The Volumetric Star Formation Law in the Almost Edge-on Galaxy NGC 4302 Revealed by ALMA

We observe the almost edge-on (i $\sim$ 90 degrees) galaxy NGC 4302 using ALMA (CO) and VLA (H I) to measure the gas disk thickness for investigating the volumetric star formation law (SFL). The recent star formation rate (SFR) is estimated based on a linear combination of IR 24 micron and H$\alpha$ emissions. The measured scale heights of CO and H I increase significantly with radius. Using the scale heights along with the vertically integrated surface densities, we derive the mid-plane volume densities of the gas ($\rho_{\rm HI}$, $\rho_{\rm H_2}$, and $\rho_{\rm gas}$ = $\rho_{\rm HI}$+$\rho_{\rm H_2}$) and the SFR ($\rho_{\rm SFR}$) and compare the volumetric SFL ($\rho_{\rm SFR} \propto \rho_{\rm gas}^n$) with the vertically integrated SFL ($\Sigma_{\rm SFR} \propto \Sigma_{\rm gas}^N$). We find tight power-law correlations between the SFR and the gas (H I, H$_2$, and the total gas) in both volume and surface densities. The power-law indices of the total gas and H I for the volumetric SFL are noticeably smaller than the indices for the vertically integrated SFL while the H$_2$ indices for both cases are similar to each other. In terms of the star formation efficiency (SFE), we find that the molecular and total gas SFEs are roughly constant, while the atomic SFE is clearly decreasing with radius in both cases.


INTRODUCTION
A power-law correlation between the SFR (Σ SFR ) and gas (e.g., Σ H2 , Σ HI , Σ gas ) based on surface densities in nearby galaxies is reported by a number of observational studies (e.g., Wong & Blitz 2002;Bigiel et al. 2008;Schruba et al. 2011;Yim et al. 2014Yim et al. , 2016) ) following the work by Kennicutt (1998), who demonstrated the correlation between Σ SFR and Σ gas (=Σ H2 +Σ HI ) averaged over the disks: Σ SFR ∝ Σ 1.4  gas .However, the volume (not surface) density is the relevant quantity for the Jeans instability and the free-fall time scale, and indeed the original Schmidt (1959) star formation law was based on volume density.Krumholz et al. (2012) showed that the volumetric star formation law is well matched to observational data including Milky Way molecular clouds, Local Group galaxies, and high-redshift galax-Corresponding author: Kijeong Yim kijeong.yim@gmail.comies.Recently, Bacchini et al. (2019) investigated the relationship between gas and the SFR volume densities estimated from hydrostatic equilibrium and found a strong correlation between them with a smaller scatter compared to the vertically integrated star formation law (Kennicutt-Schmidt law) in nearby disk galaxies.Bacchini et al. (2020) suggested that the volumetric SFL is valid even in dwarf galaxies and outer regions of spiral galaxies where the vertically integrated SFL appears to break down due to H I predominance and low metallicity in the regions.In our previous study (Yim et al. 2020), we investigated the volumetric SFL by measuring the gas disk thickness directly from the edgeon galaxies observed by the Combined Array for Research in Millimeter-wave Astronomy (CARMA) in contrast to volumetric SFL studies using the inferred disk thickness from hydrostatic equilibrium (e.g., Abramova & Zasov 2008;Bacchini et al. 2019Bacchini et al. , 2020)).However, the CO disk thickness is not well-resolved by CARMA and the angular resolution is about 3 , corresponding Yim et al. to a physical scale of ∼200 pc in the edge-on galaxy sample.For this reason, we use the Atacama Large Millimeter/submillimeter Array (ALMA), providing unprecedented sensitivity and resolution, to resolve the CO disk structure for studying the volumetric SFL.
NGC 4302 is an excellent target, allowing us to measure the disk thickness in the most straightforward way since its inclination is almost 90 • (Heald et al. 2007;Zschaechner et al. 2015).It is located at a distance of 14 Mpc based on a local flow model by Mould et al. (2000) with H 0 = 73 km s −1 Mpc −1 .NGC 4302 has a companion galaxy NGC 4298, but the disk of NGC 4302 is not obviously distorted by the interaction (Rand 1996;Howk & Savage 1999).On the other hand, Zschaechner et al. (2015) detected a faint H I bridge between NGC 4302 and NGC 4298 in a few channel maps, probably due to ram pressure stripping by the intracluster medium in the Virgo Cluster.

