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EVOLUTIONARYDESCRIPTIONOFGIANTMOLECULARCLOUDMASSFUNCTIONSONGALACTICDISKS
MASATOI.N.KOBAYASHI1,SHU-ICHIROINUTSUKA1,HIROSHIKOBAYASHI1,ANDKENJIHASEGAWA1.
1DivisionofParticleandAstrophysicalScience,GraduateSchoolofScience,NagoyaUniversity,Aichi464-8602,Japan
7
1 ABSTRACT
0
Recentradioobservationsshowthatthegiantmolecularcloud(GMC)massfunctionsnoticeablyvaryacross
2
galactic disks. High-resolution magnetohydrodynamics simulations show that multiple episodes of compres-
n
sion are required for creating a molecular cloud in the magnetized interstellar medium. In this article, we
a
J formulate the evolution equation for the GMC mass function to reproduce the observed profiles, for which
6 multiple compression are driven by the network of expanding shells due to HII regions and supernova rem-
1 nants. Weintroducethecloud-cloudcollision(CCC)termsintheevolutionequationincontrasttotheprevious
work(Inutsukaetal.2015). ThecomputedtimeevolutionsuggeststhattheGMCmassfunctionslopeisgov-
]
A erned by the ratio of GMC formation timescale to its dispersal timescale, and that the CCC effect is limited
onlyinthemassive-endofthemassfunction. Inaddition, weidentifyagasresurrectionchannelthatallows
G
thegasdispersedbymassivestarstoregenerateGMCpopulationsortoaccreteontothepre-existingGMCs.
.
h Our results show that almost all of the dispersed gas contribute to the mass growth of pre-existing GMCs in
p
arm regions whereas less than 60% in inter-arm regions. Our results also predict that GMC mass functions
-
o haveasinglepower-lawexponentinthemassrange<105.5M (whereM representsthesolarmass),which
(cid:12) (cid:12)
r iswellcharacterizedbyGMCself-growthanddispersaltimescales. MeasurementoftheGMCmassfunction
t
s
slopeprovidesapowerfulmethodtoconstrainthoseGMCtimescalesandthegasresurrectingfactorinvarious
a
[ environmentacrossgalacticdisks.
Keywords: ISM: bubbles, ISM: clouds, (ISM:) HII regions, ISM: magnetic fields, ISM: structure, Galaxy:
2
v evolution
1
8
7 1. INTRODUCTION ofdensemoleculargasisimportanttounderstandthestarfor-
3 mationwithinindividualGMCs(Inutsuka2001;Andre´etal.
Giantmolecularclouds(hereafterGMCs)aremassiveand
0 cold molecular gas reservoir for star formation ((cid:38) 104M 2010, 2011; Roy et al. 2015). The study of the GMC evo-
. (cid:12)
1 and 10 − 100 pc; see Williams et al. 2000; Kennicutt & lutionisexpectedtobefurtherproceededbasedonthisfila-
0 mentaryparadigm.
Evans 2012) and are thus essential to study star formation
7
Provided that star formation within GMCs is actively in-
1 and subsequent galaxy evolution. Especially, GMC proper-
vestigated (such as for the filament paradigm mentioned
: ties can play a pivotal role in governing star formation and
v
above), the GMC mass function (GMCMF) becomes the
i eventuallytheevolutionofgalaxies;GMCdistribution,den-
X key to understand the star formation on galactic scales be-
sity,massfunction,etc.differbetweengalacticenvironment
r (e.g. bulge/arm/inter-arm regions, galactic star formation cause the star formation rate diversity across galactic disks
a
may simply originate in the diversity of GMC populations
rates,galaxymorphologies,redshifts,etc.;seeColomboetal.
ongalacticscalesifthestarformationefficiencyinindivid-
2014a; Utomo et al. 2015; Tosaki et al. 2016; c.f. Tacconi
ual GMCs is universal, as in the local star forming clouds
et al. 2010 for giant molecular clumps at a higher redshift).
