THE INTERSTELLAR MEDIUM AND STAR FORMATION IN EDGE-ON GALAXIES. II. NGC 4157, 4565, AND 5907

The BIMA $^{12}{\rm CO}J=1\to 0$ observations were taken in 2004 toward the edge-on galaxies NGC 4157 (in B, C, and D congurations), NGC 4565 (C and D), and NGC 5907 (C and D) with a five field mosaic oriented along the major axis with a 50'' separation between the primary beam centers. In 2010 and 2011, we observed NGC 4565 and 5907 using CARMA (in C and D configurations) in order to obtain higher angular resolution. The CARMA observations are mosaics of 11 pointings toward NGC 4565 and 13 pointings toward NGC 5907 with a half beam spacing of 30''. The median baseline lengths are roughly 278 m (B), 93 m (C), and 40 m (D) for BIMA and 278 m (C) and 111 m (D) for CARMA. The instruments and configurations used are summarized in Table 2. All the CO data were calibrated and imaged using the MIRIAD package. The calibration for each track involved use of MFCAL for bandpass and gain calibration and the BOOTFLUX task for flux calibration. For NGC 4565 and 5907, we combined the separately calibrated BIMA and CARMA data using the MIRIAD task INVERT. All the CO images were produced using natural weighting to achieve the highest possible sensitivity.

Table 2. CO and Hi Observing Parameters

Galaxy	NGC 4157		NGC 4565		NGC 5907
	CO	Hi	CO	Hi	CO	Hi
Telescope	BIMA	VLA	BIMA & CARMA	VLA	BIMA & CARMA	VLA
Array	BCD	CD	CD & CD	BCD	CD & CD	CD
θ ('')	3.84 × 3.30	15.69 × 14.87	3.6 × 2.95	6.26 × 5.59	3.48 × 2.76	15.43 × 13.84
$\Delta v$ (km s−1)	10	20	10	20	10	20
Total flux (Jy km s−1)	1280	110	1030	274	1530	259
Channel noise (K)	0.23	0.67	0.16	2.73	0.15	0.81
Velocity range (km s−1)	550–990	550–990	960–1490	970–1490	420–920	410–910
Circular beam	4'' (250 pc)	15.''7 (981 pc)	3.''7 (174 pc)	6.''3 (296 pc)	3.''6 (190 pc)	15.''5 (821 pc)

Note. θ is the angular resolution. $\Delta v$ is the velocity resolution. The original beam has been convolved to a circular beam for the analysis in Section 4.1.

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The CO integrated intensity maps of NGC 4157, 4565, and 5907 are shown in Figure 1. Throughout this paper, we use x for the coordinate along the major axis (line of nodes), y for the coordinate perpendicular to x in the galaxy plane, and z for the coordinate perpendicular to the galaxy plane. The coordinate along the apparent minor axis is actually a combination of y and z (if not exactly edge-on), so we refer to it as the "minor axis offset." The angular and velocity resolutions, total CO flux in Jy km s⁻¹, and the rms noise per channel in K are presented in Table 2. For the total flux, we used a masked integrated intensity map derived from a gain corrected image cube, limited to the region where the map noise is less than twice its minimum (central) value. The masked map is obtained by blanking regions below a 3σ threshold in a smoothed cube. We used a noise-flattened map convolved to 10'' for the smoothed cube, where the noise-flattened map is the original cube divided by the normalized sensitivity image. The average rms noise is obtained from line-free edge channels of the gain corrected cube. Each map is rotated to place the blueshifted side of the disk (lower velocities than ${{V}_{{\rm sys}}}$) on the negative x-axis.

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We have obtained Hi data of the galaxies from the NRAO VLA archive. The array configurations are C and D for NGC 4157 and 5907 and B, C, and D for NGC 4565. A data reduction pipeline implemented in the CASA (Common Astronomy Software Applications) software package (McMullin et al. 2007) was built for reprocessing the whole data set in a uniform procedure, which we describe below.

For the data taken from each individual track, flagging of shadowed data was automatically performed first. Edge channels in each spectral window were also automatically flagged, if a test bandpass calibration showed their solution amplitudes falling below 0.75. Any additional bad visibilities from either interference or instrumental problems were located and examined using tasks PLOTMS and VIEWER, and flagged using FLAGCMD with mode = "manualflag" or "quack." Then the flux calibrator visibility model was set using SETJY, and we computed the bandpass and flux-scaled gain tables from the primary and secondary calibrator data, using tasks BANDPASS, GAINCAL, and FLUXSCALE. Those solution tables were later applied to the data with the task APPLYCAL.

For the calibrated visibilities of each track, we usually performed a continuum subtraction in uv-space using the task UVCONTSUB. A first order polynomial fit was used to estimate the continuum emission from line-free channels. Then the data from different tracks were combined using the task CONCAT, and we imaged and cleaned the spectral data cubes using the CASA task CLEAN, with the ROBUST weighting algorithm (the ROBUST parameter was set to R = 0.5). The clean depth went to a brightness level of 2.5 times the rms derived from the line free channels of a dirty cube.

Figure 2 shows the Hi integrated intensity maps of NGC 4157, 4565, and 5907. The observing parameters including the angular and velocity resolutions are shown in Table 2. The total flux in units of Jy km s⁻¹ has been obtained from integration of the masked cube, where the mask was constructed from the 3σ contour of a cube smoothed to 20''. The average rms noise is measured in the line-free edge channels of the original cube. The strong warp of NGC 5907 noted in previous studies (e.g., Sancisi 1976) is evident. In addition, slight Hi warps are shown in the other two galaxies (NGC 4157 and 4565). However, we do not see any evidence for warps in non-Hi data for all three galaxies.

