FOREST unbiased Galactic plane imaging survey with the Nobeyama 45 m telescope (FUGIN). VII. Molecular fraction of H i clouds

ABSTRACT

We analyze molecular-gas formation in neutral atomic hydrogen (H i) clouds using the latest CO data, obtained from the FOREST (four-beam receiver system on the 45 m telescope) unbiased Galactic plane imaging survey with the Nobeyama 45 m telescope, and using H i data taken from the Very Large Array Galactic plane survey. We applied a dendrogram algorithm to the H i data cube to identify H i clouds, and we calculated the H i mass and molecular-gas mass by summing the CO line intensity within each H i cloud. On the basis of the results, we created a catalog of 5737 identified H i clouds with local standard of rest (LSR) velocity of V_LSR ≤ −20 km s⁻¹ in galactic longitude and latitude ranges of 20° ≤ l ≤ 50° and −1° ≤ b ≤ 1°, respectively. We found that most of the H i clouds are distributed within a Galactocentric distance of 16 kpc, and most of them are in the cold neutral medium phase. In addition, we determined that the high-mass end of the H i mass function is fitted well with a power-law function with an index of 2.3. Although two sequences of self-gravitating and diffuse clouds are expected to appear in the M_tot–|$M_{\,{\rm H}_2}$| diagram according to previous works based on a plane-parallel model, the observational data show only a single sequence with large scattering within these two sequences. This implies that most of the clouds are mixtures of these two types of clouds. Moreover, we suggest the following scenario of molecular-gas formation: an H i-dominant cloud evolved with increasing H₂ mass along a path of |$M_{\,{\rm H}_2} \propto M_{\,\rm tot}^2$| by collecting diffuse gas before reaching and moving along the curves of the two sequences.

1 Introduction

Neutral atomic hydrogen (H i) and molecular hydrogen (H₂) constitute the main components of the Galactic interstellar medium (ISM). The former can be observed directly in the 1420 MHz radio line, whereas the latter is generally traced by radio lines of another molecular species such as CO (Nakanishi & Sofue 2003, 2006, 2016). Although the H₂ gas traced by the CO line is observationally found in dense gas regions, the H i line is a tracer of the diffuse gas region.

The fraction of the molecular component to the total gas is often defined as |$f_{\rm mol} = \Sigma _{{\rm H}_2}/(\Sigma _{\rm H\,{\rm {\small I}}}+\Sigma _{{\rm H}_2})$|⁠, where |$\Sigma _{\rm H\,{\rm {\small I}}}$| and |$\Sigma _{{\rm H}_2}$| are the H i and H₂ surface densities, respectively. It is observationally shown that the molecular fraction f_mol is high near the Galactic center and decreases with the Galactocentric distance (Sofue et al. 1995; Sofue & Nakanishi 2016). The molecular fraction is considered to be a useful tool for examining the physical condition of the interstellar medium because it depends on the pressure (P), ultraviolet (UV) radiation field (U), and the metallicity (Z), as theoretically suggested (Elmegreen 1993; Krumholz et al. 2008; McKee & Krumholz 2010; Sternberg et al. 2014).

According to Elmegreen (1993), the molecular fraction on the Galactic scale can be calculated by assuming that interstellar clouds are classified into diffuse and self-gravitating ones, and by summing all of the molecular components in individual clouds assuming that the cloud mass function follows a power law. Because the gas density increases with the interstellar pressure P and the shielding effect increases with the metallicity Z, the molecular fraction increases with P and Z. However, the molecular fraction decreases with U because the molecular gas is photo-dissociated by UV photons. It should be noted that the molecular fraction is sensitive to the metallicity Z because the metallicity is proportional to the amount of ISM dust, which also works as a catalyst for forming H₂ from H i gases (Honma et al. 1995).

