Statistical Study of the Star Formation Efficiency in Bars: Is Star Formation Suppressed in Gas-rich Bars?

We first select 262 nearby face-on barred galaxies from the extragalactic database HyperLeda ⁵ (Makarov et al. 2014), according to the following criteria:

• 1.  
  The morphological type is SAB or SB and the Hubble T stage is in T = 0 − 7, which corresponds to S0a–Sd. The morphological type is taken from the Third Reference Catalog of Bright Galaxies (RC3; de Vaucouleurs et al. 1991).
• 2.  
  The recessional velocity is <2000 km s⁻¹ (∼27 Mpc). The angular resolution of the images that we use, 15'', corresponds to a physical scale of less than 2.0 kpc.
• 3.  
  The inclination (i) is <70°. Because the inclinations taken from HyperLeda can be uncertain, when available, we adopt the inclinations from the catalogs of the PHANGS (Leroy et al. 2021) and the Spitzer Survey of Stellar Structure in Galaxies (S⁴G) surveys (Sheth et al. 2010). Here, we prioritize the PHANGS catalog over the S⁴G catalog.
• 4.  
  The projected major axis of a galaxy at the isophotal level 25 mag arcsec⁻² in the B-band image (D₂₅) is $\gt {2}^{{\prime} }$, and the B-band magnitude is <15.5 mag. This is used to remove apparent small and/or faint galaxies.
• 5.  
  The galactic latitude (b) is ∣b∣ > 10°. This is to minimize the contamination from the Milky Way disk.

Among the 262 galaxies, we select those with available SFR and molecular gas tracers. Because we calculate the dust attenuation–corrected Σ_SFR by combining GALEX FUV and WISE 22 μm (W4; see Section 2.2), the galaxies without FUV or W4 images (35 galaxies of 262) are excluded. We then select the galaxies with CO(1–0) and/or CO(2–1) data cubes available. First, we refer to the following catalogs of the previous CO survey projects: the Nobeyama CO Atlas of Nearby Spiral Galaxies (hereafter referred to as the NRO Atlas; Kuno et al. 2007), COMING (Sorai et al. 2019), the HERA CO Line Extragalactic Survey (HERACLES; Leroy et al. 2009), and PHANGS-ALMA (Leroy et al. 2021). For the galaxies outside these projects, we search the Atacama Large Millimeter/submillimeter Array (ALMA) archival data sets by using the python package astroquery. We extract the galaxies with available mapping data from the Atacama Compact Array (ACA) (7m+TP) from Cycle 1. Here, the galaxies where CO mapping was only performed on a part of the disk are excluded (e.g., NGC 6744). As a result, we extract 77 barred galaxies with available GALEX FUV, W4, and CO(1–0) and/or CO(2–1) data sets.

Finally, from the 77 galaxies, we select those with an apparent long-bar structure. Bar structures are often defined as an ellipse that is determined by the center, semimajor axis (or bar length; R_bar), ellipticity (_bar), and position angle (PA_bar). These parameters are calculated visually or based on an isophote with maximum ellipticity, using stellar images. In this study, we select the galaxies with R_bar ≥ 375. This selection is based on the requirement that the major axis of the bar (2R_bar) must be at least five times larger than the angular resolution of the tracers (i.e., 15''; see Sections 2.2 and 2.3) to distinguish between the center, the bar, and the bar end. For most galaxies, we adopt the catalog presented by Herrera-Endoqui et al. (2015). The authors define the bar parameters by using Spitzer 3.6 μm images. For the galaxies outside the catalog, when available, we adopt the values reported in the literature that were calculated based on near-IR images (Kuno et al. 2007; Menendez-Delmestre et al. 2007; Díaz-García et al 2021). For galaxies that did not have any corresponding literature, we visually ascertain whether R_bar is larger than 375; however, no galaxies are visually selected. Here, we manually exclude two galaxies (i.e., NGC 1055 and NGC 3556). These galaxies are listed in Herrera-Endoqui et al. (2015) as those with R_bar ≥ 375. However, the true inclination appears to be nearly edge-on, based on a visual inspection of the WISE and other optical images. Consequently, we select 35 galaxies with R_bar ≥ 375. These 35 galaxies are referred to as "long-bar sample galaxies" in this paper. Table 1 summarizes the properties of the long-bar sample galaxies.