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The IR data (IRAC 3.6 μm and MIPS 24 μm) were obtained from the Spitzer archive: Program ID 69 (PI: G. Fazio) for both 3.6 μm and 24 μm images of NGC 4157, Program ID 3 (PI: G. Fazio) for 3.6 μm images of NGC 4565 and 5907, and Program ID 20268 (PI: R. de Jong) for 24 μm images of NGC 4565 and 5907. We have downloaded the basic calibrated data (BCD) and used MOPEX (Mosaicking and Point Source Extraction) to remove background variations in the BCD images and mosaic the images. In the case of 24 μm, we have removed a residual flat field in the BCD data before background matching and mosaicking. In addition, we have downloaded 3.6 μm images along with the supplied masks from the S⁴G archive (Sheth et al. 2010) and masked out the stars on the images for stellar modeling in Section 4.2.2.

The reduced 3.6 μm and 24 μm images of NGC 4157, 4565, and 5907 are shown in Figures 3 and 4, respectively. The ring seen in the CO map of NGC 4565 is clearly shown in the 24 μm image. In the 3.6 μm images, bright stars located nearby or projected against the galaxies are blanked and replaced by values from neighboring pixels using the Groningen Image Processing System (GIPSY; van der Hulst et al. 1992) tasks BLOT and PATCH before deriving their radial distributions.

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In order to derive the radial profile of gas surface density, we employ the position–velocity diagram (PVD) method from Paper I using vertically integrated position–velocity (p-v) diagrams (Figures 5, 6, and 7) and assuming circular rotation with a flat rotation curve (ignoring the rapidly rising velocity in the central region). The vertical integration for the p-v diagram extends between 10'' above and below the midplane for CO and ±50'' for Hi. The central region in the p-v diagrams (defined as a solid box in the figures), where contributions from many different radii overlap, is excluded when deriving the radial profile. At each pixel in the p-v diagram, the radius is obtained using the position (x), the assumed circular speed (${{V}_{{\rm c}}}$), and the radial velocity (V_R ):

$$\begin{eqnarray}&&R={{V}_{{\rm c}}}\left\langle \frac{x}{{{V}_{R}}-{{V}_{{\rm sys}}}} \right\rangle \quad \quad {\rm with}\ |{{{\rm V}}_{R}}-{{{\rm V}}_{{\rm sys}}}|\lt {{{\rm V}}_{{\rm c}}},\end{eqnarray} \tag{ 1 }$$

where the angle brackets indicate the mean value of $x/({{V}_{R}}-{{V}_{{\rm sys}}})$ within the pixel (see Paper I for more details) and the assumed circular speed is 220, 250, and 240 km s⁻¹ (indicated as a dashed line in Figures 5–7) for NGC 4157, 4565, and 5907, respectively. The circular speed is obtained based on the maximum value of the rotation curve (Figure 8) that we derived using the envelope tracing method (Sofue 1996), which is based on a p-v diagram along the midplane. In this method, the projected rotation curve (${{V}_{{\rm proj}}}$) is derived from the terminal velocity (${{V}_{{\rm t}}}$) at each x offset, considering the observational velocity resolution (${{\sigma }_{{\rm obs}}}$) and the gas velocity dispersion (${{\sigma }_{{\rm g}}}$):

$$\begin{eqnarray}&&{{V}_{{\rm proj}}}={{V}_{{\rm t}}}-\sqrt{\sigma _{{\rm obs}}^{2}+\sigma _{{\rm g}}^{2}},\end{eqnarray} \tag{ 2 }$$

where the terminal velocity is selected as the highest velocity where the intensity exceeds the 3σ level. The velocity resolutions for our observations are 10 km s⁻¹ for CO data and 20 km s⁻¹ for Hi data and the assumed velocity dispersion is 8 km s⁻¹. Based on the correction term with ${{\sigma }_{{\rm obs}}}$ and ${{\sigma }_{{\rm g}}}$, we adopt an uncertainty for the rotation curve of 13 km s⁻¹ for CO and $\sim 22$ km s⁻¹ for Hi. The derived surface brightnesses of CO and Hi by the PVD method (more details in Section 4.1 of Paper I) are converted to surface mass densities using the conversion factors shown in Equation (3) for CO (Strong & Mattox 1996; Dame et al. 2001) and Equation (4) for Hi:

$$\begin{eqnarray}&&N({{{\rm H}}_{2}})\;[{\rm c}{{{\rm M}}^{-2}}]=2\times {{10}^{20}}\;{{I}_{{\rm CO}}}\;[{\rm K}\;{\rm km}\;{{{\rm s}}^{-1}}].\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&N({\rm HI})\;[{\rm c}{{{\rm M}}^{-2}}]=1.82\times {{10}^{18}}\;{{I}_{{\rm HI}}}\;[{\rm K}\;{\rm km}\;{{{\rm s}}^{-1}}].\end{eqnarray} \tag{ 4 }$$

The derived radial profiles of ${{\Sigma }_{{\rm HI}}}$, ${{\Sigma }_{{{{\rm H}}_{2}}}}$, and the total gas (${{\Sigma }_{{\rm HI}}}+{{\Sigma }_{{{{\rm H}}_{2}}}}$) are shown in Figure 9. The average value of both sides of the disk is used as the surface density profile in the figure and a factor of 1.36 for helium is included in the surface densities. For comparison between the molecular and atomic gas and to combine the two components for the total gas, the CO map has been convolved to the resolution of the Hi map. Vertical error bars show the standard deviation of the average value (of all data points) in a bin and horizontal error bars represent an uncertainty in the radius, showing obtainable maximum and minimum radii derived from varying x and V_R by the angular and the velocity resolutions. Some data points in the inner region of NGC 4565 are omitted since they are below the 3σ detection threshold, which is obtained using the rms noise in the p-v diagram. NGC 4565 shows a ring-like morphology as noticed in the CO map; NGC 4157 and 5907 exhibit small, centrally concentrated CO disks.