These theoretical models closely match the observational values on the kiloparsec scale (Honma et al. 1995; Nakanishi et al. 2006; Krumholz et al. 2009; Tanaka et al. 2014; Sofue & Nakanishi 2016). In these models the molecular component forms a core at the center of each cloud that is surrounded by an H i layer, which shields the central molecular core from the photo-dissociation owing to the UV radiation field (Elmegreen 1993; Krumholz et al. 2008; McKee & Krumholz 2010; Sternberg et al. 2014). Such models of ISM clouds are often referred to as plane-parallel photo-dissociation region (PDR) models (Tielens & Hollenbach 1985; Hollenbach & Tielens 1997), which also suggest the existence of [C ii] and [C i] layers surrounding an H₂ core in addition to H i gas.

However, such a plane-parallel model appears to be overly simplified. Recent [C i] and [C ii] observations (Kamegai et al. 2003; Kramer et al. 2008; Shimajiri et al. 2013) suggest that this model cannot completely explain the [C i] and [C ii] data, which instead imply that the CO gas is in a phase of clumpy cloudlets, as suggested by Tachihara et al. (2012). Therefore, it is necessary to verify whether most of the ISM clouds can be explained with the plane-parallel model based on observational data.

For this purpose, a wide spatial dynamic range with a high resolution and a wide field of view is essential. We used CO and H i data recorded during the four-beam receiver system on the 45 m telescope (FOREST) unbiased Galactic plane imaging survey with the Nobeyama 45 m telescope (FUGIN) and Very Large Array (VLA) Galactic plane survey (VGPS) projects (Stil et al. 2006; Umemoto et al. 2017). These data have a large spatial dynamic range and are ideal for evaluating whether the atomic and molecular components are distributed inside an H i cloud, as shown in the simple plane-parallel model described in previous research.

As Part VII of the FUGIN project paper series, we report on a study of the molecular-gas formation by comparing CO and H i data. Sections 2 and 3 summarize the data and data analysis, respectively, and section 4 details the process of obtaining the results. In section 5 we discuss the properties of identified H i clouds, such as H i cloud distribution as a function of Galactocentric distance, mass function, and H₂ gas distribution and fraction in each cloud. Finally, section 6 provides a summary of the conclusions.

2 Data

The ¹²CO(J = 1–0) data tracing the H₂ were obtained through the FUGIN project, which covered galactic longitude ranges of 10° ≤ l ≤ 50° and 198° ≤ l ≤ 236°, and a latitude range of −1° ≤ b ≤ 1°, in ¹²C¹⁶O, ¹³C¹⁶O, and ¹²C¹⁸O J = 1–0 lines (Umemoto et al. 2017).

In this study we focus on the ¹²C¹⁶O data in the longitude range of 10° ≤ l ≤ 50° to compare the molecular fraction with the high-resolution H i data. The angular resolution was 20″, whereas the angular sampling was |${8{^{\prime \prime }_{.}}5}$|⁠. The sensitivity in the antenna temperature |$T_{\rm A}^*$| was 0.24 K for the velocity resolution of 1.3 km s⁻¹. The antenna temperature was converted into the main beam temperature using a main beam efficiency of 0.43.

The H i data were taken from the archived data of the VGPS, in which a high-resolution H i survey was conducted with the VLA covering a galactic longitude range of 18° ≤ l ≤ 67° and a latitude range of |$\vert b \vert$| ≤ 1|${^{\circ}_{.}}$|3 or |$\vert b \vert$| ≤ 2|${^{\circ}_{.}}$|3. The angular resolution was 1′, and the velocity resolution was 1.56 km s⁻¹ (Stil et al. 2006). The sensitivity was 2 K per channel width of 0.824 km s⁻¹. The missing flux was recovered by single-dish observations conducted using the Green Bank 100 m telescope.

Both data were regridded so that the pixel size and velocity spacing were set at 18″ and 0.824 km s⁻¹ for comparison. Because the CO data were convolved so that the convoluted beam size matched that of the H i data (1′), the sensitivity of the convolved CO data became ∼0.2 K (¹²C¹⁶O) and ∼0.1 K (¹³C¹⁶O and ¹²C¹⁸O). Longitude–velocity diagrams of the CO and H i data are shown in the top and bottom panels of figure 1, respectively.

