CO Multi-line Imaging of Nearby Galaxies (COMING). I. Physical properties of molecular gas in the barred spiral galaxy NGC 2903

Abstract

We present simultaneous mappings of J = 1–0 emission of ¹²CO, ¹³CO, and C¹⁸O molecules toward the whole disk (8′ × 5′ or 20.8 kpc × 13.0 kpc) of the nearby barred spiral galaxy NGC 2903 with the Nobeyama Radio Observatory 45 m telescope at an effective angular resolution of 20″ (or 870 pc). We detected ¹²CO(J = 1–0) emission over the disk of NGC 2903. In addition, significant ¹³CO(J = 1–0) emission was found at the center and bar-ends, whereas we could not detect any significant C¹⁸O(J = 1–0) emission. In order to improve the signal-to-noise ratio of CO emission and to obtain accurate line ratios of ¹²CO(J = 2–1)/¹²CO(J = 1–0) (R_2−1/1−0) and ¹³CO(J = 1–0)/¹²CO(J = 1–0) (R_13/12), we performed the stacking analysis for our ¹²CO(J = 1–0), ¹³CO(J = 1–0), and archival ¹²CO(J = 2–1) spectra with velocity axis alignment in nine representative regions of NGC 2903. We successfully obtained the stacked spectra of the three CO lines, and could measure averaged R_2−1/1−0 and R_13/12 with high significance for all the regions. We found that both R_2−1/1−0 and R_13/12 differ according to the regions, which reflects the difference in the physical properties of molecular gas, i.e., density (⁠|$n_{\rm H_2}$|⁠) and kinetic temperature (T_K). We determined |$n_{\rm H_2}$| and T_K using R_2−1/1−0 and R_13/12 based on the large velocity gradient approximation. The derived |$n_{\rm H_2}$| ranges from ∼1000 cm⁻³ (in the bar, bar-ends, and spiral arms) to 3700 cm⁻³ (at the center) and the derived T_K ranges from 10 K (in the bar and spiral arms) to 30 K (at the center). We examined the dependence of star formation efficiencies (SFEs) on |$n_{\rm H_2}$| and T_K, and found a positive correlation between SFE and |$n_{\rm H_2}$| with correlation coefficient for the least-squares power-law fit R² of 0.50. This suggests that molecular gas density governs the spatial variations in SFEs.

1 Introduction

Molecular gas is one of the essential components of galaxies because it is closely related to star formation, which is a fundamental process of galaxy evolution. Thus the observational study of molecular gas is indispensable to understanding both star formation in galaxies and galaxy evolution. However, the most abundant constituent in molecular gas, H₂, cannot emit any electromagnetic wave in cold molecular gas with typical temperatures of ∼ 10 K due to the lack of a permanent dipole moment. Instead, rotational transition lines of ¹²CO, the second most abundant molecule, have been used as a tracer of molecular gas. For example, some extensive ¹²CO surveys of external galaxies, which consist of single pointings toward central regions, and some mappings along the major axis have been reported (e.g., Braine et al. 1993; Young et al. 1995; Elfhag et al. 1996). These studies provided new findings about global properties of galaxies, such as the excitation condition of molecular gas in galaxy centers and radial distributions of molecular gas across galaxy disks.

In order to understand the relationship between molecular gas and star formation in galaxies further, spatially resolved ¹²CO maps covering whole galaxy disks are necessary because star formation rates (SFRs) are often different between galaxy centers and disks. In particular, single-dish observations are essential to measure total molecular gas content in the observing beam from a dense component to a diffuse one avoiding the missing flux (e.g., Caldú-Primo et al. 2015). So far, two major surveys of wide-area ¹²CO mapping toward nearby galaxies have been performed using multi-beam receivers mounted on large single-dish telescopes.

One is the ¹²CO(J = 1–0) mapping survey of 40 nearby spiral galaxies performed with the Nobeyama Radio Observatory (NRO) 45 m telescope in the position-switch mode (Kuno et al. 2007, hereafter K07). Their ¹²CO(J = 1–0) maps cover most of the optical disks of galaxies at an angular resolution of 15″, and clearly show two-dimensional distributions of molecular gas in galaxies. K07 found that the degree of the central concentration of molecular gas is higher in barred spiral galaxies than in non-barred spiral galaxies. In addition, they found a correlation between the degree of central concentration and the bar strength adopted from Laurikainen and Salo (2002),¹ i.e., galaxies with stronger bars tend to exhibit a higher central concentration. This correlation suggests that stronger bars accumulate molecular gas toward central regions more efficiently, which may contribute the onset of intense star formation at galaxy centers (i.e., higher SFRs than disks). Using the ¹²CO(J = 1–0) data, Sorai et al. (2012) investigated the physical properties of molecular gas in the barred spiral galaxy Maffei 2. They found that molecular gas in the bar ridge regions may be gravitationally unbound, which suggests that molecular gas struggles to become dense, and to form stars in the bar.

The other survey is the Heterodyne Receiver Array CO Line Extragalactic Survey performed with the IRAM 30 m telescope (Leroy et al. 2009). They observed ¹²CO(J = 2–1) emission over the full optical disks of 48 nearby galaxies at an angular resolution of 13″, and found that the ¹²CO(J = 2–1)/¹²CO(J = 1–0) line intensity ratio (hereafter R_2−1/1−0) typically ranges from 0.6 to 1.0, with an averaged value of 0.8. In addition, Leroy et al. (2013) examined a quantitative relationship between surface densities of molecular gas and SFRs for 30 nearby galaxies at a spatial resolution of 1 kpc using the ¹²CO(J = 2–1) data. They found a first-order linear correspondence between surface densities of molecular gas and SFRs, but also found second-order systematic variations. For example, the apparent molecular gas depletion time, which is defined by the ratio of the surface density of molecular gas to that of SFR, becomes shorter with the decrease in stellar mass, metallicity, and dust-to-gas ratio; they suggest that this can be explained by a CO-to-H₂ conversion factor (X_CO) that depends on dust shielding.

