/κ, R = σ_gas/σ_{*, r}, and k is a free parameter that represents the wavenumber of the perturbation. In all cases, the instability condition is when Q < 1 (except, strictly speaking, Q_stars < 1.07). Romeo & Wiegert (2011) included disk thickness in the stability estimation by incorporating a factor related to the ratio of the vertical to radial velocity dispersion, but we settled on the Rafikov (2001) parameter for ease of comparison with other works.

We generally made similar assumptions for the components of Q as Leroy et al. (2008). We assumed σ_gas = 11 km s⁻¹, which is appropriate for the warm neutral medium, $\sigma _{\ast, \rm r} = 1.67 \sigma _{\ast, \rm z}$, and σ_{*, z} = (2π G l_* Σ_*/7.3)^0.5, where σ_{*, z} is the vertical stellar velocity dispersion and l_* is the stellar scale length (Tamburro et al. 2009; Shapiro et al. 2003; van der Kruit & Searle 1981; van der Kruit 1988). This σ_* equation assumes that the disk is isothermal in the z-direction, h_* is constant as a function of radius, and l_*/h_* = 7.3 (as observed by Kregel et al. 2002), where h_* is the stellar scale height. Note that the latter two assumptions are not in conflict with a sample like Dalcanton et al. (2004) because those authors found only a transition in dust scale height with circular velocity, not in stellar scale height. We used the scale lengths and stellar mass surface densities from Sections 2.4.4 and 3.3 to derive σ_{*, r} values between 9 and 78 km s⁻¹.

For Σ_gas, we multiplied the total-hydrogen surface density ($\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$) by a factor of 1.36 to include helium. For Σ_*, we used the value derived in Section 3.3. For the epicyclic frequency, we used the value from Section 2.2.2, evaluated at 525. For the wavenumber of the perturbation, k = 2π/λ, where λ is the wavelength of the perturbation, we used the common method of varying λ to find the minimum Q_{gas + stars} (Q_{gas + stars, min}) for the region (e.g., Yang et al. 2007; Yim et al. 2011). This typically results in smaller, less stable Q_{gas + stars} compared with Q_gas and Q_stars. We found λ = 0.5–3.8 kpc at Q_{gas + stars, min}. We propagated the uncertainties in κ, l_*, Σ_gas, and Σ_* to derive a typical uncertainty in Q_gas of 30%, in Q_stars of 60%, and in Q_{gas + stars} of 40%.

We also calculated the mid-plane pressure (P_h) with the following equation from Elmegreen (1989):

$$\begin{equation} P_{\rm h} \approx \frac{\pi }{2} \, G \, \Sigma _{\rm gas} \left(\Sigma _{\rm gas} + \frac{\sigma _{\rm gas}}{\sigma _{\ast, \rm z}} \, \Sigma _{\ast }\right). \end{equation} \tag{ 11 }$$

We propagated the uncertainties in l_*, Σ_gas, and Σ_* to derive a typical uncertainty in P_h of 40%.

In this section, we determine whether our galaxy sample follows the various versions of the Kennicutt–Schmidt law, i.e., the relation between the surface density of gas (atomic, molecular, or the sum of both) and SFR (Sections 4.1.1–4.1.3). In Section 4.2, we use these results to show that there is no transition in SFE at a circular velocity of 120 km s⁻¹ or at any other circular velocity probed by our sample (v_circ = 46–190 km s⁻¹). The circular velocities were derived from H i rotation curve fits in Paper I and we include them in Table 1 for convenience.

4.1.1. Atomic Hydrogen Kennicutt–Schmidt Law

Figure 3 shows the relationship between the SFR surface density and the atomic hydrogen surface density within the 21'' diameter circular aperture, which corresponds to physical diameters of 0.7–3.2 kpc. Red squares denote galaxies with v_circ < 120 km s⁻¹ and blue triangles denote galaxies with v_circ > 120 km s⁻¹. The vertical dashed line is at 9 M_☉ pc⁻² and represents the typical maximum H i surface density observed in most nearby galaxies (Bigiel et al. 2008). For comparison, we have shown as small dots the 750 pc diameter regions from the seven spiral galaxies studied in Bigiel et al. (2008) and Leroy et al. (2008). Consistent with our assumptions, Bigiel et al. (2008) used the same Kroupa-type IMF and do not include He in their gas surface densities. An important difference between our data sets is that Bigiel et al. (2008) derived their SFR surface densities from a combination of FUV (1350–1750 Å) and 24 μm data while we use Hα and PAH emission.