Table 1. Properties of the Long-bar Sample Galaxies

Galaxy	Morp.	D	i	References	$\mathrm{log}{M}_{\star }$	logsSFR	Rbar	PAbar	bar	References	CO(1–0)	${T}_{\mathrm{rms}}^{10}$	CO(2–1)	${T}_{\mathrm{rms}}^{21}$
		(Mpc)	(°)		(M⊙)	(yr−1)	('')	(°)				(mK)		(mK)
NGC 0613	SBbc	25.1	35.7	1	10.96	−10.03	85.1	125	0.75	5	7	46.2	⋯	⋯
NGC 1097x	SBb	13.6	48.6	2	10.76	−10.08	95.1	141	0.65	5	⋯	⋯	2	0.3
NGC 1291	SB0a	8.6	29.3	1	10.60	−11.33	95.1	165	0.41	5	10	23.2	⋯	⋯
NGC 1313	SBd	4.3	34.8	2	9.55	−9.69	54.0	17	0.71	5	⋯	⋯	10	2.1
NGC 1300x	SBbc	19.0	31.8	2	10.50	−10.44	84.4	99	0.78	5	10	6.8	2	0.8
NGC 1317	SABa	19.1	23.2	2	10.45	−10.80	42.0	150	0.24	4	10	7.9	2	0.5
NGC 1350	SBab	20.9	64.6	1	10.73	−11.05	55.7	36	0.58	5	10	6.3	⋯	⋯
NGC 1365x	SBb	19.6	55.4	2	11.06	−9.83	91.4	86	0.63	5	8	17.8	2	0.8
NGC 1433	SBab	18.6	28.6	2	10.68	−10.65	86.2	95	0.65	5	⋯	⋯	2	0.9
IC 0342x	SABcd	3.3	31.0	3	10.29	−9.67	120.0	155	0.42	3	3	34.8	⋯	⋯
NGC 1512	SBa	18.8	42.5	2	10.60	−10.49	73.5	42	0.66	5	⋯	⋯	2	0.6
NGC 1672x	SBb	19.4	42.6	2	10.78	−9.90	69.3	96	0.67	5	⋯	⋯	2	0.8
NGC 2903x	SABbc	10.0	66.8	2	10.61	−10.13	70.6	28	0.80	5	3	28.3	2	0.3
NGC 3049	SBab	30.8	58.0	1	9.97	−9.76	38.2	34	0.78	5	⋯	⋯	9	1.5
NGC 3351	SBb	10.0	45.1	2	10.28	−10.18	53.6	112	0.46	5	3	28.6	2	0.8
NGC 3359	SBc	18.8	47.2	1	10.15	−10.00	48.1	19	0.76	5	7	59.9	⋯	⋯
NGC 3368	SABab	10.9	51.1	1	10.55	−10.75	62.8	124	0.43	5	7	48.0	⋯	⋯
NGC 3627x	SABb	11.3	57.3	2	10.78	−10.20	59.1	160	0.76	5	3	32.7	2	0.4
NGC 4051x	SABbc	14.6	48.7	1	10.32	−10.01	59.1	132	0.73	5	3	33.6	⋯	⋯
NGC 4258	SABbc	7.4	68.3	1	10.55	−10.57	96.8	2	0.50	5	7	45.3	⋯	⋯
NGC 4293	SB0a	15.8	65.0	2	10.36	−10.64	70.6	75	0.75	5	⋯	⋯	2	0.8
NGC 4303x	SABbc	17.0	23.5	2	10.74	−10.01	54.0	178	0.69	5	3	35.6	2	0.7
NGC 4321x	SABbc	15.2	38.5	2	10.72	−10.17	54.6	108	0.59	5	3	10.8	2	0.5
NGC 4535x	SABc	15.8	44.7	2	10.51	−10.19	43.1	42	0.68	5	3	22.2	2	0.7
NGC 4536	SABbc	16.3	66.0	2	10.44	−9.91	39.8	77	0.50	5	3	24.7	2	0.2
NGC 4548x	SBb	16.2	38.3	2	10.52	−10.80	58.0	61	0.53	5	3	24.5	2	0.5
NGC 4579x	SABb	21.0	40.2	2	10.98	−10.66	42.2	53	0.48	5	3	33.0	2	0.5
NGC 4725	SABab	13.6	45.4	1	10.72	−10.70	130.6	45	0.68	5	⋯	⋯	9	2.4
NGC 4731	SBcd	13.3	64.0	2	9.79	−10.02	57.6	127	0.81	5	⋯	⋯	2	0.2
NGC 4941	SABab	15.0	53.4	2	10.06	−10.44	93.0	16	0.53	4	⋯	⋯	2	0.3
NGC 5236x	SABc	4.9	24.0	2	10.52	−9.89	125.5	50	0.69	5	3	44.1	2	0.6
NGC 5457x	SABcd	6.9	16.1	1	10.54	−10.00	51.0	82	0.45	4	3	24.4	9	4.6
NGC 6946x	SABcd	5.5	40.0	3	10.30	−9.79	60.0	17	0.46	4	3	26.3	9	3.3
NGC 6951x	SABbc	24.1	30.0	3	10.73	−10.19	44.0	88	0.54	6	3	13.5	⋯	⋯
NGC 7496x	SBb	18.7	35.9	2	10.10	−9.76	39.8	147	0.76	5	⋯	⋯	2	0.3

Notes. (1) ^X The gas-rich long-bar sample galaxies (Section 2.5). (2) The morphological type from RC3. (3) Distance. (4) Inclination. (5) The references for columns (3) and (4). (6) Stellar mass. (7) Specific SFR. The derivations of columns (6) and (7) are described in Section 2.1. (8) Bar length. (9) The position angle of the bar. (10) The ellipticity of the bar. (11) The references for columns (8)–(10). (12) The references for the CO(1–0) data. (13) The median rms noise of the CO(1–0) cube in 10 km s⁻¹ bins. (14) The references of the CO(2–1) data. (15) The median rms noise of the CO(2–1) cube in 10 km s⁻¹ bins.

References. (1) S⁴G (Sheth et al. 2010). (2) PHANGS-ALMA (Leroy et al. 2021). (3) The Nobeyama CO Atlas (Kuno et al. 2007). (4) Menendez-Delmestre et al. (2007). (5) Herrera-Endoqui et al. (2015). (6) Díaz-García et al (2021). (7) NRO COMING (Sorai et al. 2019). (8) Egusa et al. (2022). (9) HERACLES (Leroy et al. 2009). (10) From ALMA archival data (this work).

A machine-readable version of the table is available.

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The Σ_SFR is calculated from a linear combination of the GALEX FUV (Gil de Paz et al. 2007) and WISE W4 (Wright et al. 2010) intensities, as reported by Leroy et al. (2019).

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{SFR}}=({{\rm{\Sigma }}}_{\mathrm{SFR},\mathrm{FUV}}+{{\rm{\Sigma }}}_{\mathrm{SFR},{\rm{W}}4})\cos i,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{SFR},\mathrm{FUV}}=1.04\times {10}^{-1}\left[\displaystyle \frac{{C}_{\mathrm{FUV}}}{{10}^{-43.35}}\right]{I}_{\mathrm{FUV}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{SFR},{\rm{W}}4}=3.24\times {10}^{-3}\left[\displaystyle \frac{{C}_{{\rm{W}}4}}{{10}^{-42.7}}\right]{I}_{{\rm{W}}4},\end{eqnarray} \tag{ 3 }$$

where Σ_SFR is in units of M_⊙ yr⁻¹ kpc⁻² and I_FUV and I_W4 are the FUV and 22 μm intensities in units of MJy sr⁻¹, respectively, while the conversion factors of C_FUV and C_W4 are 10^−43.42 and ${10}^{-42.73}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{(\mathrm{erg}\,{{\rm{s}}}^{-1})}^{-1}$, respectively. The systematic uncertainty, translating from intensity to SFR, is estimated to be ≈0.1 dex. Although the SFR can be calculated using the WISE 12 μm (W3) and GALEX NUV filters, we consider using FUV and W4 to be a better method for the following reasons. The coefficient for converting the W3 intensity to the SFR estimated from the spectral energy distribution fitting shows large cell-to-cell scatter (e.g., Leroy et al. 2019), because the W3 filter is prone to strong contaminations by the 11.3 μm polycyclic aromatic hydrocarbon features (Engelbracht et al. 2005). Additionally, the NUV filter is prone to more contamination from lower-mass and older stars compared with the FUV filter. The systematic uncertainties in Σ_SFR,W4 are discussed in Section 4.2.2.