However, such global CO maps of galaxies have raised a new question: the cause of the spatial variation in star formation efficiencies (SFEs) defined as SFRs per unit gas mass.² It is reported that SFEs differ not only among galaxies (e.g., Young et al. 1996) but also within locations/regions in a galaxy (e.g., Muraoka et al. 2007); i.e., higher SFEs are often observed in galaxy mergers rather than normal spiral galaxies and are also observed in the nuclear star-forming region rather than in galaxy disks. Some observational studies based on HCN emission, an excellent dense gas tracer, suggest that SFEs increase with the increase in molecular gas density (or dense gas fraction) in galaxies (e.g., Gao & Solomon 2004; Gao et al. 2007; Muraoka et al. 2009; Usero et al. 2015), but the cause of the spatial variation in SFEs is still an open question because HCN emission in galaxy disks is too weak to obtain its map except for some gas-rich spiral galaxies (e.g., M 51: Chen et al. 2015; Bigiel et al. 2016).

Instead, isotopes of the CO molecule are promising probes of molecular gas density. In particular, ¹³CO(J = 1–0) is thought to be optically thin and thus traces denser molecular gas (∼10³⁻⁴ cm⁻³) rather than ¹²CO(J = 1–0), which is optically thick and traces relatively diffuse molecular gas (∼10²⁻³ cm⁻³). Therefore, the relative intensity between ¹³CO(J = 1–0) and ¹²CO(J = 1–0) is sensitive to the physical properties of molecular gas. For example, spatial variations in ¹³CO(J = 1–0)/¹²CO(J = 1–0) intensity ratios (hereafter R_13/12) were observed in nearby galaxy disks (e.g., Sakamoto et al. 1997; Tosaki et al. 2002; Hirota et al. 2010). Such variations in R_13/12, typically ranging from 0.05 to 0.20, are interpreted as variation in molecular gas density; i.e., R_13/12 increases with the increase in molecular gas density. However, some observations suggest that R_13/12 in central regions of nearby galaxies are lower than those in disk regions (e.g., Paglione et al. 2001; Tosaki et al. 2002; Hirota et al. 2010; Watanabe et al. 2011), although central regions of galaxies often show intense star formation activity, suggesting higher molecular gas density. The cause of the low R_13/12 in central regions is thought to be the high temperature of the molecular gas due to heating by UV radiation from a lot of young massive stars.

Such a degeneracy between density and temperature of molecular gas in a line ratio can be solved using two (or more) molecular line ratios with a theoretical calculation on the excitation of molecular gas such as the large velocity gradient (LVG) model (Scoville & Solomon 1974; Goldreich & Kwan 1974). For example, the density and kinetic temperature of giant molecular clouds (GMCs) were determined using the R_13/12 and ¹²CO(J = 3–2)/¹²CO(J = 1–0) ratios for the Large Magellanic Cloud (Minamidani et al. 2008) and M 33 (Muraoka et al. 2012), and also determined using R_13/12 and R_2−1/1−0 for the spiral arm of M 51 (Schinnerer et al. 2010). This method to determine molecular gas density is useful in investigating the cause of the variation in SFEs. Thus, the dependence of SFEs on molecular gas density should be investigated for various galaxies at high angular resolution based on multiple line ratios, including R_13/12.

In this paper, we investigate the relationship between SFE and molecular gas density within the nearby barred spiral galaxy NGC 2903 using an archival ¹²CO(J = 2–1) map combined with ¹²CO(J = 1–0) and ¹³CO(J = 1–0) maps which are newly obtained by the CO Multi-line Imaging of Nearby Galaxies (COMING) project with the NRO 45 m telescope. NGC 2903 is a gas-rich galaxy exhibiting bright nuclear star formation (e.g., Wynn-Williams & Becklin 1985; Simons et al. 1988; Alonso-Herrero et al. 2001; Yukita et al. 2012). The distance to NGC 2903 is estimated to be 8.9 Mpc (Drozdovsky & Karachentsev 2000); thus the effective angular resolution of 20″ for the on-the-fly (OTF) mapping with the NRO 45 m telescope corresponds to 870 pc. This enables us to resolve major structures within NGC 2903, such as the center, bar, and spiral arms, although its inclination of 65° (de Blok et al. 2008) is not so small. In addition, NGC 2903 is rich in archival multi-wavelength data sets; i.e., not only the ¹²CO(J = 2–1) map to examine R_2−1/1−0 but also Hα and infrared images to calculate SFRs are available. Thus this galaxy is a preferred target for examining the cause of the variation in SFE in terms of molecular gas density. The basic parameters of NGC 2903 are summarized in table 1.

Table 1.

General parameters of NGC 2903.

Morphological type*	SAB(rs)bc
Map center†
Right ascension (J2000.0)	09h32m10|${^{s}_{.}}$|1
Declination (J2000.0)	21°30′03|${^{\prime\prime}_{.}}$|0
Distance†	8.9 Mpc
Linear scale	43 pc arcsec−1
Inclination§	65°

*Morphological type from RC3.

†Map center from Jarrett et al. (2003).

†Adopted distance from Drozdovsky and Karachentsev (2000).

^§Inclination from de Blok et al. (2008).

The structure of this paper is as follows: We give an overview of the COMING project and explain the detail of the CO observations and data reduction for NGC 2903 in section 2. Then, we show results of the observations, i.e., spectra and velocity-integrated intensity maps of ¹²CO(J = 1–0) and ¹³CO(J = 1–0) emission, in section 3. We obtain averaged spectra of ¹²CO(J = 1–0), ¹³CO(J = 1–0), and ¹²CO(J = 2–1) emission for nine representative regions, and measure averaged R_13/12 and R_2−1/1−0 for each region in subsection 4.1. We determine the molecular gas density and kinetic temperature for the center, bar, bar-ends, and spiral arms using R_13/12 and R_2−1/1−0 based on the LVG approximation in subsection 4.2. Finally, we investigate the cause of the variation in SFE by examining the dependence of SFE on molecular gas density and kinetic temperature.

2 Observations and data reduction

2.1 COMING project

COMING is a project to map J = 1–0 emission of ¹²CO, ¹³CO, and C¹⁸O molecules simultaneously for a 70% area of the optical disks of 238 galaxies using the FOur-beam REceiver System on 45 m Telescope (FOREST: Minamidani et al. 2016) at NRO. The main purposes of COMING are to characterize the properties of molecular gas as a sequence of Hubble types, dynamical structures, central concentrations, and star formation activities, as well as the surrounding environments of galaxies. More detailed information on the COMING project including the current status of the survey will be reported in a forthcoming paper (K. Sorai et al. in preparation).