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The Spearman rank correlation coefficient for our data is 0.5. For comparison, the coefficient for the Bigiel et al. (2008) data is 0.4. Our data are more correlated than the Bigiel et al. (2008) data, primarily because our sample includes several galaxies with large H i surface densities.

4.1.2. Molecular Hydrogen Kennicutt–Schmidt Law

Figure 4 shows the relationship between the SFR surface density and the molecular hydrogen surface density. The symbols are as in Figure 3. We convert the Bigiel et al. (2008) $\Sigma _{\rm H_{2}}$ values to X_CO = 2.8 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ to match our assumptions. The solid line shows the Bigiel et al. (2008) fit, also converted to the above X_CO. The Spearman rank correlation coefficient for our data is 0.7, including upper limits. For comparison, the coefficient for the Bigiel et al. (2008) data is 0.8.

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Our sample of bulgeless disk galaxies, both the low- and high-v_circ objects, appears to lie offset to higher SFR surface densities relative to the Bigiel et al. (2008) fit. In what follows, we investigate the significance of and possible reasons for this offset.

Under our assumptions (He not included in the gas surface densities, X_CO = 2.8 × 10²⁰cm⁻² [K km s⁻¹]⁻¹, and the Kroupa-type IMF), the Bigiel et al. (2008) fit is log (Σ_SFR $[M_{\odot } \, {\rm yr^{-1} \, kpc^{-2}}]) = a + b\, {\rm log} \, (\Sigma _{\rm H_{2}} \, [M_{\odot } \, {\rm pc^{-2}}])$ with a = −3.3 ± 0.2 and b = 1.0 ± 0.2. We fit for the intercept of our data assuming the Bigiel slope of b = 1.0 and including our upper limits in the fit and find a = −3.0 with an rms of 0.3 dex.

We determine the significance of the offset between our sample of bulgeless disk galaxies and the Bigiel et al. (2008) fit to the molecular hydrogen star formation law by randomly selecting nineteen measurements from Bigiel et al. (2008), allowing multiple selections of the same point (Press et al. 1992). We assume a slope of 1.0, calculate the intercept, and repeat this process 10⁶ times. We find that the probability of measuring a intercept greater than or equal to −3.0 is 4 × 10⁻⁶.

We can exclude two possible reasons for the offset. First, the offset is not likely due to the measurements being central values because we confirmed that the centers of the Bigiel et al. (2008) galaxies are not offset from the general trend (this can also be seen in Figure 10 of Bigiel et al. 2008). Second, our assumption of a single CO-to-H₂ conversion factor probably does not lead to underestimated molecular surface densities because our sample includes only one galaxy for which X_CO may be underestimated because of a low oxygen abundance, as discussed in Section 3.1. Furthermore, we do not find that galaxies with lower stellar mass, and by implication lower-(O/H), are more offset from the Bigiel et al. (2008) fit. Note, however, that Schruba et al. (2011) did observe that lower oxygen abundance (down to 12 + log(O/H) = 8.25) galaxies are offset to higher molecular SFE in their study of 33 galaxies with directly measured oxygen abundances. They discussed that more observations are needed to determine whether the offset is due to X_CO variation or whether it represents true SFE variation.

The quoted Bigiel et al. (2008) intercept uncertainty (0.2 dex) takes into account variations in star formation tracers, uncertainty introduced by estimating the CO(1–0) line intensity from CO(2–1) data, and scatter in the data. The first of these is particularly relevant in comparing our data sets because we trace star formation with Hα and PAH data and Bigiel et al. (2008) trace star formation with FUV and 24 μm data. Our intercept is larger than the Bigiel et al. (2008) intercept by only 1.2σ, if we take their error bar as σ. In summary, our data are significantly offset from the Bigiel et al. (2008) fit to the molecular hydrogen star formation law in terms of statistical uncertainties, but are nearly consistent if systematic errors are included.

Kennicutt et al. (2007) discussed that offsets are expected between star formation laws derived from observations at different spatial resolution if the power-law index of the star formation law is not equal to one (Σ_SFR∝Σ^N_gas with N ≠ 1). In transitioning from high to lower resolution, the SFR and gas surface density will be decreased by approximately the same factor. If N = 1, the lower resolution data will still lie on the same relation as the higher resolution data, but at lower gas and SFR surface densities. In contrast, if N > 1 (N < 1), the lower resolution observations will be positively (negatively) offset from the higher resolution observations. This effect could contribute to our observed offset if N > 1 because our data probe up to 3.2 kpc scales and the Bigiel et al. (2008) data have 750 pc resolution. We cannot provide a firm explanation for the offset between our data and the Bigiel et al. (2008) fit, but possible reasons for the offset include the use of different star formation tracers and resolutions, X_CO variation, and true SFE differences.