3 Data analysis

First, we applied the dendrogram algorithm developed by Rosolowsky et al. (2008) to the VGPS H i data cube in order to identify the H i clouds. To avoid near–far ambiguities in the kinematic distances of clouds inside the solar circle, we analyzed clouds having only negative local standard of rest (LSR) velocities. Also, assuming that the velocity dispersion is about 10 km s⁻¹ (Malhotra 1995), we restricted our analysis to V_LSR ≤ −20 km s⁻¹ to be safe. Cloud identification was conducted for subsets of data divided into those with a longitude width of 10°. Neighboring subsets were overlapped by 5°, and double-counted H i clouds were merged into one. The dendrogram outputs bunches of voxels surrounded by the minimum contour level T_min as trunks and local maxima as leaves (“voxel” means three-dimensional pixel in the l, b, and V_LSR axes). The minimum contour level T_min was set at 5 K, and the minimum voxel number was 125. We define the “leaf” output of the dendrogram as “H i cloud” in this paper. It should be noted that segmentation methods like the dendrogram algorithm cannot recover all the H i emission including diffuse components, as is mentioned in the next section, because spatially extended emissions with broad line widths are missed.

The inner and outer rotation curves were taken from Clemens (1985) and Dehnen and Binney (1998), respectively. The Galactic constants R₀ (the Galactocentric distance of the Sun) and V₀ (the rotational velocity at the Sun) were 8 kpc and 217 km s⁻¹, respectively, as adopted by Dehnen and Binney (1998) and Nakanishi and Sofue (2016). The masses of the clouds were calculated by multiplying the sum of the brightness temperature |$\Sigma T_{\rm H\,{\rm {\small I}}}$| with the pixel size θ_pixel = 18″, velocity resolution Δ|$v$| = 0.824 km s⁻¹, the conversion factor from H i integrated intensity to H i column density |$X_{\rm H\,{\rm {\small I}}}=1.8 \times 10^{18}$| H cm⁻² (K km s⁻¹)⁻¹, and the square of the heliocentric distance D. In order to estimate the H₂ mass included in each H i cloud, we summed the brightness temperature of the ¹²CO data cube within each H i cloud. By including voxels with a brightness temperatures lower than the noise level, the diffuse emission was recovered in the same manner as using the stacking analysis (e.g., Morokuma-Matsui et al. 2015). The error was estimated by multiplying the noise in the brightness temperature by the square root of the number of voxels. For clouds in which the H₂ mass was less than three times the root-mean-squares (RMS) noise, the H₂ mass was not considered entered. The CO-to-H₂ conversion factor was set at X_CO = 1.8 × 10²⁰ H₂ cm⁻² K⁻¹ (km s⁻¹)⁻¹, as adopted from Dame, Hartmann, and Thaddeus (2001).

4 Results

The number of identified H i clouds was 5737, some of which are listed in table 1, where column (1) shows the cloud identification number; columns (2) and (3) show the galactic longitude and latitude, respectively; column (4) shows the LSR velocity; column (5) shows the heliocentric distance; column (6) shows the Galactocentric distance; column (7) shows the major- and minor-axis radii; column (8) shows the position angle; column (9) shows the velocity dispersion; column (10) shows the H i mass; and column (11) shows the H₂ mass. A complete list of these parameters is available online as digital data.

The maximum and minimum values of the H i cloud masses were 1.35 × 10⁴ M_⊙ and 14.0 M_⊙, respectively. The mean mass was calculated to be 5.59 × 10² M_⊙. The nearest and farthest H i clouds were located at heliocentric distances of 11.9 kpc and 25.1 kpc, respectively. The mean heliocentric distance was 16.4 kpc, which can be regarded as the typical distance of the H i clouds in our sample. The corresponding minimum, maximum, and mean values of the Galactocentric distances were 9.1 kpc, 18.2 kpc, and 11.3 kpc, respectively.

Considering that a minimum voxel number of 125 and a minimum contour level of T_min = 5 K were adopted as described in section 3, and that the maximum heliocentric distance was D_max = 25.1 kpc, H i clouds with mass less than |$M=125 X_{\rm H\,{\rm {\small I}}} T_{\rm min} \Delta v (D_{\rm max} {18\over 3600} {\pi \over 180})^2 {m_{\rm p}\over M_\odot }=36\, M_{\odot }$| were not fully identified (⁠|$m_{\rm p}$| is the proton mass).