We use the delivered FUV and W4 images from the z = 0 Multiwavelength Galaxy Synthesis GALEX-WISE Atlas data release 1 (Leroy et al. 2019), which are publicly available online. ⁶ We use the image atlas, which consists of a set of background-subtracted images on matched astrometry with a matched resolution, at 15'' resolution. In this study, the pixel sizes of the images are regridded from the original size of 5.5'' to half the resolution of 75, by using the Python reproject package. The Σ_SFR maps are shown in Figure 2(a). (The complete figure set (35 images) is available in the online journal.)

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2.3.1. CO(1–0) and CO(2–1) Data Cubes

2.3.2. Conversion to Σ_mol

First, we make velocity-integrated intensity maps (i.e., moment-zero maps). For the spectrum of each line of sight (pixel), we identify consecutive channels, in which the signals are above 3σ_rms. For the data cubes not observed with ALMA, we adopt the median rms noise listed in Table 1 as the σ_rms. For the ALMA data, the σ_rms is calculated in each line of sight, because the noise in the cube is nonuniform, due to the primary beam pattern or/and the difference in the integration time within the field of view (FoV). We subsequently expand these channels to include all adjacent channels, in which the signals are above 1.5σ_rms. The velocity-integrated intensity of each pixel is defined as the sum of the masked channels; Σ_mol is derived from the CO(1–0) line as follows:

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{mol}}={\alpha }_{\mathrm{CO}}{I}_{\mathrm{CO}(1-0)}\cos i,\end{eqnarray} \tag{ 4 }$$

where Σ_mol is in units of M_⊙ pc⁻², α_CO is the CO-to-H₂ conversion factor in units of ${M}_{\odot }\,{({\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2})}^{-1}$, and I_CO(1−0) is the CO(1–0) velocity-integrated intensity in units of K km s⁻¹. We adopt a constant Milky Way α_CO of $4.35\,{M}_{\odot }\,{({\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2})}^{-1}$, including a factor of 1.36, to account for the presence of helium (Bolatto et al. 2013). We discuss the uncertainties in the α_CO in Section 4.2.1.

From the CO(2–1) line, Σ_mol is derived by assuming the integrated intensity ratio (R₂₁) of CO(2–1)/CO(1–0) as follows:

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{mol}}=({\alpha }_{\mathrm{CO}}/{R}_{21}){I}_{\mathrm{CO}(2-1)}\cos i.\end{eqnarray} \tag{ 5 }$$

Here, we adopt a constant R₂₁ of 0.65, based on recent statistical studies of the ratio on a kiloparsec scale in nearby galaxies (den Brok et al. 2021; Leroy et al. 2022). We discuss the variation in the R₂₁ within the galaxy in Section 4.5. The Σ_mol maps from CO(1–0) and CO(2–1) are shown in Figures 2(c) and (d), respectively. (The complete figure set (35 images) is available in the online journal.)

The distinction between the center, bar, and bar end is important, because the star formation activity is observably different among these environments, as described in Section 1. In this study, we define the center, bar, and bar-end regions of the galaxy based on the stellar bar structure defined by the ellipse (see Section 2.1). We define a rectangle with a width of 2 × 1.25 × R_bar, a height of 2R_bar(1 − _bar), and a position angle of PA_bar, as shown in Figure 2. In this rectangle, we define the center, bar, and bar end as the regions where the distances to the minor axis of the ellipse (R) are 0.0–0.25R_bar, 0.25R_bar−0.75R_bar, and 0.75R_bar–1.25R_bar, respectively. The region outside this rectangle and inside the FoV of the CO data cube is defined as a disk. The environmental mask maps are shown in Figure 2(b). (The complete figure set (35 images) is available in the online journal.) Owing to the definition of the bar end being R/R_bar = 0.75–1.25, the peak of the Σ_SFR around R/R_bar = 1.0 and its surrounding region are excluded from the bar. The definition of the center being R/R_bar = 0−0.25 also excludes the majority of the bulge light from the bar; referring to Salo et al. (2015), who measured the effective radius (half-light radius; R_eff) of the bulge based on the Sérsic profiles for S⁴G galaxies, we compare the 0.25R_bar and R_eff for 23 galaxies in the long-bar sample. The 0.25R_bar/R_eff ranges from 1.4 to 10, with a median of 2.5. Therefore, the majority of the bulge light of the long-bar sample seems to be included within the region defined as 0–0.25R_bar.

As described in Section 1, the definition of the bar region has varied across studies. In many of the studies that focused on the individual barred galaxies, the bar region was defined as the region between the center and the bar-end regions, based on CO moment-zero, optical, or near-IR images (e.g., Handa et al. 1991; Muraoka et al. 2016; Law et al. 2018; Yajima et al. 2019; Maeda et al. 2020). By contrast, some studies included (part of) the center and the bar end in the region defined as the bar (e.g., Momose et al. 2010). In addition, the star formation in the bar region has sometimes been discussed using radial profiles (e.g., James et al. 2009; Hirota et al. 2014; Muraoka et al. 2019). Notably, our definition of the bar region does not include the center and bar-end regions, unlike recent statistical studies (Díaz-García et al 2021; Querejeta et al 2021).

