Evolutionary Description of Giant Molecular Cloud Mass Functions on Galactic Disks

The PAWS program was conducted by using PdBI and the IRAM 30 m telescope and observed the disk of the Whirlpool Galaxy M51 in ¹²CO(1–0) line over a 200 hr integration. The survey reveals cold gas kinematics with ∼40 pc spatial resolution and detects objects whose masses are ≥1.2 × 10⁵ ${M}_{\odot }$ at the 5σ level (Pety et al. 2013). The PAWS team subdivide M51 into seven regions to analyze the GMC properties in various galactic environments (see Figure 2 in Colombo et al. 2014a). One of the highlighted results is the GMCMF variation; bar and arm regions typically have shallower slopes (e.g., −1.33 in the "nuclear" bar region and −1.75 in the "density-wave spiral arm" region) whereas inter-arm regions (and an arm region on the M51 outskirts) typically have steeper slopes (e.g., −2.55 in the "downstream" region). Here the slope means −α in ${n}_{\mathrm{cl}}\propto {m}^{-\alpha }$ where n_cl is the differential number density of GMC and m represents the GMC mass. In this article, we aim to reproduce such variations based on GMC scale physics.

Rosolowsky et al. (2007) observed the CO(1–0) line emission from Galaxy M33 using the NRO 45 m telescope combined with data from the BIMA interferometer and the FCRAO 14 m telescope. The arm/inter-arm comparison shows limited differences in slopes (−1.9 and −2.2, respectively) with difficulties in defining arms. However, the innermost 2.1 kpc has a prominent cutoff at the massive end ($\gtrsim 4.5\times {10}^{5}\,{M}_{\odot }$) whereas the outer regions up to 4.1 kpc do not. Such galactocentoric radial variation is beyond our current scope but needs to be investigated in future.

As described in Section 1, recent multiphase magnetohydrodynamics simulations suggest that multiple episodes of compression are necessary to form molecular clouds from the magnetized ISM. According to these simulation results, Inutsuka et al. (2015) proposed a new scenario of GMC formation and evolution on galactic scales (hereafter, SI15 scenario), which is driven by the network of expanding shells. In the SI15 scenario, the expanding bubbles correspond to expanding H ii regions and the late phase of supernova remnants, and a dense H i shell is formed on their surface as they expand. Molecular clouds are produced at specific regions of the ISM that experience multiple episodes of supersonic shocks (i.e., swept up multiple times by different expanding shells) or where neighboring expanding shells are colliding with each other.

Based on this scenario, Inutsuka et al. (2015) formulate a continuity equation that gives the time evolution of the GMCMF. Their formulation is two-fold: GMC formation and self-growth due to multiple episodes of compression, and GMC self-dispersal due to star formation within those GMCs. Their results suggest that the ratio of typical timescales for formation and dispersal processes determines the slope of the GMCMF (see Section 4.2 for further explanation).

They estimate the formation timescale in the following manner. Successful molecular cloud formation is limited when a supersonic shock propagates at an angle less than 0.26 radian with respect to the magnetic field. Given that the supersonic shock arrives isotropically, the success rate is about $2\times {0.26}^{2}\pi /(4\pi )\sim 0.03$ per shock. Gas in the ISM typically experiences supersonic shocks due to supernovae every 1 Myr (e.g., McKee & Ostriker 1977). Thus, the time interval between consecutive shocks T_exp is somewhat smaller than 1 Myr because H ii regions also create such shocks. Overall, the typical timescale required to produce molecular clouds from WNM is given as T_exp/0.03 ∼ 10 Myr, which we opt to use as our fiducial formation timescale in this article (see Section 3.1).

The second term on the left hand side of Equation (1) represents GMC self-growth. This term is a flux term in the conservation law. The ordinary continuity equation in fluid mechanics is a simple example of an analogous conservation law, which considers mass conservation in configuration space, whereas our equation describes GMC number conservation in GMC mass space.

We consider that ${({dm}/{dt})}_{\mathrm{self}}$, the GMC self-growth speed, is determined by the multiple H i cloud compression, which depends on the shape of GMCs. Observations suggest that the GMC column density does not vary much between GMCs (e.g., typically a few times ${10}^{22}\,{\mathrm{cm}}^{-2}$: see Onishi et al. 1999; Tachihara et al. 2000); therefore, we can assume that GMCs have rather a pancake shape than perfect spherical structure, which suggests that GMC surface area is roughly proportional to mass. In addition, the amount of H i cloud accumulated onto pre-existing GMCs through multiple episodes of compression is presumably proportional to GMC surface area. Altogether, ${({dm}/{dt})}_{\mathrm{self}}$ should be proportional to GMC mass divided by a typical self-growth timescale that is independent of mass:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{dm}}{{dt}}\right)}_{\mathrm{self}}=\displaystyle \frac{m}{{T}_{\mathrm{sg}}}.\end{eqnarray} \tag{ 2 }$$

where ${T}_{\mathrm{sg}}$ is a typical timescale for GMC self-growth.

In the SI15 scenario (the bubble paradigm), GMCs are formed via multi-compressional processes, which also cause the self-growth of GMCs. Therefore, in our calculation, we adopt that ${T}_{\mathrm{sg}}$ is comparable to the typical GMC formation timescale, ${T}_{{\rm{f}}}$, which is estimated as a few 10 Myr as discussed in Section 2.2 (see Inoue & Inutsuka 2012; Inutsuka et al. 2015). The resultant ${({dm}/{dt})}_{\mathrm{self}}$ becomes

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{dm}}{{dt}}\right)}_{\mathrm{self}}=\displaystyle \frac{m}{{T}_{{\rm{f}}}}.\end{eqnarray} \tag{ 3 }$$

Equation (3) with a constant ${T}_{{\rm{f}}}=10\,\mathrm{Myr}$ indicates that GMCs grow exponentially in mass. Given the minimum mass for GMCs (${m}_{\min }$) is ∼10⁴ ${M}_{\odot }$ (see Appendix D), GMCs require at least 100 Myr for exponential growth up to 10^6.5 ${M}_{\odot }$ (see also Appendix C). With the additional 14 Myr required for destroying GMCs due to star formation (see Section 3.2), we expect that observed massive GMCs ∼10^6.5 ${M}_{\odot }$ typically have ages ≳114 Myr. 114 Myr is almost comparable or larger than a typical timescale for the half galactic rotation ∼100 Myr. Therefore, this 114 Myr indicates that massive GMCs ∼10^6.5 ${M}_{\odot }$ in inter-arm regions are not directly formed "in situ" in the inter-arm environment; they may be remnants that were originally born in the arm environment and survived the destruction process (e.g., by stellar feedback and galactic shear). Modeling such a transition from the arm environment into the inter-arm environment should be investigated further to study the observed spur features extended from spiral arms (e.g., Corder et al. 2008; Schinnerer et al. 2017) and flocculent spiral arms (e.g., in Galaxy M33). However, we leave this for future studies and focus on the GMCMF variation purely due to environmental differences. Note that 114 Myr is the "age" of large GMCs ∼ 10^6.5 ${M}_{\odot }$ but not the typical GMC "lifetime" (see Section 3.2).