Similarly, because the minimum heliocentric distance of the H i clouds was 11.9 kpc, the mass of the faintest detectable cloud was estimated to be as small as 8.1 M_⊙.

The maximum, minimum, and mean values of the mass of H₂ gas were 1.37 × 10⁴ M_⊙, 0.300 M_⊙, and 2.84 × 10² M_⊙, respectively. The H₂ gas was detected among H i clouds in the heliocentric distance range of 11.9–25.1 kpc. The mean heliocentric distance was 16.4 kpc, which is almost the same as that of the H i clouds. The corresponding minimum, maximum, and mean values of the Galactocentric distances were 9.1 kpc, 18.2 kpc, and 11.3 kpc, respectively. Because the RMS noise in the convolved ¹²CO data, the velocity resolution, and the maximum heliocentric distance of H i clouds with detected CO emission were ΔT_CO = 0.2 K, Δ|$v$| = 0.824 km s⁻¹, and D_max = 25.1 kpc, respectively, H i clouds with molecular mass less than |$M=\sqrt{125}X_{\rm CO} T_{\rm min} \Delta v (D_{\rm max} {18\over 3600} {\pi \over 180})^2 {2m_{\rm p}\over M_\odot }=26\, M_{\odot }$| could not be counted completely. Similarly in the case of H i, the mass of the faintest detectable H₂ gas was estimated to be as small as 5.8 M_⊙. It should be noted that H₂ clouds without H i gas, referred to as naked H₂ clouds, were not considered in our study.

5 Properties of identified H i clouds

5.1 Mass fraction of H i clouds to the total H i gas

While analysis with the dendrogram algorithm can identify clumpy H i clouds, it cannot recover all the diffuse H i component by definition, as mentioned in section 3. Therefore, we first discuss the fraction of H i mass recovered by our analysis to the total mass estimated by integrating the original H i spectra.

The total mass of all H i clouds identified by the dendrogram listed in table 1 amounts to 3.2 × 10⁶ M_⊙, while the total H i mass in the ranges of 20° ≤ l ≤ 50° and R = 9–19 kpc measured 1.6 × 10⁷ M_⊙. Therefore, the fraction of the H i emission recovered by the dendrogram analysis was 20%. The local fraction varies with the Galactocentric distance R, being as high as 23% around R = 12 kpc and decreasing below 20% beyond R = 13 kpc. This variation seems related to the phase transition between the cold neutral medium (CNM) and the warm neutral medium (WNM), as discussed in the next subsection.

5.2 Radial variation of H i cloud number density

Figure 3 shows the number density of the identified clouds as a function of the Galactocentric distance. This value was determined by counting the clouds within each sector of a Galactocentric radius width of 1 kpc and in a galactic longitude range of 20° ≤ l ≤ 50° and then dividing the cloud number by the area in each sector.

The maximum density occurred around a Galactocentric distance of 10 kpc, beyond which the H i cloud number density decreased monotonically with the Galactocentric distance. This peak coincides with locus of the Cygnus arm (Outer arm), which is clearly traced with the H i line, where numerous H i clouds are expected to be identified. However, few clouds were identified beyond a Galactocentric distance of 16 kpc, in spite of the existence of H i gas as shown in figure 2.

