FOREST Unbiased Galactic Plane Imaging Survey with the Nobeyama 45 m telescope (FUGIN). V. Dense gas mass fraction of molecular gas in the Galactic plane

Abstract

Recent observations of the nearby Galactic molecular clouds indicate that the dense gas in molecular clouds has quasi-universal properties on star formation, and observational studies of extra-galaxies have shown a galactic-scale correlation between the star formation rate (SFR) and the surface density of molecular gas. To reach a comprehensive understanding of both properties, it is important to quantify the fractional mass of dense gas in molecular clouds, f_DG. In particular, for the Milky Way (MW) there are no previous studies resolving f_DG disk over a scale of several kpc. In this study, f_DG was measured over 5 kpc in the first quadrant of the MW, based on the CO J = 1–0 data in l = 10°–50° obtained as part of the FOREST Unbiased Galactic plane Imaging survey with the Nobeyama 45 m telescope (FUGIN) project. The total molecular mass was measured using ¹²CO, and the dense gas mass was estimated using C¹⁸O. The fractional masses, including f_DG, in the region within ±30% of the distances to the tangential points of the Galactic rotation (e.g., the Galactic Bar, Far-3 kpc Arm, Norma Arm, Scutum Arm, Sagittarius Arm, and inter-arm regions) were measured. As a result, an averaged f_DG of |$2.9^{+2.6}_{-2.6}$|% was obtained for the entirety of the target region. This low value suggests that dense gas formation is the primary factor in inefficient star formation in galaxies. It was also found that f_DG shows large variations depending on the structures in the MW disk. In the Galactic arms, f_DG was estimated to be ∼4%–5%, while in the bar and inter-arm regions it was as small as ∼0.1%–0.4%. These results indicate that the formation/destruction processes of the dense gas and their timescales are different for different regions in the MW, leading to differences in Star formation efficiencies.

1 Introduction

Star formation in galaxies is characterized by the Kennicutt–Schmidt (KS) law (Schmidt 1959; Kennicutt 1998; Kennicutt & Evans 2012), which is a galactic-scale empirical correlation between the area-averaged star formation rate (SFR; |$\Sigma _{\rm SFR} \, [M_{\odot }\, {\rm yr^{-1}\, kpc^{-2}}]$|⁠) and gas surface density (⁠|$\Sigma _{\rm H_2+H}\, [M_{\odot }\, {\rm pc^{-2}}]$|⁠) with a power-law index of ∼1.4 (⁠|$\Sigma _{\rm SFR} \propto \Sigma _{\rm H_2+H}^N$|⁠). This correlation can be seen in the inner parts of galaxies, where H₂ is dominant (e.g., Tanaka et al. 2014; Sofue & Nakanishi 2016), indicating an index of ∼1 with scatters of approximately 0.2 dex (⁠|$\Sigma _{\rm SFR} \propto \Sigma _{\rm H_2}$|⁠; Bigiel et al. 2008). The scatter is likely due to the regional differences in star formation efficiency (SFE) in the individual galaxies (i.e., bar, arm, inter-arm, and nucleus; e.g., Momose et al. 2010). The KS law predicts the gas consumption timescale of H₂ gas to be |$\tau _{\rm con} = \Sigma _{\rm H_2}/\Sigma _{\rm SFR} \approx 1$|–2 Gyr (e.g., Bigiel et al. 2011), which is three orders of magnitude larger than a free-fall timescale of ∼1 Myr at a gas density of 100 cm⁻³. Understanding the background physics of inefficient star formation in galaxies is one of the most pressing issues in contemporary astrophysics.

While the |$\Sigma _{\rm H_2}$| used in the KS law is generally measured using the CO rotational transition emission, several observations used the HCN J = 1–0 transition with a critical density of ∼2 × 10⁶ cm⁻³ (which is in reality reduced by radiative trapping owing to its high optical depth, Gao & Solomon 2004b) to measure the mass (or luminosity) of dense molecular gas to construct a dense-gas KS law (Gao & Solomon 2004a,b; Usero et al. 2015; Bigiel et al. 2016). Their results indicate a tighter correlation with Σ_SFR rather than |$\Sigma _{\rm H_2}$|⁠.

The recent submillimeter imaging surveys of Galactic molecular clouds with the Herschel Space Observatory have made remarkable progress in understanding star formation in dense gas. The Herschel observations demonstrate that molecular filaments take up a dominant fraction of dense gas in molecular clouds (André et al. 2010; Molinari et al. 2010; Könyves et al. 2015; Arzoumanian et al. 2019). These filaments are characterized by the narrow distribution of central widths, with a full width at half maximum (FWHM) of ∼0.1 pc (Arzoumanian et al. 2011).

An important discovery made by the Herschel observations was that the majority of prestellar cores are embedded within “supercritical” filaments (André et al. 2014), for which the mass per unit length exceeds the critical line mass, |$M_{\rm line, crit} = 2 c_{\rm s}^2/G \sim 16\, M_{\odot }\, {\rm pc^{-1}}$| (e.g., Inutsuka & Miyama 1997), where c_s ∼ 0.2 km s⁻¹ is the isothermal sound speed at temperature T ∼ 10 K, and G is the gravitational constant. Given a filament width of ∼0.1 pc, |$M_{\rm line, crit} \sim 16\, M_{\odot }\, {\rm yr^{-1}}$| predicts a quasi-universal threshold for core/star formation in molecular clouds at |$\Sigma _{\rm H_2} \sim 160\, M_{\odot }\, {\rm yr^{-1}}$| in terms of the gas surface density, which corresponds to an H₂ column density |$N_{\rm H_2}$| of ∼7 × 10²¹ cm⁻² or a visual extinction A_v of ∼8 mag (assuming |$N_{\rm H_2}/A_{\rm v} = 0.94 \times 10^{21}$|⁠, Bohlin et al. 1978). Such a threshold for star formation was also discussed in independent observational studies; Onishi et al. (1998) proposed star formation of |$N_{\rm H_2} \ge 8\times 10^{21}\:$|cm⁻² based on the C¹⁸O observations of the Taurus molecular cloud. The Spitzer infrared observations of Galactic nearby clouds provided a similar threshold of |$\Sigma _{\rm H_2} \sim 130\, M_{\odot }\, {\rm yr^{-1}}$| (Heiderman et al. 2010).

Measurements of SFE in dense gas, or supercritical filaments, were performed on nearby Galactic molecular clouds (Wu et al. 2005; Lada et al. 2010, 2012; Shimajiri et al. 2017). The studies of Lada, Lombardi, and Alves (2010) and Shimajiri et al. (2017) were done on gas with A_v > 8 mag, presenting results that were consistent with the study on extra-galaxies by Gao and Solomon (2004b)—see also Bigiel et al. (2016); the gas consumption timescale of the dense gas can be computed as τ_con ∼ 20 Myr. This implies that the SFE in dense molecular gas in galaxies is quasi-universal on scales from ∼1–10 pc to >10 kpc. Lada, Lombardi, and Alves (2010) proposed that the fundamental relationship governing star formation in galaxies is |$\Sigma _{\rm SFR} \propto f_{\rm DG}\Sigma _{\rm H_2}$|⁠, where f_DG is the mass fraction of dense gas to molecular gas. Following these studies, this paper defines “dense gas” as gas with A_v > 8 mag.

For extra-galaxies, Muraoka et al. (2016) revealed the temperature and density distribution of the molecular gas in NGC 2903 using large velocity gradient analysis (Goldreich & Kwan 1974; Scoville & Solomon 1974), presenting a positive correlation between gas densities and SFE. HCN observations of extra-galaxies by Usero et al. (2015) and Bigiel et al. (2016) found that SFE in dense gases depends on galactic environment, being lower at high stellar surface densities and high H₂-to-H i mass ratio.

