Evolutionary Description of Giant Molecular Cloud Mass Functions on Galactic Disks

TL;DR

This study develops an evolution model for giant molecular cloud (GMC) mass functions in galactic disks, incorporating cloud collisions and gas resurrection, to explain observed variations and predict mass function slopes based on GMC lifecycle timescales.

Contribution

The paper introduces a new evolution equation for GMC mass functions that includes cloud-cloud collisions and gas resurrection, advancing understanding of GMC dynamics across galactic environments.

Findings

GMC mass function slope depends on formation and dispersal timescales.

Cloud-cloud collisions mainly affect the high-mass end of the GMC mass function.

GMC mass functions follow a single power-law below 10^5.5 solar masses.

Abstract

Recent radio observations show that the giant molecular cloud (GMC) mass functions noticeably vary across galactic disks. High-resolution magnetohydrodynamics simulations show that multiple episodes of compression are required for creating a molecular cloud in the magnetized interstellar medium. In this article, we formulate the evolution equation for the GMC mass function to reproduce the observed profiles, for which multiple compression are driven by the network of expanding shells due to HII regions and supernova remnants. We introduce the cloud-cloud collision (CCC) terms in the evolution equation in contrast to the previous work (Inutsuka et al. 2015). The computed time evolution suggests that the GMC mass function slope is governed by the ratio of GMC formation timescale to its dispersal timescale, and that the CCC effect is limited only in the massive-end of the mass function. In addition, we identify a gas resurrection channel that allows the gas dispersed by massive stars to regenerate GMC populations or to accrete onto the pre-existing GMCs. Our results show that almost all of the dispersed gas contribute to the mass growth of pre-existing GMCs in arm regions whereas less than 60 per cent in inter-arm regions. Our results also predict that GMC mass functions have a single power-law exponent in the mass range < 10^5.5 Msun (where Msun represents the solar mass), which is well characterized by GMC self-growth and dispersal timescales. Measurement of the GMC mass function slope provides a powerful method to constrain those GMC timescales and the gas resurrecting factor in various environment across galactic disks.

Tables

Table 1. Table 1: Studied setups

Cases	Parameters
	$T_{f,\mathrm{fid}}$	$T_d$	Cloud-Cloud Collisions	${\epsilon }_{\mathrm{res}}$	Figures
	$($ Myr $)$	$($ Myr $)$
1	10	14	yes	support	Fig. 1
2	10	14	no	support	Fig. 2
3	4.2	14	yes	support	Fig. 4
4	22.4	14	yes	support	Fig. 5
5	10	14	yes	1	Fig. 6
6	10	14	yes	0.09	Fig. 7
7	4.2	14	yes	0.0123	Fig. 9
8	22.4	14	yes	0.45	Fig. 11

Equations

$$\frac{\partial n_{\mathrm{cl}}}{\partial t}+$$

$$=$$

$$+\frac{1}{2}{\int }_0^{\infty }{\int }_0^{\infty }K(m_1,m_2)n_{\mathrm{cl},1}{n}_{\mathrm{cl},2}$$

$$\times \delta (m-m_1-m_2)\mathrm{d}{m}_1\mathrm{d}{m}_2$$

$$-{\int }_0^{\infty }K(m,m_2)n_{\mathrm{cl}}{n}_{\mathrm{cl},2}\mathrm{d}{m}_2,$$

$${\left(\frac{\mathrm{d}m}{\mathrm{d}t}\right)}_{\mathrm{self}}=\frac{m}{T_{\mathrm{sg}}}.$$

$${\left(\frac{\mathrm{d}m}{\mathrm{d}t}\right)}_{\mathrm{self}}=\frac{m}{T_{\mathrm{f}}}.$$

$$T_{\mathrm{f}}(m)=T_{\mathrm{f},\mathrm{fid}}{\left(1+\frac{m}{m_{\mathrm{trunc}}}\right)}^{\gamma }.$$

$$T_{\mathrm{d}}=T_{\ast }+T_{\mathrm{dest}},$$

$$K(m_1,m_2)={\sigma }_{\mathrm{col}1,2}{V}_{\mathrm{rel}}=c_{\mathrm{col}}\frac{m_1+m_2}{{\Sigma }_{\mathrm{mol}}}{V}_{\mathrm{rel},0}.$$