OBSERVATIONS AND DATA REDUCTION
We observed 12 CO (J = 1 → 0) emission toward NGC 4302 using ALMA (Cycle 7) 12m in array configurations C43-4 (6.7 hours) and C43-1 (1.9 hours) with a twentyfive point mosaic in 2019 October 20th -2020 March 4th.Using a pipeline script of the Common Astronomy Software Applications (CASA) package (McMullin et al. 2007) provided by ALMA, we calibrated separately the visibility data in different configurations.The calibrated data were combined using the CASA task CON-CAT and imaged using the CASA task TCLEAN with Briggs weighting (ROBUST = 0.5).A primary beam correction was applied to the cleaned image using IMP-BCOR.The combined image has an angular resolution of 1. 42 × 1. 16 (P.A. = −39.2• ) and a velocity resolution of 5 km s −1 .The rms noise per channel is ∼1.3 mJy Beam −1 (∼72.6 mK). Figure 1 (top) shows the integrated intensity map of the CO cube.In order to place the redshifted side (southern disk) on the right, the map is rotated by 90.93 • based on a position angle of -0.93 • that we measured from Spitzer 3.6 µm image using the MIRIAD task IMFIT.The galactic center (α = 12 h 21 m 42.s 32 and δ = 14 • 35 52.23 ) is located at x = 0 and z = 0 in the map.We masked the CO cube to reduce noise in the integrated intensity map by blanking regions below a 3σ threshold in a cube smoothed to FWHM = 3 .The total flux of the integrated intensity map is ∼650 Jy km s −1 .The sensitivity falls to 50% at x-offset of ∼120 .In order to examine the flux recovery, we compare the ALMA observations with 45 m single-dish NRO (Nobeyama Radio Observatory) observations (Komugi et al. 2008) in the same region over the velocity range of 990-1170 km s −1 , chosen by Komugi et al. (2008) to calculate the integrated intensity.The ALMA data are convolved to the circular beam of 16 , which is the angular resolution of the NRO observations.The integrated intensities over the same region from 990 km s −1 to 1170 km s −1 are 25 K km s −1 for NRO and 16 K km s −1 for ALMA.This indicates that the ALMA observations recover about 64% of the flux.
The H I observations were carried out with VLA in B configuration in 2009 (Zschaechner et al. 2015) and C configuration in 2005 (Chung et al. 2009).We used the CASA package tasks FLAGCMD (mode='manual') for flagging bad data, SETJY for setting the flux density of the amplitude calibrator, BANDPASS for the bandpass correction, GAINCAL for the gain calibration, FLUXSCALE for bootstrapping the flux density of the gain calibrator, APPLYCAL for applying the calibration solutions to the target, and CONCAT for combining the separately calibrated B and C data.Finally, we cleaned the combined data using the task CLEAN and subtracted continuum emission from the image using the task IMCONTSUB.In the cleaning process, we used Briggs weighting with a robustness of 0.5, which resulted in an angular resolution of 6. 56 × 5. 59 (P.A. = −87.4• ).The masked integrated intensity map of the H I cube, consisting from 940 km s −1 to 1340 km s −1 with a channel width of 20 km s −1 , is shown in Figure 1 (middle).The mask was created from the 3σ threshold of a cube smoothed to 15 .The rms noise per channel is ∼2.3K and the total flux of the integrated intensity map is 28 Jy km s −1 .For comparison, the integrated flux density from the Arecibo Legacy Fast ALFA (ALFALFA) survey is 26.81 Jy km s −1 (Giovanelli et al. 2007).The single dish flux is in good agreement with our measurement within uncertainties.
In order to estimate the SFR, we used both 24 µm and Hα emissions from dust and H II regions surrounding young massive stars, respectively.For the Spitzer MIPS 24 µm (Program ID 30945; PI: Jeffrey Kenney), we downloaded the basic calibrated data (BCD) from the Spitzer Heritage Archive and removed the residual artifacts such as latents and gradients (MIPS Instrument Handbook 1 ) in the BCD via Image Reduction and Analysis Facility (IRAF) tasks.After mosaicking the BCD images using MOPEX (Mosaicking and Point Source Extraction), we employed the Groningen Image Processing System (GIPSY; van der Hulst et al. 1992) task BLOT to mask out bright stars of the 24 µm image in Figure 1 (third panel).The Hα image in the bottom panel was observed by Rand (1996) with the 2.1 m Kitt Peak National Observatory (KPNO).The continuum subtraction was carried out in the image and there is no contamination by the [N II] lines (Rand 1996).