We should note here that the above formulation over-estimates the growth rate of very massive GMCs whose mass is comparable to the mass of a shell swept up by an expanding bubble. Once the GMC mass is comparable to or larger than the typical mass of a swept-up shell, the growth in mass should saturate, because the dense gas that can be used to form a cloud is limited by the amount of total mass in the expanding shell. Indeed, observations have revealed that GMCs exist only up to 10⁸ ${M}_{\odot }$ (e.g., Rosolowsky et al. 2007; Colombo et al. 2014a). For modeling such gas shortage, we modify ${T}_{{\rm{f}}}$ by applying a growing factor with a truncation mass ${m}_{\mathrm{trunc}}$ as

$$\begin{eqnarray}&&{T}_{{\rm{f}}}(m)={T}_{{\rm{f}},\mathrm{fid}}{\left(1+\displaystyle \frac{m}{{m}_{\mathrm{trunc}}}\right)}^{\gamma }.\end{eqnarray} \tag{ 4 }$$

Here the subscript fid stands for the fiducial constant value (i.e., ${T}_{{\rm{f}},\mathrm{fid}}=10\,\mathrm{Myr}$) and the exponent γ determines the gas-deficient efficiency. The Taylor series expansion of Equation (4) gives ${T}_{{\rm{f}}}(m)\simeq {T}_{{\rm{f}},\mathrm{fid}}(1+\gamma m/{m}_{\mathrm{trunc}})$. Therefore, when GMCs grow up to ${m}_{\mathrm{crit}}\sim {m}_{\mathrm{trunc}}/\gamma$, ${T}_{{\rm{f}}}$ deviates longer than ${T}_{{\rm{f}},\mathrm{fid}}$ so that the choices of ${m}_{\mathrm{trunc}}$ and γ modify the massive end of the GMCMF. Essentially, ${m}_{\mathrm{crit}}$ represents the typical maximum GMC mass that can be created in the SI15 scenario. In this scenario, GMCs are created from the ISM swept up by supersonic shock compression, thus it is less likely that GMCs will form that are more massive than the total mass that a single supernova remnant can sweep. The total mass initially contained within a sphere of 100 pc radius with H i density $10\,{\mathrm{cm}}^{-3}$ is about $7.7\times {10}^{5}\,{M}_{\odot }$. Thus ${m}_{\mathrm{crit}}=7.7\times {10}^{5}\,{M}_{\odot }$ and this gives ${m}_{\mathrm{trunc}}=7.7\times {10}^{6}\,{M}_{\odot }$ given that we opt to use γ = 10.

Detailed modeling of ${m}_{\mathrm{trunc}}$ and γ changes the relative importance of GMC self-growth/dispersal and CCC. In addition, rapid star formation triggered by CCC also needs to be properly modeled if CCC becomes effective (see Section 4.3). However, these details do not have a large impact as long as we focus on the GMCMF slope (see Section 4.1). Therefore, we will reserve this detailed investigation for future works.

Note that the second term on the left hand side of Equation (1) has its boundary condition at $m={m}_{\min }={10}^{4}\,{M}_{\odot }$. The flux at this boundary in our formulation corresponds to the minimum-mass GMC production rate. In later sections, we will explain that this rate differs between setups (see the first paragraph in Section 4 and the second paragraph in Section 5).

The first term on the right hand side gives the GMC self-dispersal rate. Here, self-dispersal means that massive stars born within GMCs destroy their parental GMCs by any means (ionization, dissociation, heating, blowing-out, etc.). The characteristic self-dispersal timescale, ${T}_{d}$, is given as

$$\begin{eqnarray}&&{T}_{d}={T}_{* }+{T}_{\mathrm{dest}},\end{eqnarray} \tag{ 5 }$$

where ${T}_{* }$ is the typical timescale for star formation after GMC birth and ${T}_{\mathrm{dest}}$ is the typical timescale for GMC destruction once stars become main-sequence stars. We assume the typical star formation timescale within GMCs is $\sim {T}_{{\rm{f}},\mathrm{fid}}$ and thus employ ${T}_{* }=10\,\mathrm{Myr}$. ${T}_{\mathrm{dest}}$ can be estimated by line-radiation magnetohydrodynamics simulations. Inutsuka et al. (2015) reported one such simulation result, which updated the work in Hosokawa & Inutsuka (2006) by including magnetic fields. Their result indicates that, once a massive star with mass >30 ${M}_{\odot }$ is formed in a GMC, the star destroys more than 10⁵ ${M}_{\odot }$ surrounding molecular hydrogen within 4 Myr. Therefore, typical ${T}_{d}$ is $10+4=14\,\mathrm{Myr}$.

${T}_{d}$ can be understood as the typical GMC "lifetime" averaged over all populations because ${T}_{d}$ is the typical timescale within which GMCs disperse in the system. This is consistent with the short lifetime indicated by observations (e.g., in LMC (Kawamura et al. 2009) and in M51 (Meidt et al. 2015)).

${T}_{\mathrm{dest}}$ is basically independent of GMC mass because, for example, 10 times more massive GMCs generate 10 times more $\gt 30\,{M}_{\odot }$ stars, which in turn results in destroying 10 times more molecular gas (see Inutsuka et al. 2015). Therefore, if ${T}_{* }$ is irrespective of GMC mass, ${T}_{d}$ does not depend on GMC mass. Thus, we opt to use ${T}_{d}=14\,\mathrm{Myr}$ throughout the current article. Note that this argument of mass independence is valid only if parental GMCs are >10⁵ ${M}_{\odot }$ because this is the minimum mass required to generate stars $\gt 30\,{M}_{\odot }$ if we adopt the Salpeter initial stellar mass function (Salpeter 1955). When a GMC is <10⁵ ${M}_{\odot }$, the self-dispersal may be less effective (see Inutsuka et al. 2015) and more careful treatment is desired. In addition, ${T}_{* }$ can be one order of magnitude shorter if CCC triggers rapid formation of massive stars (see Fukui et al. 2016) (see also Section 4.3). For simplicity, however, we do not consider any ${T}_{* }$ and ${T}_{\mathrm{dest}}$ variation in the current article (see also Section 4.3).

The second and third terms on the right hand side represent the collisional coagulation between GMCs (i.e., the CCC process). The second term on the right hand side produces a GMC with mass $m(={m}_{1}+{m}_{2})$ through CCC between GMCs with their mass m₁ and m₂. Similarly, the last term on the right hand side creates a GMC with mass $m+{m}_{2}$ through CCC between GMCs with their mass m and m₂, so that the negative sign indicates that this collision decreases the number density of GMCs with mass m. $K({m}_{1},{m}_{2})$ is the product of the total GMC collisional cross section between GMCs with mass m₁ and m₂, ${\sigma }_{\mathrm{col}\mathrm{1,2}}$, and the relative velocity between GMCs, V_rel:

$$\begin{eqnarray}&&K({m}_{1},{m}_{2})={\sigma }_{\mathrm{col}\mathrm{1,2}}\,{V}_{\mathrm{rel}}={c}_{\mathrm{col}}\displaystyle \frac{{m}_{1}+{m}_{2}}{{{\rm{\Sigma }}}_{\mathrm{mol}}}{V}_{\mathrm{rel},0}.\end{eqnarray} \tag{ 6 }$$

Here, c_col is a correction parameter, ${{\rm{\Sigma }}}_{\mathrm{mol}}$ is GMC column density, and ${V}_{\mathrm{rel},0}$ is the fiducial relative velocity between GMCs. c_col reflects various effects (e.g., geometrical structure, gravitational focusing, relative velocity variation) and is expected to be on the order of unity (see discussions in Appendix A). For simplicity, we use ${c}_{\mathrm{col}}=1$ throughout this article. For the column density and the relative velocity, we opt to employ an observed constant value ${{\rm{\Sigma }}}_{\mathrm{mol}}=2\times {10}^{22}\,\mu {m}_{{\rm{H}}}\,{\mathrm{cm}}^{-2}$ (see Section 3.1) and a typical expanding speed of the ionization–dissociation front ${V}_{\mathrm{rel},0}=10\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (see Appendix B.3) respectively, where μ is the mean molecular weight and m_H is the atomic hydrogen mass.