2.2 Observations

¹²CO(J = 1–0), ¹³CO(J = 1–0), and C¹⁸O(J = 1–0) emission observations of NGC 2903 were performed using the NRO 45 m telescope from 2015 April to May, employing the OTF mapping mode. The observed area is about 8′ × 5′, which corresponds to 20.8 kpc × 13.0 kpc at the distance of 8.9 Mpc, as indicated in figure 1. The total time for the observations was 13 hr.

Fig. 1.

Observed 8′ × 5′ area of NGC 2903 with the FOREST mounted on the NRO 45 m telescope, indicated by a large rectangle, superposed on a Spitzer/IRAC 8 μm image (Kennicutt et al. 2008).

We used a new 2 × 2 focal-plane dual-polarization sideband-separating SIS mixer receiver for the single side band (SSB) operation of FOREST, which provides eight intermediate frequency (IF) paths (i.e., four beams × two polarizations) independently. Owing to the wide IF range of 4–12 GHz, we could simultaneously observe ¹²CO(J = 1–0) emission at 115 GHz (IF = 10 GHz) and ¹³CO(J = 1–0) and C¹⁸O(J = 1–0) emission at 110 GHz (IF = 5 GHz) when the frequency of the local oscillator was set to 105 GHz. The backend was an FX-type correlator system, SAM45, which consists of 16 arrays with 4096 spectral channels each. We set the frequency coverage and resolution for each array to 2 GHz and 488.24 kHz, respectively, which gives a velocity coverage and resolution of 5220 km s⁻¹ and 1.27 km s⁻¹ at 115 GHz, and 5450 km s⁻¹ and 1.33 km s⁻¹ at 110 GHz. We assigned 8 of the 16 arrays to the 115 GHz band (i.e., IF = 9–11 GHz for ¹²CO(J = 1–0) emission) and the other 8 arrays to the 110 GHz band [i.e., IF = 4–6 GHz for ¹³CO(J = 1–0) and C¹⁸O(J = 1–0) emission]. The half-power beam widths of the 45 m telescope with FOREST were ∼14″ at 115 GHz and ∼15″ at 110 GHz. The system noise temperatures were 300–500 K at 115 GHz and 200–250 K at 110 GHz during the observing run.

We performed the OTF mapping along the major and minor axes of the galaxy disk whose position angle was 25° (K07). The separation between the scan rows was set to 5|${^{\prime\prime}_{.}}$|0, and the spatial sampling interval was < 2|${^{\prime\prime}_{.}}$|4, applying a dump time of 0.1 s and a scanning speed of < 24″ s⁻¹. The data sets scanned along two orthogonal axes were co-added by the basket-weave method (Emerson & Graeve 1988) to remove any effects of scanning noise. In order to check the absolute pointing accuracy, every hour we observed an SiO maser source, W-Cnc, using a 43 GHz band receiver. The accuracy was better than 6″ (peak-to-peak) throughout the observations. In addition, we observed ¹²CO(J = 1–0) and ¹³CO(J = 1–0) emission of W 3 and IRC +10216 every day to obtain the scaling factors for converting the observed antenna temperature to the main beam temperature for each IF. Note that these scaling factors not only correct the main-beam efficiency (η_MB) of the 45 m antenna but also compensate for the decrease in line intensity due to the incompleteness of the image rejection for the SSB receiver (e.g., Nakajima et al. 2013). The absolute error of the temperature scale for each CO spectrum was about ±20%, mainly due to variations in η_MB and the image rejection ratio of FOREST.

2.3 Data reduction

Data reduction was performed using the software package NOSTAR, which comprises tools for OTF data analysis developed by NRO (Sawada et al. 2008). We excluded bad spectra, which includes strong baseline undulation and spurious lines, from the raw data. Then, linear baselines were subtracted, and the raw data were regridded to 6″ per pixel with an effective angular resolution of approximately 20″ (or 870 pc). We binned the adjacent spectral channels to a velocity resolution of 10 km s⁻¹ for the spectra. Finally, we created three-dimensional data cubes in ¹²CO(J = 1–0), ¹³CO(J = 1–0), and C¹⁸O(J = 1–0) emission. The resultant r.m.s. noise levels (1 σ) were 60 mK, 39 mK, and 40 mK for ¹²CO(J = 1–0), ¹³CO(J = 1–0), and C¹⁸O(J = 1–0), respectively.

3 Results

3.1 ¹²CO(J = 1–0) emission

Figure 2 shows ¹²CO(J = 1–0) spectra of the whole optical disk in NGC 2903. As is the case in earlier studies (e.g., Helfer et al. 2003; K07; Leroy et al. 2009), strong ¹²CO(J = 1–0) emission, whose peak temperature was ∼0.6 K, was found at the center and significant ¹²CO(J = 1–0) emission was detected in the bar (∼0.3 K), bar-ends (0.4–0.5 K), and spiral arms (0.1–0.3 K).

Fig. 2.

Global spectra of ¹²CO(J = 1–0) emission in NGC 2903. The grid spacing was set to 12″ in order to display the spectrum in each pixel clearly. The temperature scale of the spectra is indicated by the small box inserted in the lower left corner.

We calculate velocity-integrated ¹²CO(J = 1–0) intensities (I_12CO(1–0)) from the spectra. In order to obtain more accurate line intensities (in other words, to minimize the effects of the noise and the undulation of the baseline for weak lines), we defined the “line channels,” which are successive velocity channels where significant emission exists, in advance for each pixel as described below.

In order to define the “line channels” we utilized ¹²CO(J = 2–1) data (Leroy et al. 2009), which was regridded and convolved to 20″ to match our ¹²CO(J = 1–0) spectra. Since the 1 σ r.m.s. of ¹²CO(J = 2–1) data of 6 mK at 20″ and 10 km s⁻¹ resolutions was 10 times better than that of our ¹²CO(J = 1–0) data, the ¹²CO(J = 2–1) spectra are appropriate for choosing “line channels” in each pixel. We first identified a velocity channel exhibiting the peak ¹²CO(J = 2–1) temperature and defined the channel as the “CO peak channel” for each pixel. Then, successive channels whose ¹²CO(J = 2–1) emission consistently exceeds 1 σ including the “CO peak channel” are defined as “line channels.” Finally, we calculated I_12CO(1–0) for the specified “line channels” in each pixel.