(c.f.,Izumietal.inprep).TheGMCMFisactivelystudiedin
Therefore, complete understanding of galaxy evolution re-
thesolarneighborhood(e.g.,Yonekuraetal.1997;Williams
quires proper model of the GMC formation and subsequent
& McKee 1997; Kramer et al. 1998; also see the review by
star formation in different galactic environment. To inves-
Heyer & Dame 2015). On the other hand, such statistical
tigate the life cycle of GMCs (e.g., their formation process,
studywasdifficulttoconductinexternalgalaxiesbecauseof
lifetime, starformationwithinGMCs, etc.), variousphysics
two reasons: the difficulties to identify individual GMCs in
needtobeunderstood(e.g. selfgravity,magneticfields,cos-
distantgalaxies,andtheenormousexposuretimerequiredto
micrays,andsoon). Recenttheoriesandobservationsinthe
detectweakmolecularlineemissionsfromlowmassGMCs
Galaxyadvanceourknowledgethatthefilamentarystructure
(cid:46)106M .
(cid:12)
Recently,however,largeradioobservationswithexquisite
resolutionstartedtomapthewholedisksofnearbygalaxies
[email protected]
2 M.I.N.KOBAYASHIETAL.
in detail and to shed lights on GMC’s statistical properties studiedthemodelsofGMCgrowth,inwhichthecoagulation
on galactic scales (e.g. Engargiola et al. 2003; Rosolowsky due to cloud-cloud collision (hereafter CCC) alone governs
et al. 2003, 2007; Koda et al. 2009, 2011, 2012; Schinnerer GMC growth, and the authors do not take into account HI
etal.2013;Colomboetal.2014a,b). Especially,thePlateau cloud accumulation expected with magnetic fields. There-
de Bure Interferometer(PdBI) Arcsecond Whirlpool Survey fore,theresultantGMCMFslopemaybedeterminedmerely
(PAWS) program (Schinnerer et al. 2013) demonstrate that by the dependence of the CCC rate on GMC mass, so that
the GMCMF varies on galactic scales: shallow slopes in theydonotnecessarilycorrespondtotheobservedones(see
armandcentralbarregionswhereassteepslopesininter-arm Section6.2).
regions (Colombo et al. 2014a), which indicates that mas- Recently,Inutsukaetal.(2015)haveproposedanewsce-
siveGMCsarelesslikelytobeformedininter-armregions. nario of GMC formation and evolution on galactic scales,
TherearealsosomereportsontheGMCMFvariationalong which is driven by a network of expanding HII regions and
thegalactocentricradii(e.g.,intheGalaxy(Riceetal.2016) supernovae. Inutsuka et al. (2015) formulate a continu-
and in M33 (Rosolowsky et al. 2003, 2007); c.f. the outer ity equation to describe GMC self-growth through multiple
Galaxy (Heyer et al. 2001), LMC(Fukui et al. 2001), and episodes of WNM compression and GMC self-dispersal by
M83(Hirotaetal.inprep.)) Presumably,thistypeofstatis- massive stars that are born in those GMCs. Their formu-
ticalGMCstudieswillproceedfurtherdowntobothsmaller lation suggests that the observed GMCMF slope variation
mass scales (cid:46) 104M and smaller spatial scales thanks to mayoriginateinthevariationofGMCself-growth/dispersal
(cid:12)
ongoinglatestobservations(e.g.,inAtacamaLargeMillime- timescales(seealsoSections2.2and4.2).
ter/SubmillimeterArray(ALMA);c.f.,Tosakietal.2016). The bubble paradigm proposed by Inutsuka et al. (2015)
On the theoretical side, supersonic shock compression inherently include CCC as the collision between GMCs on
is one of the key processes that forms molecular clouds. the surface of neighboring expanding shells. The authors
Theinterstellarmedium(ISM)hasthermallybistableatomic mentioned the possible CCC contribution to the GMCMF
phases where radiative cooling balance photo-electric heat- evolution.However,theirformulationdoesnotincludeCCC.
ing and partially cosmic ray heating (Field et al. 1969; ThereareclaimsthatCCCpossiblydrivesthemostofmas-
Wolfireetal.1995,2003);onephaseiswarmneutralmedium sive star formation within the Galaxy (c.f. Tan 2000; Naka-
(WNM), which is a diffuse HI gas with the number den- mura et al. 2012; Fukui et al. 2014; Torii et al. 2015; Fukui
sity ∼ 0.1cm−3 and the temperature ∼ 6000 K, and the etal.2015b,2016). Therefore, theGMCMFtimeevolution
other phase is cold neutral medium (CNM), which is an HI may also be modified by CCC. In addition, Inutsuka et al.
cloud with the number density 100cm−3 and the tempera- (2015) do not consider how the dispersed gas are recycled
ture ∼ 100 K. Supersonic shock causes the transition be- into the ISM, pre-existing GMCs, and newer generation of
tween these two phases (e.g., Hennebelle & Pe´rault 1999, GMCs.