To compare the SFEs between the bar and disk regions, we select 18 galaxies with gas-rich bars and disks from the long-bar sample. In our CO data cubes, the rms noise exhibited a large variation. The rms noise is much higher in the CO(1–0) cube than in the CO(2–1) cube (see Table 1 and Figure 2); the typical surface densities corresponding to the 3σ upper limit of CO(1–0) and CO(2–1) are ∼5.0 M_⊙ pc⁻² and ∼1.0 M_⊙ pc⁻², respectively. Therefore, for a fair comparison, we focus on the region where Σ_mol ≧ 5 M_⊙ pc⁻². According to this, in this study, we define a gas-rich bar as a bar region where more than 40% of the area is Σ_mol ≧ 5 M_⊙ pc⁻². This condition is represented as follows:

$$\begin{eqnarray}&&{f}_{\mathrm{bar}}({{\rm{\Sigma }}}_{\mathrm{mol}}\geqq 5)\equiv {A}_{\mathrm{bar}}({{\rm{\Sigma }}}_{\mathrm{mol}}\geqq 5)/{A}_{\mathrm{bar}}\geqq 0.40,\end{eqnarray} \tag{ 6 }$$

where A_bar(Σ_mol ≧ 5) is the area of the region where Σ_mol ≧ 5 M_⊙ pc⁻² in the bar region, and A_bar is the total area of the bar region. The f_bar(Σ_mol ≧ 5) of each CO cube is shown in Figure 2. We select the CO(1–0) and CO(2–1) cubes that satisfy this condition. This condition allows us to select those galaxies with a continuous distribution of molecular gas with Σ_mol ≧ 5 M_⊙ pc⁻² from the center to the bar-end region. We select 22 galaxies with a gas-rich bar from the long-bar sample galaxies.

To compare the SFEs between the bar and disk, we require sufficient pixels with Σ_mol ≧ 5 M_⊙ pc⁻² in the disk. From visual inspection, we thus exclude NGC 0613, NGC 1317, NGC 4536, and NGC 4941, as these galaxies have few or no pixels with Σ_mol ≧ 5 M_⊙ pc⁻² in the disk (see Figure 2). As for NGC 1365, we exclude only CO(2–1), due to the small FoV. Consequently, we select 18 galaxies as gas-rich long-bar sample galaxies; these are shown with X marks in Table 1. We use the CO(1–0) and CO(2–1) data cubes for 13 and 14 galaxies, respectively. We use both CO lines for nine galaxies. The angular resolution of 15'' corresponds to 0.3–1.8 kpc of the gas-rich long-bar sample galaxies. In the disks of most gas-rich long-bar sample galaxies, the regions where Σ_mol ≧ 5 M_⊙ pc⁻² correspond to the regions where Σ_SFR ≧ 10^−2.5 M_⊙ yr⁻¹ kpc⁻². Although the 3σ upper limit of the CO(1–0) cube is much higher than 5 M_⊙ pc⁻² in IC 0342, NGC 4303, and NGC 5236, they are included in the gas-rich long-bar sample, because CO(1–0) is detected in most of the FoV. In the following section, we investigate the SFEs in the bar regions of the 18 gas-rich long-bar sample galaxies.

Figure 3 shows the f_bar(Σ_mol ≧ 5) of the target sample galaxies on the stellar mass (M_star) versus specific SFR (sSFR) diagram. The M_star and sSFR are listed in Table 1. The SFRs of the host galaxies are calculated by integrating the region within the radius of D₂₅ in the Σ_SFR map. Similar to the SFR, M_star is calculated using the stellar mass surface density (Σ_star) map. We follow the calculation of the Σ_star by Leroy et al. (2019), as ${{\rm{\Sigma }}}_{\mathrm{star}}=330({{\rm{\Upsilon }}}_{* }^{3.4}/0.5){I}_{{\rm{W}}1}\cos i$, where Σ_star is in units of M_⊙ pc⁻² and I_W1 is the intensities of the WISE 3.4 μm image in units of MJy sr⁻¹. ${{\rm{\Upsilon }}}_{* }^{3.4}$ is the near-IR mass-to-light ratio in units of ${M}_{\odot }\,{L}_{\odot }^{-1}$. We adopt $0.35\,{M}_{\odot }\,{L}_{\odot }^{-1}$, which is the average value in the PHANGS-ALMA sample (Leroy et al. 2021).

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In Figure 3, the galaxies in the gas-rich long-bar sample are represented by the square symbols. The other long-bar sample galaxies are represented by circles. We find that f_bar(Σ_mol ≧ 5) depends on the M_star and sSFR; for barred galaxies with low stellar mass (M_star ≦ 10¹⁰ M_⊙), or those located on the lower side of the main sequence, f_bar(Σ_mol ≧ 5) is small (<0.4), and therefore the Σ_mol in the bar tends to be low. By contrast, the galaxies with gas-rich bars (f_bar(Σ_mol ≧ 5) > 0.4) tend to be located in the region where M_star ≧ 10¹⁰ M_⊙ and the upper side of the main sequence. Most of the gas-rich long-bar sample galaxies are located in this region. Such a dependence of the f_bar(Σ_mol ≧ 5) is possibly related to the evolution of the barred galaxies, such as the bar-quenching process. Investigating this relationship could be interesting; however, this is beyond the scope of this study and could be a future study. The sample bias in our study is discussed in Section 4.3.

Figure 4 shows the profiles of Σ_mol, Σ_SFR, and SFE as a function of R/R_bar for the gas-rich long-bar sample galaxies. The SFE is expressed as Σ_SFR/Σ_mol. We divide R/R_bar =0.0–1.25 into 10 bins and derive the median (the circles in the figure), the mean (the squares), and the CO flux-weighted mean (the triangles) in each bin. The bar with the circles shows the range of the 75th–25th percentile, which is known as the interquartile range (IQR). The values in the disk region are shown at R/R_bar = 1.3, for convenience. Additionally, the SFEs in the disk are represented using horizontal lines. In addition, we obtained the Σ_mol by stacking the CO profiles in each bin, which are represented using the cross symbols. The stacking method followed in this study is the same as that in Maeda et al. (2020; refer to Section 3.3 for details). In this case, the mean Σ_SFR is used in the SFE calculation. In all the gas-rich long-bar sample galaxies, significant variations are observed in the SFE, from the center to the bar and bar end; the SFEs tend to be higher at the center and the bar end than at the bar. The dip in the SFE profile tends to be located at around R/R_bar = 0.5, which corresponds to the midpoint between the center and the bar end. These results clearly demonstrate the importance of the distinction between the center, bar, and bar end in SFE analysis.