This similarity can be explained by considering the two-phase medium model (Wolfire et al. 1995) such that clumpy and diffuse H i clouds are CNM and WNM components, respectively. Figure 4 shows a thermal equilibrium state of the ISM in the density–pressure diagram. This was calculated on the basis of a model presented by Inoue, Inutsuka, and Koyama (2006), in which the volume density and pressure of the H i gas shown in figure 3 are also plotted. The pressure was calculated assuming that the temperature was 7500 K, considering Sofue and Nakanishi (2016), who reported a WNM temperature of 7000 K by comparing the observational molecular fraction with a theoretical curve, and Heiles and Troland (2003), who suggested that the temperature of a thermally stable WNM is expected to be 8000 K. This diagram implies that the H i volume density beyond a Galactocentric distance of 16 kpc is lower than 0.01 cm⁻³, and its physical state is quite far from the WNM–CNM transition at about n = 1–10 cm⁻³. According to Inoue, Inutsuka, and Koyama (2006), the thermal pressure of ISM with a volume density of 1 cm⁻³ can increase 10–10^1.5 times due to an external perturbation, and the physical state can move from the WNM phase to the CNM phase. However, the thermal pressure of the H i gas with density n < 0.01 cm⁻³ in the outermost region of (R > 16 kpc) cannot exceed p/k_B = 10⁴ K cm⁻³ to cross the transition region even if it increases by 100 times. Therefore, the H i gas at a Galactocentric distance of >16 kpc is considered to remain in the diffuse WNM phase without forming H i clouds or molecular clouds. However, the thermal pressure of the WNM can reach p/k_B ≥ 1000 K cm⁻³ for the inner region with n ≥ 0.1 cm⁻³, where the phase of the ISM can jump into the CNM regime when it is compressed by the external perturbation, which leads to the formation of molecules. Such CNM clouds are known as standard clouds (Field et al. 1969; Wolfire et al. 1995), and have been likened to raisins in the “raisin pudding” model (Clark 1964; Field et al. 1969), although a recent study suggested that these clouds appear in sheets resembling “steamrolled raisins” (Kalberla et al. 2016).

5.3 Volume densities of neutral hydrogen in clouds

The mean volume densities of neutral hydrogen in the identified H i clouds were calculated by dividing the total gas masses M_tot by the volume |$V={4\pi \over 3}r^3$|⁠, where the radius of a cloud r was defined as |$r=\sqrt{r_{\rm min} r_{\rm maj}}$|⁠, as shown in the histogram in figure 5. The mean volume densities of most of the H i clouds (more than 99.7%) were above the density of n = 3 H cm⁻³, which is nearly equal to the threshold density between WNM and CNM (figure 4). The fact that most of the clouds have higher gas density than 3 H cm⁻³ indicates that these clouds are in the CNM phase. However, it should also be mentioned that clouds exist with low mean densities even though the number is small. A possible interpretation is that these are mixtures of CNM- and WNM-phase gases, though such clouds are rare.

5.4 Mass function

Figure 6 shows the mass functions of the H i clouds, which was obtained by counting the number of H i clouds in each mass interval (⁠|$\Delta \log_{10} M$|⁠) of 0.1. Because H i clouds with mass less than 36 M_⊙ are incompletely identified, as explained in section 4, the mass functions for such H i clouds are plotted with a black line; others are plotted with a gray line.

First, it should be mentioned that the minimum mass of the H i clouds was 14.0 M_⊙, whereas the minimum detectable mass was 8.1 M_⊙. This lower limit is close to the mass of WNM within a sphere of acoustic scale l_a ∼ 10 pc, which is calculated as l_a = c_st_c using the sound speed c_s and cooling timescale t_c (Inoue et al. 2006). This is reasonable because the mass of WNM within a sphere of diameter l_a is estimated as |$M=m_{\rm H\,{\rm {\small I}}} n (4\pi /3)(l_{\rm a}/2)^3=12\, M_{\odot }$|⁠, assuming an H i density of n = 1 cm⁻³. However, this lower limit does not match the CNM mass within a sphere of CNM at an acoustic scale of ∼0.1 pc or the Field length of ∼10⁻² pc, as well as a WNM mass within a sphere of WNM Field length ∼10⁻¹ pc. Therefore, the lower limit appears to be consistent with the scenario of H i clouds forming in shocked WNM owing to thermal instability, as suggested by Koyama and Inutsuka (2002).

5.5 H₂ mass versus total mass and H i mass

In order to study the amount of molecular gas formed in each cloud, the H₂ mass |$M_{\,{\rm H}_2}$| in each cloud is plotted against the total gas mass M_tot in figure 7, which also shows the curves of diffuse and self-gravitating clouds discussed by Elmegreen (1993).