4.1.3. Total Hydrogen Kennicutt–Schmidt Law

Figure 5 shows the relationship between the SFR surface density and the total hydrogen surface density. The vertical dashed line again shows the typical maximum H i density that is observed in most nearby galaxies. The Spearman rank correlation coefficient for our data and also the Bigiel et al. (2008) data is 0.8. This correlation is stronger than the correlation between SFR surface density and either atomic or molecular hydrogen surface density. In their study of star formation in the atomic-dominated regime, Schruba et al. (2011) found that the rank correlation coefficient between SFR surface density and total hydrogen surface density is similar to or somewhat larger than the value for the correlation with the molecular hydrogen surface density. However, they noted that data can be correlated with a strong rank statistic even if the parameters are related by different functions in the atomic- and molecular-dominated regimes. As in the Schruba et al. study, our stronger total hydrogen correlation is not likely related to fundamental physics.

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The solid black line shows the Krumholz et al. (2009) model with 0.4 Z_☉ metallicity. The model assumes a clumping factor, c, that is the inverse of the filling factor of ∼100 pc sized atomic–molecular complexes in the beam. This value is not constrained well by data, so we assume the same value as Krumholz et al. (2009): c = 5. The magenta dot-dashed line shows the Kennicutt (1998) fit to a sample of normal and starburst galaxies, where the measurements are averages over the entire optical disk. The blue dotted line shows the Kennicutt et al. (2007) fit to regions in M51 that were studied at 520 pc resolution. Kennicutt et al. (2007) attribute the offset between the blue and magenta lines to dilution effects when the power-law index of the star formation law is not 1.0 (as discussed in Section 4.1.2).

In Section 1, we discussed that there could be an SFE transition in bulgeless disk galaxies at v_circ = 120 km s⁻¹, depending on the star formation model assumed. In this section, we show that there is no offset between the low- and high-v_circ galaxies on the star formation law. In Section 5, we interpret this result to discuss the relationship between SFE and the scale height of the cold ISM.

In Figures 3–5, there is a slight trend for low-v_circ galaxies to lie at lower surface densities compared with high-v_circ galaxies, with average low- versus high-v_circ surface densities as follows: ${\rm log} \langle \Sigma _{\rm H\,\mathsc{i}} \rangle = 0.9$ versus 1.0, ${\rm log} \langle \Sigma _{\rm H_{2}} \rangle < 0.6$ versus = 0.9, ${\rm log} \langle \Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}} \rangle = 1.1$ versus 1.2, and log〈Σ_SFR〉 = −2.4 versus −2.2. However, the differences are not significant; the Kolmogorov–Smirnov (K-S) test probabilities that the low- versus high-v_circ galaxies are drawn from the same population in $\Sigma _{\rm H_{2}}$, $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$, and Σ_SFR are 0.04, 0.2, and 0.2, respectively.

Figure 6 shows the distribution of SFE in the low-v_circ (red solid) and high-v_circ (blue dashed) galaxies. If the slope of the star formation law is not 1.0, a sample that follows the relation would have SFE that varies as a function of $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$, which will tend to broaden the distributions. However, our low- and high-v_circ objects have similar $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ distributions, so this should not lead to spurious offsets between the samples. Furthermore, the $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ range is not so great that it would hide any differences between the samples. We find no significant difference between the SFE distributions, with a K-S test probability of 0.8 that the low- and high-v_circ galaxies are drawn from the same population.

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We conclude that our sample of bulgeless disk galaxies does not show a strong transition in SFE at v_circ = 120 km s⁻¹, where Dalcanton et al. (2004) concluded that there is a strong transition in dust scale height. In Figure 7, we show that there is no circular velocity at which there is a transition in SFE. Furthermore, there is no strong trend of SFE with circular velocity within the range we probe. Finally, we find no difference in the ratio of the molecular to atomic surface density between the low- and high-v_circ galaxies (the K-S test probability that the samples are drawn from the same R_mol population is 0.4) nor do we find a transition in R_mol at any circular velocity (Figure 8).