These studies emphasize the importance of measuring f_DG in various galactic environments and understanding its relationship with star formation. In the MW, Battisti and Heyer (2014) measured the H₂ mass fractions of dense gas components in Giant Molecular Clouds (GMCs) in the Galactic plane. The masses of the GMCs were calculated using the ¹³CO J = 1–0 data for l = 18°–56° taken by the Five College Radio Astronomical Observatory (FCRAO), while those of the dense gas were calculated from the Bolocam Galactic Plane Survey (BGPS) 1.1 mm dust continuum images. The radii and masses of their GMC samples covered ∼5–20 pc and 10³–|$10^{5}\, M_{\odot }$|⁠, respectively. The authors obtained a low averaged fractional mass of 11|$^{+12}_{-06}$|%, and the derived mass fractions are independent of the GMC masses. Since the GMC masses were derived in ¹³CO, they cannot be directly compared to the KS law in extra-galaxies, in which |$\Sigma _{\rm H_2}$| is usually measured using ¹²CO. Based on the ¹²CO and CS observations toward a 2 deg² area at l ∼ 44.°1–46.°3, Roman-Duval et al. (2016) obtained an f_DG of ∼14% over ∼200–300 pc at R of 6–8.5 kpc. To date, f_DG for the Galactic plane has not been derived for kpc scales.

In this study, ¹²CO, ¹³CO, and C¹⁸O J = 1–0 data obtained for l = 10°–50° using the Nobeyama 45 m radio telescope were analyzed to measure the H₂ mass (⁠|$M_{\rm H_2}$|⁠) of molecular gas detected independently in the three CO isotopologues. The J = 1–0 transition of CO has a critical density of ∼2 × 10³ cm⁻³, and the differences in the abundance ratios of the three CO isotopologues (which lead to differences in the optical depths along the line of sight) allow us to probe different ranges of |$N_{\rm H_2}$| in the molecular clouds.

Our analyses include the Galactic bar, Far-3 kpc Arm, Norma Arm, Scutum-Centaurus Arm, and Sagittarius Arm, as well as the inter-arm regions between these arms. The fractional masses of dense molecular gas in these regions were first measured by taking the mass ratios of the C¹⁸O emitting gas to the ¹²CO emitting gas.

The J = 1–0 transition of ¹²CO is known as a tracer of the total |$M_{\rm H_2}$| of molecular clouds, although it is consistently found to be optically thick. Comparisons between the ¹²CO integrated intensities W(¹²CO) with other |$M_{\rm H_2}$| tracers (e.g., virial mass, gamma-ray emission, and dust emission) indicated a surprisingly close correlation between W(¹²CO) and the |$N_{\rm H_2}$| of the molecular clouds (e.g., Dickman 1978; Sanders et al. 1984; Solomon et al. 1987; Strong & Mattox 1996; Dame et al. 2001; Planck Collaboration 2011). This provides a CO-to-H₂ conversion factor in the inner Galaxy at the galactocentric radius R of 1–9 kpc (Bolatto et al. 2013) as reported and expressed by equation (1),

$$\begin{equation} X({\rm CO}) \ = \ \frac{N_{\rm H_2}}{W({\rm ^{12}CO})} \ = \ 2 \times 10^{20} {\rm (K\, km\, s^{-1})^{-1}\, cm^{-2}}. \end{equation}$$

(1)

The uncertainty of the X(CO) was discussed to be as small as a factor of 1.3 by Bolatto, Wolfire, and Leroy (2013). X(CO) has been utilized to measure |$\Sigma _{\rm H_2}$| in other galaxies; therefore, it is important to measure the |$M_{\rm H_2}$| of the molecular gas in the MW from the ¹²CO J = 1–0 data and X(CO) to evaluate f_DG.

Although ¹²CO J = 1–0 is essentially a good |$M_{\rm H_2}$| tracer of molecular clouds, it barely works as an |$N_{\rm H_2}$| tracer in the dense parts of molecular clouds that have large |$N_{\rm H_2}$| due to intensity saturation by the opacity effect, and therefore the less abundant ¹³CO and C¹⁸O are typically used to measure more accurately |$M_{\rm H_2}$| and |$N_{\rm H_2}$| in these parts. Low-J transitions of ¹³CO have been used in observations of nearby GMCs, which have a typical density of ∼10³ cm⁻³ (e.g., Mizuno et al. 1995; Nagahama et al. 1998; Goldsmith et al. 2008; Narayanan et al. 2008; Nishimura et al. 2015). C¹⁸O is optically thin even in the higher-|$N_{\rm H_2}$| parts of the molecular clouds, which have dominant filamentary structures of width 0.1 pc (e.g., Onishi et al. 1996, 1998; Hacar et al. 2013; Nishimura et al. 2015; Tokuda et al. 2018; Arzoumanian et al. 2019). Therefore, C¹⁸O can be used to measure the |$M_{\rm H_2}$| of the dense gas to evaluate f_DG in the MW.

Here, it is noteworthy that CO molecules sometimes do not trace |$N_{\rm H_2}$| in dense cores at the innermost parts of molecular clouds owing to the heavy opacities and/or molecular depletion of dust grains (e.g., Caselli et al. 1999; Bergin & Tafalla 2007). However, as the Herschel observations revealed, the ratio of the mass in the dense star-forming cores to the mass of the parental filament is less than ∼15% on average (André et al. 2014; Könyves et al. 2015). This means that possible depletion of molecules in the dense regions has only a limited effect on the estimation of the total mass in the filaments. Likewise, molecular depletion is not expected to be significant in our mass estimate from the C¹⁸O data with pc-scale resolution in this study.

The remainder of this paper is organized as follows. Section 2 describes the CO J = 1–0 dataset used in this study. Section 3 presents the target region of the current analyses. Section 4 presents the main results of analyzing the CO dataset. The results are discussed in section 5, and a summary is presented in section 6.

2 Dataset

The ¹²CO, ¹³CO, and C¹⁸O J = 1–0 datasets obtained by the FOREST Unbiased Galactic plane Imaging survey using the Nobeyama 45 m telescope (FUGIN; see Umemoto et al. 2017 for a full description of the observations and data reduction) were analyzed. FUGIN involved a large-scale Galactic plane survey using the FOur-beam REceiver System on the 45 m Telescope (FOREST; Minamidani et al. 2016): the four-beam, dual-polarization, two-sideband receiver installed in the Nobeyama 45 m telescope. This study utilized the FUGIN dataset obtained at l = 10°–50° in the first quadrant of the Galactic plane. The typical system temperatures were ∼250 K for ¹²CO J = 1–0 (115.271 GHz) and ∼150 K for ¹³CO J = 1–0 (110.201 GHz) and C¹⁸O J = 1–0 (109.782 GHz). The backend system was the digital spectrometer “SAM45” (Kuno et al. 2011; Kamazaki et al. 2012), which provided a bandwidth of 1 GHz and a resolution of 244.14 kHz. These figures corresponded to 2600 km s⁻¹ and 0.65 km s⁻¹ at 115 GHz, respectively. The observations were made in the On-The-Fly mode with a unit map size of 1° × 1°. The pointing accuracy was checked almost every hour to keep within 3″ by observing SiO maser sources. The output data were formatted into a size of 1° × 2° with spatial and velocity grid sizes of |${8{^{\prime\prime}_{.}}5}$| and 0.65 km s⁻¹, respectively. Absolute intensity calibrations were performed by adopting main beam efficiencies of 0.45 ± 0.02 and 0.43 ± 0.02 at 110 and 115 GHz, respectively (Umemoto et al. 2017).

The output CO data, particularly ¹²CO data, suffer from scan effects and spurious structures. To remove these features, as well as to improve the sensitivity, the following post-processes were applied to the output data cube: (1) one-dimensional median filtering to the velocity axis with a kernel of 3 ch, (2) two-dimensional median filtering to the spatial axes with a kernel of 3 × 3 ch, and (3) two-dimensional spatial smoothing with a Gaussian function to achieve a spatial resolution of 40″. Figure 1 presents the root mean square (rms) 1 ch noise σ distributions of the post-processed ¹²CO, ¹³CO, and C¹⁸O data. Although the post-processing efficiently removes the scan effects and spurious structures, some of these features still remain in some tiles. These remaining features were finally removed in identifying the CO sources (subsection 4.2).