Our calculation focuses only on collisional coagulation and ignores fragmentation. This assumption is based on the observational fact that cloud-to-cloud velocity dispersion ($\sim 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$: e.g., Stark & Lee 2005) does not largely exceed the sound velocity ($\sim 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$) of the inter-cloud medium (i.e., WNM). However, fragmentation may have a severe impact under the CCC-dominated phase (i.e., the GMC number density is high) such as in the Galactic Center (Tsuboi et al. 2015). This is beyond our current scope and we will report findings in our forthcoming paper.

Note that ${T}_{d}$ in principle should be the ensemble average of all the destructive processes including, for example, shear (see Koda et al. 2009). However, shear may be sub-dominant in disk regions typically >5 kpc (see Dobbs & Pringle 2013). For simplicity, we assume that the inclusion of these processes would not vary ${T}_{d}$ significantly.

Figures 1 and 2 show our fiducial calculations (i.e., ${T}_{{\rm{f}},\mathrm{fid}}=10\,\mathrm{Myr}$, ${T}_{d}=14$ Myr), which correspond to Case 1 and Case 2, respectively. The only difference between these two cases is whether the calculation includes the CCC terms (Figure 1) or not (Figure 2). Both computed GMCMFs demonstrate almost the same slope −α ∼ −1.7 (where ${n}_{\mathrm{cl}}\propto {m}^{-\alpha }$), which successfully fits into the middle of the observed range. This result indicates that CCC does not impact the GMCMF slope significantly whereas the shape of the massive end is modified by the relative importance between self-growth/dispersal and CCC.

To examine the relative contribution of CCC, we compute the collision timescale, ${T}_{\mathrm{col}}$, as

$$\begin{eqnarray}&&{T}_{\mathrm{col}}=\displaystyle \frac{{n}_{\mathrm{cl}}(m)}{{(\partial {n}_{\mathrm{cl}}/\partial t)}_{\mathrm{CCC}}},\end{eqnarray} \tag{ 7 }$$

where the denominator is the time evolution due to CCC:

$$\begin{eqnarray}{\left(\displaystyle \frac{\partial {n}_{\mathrm{cl}}}{\partial t}\right)}_{\mathrm{CCC}} & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{0}^{\infty }{\displaystyle \int }_{0}^{\infty }K({m}_{1},{m}_{2}){n}_{\mathrm{cl},1}{n}_{\mathrm{cl},2}\\ & & \times \delta (m-{m}_{1}-{m}_{2}){{dm}}_{1}{{dm}}_{2}\\ & & -{\displaystyle \int }_{0}^{\infty }K(m,{m}_{2}){n}_{\mathrm{cl}}{n}_{\mathrm{cl},2}{{dm}}_{2}.\end{eqnarray} \tag{ 8 }$$

Figure 3 shows the time evolution of computed ${T}_{\mathrm{col}}$, with ${T}_{{\rm{f}}}$ and ${T}_{d}$ overplotted together. The figure indicates that the GMC mass growth is determined by GMC self-growth in the low-mass regime where ${T}_{{\rm{f}}}$ ($\sim { \mathcal O }(1)$ Myr) is one order of magnitude shorter than ${T}_{\mathrm{col}}$ ($\sim { \mathcal O }(2)$ Myr), and is determined by CCC at the high-mass end where ${T}_{\mathrm{col}}$ ($\sim { \mathcal O }(1)$ Myr) becomes one order of magnitude smaller than ${T}_{{\rm{f}}}$ ($\sim { \mathcal O }(2)$ Myr). This indicates that the GMCMF slope in m ≲ 10^5.5 ${M}_{\odot }$ is well characterized by the combination of GMC self-growth and dispersal. In the next section, we are going to focus only on the GMCMF slopes and will discuss the massive-end behavior in Section 4.3.

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As shown in Section 4.1, CCC does not modify the GMCMF evolution significantly. Especially in the lower mass range (e.g., m < 10^5.5 ${M}_{\odot }$) CCC does not affect the mass function, and hence our formulation can be rewritten without the CCC terms. Therefore, the evolution of differential number density of GMCs with mass m, n_cl, is now simply given as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{\mathrm{cl}}}{\partial t}+\displaystyle \frac{\partial }{\partial m}\left({n}_{\mathrm{cl}}\displaystyle \frac{m}{{T}_{{\rm{f}}}}\right)=-\displaystyle \frac{{n}_{\mathrm{cl}}}{{T}_{d}}.\end{eqnarray} \tag{ 9 }$$

This formulation has been already (and originally) proposed by Inutsuka et al. (2015). One can obtain the steady state solution of this differential equation for $m\lesssim {m}_{\mathrm{crit}}$ with a constant ${T}_{{\rm{f}}}$ as

$$\begin{eqnarray}&&{n}_{\mathrm{cl}}(m)={n}_{0}{\left(\displaystyle \frac{m}{{M}_{\odot }}\right)}^{-1-\tfrac{{T}_{{\rm{f}}}}{{T}_{d}}},\end{eqnarray} \tag{ 10 }$$

where n₀ is the differential number density normalized at $m=1\,{M}_{\odot }$. This predicts that GMCMFs have slopes with a single exponent, which is well characterized by $-1-{T}_{{\rm{f}}}/{T}_{d}$. Indeed, the computed GMCMF shows the slope consistent with this exponent $-1-{T}_{{\rm{f}}}/{T}_{d}$ without CCC (see Figure 2), and in the lower mass (m < 10^5.5 ${M}_{\odot }$) part even in the case with CCC (see Figure 1).

To examine the validity of the $-1-{T}_{{\rm{f}}}/{T}_{d}$ prediction, we conduct the calculation again by employing various ${T}_{{\rm{f}}}$. Figures 4 and 5 are two representative results and both cases exhibit a slope well characterized by $-1-{T}_{{\rm{f}}}/{T}_{d}$. Figure 4 employs a shorter formation timescale, ${T}_{{\rm{f}},\mathrm{fid}}=4.2\,\mathrm{Myr}$, and reproduces the observed shallow slope in arm regions. Similarly, Figure 5 shows the result with ${T}_{{\rm{f}},\mathrm{fid}}=22.4\,\mathrm{Myr}$, which reproduces the observed steep slope in inter-arm regions. Therefore, the arm regions presumably have a larger number of massive stars forming H ii regions and supernova remnants thus experience recurrent supersonic compression twice more frequently than the fiducial case, and the inter-arm regions typically have a smaller number of massive stars thus experience less frequent supersonic compression, which results in about a factor two longer formation timescale than the fiducial case. This may also explain the observed steep slope on the outskirts of galaxies (e.g., the observed steep slope in the CO luminosity function in Galaxy M33; see Gratier et al. 2012) where star formation is less active compared with normal disk regions. Note that the above argument might be modified, if the effect of large-scale dynamics (e.g., interaction with shock waves or strong shear flows) plays a more important role in the destruction of GMCs than stellar feedback, especially on the outskirts of galaxies with prominent spiral structures.