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In this section, we address whether our sample shows a transition or trends in stability or mid-plane pressure, using the Q_gas, Q_stars, Q_{gas + stars, min}, and P_h values calculated in Section 3.4. Figure 9 shows the SFE versus Q_gas (left), Q_stars (middle), and Q_{gas + stars, min} (right). We find no unstable regions. The SFE generally decreases with larger, more stable Q values, although the correlation is not strong. For SFE versus Q_gas, Q_stars, and Q_{gas + stars, min}, we find Spearman rank correlation coefficients of −0.2, −0.3, and −0.2, respectively. The sign of this correlation is as expected, but one might instead have expected a sharp decrease in SFE when Q rises above 1. These results are qualitatively similar to those in Leroy et al. (2008).

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Dalcanton et al. (2004) concluded that galaxies with v_circ < 120 km s⁻¹ are generally stable while galaxies with v_circ > 120 km s⁻¹ are generally unstable, especially in the central (r < l_*) regions. Contrary to these results, we see no evidence for a transition in stability at v_circ = 120 km s⁻¹; the K-S test probability that the low- and high-v_circ galaxies are drawn from the same population of Q_gas, Q_stars, and Q_{gas + stars, min} is 0.5, 0.2, and 0.8, respectively.

There are two principal differences between our assumptions for the stability inputs and those in Dalcanton et al. (2004). First, we use a constant gas velocity dispersion and they used the quadrature sum of the velocity dispersion of atomic and molecular gas (10 and 5 km s⁻¹, respectively), weighted by the relative mass surface densities of the components. We use these assumptions and still find no difference in stability between the low- and high-v_circ galaxies. Second, Dalcanton et al. (2004) estimated the molecular hydrogen surface densities for their sample from a scaling with circular velocity ($\Sigma _{\rm H_{2}} = (v_{\rm circ}/47.1 \, {\rm km \, s^{-1}})^{2.49}$). This scaling relation was derived from a similar sample of galaxies with $\Sigma _{\rm H_{2}}$ values from Rownd & Young (1999) and v_circ estimated from single-dish H i data. Molecular hydrogen surface densities calculated using the scaling with v_circ are too large by a factor of six for our sample of high-v_circ galaxies and too large by a factor of two for our sample of low-v_circ galaxies. This discrepancy leads to smaller, less stable Q_gas and Q_{gas + stars} in the high-v_circ galaxies relative to the low-v_circ galaxies. However, even under these assumptions there is no significant difference between the low- and high-v_circ galaxies in their Q values. Note that the difference in our molecular hydrogen surface densities compared with the values derived from the scaling with v_circ may also be related to resolution differences. Our $\Sigma _{\rm H_{2}}$ measurements are within the central 21'' while the Rownd & Young (1999) data have 45'' resolution and a similar distance distribution.

We test whether a stability transition occurs at any circular velocity in Figure 10, where we plot Q_{gas + stars, min} versus v_circ. There is no circular velocity below which the regions are stable and above which the regions are unstable. If any trend is present, it is that higher circular velocity objects are more stable.

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We plot the SFE versus the mid-plane pressure in Figure 11. As seen in Leroy et al. (2008), we find an increase in SFE with increasing mid-plane pressure, but we do not probe high enough P_h values or have enough data points to sample the constant-SFE region of the diagram that is clear in Leroy et al. (2008). We find no difference between the low- and high-v_circ galaxies in their P_h distributions, with a K-S test probability of 0.7 that they are drawn from the same population. Even focusing only on galaxies with the same $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ (within the uncertainty in the parameter), the median pressure is the same in low- and high-v_circ galaxies.

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In this section, we use the oxygen abundance estimated from the stellar mass to study how star formation depends on metallicity and compare these results to recent theoretical work by Krumholz et al. (2009). At a given total gas surface density, the Krumholz et al. (2009) model predicts lower SFE at lower metallicity because H₂ survival requires a higher column density as H₂ self-shielding becomes more important than shielding by dust. This metallicity dependence distinguishes the model from other leading models, such as the model where mid-plane pressure determines where the ISM is molecular.

The left panel of Figure 12 shows the total hydrogen star formation law, as plotted in Figure 5, except the galaxies are divided into low-(O/H) and high-(O/H), with the division at 12 + log(O/H) = 8.7. The orange solid and purple dashed lines show the model of Krumholz et al. (2009) at 0.1 Z_☉ and 1.4 Z_☉, respectively. These metallicities correspond to the extrema of our data set, assuming Z/Z_☉ = (O/H)/(O/H)_☉ and 12 + log(O/H)_☉ = 8.86 (Delahaye & Pinsonneault 2006). We see no evidence that the low- and high-(O/H) galaxies cluster toward the low- and high-Z models, respectively.