Figure 3 shows the I_12CO(1–0) map of NGC 2903. The strongest I_12CO(1–0) of 92 K km s⁻¹ is observed at the center, and the secondary peak of 55 K km s⁻¹ is at the northern bar-end. The total molecular gas mass for the observed area in NGC 2903 is estimated to be (2.8 ± 0.6) × 10⁹ M_⊙ under the assumptions of constant X_CO of 1.8 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ (Dame et al. 2001) over the disk and an uncertainty of 20% in the brightness temperature scale of the CO line. This value is consistent with 3.2 × 10⁹ M_⊙ obtained by K07, which was recalculated using the same distance and X_CO. We also compared the I_12CO(1–0) obtained by COMING with those obtained by K07 to confirm the validity of our ¹²CO(J = 1–0) data. We examined the pixel-by-pixel comparison for the two I_12CO(1–0) maps at the same angular resolution of 20″, as shown in figure 4, and confirmed that both I_12CO(1–0) are well correlated with each other. The median and the standard deviation in I_12CO(1–0) are 12.9 K km s⁻¹ and 12.4 K km s⁻¹ for the COMING dataset, and 15.2 K km s⁻¹ and 12.6 K km s⁻¹ for the K07 dataset.

Fig. 3.

Map of I_12CO(1–0) for NGC 2903 obtained by COMING. The contour levels are 10, 20, 30, 40, 60, and 80 K km s⁻¹.

Fig. 4.

Pixel-by-pixel comparison between I_12CO(1–0) in NGC 2903 obtained by COMING and those obtained by K07. The vertical and horizontal lines indicate the 2 σ of I_12CO(1–0) for COMING and K07 data, respectively. The diagonal solid line indicates a ratio of I_12CO(1–0) for COMING to K07 of unity, and the dashed lines indicate ratios of 0.5 and 2.0. Both I_12CO(1–0) are well correlated with each other.

3.2 ¹³CO(J = 1–0) and C¹⁸O(J = 1–0) emission

Figure 5a shows the global ¹³CO(J = 1–0) spectra, which are overlaid with the ¹²CO(J = 1–0) spectra for comparison. In addition, figures 5b–5d show the magnified ¹³CO(J = 1–0) spectra at the northern bar-end, the center, and the southern bar-end, respectively. We found significant ¹³CO(J = 1–0) emission at the center and both bar-ends. However, we could not detect any significant C¹⁸O(J = 1–0) emission.

Fig. 5.

(a) Global spectra of ¹³CO(J = 1–0) emission in NGC 2903. For comparison, spectra of ¹²CO(J = 1–0) emission multiplied by 0.5 are overlaid with a red line. As in figure 2, the grid spacing was set to 12″. The temperature scale of the spectra is indicated by the small box inserted in the lower left corner. (b) Magnified spectra with grid spacing of 6″ at the northern bar-end. (c) As (b), but at the center. (d) As (b), but at the southern bar-end.

We calculated the velocity-integrated ¹³CO(J = 1–0) intensities (I_13CO(1–0)). As for ¹²CO(J = 1–0), we utilized the “line channels” defined by the ¹²CO(J = 2–1) spectra. Figure 6 shows the spatial distribution of I_13CO(1–0) in pseudo-color overlaid by I_12CO(1–0) in contour. The global distribution of ¹³CO(J = 1–0) is similar to ¹²CO(J = 1–0); several peaks whose I_13CO(1–0) exceeds 5 K km s⁻¹ are observed at the center, bar-ends, and in spiral arms.

Fig. 6.

A color map of I_13CO(1–0) superposed on the contour map of I_12CO(1–0) for NGC 2903. The contour levels of I_12CO(1–0) are the same as figure 3.

3.3 Line intensity ratios R_2−1/1−0 and R_13/12

Intensity ratios of two (or more) molecular lines provide important clues for estimating physical properties of molecular gas, such as density and temperature. We examined the spatial variations in line intensity ratios among ¹²CO(J = 1–0), ¹³CO(J = 1–0), and ¹²CO(J = 2–1) emission. Figure 7 shows spatial distributions of R_2−1/1−0 and R_13/12 over the disk of NGC 2903. In these maps, we displayed pixels with each line intensity exceeding 2 σ [5 K km s⁻¹, 3 K km s⁻¹, and 0.5 K km s⁻¹ for ¹²CO(J = 1–0), ¹³CO(J = 1–0), and ¹²CO(J = 2–1), respectively]. We found some local peaks of R_2−1/1−0 (∼1.0) near the center and at the downstream side of the northern spiral arm, whereas lower R_2−1/1−0 (∼0.5–0.6) was observed in the bar. The spatial distribution of R_13/12 is unclear due to the poor signal-to-noise (S/N) ratio of the ¹³CO(J = 1–0) emission.

Fig. 7.

Spatial distributions of R_2−1/1−0 (left) and R_13/12 (right) superposed on the contour map of I_12CO(1–0) for NGC 2903. The contour levels of I_12CO(1–0) are the same as figure 3. There are some local peaks of R_2−1/1−0 (∼1.0) near the center and at the downstream side of the northern spiral arm, whereas lower R_2−1/1−0 (∼0.5–0.6) was observed in the bar. The spatial distribution of R_13/12 is unclear due to the poor S/N of the ¹³CO(J = 1–0) emission.

4 Discussion

4.1 Velocity axis alignment stacking of CO spectra

As described in subsection 3.3, the spatial distribution of R_13/12 seems noisy and unclear due to the poor S/N, although we could obtain the spatial distribution of I_13CO(1–0). In order to improve the S/N of weak emissions such as ¹³CO(J = 1–0), the stacking analysis of CO spectra with velocity axis alignment seems a promising method.

The stacking technique for CO spectra in external galaxies was originally demonstrated by Schruba et al. (2011, 2012). Since the observed velocities of each position within a galaxy are different due to its rotation, simple stacking causes a smearing of the spectrum. In order to overcome such a difficulty, Schruba et al. (2011) demonstrated the velocity axis alignment of CO spectra in different regions in a galaxy disk according to mean H i velocity. They stacked velocity-axis-aligned CO spectra, and successfully confirmed very weak ¹²CO(J = 2–1) emission (< 1 K km s⁻¹) with high significance in H i-dominated outer disk regions of nearby spiral galaxies. In addition, Schruba et al. (2012) applied this stacking technique to perform a sensitive search for weak ¹²CO(J = 2–1) emission in dwarf galaxies. Furthermore, Morokuma-Matsui et al. (2015) applied the stacking technique to ¹³CO(J = 1–0) emission in the optical disk of the nearby barred spiral galaxy NGC 3627. By stacking with velocity axis alignment based on mean ¹²CO(J = 1–0) velocity, they obtained high-S/N¹³CO(J = 1–0) spectra which were improved by a factor of up to 3.2 compared to the normal (without velocity axis alignment) stacking analysis.