Fig. 1.

Root mean square noise (σ) distributions of the ¹²CO, ¹³CO, and C¹⁸O data. (Color online)

3 Region selection

The vertical distributions of ¹²CO in the inner Galaxy were measured to be ∼50–100 pc at FWHM (e.g., Nakanishi & Sofue 2006). Thus, it is necessary to cover ∼200 pc in b to accurately measure the total |$M_{\rm H_2}$| in ¹²CO. It is also important to achieve a spatial resolution of less than a few pc to detect the dense gas components in molecular clouds (e.g., Bergin & Tafalla 2007). Furthermore, in order to estimate f_DG accurately, it is of primary importance to reduce the errors on estimating the distances of molecular clouds, as the above requirements for the b coverage and spatial resolution cannot be guaranteed otherwise.

Considering that these conditions required measuring f_DG in the Galactic plane, the focus was placed on the tangential points of the Galactic rotation relative to the local standard of rest (LSR), at which the radial velocities of the CO emissions v_LSR correspond to the terminal radial velocities of the Galactic rotation v_term, and one unique solution in kinematic distance can be given. The kinematic distance to the tangential points are referred to as d_tan, which depends only on l, from here on. Assuming the IAU standard parameters (distance to the Galactic center R₀ = 8.5 kpc, LSR rotational velocity Θ₀ = 220 km s⁻¹), the l coverage of the FUGIN observations (l = 10°–50°) corresponds to d_tan of ∼5.5–8.4 kpc and galactocentric distances to the tangential points R_tan of ∼1.5–6.5 kpc.

In figure 2a, the thick black line plotted on an illustration of the face-on view of the MW indicates the tangential points included in l = 10°–50°. The dashed black lines and solid green area indicate the area at distances within ±30% of d_tan, at which the |$M_{\rm H_2}$| of molecular gases were measured in this study. This target area, measuring ∼25.7 kpc², was defined to include the Galactic bar, Far-3 kpc Arm, Norma Arm, Scutum Arm, and Sagittarius Arm, as well as to satisfy the required conditions for measuring f_DG by quantifying the |$M_{\rm H_2}$| traced by ¹²CO, ¹³CO, and C¹⁸O as discussed above. At d_tan ∼5.5–8.4 kpc in this area, the b coverage of the FUGIN data (|b| ≤ ±1°) corresponds to ∼192–293 pc (or 134–380 pc including the ±30% error of d_tan), and the spatial resolutions of the post-processed FUGIN data were calculated as ∼1.1–1.6 pc (or 0.7–2.1 pc including the ±30% error of d_tan). In the majority of the target area, the vertical coverage and spatial resolution satisfied the required conditions for measuring f_DG; however, in some parts at higher l the vertical coverage was less than 200 pc. Thus, the vertical extent of the ¹²CO emission may not be fully covered in these parts.

Fig. 2.

(a) The target region of this study is plotted on an illustration of the face-on view of the MW [NASA/JPL-Caltech/R. Hurt (SSC/Caltech)]. The thick black line indicates the tangential points of the Galactic rotation relative to LSR, whose distances from the Sun (d_tan) are used to compute the physical parameters of CO emissions, while the two thick dashed black lines show the lines of ±30% of d_tan. The solid green area indicates the target region of the present analyses. Thin black dashed lines show the coverage of the FUGIN observations in the first quadrant, which are plotted every 10° from l = 10° to 50°. Dotted lines show isodistance contours from the Sun. Colored lines denote loci of the Galactic arms constructed by Reid et al. (2016). (b) As panel (a), but without the background image and with the plots of the sources whose distances are determined by trigonometry (Hou & Han 2014), where the circles, squares, and triangles depict the GMCs, maser sources, and H ii regions, respectively. (Color online)

Figure 3 shows the l–v diagram of the FUGIN ¹²CO J = 1–0 data integrated over ±1° in b. The thick black line shows the curve of v_term. The two dashed lines define the target velocity ranges of the present analyses; the dashed line plotted below v_term indicates the velocities which correspond to ±30% of d_tan, while the other shows a +30 km s⁻¹ margin from v_term, which is set to cover the CO features with v_LSR higher than v_term. In figures 2b and 3 the sources whose distances were determined by trigonometry are plotted; this data was compiled by Hou and Han (2014). Many of the sources within d_tan ± 30% (open symbols) are distributed within the target velocity ranges indicated as the unmasked area in figure 3, and the numbers of false positives (sources within d_tan ± 30% that are distributed outside target velocities) and false negatives (sources outside d_tan ± 30% (closed symbols) that are located within the target velocities) are small. The small number of false negatives supports the assumption of flat rotation in the target area. Here it is notable that the distance-determined sources were not catalogued in the Galactic Bar region (figure 2), and the assumption of flat rotation may not apply in this region owing to the non-circular rotation of the Galactic Bar (e.g., Regan et al. 1999; Sorai et al. 2012). This results in some fraction of the molecular gas distributed in d_tan ± 30% not having v_LSR within the target velocities. However, it is still probably rare for molecular gas outside the bar region to contaminate the target velocities plotted in figure 3.

Fig. 3.

l–v diagram of the FUGIN ¹²CO J = 1–0 data. The integration range in b is from −1° to +1°. The thick black line shows the curve of the terminal velocities. The two dashed lines define the target velocity ranges of the present analyses; the one plotted below the curve of the terminal velocities indicates the velocities which correspond to +30% or −30% of d_tan, while the other shows a +30 km s⁻¹ margin from the terminal velocities. The masked area in this figure is not used for the |$M_{\rm H_2}$| estimates in this study. The colored lines show the loci of the Galactic arms constructed by Reid et al. (2016). The four horizontal arrows show the rough extents of the four regions. Regions A–D, of this study. The circles, squares, and triangles show the GMCs, maser sources, and H ii regions of the Hou and Han (2014) samples, respectively. The open and closed symbols indicate the sources within and outside d_tan ± 30% in figure 2, respectively. (Color online)

In figure 3, the loci of the Galactic arms constructed by Reid et al. (2016) are plotted in colored lines. Two famous massive star-forming regions—W|$\, 51$| and W|$\, 43$|—are distributed around the tangential points of the Sagittarius Arm and Scutum Arm, respectively (e.g., Mehringer 1994; Carpenter & Sanders 1998; Motte et al. 2014; Sofue et al. 2019). The 3 kpc Arm is thought to be distributed at the same R as the major axis of the Galactic Bar (figure 2a). Although the location of the tangential point of the 3 kpc Arm has not been confirmed, Green et al. (2011) proposed that it may be around l ∼ 20°–22° in the first quadrant. Given the distributions of these components in l, the target region of this study can be roughly classified into four subregions: Region A, the Galactic Bar and Far-3 kpc Arm (l < 20°); Region B, the Norma Arm and the Scutum Arm (l ∼ 22°–33°); Region C, the inter-arm region between the Scutum Arm and the Sagittarius Arm (l ∼ 35°–45°); and Region D, the Sagittarius Arm (l > 47°). Note that these classifications remain ambiguous, as the distributions of the Galactic Bar and arms are not fully understood.

Figures 4, 5, and 6 show the ¹²CO, ¹³CO, and C¹⁸O intensity distributions, respectively, integrated over the target velocity ranges of figure 3. These distributions are denoted W(¹²CO), W(¹³CO), and W(C¹⁸O) from here on. It can be seen from figures 4–6 that the vertical extents of the CO emission are well covered within the |b| < 1° coverage of the FUGIN observations. More detailed b distributions of W(¹²CO) are shown in figure 7, where the W(¹²CO) profiles along b are plotted at every 1° in l, with the intensity-weighted average velocities and |$\pm 1 \, \sigma$| velocity dispersions plotted with pink lines. This figure indicates that the vertical distributions of the ¹²CO emission are sufficiently covered to estimate the total |$M_{\rm H_2}$| in all the regions except for l ∼ |$40{^{\circ}_{.}}5-48{^{\circ}_{.}}5$|⁠, where the W(¹²CO) distribution is shifted toward the negative direction in b by ∼|$0{^{\circ}_{.}}2-0{^{\circ}_{.}}3$|⁠, running off the edge at b = −1°. As the |$\pm 1 \, \sigma$| dispersions were covered even in these regions, it is expected that approximately 80%–90% of the total |$M_{\rm H_2}$| is included within the present b coverage.