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The right panel of Figure 12 shows the offset of our data from the 0.4 Z_☉ Krumholz et al. (2009) model, which provides the best fit to our data set as a whole (although note that we would expect the best fit to be the 0.7 Z_☉ model, as that corresponds to the average (O/H) of our full sample). The orange solid and purple dashed lines show the distributions of our low-(O/H) and high-(O/H) galaxies, as divided in the left panel. If our data clearly followed the Krumholz et al. (2009) model, high-(O/H) galaxies would be positively offset from the 0.4 Z_☉ model and low-(O/H) galaxies would be negatively offset. We find no significant difference between the offset of the low- versus high-(O/H) galaxies (the K-S test probability that the samples are drawn from the same offset population is 0.14). A second-order effect is that the offset from the 0.4 Z_☉ model is expected to decrease with increasing $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$. This may be related to the narrower offset distribution for the high-(O/H) galaxies, which tend to have higher $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$.

Figure 13 shows that there is a correlation between SFE and oxygen abundance in our data. For comparison, the line represents the Krumholz et al. (2009) model, where the SFE depends on metallicity, Σ_gas, and the filling factor. We have plotted the Krumholz et al. (2009) model with ${\rm log} \, (\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}) = 0.8$ (no He). This is a low $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ value compared to our data. Higher $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ models reach constant SFE at lower metallicity. The mismatch between the average $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ in the data and the best-fit $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ used in the model is likely due to our assumptions for X_CO and the IMF in the data, our filling factor assumption in the model, and to uncertainties in the metallicity scale. For the latter, various strong-line metallicity methods return 12 + log(O/H) values as disparate as 0.5 dex (Kewley & Ellison 2008), and it is unknown which calibration best aligns with the solar value. Therefore, the model line in Figure 13 only represents the expected trend of lower SFE at lower metallicity, at a given Σ_gas. At this level, the data do show the expected trend, with a Spearman correlation coefficient of 0.6. However, upon closer inspection, this agreement is mainly due to the fact that lower-(O/H) galaxies tend to have lower $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ (this is most obvious in Figure 12, although also note that the K-S test probability is 0.05 that the $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ distributions of the low- and high-(O/H) samples are drawn from the same population, which is not strong evidence for a difference). Because of their lower $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ values, the low-(O/H) galaxies are closer to the atomic-dominated regime of the star formation law where SFEs are lower. This is of course consistent with the Krumholz et al. (2009) model, but is not a discriminating test of the model because many other properties correlate with gas surface density (for examples, see Leroy et al. 2008). A sample with a range of directly measured metallicities within a small range of gas surface density would provide a more discriminating test. In summary, we see no clear evidence to support the Krumholz et al. (2009) model, but our data are also not inconsistent with it.

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In this section, we investigate the argument that the dust structure transition observed by Dalcanton et al. (2004) is due to a transition in dust scale height rather than due to a transition in the amount of dust present. Dalcanton et al. (2004) came to this conclusion because the dust structure transition occurs over a narrow range in circular velocity, where a large change in the DGR, and therefore dust content, is unexpected. All but one galaxy in our sample are detected at Infrared Astronomical Satellite (IRAS) 60 and 100 μm, which indicates that there is at least some dust in the low-v_circ galaxies. We estimated a DGR proxy as the ratio of the total infrared luminosity (L_TIR), calculated with IRAS 25, 60, and 100 μm data (Moshir et al. 1990) and the Dale & Helou (2002) relation, to the combined atomic and molecular hydrogen mass. We find no transition in this DGR proxy at v_circ = 120 km s⁻¹, nor do we find any correlation between the DGR proxy and circular velocity. L_TIR is not necessarily proportional to the dust mass because the dust temperature may not be the same for all the galaxies; nevertheless, there is clearly a significant amount of dust in the low-v_circ galaxies.

Two recent studies have found convincing evidence that low-v_circ spirals have large dust scale heights. Seth et al. (2005) carried out a resolved stellar population study of six edge-on, late-type spirals with v_circ = 67–131 km s⁻¹ and found that the scale height of the young (≲ 10⁸ yr) stellar population, which presumably formed from the cold ISM, is larger than in the Milky Way. MacLachlan et al. (2011) modeled the spectral energy distribution (SED) of three edge-on, low-surface brightness galaxies with v_circ = 88–105 km s⁻¹ and found that a significant amount of dust must be present to account for the FIR (70 and 160 μm) emission, but the dust must have a large scale height such that it does not significantly obscure the optical emission. The authors concluded that the galaxies have dust scale heights greater than or equal to the stellar scale heights. This is in contrast to modeling studies of high surface brightness galaxies, which concluded that the dust scale height is about half the stellar scale height (e.g., Xilouris et al. 1999).