In the case of a diffuse cloud of mass M, the molecular mass |$M_{\,{\rm H}_2}$| was calculated with |$M_{\,{\rm H}_2}={{4\pi \over 3}R_{\rm m}^3n\mu }$| using the mean molecular weight μ, the gas number density n = P/kT, the interstellar pressure P, the Boltzmann constant k, the temperature T, the radius of the molecular core R_m = R_e(1 − 3S^−3/2)^1/3, the cloud radius |$R_{\rm e}=\root 3 \of {3M/4\pi n\mu }$|⁠, the shielding function S = (Zφ_e/Z₀φ_{e, 0})(n_e/n_{e, 0})^5/3(R_e/R_{e, 0})^2/3, the metallicity Z, and the gas number density n_e and radiation intensity φ_e at the edge of a cloud. The subscript 0 indicates the value at a Galactocentric distance of R = R₀. The interstellar pressure, metallicity, and radiation intensity were taken to be the values at R = R₀.

In the case of a self-gravitating cloud with mass M, the molecular mass |$M_{\,{\rm H}_2}$| was calculated with |$M_{\,{\rm H}_2}={{4\pi \over 3}R_{\rm m}^3\langle n\rangle \mu }$| using the mean molecular weight μ, the mean gas number density 〈n〉 = 3n_e(R_e/R_m)², the gas number density at the edge of the cloud |$n_{\rm e}=217\sqrt{P}/R_{\rm e}$|⁠, the radius of the cloud |$R_{\rm e}=\sqrt{M/(190\sqrt{P})}$|⁠, the interstellar pressure P, and the radius of the molecular core R_m, which was calculated by numerically solving the equation (R_m/R_e)^−3/2 − (R_m/R_e)^−1/2 = S^−3/2 using the shielding function S = (Zφ_e/Z₀φ_{e, 0})(n_e/n_{e, 0})^5/3(R_e/R_{e, 0})^2/3, the metallicity Z, and the radiation intensity φ_e at the edge of the cloud.

Because previous works have suggested that two types of clouds exist, as mentioned above, two sequences are expected to appear in this diagram. Moreover, if molecular gas is rapidly formed in turbulent ISM on a timescale of 1–2 Myr, as suggested by Glover and Mac Low (2007), all clouds would be found along the sequence of self-gravitating clouds because such clouds are H₂ dominant.

However, only a single sequence with large scattering was found, as shown in figure 7. The least-squares fit shows that the H₂ mass can be expressed with the power-law function |$M_{\,{\rm H}_2}= 10^{-0.75\pm 0.08}\, M_{\rm tot}^{(1.06\pm 0.03)}$|⁠, which implies that the molecular mass is almost linearly proportional to the total gas mass, and that the typical molecular fraction is |$10^{-0.75} \sim 17\%$| of the total gas mass.

Interestingly, the M_tot–|$M_{\,{\rm H}_2}$| diagram shows that the relation between |$M_{\,{\rm H}_2}$| and M_tot is scattered near the linear relation with larger dispersion than the errors. In addition, most of the points are distributed between the two curves of self-gravitating and diffuse clouds. A possible explanation is that each cloud is a mixture of these two clouds, as discussed in the following section.

6 Discussion

6.1 Possible paths of molecular-gas evolution

To better understand the M_tot–|$M_{\,{\rm H}_2}$| diagram, let us compare it with a theoretical model from a three-dimensional magnetohydrodynamic (MHD) simulation given by Inoue and Inutsuka (2012). They studied molecular-cloud formation within a box of (20 pc)³ with a resolution of 0.02 pc. An initial H i medium was set, with fluctuations obeying a power law with the Komorogorov spectral index, and was put in a continuous flow with a velocity of 20 km s⁻¹ along a magnetic field of 5 μG. The time evolution of molecular clouds was examined by solving the MHD equations, including the effects of chemical reactions, radiative cooling/heating, and thermal conductions.