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Note that here we assume that ${T}_{d}$ variation between different regions is limited compared with ${T}_{{\rm{f}}}$ variation. This is because ${T}_{d}$ is basically independent of GMC mass as explained in Section 3.2 and also because observations suggest that star formation efficiency is almost the same throughout the galactic disks in nearby galaxies (e.g., Schruba et al. 2011) and in the Galaxy (N. Izumi et al. 2017, in preparation). However, ${T}_{d}$ might be longer in inter-arm regions compared with arm regions due to its lower star-forming activity where ${T}_{d}$ can be up to ∼30 Myr (the upper limit measured in M51 by Meidt et al. 2015), or might be a factor longer in smaller clouds ≲10⁵ ${M}_{\odot }$ where it is invalid to apply our mass-independent assumption on the cloud destruction rate due to massive stars. The significance of ${T}_{d}$ variation should be investigated more and we will leave this for future work.

Our results suggest that the variation of the GMCMF slopes is governed by ${T}_{{\rm{f}}}/{T}_{d}$ diversity in different environments on galactic scales. We predict that future large radio surveys will reveal that the GMCMFs have a single power-law exponent with which the ratio ${T}_{{\rm{f}}}/{T}_{d}$ may be uniquely constrained.

The shape of the GMCMF massive end, typically >10⁶ ${M}_{\odot }$, is determined by the relative importance of GMC self-growth and CCC as shown in previous sections. The quantitative discussion involves many uncertainties from our modeling; for example, a more detailed prescription for γ is necessary for GMC self-growth (i.e., ${T}_{{\rm{f}}}$), and it is also required to model T_* variation due to the drastic star formation invoked by CCC, which is inferred by observations in the Galaxy (see Fukui et al. 2014; also see Section 6.1). These are beyond our current scope and to be discussed in our forthcoming paper.

Despite these limitation, our results indicate that, if CCC becomes effective, another structure may appear in the massive end. This possibly explains the extra power-law feature observed in some regions from the PAWS survey (e.g., "Material Arms"; Colombo et al. 2014a).

For all the time evolutions of the GMCMF we have shown in the previous sections, we assume that minimum-mass molecular clouds are continuously provided and the dispersed gas is removed from the system and never restored into the system.

However, in reality, dispersed gas should return to the ISM and become either seeds of a newer generation of molecular clouds or mass accreting onto the pre-existing GMCs. To establish a complete picture of gas resurrecting processes in the ISM, we need to evaluate the fate of such dispersed gas. This aspiration requires detailed numerical simulations, ideally, three-dimensional radiation magnetohydrodynamics simulations, which are however computationally too expensive to conduct. Instead in this article, we quantify the amount of dispersed gas by introducing the "resurrecting factor," ${\varepsilon }_{\mathrm{res}}$, which is the mass fraction out of the total dispersed gas that is transformed to a newer generation of minimum-mass GMCs. In Section 4, we artificially set the production rate of minimum-mass GMCs to keep its number density constant. Here instead, we evaluate this production rate due to gas resurrection as

$$\begin{eqnarray}&&{\displaystyle \frac{\partial ({n}_{\mathrm{cl}}m)}{\partial t}|}_{\mathrm{res}}={\varepsilon }_{\mathrm{res}}{\dot{M}}_{\mathrm{total},\mathrm{disp}}\delta (m-{m}_{\min }),\end{eqnarray} \tag{ 11 }$$

where ${\dot{M}}_{\mathrm{total},\mathrm{disp}}$ is the total dispersed gas mass produced from the system per unit time, ${m}_{\min }$ is the minimum mass of GMCs (i.e., 10⁴ ${M}_{\odot }$ in this study), and δ is the Dirac delta function. By this definition, ${\varepsilon }_{\mathrm{res}}$ can be considered as the probability that dispersed gas becomes minimum-mass GMCs before they accrete onto the pre-existing GMCs and thus the $1-{\varepsilon }_{\mathrm{res}}$ fraction of dispersed gas is consumed for the self-growth of pre-existing GMCs if a GMCMF is in a steady state.

From now, we are going to focus only on steady state GMCMFs for simplicity. We may assume that the GMCMFs are quasi-steady in the Galaxy and nearby massive spiral galaxies because they presumably have already undergone an active star formation phase at redshift ∼2 and have relatively constant star formation activity at the present day (e.g., Madau & Dickinson 2014). Note that, although we have already proved that the CCC effect is limited, we fully solve Equation (1) including the CCC terms to compute the time evolution in all the following results. We mainly analyze Cases 5 to 8 in Table 1. The spikes and kinks that appear in the massive end (especially in Figures 6, 9, and 11) are due to the numerical effect by the CCC turn-off procedure as explained in the second paragraph in Section 4.

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Here, we examine the fiducial timescales ${T}_{{\rm{f}},\mathrm{fid}}=10\,\mathrm{Myr}$ and ${T}_{d}=14\,\mathrm{Myr}$. First, we employ an extreme case where ${\varepsilon }_{\mathrm{res}}=1;$ all the dispersed gas is consumed to form minimum-mass GMCs (Case 6) and Figure 6 shows the result. Initially the GMCMF exhibits slopes slightly steeper than −1.7 predicted by Equation (10). However the slope becomes even steeper in the low-mass regime and the total mass in the system keeps increasing so that the GMCMF does not show any steady state (see Figure 8). Therefore, the slopes are provisional and transitional. Indeed, the CCC becomes significantly effective after 60 Myr so that GMCs more massive than observed ones are instantly created after 60 Myr, which do not reproduce the observed GMCMFs.

The steepened slope observed in the low-mass end in Figure 6 appears due to overloading ${\varepsilon }_{\mathrm{res}}$, because pre-existing GMCs cannot acquire the excessive amount of resurrecting gas faster than a given self-growth timescale (in this case 10 Myr). To achieve a steady state with a shallower slope, we need to reduce ${\varepsilon }_{\mathrm{res}}$. Figure 7 shows a result with ${\varepsilon }_{\mathrm{res}}=0.09$ where we successfully reproduce a slope −α = −1.7 throughout all the mass range and achieve a steady state GMCMF. To check the steadiness, we compute the total mass in the system as a function of time with various ${\varepsilon }_{\mathrm{res}}$ and Figure 8 shows the result. ${\varepsilon }_{\mathrm{res}}=0.09$ indicates that the GMCMF settles down on the steady state with a slight decrement in its total mass before the calculation ends at 200 Myr. Contrarily, excessive input ${\varepsilon }_{\mathrm{res}}\gg 0.09$ leads to growth of the total mass and the GMCMF does not become steady by 200 Myr as, for example, already seen in Figure 6. Less input ${\varepsilon }_{\mathrm{res}}\ll 0.09$ leads to too much gas dispersal into the ISM and eventually decreases the total mass in the GMC phase, which would not reach any steady state before 200 Myr. In Section 5.5, we will investigate an analytical justification why ${\varepsilon }_{\mathrm{res}}\sim 0.09$ reaches a steady state and present the possible range of ${\varepsilon }_{\mathrm{res}}$ that provides a steady GMCMF.

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We now focus on the shorter formation timescale ${T}_{{\rm{f}},\mathrm{fid}}=4.2\,\mathrm{Myr}$ (Case 7) to examine what value of ${\varepsilon }_{\mathrm{res}}$ can reproduce the slopes observed in arm regions. Figure 9 shows the GMCMF time evolution with ${\varepsilon }_{\mathrm{res}}=0.0123$, which successfully reproduces the observed shallow slope ∼−1.3. Again, to check the steadiness, we compute the total mass in the system and Figure 10 shows the result with ${\varepsilon }_{\mathrm{res}}$ from 5.8 × 10⁻³ to 0.021. The figure also confirms that ${\varepsilon }_{\mathrm{res}}=0.0123$ produces a steady state GMCMF. This factor 0.0123 indicates that almost 99 per cent of dispersed gas is accreting onto and fueling pre-existing GMCs due to the multiple episodes of compression and that only 1 per cent of dispersed gas is turned to form a newer generation of GMCs. This extreme fraction is expected due to massive GMCs; massive GMCs have a larger surface area than less massive GMCs and collect a large amount of diffuse ISM gas to grow. Therefore, small ${\varepsilon }_{\mathrm{res}}$, or large $1-{\varepsilon }_{\mathrm{res}}$, is likely to be realized in arm regions, which have many massive GMCs. We will discuss the range of ${\varepsilon }_{\mathrm{res}}$ around 0.0123 that also keeps the GMCMF steady in Section 5.5.