$$T_{\mathrm{col}}=\frac{n_{\mathrm{cl}}(m)}{{\left(\partial n_{\mathrm{cl}}/\partial t\right)}_{\mathrm{CCC}}},$$

$${\left(\frac{\partial n_{\mathrm{cl}}}{\partial t}\right)}_{\mathrm{CCC}}=$$

$$\times \delta (m-m_1-m_2)\mathrm{d}{m}_1\mathrm{d}{m}_2$$

$$-{\int }_0^{\infty }K(m,m_2)n_{\mathrm{cl}}{n}_{\mathrm{cl},2}\mathrm{d}{m}_2.$$

$$\frac{\partial n_{\mathrm{cl}}}{\partial t}+\frac{\partial }{\partial m}\left(n_{\mathrm{cl}}\frac{m}{T_{\mathrm{f}}}\right)=-\frac{n_{\mathrm{cl}}}{T_{\mathrm{d}}}.$$

$$n_{\rm cl}(m)=n_{0}\left(\frac{m}{\mbox{${\rm M_{\odot}}$}}\right)^{-1-\frac{T_{\rm f}}{T_{\rm d}}}\,,$$

$${\frac{\partial \left(n_{\mathrm{cl}}m\right)}{\partial t}|}_{\mathrm{res}}={\epsilon }_{\mathrm{res}}{\dot{M}}_{\mathrm{total},\mathrm{disp}}\delta (m-m_{\min }),$$

$$\frac{\partial m{n}_{\mathrm{cl}}}{\partial t}+\frac{\partial }{\partial m}F(m)=0.$$

$$F(m)=\frac{m^2{n}_{\mathrm{cl}}}{T_{\mathrm{f}}(m)}-{\int }_{m_{\min }}^m\frac{m{n}_{\mathrm{cl}}\mathrm{d}m}{T_{\mathrm{f}}(m)}+{\int }_{m_{\min }}^m\frac{m{n}_{\mathrm{cl}}\mathrm{d}m}{T_{\mathrm{d}}},$$

$$F(m)$$

$$+A{m}_{\min }^{2-\alpha }\left(\frac{1}{(2-\alpha )T_{\mathrm{f}}}-\frac{1}{(2-\alpha )T_{\mathrm{d}}}\right).$$

$$\alpha =1+\frac{T_{\mathrm{f}}}{T_{\mathrm{d}}}.$$

$$F_{\mathrm{in}}$$

$$F_{\mathrm{out}}$$

$${\dot{M}}_{\mathrm{grow}}$$

$${\dot{M}}_{\mathrm{disp}}$$

$$F_{\mathrm{in}}=F_{\mathrm{out}}-{\dot{M}}_{\mathrm{grow}}+{\dot{M}}_{\mathrm{disp}}.$$

$${\epsilon }_{\mathrm{res}}=\frac{F_{\mathrm{in}}}{F_{\mathrm{out}}+{\dot{M}}_{\mathrm{disp}}}.$$

$${\epsilon }_{\mathrm{res},\mathrm{std}}$$

$$=\frac{F_{\mathrm{in}}}{F_{\mathrm{in}}+{\dot{M}}_{\mathrm{grow}}}$$

$$=\{\begin{array}{ll}& {\left\{1+\frac{1}{2-\alpha }\left[{\left(\frac{m_{\max }}{m_{\min }}\right)}^{2-\alpha }-1\right]\right\}}^{-1}(\text{for α=2}),\\ & {\left[1+\ln (m_{\max }/m_{\min })\right]}^{-1}(\text{for α=2}).\end{array}$$

$${\epsilon }_{\mathrm{res},\mathrm{std}}$$

$$\frac{d{M}_{\mathrm{total}}}{dt}$$

$$M_{\mathrm{total}}={\int }_{m_{\min }}^{m_{\max }}m{n}_{\mathrm{cl}}\mathrm{d}m,$$

$${\dot{M}}_{\mathrm{grow}}=-\left(1-\frac{1}{{\epsilon }_{\mathrm{res},\mathrm{std}}}\right)F_{\mathrm{in}}.$$

$$\frac{d{M}_{\mathrm{total}}}{dt}=\left(\frac{1}{{\epsilon }_{\mathrm{res},\mathrm{std}}}-\frac{1}{{\epsilon }_{\mathrm{res}}}\right)F_{\mathrm{in}}.$$