Hunter & Elmegreen (2006) provided an alternative explanation for the dust structure transition observed by Dalcanton et al. (2004). The authors studied a large sample of irregular galaxies and found that the average B-band surface brightness is smaller than in higher-mass spiral galaxies. Based on this result, they suggested that lower stellar surface density is the cause of the larger scale heights in low-v_circ galaxies. They discussed that disk stability could correlate with the dust structure but not be the cause of the transition because the gas scale height (h_gas∝(σ²_gas/G ρ), where ρ is the mass volume density of gas and stars; Blitz & Rosolowsky 2004) shares many of the same parameters with the stability parameter.

The authors discussed that dust opacity may also contribute to the observed difference in dust structure because of two effects. First, lower-v_circ objects should have lower metallicity, and therefore lower dust content. Second, the scale length of a lower-mass galaxy is smaller and therefore a low-v_circ, edge-on galaxy will have less depth from the edge to the center over which to accumulate dust column density compared with a high-v_circ galaxy. Note that no scale height transition is needed in this interpretation. However, dust opacity is not likely the sole cause of the dust structure transition because of the reasons listed above. Hunter & Elmegreen (2006) suggested that the observed dust structure transition is likely due to a combination of stellar surface density and dust opacity. The authors agreed that a scale height transition does occur, so if this interpretation is correct, we can still constrain the effect of scale height differences on SFE.

In Sections 5.2 and 5.3, we assume that the dust scale heights in low-v_circ galaxies are a factor of two larger than in high-v_circ galaxies. Dalcanton et al. (2004) estimated this factor by examining the dust morphology in Hubble Space Telescope images of a couple edge-on, late-type galaxies. This factor is consistent with the SED modeling results of MacLachlan et al. (2011) and Xilouris et al. (1999) if the stellar scale height distributions of low- and high-v_circ galaxies have significant overlap, which Dalcanton et al. (2004) found to be true (although there are low-v_circ galaxies where the stellar scale height is about half the value in high-v_circ galaxies, in which case the dust scale heights may be comparable).

In Section 4.2, we found no transition in SFE at v_circ = 120 km s⁻¹ or at any circular velocity probed by our sample. In this section, we interpret this result to discuss the relationship between SFE and the scale height of the cold ISM. For this discussion, we make a number of assumptions. First, we assume that our sample of bulgeless disk galaxies is similar to that of Dalcanton et al. (2004) in that galaxies with v_circ > 120 km s⁻¹ have narrow dust lanes while galaxies with v_circ < 120 km s⁻¹ have no obvious dust lanes. This is a reasonable assumption because it was our main consideration in choosing the sample, but because our galaxies are moderately inclined rather than edge-on, we are unable to directly measure the vertical dust structure. Second, as in Dalcanton et al. (2004), we assume that the dust scale heights in the low-v_circ galaxies are larger than in the high-v_circ galaxies (Section 5.1 addresses this assumption). Finally, we assume that the molecular gas and dust scale heights are comparable.

Dalcanton et al. (2004) suggested that there may be an SFE transition associated with the dust scale height transition at v_circ = 120 km s⁻¹. The authors gave an example that assumes the true star formation law is a correlation between the volume density of gas (ρ_gas) and the SFR volume density (ρ_SFR): ρ_SFR∝ρ^N_gas. The larger scale heights of low-v_circ galaxies lead to lower gas volume densities. Depending on the index, N, a low-v_circ galaxy with the same gas surface density as a high-v_circ galaxy can have a lower SFR surface density and therefore a lower SFE. To consider this point, we first assume the same Σ_gas for a low- and high-v_circ galaxy and the following relationships between the surface and volume density of gas and SFR: Σ_gas∝ρ_gas h, and Σ_SFR∝ρ_SFR h, where h is the scale height of the star-forming gas and newly formed stars and the proportionality constant is the same for both relationships. We set β equal to the ratio of dust scale heights in low-v_circ (lv) versus high-v_circ (hv) galaxies: β = h_lv/h_hv. Dalcanton et al. (2004) very approximately estimated this ratio to be about two. Under these assumptions, Σ_{SFR, hv} = β^{N − 1} Σ_{SFR, lv}, where Σ_{SFR, hv} and Σ_{SFR, lv} are the SFR surface densities of the low- and high-v_circ galaxy with the same Σ_gas. Dalcanton et al. (2004) discussed the case where β = 2 and N = 1.5, where we expect the high-v_circ galaxy to have a Σ_SFR that is a factor of 1.4 larger than the low-v_circ galaxy. Note that if N = 1 there is no expected difference in SFR surface density.