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For comparison purposes, we also plot the radial profile of ${{\Sigma }_{{{{\rm H}}_{2}}}}$ obtained from the GIPSY task RADPROF (see Section 3.2 for details) as a magenta line in Figure 9. The RADPROF profile is in good agreement with the PVD profile of ${{\Sigma }_{{{{\rm H}}_{2}}}}$, although NGC 4565 exhibits differences between them in the central region, where CO emission is deficient and the uncertainties of the PVD profile are large. As discussed in Paper I, RADPROF is also subject to larger errors in the central region due to emission from outer radii seen in projection. In addition, it is unable to reproduce sharp gradients in the radial profile, such as those caused by gaps or rings.

We have used the RADPROF task to obtain radial distributions of the stellar and SFR surface densities since the IR data lack velocity information. The RADPROF task uses an Abel inversion technique which assumes axisymmetry and requires a strip integral for each side of a galaxy, along with position angle, inclination (the values in Table 1 are used), beam size, and an initial function for the radial profile, which we assume to be exponentially decreasing for the 3.6 μm, and a Gaussian function for the 24 μm data. Before employing the RADPROF task, all maps are smoothed to the lowest resolution among the data set. For better accuracy in comparison between SFR and gas, the 24 μm maps were first convolved to a Gaussian beam of 7'' using the IDL code and the kernel provided by Aniano et al. (2011) and then convolved to the Hi beam except for NGC 4565. Since the Hi beam size of NGC 4565 is less than 7'', other maps are convolved to the 24 μm Gaussian beam.

The derived 3.6 μm radial profile from RADPROF is converted to stellar mass density using a conversion factor empirically derived by Leroy et al. (2008):

$$\begin{eqnarray}&&{{\Sigma }_{*}}\;[{{M}_{\odot }}\;{\rm p}{{{\rm c}}^{-2}}]=280\;({\rm cos} \;i)\;{{I}_{3.6}}\;[{\rm MJy}\;{\rm s}{{{\rm r}}^{-1}}],\end{eqnarray} \tag{ 5 }$$

where the inclination i is zero since the 3.6 μm intensity (${{I}_{3.6}}$) obtained from RADPROF is the face-on surface density. The assumed mass-to-light ratio (K band) for the conversion factor is 0.5 and the uncertainty quoted by Leroy et al. (2008) is about 0.1–0.2 dex depending on galaxy color. The stellar radial distributions from RADPROF are shown as a blue line in Figure 10.

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In order to estimate the effect of extinction on the stellar surface density, we have estimated the extinction at 3.6 μm from the ratio of gas column density ${{N}_{{\rm H}}}$ to color excess $E(B-V)$ given by Bohlin et al. (1978) and the ratio of the extinction A_V to $E(B-V)~\sim$ 3.1 (Rieke & Lebofsky 1985):

$$\begin{eqnarray}&&\frac{{{N}_{{\rm H}}}}{{{A}_{V}}}\approx 1.87\times {{10}^{21}}\;{\rm atoms}\;{\rm c}{{{\rm M}}^{-2}}\;{\rm ma}{{{\rm g}}^{-1}}.\end{eqnarray} \tag{ 6 }$$

Using the ratios ${{A}_{3.6}}/{{A}_{K}}=0.56$ (Indebetouw et al. 2005) and ${{A}_{K}}/{{A}_{V}}=0.112$ (Rieke & Lebofsky 1985), the dust extinction at 3.6 μm is:

$$\begin{eqnarray}&&{{A}_{3.6}}\approx 3.35\times {{10}^{-23}}\;{{N}_{{\rm H}}}\;{\rm mag}.\end{eqnarray} \tag{ 7 }$$

Using the combined map of CO and Hi as a proxy for gas column density, we have obtained an extinction corrected map and derived a radial profile from the map. The largest extinctions we measured through the midplane are $\sim 1$ mag for all the galaxies. By comparing the radial profile (Figure 10) with the extinction corrected profile, we noticed that the differences between two profiles are a factor of $\sim 2$ in the central regions for all the galaxies. In the outer regions, the difference is almost negligible. Note that the extinction correction is not applied to the stellar profile (Figure 10).

In addition to the RADPROF-derived stellar density profile, we also plot a fitted exponential disk model (red dashed line in the figure). The disk model we use is a self-gravitating and isothermal disk model provided by van der Kruit & Searle (1981):

$$\begin{eqnarray}&&L(R,z)={{L}_{0}}\;{{e}^{-R/l}}\;{\rm sec}{{{\rm H}}^{2}}\;\left( \frac{z}{{{z}_{*}}} \right),\end{eqnarray} \tag{ 8 }$$

where L₀ is the space luminosity density at the center and l is the scale length. The vertically integrated luminosity density L(R) is converted to the mass density using Equation (5). For fitting, we use the projected intensity distribution for an edge-on galaxy, obtained by integrating the disk model:

$$\begin{eqnarray}&&\mu (x,z)=\mu (0,0)\left( \frac{x}{l} \right){{K}_{1}}\left( \frac{x}{l} \right){\rm sec}{{{\rm H}}^{2}}\left( \frac{z}{{{z}_{*}}} \right),\end{eqnarray} \tag{ 9 }$$

where $\mu (0,0)=2l{{L}_{0}}$ and K₁ is the modified Bessel function of the second kind of order 1. When fitting to the 3.6 μm maps, the central regions (including the stellar bulge) are excluded: $|x|\lt 10^{\prime\prime}$ for NGC 4157, $|x|\lt 80^{\prime\prime}$ for NGC 4565, and $|x|\lt 30^{\prime\prime}$ for NGC 5907. The scale lengths obtained by fitting Equation (9) are $\sim 32^{\prime\prime}$ or $\sim 2$ kpc (NGC 4157), $\sim 85^{\prime\prime}$ or $\sim 4$ kpc (NGC 4565), and $\sim 57^{\prime\prime}$ or $\sim 3$ kpc (NGC 5907). The scale heights (${{z}_{*}}$) are $\sim 12^{\prime\prime}$ or $\sim 780$ pc (NGC 4157), $\sim 14^{\prime\prime}$ or $\sim 640$ pc (NGC 4565), and $\sim 13^{\prime\prime}$ or $\sim 670\;{\rm pc}$ (NGC 5907). The vertical error bar in the lower left corner represents the uncertainty of the radial profile and is obtained from the largest difference between the RADPROF and the disk model profiles: 0.25 dex (NGC 4157), 0.10 dex (NGC 4565), and 0.19 dex (NGC 5907). Note that the excluded central regions are not considered when finding the largest difference for the uncertainty estimate.