Fig. 4.

Integrated intensity distributions of the FUGIN ¹²CO J = 1–0 data for the velocity ranges plotted in figure 3 [W(¹²CO)]. The vertical arrows indicate the ±50 pc heights at d_tan = 5.5, 6, 6.5, 7, 7.5, and 8 kpc. (Color online)

Fig. 5.

As figure 4, but for W(¹³CO). (Color online)

Fig. 6.

As figure 4, but for W(C¹⁸O). (Color online)

Fig. 7.

The W(¹²CO) distributions along the Galactic latitude are plotted with a bin size of l = 1°. The peak W(¹²CO) of each plot is normalized to 1, and the peak value is presented at the top right of the plot in K km s⁻¹. The pink lines indicate the intensity-weighted average velocities and |$\pm 1 \, \sigma$| velocity dispersions. (Color online)

4 Methods

4.1 Mass calculations

The H₂ column density measured by ¹²CO, N_H2(¹²CO), was estimated using X(CO), and this study adopted a uniform value of 2.0 × 10²⁰ (K km s⁻¹)⁻¹ cm⁻² [equation (1)]. The ¹³CO column density N₁₃ was estimated by assuming local thermodynamic equilibrium (LTE). The excitation temperature of the ¹³CO emission, T_{ex, 13}, was derived in each line of sight using the peak brightness temperature of the optically thick ¹²CO emission, T_peak(¹²CO), assuming a common excitation temperature between ¹²CO and ¹³CO:

$$\begin{equation} T_{\rm ex,13}\ = \ \frac{5.53}{\ln \lbrace 1 + 5.53 / [T_{\rm peak}(^{12}{\rm CO}) + 0.819] \rbrace }. \end{equation}$$

(2)

Then, the ¹³CO optical depth τ₁₃ at each voxel can be computed by the following equation:

$$\begin{equation} \tau _{13} \ = \ -\ln \left\lbrace 1 - \frac{T(^{13}{\rm CO})}{5.29 [J_{13}(T_{\rm ex,13}) - 0.164] } \right\rbrace , \end{equation}$$

(3)

where J₁₃(T) ≡ 5.29/[exp (5.29/T) − 1] and T(¹³CO) is the brightness temperature of the ¹³CO emission in each voxel. N₁₃ was finally computed by integrating τ₁₃ along the given velocity ranges as follows:

$$\begin{equation} N_{13}\ = \ 2.42 \times 10^{14} \frac{T_{\rm ex,13} + 0.87}{1- \exp (-5.29/T_{\rm ex,13})} \int \tau _{13}\, dv. \end{equation}$$

(4)

The C¹⁸O column density N₁₈ was also estimated assuming LTE. As the optically thick ¹²CO traces different parts of the molecular clouds than the optically thin C¹⁸O, it is difficult to assume a common excitation temperature between ¹²CO and C¹⁸O. Further, it is difficult to estimate the excitation temperature of C¹⁸O, T_{ex, 18}, from the ¹³CO spectra, because ¹³CO emission is not always optically thick toward the identified C¹⁸O sources. Thus, in this study a uniform T_{ex, 18} of 10 K was assumed as the typical temperature of dense gas (e.g., Onishi et al. 1996; Schneider et al. 2016). Then, the C¹⁸O optical depth, τ₁₈, and N₁₈ were derived as follows:

$$\begin{equation} \tau _{18} \ = \ -\ln \left\lbrace 1 - \frac{T({\rm C^{18}O})}{5.27 [J_{18}(T_{\rm ex,18}) - 0.166] } \right\rbrace , \end{equation}$$

(5)

$$\begin{equation} N_{18} \ = \ 2.54 \times 10^{14} \frac{T_{\rm ex,18} + 0.87}{1- \exp (-5.27/T_{\rm ex,18})} \int \tau _{18} \, dv, \end{equation}$$

(6)

where J₁₈(T) ≡ 5.27/[exp (5.27/T) − 1] and T(C¹⁸O) is the brightness temperature of the C¹⁸O emission.

The derived N₁₃ and N₁₈ in equations (4) and (6) were then converted into the H₂ column densities, N_H2(¹³CO) and |$N_{\rm H_2}({\rm C^{18}O})$|⁠, respectively, by adopting the abundance ratios of the CO isotopologues. This study adopted the following relationships constructed by Wilson and Rood (1994):

$$\begin{equation} [{\rm ^{12}C}]/[{\rm ^{13}C}] \ = \ (7.5\pm 1.9) R + (7.6\pm 12.9), \end{equation}$$

(7)

$$\begin{equation} [{\rm ^{16}O}]/[{\rm ^{18}O}] \ = \ (58.8\pm 11.8) R + (37.1\pm 82.6). \end{equation}$$

(8)

The slope of [¹²C]/[¹³C] is consistent with the fits for the CO and CN data by Milam et al. (2005). The measurements of the abundance ratios are not enough at R < 4 kpc, while a [¹²C]/[¹³C] of 20 and a [¹⁶O]/[¹⁸O] of 250 were measured in the Galactic center at R < 150 pc (Wilson & Rood 1994). Thus, the lower limits for equations (7) and (8) were set as 20 and 250 at R < 4 kpc, respectively. A [H₂]|$/$|[¹²CO] ratio of 10⁴ was adopted (e.g., Frerking et al. 1982; Leung et al. 1984). N_H2(¹³CO) and |$N_{\rm H_2}({\rm C^{18}O})$| were finally derived based on [¹²C]/[¹³C] and [¹⁶O]/[¹⁸O] calculated at R_tan in each l.

The obtained N_H2(¹²CO), N_H2(¹³CO), and |$N_{\rm H_2}({\rm C^{18}O})$| were then used to calculate the H₂ masses, M_H2(¹²CO), |$M_{\rm H_2}{\rm (^{13}CO)}$|⁠, and |$M_{\rm H_2}{\rm (C^{18}O)}$|⁠, respectively:

$$\begin{equation} M_{\rm H_2} = 2.8 m_{\rm H} d_{\rm tan}^2 \Delta _{\rm grid}^2 N_{\rm H_2}, \end{equation}$$

(9)

where m_H is the mass of the hydrogen and Δ_grid is the spatial grid size of the CO data: |${8{^{\prime\prime}_{.}}5}$|⁠.

4.2 Identification of the CO sources

The σ of the FUGIN CO data displays a large variation for each region and for each CO line (figure 1), and as presented in the histograms of σ in all the 1° × 2° regions in figures 15–19 in appendix 1, in many regions the σ distribution does not show symmetric profiles with respect to the average values, particularly in ¹²CO (figures 15–19); therefore it is difficult to estimate |$M_{\rm H_2}$| by simply summing all the pixel values included in the target velocity ranges. Hence, this study identified CO sources used to estimate |$M_{\rm H_2}$| in the following two steps: (1) identify every local maximum by drawing contours in the data cube at a brightness temperature of T_min; (2) remove the identified structures with voxel numbers less than N_min (the remaining structures are counted as CO sources). The second step is useful for removing spurious structures; for instance, if N_min is sufficiently larger than the voxel number of the spurious structures (Rice et al. 2016). As the post-processed FUGIN CO data have a beamsize of 40″ with a grid size of |${8{^{\prime\prime}_{.}}5}$|⁠, approximately 20 pixels are included in the beam on the l–b plane. Considering a narrow width (1–2 pixels) of the spurious features in velocity, an N_min value of 40 was applied in the present identifications. We found that this choice effectively removes many spurious structures that still remain after the post-processing (section 2). Then T_min in the first step was determined with a fixed N_min of 40.