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We now focus on the longer formation timescale ${T}_{{\rm{f}},\mathrm{fid}}\,=22.4\,\mathrm{Myr}$ (Case 8) to examine what value of ${\varepsilon }_{\mathrm{res}}$ can reproduce the slopes observed in inter-arm regions. Figure 11 shows the GMCMF time evolution with ${\varepsilon }_{\mathrm{res}}=0.45$. This GMCMF has its slope ∼−2.4, shallower than the observed slope ∼−2.6 due to the coagulation by CCC, which is also observed in Figure 5. We compute the total mass in the system and Figure 12 shows the result with ${\varepsilon }_{\mathrm{res}}$ from 0.133 to 0.713. The figure confirms that ${\varepsilon }_{\mathrm{res}}=0.45$ produces a steady state GMCMF. The initial condition is coincidently close enough to the steady state with ${\varepsilon }_{\mathrm{res}}=0.45$ thus the GMCMF with ${\varepsilon }_{\mathrm{res}}=0.45$ keep its total mass over the whole 200 Myr. This factor 0.45 indicates that about 55 per cent of dispersed gas is accreting onto and fueling pre-existing GMCs due to the multiple episodes of compression and that 45 per cent of dispersed gas is turned to form a newer generation of GMCs. This is naturally expected because the inter-arm regions do not contain many massive GMCs that may collect diffuse ISM to grow. We will discuss the range of ${\varepsilon }_{\mathrm{res}}$ around 0.45 that also keeps the GMCMF steady in Section 5.5.

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In this section, we derive an analytical estimation for the steady state ${\varepsilon }_{\mathrm{res}}$ with its possible variation as a function of the GMCMF slope to explain the reason why the values we choose for the resurrecting factor in Sections 5.3 and 5.4 can successfully reproduce the observed variation of the GMCMFs. Here, we assume that the GMCMF in a mass range from ${m}_{\min }$ to ${m}_{\max }$ has a steady power-law state ${n}_{\mathrm{cl}}={{Am}}^{-\alpha }$, where A is a constant. We choose ${m}_{\min }={10}^{4}\,{M}_{\odot }$ in our calculation (see Appendix D for the choice of ${m}_{\min }$ value), whereas ${m}_{\max }\,\approx {m}_{\mathrm{trunc}}$ as shown in the figures from our calculation.

To derive ${\varepsilon }_{\mathrm{res}}$ in a steady state GMCMF, we multiply m by Equation (1) without the CCC term and modify the dispersal term into a form of $\partial /\partial m$:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {{mn}}_{\mathrm{cl}}}{\partial t}+\displaystyle \frac{\partial }{\partial m}F(m)=0.\end{eqnarray} \tag{ 12 }$$

Here,

$$\begin{eqnarray}&&F(m)=\displaystyle \frac{{m}^{2}{n}_{\mathrm{cl}}}{{T}_{{\rm{f}}}(m)}-{\int }_{{m}_{\min }}^{m}\displaystyle \frac{{{mn}}_{\mathrm{cl}}{dm}}{{T}_{{\rm{f}}}(m)}+{\int }_{{m}_{\min }}^{m}\displaystyle \frac{{{mn}}_{\mathrm{cl}}{dm}}{{T}_{d}},\end{eqnarray} \tag{ 13 }$$

is the net mass flux at m across the mass coordinate. Because we consider a steady state GMCMF at $m\leqslant {m}_{\mathrm{trunc}}$, ${T}_{{\rm{f}}}$ will be treated as a constant in the following analysis. Then F(m) can be rewritten as

$$\begin{eqnarray}F(m) & = & {{Am}}^{2-\alpha }\left(\displaystyle \frac{1}{{T}_{{\rm{f}}}}-\displaystyle \frac{1}{(2-\alpha ){T}_{{\rm{f}}}}+\displaystyle \frac{1}{(2-\alpha ){T}_{d}}\right)\\ & & +{{Am}}_{\min }^{2-\alpha }\left(\displaystyle \frac{1}{(2-\alpha ){T}_{{\rm{f}}}}-\displaystyle \frac{1}{(2-\alpha ){T}_{d}}\right).\end{eqnarray} \tag{ 14 }$$

This mass flux is useful to evaluate the mass evolution in the system as we prove here, and is also complementary to the original number density analysis because $\partial F(m)/\partial m=0$ reproduces the steady state slope predicted in Equation (10):

$$\begin{eqnarray}&&\alpha =1+\displaystyle \frac{{T}_{{\rm{f}}}}{{T}_{d}}.\end{eqnarray} \tag{ 15 }$$

We introduce four characteristic quantities that control ${\varepsilon }_{\mathrm{res}};$ the incoming flux at the minimum-mass end, ${F}_{\mathrm{in}}$, the outgoing flux at the high-mass end, ${F}_{\mathrm{out}}$, the total mass growth per unit time by all the pre-existing GMCs, ${\dot{M}}_{\mathrm{grow}}$, and the total dispersed mass generated per unit time from the system, ${\dot{M}}_{\mathrm{disp}}$. They are given as

$$\begin{eqnarray}&&{F}_{\mathrm{in}}=F({m}_{\min })={m}_{\min }^{2}{n}_{\mathrm{cl}}/{T}_{{\rm{f}}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{F}_{\mathrm{out}}={m}_{\max }^{2}{n}_{\mathrm{cl}}/{T}_{{\rm{f}}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{grow}}={\int }_{{m}_{\min }}^{{m}_{\max }}({n}_{\mathrm{cl}}m/{T}_{{\rm{f}}}){dm},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{disp}}={\int }_{{m}_{\min }}^{{m}_{\max }}({n}_{\mathrm{cl}}m/{T}_{d}){dm}.\end{eqnarray} \tag{ 19 }$$

With these quantities, $F({m}_{\max })$ corresponds to ${F}_{\mathrm{out}}-{\dot{M}}_{\mathrm{grow}}\,+{\dot{M}}_{\mathrm{disp}}$. In the steady state, F(m) is independent of m. Thus $F({m}_{\min })=F(m)=F({m}_{\max })$ and this can be rewritten as

$$\begin{eqnarray}&&{F}_{\mathrm{in}}={F}_{\mathrm{out}}-{\dot{M}}_{\mathrm{grow}}+{\dot{M}}_{\mathrm{disp}}.\end{eqnarray} \tag{ 20 }$$

For massive GMCs, the power-law relation $n={{Am}}^{-\alpha }$ is broken: the treatment is valid if we take ${m}_{\max }\approx {m}_{\mathrm{trunc}}$. For $m\gtrsim {m}_{\max },{T}_{{\rm{f}}}$ rapidly increases with mass and the dispersal becomes more important than the growth so that GMCs with $m\gtrsim {m}_{\max }$ are eventually dispersed. Therefore, ${F}_{\mathrm{out}}$ is interpreted as the mass dispersing rate at $m\geqslant {m}_{\max }$ and the total dispersal rate of GMCs is given by ${\dot{M}}_{\mathrm{disp}}+{F}_{\mathrm{out}}$. On the other hand, ${F}_{\mathrm{in}}$ means the formation rate of minimum-mass GMCs. Therefore, the resurrecting factor is given by