In the star formation law plots of Figures 3–5, we are sensitive to offsets in intercept between the low- and high-v_circ samples that are greater than the uncertainty in the intercept, which is ∼0.3–0.4 dex, assuming an uncertainty in Σ_SFR of 0.2 dex and the number of galaxies in our low- and high-v_circ samples. In the case discussed by Dalcanton et al. (2004), we expect the low-v_circ galaxies to be offset to lower Σ_SFR by 0.15 dex. Therefore, we cannot exclude offsets at the level expected by Dalcanton et al. (2004).

The star formation law assumed above is not likely correct given that recent studies have found a strong molecular star formation law and no atomic gas star formation law. However, we can also determine the expected Σ_SFR offset if the molecular fraction is set by the mid-plane pressure and the molecular SFE is constant (see also Section 1). We assume the same $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ for a low- and high-v_circ galaxy, R_mol∝P_h, and $P_{\rm h} \propto \rho _{\rm H\,\mathsc{i}+{\rm H}_{2}} \sigma _{\rm gas}^{2}$, where $\rho _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ is the total hydrogen volume density and the velocity dispersion of the gas is constant. In this scenario, $\rho _{\rm H\,\mathsc{i}+{\rm H}_{2}}$, P_h, and R_mol are lower by a factor of β in the low-v_circ galaxy compared to the high-v_circ galaxy. The expected offset in Σ_SFR depends on R_mol, which varies from 0.1 to 2.7 in our sample. If β = 2 and R_mol of the high-v_circ galaxy is 0.1 (2.7), we expect Σ_SFR to be lower by 0.3 dex (0.1 dex) in the low-v_circ galaxy compared with the high-v_circ galaxy. Given our uncertainties, this level of offset would also be difficult to detect. However, we can reject our assumption that P_h is lower by a factor of two in low-v_circ galaxies. This is evident from Section 4.3, where we found no significant difference in the mid-plane pressure distributions of the low- and high-v_circ galaxies, even when considering only objects with the same $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$. Furthermore, we found no difference in the R_mol distributions of the low- versus high-v_circ galaxies (Section 4.2).

The two scenarios explored above predict offsets in intercept between the low- and high-v_circ samples that are less than the uncertainty. Therefore, we cannot clearly exclude these options. Nonetheless, our data show no evidence for a strong transition in SFE at any circular velocity. A simple interpretation that is consistent with our data is that low-v_circ galaxies have a lower number of molecular clouds per unit volume compared with high-v_circ galaxies at the same $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$, but lower only by the ratio of the cold ISM scale heights in low- versus high-v_circ galaxies. This results in the same total number of molecular clouds within the beam for a low- and high-v_circ galaxy at the same $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$ and thus gives the same $\Sigma _{\rm H_{2}}$ and Σ_SFR (assuming the molecular clouds have the same density and that we average over evolutionary effects). Note that the above applies in the molecular-dominated regime. We have few data points below $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}} \sim 9 \, M_{\odot } \, {\rm pc}^{-2}$, but expect that Σ_SFR can vary substantially for galaxies with the same $\Sigma _{\rm H\,\mathsc{i}+{\rm H}_{2}}$, depending on the physical processes that affect the molecular fraction (e.g., those processes discussed in Krumholz et al. 2009 and Ostriker et al. 2010).

In conclusion, we interpret our result that there is no transition in SFE at any circular velocity as evidence that scale height differences at the level of about a factor of two do not significantly affect the molecular fraction or SFE in bulgeless disk galaxies. However, offsets in SFE below our uncertainty level are still possible.

Dalcanton et al. (2004) very approximately estimated that the dust scale heights of low-v_circ galaxies are about a factor of two larger than high-v_circ galaxies. Assuming our sample has a similar range in scale height, our results indicate that these scale height differences, which lead to gas volume density differences also at the level of a factor of about two, do not lead to obvious differences in the SFE. Our results favor star formation models where small scale physical processes are more important than processes that act on larger scales, of order the dust and cold gas scale height (10 s to 100 pc). Based on their comparison to many star formation models without an obvious favorite, Leroy et al. (2008) discussed that physics below their resolution of 750 pc is likely most important for determining the SFE. We contribute with a further constraint that the SFE is likely affected primarily by processes that act on scales smaller than the cold gas and dust scale height.