Full text

ISM: bubbles, ISM: clouds, (ISM:) HII regions, ISM: magnetic fields, ISM: structure, Galaxy: evolution

1 INTRODUCTION

Giant molecular clouds (hereafter GMCs) are massive and cold molecular gas reservoir for star formation (\gtrsim 10^{4}\mbox{{\rm M_{\odot}}} and $10-100$ pc; see Williams et al. 2000; Kennicutt & Evans 2012) and are thus essential to study star formation and subsequent galaxy evolution. Especially, GMC properties can play a pivotal role in governing star formation and eventually the evolution of galaxies; GMC distribution, density, mass function, etc. differ between galactic environment (e.g. bulge/arm/inter-arm regions, galactic star formation rates, galaxy morphologies, redshifts, etc.; see Colombo et al. 2014a; Utomo et al. 2015; Tosaki et al. 2016; c.f. Tacconi et al. 2010 for giant molecular clumps at a higher redshift). Therefore, complete understanding of galaxy evolution requires proper model of the GMC formation and subsequent star formation in different galactic environment. To investigate the life cycle of GMCs (e.g., their formation process, lifetime, star formation within GMCs, etc.), various physics need to be understood (e.g. self gravity, magnetic fields, cosmic rays, and so on). Recent theories and observations in the Galaxy advance our knowledge that the filamentary structure of dense molecular gas is important to understand the star formation within individual GMCs (Inutsuka 2001; André et al. 2010, 2011; Roy et al. 2015). The study of the GMC evolution is expected to be further proceeded based on this filamentary paradigm.

Provided that star formation within GMCs is actively investigated (such as for the filament paradigm mentioned above), the GMC mass function (GMCMF) becomes the key to understand the star formation on galactic scales because the star formation rate diversity across galactic disks may simply originate in the diversity of GMC populations on galactic scales if the star formation efficiency in individual GMCs is universal, as in the local star forming clouds (c.f., Izumi et al. in prep). The GMCMF is actively studied in the solar neighborhood (e.g., Yonekura et al. 1997; Williams & McKee 1997; Kramer et al. 1998; also see the review by Heyer & Dame 2015). On the other hand, such statistical study was difficult to conduct in external galaxies because of two reasons: the difficulties to identify individual GMCs in distant galaxies, and the enormous exposure time required to detect weak molecular line emissions from low mass GMCs \lesssim 10^{6}\mbox{{\rm M_{\odot}}}.

Recently, however, large radio observations with exquisite resolution started to map the whole disks of nearby galaxies in detail and to shed lights on GMC’s statistical properties on galactic scales (e.g. Engargiola et al. 2003; Rosolowsky et al. 2003, 2007; Koda et al. 2009, 2011, 2012; Schinnerer et al. 2013; Colombo et al. 2014a, b). Especially, the Plateau de Bure Interferometer(PdBI) Arcsecond Whirlpool Survey (PAWS) program (Schinnerer et al. 2013) demonstrate that the GMCMF varies on galactic scales: shallow slopes in arm and central bar regions whereas steep slopes in inter-arm regions (Colombo et al. 2014a), which indicates that massive GMCs are less likely to be formed in inter-arm regions. There are also some reports on the GMCMF variation along the galactocentric radii (e.g., in the Galaxy (Rice et al. 2016) and in M33 (Rosolowsky et al. 2003, 2007); c.f. the outer Galaxy (Heyer et al. 2001), LMC(Fukui et al. 2001), and M83 (Hirota et al. in prep.)) Presumably, this type of statistical GMC studies will proceed further down to both smaller mass scales \lesssim 10^{4}\mbox{{\rm M_{\odot}}} and smaller spatial scales thanks to ongoing latest observations (e.g., in Atacama Large Millimeter/Submillimeter Array (ALMA); c.f., Tosaki et al. 2016).