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As reported in other studies (e.g., Leroy et al. 2022), the NRO Atlas data cubes suffer from visible mapping artifacts and poor baselines. Therefore, we compare the CO(1–0) fluxes from the NRO Atlas with those from various independent observations, when available. Seven galaxies—namely NGC 2903, NGC 3627, NGC 4303, NGC 4321, NGC 4535, NGC 5236, and NGC 6951—were compared; CO(1–0) was detected in the COMING project (Sorai et al. 2019) for NGC 2903, NGC 3627, and NGC 4303. The difference is within a factor of 1.5. The CO(1–0) in NGC 4321 was observed using ALMA, under the project 2011.0.00004.SV in Cycle 0, as science verification data, and both the CO fluxes are comparable. We confirm that the CO fluxes in the bar of NGC 5236 reported by Hirota et al. (2014) and those in the NRO Atlas are comparable. We compare the CO fluxes in NGC 4535 and NGC 6951 with those reported by Díaz-García et al (2021). The differences in the CO fluxes in NGC 4535 are within a factor of 1.5. However, the CO fluxes in the bar of NGC 6951 reported by Díaz-García et al (2021) are a factor of 6 higher than those in the NRO Atlas. Owing to the lack of consensus on the CO(1–0) flux, NGC 6951 is not included in the remaining study. We also compare the CO(2–1) fluxes between PHANGS-ALMA and HERACLES for the galaxies for which both sets of data available (i.e., NGC 2903, NGC 3627, NGC 4321, and NGC 4579). The difference is within a factor of 1.2, which is consistent with the values reported by Leroy et al. (2022).

Figure 5 displays our main results, i.e., the SFE profiles from R/R_bar = 0.0 to 1.25 that are normalized by the SFE in the disk of the gas-rich long-bar sample galaxies. The results obtained using CO(1–0) are shown in Figures 5(a), (c), (e), and (g), and those obtained using CO(2–1) are shown in Figures 5(b), (d), (f), and (h). Panels (a) and (b) show the median SFE in each bin, which is normalized by the median SFE in the disk. The median values and IQRs of all the samples in each bin are shown as the red squares and bars, respectively. The results when we use the SFE derived by the stacking method, the mean SFE, and the CO flux-weighted SFE are shown in panels (c)–(h). Table 2 summarizes the SFE/SFE_disk for each method. This table shows the median values and IQRs of all the gas-rich long-bar sample galaxies in each R/R_bar bin, which correspond to the red squares and bars in Figure 5, respectively.

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Table 2. Normalized SFEs from R/R_bar = 0.00 to 1.25

		SFE/SFEdisk [CO(1–0)]				SFE/SFEdisk [CO(2–1)]
Mask	R/Rbar	Median	Stacking	Mean	Weighted	Median	Stacking	Mean	Weighted
Center	0.000–0.125	${1.70}_{-0.31}^{+0.86}$	${2.24}_{-0.78}^{+2.04}$	${2.00}_{-0.78}^{+0.51}$	${2.16}_{-0.87}^{+1.65}$	${1.02}_{-0.24}^{+0.39}$	${1.20}_{-0.44}^{+0.38}$	${1.07}_{-0.39}^{+0.30}$	${1.16}_{-0.41}^{+0.25}$
Center	0.125–0.250	${1.11}_{-0.23}^{+0.54}$	${1.14}_{-0.31}^{+0.59}$	${1.02}_{-0.28}^{+0.67}$	${1.05}_{-0.25}^{+0.47}$	${0.90}_{-0.18}^{+0.35}$	${0.92}_{-0.25}^{+0.45}$	${0.88}_{-0.27}^{+0.34}$	${0.90}_{-0.25}^{+0.36}$
Bar	0.250–0.375	${0.71}_{-0.12}^{+0.12}$	${0.59}_{-0.08}^{+0.32}$	${0.59}_{-0.10}^{+0.15}$	${0.56}_{-0.08}^{+0.23}$	${0.65}_{-0.07}^{+0.21}$	${0.61}_{-0.11}^{+0.14}$	${0.67}_{-0.13}^{+0.25}$	${0.59}_{-0.08}^{+0.14}$
Bar	0.375–0.500	${0.68}_{-0.22}^{+0.13}$	${0.57}_{-0.07}^{+0.14}$	${0.56}_{-0.06}^{+0.23}$	${0.52}_{-0.06}^{+0.19}$	${0.63}_{-0.07}^{+0.20}$	${0.58}_{-0.10}^{+0.11}$	${0.63}_{-0.08}^{+0.11}$	${0.55}_{-0.09}^{+0.14}$
Bar	0.500–0.625	${0.75}_{-0.18}^{+0.09}$	${0.62}_{-0.11}^{+0.21}$	${0.70}_{-0.19}^{+0.09}$	${0.66}_{-0.19}^{+0.12}$	${0.75}_{-0.16}^{+0.13}$	${0.66}_{-0.17}^{+0.19}$	${0.72}_{-0.19}^{+0.16}$	${0.63}_{-0.13}^{+0.19}$
Bar	0.625–0.750	${0.83}_{-0.22}^{+0.17}$	${0.62}_{-0.07}^{+0.35}$	${0.71}_{-0.16}^{+0.18}$	${0.65}_{-0.09}^{+0.25}$	${0.75}_{-0.12}^{+0.07}$	${0.72}_{-0.20}^{+0.14}$	${0.74}_{-0.12}^{+0.18}$	${0.69}_{-0.16}^{+0.12}$
Bar end	0.750–0.875	${1.01}_{-0.27}^{+0.49}$	${0.95}_{-0.29}^{+0.37}$	${0.97}_{-0.32}^{+0.44}$	${0.93}_{-0.30}^{+0.38}$	${0.86}_{-0.12}^{+0.31}$	${0.81}_{-0.24}^{+0.32}$	${0.88}_{-0.26}^{+0.31}$	${0.79}_{-0.19}^{+0.31}$
Bar end	0.875–1.000	${1.22}_{-0.42}^{+0.66}$	${1.02}_{-0.35}^{+0.59}$	${1.20}_{-0.49}^{+0.48}$	${1.14}_{-0.44}^{+0.63}$	${0.99}_{-0.23}^{+0.37}$	${0.94}_{-0.28}^{+0.46}$	${0.97}_{-0.32}^{+0.46}$	${0.91}_{-0.22}^{+0.39}$
Bar end	1.000–1.125	${1.33}_{-0.46}^{+0.53}$	${1.41}_{-0.64}^{+0.55}$	${1.22}_{-0.48}^{+0.68}$	${1.27}_{-0.53}^{+0.42}$	${1.12}_{-0.32}^{+0.18}$	${1.14}_{-0.37}^{+0.34}$	${1.07}_{-0.29}^{+0.46}$	${1.10}_{-0.33}^{+0.27}$
Bar end	1.125–1.250	${1.21}_{-0.30}^{+0.22}$	${0.93}_{-0.16}^{+0.69}$	${1.06}_{-0.21}^{+0.25}$	${1.10}_{-0.27}^{+0.26}$	${1.20}_{-0.29}^{+0.26}$	${1.05}_{-0.17}^{+0.36}$	${1.14}_{-0.29}^{+0.34}$	${1.04}_{-0.19}^{+0.30}$