Position-Velocity Diagrams
The position-velocity (p-v) diagram allows us to obtain the rotation curve using the envelope tracing method which uses the terminal velocity corrected for the instrumental resolution and gas velocity dispersion (Sofue & Rubin 2001; see Section 3.3) and the radial density distribution based on the PVD method which assumes axisymmetry (Yim et al. 2011; see Section 4.1).We obtained the p-v diagrams (Figure 2) by integrating the data cubes over ±10 for CO and ±30 for H I from the mid-plane.The horizontal dotted line indicates the systemic velocity V sys of 1140 km s −1 that we adopted based on both the consistency of the redshifted and blueshifted rotation curves and HyperLEDA2 (Makarov et al. 2014).
The p-v diagram of CO (top panel of Figure 2) shows a central elongated feature followed by a gap, possibly caused by a bar such as in NGC 891 (Yim et al. 2011) and NGC 4013 (Yim et al. 2020).The maximum radial velocity of the central feature (∼1280 km s −1 ) is lower than the almost flat velocity (∼1320 km s −1 ) beyond ∼30 in the redshifted side.The lower velocity compared to the flat velocity suggests an end-on bar (e.g.NGC 4013) while a higher velocity in the central feature suggests a side-on bar as in NGC 891 when we are seeing the x 2 periodic orbits in the center (Athanassoula & Bureau 1999).Therefore, we suggest that a bar of NGC 4302 is located close to the line of sight (i.e.end-on bar).Unlike the CO p-v diagram, the elongated feature in the central region does not appear in the H I p-v diagram (middle panel of Figure 2); this is perhaps because the central region has little H I emission.The p-v diagram clearly shows that the emission in the blueshifted side is much further extended than the emission in the redshifted side.Chung et al. (2007) reported the extended emission as a long one-sided H I tail and they suggest that the H I tail could be caused by ram pressure stripping.

Rotation Curve
We derived the CO and H I rotation curves using the p-v diagrams along the mid-plane through the envelope tracing method.This method calculates the rotation velocities (V rot ) based on the terminal velocities (V ter ) of the p-v diagram, the observational velocity resolution (σ obs ) and the velocity dispersion of gas (σ g ): where σ 2 obs for CO and H I is 2.2 km s −1 and 8.5 km s −1 based on the channel resolutions, respectively.The adopted velocity dispersions of CO and H I are 4 km s  The density distributions as a function of radius are necessary to compare the gas with the SFR for investigating the star formation law.We obtained the radial profiles using the PVD method for the gas and the GIPSY task RADPROF for the SFR.The CO, H I, 24 µm, and Hα maps are convolved to a circular beam of 6.56 × 6.56 based on the beam size of H I before deriving the radial profiles in order to compare and/or combine them.When we convolve the 24 µm and Hα maps, we use the kernels provided by Aniano et al. (2011).The assumed original point spread functions for 24 µm and Hα are 5.9 and 1.5 (Rand 1996), respectively.

Molecular and Atomic Gas
We use the vertically integrated p-v diagrams in Figure 2 to obtain the radial profiles of Σ HI and Σ H2 using the PVD method, which is based on an assumption of circular rotation.Each pixel position in the p-v diagram allows us to derive a galactic radius and the face-on surface brightness at the radius: where V c is the assumed circular speed obtained from the rotation curve (bottom panel of Figure 2) and V los is the observed line-of-sight velocity at each position x.
The angle brackets denote the mean value within a pixel.We average the surface mass densities within each 10 bin and plot the average values of H 2 (red squares) and H I (blue circles) as a function of radius in Figure 3 (top).The molecular and atomic gas densities include a factor of 1.36 for the helium correction.This figure shows that the molecular gas (Σ H2 ) is centrally concentrated while the atomic gas (Σ HI ) is almost uniformly distributed over the disk even beyond the CO disk.The surface density of the total gas (Σ gas ) is the combination of the molecular and atomic gas densities.The vertical error bars show the standard deviation of the average data within each bin.We also estimate the uncertainty from error propagation using the rms noise in the p-v diagram; this uncertainty is significantly smaller than the standard deviation.The uncertainties from the center outward are 0.14-0.01M pc −2 for H 2 and 2.15-0.04M pc −2 for H I. The horizontal error bars indicate the maximum and minimum radius obtained by using the angular and velocity resolutions in the calculation of radius (Equation 2).Since the circular speed could be affected by a bar, we do not exactly know the rotation curve near the center.Therefore, we also use a circular speed of 180 km s −1 for V c instead of the rotation curve to examine how the radial profiles vary in the central regions within ∼40 , where the rotation curve does not match to the assumed flat velocity of 180 km s −1 .The maximum difference between the surface densities using the adopted rotation curve and the assumed flat rotation curve (180 km s −1 ) in the central regions is a factor of ∼2 for CO and ∼5 for H I. However, the surface densities using the different V c values are consistent with each other beyond R ∼40 .