The SFR surface density is determined from the Spitzer 24 μm image by adopting the calibration given by Calzetti et al. (2007):

$$\begin{eqnarray}&&\frac{{{\Sigma }_{{\rm SFR}}}}{{{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}}=1.56\times {{10}^{-35}}\;{{\left( \frac{{{S}_{24\,\mu {\rm M}}}}{{\rm erg}\;{{{\rm s}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}} \right)}^{0.8104}},\end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray}&&\frac{{{S}_{24\,\mu {\rm M}}}}{{\rm erg}\;{{{\rm s}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}}=1.5\times {{10}^{40}}\;\left( \frac{{{I}_{24}}}{{\rm MJy}\;{\rm s}{{{\rm r}}^{-1}}} \right),\end{eqnarray} \tag{ 11 }$$

and I₂₄ is the SFR (24 μm) surface brightness derived from RADPROF.

Figure 11 shows the SFR radial profiles of the galaxies. The error bar, showing a factor of 2 uncertainty, is based on the largest difference between radial profiles from two different methods (RADPROF and ELLINT) for several face-on galaxies (observed by Spitzer at 24 μm) as described in Paper I. Before deriving the SFR radial profile of NGC 4565, the central compact source (possibly an AGN; Laine et al. 2010) in the 24 μm map has been removed using the GIPSY tasks BLOT and PATCH. The SFR in these galaxies seems relatively low compared to the SFR in NGC 891 (Paper I). Dumke et al. (1997) also found that NGC 4565 and 5907 show lower star formation than NGC 891. The far-infrared luminosity at 40–400 μm by IRAS (often used as a tracer for SFR) is 10.18, 10.03, 9.61, 9.80 in ${{{\rm log} }_{10}}\;(L/{{L}_{\odot }})$ for NGC 891, 4157, 4565, and 5907, respectively (Sanders et al. 2003).

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Unlike NGC 891, whose inclination is known to be $\gt 89{}^\circ$ (Oosterloo et al. 2007), the other galaxies in our sample (NGC 4157, 4565, and 5907) appear less inclined. Since the inclination affects the apparent disk thickness, it is important to determine the inclination of the galaxies in order to obtain correct values for the scale height and the velocity dispersion.

We have derived inclinations and scale heights of these less edge-on galaxies via Ollingʼs method (Olling 1996) which is described in detail in the Appendix. The derived inclinations with radius are shown in Figure 12 as blue solid and red open circles for CO and green solid and magenta open triangles for Hi. Different colors indicate different halves of the galaxy. We show a weighted mean value of the inclination points as dotted (CO) and dashed (Hi) lines and adopt a representative value of $i=86.5{}^\circ$ for NGC 4565 and $i=86{}^\circ$ for NGC 5907 since their mean CO and Hi values are very close. However, in the case of NGC 4157, the CO (84°) and Hi (83°) values show a difference, so we adopt the CO inclination value since the Hi derived value may be affected by the outer warp. Even though there is a 1° difference between CO and Hi inclinations in NGC 4157, the values match well in the region where both tracers can be analyzed. We also note that the rotation curves of CO and Hi in this galaxy show a mismatch (especially in the region of 30''–70'') unlike the other galaxies. Since the inclinations are determined using the distance ${{y}_{{\rm off}}}$ (see the Appendix), which depends on the velocity field map, the mismatch between the rotation curves may contribute to the differences between the inclinations derived from CO and Hi.

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The measured scale heights (see the Appendix) considering the projection effects for the less edge-on galaxies are shown in Figure 13. We use the Gaussian width (0.42 × FWHM) for the scale height. The scale heights for NGC 891 presented in Paper I have been included in this figure for comparison purposes. The lines show the linear least-squares fits to the data points and we use the best fit line to derive the radial variation in the volume density and velocity dispersion in Sections 4.4 and 4.5. For the inner and outer regions where no data points are available, we assume a constant scale height within the innermost measured point when deriving the total midplane density and the total gas dispersion for the gravitational instability parameter Q (Section 5.2) and the interstellar pressure (Section 6).

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The thickness of the gas disk may be related to the star formation activity (Dumke et al. 1997 and references therein). From comparison between the CO disk thickness and the SFR surface density profile (Figure 11), we note that the galaxies with higher SFR (NGC 891 and NGC 4157) show a thicker CO disk. In order to investigate the relationship between the CO disk thickness and SFR, we plot the average CO scale height as a function of average ${{\Sigma }_{{\rm SFR}}}$ in Figure 14. The average ${{\Sigma }_{{\rm SFR}}}$ values are obtained by integrating ${{\Sigma }_{{\rm SFR}}}$ only over the CO disk range (i.e., the radial range over which estimates of the CO scale height are available) and normalizing by the disk area. This figure suggests a weak correlation between the galaxy-averaged CO disk thickness and the SFR. Within a galaxy, however, the two values are anti-correlated, with the CO disk being thicker in the outer galaxy where the SFR is lower.

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Visual inspection of the 3.6 μm images does not reveal clear indications of an increase in the scale height with radius. This may be due, however, to the rapidly decreasing surface brightness with radius and a viewing angle that is not exactly edge-on. We have therefore adopted an iterative approach to modeling the variation in stellar scale height.