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We find the SFE in the bar to be systematically lower than that in the disk, regardless of whether Σ_mol is measured using CO(1–0) or CO(2–1). The median normalized SFE (SFE/SFE_disk) is 0.6–0.8 in R/R_bar = 0.25–0.75, regardless of the calculation methods. The SFE/SFE_disk is at a minimum at around R/R_bar = 0.5. These results suggest that massive star formation in the bar region is systematically suppressed in comparison with the disk region in massive (M_star ≧ 10¹⁰ M_⊙) high-sSFR (i.e., upper side of the main sequence) galaxies with gas-rich bars and disks. Our results are consistent with those reported in previous studies that observed individual barred galaxies (Handa et al. 1991; Momose et al. 2010; Hirota et al. 2014; Muraoka et al. 2016; Pan & Kuno 2017; Law et al. 2018; Yajima et al. 2019; Maeda et al. 2020). Although the SFE in the bar region (R/R_bar = 0.25–0.75) is systematically suppressed, its scatter is approximately 0.5 dex; some areas possess SFEs comparable to those in the disk, whereas others possess significantly lower SFEs than those in the disk. Additionally, the degree of suppression of star formation appears to vary among galaxies and within a galaxy.

In the center (R/R_bar = 0.0–0.25), the SFE is higher or comparable to that in the disk. The SFE/SFE_disk that is obtained from CO(1–0) is higher than that obtained from CO(2–1). This is because the Σ_mol in the center would be overestimated when we use the CO(2–1) line, because of the high R₂₁ of >0.65 in the center (refer to Section 4.5). In the bar end (R/R_bar = 0.75–1.25), the SFE/SFE_disk are scattered in the range of 1.0–1.3, with peaks at R/R_bar ∼ 1.0. The star formation in the bar end appears to be slightly enhanced in comparison with that in the disk.

Although we focus on the region where Σ_mol ≧ 5 M_⊙ pc⁻², which corresponds to the typical 3σ upper limit of the CO(1–0) data cubes, our conclusion does not depend on this threshold surface density. For the PHANGS galaxies in the gas-rich long-bar sample, we remeasure the SFE/SFE_disk by focusing on the region where Σ_mol ≧ 1 M_⊙ pc⁻², which corresponds to the typical 3σ upper limit of the PHANGS data. As a result, the SFE/SFE_disk changes slightly. The differences are within 10%.

The trend of obtaining a lower SFE in the bar region is additionally observed in the Kennicutt–Schmidt diagram or the Σ_mol versus Σ_SFR diagram, as shown in Figure 6. The large colored circles correspond to the median values in each R/R_bar bin for each galaxy. The gray data points correspond to the pixel values in the disk regions, the best-fitting line of which is represented by the black line. We fit the data points in the disk region using the ordinary least-squares bisector method (Isobe et al. 1990). The best-fitting relation is described as ${{\rm{\Sigma }}}_{\mathrm{SFR}}={10}^{-3.72}{{\rm{\Sigma }}}_{\mathrm{mol}}^{1.28}$ for CO(1–0) and ${{\rm{\Sigma }}}_{\mathrm{SFR}}={10}^{-3.20}{{\rm{\Sigma }}}_{\mathrm{mol}}^{0.99}$ for CO(2–1). These slopes are consistent with those reported in previous studies (e.g., Bigiel et al. 2011; Yajima et al. 2021). As shown in the middle panels, the median values in the bar regions are systematically located below the best-fitting lines, unlike those in the center and bar-end regions. Similar to Figure 5, this result suggests that in the galaxies with gas-rich bars and disks, star formation tends to be suppressed in the bars, in comparison with that in the disks.

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We investigate the effect of angular resolution on the SFE in the bar. We remeasure the SFE using images convoluted to beam sizes of 25'' and 35''. In the case of the beam size of 25'' (35''), the bar lengths of approximately 50 % (70 %) galaxies in the gas-rich long-bar sample are less than five times the beam size. Figure 7 displays the normalized SFE profiles when beam size is 15'' (orange), 25'' (green), and 35'' (red). Here, we use the pixels with Σ_mol ≧ 5 M_⊙ pc⁻² in the images with a beam size of 15''. As the beam size increases, the SFE profile is smoothed and becomes constant. This result indicates that the SFE in the bars is possibly overestimated, if the sample contains galaxies with bar lengths that are less than five times the beam size. One reason for the constant SFE radial profile or the nonenvironmental dependence of the SFE reported in the recent statistical studies could be due to the beam size being larger compared to the bar lengths of the sample galaxies. In the studies by Muraoka et al. (2016), Díaz-García et al (2021), and Querejeta et al (2021), the bar lengths of about 50%, 80%, and 40% sample galaxies are less than five times the beam sizes of the images used in these studies (17'', 215, and 15'', respectively). Therefore, the SFE radial profiles and maps would be smoothed and become constant.