Figure 8 shows examples of the identification of the ¹²CO, ¹³CO, and C¹⁸O sources in the 1° × 2° region at l = 32°–33°. Here, CO sources were identified using two T_min of |$3 \, \sigma _{\rm med}$| and |$5 \, \sigma _{\rm med}$|⁠, where σ_med is the median value of the σ in this region. As σ increase at b < |$-0{^{\circ}_{.}}5 $| in this region (figure 1), many tiny structures were identified at b < |$-0{^{\circ}_{.}}5 $| in all three CO isotopologues at |$T_{\rm min} = 3 \, \sigma _{\rm med}$| (gray contours), which are misidentifications of the CO sources. On the other hand, when |$T_{\rm min} = 5 \, \sigma _{\rm med}$| (red contours), these noise structures were removed, and the CO sources were properly identified.

Fig. 8.

Examples of CO source identifications at l = 32°–33° for (a) ¹²CO, (b) ¹³CO, and (c) C¹⁸O data. The gray and red contours show the outlines of the identified structures at |$T_{\rm min} = 3 \, \sigma _{\rm med}$| and |$5 \, \sigma _{\rm med}$|⁠, respectively. The |$1 \, \sigma _{\rm med}$| of the ¹²CO, ¹³CO, and C¹⁸O data were computed to be ∼0.38 K, ∼0.16 K, and ∼0.16 K, respectively. (Color online)

Figure 9 shows histograms of σ for the ¹²CO, ¹³CO, and C¹⁸O data in this region. At the data points included in the orange areas the CO emissions were identified at |$\gt 3 \, \sigma$| when |$T_{\rm min} = 5 \, \sigma _{\rm med}$|⁠, and these data points account for 98.97%, 99.48%, and 99.35% of all the data points in this region for ¹²CO, ¹³CO, and C¹⁸O, respectively, while these fractions decreased to 59.61%, 79.63%, and 78.70% with |$T_{\rm min} = 3 \, \sigma _{\rm med}$|⁠, respectively. CO sources could be identified at less than |$3 \, \sigma$| for many data points. The σ distributions in all the 1° × 2° regions in figures 15–19 in appendix 1 indicate that the |$5 \, \sigma _{\rm med}$| threshold can be used for the effective identification of CO sources. Therefore, in this study |$T_{\rm min} = 5 \, \sigma _{\rm med}$| was uniformly adopted in a given region. After applying this algorithm, the results of the identified structures were visually confirmed, and if the artificial structures due to the scanning effects were identified as CO sources, these structures were removed by hand.

Fig. 9.

Histograms of σ for the ¹²CO (top), ¹³CO (center), and C¹⁸O data in the region at l = 32°–33° (figure 8). The vertical solid and dotted lines respresent σ_med and |$5 \, \sigma _{\rm med}$|⁠, respectively. The orange area indicates the data points with |$3 \, \sigma \lt 5 \, \sigma _{\rm med}$|⁠, and the fraction of the data points in the orange area is shown at the top right of the panel. The histograms of all the 1° × 2° regions are presented in figures 15–19 in appendix 1. (Color online)

4.3 Constructing the longitudinal distribution of M_H2

The target 40° × 2° region was divided into 40 tiles (1° × 2°), and these tiles each have a different |$5 \, \sigma _{\rm med}$|⁠. It is important to apply a uniform T_min to make fair comparisons between different tiles, and a T_min of 1 K was uniformly adopted in this study. The |$5\, \sigma _{\rm med}$| levels of the ¹³CO and C¹⁸O data were lower than 1 K in every tile, and the |$M_{\rm H_2}{\rm (^{13}CO)}$| and |$M_{\rm H_2}{\rm (C^{18}O)}$| could be measured directly by applying T_min = 1 K. However, the |$5 \, \sigma _{\rm med}$| levels of the ¹²CO data exceeded 1 K in all the tiles except for those with relatively low σ at l ∼ 38°–43°. A reasonable way of deriving the M_H2(¹²CO) at T_min = 1 K in the data for tiles with high σ is to apply an extrapolation technique. The M_H2(¹²CO) of the regions with |$5\, \sigma _{\rm med} \gt 1\:$|K at T_min = 5, 6, 7, 8, 9, 10, and 11 × σ_med were derived, and plots of the derived M_H2(¹²CO) were made as functions of T_min. Figure 10 shows an example at l = 32°–33°, where the vertical axis shows M_H2(¹²CO) divided by max|${}[M_{\rm H_2}{\rm (^{12}CO)}]$|⁠, which is the maximum M_H2(¹²CO) measured at |$T_{\rm min} = 3\, \sigma _{\rm med}$|⁠. The plots were extrapolated by making linear fits using the three data points at T_min = 5, 6, and 7 × σ_med to estimate the M_H2(¹²CO) at T_min = 1 K. The results of the extrapolations in all the 1° × 2° regions are presented in figures 20–22 in appendix 2. The resulting M_H2(¹²CO) was increased by a factor of ∼1.1–2.0 from max|${}[M_{\rm H_2}{\rm (^{12}CO)}]$|⁠. If two or four data points were used for the fits instead of three, the obtained factors changed by ±1%–7% depending on the temperature difference |$\Delta T\, {\rm (K)}= 5 \, \sigma _{\rm med}-T_{\rm min}$|⁠.

It would be best if |$M_{\rm H_2}$| at T_min = 0 K were calculated using this extrapolation technique; however, the errors in this case could become significantly large because of large ΔT. Therefore, |$T_{\rm min} = 1\,$|K was applied in this study to suppress the errors due to the extrapolations.

Fig. 10.

Example of the extrapolation to derive the M_H2(¹²CO) at T_min = 1 K at l = 32°–33°. The blue circles indicate the normalized values of M_H2(¹²CO) measured at T_min = 3, 4, 5, 6, 7, 8, 9, 10, and 11 × σ_med. The normalizations are generated by dividing M_H2(¹²CO) by max|$[M_{\rm H_2}{\rm (^{12}CO)}]$|⁠, which is the maximum M_H2(¹²CO). The arrow indicates the resulting value of the extrapolation at T_min = 1 K. The plots of all the 1° × 2° regions are presented in figures 20–22 in appendix 2. (Color online)

4.4 Uncertainties

4.4.1 M  _H2(¹²CO)

The uncertainty of the derived M_H2(¹²CO) was calculated from the uncertainties of W(¹²CO), X(CO), and extrapolation. Umemoto et al. (2017) discussed that the FUGIN data has uncertainties in the observed brightness temperatures of ±10%–20% for ¹²CO and ±10% for ¹³CO and C¹⁸O (Umemoto et al. 2017). The uncertainty in X(CO) was uniformly set as ±30%, as reported by Bolatto, Wolfire, and Leroy (2013) for R = 1–9 kpc. Note that the assumption of uniform X(CO) possibly led to overestimates of the derived M_H2(¹²CO) at R < ∼ 2–3 kpc by up to a factor of ∼2, as it has been reported that X(CO) decreases in the Galactic center region at R < 1 kpc, e.g., 0.24 × 10²⁰ (K km s⁻¹)⁻¹ cm⁻² (Oka et al. 1998) at R < 0.1 kpc and 0.7 × 10²⁰ (K km s⁻¹)⁻¹ cm⁻² at R ∼ 0.7 kpc (Torii et al. 2010). As it is difficult to estimate the uncertainty of the extrapolations in subection 4.3, including the choice of the fitting function, a uniform error of ±20% was assumed. In addition, the derived |$M_{\rm H_2}$| is affected by a distance error of 30% as shown in figure 2, which can be canceled in taking ratios among M_H2(¹²CO), |$M_{\rm H_2}{\rm (^{13}CO)}$|⁠, and |$M_{\rm H_2}{\rm (C^{18}O)}$|⁠.