2000;Koyama&Inutsuka2000). To evaluate the combining contribution both from multi-
Previous studies with hydrodynamics simulations investi- pleepisodesofsupersoniccompressionandCCC,we,inthis
gated the CNM formation due to the thermal instability in article, first evaluate the supersonic compression as in Inut-
betweentwocollidingWNMflowsasaprecursorofmolecu- suka et al. (2015) and additionally analyze the CCC effect.
larclouds(e.g.Walder&Folini1998a,b;Koyama&Inutsuka Basedonthisformulation,wecomputethetimeevolutionof
2002; Audit & Hennebelle 2005, 2008; Heitsch et al. 2005, theGMCMFandcomparewithobservations,whichindicate
2006; Va´zquez-Semadeni et al. 2006; Hennebelle & Audit the possibility for future large radio surveys to put unique
2007; Hennebelle et al. 2007). However, recent magneto- constraints on relevant GMC timescales on galactic scales.
hydrodynamics simulations reveal that the molecular cloud Wealsoadditionallyintroduceagasresurrectionchanneland
formation is significantly retarded in the magnetized ISM suggesttheimportanceofgasresurrectingprocessesregulat-
(e.g. Inoue & Inutsuka 2008, 2009; Heitsch et al. 2009; In- ingtheGMCevolution.
oue & Inutsuka 2012; Ko¨rtgen & Banerjee 2015; Valdivia This article is organized as follows. In Section 2, we re-
et al. 2016). Inoue & Inutsuka (2008, 2009, 2012) demon- view recent radio observational studies and Inutsuka et al.
strate that such prevention occurs unless supersonic shock (2015) model in more detail to endorse our formulation of
propagates along the magnetic fields. In the ISM however, evolutionequation,whichispresentedinSection3. Wewill
theshocksmaycomefromvariousdirectionsanddonotnec- compute the time-integration of the evolution equation and
essarily propagate along the magnetic filed. Therefore, it is present the GMCMF time evolution without CCC and with
expectedthattheshockpropagationalongthemagneticfield CCCrespectively(Section4). InSection5,wewillalsofur-
occurs on average once out of a few 10 times shocks. This ther focus on the fate of dispersed gas to (1) discover the
suggests that multiple episodes of supersonic WNM com- relation between the GMCMF slope and the amount of res-
pression is essential for successful molecular cloud forma- urrectinggasand(2)insisttheimportanceofgasresurrecting
tion. processes. InSection6,we(1)givethepossibleexplanation
Kwan(1979);Scoville&Hersh(1979);Tomisaka(1986) thatreconcilesourresultsandtheCCCimportanceindicated
TIMEEVOLUTIONOFGMCMASSFUNCTIONS 3
byrecentradioobservationsand(2)exploretheotherpossi- expand. Molecular clouds are produced at specific regions
blemechanismstoreproducetheobservedGMCMFslopes. oftheISMthatexperiencesmultipleepisodesofsupersonic
Finally, we summarize our study in Section 7. The further shocks(i.e. sweptupmultipletimesbydifferentexpanding
explanationofourmodelingisreportedinAppendices. shells) or where neighboring expanding shells are colliding
witheachother.
2. GMC:OBSERVATIONSANDMODELING
Based on this scenario, Inutsuka et al. (2015) formulate
To add detailed information to Section 1, we summarize a continuity equation that gives the time evolution of the
twopreviousradiosurveys(PAWSandNobeyamaRadioOb- GMCMF. Their formulation is two-folds: GMC forma-
servatory (NRO)) and review the GMC formation scenario tion and self-growth due to multiple episodes of compres-
which we are going to be based on extensively throughout sion and GMC self-dispersal due to star formation within
thisarticle. those GMCs. Their results suggest that the ratio of typical
timescalesforformationanddispersalprocessesdetermines
2.1. Observedmassfunctions
theslopeoftheGMCMF(seeSection4.2forthefurtherex-
The PAWS program was conducted by using PdBI and planation).