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4.2.1. CO-to-H₂ Conversion Factor

4.2.2. SFR

The SFR derived from IR emissions includes various contaminants. One is the dust emissions from the old stellar population, which are known as IR cirrus. The contribution from IR cirrus has been reported to be 30%–60% when Σ_SFR is less than 10⁻² M_⊙ yr⁻¹ kpc⁻² (Leroy et al. 2012). On average, the Σ_SFR that are less than 10⁻² M_⊙ yr⁻¹ kpc⁻² obtained from GALEX FUV and WISE W4 are 20%–30% higher than those obtained from the Balmer decrement–corrected Hα in the PHANGS-ALMA sample galaxies (Leroy et al. 2021). Therefore, the Σ_SFR in this study may be overestimated. However, the IR cirrus does not seem to change our conclusion. Because the Σ_SFR are within similar ranges in the bar and disk regions (Figure 6), the contribution from IR cirrus is similar between the bar and disk regions, and the SFE/SFE_disk in the bar region is not expected to change significantly.

Another possible contamination is active galactic nuclei (AGNs), which would contribute to strong nuclear IR emission (e.g., Catalan-Torrecilla et al. 2015). The Σ_SFR in the center may be overestimated, although the contribution to the Σ_SFR in the bar region is considered to be small, because we select galaxies with an apparently large bar length. We measure the SFE by distinguishing non-AGNs from AGNs in the gas-rich long-bar sample galaxies. The Seyferts or LINERs that are extracted based on the catalogs by Ho et al. (1997) and Veron-Cetty & Veron (2010) are as follows: NGC 1097, NGC 1365, NGC 1672, NGC 3627, NGC 4051, NGC 4303, NGC 4321, NGC 4548, NGC 4579, NGC 6951, and NGC 7496. Figure 8 shows the normalized SFE profiles of the non-AGN and AGN samples. The SFEs in the centers of both samples are comparable, which suggests that the contamination by AGNs is small. Regardless of the presence of AGNs, the suppression of star formation is commonly observed. Interestingly, in the bar-end region, the SFE/SFE_disk tends to be >0 in the AGN sample and <0 in the non-AGN sample.

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The environmental dependence of the SFE that is found in this study is inconsistent with that reported in recent statistical studies (Muraoka et al. 2019; Díaz-García et al 2021; Querejeta et al 2021). Here, we summarize four possible causes of this inconsistency.

(1) Bar definition. In these studies, (parts of) the center and bar-end regions are included in the region defined as the bar region, which would make the SFE in the bars high. Our results showing that the SFE/SFE_disk in the center and bar-end regions are higher than that in the bar region (Figure 5) support this possibility. In fact, the median SFE/SFE_disk in the range of R/R_bar = 0.0–1.0 in all the gas-rich long-bar sample is close to unity; ${0.91}_{-0.09}^{+0.18}$ for CO(1–0) and ${0.88}_{-0.15}^{+0.05}$ for CO(2–1).

(2) Spatial resolution. As described in Section 4.1, the bar lengths of about half the sample galaxies in these studies are less than five times the beam sizes of the images that they used, which would smooth the SFE profiles.

(3) CO-to-H₂ conversion factor. As already mentioned in Section 4.2.1, Querejeta et al (2021) tested the impact of an adopted α_CO on the SFE and found that the bar SFE becomes smaller with the constant α_CO than with the metallicity-dependent α_CO. However, Muraoka et al. (2019) and Díaz-García et al (2021) adopted the constant α_CO and reported the SFE as being independent of the environment. Therefore, the inconsistency between these two studies should be attributed to other factors than the α_CO.

(4) Sample bias. The difference between the sample galaxies in this study and the recent statistical studies is possibly the cause of this inconsistency. We mainly focus on the galaxies that are located on the upper side of the main sequence, with M_star ≧ 10¹⁰ M_⊙ in the M_star versus sSFR diagram (Figure 3). However, the sample galaxies in the recent statistical studies are located within, as well as outside, the region, as shown in the figure. Some of the sample galaxies in Muraoka et al. (2019) possess a high sSFR (≧10^−9.6 yr⁻¹). Many of the sample galaxies in Querejeta et al (2021) and Díaz-García et al (2021) are located in the region where M_star ≦ 10¹⁰ M_⊙ and on the lower side of the main sequence. The SFE profiles of the galaxies located in these regions may differ from those of our gas-rich long-bar sample galaxies. The SFEs in bars with a low stellar mass of M_star ≦ 10¹⁰ M_⊙ may be comparable to those in the disk. The gas in the galaxies on the lower side of the main sequence may be depleted in the bar and the disk, as shown by the blue symbols in Figure 3, and the star formation may be quenched in the entire disk, which may result in the SFE being constant. Investigating the relationship between the locations of the host galaxies in the main sequence and the changes in SFE within the galaxies will be important. CO and SFR tracer observations, with high angular resolution and sensitivity, are required for the further examination of the SFEs of low-M_star and low-sSFR galaxies, using the same methodology as that used in this study. This is because the apparent bar and disk sizes of low–stellar mass galaxies are small, and the Σ_mol and Σ_SFR of the galaxies located on the lower side of the main sequence are thought to be much lower than 5 M_⊙ pc⁻² and 10^−2.5 M_⊙ yr⁻¹ kpc⁻², respectively.

Note that the method of SFR calculation is not the source of the inconsistency, because the SFR is derived from WISE W4 and GALEX FUV in all statistical studies, as in our study. ¹¹

As mentioned in Section 3.2, the degree of star formation suppression seems to vary among galaxies and within a galaxy. What determines the degree of the suppression? One promising parameter is the strength of the noncircular motion in the bar region. Star formation has arguably been suppressed by some dynamical effects, such as strong shocks, large shears, and fast cloud–cloud collisions, which are caused by the noncircular motion (e.g., Tubbs 1982; Athanassoula 1992; Reynaud & Downes 1998; Fujimoto et al. 2014a, 2020; Emsellem et al. 2015; Maeda et al. 2021). Additionally, a number of CO observations have reported that the CO line width in the bar is larger than the line widths in the bar-end and arm regions, which would support the presence of a large noncircular motion in the bar region (e.g., Reynaud & Downes 1998; Regan et al. 1999; Watanabe et al. 2011; Sorai et al. 2012; Morokuma-Matsui et al. 2015; Muraoka et al. 2016; Maeda et al. 2018; Sun et al. 2018; Yajima et al. 2019).