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{res}}=\displaystyle \frac{{F}_{\mathrm{in}}}{{F}_{\mathrm{out}}+{\dot{M}}_{\mathrm{disp}}}.\end{eqnarray} \tag{ 21 }$$

We consider that the GMCMF from ${m}_{\min }$ to ${m}_{\max }$ becomes a steady state when ${\varepsilon }_{\mathrm{res}}={\varepsilon }_{\mathrm{res},\mathrm{std}}$. Combined with Equation (20), ${\varepsilon }_{\mathrm{res},\mathrm{std}}$ is given as

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{res},\mathrm{std}}\\ &&\quad =\displaystyle \frac{{F}_{\mathrm{in}}}{{F}_{\mathrm{in}}+{\dot{M}}_{\mathrm{grow}}}\\ &&\quad =\,\left\{\begin{array}{l}{\left\{1+\tfrac{1}{2-\alpha }\left[{\left(\tfrac{{m}_{\max }}{{m}_{\min }}\right)}^{2-\alpha }-1\right]\right\}}^{-1}\quad (\mathrm{for}\ \alpha \ne 2),\\ {[1+\mathrm{ln}({m}_{\max }/{m}_{\min })]}^{-1}\quad (\mathrm{for}\ \alpha =2).\end{array}\right.\end{eqnarray} \tag{ 22 }$$

Table 2 shows ${\varepsilon }_{\mathrm{res},\mathrm{std}}$ computed by Equation (22). Here, as a representative case of ${m}_{\max }\lt {m}_{\mathrm{trunc}}$, we opt to use ${m}_{\max }={10}^{6.2}\,{M}_{\odot }$, which is also consistent with the truncated mass scale observed in our computed GMCMF (see also Figure 13). Because ${m}_{\max }\gg {m}_{\min }$, Equation (22) can be simplified as

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{res},\mathrm{std}}=\left\{\begin{array}{l}(2-\alpha ){\left(\tfrac{{m}_{\min }}{{m}_{\max }}\right)}^{2-\alpha }\quad (\mathrm{for}\ \alpha \lt 2),\\ {[\mathrm{ln}({m}_{\max }/{m}_{\min })]}^{-1}\quad (\mathrm{for}\ \alpha =2),\\ \tfrac{\alpha -2}{\alpha -1}\quad (\mathrm{for}\ \alpha \gt 2).\end{array}\right.\end{eqnarray} \tag{ 23 }$$

Note that the accuracy of the approximation ${m}_{\max }\lt {m}_{\mathrm{trunc}}$ differs between cases, and thus the computed GMCMFs exhibit resurrecting factors that slightly deviate by a factor 0.6–1.2 from Equation (22) (i.e., Table 2).

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Table 2.  Estimated Resurrecting Factor as a Function of Growth Timescale

	Estimated ${\varepsilon }_{\mathrm{res}}$
$\mathrm{log}({m}_{\max })$	${T}_{{\rm{f}},\mathrm{fid}}=4.2$	${T}_{{\rm{f}},\mathrm{fid}}=10$	${T}_{{\rm{f}},\mathrm{fid}}=22.4$
(${M}_{\odot }$)	(Myr)	(Myr)	(Myr)
6.5	0.0201	0.0798	0.383

Note. Predicted ${\varepsilon }_{\mathrm{res},\mathrm{std}}$ for three timescales: ${T}_{{\rm{f}},\mathrm{fid}}=4.2,10$, and 22.4 Myr based on Equation (22). All the cases have a constant ${T}_{d}=14\,\mathrm{Myr}$.

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5.5.1. Range of the Resurrecting Factor

Integration of Equation (12) over m from ${m}_{\min }$ to ${m}_{\max }$ results in

$$\begin{eqnarray}\displaystyle \frac{{{dM}}_{\mathrm{total}}}{{dt}} & = & F({m}_{\min })-F({m}_{\max }),\\ & = & {F}_{\mathrm{in}}-{F}_{\mathrm{out}}-{\dot{M}}_{\mathrm{disp}}+{\dot{M}}_{\mathrm{grow}},\\ & = & \left(1-\displaystyle \frac{1}{{\varepsilon }_{\mathrm{res}}}\right){F}_{\mathrm{in}}+{\dot{M}}_{\mathrm{grow}}.\end{eqnarray} \tag{ 24 }$$

where ${M}_{\mathrm{total}}$ is the total mass of GMCs from ${m}_{\min }$ to ${m}_{\max }$:

$$\begin{eqnarray}&&{M}_{\mathrm{total}}={\int }_{{m}_{\min }}^{{m}_{\max }}{{mn}}_{\mathrm{cl}}{dm},\end{eqnarray} \tag{ 25 }$$

In the steady state, ${{dM}}_{\mathrm{total}}/{dt}=0$ and ${\varepsilon }_{\mathrm{res}}={\varepsilon }_{\mathrm{res},\mathrm{std}}$. Therefore,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{grow}}=-\left(1-\displaystyle \frac{1}{{\varepsilon }_{\mathrm{res},\mathrm{std}}}\right){F}_{\mathrm{in}}.\end{eqnarray} \tag{ 26 }$$

The variation of ${\varepsilon }_{\mathrm{res}}$ changes ${n}_{\mathrm{cl}}$ and hence ${\dot{M}}_{\mathrm{grow}}$. However for simplicity, we here ignore the dependence of ${\dot{M}}_{\mathrm{grow}}$ on ${\varepsilon }_{\mathrm{res}}$ by considering ${\varepsilon }_{\mathrm{res}}\sim {\varepsilon }_{\mathrm{res},\mathrm{std}}$. From Equations (24) and (26), we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{\mathrm{total}}}{{dt}}=\left(\displaystyle \frac{1}{{\varepsilon }_{\mathrm{res},\mathrm{std}}}-\displaystyle \frac{1}{{\varepsilon }_{\mathrm{res}}}\right){F}_{\mathrm{in}}.\end{eqnarray} \tag{ 27 }$$

If the GMCMF is steady and the total mass in the system ${M}_{\mathrm{total}}$ does not increase/reduce by factor β within a certain timescale ${T}_{\mathrm{steady}}$,

$$\begin{eqnarray}&&\left|\displaystyle \frac{\beta {M}_{\mathrm{total}}}{{\dot{M}}_{\mathrm{total}}}\right|\gtrsim {T}_{\mathrm{steady}}.\end{eqnarray} \tag{ 28 }$$

Combining with Equation (26), the range of ${\varepsilon }_{\mathrm{res}}$ that leads to steady state becomes

$$\begin{eqnarray}&&\left(1-\displaystyle \frac{{\varepsilon }_{\mathrm{res},\mathrm{std}}}{{\varepsilon }_{\mathrm{res},\mathrm{std}}+a}\right)\lesssim \displaystyle \frac{{\varepsilon }_{\mathrm{res}}}{{\varepsilon }_{\mathrm{res},\mathrm{std}}}\lesssim \left(1+\displaystyle \frac{{\varepsilon }_{\mathrm{res},\mathrm{std}}}{a-{\varepsilon }_{\mathrm{res},\mathrm{std}}}\right),\end{eqnarray} \tag{ 29 }$$

where $a={F}_{\mathrm{in}}{T}_{\mathrm{steady}}/({M}_{\mathrm{total}}\beta )$.