The MHD simulations show that the diffuse H i gas accumulates onto H i clouds initially formed, and that each cloud increases its mass in the flow. According to figure 5 of Inoue and Inutsuka (2012), the total mass increases approximately in a power law [|$M_{\rm tot}=M_{{\rm tot}_0}(t/t_0)^{\alpha _{\rm tot}}$|] from ∼4 × 10³ M_⊙ to ∼3 × 10⁴ M_⊙ in a time range of 1–10 Myr. The molecular mass also increases approximately in a power law [|$M_{\,{\rm H}_2}=M_{{{\rm H}_2}0}(t/t_0)^{\alpha _{\rm mol}}$|] from ∼3 × 10² M_⊙ to ∼2 × 10⁴ M_⊙ in the same time range of 1–10 Myr. Therefore, the indices α_tot and |$\alpha _{{\rm H}_2}$| are log₁₀[3 × 10⁴/(4 × 10³)] = 0.9 and log₁₀[2 × 10⁴/(3 × 10²)] = 1.8, respectively. Therefore, it is found that the mass of molecular clouds evolves as |$M_{\,{\rm H}_2}\propto M_{\rm tot}^{\alpha _{{\rm H}_2}/\alpha _{\rm tot}}\propto M_{\,\rm tot}^2$| by eliminating t from these expressions, though this relation is not explicitly mentioned in Inoue and Inutsuka (2012).

Considering the observations mentioned above, figure 8 suggests possible paths along which clouds evolve and experience increases in molecular mass. In this figure, the same data as in figure 7 are plotted with gray dots; the cases of self-gravitating and diffuse clouds are shown by solid curves, and the predicted evolution paths of |$M_{\,{\rm H}_2} = (M_{\rm total}/M_0)^2$|⁠, as described above, where M₀ denotes the total gas mass of a cloud containing molecular mass of |$M_{\,{\rm H}_2}=1\, M_{\odot }$|⁠, are also overlain by dashed lines. Three cases of mass are presented as examples: M₀ = 7 M_⊙, 70 M_⊙, and 700 M_⊙.

If the molecular fraction of clouds is less than unity, the mass of molecular gas increases along the lines of |$M_{\,{\rm H}_2} = (M_{\rm H \rm{\small {I}}}/M_0)^2$|⁠, as indicated in figure 8 by the three arrows aligned with dashed lines. If a cloud reaches the self-gravitating cloud phase by increasing the molecular mass, the cloud would evolve along the line of the self-gravitating phase with no further increase in the molecular mass. This path is indicated in the figure by the arrow aligned with the self-gravitating cloud line. Similarly, if a cloud reaches the curve of the diffuse cloud phase, the molecular fraction will increase rapidly, as indicated by the arrow aligned with the curve of the diffuse cloud phase.

6.2 H i and CO distributions of the four most massive H i clouds

Figure 9 shows the H i and H₂ distributions in the four most massive H i clouds, which are presented by gray images and contours, respectively. The thick-dashed lines indicate the edges of the H i clouds identified by using the dendrogram. The same physical parameters shown in table 1 are listed in table 2 for these four H i clouds. As shown in figure 9, the H i gas was distributed asymmetrically in all cases.

Previous researches indicate that ISM clouds consist of molecular cores at the centers that are surrounded by spherical H i envelopes, resembling jam doughnuts (Elmegreen 1993; Krumholz et al. 2008; McKee & Krumholz 2010; Sternberg et al. 2014). However, they actually consist of small molecular clouds distributed over large H i clouds, resembling chocolate-chip scones.

Although we do not consider the theoretical model given by the previous research to be vastly different from the actual conditions because the model has been able to roughly explaining the observations thus far, we suggest that the model requires a modification. Considering that the relation between molecular-gas and total gas masses is restricted to the range of theoretically predicted self-gravitating and diffuse clouds, as discussed in the previous section, a large molecular cloud likely consists of an assemblage of smaller self-gravitating clouds.

As shown in figure 9, clouds 2895 and 4443 exhibit the most typical characteristics of the aforementioned chocolate-chip scone model. Clouds 5584 and 5704 contain more molecular gas, which can be seen at the edges as well as the centers of the H i clouds. This seems to indicate that the cloud previously collided with another cloud or experienced ram pressure owing to gas flow to form molecular gas in a triggered shock front.

These images effectively illustrate the chocolate-chip scone model, in which the CNM cloud consists of small H₂-dominant self-gravitating clouds (chocolate chips) and cold H i-dominant diffuse clouds (plain scone). Both self-gravitating and diffuse clouds are essentially in the CNM phase because the volume density is in the range of the CNM phase, as shown in subsection 5.3. Such CNM clouds are distributed inside a “soup” of H i-dominant WNM. A schematic diagram of the chocolate-chip scone model described above is shown in figure 10.