Here, we investigate the relationship between the velocity width of the CO spectrum and the SFE in the bar, the bar-end, and the disk regions of the gas-rich long-bar sample galaxies. Using the CO spectrum obtained by the stacking method described in Section 3.1, we measure the effective width as a proxy for the line width. We follow the definition of the effective width by Sun et al. (2018), as ${\sigma }_{v}={I}_{\mathrm{CO}}/(\sqrt{2\pi }{T}_{\mathrm{peak}})$, where T_peak is the peak temperature of the spectrum. The effective width is less sensitive to noise in the line wings than in the second moment.

Figure 9 shows the relationship between the normalized velocity width of the CO(1–0) spectrum and the normalized SFE in the bar and bar-end regions of the gas-rich long-bar sample galaxies. The velocity width is normalized by that in the disk region of the galaxy. We find negative correlations between the normalized velocity width and SFE, as indicated by the Spearman's rank correlation coefficients of r_s = − 0.612 and −0.465 for CO(1–0) and CO(2–1), respectively. This trend is consistent with that reported by Yajima et al. (2019), who investigated this relationship in NGC 4303. The σ_v in the bar end is ∼0.8–1.2 times larger than that in the disk, whereas that in the bar is ∼1.2–4.0 times larger than that in the disk. This negative correlation would support the idea that the larger the noncircular motion, the lower the SFE.

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This result can be interpreted as follows. In the bar-end regions, gas accumulates not only because of the stagnation of the gas in the elongated elliptical orbit, due to the bar potential, but also because of the inflow of the gas rotating in the disk (e.g., Downes et al. 1996), resulting in the moderate velocity width. Such orbital crowding increases the probability of cloud–cloud collisions, leading to an increased gas density and SFE (e.g., Renaud et al. 2015). In fact, the high gas density and SFE in the bar-end regions, compared to those in the arm and bar regions, have been reported in NGC 4303 (Yajima et al. 2019). Furthermore, frequent cloud–cloud collisions in the W43 giant molecular cloud (GMC) complex, which is considered to be located in the bar-end of the Milky Way, have been suggested (Kohno et al. 2021).

On the other hand, in the bar regions, the large velocity width suggests the presence of strong shocks, large shears, and fast cloud–cloud collisions, compared to those in the bar-end region, leading to the low SFE. Some simulations have shown that strong shocks and/or large shears due to the noncircular gas motion by the bar potential destroy the molecular clouds and/or suppress the molecular cloud formation, leading to the suppression of star formation (e.g., Tubbs 1982; Athanassoula 1992; Emsellem et al. 2015; Renaud et al. 2015). In fact, CO observations toward NGC 1530 have suggested that intense shocks with high-velocity jumps and a large shear suppress the star formation by destroying molecular clouds (Reynaud & Downes 1998). In terms of the cloud–cloud collisions, subparsec-scale simulations (Takahira et al. 2014, 2018) have shown that a faster collision can shorten the gas accretion phase of the cloud cores formed, leading to the suppression of the core growth and massive star formation. Simulations of the cloud motions within a barred galaxy by Fujimoto et al. (2020) have shown that the collision velocities between the clouds in the bar regions are larger than those in the other regions, which may be due to the motions of the clouds being perturbed to elliptical gas orbits, by gravitational interaction between clouds. Based on the subparsec-scale simulations, the authors have proposed that fast collisions in the bar regions suppress massive star formation (see also Fujimoto et al. 2014a, 2014b).

Because the velocity width is affected by the molecular gas distribution, the relative velocities among the molecular clouds, and the gradient of the velocity field in the beam, it is unclear which of the above dynamical effects (shocks, shears, and cloud–cloud collisions) are dominant in the velocity width in the bar region, based on kiloparsec-scale measurements only. Therefore, further observations at higher angular resolutions are required. For example, Maeda et al. (2021) examined the motions of the GMCs in NGC 1300 on a spatial resolution of 40 pc and found that the dispersion of the line-of-sight velocity among the GMCs was larger in the bar than in the arm. Further, using the velocity field model, the authors suggested that the fast cloud–cloud collisions in the bar region, which were caused by noncircular motions owing to the bar potential, suppressed star formation. Therefore, the large velocity width of the CO spectrum in the bar may reflect fast cloud–cloud collisions.

The CO(2–1)/CO(1–0) line ratio, R₂₁, is dependent on gas conditions, such as the density and/or temperature, and the systematic variations in R₂₁ on a kiloparsec scale have been observed in many galaxies (e.g., Leroy et al. 2009; Koda et al. 2012; Leroy et al. 2013; Muraoka et al. 2016; Koda et al. 2020; Maeda et al. 2020; den Brok et al. 2021; Yajima et al. 2021; Leroy et al. 2022). Figure 10(a) shows the R₂₁( =I_CO(2−1)/I_CO(1−0)) profiles of the gas-rich long-bar sample galaxies that have both CO(1–0) and CO(2–1) data available. Here, we stack the CO profiles of all pixels in which both CO(1–0) and CO(2–1) lines are detected. We observe environmental dependence; the R₂₁ in the center is the highest (0.6–1.1), followed by those in the bar end (0.6–0.8) and the bar (0.4–0.6). Previous independent studies have reported the same trend in the R₂₁ profiles of NGC 1300, NGC 2903, and NGC 5236 (Muraoka et al. 2016; Koda et al. 2020; Maeda et al. 2020). However, we emphasize that the range of R₂₁ in the disk is roughly comparable to that in the bar region, which suggests that the SFE/SFE_disk in the bar that was obtained from CO(2–1) does not strongly depend on R₂₁.

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Figure 10(b) shows the relationship between SFE and R₂₁. As indicated by the high r_s of 0.710, a strong positive correlation is observed. This result is consistent with that reported by Maeda et al. (2020), who find the same trend in NGG 1300. The authors further find a negative correlation between the SFE and the fraction of diffuse extended molecular gas, which is missed in interferometer observations and would not directly contribute to the current star formation activity. They concluded that the SFE is roughly controlled by the amount of diffuse molecular gas. Our results would support this idea. However, our results are inconsistent with those reported by Querejeta et al (2021), who reported no clear trend in the R₂₁ from the center to the bar and bar end, and no correlation between the SFE and R₂₁. The causes of these differences remain unclear. The possible causes include different sample galaxies and spatial resolutions (Section 4.3). Therefore, increasing the sample number of R₂₁ maps with sufficient angular resolutions to resolve the environments will be important.