We cannot exclude all star formation models that include large scale physics because there are processes that may affect star formation but are neither affected by nor affect the scale height. For example, our sample has no power to constrain the effects of large scale radial processes, like shear (see, e.g., Hunter et al. 1998), on star formation. In addition, star formation may be affected by environmental properties that depend on the gas volume density but also depend on other variables that counteract a variable volume density. For example, the pressure in the ISM is related to the gas volume density and velocity dispersion: P∝ρ_gas σ²_gas. While we expect the volume density to be lower in low-v_circ galaxies, the velocity dispersion may be larger. With the right combination of ρ_gas and σ_gas, there could be no difference in pressure between low- and high-v_circ galaxies. While the latter argument illustrates our limitations, we note that there is currently no strong observational reason to assume different gas velocity dispersions between the low- and high-v_circ galaxies. Tamburro et al. (2009) found some variation in the central H i velocity dispersion in 11 galaxies ranging from early-type spirals to irregulars, but more observations are needed.

Would leading star formation models have predicted a difference in SFE in galaxies with scale heights that differ by about a factor of two? In the Krumholz et al. (2009) model, Σ_SFR is a function of metallicity, Σ_gas, and the beam filling factor of ∼100 pc sized atomic–molecular complexes. There is no direct dependence on the scale height, so we would not expect a transition in SFE unless there is a transition in the metallicity or filling factor with scale height. We find no transition in oxygen abundance at any circular velocity in our data, but there should be a correlation between these two properties given the mass–metallicity relation (e.g., Tremonti et al. 2004), which we also do not see. The filling factor of star-forming complexes is not well constrained, although there is no a priori reason to suppose that it would be different in low- versus high-v_circ objects. Krumholz et al. (2009) do not predict a difference in SFE in galaxies with different scale heights and the fact that we did not find a transition in SFE at v_circ = 120 km s⁻¹ is not a strong constraint on this model.

In the model where the mid-plane pressure sets the molecular fraction and the molecular SFE is constant, the pressure is proportional to the gas volume density, which we expect to vary between the low- and high-v_circ galaxies. If the gas velocity dispersion is fixed, we would expect the low-v_circ galaxies to have lower molecular to atomic surface density ratios and lower SFEs relative to the high-v_circ galaxies. However, we found no difference between the mid-plane pressure, molecular to atomic surface density ratio, or SFE distributions of the low- and high-v_circ galaxies. If our assumptions are correct, our result is inconsistent with this model. However, the offset in SFE expected for galaxies with cold gas scale heights that differ by about a factor of two may be less than our SFE uncertainties. Furthermore, the Ostriker et al. (2010) model relates the molecular fraction (or in their terms, the fraction of gas in gravitationally bound complexes) to the pressure of the diffuse component of the ISM. We have a constraint only on the scale height and volume density of the cold component of the ISM; therefore the Ostriker et al. (2010) model may not predict molecular fraction and SFE differences in our sample. In general, our results are somewhat more consistent with local models of star formation, like the Krumholz et al. (2009) model, but we do not find conclusive evidence for or against either the Krumholz et al. (2009) or Ostriker et al. (2010) model.

One final matter to address is whether central measurements are sufficient to determine if there is a transition in molecular fraction, SFE, and/or stability at the dust structure transition of v_circ = 120 km s⁻¹. One might question the use of central measurements because dust structure, SFE, and stability may be affected by the higher gas and stellar densities and shorter dynamical times characteristic of these regions. The only approach that will fully address this concern is to obtain off-center measurements of the above properties. Our single-beam CO data currently limit us from carrying out this analysis. Meanwhile, there is some evidence that our central pointings are sufficient to address these questions. First, our measurements trace a significant fraction of the disk: the 21'' aperture probes physical scales of 0.7–3.2 kpc, which are similar to the disk scale lengths of our sample (0.7–3.4 kpc). Second, the central regions of bulgeless galaxies are more morphologically and kinematically similar to the outskirts than in a galaxy with a bulge. Finally, we expect a galaxy to be less stable against gravitational collapse in the center compared to the outer disk because the gas and stellar surface densities are larger. However, we find that both the low- and high-v_circ galaxies are stable. This suggests that there would also not be a stability transition at v_circ = 120 km s⁻¹ in off-center measurements because both the low- and high-v_circ galaxies would be more stable.