4.2.1. Initial Estimates

4.2.2. Iterative Modeling

We characterize the radial variation in scale height using a linear function h(R) expressed as:

$$\begin{eqnarray}&&h=a+b\times R,\end{eqnarray} \tag{ 12 }$$

where the intercept (a) and the slope (b) with their uncertainties are summarized in Table 3. For stars, we show the parameters used to generate the best model. For ease of presentation, the slope is quoted in pc kpc⁻¹ rather than in dimensionless form. Unlike the scale heights of Hi, the uncertainties in the slopes of the CO scale heights are very large in the three galaxies NGC 4157, 4565, and 5907 and in the case of NGC 4565 the measured gradient is not significant. However, most galaxies show increases with radius in the scale heights of CO, Hi, and stars.

Using the derived surface mass densities and the scale heights from the previous sections, we have obtained the midplane volume density (ρ₀) profiles with radius:

$$\begin{eqnarray}\begin{array}{rcl} {{\rho }_{0{{{\rm H}}_{2}}}} & = & \frac{{{\Sigma }_{{{{\rm H}}_{2}}}}}{{{h}_{{{{\rm H}}_{2}}}}\sqrt{2\pi }},\qquad {{\rho }_{0{\rm HI}}}=\frac{{{\Sigma }_{{\rm HI}}}}{{{h}_{{\rm HI}}}\sqrt{2\pi }}, \\ {{\rho }_{0*}} & = & \frac{{{\Sigma }_{*}}}{2{{h}_{*}}}. \\ \end{array}\end{eqnarray} \tag{ 13 }$$

Here we assume a Gaussian distribution for the gas (H₂ and Hi) and a ${\rm sec}{{{\rm H}}^{2}}\;(z/{{h}_{*}})$ function for stars. The estimated volume densities are shown in Figure 15. The uncertainties shown in the figure include errors in the radial profile but not in the scale height or the CO-to-H₂ conversion factor. A factor of 2 variation in the ${{X}_{{\rm CO}}}$ factor would lead to a similar variation in the volume density. This midplane density profile allows us to explore the role that the turbulent interstellar pressure (${{\rho }_{0}}\sigma _{{\rm g}}^{2}$) plays in controlling the molecular to atomic (volume) gas density ratio, in comparison with the relationship between the hydrostatic midplane pressure and the H₂/Hi ratio based on surface mass density. In addition, the density profile will allow us to investigate the SFL in terms of volume density instead of surface density, although we defer this analysis to the next paper in this series.

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Since it is not easy to measure a gaseous vertical velocity dispersion directly, some earlier studies had suggested fairly constant values (Combes & Becquaert 1997; Petric & Rupen 2007). As an alternative, we have inferred the velocity dispersions as a function of radius using a numerical solution to the Poisson equation for a multi-component disk (Narayan & Jog 2002):

$$\begin{eqnarray}&&\sigma _{i}^{2}=\frac{4\pi G{{\rho }_{0,{\rm tot}}}{{\rho }_{0i}}}{-{{({{d}^{2}}{{\rho }_{i}}/d{{z}^{2}})}_{z=0}}},\end{eqnarray} \tag{ 14 }$$

where the subscript i can refer to either g (gas) or * (stars):

$$\begin{eqnarray}\begin{array}{rcl} {{d}^{2}}\;{{\rho }_{{\rm g}}}/d{{z}^{2}} & = & -{{\rho }_{0{\rm g}}}/h_{{\rm g}}^{2},\qquad {{d}^{2}}\;{{\rho }_{*}}/d{{z}^{2}}=-2{{\rho }_{0*}}/h_{*}^{2} \\ {\rm at}\ z & = & 0, \\ \end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\Rightarrow \;{{\sigma }_{{\rm g}}}=\sqrt{4\pi Gh_{{\rm g}}^{2}{{\rho }_{0,{\rm tot}}}},\qquad {{\sigma }_{*}}=\sqrt{2\pi Gh_{*}^{2}{{\rho }_{0,{\rm tot}}}}.\end{eqnarray} \tag{ 16 }$$

Here the total midplane density is

$$\begin{eqnarray}&&{{\rho }_{0,{\rm tot}}}={{\rho }_{0{{{\rm H}}_{2}}}}+{{\rho }_{0{\rm HI}}}+{{\rho }_{0*}}.\end{eqnarray} \tag{ 17 }$$

We ignore the dark matter halo under the assumption that the dark matter density in the disk is relatively small compared to that of the stars. The boundary conditions we have used are ${{\rho }_{i}}={{\rho }_{0i}}$ and $d{{\rho }_{i}}/dz=0$ at the midplane (z = 0).

The velocity dispersions of CO, Hi, and stars for the galaxies including NGC 891 are shown in Figure 16. The two galaxies NGC 4565 and 5907 show very low values of the CO velocity dispersion compared to the other galaxies. The two main reasons for the low velocity dispersions are because of low values in (1) the total midplane density and (2) the scale height of CO (see Equation 16). On the other hand, the velocity dispersions of Hi and stars show similar values (within a factor of $\sim 2$) and trends for all galaxies. Note that when we estimate the midplane total density to derive the Hi and stellar velocity dispersions, the molecular densities in the regions outside the CO field of view are assumed to be zero—a reasonable approximation because these regions are strongly Hi dominated (with the possible exception of NGC 4565). In general, our results show that the velocity dispersions of CO, Hi, and stars decrease as a function of radius.

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In order to verify the derived scale heights and velocity dispersions of the gas, we have built CO and Hi models of NGC 5907 (as a representative example) using the TiRiFiC (Józsa et al. 2007) and compared the models with the data. The main input parameters we used for modeling in TiRiFiC are the derived rotation curve, density profile, scale height, velocity dispersion, and inclination. From the comparisons of the midplane and z-integrated p-v diagrams and the integrated intensity maps (Figure 17), we confirmed that the gas models match the data reasonably well although the model does not include the Hi warp. It suggests that the derived CO and Hi scale heights and velocity dispersions as well as the rotation curves are consistent with the data.