These earlier studies clearly suggest that the stacking analysis is very useful for detecting a weak molecular line. In this section, we employ the same stacking technique as Morokuma-Matsui et al. (2015) to improve the S/N of the ¹³CO(J = 1–0) emission and to obtain more accurate line ratios. Based on our I_12CO(1–0) image (figure 3), we have separated NGC 2903 into nine regions according to its major structures: center, northern bar, southern bar, northern bar-end, southern bar-end, northern arm, southern arm, inter-arm, and outer-disk. The left panel of figure 8 shows the separation of each region overlaid on a gray-scale map of I_12CO(1–0). For each region, we stacked ¹²CO(J = 1–0), ¹³CO(J = 1–0), and ¹²CO(J = 2–1) spectra with velocity axis alignment based on the intensity-weighted mean velocity field calculated from our ¹²CO(J = 1–0) data (right panel of figure 8). We successfully obtained the stacked CO spectra shown in figure 9. The S/N of each CO emission is dramatically improved, and thus we could confirm significant ¹³CO(J = 1–0) emission for all the regions. We found the difference in the line shape of the stacked CO spectra according to the regions. In particular, the stacked ¹²CO spectra in the bar show a flat peak over the velocity width of 100–150 km s⁻¹. This is presumably due to the rapid velocity change in the bar, which makes the velocity axis alignment difficult.

Fig. 8.

(Left) Separation of representative regions for stacking analysis superposed on a gray-scale map of I_12CO(1–0) for NGC 2903. The green frame indicates the center, the purple and orange indicate the northern and southern bars, the blue and red indicate the northern and southern bar-ends, the cyan and magenta indicate the northern and southern arms, the yellow indicates the inter-arm, and the gray indicates the outer-disk. (Right) Intensity-weighted mean velocity field calculated from ¹²CO(J = 1–0) data superposed on the contour map of I_12CO(1–0). The contour levels of I_12CO(1–0) are the same as figure 3.

Fig. 9.

Stacked CO spectra for each region in NGC 2903. The black line indicates ¹²CO(J = 1–0) emission multiplied by 0.5, the red indicates ¹²CO(J = 2–1) emission multiplied by 0.5, and the blue indicates ¹³CO(J = 1–0) emission. The S/N of each CO emission is dramatically improved, and thus significant ¹³CO(J = 1–0) emission is confirmed for all the regions.

We summarize the averaged line intensities and line ratios for each region in table 2. We found that the averaged R_2−1/1−0 shows the highest value of 0.92 at the center, and a moderate value of 0.7–0.8 at both bar-ends and in the northern arm. A slightly lower R_2−1/1−0 of 0.6–0.7 is observed in the bar, southern arm, inter-arm, and outer-disk. Such a variation in R_2−1/1−0 ranging from 0.6 to 1.0 in NGC 2903 is quite consistent with those observed in nearby galaxies (e.g., Leroy et al. 2009). However, the highest R_13/12 of 0.19 is observed not at the center but in the northern arm. The central R_13/12 of 0.11 is similar to those in other regions (0.08–0.13), except for the northern arm and outer-disk (∼0.04). The typical R_13/12 of ∼0.1 is frequently observed in nearby galaxies (e.g., Paglione et al. 2001; Vila-Vilaro et al. 2015), but slightly higher than the averaged R_13/12 in representative regions of NGC 3627 of 0.04–0.09 (Morokuma-Matsui et al. 2015).

Table 2.

Averaged line intensity and line ratios based on the stacking analysis.

	I12CO(1–0)	I12CO(2–1)	I13CO(1–0)	R2−1/1−0	R13/12
	[K km s−1]	[K km s−1]	[K km s−1]
Center	46.7 ± 0.7	43.0 ± 0.1	5.1 ± 0.2	0.92 ± 0.02	0.11 ± 0.01
Northern bar	26.4 ± 0.8	15.6 ± 0.1	2.2 ± 0.6	0.59 ± 0.02	0.08 ± 0.02
Southern bar	26.6 ± 0.8	18.1 ± 0.1	3.5 ± 0.7	0.68 ± 0.02	0.13 ± 0.03
Northern bar-end	36.8 ± 0.8	27.3 ± 0.1	3.6 ± 0.6	0.74 ± 0.02	0.10 ± 0.02
Southern bar-end	28.6 ± 0.6	21.2 ± 0.1	2.4 ± 0.6	0.74 ± 0.02	0.08 ± 0.02
Northern arm	19.8 ± 0.5	15.0 ± 0.1	3.8 ± 0.6	0.76 ± 0.02	0.19 ± 0.02
Southern arm	19.6 ± 0.5	11.9 ± 0.1	2.4 ± 0.4	0.61 ± 0.02	0.12 ± 0.02
Inter-arm	11.8 ± 0.3	7.2 ± 0.1	1.2 ± 0.3	0.61 ± 0.02	0.10 ± 0.02
Outer-disk	5.0 ± 0.2	3.2 ± 0.1	0.2 ± 0.1	0.64 ± 0.02	0.04 ± 0.02

4.2 Derivation of physical properties and their comparison with star formation

4.2.1 LVG calculation for stacked CO spectra

Using R_2−1/1−0 and R_13/12, we derive averaged physical properties of molecular gas, its density (⁠|$n_{\rm H_2}$|⁠), and kinetic temperature (T_K) in seven regions (center, northern bar, southern bar, northern bar-end, southern bar-end, northern arm, and southern arm) of NGC 2903 based on the LVG approximation. Some assumptions are required to perform the LVG calculation: the molecular abundances Z(¹²CO) = [¹²CO]/[H₂], [¹²CO]/[¹³CO], and the velocity gradient dv/dr. First, we fixed the Z(¹²CO) of 1.0 × 10⁻⁵ and dv/dr of 1.0 km s⁻¹ pc⁻¹; i.e., the ¹²CO abundance per unit velocity gradient Z(¹²CO)/(dv/dr) was assumed to be 1.0 × 10⁻⁵ (km s⁻¹ pc⁻¹)⁻¹. This is the same as the assumed Z(¹²CO)/(dv/dr) for the GMCs in M 33 (Muraoka et al. 2012).