4.4.2 M  _H2(¹³CO) and M_H2(C¹⁸O)

The uncertainties in N₁₃ and N₁₈ were estimated from the uncertainties in T(¹³CO) and T(C¹⁸O) (±10%; Umemoto et al. 2017) and the uncertainties in T_{ex, 13} and T_{ex, 18}, respectively—see equations (2)–(6). Here, a uniform error of ±50% was assumed for T_{ex, 13} and T_{ex, 18}, as it is not easy to evaluate these uncertainties in the large target area of this study. Then, the ±30% uncertainties of N₁₃ and N₁₈ were derived. In converting N₁₃ and N₁₈ into N_H2(¹³CO) and N_H2(¹³CO), the uncertainties on the abundance ratios among the CO isotopologues were significant, which can be calculated as ∼± 40% and ∼± 30% for [¹²C]/[¹³C] and [¹⁶O]/[¹⁸O], respectively—equations (7) and (8).

In addition to these statistical uncertainties, a systematic error of +20% was considered for N₁₃ and N₁₈, as the LTE assumption may overestimate the true column densities due to the subthermal excitation of higher rotational transitions of CO (Harjunpää et al. 2004). Furthermore, in each l the present analysis includes uncertainties in R in equations (7) and (8), due to the ±30% error of d_tan, which provide additional +10–+30% uncertainties for the abundance ratios.

5 Results

Figure 11 shows the longitudinal distributions of (a) the number of voxels N_vox and (b) W(CO) for the CO sources identified at T_min = 1 K. The blue, green, and red bars indicate the derived values of ¹²CO, ¹³CO, and C¹⁸O, respectively. In addition, the gray bar in figure 11 shows the total number of voxels included in the target l and v ranges (figure 3). Here, the N_vox and W(CO) for the ¹²CO data [hereafter N_vox(¹²CO) and W(CO)(¹²CO)] were derived by extrapolating to T_min = 1 K, following the method described in subsection 4.3. ¹²CO and ¹³CO sources were detected in the entirety of the target region, while no C¹⁸O sources were detected in the l range of 14°–17° at T_min = 1 K. The two peaks of W(CO) at l = 30°–31° and 49°–50° correspond to the regions that include W|$\, 43$| and W|$\, 51$|⁠, respectively.

Fig. 11.

Longitudinal distributions of (a) N_vox and (b) W(CO). The blue, green, and red bars show the ¹²CO, ¹³CO, and C¹⁸O distributions, respectively. The horizontal red axis plotted in (a) indicates R_tan. C¹⁸O is not detected significantly in l = 14°–17°. In (a) the gray bar indicates the total number of voxels included in the target l and v ranges (figure 3). (Color online)

Figure 12 shows the fractions of N_vox and W(CO) plotted in figures 11a and 11b [hereafter f(N_vox) and f(W)], respectively. In figure 12a the f(N_vox) of ¹³CO (green) and C¹⁸O (red) to ¹²CO are presented [|$f_{\rm ^{13}CO}({N_{\rm vox}})$| and |$f_{\rm C^{18}O}({N_{\rm vox}})$|⁠, respectively], while in figure 12b the f(W) of ¹³CO (green) and C¹⁸O (red) to ¹²CO are plotted [|$f_{\rm ^{13}CO}(W)$| and |$f_{\rm C^{18}O}(W)$|⁠, respectively]. f(N_vox) and f(W) overall show similar distributions, with f(N_vox) being slightly larger (by a factor of ∼2–3) than f(W). Figure 12 indicates that the ¹³CO, and particularly the C¹⁸O, emissions were detected in a small portion of the ¹²CO emitting regions. |$f_{\rm ^{13}CO}({N_{\rm vox}})$| and |$f_{\rm ^{13}CO}(W)$| range from 1% to several 10%, while f₁₈(N_vox) and f₁₈(W) range from 0.01 pc to 1% with large variations.

Fig. 12.

Longitudinal distributions of (a) |$f_{N_{\rm vox}}$| and (b) f_W(CO). The green circles indicate the ratios of ¹³CO to ¹²CO, while the red circles show the ratios of C¹⁸O to ¹²CO. The horizontal red axis plotted in (a) indicates R_tan. (Color online)

Figure 13a shows the longitudinal distribution of |$M_{\rm H_2}$| in the same manner as figure 11 but with error bars. |$M_{\rm H_2}$| for the dense gas |$M_{\rm H_2}{\rm (DG)}$| was calculated using the subregions of the identified C¹⁸O sources at which |$N_{\rm H_2}{\rm (C^{18}O)} \ge 7\times 10^{21}\:$|cm⁻² (or A_v ≥ 8) and is plotted with gray bars in figure 13b. The dense gas at which the filaments of 0.1 pc width are dominant was not detected in l = 13°–14°.

Fig. 13.

(a) The longitudinal distribution of |$M_{\rm H_2}$| is shown in the same manner as figure 11 but with error bars. (b) The distributions of |$M_{\rm H_2}{\rm (C^{18}O)}$| and |$M_{\rm H_2}{\rm (DG)}$| are plotted with the red and gray bars, respectively. (c) The distributions of |$f_{\rm ^{13}CO}$|⁠, |$f_{\rm C^{18}O}$|⁠, and f_DG are plotted in green, red, and black, respectively. (Color online)

Figure 13c shows the fractions of |$M_{\rm H_2}{\rm (^{13}CO)}$| (green), |$M_{\rm H_2}{\rm (C^{18}O)}$| (red), and |$M_{\rm H_2}{\rm (DG)}$| (black) to M_H2(¹²CO) in percent (⁠|$f_{\rm ^{13}CO}$|⁠, |$f_{\rm C^{18}O}$|⁠, and f_DG, respectively). |$f_{\rm ^{13}CO}$| in the inter-arm regions shows slightly lower values than the Galactic arms, maintaining values of ∼20%–40%, while in Region A, which includes the Galactic Bar and Far-3 kpc Arm, |$f_{\rm ^{13}CO}$| begins decreasing to ∼4%–10%.

On the other hand, the four regions show high variations in |$f_{\rm C^{18}O}$|⁠, and particularly in f_DG. The fractions are relatively high in Regions B and D, ranging from ∼2% to ∼8%, while the fractions are typically as low as <1% in Regions A and D, and some tiles have a very low f_DG of less than 0.1%.

Region B shows two peaks in f_DG at l ∼ 30° and l ∼ 24°. The former corresponds to W|$\, 43$|⁠, while the latter includes a GMC associated with the infrared ring N35, which is an active star-forming region (Torii et al. 2018). These two star-forming regions are probably located near the tangential points of the Scutum and Norma Arms, respectively, as seen in the l–v diagram of figure 3. In addition, |$f_{\rm C^{18}O}$| and f_DG increase in l = 49°–50° in Region D, where another active star-forming region W|$\, 51$| is distributed around the tangential point of the Sagittarius Arm.

The total |$M_{\rm H_2}$| of the three CO isotopologues in l = 10°–50° and their fractional masses are summarized in table 1; the derived M_H2(¹²CO), |$M_{\rm H_2}{\rm (^{13}CO)}$|⁠, |$M_{\rm H_2}{\rm (C^{18}O)}$|⁠, and |$M_{\rm H_2}{\rm (DG)}$| are |${\sim } 10^{8.1}\, M_{\odot }$|⁠, |${\sim } 10^{7.4}\, M_{\odot }$|⁠, |${\sim } 10^{6.6}\, M_{\odot }$|⁠, and |${\sim } 10^{6.5}\, M_{\odot }$|⁠, respectively. Given the surface area of 25.7 kpc² of the target area of this study (figure 2), the corresponding surface mass densities are |${\sim } 4.36\, M_{\odot }\:$|pc⁻², |${\sim } 1.02\, M_{\odot }\:$|pc⁻², |${\sim } 0.16\, M_{\odot }\:$|pc⁻², and |${\sim } 0.13\, M_{\odot }\:$|pc⁻², respectively. The averaged |$f_{\rm ^{13}CO}$|⁠, |$f_{\rm C^{18}O}$|⁠, and f_DG are calculated as 23.7%, 3.7%, and 2.9%, respectively.

Table 1.