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In Figure 18, we plot ${{\Sigma }_{{\rm SFR}}}$ versus ${{\Sigma }_{{{{\rm H}}_{2}}}}$ (left) and ${{\Sigma }_{{\rm gas}}}$ (right) for our galaxy sample including NGC 891 in order to examine the strength of the relationships and compare their slopes. The uncertainties on the SFR and the gas surface densities are shown in their radial profiles in Section 3. In addition, about a factor of 2 uncertainty in the CO-to-H₂ conversion factor (e.g., Bolatto et al. 2013) and the 24 μm SFR calibration (e.g., Murphy et al. 2012) are suggested by the literature. We use the ordinary least-squares (OLS) bisector (Isobe et al. 1990) to fit the relationships between log quantities. The obtained power-law index and rms scatter (dex) around the fit for all galaxies are presented in Table 4. Note that the index of NGC 891 is a bit different from Paper I since we use the OLS bisector instead of the least-squares fitting used in Paper I. In addition, the index of NGC 891 for the total gas is much higher (by a factor of 2) than the index in Paper I because we use gas data points up to R = 400'', instead of 300'' in Paper I. Our results suggest that the correlation between ${{\Sigma }_{{\rm SFR}}}$ and ${{\Sigma }_{{{{\rm H}}_{2}}}}$ is better than the correlation between ${{\Sigma }_{{\rm SFR}}}$ and ${{\Sigma }_{{\rm gas}}}$ when considering the rms scatter around the relationship. In addition, most individual relations between ${{\Sigma }_{{\rm SFR}}}$ and ${{\Sigma }_{{\rm gas}}}$ appear to show distinct slopes at the low and high ends; this is most noticeable in NGC 4565.

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The most frequently cited relation given by Kennicutt (1998) suggests an index of 1.4: ${{\Sigma }_{{\rm SFR}}}\propto \Sigma _{{\rm gas}}^{1.4}$. Wong & Blitz (2002) obtained weighted average indices of 0.78 (molecular gas) and 1.12 (total gas) from seven spiral galaxies. Leroy et al. (2005) found an index of 1.3 for the relationship between SFR (traced by the radio continuum) and H₂ in dwarf galaxies. Bigiel et al. (2008) derived average values of 0.96 for H₂ and 1.85 for Hi + H₂ from seven spiral galaxies. Bigiel et al. (2008) also obtained a smaller scatter for the molecular gas relation (0.2 dex) than for the total gas relation (0.3 dex). The overall tendency toward a steeper index in the correlation between total gas and SFR compared to the "molecular SFL" (${{\Sigma }_{{\rm SFR}}}\propto {{\Sigma }_{{{{\rm H}}_{2}}}}{{}^{{\rm n}}}$) is also shown in our results since the relatively constant values in the atomic gas density profile steepen the total gas SFL (${{\Sigma }_{{\rm SFR}}}\;\propto {{\Sigma }_{{\rm gas}}}{{}^{{\rm n}}}$). However, the derived "molecular" SFL indices are generally smaller (0.32–0.91) than found in previous studies, especially in NGC 4565 and 5907. Among the reasons for the shallow slopes in our results might be optical depth effects in the 24 μm emission which decreases the 24 μm brightness in bright regions or contribution from diffuse starlight heating which increases the 24 μm brightness in faint regions. As mentioned in Section 3.2, we have explored the opacity issue on 3.6 μm and obtained the uncertainty of a factor of 2 by the extinction effects. Since the 24 μm opacity should be less than the 3.6 μm opacity, we expect less than a factor of 2 increase in the central region at the 24 μm radial profile. When we applied the uncertainty by the extinction at 3.6 μm to the SFR radial profile, the indices for H₂ are increased by a factor of $\sim 1.2$ (NGC 4157), $\sim 1.4$ (NGC 4565), and $\sim 1.2$ (NGC 5907). Nevertheless, the index of NGC 4565 is still smaller, but it is not unexpected since the CO emission of NGC 4565 is deficient in the central regions as shown in the CO map (ring-like morphology) and the gradient of the 24 μm profile is flatter than the others. In addition, NGC 4565 has an AGN, which can heat the dust to produce 24 μm even without much star formation. Rahman et al. (2011, 2012) have suggested that one possible reason for the flatter slopes in some galaxies among their sample might be a greater contribution from diffuse emission to the 24 μm emission. For this to explain the trends we see in NGC 4565 and NGC 5907, they would need to suffer a factor of $\sim 2$ more diffuse 24 μm contamination than the other galaxies whose slopes are consistent with that of NGC 4254 without diffuse emission (Rahman et al. 2011).

In order to check whether the difference between RADPROF (SFR) and PVD (CO) methods is responsible for the flatter slope in the SFR versus molecular gas relation, we can also investigate the relationship between ${{\Sigma }_{{\rm SFR}}}$ and ${{\Sigma }_{{{{\rm H}}_{2}}}}$ on a pixel-by-pixel basis. For the pixel-by-pixel comparison, we use the masked CO intensity maps. Since most of the galaxies have very low resolution in Hi compared to CO, we do not attempt a pixel-by-pixel comparison between ${{\Sigma }_{{\rm SFR}}}$ and ${{\Sigma }_{{\rm gas}}}$ and focus on the "molecular SFL" with the CO map convolved to the 24 μm Gaussian beam (7'') for NGC 4157, 4565, and 5907. The CO beam of NGC 891 is 7''. Using the beam matched maps, we extract data on a pixel-by-pixel basis using the MIRIAD task IMCMP. All the extracted data are placed in 0.1 dex bins and the number of data points in each bin is mapped to the color scale in Figure 19. The blue solid points in the figures show the correlations from the radial profile analysis for comparison purposes. Most of the radial profile points for each galaxy follow the mean trends defined by the pixel-by-pixel plot (Figure 19), but their values are generally lower compared to the contours since the projection effect of an edge-on galaxy leads to higher values in the pixel-by-pixel method. In particular, NGC 4565 shows much lower values for the radial profile, due to its ring-like morphology leading to lower surface densities, especially in the central region. Note in our pixel-by-pixel analysis, unlike most K–S analyses, no deprojection is done.