Then, we determined the [¹²CO]/[¹³CO] abundance ratio to be assumed in this study by considering earlier studies. Langer and Penzias (1990) found a systematic gradient in the ¹²C/¹³C isotopic ratio across our Galaxy, from ∼30 in the inner part at 5 kpc to ∼70 at 12 kpc with a galactic center value of 24. For external galaxies, the reported ¹²C/¹³C isotopic ratios in their central regions are 40 for NGC 253 (Henkel et al. 1993), 50 for NGC 4945 (Henkel et al. 1994), >40 for M 82 and >30 for IC 342 (Henkel et al. 1998). Mao et al. (2000) reported a higher [¹²CO]/[¹³CO] abundance ratio of 50–75 in the central region of M 82. Martín et al. (2010) also reported higher ¹²C/¹³C isotopic ratios of >50–100 in the central regions of M 82 and NGC 253. In summary, reported ¹²C/¹³C isotopic (and [¹²CO]/[¹³CO] abundance) ratios in nearby galaxy centers (30–100) are typically higher than that in the inner 5 kpc of our Galaxy (24–30), but the cause of such discrepancies in ¹²C/¹³C and [¹²CO]/[¹³CO] between our Galaxy and external galaxies is still unresolved. Here, we assumed an intermediate [¹²CO]/[¹³CO] abundance ratio of 50 in NGC 2903 without any gradient across its disk for our LVG calculation. Note that we performed an additional LVG calculation for the center of NGC 2903 assuming [¹²CO]/[¹³CO] abundance ratios of 30 and 70 to evaluate the effect of the variation in the assumed [¹²CO]/[¹³CO] abundance ratio on results of LVG calculation.

Figure 10 shows the results of the LVG calculation for each region in NGC 2903. The thin line indicates a curve of constant R_2−1/1−0 as functions of |$n_{\rm H_2}$| and T_K, and the thick line indicates that of constant R_13/12. We can determine both |$n_{\rm H_2}$| and T_K at the point where the two curves intersect each other. Under the assumption of a [¹²CO]/[¹³CO] abundance ratio of 50, the derived |$n_{\rm H_2}$| ranges from ∼1000 cm⁻³ (in the disk; i.e., bar, bar-ends, and spiral arms) to 3700 cm⁻³ (at the center), and the derived T_K ranges from 10 K (in the spiral arms) to 30 K (at the center). Note that both |$n_{\rm H_2}$| and T_K vary depending on the assumption of the [¹²CO]/[¹³CO] abundance ratio; at the center of NGC 2903, an abundance ratio of 30 yields lower |$n_{\rm H_2}$| of 1800 cm⁻³ and higher T_K of 38 K, whereas the abundance ratio of 70 yields higher |$n_{\rm H_2}$| of 5900 cm⁻³ and intermediate T_K of 29 K. It seems that |$n_{\rm H_2}$| is proportional to the [¹²CO]/[¹³CO] abundance ratio. This trend for |$n_{\rm H_2}$| can be explained naturally if we consider the optical depth of ¹²CO and ¹³CO emission. ¹²CO is always optically thick and thus its emission emerges from the diffuse envelope of dense gas clouds, while ¹³CO emission emerges from further within these dense gas clouds due to its lower abundance. Since the increase in the assumed [¹²CO]/[¹³CO] abundance ratio means that ¹³CO becomes more optically thin, ¹³CO emission emerged from deeper within the dense gas clouds and thus it probes denser gas. The derived physical properties, |$n_{\rm H_2}$| and T_K, are summarized in table 3.

Fig. 10.

Curves of constant R_2−1/1−0 (thin line) and R_13/12 (thick line) as functions of molecular gas density |$n_{\rm H_2}$| and kinetic temperature T_K. The ¹²CO fractional abundance per unit velocity gradient Z(¹²CO)/(dv/dr) was assumed to be 1.0 × 10⁻⁵ (km s⁻¹ pc⁻¹)⁻¹. The [¹²CO]/[¹³CO] abundance ratio was assumed to be a fixed value of 50 for the bar, bar-ends, and spiral arms, but three different [¹²CO]/[¹³CO] abundance ratios of 30, 50, and 70 were assumed for the center. Dashed lines indicate the ±1 σ error for each line ratio.

Table 3.

Derived |$n_{\rm H_2}$| and T_K, and SFE in each region of NGC 2903.

Region	[12CO]/[13CO]*	|$n_{\rm H_2}$|	TK	SFE
		[103 cm−3]	[K]	[109 yr−1]
Center	50	|$3.7^{+1.2}_{-0.9}$|	30 ± 3	6.8 ± 0.8
Northern bar	50	|$0.96^{+0.22}_{-0.28}$|	|$14^{+5}_{-4}$|	3.2 ± 0.4
Southern bar	50	|$1.5^{+0.2}_{-0.3}$|	|$12^{+4}_{-3}$|	2.0 ± 0.3
Northern bar-end	50	1.4 ± 0.2	|$18^{+4}_{-3}$|	1.6 ± 0.2
Southern bar-end	50	1.2 ± 0.3	|$20^{+10}_{-4}$|	1.5 ± 0.2
Northern arm	50	|$2.1^{+0.4}_{-0.3}$|	12 ± 3	3.6 ± 0.5
Southern arm	50	1.3 ± 0.2	|$10^{+3}_{-2}$|	2.8 ± 0.4
Center	30	|$1.8^{+0.4}_{-0.2}$|	38 ± 3	6.8 ± 0.8
Center	70	|$5.9^{+3.9}_{-1.8}$|	|$29^{+5}_{-4}$|	6.8 ± 0.8

*The assumed [¹²CO]/[¹³CO] abundance ratio for the LVG calculation.

We compared the derived |$n_{\rm H_2}$| and T_K in NGC 2903 with those determined in other external galaxies. Muraoka et al. (2012) determined |$n_{\rm H_2}$| and T_K for GMCs associated with the giant H ii region NGC 604 in M 33 at a spatial resolution of 100 pc using three molecular lines, ¹²CO(J = 1–0), ¹³CO(J = 1–0), and ¹²CO(J = 3–2), based on the LVG approximation. The derived |$n_{\rm H_2}$| and T_K are 800–2500 cm⁻³ and 20–30 K, respectively, which are similar to our study for NGC 2903 in spite of the difference in the spatial resolution.