On the theoretical side, supersonic shock compression is one of the key processes that forms molecular clouds. The interstellar medium (ISM) has thermally bistable atomic phases where radiative cooling balance photo-electric heating and partially cosmic ray heating (Field et al. 1969; Wolfire et al. 1995, 2003); one phase is warm neutral medium (WNM), which is a diffuse Hi gas with the number density $\sim 0.1\,\mathrm{cm^{-3}}$ and the temperature $\sim 6000$ K, and the other phase is cold neutral medium (CNM), which is an Hi cloud with the number density $100\,\mathrm{cm^{-3}}$ and the temperature $\sim 100$ K. Supersonic shock causes the transition between these two phases (e.g., Hennebelle & Pérault 1999, 2000; Koyama & Inutsuka 2000).

Previous studies with hydrodynamics simulations investigated the CNM formation due to the thermal instability in between two colliding WNM flows as a precursor of molecular clouds (e.g. Walder & Folini 1998a, b; Koyama & Inutsuka 2002; Audit & Hennebelle 2005, 2008; Heitsch et al. 2005, 2006; Vázquez-Semadeni et al. 2006; Hennebelle & Audit 2007; Hennebelle et al. 2007). However, recent magnetohydrodynamics simulations reveal that the molecular cloud formation is significantly retarded in the magnetized ISM (e.g. Inoue & Inutsuka 2008, 2009; Heitsch et al. 2009; Inoue & Inutsuka 2012; Körtgen & Banerjee 2015; Valdivia et al. 2016). Inoue & Inutsuka (2008, 2009, 2012) demonstrate that such prevention occurs unless supersonic shock propagates along the magnetic fields. In the ISM however, the shocks may come from various directions and do not necessarily propagate along the magnetic filed. Therefore, it is expected that the shock propagation along the magnetic field occurs on average once out of a few 10 times shocks. This suggests that multiple episodes of supersonic WNM compression is essential for successful molecular cloud formation.

Kwan (1979); Scoville & Hersh (1979); Tomisaka (1986) studied the models of GMC growth, in which the coagulation due to cloud-cloud collision (hereafter CCC) alone governs GMC growth, and the authors do not take into account Hi cloud accumulation expected with magnetic fields. Therefore, the resultant GMCMF slope may be determined merely by the dependence of the CCC rate on GMC mass, so that they do not necessarily correspond to the observed ones (see Section 6.2).

Recently, Inutsuka et al. (2015) have proposed a new scenario of GMC formation and evolution on galactic scales, which is driven by a network of expanding Hii regions and supernovae. Inutsuka et al. (2015) formulate a continuity equation to describe GMC self-growth through multiple episodes of WNM compression and GMC self-dispersal by massive stars that are born in those GMCs. Their formulation suggests that the observed GMCMF slope variation may originate in the variation of GMC self-growth/dispersal timescales (see also Sections 2.2 and 4.2).

The bubble paradigm proposed by Inutsuka et al. (2015) inherently include CCC as the collision between GMCs on the surface of neighboring expanding shells. The authors mentioned the possible CCC contribution to the GMCMF evolution. However, their formulation does not include CCC. There are claims that CCC possibly drives the most of massive star formation within the Galaxy (c.f. Tan 2000; Nakamura et al. 2012; Fukui et al. 2014; Torii et al. 2015; Fukui et al. 2015b, 2016). Therefore, the GMCMF time evolution may also be modified by CCC. In addition, Inutsuka et al. (2015) do not consider how the dispersed gas are recycled into the ISM, pre-existing GMCs, and newer generation of GMCs.

To evaluate the combining contribution both from multiple episodes of supersonic compression and CCC, we, in this article, first evaluate the supersonic compression as in Inutsuka et al. (2015) and additionally analyze the CCC effect. Based on this formulation, we compute the time evolution of the GMCMF and compare with observations, which indicate the possibility for future large radio surveys to put unique constraints on relevant GMC timescales on galactic scales. We also additionally introduce a gas resurrection channel and suggest the importance of gas resurrecting processes regulating the GMC evolution.