Star Formation Rate
We use a linear combination of Hα and 24 µm (Calzetti et al. 2007;Schruba et al. 2011) to estimate the recent SFR density: Σ SFR [M kpc −2 yr −1 ] = 0.029 I Hα + 0.0025 I 24 µm , (5) where the intensities of Hα and 24 µm in units of MJy sr −1 are obtained from RADPROF, which provides the radial profile in a face-on disk.The linear combination is meant to capture both obscured and unobscured star formation.Calzetti et al. (2007) derived the calibration from a sample of 33 nearby galaxies obtained by the SINGS (Spitzer Infrared Nearby Galaxies Survey; Kennicutt et al. 2003) project.
Since the data do not include the velocity information, the PVD method is not usable to obtain the SFR radial profile.Instead, we use the GIPSY task RADPROF, requiring integrated intensity strips and some input parameters (position angle, inclination, beam size, sigma, and initial estimate for the density distribution) and providing the radial surface density distribution based on the Lucy iterative scheme assuming axisymmetry (Lucy 1974) developed by Warmels (1988).The SFR surface density (Σ SFR ) from RADPROF is shown as magenta diamonds in Figure 3 (top).The SFR density profile is dominated by the 24 µm term in Equation 5.The vertical error bars show the standard deviation of data in a bin of 10 .The SFR density is also centrally concentrated like the CO density, but the emission is extended beyond the CO disk, up to the H I disk.We also derive another SFR estimate using the VLA 1.4 GHz radio continuum data (Vollmer et al. 2013) based on the method given by Murgia et al. (2002).We have compared the SFR radial profiles from the linear combination of Hα and 24 µm and the radio continuum at the same resolution of 22 in Figure 3 (bottom).Overall, they are very consistent with each other except for the central regions within 40 , where the largest difference is a factor of 1.9.Vargas et al. (2018) found that edge-on galaxies have a measurable optical depth at 24 µm and have lower flux ratios of 25/100 µm flux ratios compared to less inclined galaxies by a factor of 1.36.

GAS DISK THICKNESS
NGC 4302 is the almost edge-on galaxy that allows us to measure the disk thickness directly.First, we integrate the velocity channels over 950-980 km s −1 for the redshifted side and 1300-1330 km s −1 for the blueshifted side to obtain the terminal-velocity integrated intensity maps using the CO and H I masked cubes.The radial ranges where the measurements are available are about 25 -80 for CO and about 25 -140 for H I. Second, we fit a Gaussian function to vertical density profiles of the integrated intensity map at each x-offset (which approaches a galactic radius at the terminal velocity).Finally, we use the averaged Gaussian widths in radial bins of 3 for CO and 7 for H I as the scale heights, where the beam size is deconvolved.Figure 4 (left) shows the scale heights of H I (blue circles) and CO (red squares) as open (blueshifted side) and filled (redshifted side) circles with vertical error bars, representing the uncertainties of the Gaussian fits.The right three panels in the figure show the Gaussian fits to the vertical density profiles at different x-offsets.The uncertainty of the CO scale height is clearly smaller than that of the edge-on galaxy NGC 4013 (i ∼ 90 • ) using CARMA (Yim et al. 2020) thanks to the unprecedented sensitivity of ALMA.The blue and red lines are the linear least-squares fits to the average data points and the best fits for the scale heights (h), h HI = (39.6 ± 0.7) × R [kpc] + (78.5 ± 3.7) [pc], (6) h H2 = (18.2± 0.3) × R [kpc] + (2.6 ± 0.9) [pc], (7) will be used to derive the gas volume density.The uncertainties of the best fits are shown as the shaded (cyan) region around the lines.Both the CO and H I scale heights increase as a function of radius, but the H I gradient is much steeper than the CO gradient.NGC 4302 has a significantly steeper slope in the gas scale height compared to the other edge-on galaxies presented in the previous studies (Yim et al. 2014(Yim et al. , 2020)); steeper by a factor of ∼4 for CO and a factor of ∼2 for H I compared to the average CO and H I gradients of the other galaxies.