However, Schinnerer et al. (2010) obtained different physical properties for GMCs in spiral arms of M 51. They performed the LVG analysis using R_13/12 and R_2−1/1−0 at a spatial resolution of 120–180 pc. For the case of constant dv/dr = 1.0 km s⁻¹ pc⁻¹, the derived T_K ranges from 10 to 50 K, which is similar to our study for NGC 2903, whereas |$n_{\rm H_2}$| ranges from 100 to 400 cm⁻³, which is 5–10 times lower than that in the disk of NGC 2903 in spite of the values of R_2−1/1−0 and R_13/12 in M 51 being not so different from those in NGC 2903. This is presumably due to differences in the assumed Z(¹²CO) and [¹²CO]/[¹³CO] abundance ratio. The authors assumed Z(¹²CO) of 8.0 × 10⁻⁵, which is higher than that assumed in our study, and a lower [¹²CO]/[¹³CO] abundance ratio of 30. Under the LVG approximation with the assumption of Z(¹²CO) of 8.0 × 10⁻⁵, we found that the derived |$n_{\rm H_2}$| is typically ∼3 times lower than that with an assumption of Z(¹²CO) of 1.0 × 10⁻⁵. Physically, high Z(¹²CO) means abundant ¹²CO molecules among the molecular gas. In this condition, the optical depth of the ¹²CO line also increases, and thus the photon-trapping effect in molecular clouds becomes effective. Since this effect contributes to the excitation of the ¹²CO molecule, the effective critical density of the ¹²CO line decreases. In other words, since ¹²CO is easily excited to upper J levels even at a low molecular gas density, |$n_{\rm H_2}$| at a given R_2−1/1−0 decreases. As a result, LVG analysis with the assumption of Z(¹²CO) of 8.0 × 10⁻⁵ yields lower |$n_{\rm H_2}$|⁠. In addition, the low [¹²CO]/[¹³CO] abundance ratio of 30 causes a decrease in the derived molecular gas density as described above. Therefore, the difference in the derived |$n_{\rm H_2}$| between NGC 2903 and M 51 can be explained by the difference in the assumed Z(¹²CO) and the [¹²CO]/[¹³CO] abundance ratio.

4.2.2 Comparison of SFE with density and kinetic temperature of molecular gas

As described in section 1, SFEs often differ between galaxy centers and disks. Since NGC 2903 has a bright star-forming region at the center, its SFE is expected to be higher than those in other regions. Here, we calculate SFEs for seven regions where averaged physical properties of molecular gas were obtained, and compare SFE with |$n_{\rm H_2}$| and T_K in each region to examine what parameter controls SFE in galaxies.

SFE is expressed using the surface density of SFR (Σ_SFR) and that of molecular hydrogen (⁠|$\Sigma _{\rm H_2}$|⁠) as follows:

\begin{eqnarray} \left[ \frac{\rm SFE}{\rm yr^{-1}} \right]= \left( \frac{\Sigma _{\rm SFR}}{M_{\odot }\,{\rm yr^{-1}\,pc^{-2}}} \right) {\displaystyle \biggl /} \left( \frac{\Sigma _{\rm H_2}}{M_{\odot }\,{\rm pc^{-2}}} \right). \end{eqnarray}

(1)

We calculated extinction-corrected SFRs from a linear combination of Hα and Spitzer/MIPS 24 μm luminosities using the following formula (Kennicutt 1998a, 1998b; Calzetti et al. 2007):

\begin{eqnarray} &&\left[ \frac{\Sigma_{\rm SFR}}{M_{\odot}\>{\rm yr^{-1}\,pc^{-2}}} \right]\nonumber\\ &&\quad= 7.9 \times 10^{-42} \left( \frac{L_{{\rm H} \alpha } + 0.031 \times L_{24 \mu {\rm m}}}{{\rm erg} \, {\rm s}^{-1}} \right) \frac{{\rm cos} \ i}{\Omega }, \end{eqnarray}

(2)

where L_Hα and L_24μm mean Hα and 24 μm luminosities, respectively; i is the inclination of 65° for NGC 2903 and Ω is the covered area for each region (in units of pc²). We used archival continuum-subtracted Hα and 24 μm images of NGC 2903 obtained by Hoopes, Walterbos, and Bothun (2001) and the Local Volume Legacy survey project (Kennicutt et al. 2008; Dale et al. 2009), respectively. In addition, we calculated |$\Sigma _{\rm H_2}$| using I_12CO(1–0) as follows:

\begin{eqnarray} \left[ \frac{\Sigma _{\rm H_2}}{M_{\odot }\,{\rm pc^{-2}}} \right] &=& 2.89 \times {\rm cos} \ i \left( \frac{I_{\rm 12CO(1-0)}}{{\rm K\,km\,s^{-1}}} \right)\nonumber\\ &&\times\, \left\lbrace \frac{X_{\rm CO}}{1.8 \times 10^{20}\,{\rm cm}^{-2}\,({\rm K\,km\,s^{-1}})^{-1}} \right\rbrace . \end{eqnarray}

(3)

Here, we adopted a constant X_CO value of 1.8 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ (Dame et al. 2001). We found that SFE at the center, 6.8 × 10⁻⁹ yr⁻¹, is 2–4 times higher than those in other regions. The calculated SFEs are listed in table 3.

We examined the dependence of SFE on |$n_{\rm H_2}$| and T_K as shown in figure 11. We found that SFE positively correlates with both |$n_{\rm H_2}$| and T_K. However, the trend of these correlations might change because it is possible that variations in the [¹²CO]/[¹³CO] abundance ratio and X_CO affect the estimates of |$n_{\rm H_2}$|⁠, T_K, and SFE. In fact, both the [¹²CO]/[¹³CO] abundance ratio and X_CO often differ between galaxy centers and disks. Therefore, we examine how variations in the [¹²CO]/[¹³CO] abundance ratio and X_CO alter the estimates of |$n_{\rm H_2}$|⁠, T_K, and SFE at the center of NGC 2903.

Fig. 11.

Correlation between SFE and |$n_{\rm H_2}$| (left) and that between SFE and T_K (right) in each region of NGC 2903.