This article is organized as follows. In Section 2, we review recent radio observational studies and Inutsuka et al. (2015) model in more detail to endorse our formulation of evolution equation, which is presented in Section 3. We will compute the time-integration of the evolution equation and present the GMCMF time evolution without CCC and with CCC respectively (Section 4). In Section 5, we will also further focus on the fate of dispersed gas to (1) discover the relation between the GMCMF slope and the amount of resurrecting gas and (2) insist the importance of gas resurrecting processes. In Section 6, we (1) give the possible explanation that reconciles our results and the CCC importance indicated by recent radio observations and (2) explore the other possible mechanisms to reproduce the observed GMCMF slopes. Finally, we summarize our study in Section 7. The further explanation of our modeling is reported in Appendices.

2 GMC: observations and modeling

To add detailed information to Section 1, we summarize two previous radio surveys (PAWS and Nobeyama Radio Observatory (NRO)) and review the GMC formation scenario which we are going to be based on extensively throughout this article.

2.1 Observed mass functions

The PAWS program was conducted by using PdBI and IRAM 30 m telescope and observed the disk of the Whirlpool Galaxy M51 in 12CO(1-0) line over the 200 hours integration. The survey reveals the cold gas kinematics with the $\sim 40$pc spatial resolution and detects objects whose mass are \geq 1.2\times 10^{5}\mbox{{\rm M_{\odot}}} at the $5\sigma$ level (Pety et al. 2013). The PAWS team subdivide M51 into seven regions to analyze the GMC properties in various galactic environment (see Figure 2 in Colombo et al. 2014a). One of the highlighting 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 at M51 outskirts) typically have steeper slopes (e.g., $-2.55$ in the “downstream” region). Here the slope means $-\alpha$ in $n_{\rm cl}\propto m^{-\alpha}$ where $n_{\rm cl}$ is the differential number density of GMC and $m$ represents the GMC mass. In this article, we aim to reproduce such variation based on GMC scale physics.

Rosolowsky et al. (2007) observed the CO(1-0) line emission from Galaxy M33 using the NRO 45m telescope combined with the data from the BIMA interferometer and the FCRAO 14m telescope. The arm/inter-arm comparison shows limited differences in the 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}\,\mbox{{\rm 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.

2.2 GMC formation and evolution driven by the network of expanding shells

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 SI15 scenario, the expanding bubbles  correspond to expanding Hii regions and the late phase of supernova remnants, and dense Hi shell is formed on their surface as they expand. Molecular clouds are produced at specific regions of the ISM that experiences 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-folds: 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 the further explanation).

They estimate the formation timescale in the following manner. The successful molecular cloud formation is limited when a supersonic shock propagates in 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 $T_{\rm exp}$ is somewhat smaller than 1 Myr because Hii regions also create such shocks. Overall, the typical timescale required to produce molecular clouds from WNM is given as $T_{\rm exp}/0.03\sim 10$ Myr, which we opt to use our fiducial formation timescale in this article (see Section 3.1).

3 COAGULATION EQUATION

Based on SI15 scenario described in Section 2.2, we now introduce our formulation including the CCC term to compute the time evolution of the GMCMF. The evolution of differential number density of GMCs with mass $m$, $n_{\rm cl}$, is given as:

[TABLE]

where $({\rm d}m/{\rm d}t)_{\rm self}$ is the mass gain rate of GMCs due to their self-growth, $T_{\rm d}$ is the timescale of GMC self-dispersal, $n_{{\rm cl},1}$ and $n_{{\rm cl},2}$ are the differential number densities of GMCs with mass $m_{1}$ and $m_{2}$ respectively, $K(m_{1},m_{2})$ is the kernel function on the collision between GMCs with $m_{1}$ and $m_{2}$, and $\delta$ is the Dirac delta function.

For each term in Equation (1), we give ample description in the following subsections. We also discuss the detailed variation from this fiducial formulation in Appendix A.

3.1 Self-growth term

The second term on the left hand side of Equation (1) represents the 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 the mass conservation in configuration space, whereas our equation describes GMC number conservation in GMC mass space.