Figure 13 shows ${\varepsilon }_{\mathrm{res},\mathrm{std}}$ and ${\varepsilon }_{\mathrm{res}}$ variation based on Equations (22) and (29) with ${m}_{\max }={10}^{6.2}\,{M}_{\odot }$. Here, we assume that the GMCMF can be steady and localized within the half galactic rotation (in a case of two spiral arms in a galactic disk) so that ${T}_{\mathrm{steady}}=100\,\mathrm{Myr}$ and opt to use $\beta =0.5$. The figure shows that ${\varepsilon }_{\mathrm{res}}$ increases with α. This trend is naturally expected because the inter-arm regions have fewer massive GMCs that can sweep up dispersed gas compared with the arm regions and dispersed gas easily produces minimum-mass GMCs when they experience multiple episodes of compression.

On the one hand, overwhelming resurrection more than the maximum ${\varepsilon }_{\mathrm{res}}$ increases the number density of GMCs, which ends up with the CCC-dominated regime after a long time. On the other hand, the GMC number density decreases so that the GMCMF decays with a resurrecting factor less than the minimum ${\varepsilon }_{\mathrm{res}}$.

5.5.2. Unseen Gas

Our results indicate that CCC is limited only at the massive end and does not alter GMCMF time evolution significantly. This seems contrary to the CCC importance suggested by galactic simulations and observations, but is still consistent. The collision timescale typically varies from 1 to 100 Myr (see Figure 3), which is consistent with global galactic simulations (c.f., 25 Myr by Tasker & Tan (2009), 8–28 Myr by Dobbs et al. 2015). Figure 3 shows that GMCs > 10⁶ ${M}_{\odot }$ experience CCC much more frequently than GMCs < 10⁵ ${M}_{\odot }$ as expected by the kernel function (Equation (6)). The lack of star cluster formation during CCC in our calculation generates many massive GMCs at the massive end (see Tasker & Tan 2009). However, we expect that such drastic star cluster formation takes place only when larger GMCs collide with each other (Fukui et al. 2014), which may affect the massive end but not the slope. Therefore, our results indicate that inclusion of star cluster formation still does not impact the GMCMF slope significantly.

In this section, we are going to investigate whether or not the cascade collisional coagulation alone can create a steady state GMCMF, which we investigated so far. Let us consider the mass flux as a function of GMC mass m. Suppose that the CCC kernel function has its mass dependence as $K\propto {m}^{p}$ and the resultant differential number density has spectra $n\propto {m}^{-\alpha }$, then the mass flux F due to the cascade collisional coagulation becomes (c.f., Kobayashi & Tanaka 2010)

$$\begin{eqnarray}&&F\propto {m}^{3}{n}^{2}K\propto {m}^{-2\alpha +p+3}.\end{eqnarray} \tag{ 30 }$$

In a steady state, F is constant across the whole mass range (i.e., F ∝ m⁰), therefore

$$\begin{eqnarray}&&p=2\alpha -3.\end{eqnarray} \tag{ 31 }$$

If the mass dependence of the kernel function has a variation with $p\in [-0.4,2.2]$ the observed slopes can be reproduced, which typically varies as $-\alpha =-1.3\,\mathrm{to}\,-2.6$. Thus, the regions where the GMC number density is large (e.g., the Galactic Center) may form this type of CCC-driven GMCMF (see Tsuboi et al. 2015). However, the collision alone reduces the number of minimum-mass GMCs quickly and also creates infinitely massive GMCs. Therefore, we still need some proper prescription for the creation process at the low-mass end and the dispersal process at the high-mass end. We will investigate such a formulation in our upcoming paper. Also note that the GMCMF may have the same slope when collision is not a cascade but always occurs only with the minimum-mass bin, which we will also report in our forthcoming paper.

For simplicity, let us ignore the thickness of GMCs, which makes ${c}_{\mathrm{col}}$ a factor larger, and assume that GMCs always coagulate even when their peripheries alone touch each other.

Given that GMCs have a uniform column density ${{\rm{\Sigma }}}_{\mathrm{mol}}$, the cross section for face-on collisional coagulation between GMCs whose masses are m₁ and m₂ (${m}_{1}\geqslant {m}_{2}$), ${\sigma }_{\mathrm{col}\mathrm{peri}\mathrm{1,2}}$, is described as

$$\begin{eqnarray}{\sigma }_{\mathrm{col}\mathrm{peri}\mathrm{1,2}} & = & \pi {({R}_{1}+{R}_{2})}^{2}\\ & = & \pi {\left(\sqrt{\displaystyle \frac{{m}_{1}}{\pi {{\rm{\Sigma }}}_{\mathrm{mol}}}}+\sqrt{\displaystyle \frac{{m}_{2}}{\pi {{\rm{\Sigma }}}_{\mathrm{mol}}}}\right)}^{2}\\ & = & \displaystyle \frac{{(\sqrt{{m}_{1}}+\sqrt{{m}_{2}})}^{2}}{{{\rm{\Sigma }}}_{\mathrm{mol}}},\end{eqnarray} \tag{ 32 }$$

where R₁ and R₂ are the radii of GMCs with masses m₁ and m₂ respectively (see the left panel of Figure 14 for the schematic configuration). In this case, the correction factor becomes

$$\begin{eqnarray}&&{c}_{\mathrm{col}}=\displaystyle \frac{{\sigma }_{\mathrm{col}\mathrm{peri}\mathrm{1,2}}}{{\sigma }_{\mathrm{col}\mathrm{1,2}}}=1+\displaystyle \frac{2\sqrt{{m}_{1}}\sqrt{{m}_{2}}}{{m}_{1}+{m}_{2}}.\end{eqnarray} \tag{ 33 }$$

The second term gives the deviation from unity and thus ${c}_{\mathrm{col}}$ has its maximum value 2 when ${m}_{1}={m}_{2}$. In reality, GMCs are less likely to coagulate together when their peripheries alone touch each other so that Equation (33) overestimates the collision frequency and ${c}_{\mathrm{col}}$ should be smaller by some factor. Therefore, we can assume ${c}_{\mathrm{col}}=1$ for simplicity.

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An edge-on collision can be similarly evaluated. Suppose that GMCs collide with an angle θ, the cross section becomes $2{R}_{1}\times 2{R}_{2}\times \sin \theta$ (see the right panel of Figure 14). ${\sigma }_{\mathrm{col}\mathrm{peri}\mathrm{1,2}}$ can be obtained by averaging this cross section over θ:

$$\begin{eqnarray}{\sigma }_{\mathrm{col}\mathrm{peri}\mathrm{1,2}} & = & \displaystyle \frac{{\int }_{0}^{\pi }2{R}_{1}2{R}_{2}\sin \theta }{\pi }\\ & = & \displaystyle \frac{8}{\pi }{R}_{1}{R}_{2}\\ & = & \displaystyle \frac{8}{{\pi }^{2}{{\rm{\Sigma }}}_{\mathrm{mol}}}\sqrt{{m}_{1}{m}_{2}}.\end{eqnarray} \tag{ 34 }$$

Therefore, the correction factor becomes

$$\begin{eqnarray}&&{c}_{\mathrm{col}}=\displaystyle \frac{8}{{\pi }^{2}}\displaystyle \frac{\sqrt{{m}_{1}{m}_{2}}}{{m}_{1}+{m}_{2}},\end{eqnarray} \tag{ 35 }$$

whose maximum is 0.57 when ${m}_{1}={m}_{2}$. Therefore, edge-on collision is less frequent than our fiducial cross section in Equation (6). Less frequent collision may impact on the shape of the massive end of the GMCMF. However, the slope calculated with overestimated ${c}_{\mathrm{col}}=1$ is not affected by CCC as shown in the figures by now. Therefore, for simplicity we can assume ${c}_{\mathrm{col}}=1$ even for the edge-on collision if we restrict ourselves to the slope of the GMCMF.