Total |$M_{\rm H_2}$| and fractional masses.*

Region	l range	|$\log _{10} (M_{\rm H_2}) \ [M_{\odot }]$|					f [%]				f  13 [%]			f  18 [%]
		12CO	13CO	C18O	Dense gas		13CO	C18O	Dense gas		C18O	Dense gas		Dense gas
(1)	(2)	(3)	(4)	(5)	(6)		(7)	(8)	(9)		(10)	(11)		(12)
All	10°–50°	|$8.05^{+0.06}_{-0.07}$|	|$7.42^{+0.10}_{-0.11}$|	|$6.62^{+0.12}_{-0.14}$|	|$6.51^{+0.12}_{-0.15}$|		|$23.7^{+10.8}_{-10.9}$|	|$3.7^{+3.0}_{-3.0}$|	|$2.9^{+2.6}_{-2.6}$|		|$15.7^{+8.5}_{-8.6}$|	|$12.1^{+7.3}_{-7.3}$|		|$77.4^{+11.1}_{-11.1}$|
A	10°–20°	|$6.81^{+0.09}_{-0.12}$|	|$5.55^{+0.15}_{-0.16}$|	|$4.36^{+0.19}_{-0.19}$|	|$3.89^{+0.28}_{-0.31}$|		|$5.5^{+2.8}_{-2.8}$|	|$0.4^{+0.3}_{-0.3}$|	|$0.1^{+0.2}_{-0.2}$|		|$6.4^{+5.1}_{-5.2}$|	|$2.2^{+3.5}_{-3.5}$|		|$35.6^{+28.3}_{-28.3}$|
B	23°–32°	|$7.75^{+0.10}_{-0.14}$|	|$7.23^{+0.14}_{-0.18}$|	|$6.54^{+0.13}_{-0.17}$|	|$6.44^{+0.14}_{-0.18}$|		|$29.9^{+10.1}_{-10.5}$|	|$6.1^{+2.3}_{-2.5}$|	|$4.8^{+2.3}_{-2.3}$|		|$20.5^{+7.0}_{-7.4}$|	|$16.2^{+6.2}_{-6.5}$|		|$78.8^{+9.5}_{-9.5}$|
C	35°–45°	|$7.38^{+0.09}_{-0.12}$|	|$6.61^{+0.11}_{-0.14}$|	|$5.20^{+0.12}_{-0.16}$|	|$4.99^{+0.12}_{-0.16}$|		|$17.2^{+6.7}_{-6.9}$|	|$0.7^{+0.5}_{-0.5}$|	|$0.4^{+0.4}_{-0.4}$|		|$3.9^{+2.7}_{-2.7}$|	|$2.4^{+1.7}_{-1.8}$|		|$61.9^{+12.1}_{-12.1}$|
D	49°–50°	|$6.41^{+0.24}_{-0.57}$|	|$6.02^{+0.26}_{-0.71}$|	|$5.03^{+0.25}_{-0.62}$|	|$4.99^{+0.25}_{-0.62}$|		|$40.6^{+26.3}_{-27.5}$|	|$4.2^{+2.3}_{-2.6}$|	|$3.9^{+2.3}_{-2.4}$|		|$10.4^{+6.8}_{-7.4}$|	|$9.5^{+6.2}_{-6.8}$|		91.5

*(1) Region name. (2) Galactic longitude range used for the calculations of the average mass and fractional mass. (3)–(6) Logarithm of |$M_{\rm H_2}$|⁠. (7–9) Fractional mass with M_H2(¹²CO) as denominator. (10, 11) Fractional mass with |$M_{\rm H_2}{\rm (^{13}CO)}$| as denominator. (12) Fractional mass with M_H2(¹²CO) as denominator and |$M_{\rm H_2}{\rm (DG)}$| as numerator.

The total |$M_{\rm H_2}$| and fractional masses in the four regions, Regions A–D, are also summarized in table 1, where the borders of the two neighboring regions are removed. In Regions A, B, C, and D, |$f_{\rm ^{13}CO}$| has the average values of 5.5%, 29.9%, 17.2%, and 40.6%, while f_DG has the average values of 0.1%, 4.8%, 0.4%, and 3.9%, respectively. Region D has only one bin at l = 49°–50°, resulting in a relatively large error on the averaged |$f_{\rm C^{18}O}$| and f_DG. Additional observations at l ≥ 50° are needed to obtain more reliable representative values of the fractional masses in the Sagittarius Arm. In addition, as shown in figure 7, the vertical extents of the ¹²CO emission are not fully covered at l ∼ 40°–49° in Region C, which may lead to overestimating the obtained fractional masses by ∼10–20%.

Figure 14 shows the fractions of |$M_{\rm H_2}{\rm (C^{18}O)}$| (red) and |$M_{\rm H_2}{\rm (DG)}$| (black) to |$M_{\rm H_2}{\rm (^{13}CO)}$| (⁠|$f^{13}_{\rm C^{18}O}$| and |$f^{13}_{\rm DG}$|⁠, respectively) and the fractions of |$M_{\rm H_2}{\rm (DG)}$| to |$M_{\rm H_2}{\rm (C^{18}O)}$| (⁠|$f^{18}_{\rm DG}$|⁠). The averaged |$f^{13}_{\rm C^{18}O}$|⁠, |$f^{13}_{\rm DG}$|⁠, and |$f^{18}_{\rm DG}$| in the entirety of l = 10°–50° and the four regions are summarized in table 1. The averaged |$f^{13}_{\rm C^{18}O}$| and |$f^{13}_{\rm DG}$| in l = 10°–50° are estimated as 15.7% and 12.1%, respectively, while Regions A, B, C, and D have averaged |$f^{13}_{\rm C^{18}O}$| of 6.4%, 20.5%, 3.9%, and 10.4% and |$f^{13}_{\rm DG}$| of 2.2%, 16.2%, 2.4%, and 9.5%, respectively. |$f^{18}_{\rm DG}$| appears to be relatively stable compared with f_DG and |$f^{13}_{\rm DG}$|⁠. Although some regions in the Galactic Bar and inter-arm region show small |$f^{18}_{\rm DG}$| of ∼10%, while the other regions typically have high |$f^{18}_{\rm DG}$| of ∼70%–90%.

Here, it is noted that in this study the CO sources were detected at T_min = 1 K, and the derived fractions whose numerators are |$M_{\rm H_2}{\rm (DG)}$| (i.e., f_DG, |$f^{13}_{\rm DG}$|⁠, and |$f^{18}_{\rm DG}$|⁠) gave the upper limits, as lower T_min would provide larger values for M_H2(¹²CO), |$M_{\rm H_2}{\rm (^{13}CO)}$|⁠, and |$M_{\rm H_2}{\rm (C^{18}O)}$|⁠, while the |$M_{\rm H_2}{\rm (DG)}$| was not changed.

Fig. 14.

Longitudinal distributions of |$f^{13}_{\rm C^{18}O}$| (red) and |$f^{13}_{\rm DG}$| (black) are plotted in panel (a), while the distribution of f¹⁸ is shown in panel (b). (Color online)

6 Discussion

The |$M_{\rm H_2}{\rm (DG)}$| derived in this study is expected to be nearly consistent with the masses of the supercritical filaments (André et al. 2014), whose SFE has been found to be quasi-universal throughout the galaxies (e.g., Lada et al. 2010; Shimajiri et al. 2017). Analysis of the FUGIN CO data provides an average f_DG value of ∼2.9% over ∼5 kpc in the Galactic plane in the first quadrant (table 1). This figure is consistent with the gap between the gas consumption timescale of ∼1–2 Gyr given by the KS law and the dense gas consumption timescale of ∼20 Myr. This suggests that the formation of dense gas in molecular clouds is the primary cause of inefficient star formation in galaxies, which is consistent with the discussion of Lada, Lombardi, and Alves (2010) that |$\Sigma _{\rm SFR} \propto f_{\rm DG}\Sigma _{\rm H_2}$| is the fundamental relationship governing star formation.