IRAM30mtelescopeandobservedthediskoftheWhirlpool They estimate the formation timescale in the following
Galaxy M51 in 12CO(1-0) line over the 200 hours integra- manner. The successful molecular cloud formation is lim-
tion. The survey reveals the cold gas kinematics with the ited when a supersonic shock propagates in an angle less
∼ 40pc spatial resolution and detects objects whose mass than 0.26 radian with respect to the magnetic field. Given
are ≥ 1.2×105M at the 5σ level (Pety et al. 2013). The that the supersonic shock arrives isotropically, the success
(cid:12)
PAWSteamsubdivideM51intosevenregionstoanalyzethe rate is about 2 × 0.262π/(4π) ∼ 0.03 per shock. Gas in
GMC properties in various galactic environment (see Fig- the ISM typically experiences supersonic shocks due to su-
ure 2 in Colombo et al. 2014a). One of the highlighting re- pernovaeevery1Myr(e.g.,McKee&Ostriker1977). Thus,
sultsistheGMCMFvariation;barandarmregionstypically the time interval between consecutive shocks T is some-
exp
haveshallowerslopes(e.g.,−1.33inthe“nuclear”barregion whatsmallerthan1MyrbecauseHIIregionsalsocreatesuch
and−1.75inthe“density-wavespiralarm”region)whereas shocks. Overall, the typical timescale required to produce
inter-arm regions (and an arm region at M51 outskirts) typ- molecular clouds from WNM is given as T /0.03 ∼ 10
exp
ically have steeper slopes (e.g., −2.55 in the “downstream” Myr,whichweopttouseourfiducialformationtimescalein
region). Heretheslopemeans−αinn ∝ m−α wheren thisarticle(seeSection3.1).
cl cl
isthedifferentialnumberdensityofGMCandmrepresents
3. COAGULATIONEQUATION
the GMC mass. In this article, we aim to reproduce such
variationbasedonGMCscalephysics. BasedonSI15scenariodescribedinSection2.2, wenow
Rosolowskyetal.(2007)observedtheCO(1-0)lineemis- introduce our formulation including the CCC term to com-
sion from Galaxy M33 using the NRO 45m telescope com- putethetimeevolutionoftheGMCMF.Theevolutionofdif-
bined with the data from the BIMA interferometer and ferentialnumberdensityofGMCswithmassm,ncl,isgiven
the FCRAO 14m telescope. The arm/inter-arm comparison as:
(cid:18) (cid:18) (cid:19) (cid:19)
shows limited differences in the slopes (−1.9 and −2.2 re- ∂n ∂ dm
cl+ n
spectively) with difficulties in defining arms. However, the ∂t ∂m cl dt
self
innermost2.1kpchasaprominentcutoffatthemassiveend n
=− cl
((cid:38) 4.5×105M(cid:12)) whereas the outer regions up to 4.1 kpc Td
do not. Such galactocentoric radial variation is beyond our 1(cid:90) ∞(cid:90) ∞ (1)
+ K(m ,m )n n
currentscopebutneedstobeinvestigatedinfuture. 2 1 2 cl,1 cl,2
0 0
×δ(m−m −m )dm dm
2.2. GMCformationandevolutiondrivenbythenetworkof 1 2 1 2
expandingshells (cid:90) ∞
− K(m,m )n n dm ,
2 cl cl,2 2
As described in Section 1, recent multiphase magnetohy- 0
drodynamics simulations suggest that multiple episodes of where(dm/dt) isthemassgainrateofGMCsduetotheir
self
compression are necessary to form molecular clouds from self-growth,T isthetimescaleofGMCself-dispersal,n
d cl,1
themagnetizedISM. Accordingtothesesimulationresults, andn arethedifferentialnumberdensitiesofGMCswith
cl,2
Inutsukaetal.(2015)proposedanewscenarioofGMCfor- massm andm respectively,K(m ,m )isthekernelfunc-
1 2 1 2
mationandevolutionongalacticscales(hereafter,SI15sce- tiononthecollisionbetweenGMCswithm andm ,andδ
1 2
nario), which is driven by the network of expanding shells. istheDiracdeltafunction.