We first consider the effect of the variation in [¹²CO]/[¹³CO] abundance ratio on the estimates of |$n_{\rm H_2}$| and T_K. As described in sub-subsection 4.2.1, it is reported that the ¹²C/¹³C abundance ratio in our Galaxy increases with the galactocentric radius (Langer & Penzias 1990). Thus we examine the case of a lower [¹²CO]/[¹³CO] abundance ratio at the center of NGC 2903. If we adopt a [¹²CO]/[¹³CO] abundance ratio of 30 at the center, |$n_{\rm H_2}$| and T_K are estimated to be 1800 cm⁻³ and 38 K, respectively. This |$n_{\rm H_2}$| value is slightly lower than that in the northern arm, whereas the positive correlation between SFE and |$n_{\rm H_2}$| is not destroyed. Similarly, the T_K of 38 K does not destroy the positive correlation between SFE and T_K.

Next, we consider the effect of variation in X_CO on the estimate of SFE. In central regions of disk galaxies, X_CO drops (i.e., CO emission becomes luminous at a given gas mass) by a factor of 2–3 or more (e.g., Nakai & Kuno 1995; Regan 2000), including the Galactic Center (e.g., Oka et al. 1998; Dahmen et al. 1998). Such a trend is presumably applicable to NGC 2903 considering the relationship between X_CO and metallicity, 12 + log(O/H). In general, X_CO decreases with an increase in metallicity because the CO abundance should be proportional to the carbon and oxygen abundances (e.g., Arimoto et al. 1996; Boselli et al. 2002). In addition, it is reported that metallicity decreases with the galactocentric distance in NGC 2903 (e.g., Dack & McCall 2012; Pilyugin et al. 2014). These observational facts suggest a smaller X_CO by a factor of 1.5–2 at the center than in the disk of NGC 2903, which yields a smaller gas mass, providing a higher SFE than the present one shown in table 3 and figure 11. However, even if a higher SFE by a factor of two is adopted for the center, the global trend of the correlations shown in figure 11 does not change so much because the original SFE at the center is already the highest in NGC 2903. Therefore, we concluded that variations in the [¹²CO]/[¹³CO] abundance ratio and X_CO do not affect the correlations of SFE with |$n_{\rm H_2}$| and T_K in NGC 2903.

Note that a smaller X_CO corresponds to a larger Z(¹²CO) at the center of NGC 2903, but a larger dv/dr is also suggested because the typical velocity width at the center (250–300 km s⁻¹) is wider than those in other regions (150–200 km s⁻¹) due to the rapid rotation of molecular gas near the galaxy center. Thus we consider that Z(¹²CO)/(dv/dr) itself does not differ between the center and the disk in NGC 2903, even if the Z(¹²CO) at the center is larger than that in the disk.

Finally, we examine the correlation coefficient for the least-squares power-law fit R² between SFE and |$n_{\rm H_2}$| and that between SFE and T_K shown in figure 11. We found that the former is 0.50 and the latter is 0.08. The significant correlation between SFE and |$n_{\rm H_2}$| with R² of 0.50 suggests that molecular gas density governs the spatial variations in SFE. This speculation is well consistent with earlier studies based on HCN emission (e.g., Gao & Solomon 2004; Gao et al. 2007; Muraoka et al. 2009; Usero et al. 2015). In order to confirm whether such a relationship between SFE and |$n_{\rm H_2}$| is applicable to other galaxies or not, we will perform further analysis toward other COMING sample galaxies, considering variations in the [¹²CO]/[¹³CO] abundance ratio, X_CO, and Z(¹²CO)/(dv/dr), in forthcoming papers.

5 Summary

We have performed simultaneous mappings of J = 1–0 emission of ¹²CO, ¹³CO, and C¹⁸O molecules toward the whole disk (8′ × 5′ or 20.8 × 13.0 kpc at a distance of 8.9 Mpc) of the nearby barred spiral galaxy NGC 2903 with the NRO 45 m telescope equipped with FOREST at an effective angular resolution of 20″ (or 870 pc). A summary of this work is as follows:

1. We detected ¹²CO(J = 1–0) emission over the disk of NGC 2903. In addition, significant ¹³CO(J = 1–0) emission was found at the center and bar-ends, whereas we could not detect any significant C¹⁸O(J = 1–0) emission.

2. In order to improve the S/N of CO emission and to measure R_2−1/1−0 and R_13/12 with high significance, we performed a stacking analysis for our ¹²CO(J = 1–0), ¹³CO(J = 1–0), and archival ¹²CO(J = 2–1) spectra with velocity axis alignment in nine representative regions (i.e., center, northern bar, southern bar, northern bar-end, southern bar-end, northern arm, southern arm, inter-arm, and outer-disk) of NGC 2903. We successfully obtained stacked CO spectra with highly improved S/N, and thus we could confirm significant ¹³CO(J = 1–0) emission for all the regions.

3. We examined the averaged R_2−1/1−0 and R_13/12 for the nine regions, and found that the averaged R_2−1/1−0 shows the highest value of 0.92 at the center, and moderate or lower values of 0.6–0.8 are observed in the disk. However, the highest R_13/12 of 0.19 is observed not at the center but in the northern arm. The central R_13/12 of 0.11 is similar to those in other regions (0.08–0.13) except for the northern arm and outer-disk (∼0.04).

4. We determined |$n_{\rm H_2}$| and T_K of molecular gas using R_2−1/1−0 and R_13/12 based on the LVG approximation. Under the assumption of a [¹²CO]/[¹³CO] abundance ratio of 50, the derived |$n_{\rm H_2}$| ranges from ∼1000 cm⁻³ (in the bar, bar-ends, and spiral arms) to 3700 cm⁻³ (at the center) and the derived T_K ranges from 10 K (in the bar and spiral arms) to 30 K (at the center).

5. We examined the dependence of SFE on |$n_{\rm H_2}$| and T_K of molecular gas, and found a positive correlation between SFE and |$n_{\rm H_2}$| with correlation coefficient for the least-squares power-law fit R² of 0.50. This suggests that molecular gas density governs the spatial variations in SFE.

We thank the referee for invaluable comments, which significantly improved the manuscript. We are indebted to the NRO staff for the commissioning and operation of the 45 m telescope and their continuous efforts to improve the performance of the instruments. This work is based on observations at NRO, which is a branch of the National Astronomical Observatory of Japan, National Institutes of Natural Sciences. This research has made use of the NASA/IPAC Extragalactic Database, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

1

Laurikainen and Salo (2002) estimated the maxima of the tangential force and the averaged radial force at each radius in a bar using JHK-band images, and they defined the maximum of the ratio between the two forces as the bar strength.

2

In this paper, a correction factor to account for helium and other heavy elements is not included in the calculation of molecular gas mass and SFE.