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The best-fit line in the figures is obtained using the OLS bisector and the fitted power-law index (K–S law index) is shown in the upper-left corner of the figure. The obtained indices from the pixel-by-pixel analysis are in the range of 0.43–0.68 for the molecular SFL. Those values are similar to the indices from the radial profile analysis. This is in contrast to the results of Bigiel et al. (2008) who obtained a power-law index of about unity (shown as red dashed line in Figure 19) for the H₂ star formation relation based on a pixel-by-pixel comparison for seven spiral galaxies.

Figure 20 shows the star formation efficiency (SFE) as a function of radius (normalized by R₂₅) in terms of total gas (${\rm SFE}={{\Sigma }_{{\rm SFR}}}/{{\Sigma }_{{\rm gas}}}$) and the molecular gas (${{\Sigma }_{{\rm SFR}}}/{{\Sigma }_{{{{\rm H}}_{2}}}}$). The inverse of this quantity (SFE⁻¹) represents the gas depletion time. Overall, most galaxies show declining SFE (in terms of total gas) with radius even though fluctuations in SFE exist in some galaxies. On the other hand, the molecular SFE in Figure 20 (right) appears to increase with radius in some galaxies. This is caused by a steeper drop in the ${{\Sigma }_{{{{\rm H}}_{2}}}}$ profile compared to that of SFR and this situation could be due in part to the 24 μm opacity and diffuse emission issues. Once again this contrasts with previous results which find a roughly constant SFE in the molecular gas. Rownd & Young (1999) observed the constancy of SFE in a study of the SFE within CO-emitting galaxies. Bigiel et al. (2008) and Leroy et al. (2008) showed a constant SFE in the H₂ dominated region and decreasing SFE with radius in the Hi dominated region.

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Since both gas and stars can contribute to the gravitational instability of a galactic disk, many proposals have been made to evaluate stability using a single parameter ${{Q}_{{\rm tot}}}$ analogous to the Toomre (1964) Q parameter but including the properties of both gaseous and stellar disks (e.g., Jog & Solomon 1984; Rafikov 2001; Romeo & Wiegert 2011). A link between gravitational instability and star formation has been frequently suggested (Kennicutt 1989; Martin & Kennicutt 2001; Li et al. 2005) but has been difficult to confirm observationally (Wong & Blitz 2002; Leroy et al. 2008). Recently, Elmegreen (2011) has suggested that ${{Q}_{{\rm tot}}}$ primarily governs the spiral structure on large scales and has little direct influence on local collapse leading to star formation. He also suggests that gas dissipation can increase the stability threshold from $\sim 1$ to 2–3 and allow small-scale instability to persist even at large ${{Q}_{{\rm tot}}}$. In order to investigate the relationship between the gravitational instability parameter Q and massive star formation, we obtain the parameter ${{Q}_{{\rm tot}}}$ using a modified version (to treat Hi and H₂ separately) of the equation provided by Rafikov (2001), assuming two components of collisional gas and collisionless stars in a thin rotating galactic disk:

$$\begin{eqnarray}\begin{array}{rcl} \frac{1}{{{Q}_{{\rm tot}}}} & = & \frac{2}{{{Q}_{{\rm HI}}}}{{\Gamma }_{{{\sigma }_{{\rm HI}}}}}\frac{q}{1+{{q}^{2}}\Gamma _{{{\sigma }_{{\rm HI}}}}^{2}}+\frac{2}{{{Q}_{{{{\rm H}}_{2}}}}}{{\Gamma }_{{{\sigma }_{{{{\rm H}}_{2}}}}}}\frac{q}{1+{{q}^{2}}\Gamma _{{{\sigma }_{{{{\rm H}}_{2}}}}}^{2}} \\ {} & {} & +\frac{2}{{{Q}_{{\rm star}}}}\frac{1}{q}[1-{{e}^{-{{q}^{2}}}}{{I}_{0}}({{q}^{2}})]\gt 1, \\ \end{array}\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}\begin{array}{rcl} {{Q}_{{\rm HI}}} & = & \frac{\kappa {{\sigma }_{{\rm HI}}}}{\pi G{{\Sigma }_{{\rm HI}}}},\qquad {{Q}_{{{{\rm H}}_{2}}}}=\frac{\kappa {{\sigma }_{{{{\rm H}}_{2}}}}}{\pi G{{\Sigma }_{{{{\rm H}}_{2}}}}}, \\ {{Q}_{{\rm star}}} & = & \frac{\kappa {{\sigma }_{*,R}}}{\pi G{{\Sigma }_{*}}}, \\ \end{array}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{{\Gamma }_{{{\sigma }_{{\rm HI}}}}}={{\sigma }_{{\rm HI}}}/{{\sigma }_{*,R}},\qquad {{\Gamma }_{{{\sigma }_{{{{\rm H}}_{2}}}}}}={{\sigma }_{{{{\rm H}}_{2}}}}/{{\sigma }_{*,R}},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&q=\frac{k{{\sigma }_{*,R}}}{\kappa },\qquad \kappa =\frac{{{V}_{{\rm rot}}}}{R}\sqrt{2\left( 1+\frac{R}{{{V}_{{\rm rot}}}}\frac{d{{V}_{{\rm rot}}}}{dR} \right)}.\end{eqnarray} \tag{ 21 }$$