One caveat here is that we focus only on collisions that are in face-on and edge-on. The other orientations leads to cross sections smaller than the configuration we analyze here and would not greatly modify our GMCMF calculation results. Thus, we ignore such orientations for simplicity and leave them for other works. Further investigation, especially three-dimensional hydrodynamics simulation, is needed.

During a close-by encounter, two GMCs experience gravitational attraction toward each other (i.e., gravitational focusing effect), which effectively enlarges the cross section so that the rate for collisional coagulation increases. To evaluate the gravitational focusing effect, let us consider an example where a GMC whose mass and radius are m₂ and r₂ initially flies at a speed ${v}_{{\rm{i}}}$ with an impact parameter b with respect to another GMC with mass m₁ and radius r₁. Combining angular momentum conservation and energy conservation, one can derive the ratio of the effective cross section b² against the area ${({r}_{1}+{r}_{2})}^{2}$ as

$$\begin{eqnarray}\displaystyle \frac{{b}^{2}}{{({r}_{1}+{r}_{2})}^{2}} & = & 1+{\rm{\Theta }}\\ & = & 1+\displaystyle \frac{{v}_{\mathrm{esc}}^{2}}{{v}_{{\rm{i}}}^{2}}\\ & = & 1+\displaystyle \frac{2{\rm{G}}({m}_{1}+{m}_{2})}{({r}_{1}+{r}_{2}){v}_{{\rm{i}}}^{2}},\end{eqnarray} \tag{ 36 }$$

where ${v}_{\mathrm{esc}}$ is the escape velocity when the GMC peripheries touch each other. Θ is the so called gravitational focusing factor, which characterizes the relative increment of cross section due to gravitational attraction.

The condition where the gravitational focusing effect outnumbers the cross section would be ${\rm{\Theta }}\gtrsim 1$. This condition with our ordinal assumption that GMCs have constant column density Σ and ${v}_{{\rm{i}}}=10\,\mathrm{km}\,{{\rm{s}}}^{-1}$ leads to

$$\begin{eqnarray}\displaystyle \frac{{m}_{1}+{m}_{2}}{{r}_{1}+{r}_{2}} & \gtrsim & 1.16\times {10}^{4}\,{M}_{\odot }\,{\mathrm{pc}}^{-1}\\ \Rightarrow \displaystyle \frac{{m}_{1}+{m}_{2}}{\sqrt{{m}_{1}}+\sqrt{{m}_{2}}} & \gtrsim & \displaystyle \frac{1.16\times {10}^{4}\,{M}_{\odot }\,{\mathrm{pc}}^{-1}}{\sqrt{\pi {\rm{\Sigma }}}}.\end{eqnarray} \tag{ 37 }$$

The left hand side of Equation (37) has its minimum when ${m}_{2}=0.17{m}_{1}$ and maximum when ${m}_{2}={m}_{1}$. The most conservative estimation can be obtained by employing the maximum case ${m}_{2}={m}_{1}$, which results in ${m}_{1}\gtrsim 5\times {10}^{6}\,{M}_{\odot }$. This lower boundary may increase by another factor 3 to 5 because other collisions with ${m}_{2}\lt {m}_{1}$ generally exist and also ${v}_{\mathrm{esc}}\gtrsim { \mathcal O }(10){v}_{{\rm{i}}}$ is required for CCC to overwhelm GMC self-growth/dispersal. Therefore, we can conclude that the gravitational focusing effect may be involved only in the mass range beyond ${10}^{7}\,{M}_{\odot }$. As shown in Sections 4 and after that, all the GMCMFs have already exhibited their well-defined slope below ${10}^{7}\,{M}_{\odot }$ thus treating the correction factor ${c}_{\mathrm{col}}=1$ is a decent approximation to compute the GMCMF slope correctly.

Note that our fiducial cross section is given as the sum of geometrical cross sections (i.e., $\pi {r}_{1}^{2}+\pi {r}_{2}^{2}$) rather than $\pi {({r}_{1}+{r}_{2})}^{2}$ that we studied here. However, the former is smaller than the latter by only a factor one to two, thus the criterion ${m}_{1}\gtrsim 5\times {10}^{6}\,{M}_{\odot }$ remains valid in our setup. Overall, the gravitational focusing effect can increase the cross section by some factors but we can still assume ${c}_{\mathrm{col}}=1$ for the mass range defining the GMCMF slope.

The relative velocity between clouds is another component that may vary the collisional kernel. Because our model assumes that GMCs are swept up by expanding shells (e.g., H ii regions and supernova remnants), we expect that the relative velocity between GMCs is comparable to a typical shock expanding speed driven by the ionization–dissociation front (Hosokawa & Inutsuka 2006) and does not have any strong mass dependence. Even a simple setup can derive this estimation; the Rankine–Hugoniot relation (see Landau & Lifshitz 1959), for example, predicts that velocity change across the shock front is described as ${v}_{2}={v}_{1}\times \{{(\gamma -1){{ \mathcal M }}_{1}^{2}+2\}\times ((\gamma +1){{ \mathcal M }}_{1}^{2})}^{-1}$, where v₁ and v₂ are the fluid speed in the pre-shock and post-shock regions respectively, ${{ \mathcal M }}_{1}$ is the Mach number in the pre-shock region, and γ is the polytropic index for the fluid. With a strong shock ${ \mathcal M }\gg 1$, the cooling time can be longer than the dynamical time for the shock passing through a GMC, which leads to $\gamma \gt 1$ rather than $\gamma =1$ even for a molecular cloud. Then the velocity change becomes ${v}_{2}={ \mathcal O }(0.1)\,{v}_{1}$ thus GMCs in the post-shock regions may have a relative velocity ${ \mathcal O }(0.1)\times 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Semi-analytic studies also demonstrate that a molecular cloud moves slower than the shock itself (Iwasaki et al. 2011a, 2011b), thus the GMC relative velocity would be somewhat smaller than $10\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Indeed, observations show that the mean intercloud dispersion in the Galaxy is a few $\mathrm{km}\,{{\rm{s}}}^{-1}$ (Stark & Brand 1989; Stark & Lee 2005, 2006). Therefore, we opt to use ${V}_{\mathrm{rel},0}=10\,\mathrm{km}\,{{\rm{s}}}^{-1}$ irrespective of GMC mass, which may reduce ${c}_{\mathrm{col}}$ by a factor of a few. This compensates the increasing trend discussed in the previous sections so that ${c}_{\mathrm{col}}$ remains on the order of unity.

Note that the shear motion in the galactic disk would not outnumber the velocity dispersion produced by shocks. For example, suppose that two clouds at R in galactocentric coordinates have a separation d and the speed of the galactic rotation is ${V}_{\mathrm{rot}}$, then the shear speed can be estimated about $\sim {V}_{\mathrm{rot}}{{dR}}^{-1}$. With ${V}_{\mathrm{rot}}=200\,\mathrm{km}\,{{\rm{s}}}^{-1}$, $d=100\,\mathrm{pc}$, and $R\,=8\mathrm{kpc}$ (e.g., solar neighbors), the shear speed becomes $2.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$. At galactic centers with smaller R, the shear motion definitely contributes more to the GMC relative velocity, but the significance is non-trivial because GMCs are closer to each other than disk regions so they may have smaller d. Because we now focus on disk regions, the shear motion does not alter the relative velocity appreciably in the conditions we present in the current article.