THE VOLUMETRIC STAR FORMATION LAW
We derive the mid-plane volume densities using the surface densities and the scale heights under assumptions of Gaussian (gas) and exponential (SFR) distributions along the vertical direction z.The surface densities are (8) Therefore, the volume densities are where we use the CO scale height for h SFR under the assumption that the molecular gas should have about the same scale height as the SF.The 24 µm scale height is likely not the SF scale height because of high-latitude dust being heated by a thinner SF layer.We plot ρ SFR against ρ HI , ρ H2 , and ρ gas and Σ SFR against Σ HI , Σ H2 , and Σ gas in the top panels of Figure 5 to investigate and compare the star formation laws based on volume and surface densities.We find tight power-law correlations between the SFR and H I, H 2 , and the total gas by fitting the ordinary least-squares bisector (Isobe et al. 1990) on a log-log scale: (11) The molecular power-law index of the volumetric SFL is comparable to that of the vertically integrated SFL while the atomic and total gas indices of the volume density are noticeably flatter than the indices of the surface density.When we correct the SFR for the extinction in the central region (R ≤ 30 ) by a factor of 1.9 that we obtained from the comparison between the radio continuum and the linear combination of Hα and 24 µm, the both SFLs have a bit steeper slopes: 1.00 for the both molecular slopes, 1.09 (volume) and 1.57 (surface) for the total gas, and 2.44 (volume) and 3.82 (surface) for the atomic gas.The main reason for quite different indices between surface and volume densities for the atomic and total gas is that the H I scale height increases with radius rapidly, causing a big difference in the gas surface and volume densities.In both cases, the H I power-law index is obviously larger compared to the indices for the molecular and total gas since the H I density is roughly constant until it decreases slowly in the outer regions, whereas the SFR density decreases rapidly with radius.On the other hand, a tight correlation between SFR and H I in volume and surface densities is not shown in many previous studies (e.g., Bigiel et al. 2008;Schruba et al. 2011;Yim et al. 2020) though in a recent study, Bacchini et al. (2019) present a strong correlation between ρ SFR and ρ HI .The correlation shown in H I, unlikely from the previous study of Yim et al. (2020), is caused by the different radial density distributions: this study using the derived rotation curve and the previous study assuming a flat rotation curve.
In terms of the star formation efficiency (SFE = ρ SFR /ρ gas or Σ SFR /Σ gas ) in the bottom panels of Figure 5, the SFE H2 and SFE gas appear to be roughly constant over the disk, while the atomic gas SFE HI decreases as a function of radius in both volume and surface densities.The molecular gas depletion time (SFE −1 H2 ) is in ranges of 1.8-6.6Gyr (volume) and 2.2-8.3Gyr (surface) and the total gas depletion time is in ranges of 2.4-7.4Gyr (volume) and 3.8-14.1 Gyr (surface).

SUMMARY AND CONCLUSION
Using ALMA ( 12 CO J = 1 → 0) and VLA (H I), we derive the molecular and atomic gas surface densities of the edge-on galaxy NGC 4302.We also estimate the SFR surface density from the linear combination of 24 µm and Hα emissions.In order to obtain the midplane volume density, we measure the CO and H I scale heights by fitting the Gaussian function to the vertical profiles and find that the scale heights increase significantly as a function of radius.Uncertainty of the CO scale height using ALMA is smaller than the uncertainty using CARMA given by Yim et al. (2020): the gradient uncertainty of the ALMA scale height is smaller by a factor of ∼7.
Using the measured scale heights and surface densities, we infer the volume densities of CO, H I, and SFR (24 µm ± Hα) to investigate the volumetric star formation law and compare the volumetric SFL with the vertically integrated SFL.we find strong power-law correlations between ρ HI , ρ H2 , ρ gas and ρ SFR with the indices of 2.15, 0.86, and 0.96, respectively.These volumetric power-law indices for H I and the total gas (H I+H 2 ) are quite smaller than the indices based on the surface densities while the H 2 indices for both volume and surface densities are similar to each other.We also find that the SFE H2 and SFE gas are roughly constant, while the SFE HI decreases as a function of radius in both volume and surface densities.