We consider that $({\rm d}m/{\rm d}t)_{\rm self}$, the GMC self-growth speed, is determined by the multiple Hi 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 $10^{22}\,{\rm cm^{-2}}$: c.f. Onishi et al. 1999; Tachihara et al. 2000); therefore, we can assume that GMCs have rather pancake shape than perfect spherical structure, which suggests that the GMCs’ surface area is roughly proportional to their mass. In addition, the amount of Hi cloud accumulated onto pre-existing GMCs through the multiple episodes of compression is presumably proportional to the GMC’s surface area. Altogether, $({\rm d}m/{\rm d}t)_{\rm self}$ should be proportional to GMC mass divided by a typical self-growth timescale that is independent of mass:

[TABLE]

where $T_{\rm sg}$ is a typical timescale for the GMC self-growth.

In 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 $T_{\rm sg}$ is comparable to the typical GMC formation timescale, $T_{\rm f}$, which is estimated as a few 10 Myrs as discussed in Section 2.2 (c.f. Inoue & Inutsuka 2012; Inutsuka et al. 2015). The resultant $({\rm d}m/{\rm d}t)_{\rm self}$ becomes:

[TABLE]

Equation (3) with a constant $T_{\rm f}=10$ Myr indicates that GMCs grow exponentially in mass. Given the minimum mass for GMCs ($m_{\rm min}$) are \sim 10^{4}\mbox{{\rm M_{\odot}}} (see Appendix D), GMCs require at least 100 Myr for the exponential growth up to 10^{6.5}\mbox{{\rm M_{\odot}}} (see also Appendix C). With additional $14$ Myr required for destroying GMCs due to star formation (see Section 3.2), we expect that observed massive GMCs \sim 10^{6.5}\mbox{{\rm M_{\odot}}} typically have their ages $\gtrsim 114$ Myr. $114$ Myr is almost comparable or larger than a typical timescale for the half galactic rotation $\sim 100$ Myr. Therefore, this $114$ Myr indicates that massive GMCs \sim 10^{6.5}\mbox{{\rm M_{\odot}}} in inter-arm regions are not directly formed “in-situ” in inter-arm environment; they may be remnants that were originally born in arm environment and survived the destructing processes (e.g., by stellar feedback and galactic shear). Modeling such transition from arm environment into 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 the environmental differences. Note that $114$ Myr is the “age” of large GMCs \sim 10^{6.5}\mbox{{\rm 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 GMC 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 saturates, 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  10^{8}\mbox{{\rm 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 $m_{\rm trunc}$ as:

[TABLE]

Here the subscript ${\rm fid}$ stands for the fiducial constant value (i.e., $T_{\rm f,fid}=10\ \mathrm{Myr}$) and the exponent $\gamma$ determines the gas-deficient efficiency. The Taylor series expansion of Equation (4) gives $T_{\rm f}(m)\simeq T_{\rm f,fid}(1+\gamma m/m_{\rm trunc})$. Therefore, when GMCs grow up to $m_{\rm crit}\sim m_{\rm trunc}/\gamma$, $T_{\rm f}$ deviates longer than $T_{\rm f,fid}$ so that the choice of $m_{\rm trunc}$ and $\gamma$ modify the massive end of the GMCMF. Essentially, $m_{\rm crit}$ represents the typical maximum GMC mass that can be created in SI15 scenario. In this scenario, GMCs are created from interstellar medium swept up by supersonic shock compression, thus it is less likely to form GMCs 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 Hi density $10\,\mathrm{cm^{-3}}$ is about 7.7\times 10^{5}\mbox{{\rm M_{\odot}}}. Thus m_{\rm crit}=7.7\times 10^{5}\mbox{{\rm M_{\odot}}} and this gives m_{\rm trunc}=7.7\times 10^{6}\mbox{{\rm M_{\odot}}} given that we opt to use $\gamma=10$.

The detailed modeling of $m_{\rm trunc}$ and $\gamma$ change 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 would not largely impact if we focus on the GMCMF slope (see Section 4.1). Therefore, we will reserve the 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_{\rm min}=10^{4}\mbox{{\rm 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).