Here ${{\sigma }_{*,R}}$ is the stellar velocity dispersion in the radial direction, estimated from the vertical velocity dispersion: ${{\sigma }_{*,R}}={{\sigma }_{*}}$/0.6 (Bottema 1993). I₀ is the Bessel function of zero order, k is the wavenumber ($2\pi /\lambda$), κ is the epicyclic frequency, and ${{V}_{{\rm rot}}}$ is the rotational velocity, obtained using Equation (2) with a correction for the inclination: ${{V}_{{\rm rot}}}={{V}_{{\rm proj}}}/{\rm sin} i$. Note that ${{Q}_{{\rm tot}}}$ is the minimum among those values we can obtain with a range of λ; usually the selected λ (providing the minimum ${{Q}_{{\rm tot}}}$) is up to about 4 kpc. In this procedure, we use the inferred vertical velocity dispersion as a function of radius (Figure 16) to derive the parameter ${{Q}_{{\rm tot}}}$ shown in the left panel of Figure 21 for the galaxies including NGC 891 from Paper I. The error bars of ${{Q}_{{\rm tot}}}$ show maximum and minimum values implied by the uncertainties in ${{\Sigma }_{{{{\rm H}}_{2}}}}$, ${{\Sigma }_{{\rm HI}}}$, and ${{\Sigma }_{*}}$. Since the differences between the maximum and minimum values are very small, the error bars are not very noticeable. The unstable condition is ${{Q}_{{\rm tot}}}\lt 1$. The results show that most galaxies are marginally stable ($Q\sim 1$) in most regions, supporting the self-regulation of star formation (e.g., Ostriker et al. 2010; Kim et al. 2011). The ${{Q}_{{\rm tot}}}$ value is increasing toward the center, contradicting the general expectation of an unstable central region. This tendency is also shown in a study of Leroy et al. (2008) even though they used a constant velocity dispersion for gas (11 km s⁻¹) and varying velocity dispersion of stars proportional to $\Sigma _{*}^{0.5}$. The increase of Q toward the center appears to be due to the epicyclic frequency associated with the steeply rising slope of the rotation curve. It is well known that a strong central mass concentration makes Q increase in the central regions; this is called the "Q barrier" and is discussed by Sellwood (1985) and others. Generally, the ${{Q}_{{\rm tot}}}$ profiles beyond the central regions are consistent with the total gas SFE; Q is rising as the SFE decreases.

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In Figure 21 (top right), we obtain another ${{Q}_{{\rm tot}}}$ profile with the vertical velocity dispersions assumed to be constant (${{\sigma }_{*}}$ = 21 km s⁻¹ and ${{\sigma }_{{\rm g}}}$ = 8 km s⁻¹), which is commonly assumed in the literature (e.g., Combes & Becquaert 1997; Rafikov 2001). In this regime, two galaxies (NGC 891 and 4157) show unstable regions predicting star formation in the inner disk, while the other galaxies (NGC 4565 and 5907) show a marginally stable disk. Considering the relatively low SFR in NGC 4565 and 5907 compared to the SFR in NGC 891 and 4157, the ${{Q}_{{\rm tot}}}$ profile with constant velocity dispersions shows a reasonable prediction for the massive star formation. However, the assumed constant values are in conflict with the inferred radial variation in the velocity dispersion. The increase toward the center is also shown in this profile.

The different ${{Q}_{{\rm tot}}}$ profiles for constant and varying velocity dispersions show how much the adopted velocity dispersions affect the parameter. The variable velocity dispersions bring the Q versus R trends into better agreement galaxy-to-galaxy. By plotting individual Q profiles for gas and stars in both cases, it appears that ${{Q}_{{\rm tot}}}$ is affected mostly by ${{Q}_{{\rm star}}}$. The range and values of ${{Q}_{{\rm gas}}}$ are very large compared to ${{Q}_{{\rm star}}}$ and the profile shape of ${{Q}_{{\rm tot}}}$ follows that of ${{Q}_{{\rm star}}}$. One can see the individual Q profiles in Paper I (Figure 15) for NGC 891.

For comparison purposes, we also applied the recent approximation considering realistically thick disks given by Romeo & Wiegert (2011):

$$\begin{eqnarray*}\frac{1}{{{Q}_{{\rm tot}}}}=\left\{ \begin{array}{ccccccccccccccc} \frac{W}{{{T}_{*}}{{Q}_{{\rm star}}}}+\frac{1}{{{T}_{{\rm g}}}{{Q}_{{\rm gas}}}} & {\rm if}\;\;{{T}_{*}}{{Q}_{{\rm star}}}\;\geqslant \;{{T}_{{\rm g}}}{{Q}_{{\rm gas}}}, \\ \frac{1}{{{T}_{*}}{{Q}_{{\rm star}}}}+\frac{W}{{{T}_{{\rm g}}}{{Q}_{{\rm gas}}}} & {\rm if}\;\;{{T}_{{\rm g}}}{{Q}_{{\rm gas}}}\;\geqslant \;{{T}_{*}}{{Q}_{{\rm star}}}, \\ \end{array} \right.\end{eqnarray*}$$

where

$$\begin{eqnarray}&&W=\frac{2s}{1+{{s}^{2}}},\qquad s=\frac{{{\sigma }_{{\rm g}}}}{{{\sigma }_{*,R}}},\qquad T\approx 0.8+0.7\left( \frac{{{\sigma }_{z}}}{{{\sigma }_{R}}} \right).\end{eqnarray} \tag{ 22 }$$

Here the ratio of vertical to radial velocity dispersion (${{\sigma }_{z}}/{{\sigma }_{R}}$) is 0.6 for stars and 1 for gas. The ${{Q}_{{\rm tot}}}$ profiles using varying and constant σ are shown in the bottom panels of Figure 21. Even though there is not much difference between the two different approaches provided by Rafikov (2001) and Romeo & Wiegert (2011), the Rafikov ${{Q}_{{\rm tot}}}$ values using varying σ seem consistently lower than the Romeo and Wiegert values by enough that the Rafikov values are $\sim 1$ but the Romeo and Wiegert values are usually $\geqslant 1$. Overall, they show similar behavior in the profiles although the values are slightly different.