Figure 1 .
Figure1.CO and H I integrated intensity maps.The northern disk is placed in negative x. Contour levels are 10.0 × 2.5 n K km s −1 , with n=0, 1, 2, 3, 4 for CO and 750.0 × 1.6 n K km s −1 , with n=0, 1, 2, 3 for H I. The lowest contour levels are ∼10σ for CO and ∼5σ for H I. The synthesized beam is shown in the lower right corner of each panel.The Spitzer 24 µm contour levels are 0.2 × 3.4 n MJy sr −1 , with n = 0, 1, 2, and 3 and the Hα contour levels are 0.0003 × 2.5 n MJy sr −1 , with n = 0, 1, and 2. The lowest contours are ∼10σ for 24 µm and ∼3σ for Hα.The point spread function (PSF) FWHM is shown in the lower right corner of each panel: 5.9 for 24 µm and 1.5 for Hα.

Figure 2 .
Figure 2. Position-velocity diagrams integrated vertically for CO (top) and H I (middle).CO contours are 1.0 × 1.8 n K arcsec, with n=0, 1, 2, 3, 4. H I contour levels are 175.0×1.5 n K arcsec, with n=0, 1, 2, 3, 4. The lowest contour level is 5σ.The horizontal dotted lines indicate the heliocentric systemic velocity of 1140 km s −1 .Bottom: Rotation curves of CO (red squares) and H I (blue circles) obtained from p-v diagrams along the mid-plane.The rotation velocities are overlaid on the p-v diagrams as red filled circles.The uncertainties from the correction term in Equation 1 are shown in the upper left and right corners.
−1(Wilson et al. 2011;Mogotsi et al. 2016;Marasco et al. 2017) and 8 km s −1(Shostak & van der Kruit 1984;Blitz & Rosolowsky 2006), respectively.The adopted values do not noticeably affect the results.The terminal velocity is defined as the highest velocity at the 3σ contour level on the mid-plane p-v diagram.The obtained rotation velocities are shown as red circles overlaid on the integrated p-v diagrams in Figure2.The rotation curve is very uncertain near the center due to the existence of a bar.The blueshifted side is in good agreement with the redshifted side for both CO and H I. The rotation curves in the bottom panel of Figure2show average values of the blueshifted and redshifted velocities as red open circles for CO and blue crosses for H I. The differences between the blueshifted and redshfited velocities are used as the error bars on the averages.In addition, the uncertainties from the correction term ( σ 2 obs + σ 2 g ) in Equation 1 are shown in the upper left and right corners in the figure.

Figure 3 .
Figure 3. Top: radial surface density profiles of H2 (red squares), H I (blue circles), SFR (magenta diamonds), and the total gas (black triangles).The vertical error bars represent the standard deviation of data in a radial bin.Bottom: SFR radial profiles from the linear combination of Hα and 24 µm (magenta diamonds) and the radio continuum (blue circles).The vertical error bars are the standard deviation of the average in each bin of 25 .
When we use the p-v diagram, we exclude the central regions (|x| < 40 and |V los − V sys | < 40 km s −1 ) to avoid non-circular motions and confusion caused by the line of sight blending of emission from many different radii in the regions.Using the rotation curve, we find a radius corresponding to a position of x and V los in the p-v diagram.Finally, we convert the flux in the pixel into the face-on surface brightness and obtain the surface mass densities from the surface brightnesses using a standard Galactic value for the CO-to-H 2 conversion factor (Strong & Mattox 1996; Dame et al. 2001) and assuming optically thin H I emission: Σ H2 [M pc −2 ] = 3.2I CO [K km s −1 ], (3) Σ HI [M pc −2 ] = 0.0146I HI [K km s −1 ].(4)

Figure 4 .
Figure 4. Scale heights of H I (blue circles) and CO (red squares) as functions of radius.The open (blueshifted side) and filled (redshifted side) circles are average values in bins of 3 (CO) and 7 (H I).The lines are the linear least-squares fit to the average data points.The shaded regions around the best fit lines show uncertainties of the best-fits and the vertical error bars show uncertainties of the Gaussian fitting for the scale height.The right three panels show Gaussian fits (line) to the vertical profiles (filled circles) of CO at x-offsets of −34 , 56 , and 78 .

Figure 5 .
Figure 5. Top panels: SFR volume density ρSFR as a function of ρHI (blue circles), ρH 2 (red squares), and ρgas (black triangles).SFR surface density ΣSFR as a function ΣHI, ΣH 2 , and Σgas.The gas depletion time (SFE −1 ) is labeled on the corresponding dashed lines.Bottom panels: H2, H I, and the total gas SFEs as a function of radius for volume (left) and surface (right) densities.The vertical error bars represent the standard deviation of data in each bin.