6.3 Possible systematic errors in the analysis

In addition to the errors due to the random noise in the observational data considered above, we discuss how possible systematic errors, such as kinematic distances and CO-to-H₂ conversion factors, affect the main result shown in figure 7. As shown below, these systematic errors do not change the appearance of the M_tot–|$M_{\,{\rm H}_2}$| diagram and do not affect our result.

6.3.1 Kinematic distance

The heliocentric distance D of each cloud was calculated based on the kinematic distance, which depends on the choice of the rotation curve and the Galactic constants R₀ and V₀. These would systematically affect the estimation of mass by the factor of D². In order to test how much this affects the M_tot–|$M_{\,{\rm H}_2}$| diagram, let us check the case of adopting a flat rotation curve and another set of Galactic constants: R₀ = 10 kpc and V₀ = 250 km s⁻¹. The left panel of figure 11 shows the M_tot–|$M_{\,{\rm H}_2}$| diagram when adopting this kinematic distance. While the kinematic distance is estimated to be larger, figure 11 has almost the same appearance as figure 7 because both H i and H₂ masses are estimated similarly and the plot moves upward from left to right in the diagram but does not change in appearance.

6.3.2 CO-to-H₂ conversion factor

The H₂ mass depends on the choice of the CO-to-H₂ conversion factor. While it is suggested that the conversion factor varies with Galactocentric distance (Arimoto et al. 1996), we adopted a constant conversion factor in the above discussions. Therefore, similarly in the case of kinematic distance, let us test how much the choice of conversion factor affects the M_tot–|$M_{\,{\rm H}_2}$| diagram. The right panel of figure 11 shows the M_tot–|$M_{\,{\rm H}_2}$| diagram when adopting twice as large a conversion factor. Similarly in the previous case, the appearance of figure 11 is almost the same as figure 7, because the conversion factor makes both H₂ and the total masses larger and the plot moves upward from left to right in the diagram but does not change the appearance.

7 Conclusion

We applied the dendrogram analysis to the VGPS H i data cube to create a catalog of H i clouds of the outer Galaxy, including the masses of H₂ gas within each H i cloud, which were estimated by using FUGIN CO survey data. The ranges of the detected H i and H₂ masses are 14.0–1.35 × 10⁴ M_⊙ and 0.300–1.37 × 10⁴ M_⊙, respectively. The radial distribution of the H i cloud number density correlated strongly with the H i volume density, and most of the identified H i clouds were found to be in the CNM phase. The high-mass end of the mass function of the H i clouds was fitted well with a power-law function with an index of 2.3. Previous theoretical studies based on the jam doughnut model indicate that clouds can be divided into two sequences, self-gravitating and diffuse. However, the M_tot–|$M_{\,{\rm H}_2}$| diagram shows only a single sequence with large scattering rather than two sequences. Considering that the M_tot–|$M_{\,{\rm H}_2}$| relation of each cloud is found between these two sequences, we suggest that most of the clouds can be modeled by using the chocolate-chip scone model, which contains a mixture of self-gravitating clouds resembling chocolate chips and diffuse clouds resembling a scone. This description is consistent with the images shown in the CO and H i maps of the four most massive clouds.

Considering that the H₂ gas increases with |$M_{\,{\rm H}_2} \propto M_{\,\rm tot}^2$|⁠, as implied in a recent work, we suggest that the molecular clouds first evolved along the path of |$M_{\,{\rm H}_2} \propto M_{\,\rm tot}^2$| by collecting diffuse gas and then moved along the curves of self-gravitating and diffuse clouds.

Acknowledgements

We thank the FUGIN project members and staffs at the Nobeyama radio observatory for conducting and supporting the observations. We would also like to thank anonymous referee for carefully reading the manuscript. This work was financially supported Grants-in-Aid for Scientific Research (KAKENHI) of the Japanese society for the Promotion for Science (JSPS); grant numbers 15H05694 and 17H06740.

The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

References