On the other hand, analyses of the FUGIN data revealed that there are huge variations of f_DG depending on the structures of the MW disk. In the regions including the Galactic arms (i.e., Regions B and D), f_DG is as high as ∼4%–5%, while in the Galactic Bar and inter-arm regions (i.e., Regions A and C) it becomes quite small at ∼0.1%–0.4%. As f_DG is an indicator of the dense gas formation speed in the steady state, and the SFE in dense gas is likely quasi-universal, these large gaps in f_DG may result in SFR/M_H2(¹²CO) differences between these regions. Indeed, studies of extra-galaxies indicate a systematic offset of ∼50% in the |$\Sigma _{\rm SFR}/\Sigma _{\rm H_2}$| among the arms, inter-arm, and bar (e.g., Bigiel et al. 2008; Momose et al. 2010). This is qualitatively consistent with our results, although the 50% difference is smaller than the one order of magnitude difference found in this study. Therefore, it is important to directly quantify the SFRs in the present target regions in the MW based on infrared/radio observations.

The variations of the fractional masses may be attributed to differences in the formation and destruction processes of the dense gas in the molecular clouds. Although there are many theoretical studies on the formation process of supercritical filaments (e.g., Hennebelle et al. 2008; Federrath et al. 2010; Inutsuka et al. 2015), the fractional mass of the dense gas to total molecular gas has not yet been quantified. The analyses first resolved the fractional masses over 5 kpc in the Galactic plane, which will encourage theoretical developments to understand the detailed process of dense gas formation in molecular clouds in the Galactic plane.

A reasonable process for dense gas formation is compression by shock wave. Inutsuka et al. (2015) proposed a scenario of star formation for scales of ∼100 pc, in which the multiple compression of gas powered by the feedback of the massive stars regulates star formation in galaxies. Kobayashi et al. (2017, 2018) constructed a semi-analytical model of GMC formation, including multiple compressions driven by a network of expanding shells due to H ii regions and supernova remnants, which resulted in finding slopes of the GMC mass functions similar to those observed in spiral galaxies.

The roles of galactic-scale gas motion have been discussed previously in studies of extra-galaxies. According to the spiral density wave theory (Shu 2016), the gas in the arms is affected by the strong compression caused by galactic shocks or cloud–cloud collisions. This mechanism can be expected in the other models, such as the non-steady spiral arm model (Wada et al. 2011; Baba et al. 2013; Dobbs & Baba 2014). It has been suggested that the decrease in gas density observed in the bar regions of extra-galaxies can be attributed to the gravitationally unbound conditions of molecular clouds (e.g., Sorai et al. 2012; Meidt et al. 2013): these conditions may be caused by the shear motion and/or cloud–cloud collisions (e.g., Fujimoto et al. 2014). Yajima et al. (2019) suggested that the large velocity dispersion at >100 km s⁻¹ in the galactic bars may disperse GMCs. The decrease in |$f_{\rm ^{13}CO}$|⁠, |$f_{\rm C^{18}O}$|⁠, and f_DG in Region A may possibly be interpreted by these mechanisms.

Analyses of the FUGIN data have found no significant differences between the fractional masses of Regions A and C, except for |$f_{\rm ^{13}CO}$|⁠, which showed average values of ∼6% and ∼17% in Regions A and C, respectively (figure 13c and table 1). This may suggest that formation of the relatively dense gas traced in ¹³CO is more efficient in the inter-arm regions rather than in the Galactic Bar, although there are no significant differences in |$f_{\rm C^{18}O}$| and f_DG between these two regions. Observations of extra-galaxies have suggested that moderate shear motion in the arms may allow GMCs to stream into the inter-arm regions, while GMCs hardly survive in the Galactic Bar (e.g., Koda et al. 2009; Miyamoto et al. 2014). This may lead to higher |$f_{\rm ^{13}CO}$| in the inter-arm region compared to the Galactic Bar.

To reach to a comprehensive understanding of the dense gas and star formation in the MW, it is important to perform additional analyses of the FUGIN CO dataset to identify and quantify the various structures of molecular gas in various spatial scales from 1 pc to kpc, which will also allow us to make direct comparisons with future large-scale observations of extra-galaxies with pc-scale resolutions.

7 Summary

The conclusions of the present study are summarized as follows:

1. The CO J = 1–0 data, obtained as part of the FUGIN project using the Nobeyama 45 m telescope, was analyzed to construct the longitudinal distributions of |$M_{\rm H_2}$| traced by ¹²CO, ¹³CO, and C¹⁸O emissions with a bin size of l = 1°.

2. |$M_{\rm H_2}$| was measured in the region within d_tan ± 30% by choosing the corresponding v_LSR ranges in the l–v diagram. The target region included the Galactic Bar, Far-3 kpc Arm, Norma Arm, Scutum Arm, Sagittarius Arm, and inter-arm regions.

3. M_H2(¹²CO) for these regions was measured assuming constant X(CO), and |$M_{\rm H_2}{\rm (^{13}CO)}$| and |$M_{\rm H_2}{\rm (C^{18}O)}$| were estimated assuming LTE. |$M_{\rm H_2}{\rm (DG)}$| was measured using the subregions of the C¹⁸O sources at which A_v > 8 mag.

4. The derived |$M_{\rm H_2}{\rm (DG)}$| and M_H2(¹²CO) were then used to calculate f_DG, and the derived f_DG showed large variations depending on the structures of the MW disk: the regions including the Galactic arms have high f_DG of ∼4%–5%, while f_DG in the Galactic Bar and inter-arm regions is small at ∼0.1%–0.4%. The averaged f_DG over the entirety of the target region (∼5 kpc) is ∼2.9%. This figure is consistent with the gap between the gas consumption timescale observed in the KS law (∼1–2 Gyr) and the dense gas consumption timescale (∼20 Myr), indicating that the formation of dense gas is the primary bottleneck of star formation in the MW.

5. Other mass ratios such as |$f^{13}_{\rm DG}$| and |$f^{18}_{\rm DG}$| were also measured; it was demonstrated that every mass ratio tends to increase in the arm regions as opposed to the inter-arm and bar regions. Only |$f_{\rm ^{13}CO}$| showed moderate differences between the arms and inter-arms, while still showing significantly small values in the bar region.

6. The analyses first resolved f_DG and other mass ratios over ∼5 kpc in the Galactic plane, which provided crucial information on dense gas and star formation in the MW. It is expected that these results will encourage future theoretical and observational studies.

Acknowledgments

This work was financially supported by Grants-in-Aid for Scientific Research (KAKENHI) of the Japanese Society for the Promotion of Science (JSPS, grant numbers 15H05694, 18K13580, 18K13582, 17H06740, 15K17607, 24224005, 26247026, and 23540277). Data analysis of the CO emissions was in part carried out on the open use data analysis computer system at the Astronomy Data Center, ADC, of the National Astronomical Observatory of Japan.

Appendix 1 Noise distributions of the FUGIN CO data

In figures 15–19 the histograms of the post-processed CO data are presented for all the 1° × 2° regions in the same manner as in figure 9.

Fig. 15.

Histograms of σ for the ¹²CO (top), ¹³CO (center), and C¹⁸O data in all the 1° × 2° regions analyzed in this study. The vertical solid and dotted lines indicate σ_med and |$\, 5\sigma _{\rm med}$|⁠, respectively. The orange area indicates the data points with |$3 \, \sigma \lt 5 \, \sigma _{\rm med}$|⁠, and the fraction of data points in the orange area is shown at the top right of the panel. (Color online)

Fig. 16.

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Fig. 17.

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Fig. 18.

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Fig. 19.

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Appendix 2 Mass estimates from the ¹²CO data by extrapolation

An extrapolation technique was adopted in this study to estimate M_H2(¹²CO) at T_min = 1 K (see subsection 4.3). In figures 20–22 the results of the extrapolations in all the 1° × 2° regions analyzed in this study are presented in the same manner as in figure 10.

Fig. 20.

Results of the extrapolations to derive M_H2(¹²CO) at T_min = 1 K. The blue circles indicate the normalized values of M_H2(¹²CO) measured at T_min = 3, 4, 5, 6, 7, 8, 9, 10, and 11 × σ_med. The normalizations are made by dividing M_H2(¹²CO) by max|$[M_{\rm H_2}{\rm (^{12}CO)}]$|⁠, which is the maximum M_H2(¹²CO). (Color online)

Fig. 21.

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Fig. 22.

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