In SI15 scenario, the expanding bubbles correspond to ex- For each term in Equation (1), we give ample description
panding HII regions and the late phase of supernova rem- in the following subsections. We also discuss the detailed
nants, and dense HI shell is formed on their surface as they variationfromthisfiducialformulationinAppendixA.
4 M.I.N.KOBAYASHIETAL.
3.1. Self-growthterm Schinnerer et al. 2017) and flocculent spiral arms (e.g., in
GalaxyM33). However,weleavethisforfuturestudiesand
ThesecondtermonthelefthandsideofEquation(1)rep-
focus on the GMCMF variation purely due to the environ-
resentstheGMCself-growth. Thistermisafluxterminthe
mental differences. Note that 114 Myr is the “age” of large
conservation law. The ordinary continuity equation in fluid
GMCs∼ 106.5M butnotthetypicalGMC“lifetime”(see
mechanicsisasimpleexampleofananalogousconservation (cid:12)
Section3.2).
law,whichconsidersthemassconservationinconfiguration
We should note here that the above formulation over-
space,whereasourequationdescribesGMCnumberconser-
estimatesthegrowthrateofverymassiveGMCwhosemass
vationinGMCmassspace.
is comparable to the mass of a shell swept up by an ex-
Weconsiderthat(dm/dt) ,theGMCself-growthspeed,
self
panding bubble. Once the GMC mass is comparable to or
is determined by the multiple HI cloud compression, which
larger than the typical mass of a swept-up shell, the growth
depends on the shape of GMCs. Observations suggest that
in mass should saturates, because the dense gas that can be
theGMCcolumndensitydoesnotvarymuchbetweenGMCs
(e.g. typicallyafewtimes1022cm−2:c.f.Onishietal.1999; usedtoformacloudislimitedbytheamountoftotalmassin
theexpandingshell. Indeed,observationshaverevealedthat
Tachiharaetal.2000);therefore,wecanassumethatGMCs
GMCsexistonlyupto 108M (e.g.,Rosolowskyetal.2007;
have rather pancake shape than perfect spherical structure, (cid:12)
Colomboetal.2014a). Formodelingsuchgasshortage,we
whichsuggeststhattheGMCs’surfaceareaisroughlypro-
modify T by applying a growing factor with a truncation
portionaltotheirmass. Inaddition, theamountofHI cloud f
massm as:
accumulated onto pre-existing GMCs through the multiple trunc
episodes of compression is presumably proportional to the (cid:18) m (cid:19)γ
GMC’ssurfacearea. Altogether,(dm/dt)self shouldbepro- Tf(m)=Tf,fid 1+ m . (4)
trunc
portional to GMC mass divided by a typical self-growth
timescalethatisindependentofmass: Here the subscript fid stands for the fiducial constant value
(cid:18)dm(cid:19) m (i.e., Tf,fid = 10 Myr) and the exponent γ determines
= . (2) the gas-deficient efficiency. The Taylor series expansion of
dt T
self sg Equation(4)givesT (m) (cid:39) T (1+γm/m ). There-
f f,fid trunc
whereTsg isatypicaltimescalefortheGMCself-growth. fore,whenGMCsgrowuptomcrit ∼mtrunc/γ,Tf deviates
InSI15scenario(thebubbleparadigm),GMCsareformed longerthanT sothatthechoiceofm andγ modify
f,fid trunc
viamulti-compressionalprocesses,whichalsocausetheself- the massive end of the GMCMF. Essentially, m repre-
crit
growthofGMCs.Therefore,inourcalculation,weadoptthat sentsthetypicalmaximumGMCmassthatcanbecreatedin
Tsg is comparable to the typical GMC formation timescale, SI15scenario.Inthisscenario,GMCsarecreatedfrominter-
Tf,whichisestimatedasafew10MyrsasdiscussedinSec- stellar medium swept up by supersonic shock compression,
tion 2.2 (c.f. Inoue & Inutsuka 2012; Inutsuka et al. 2015). thus it is less likely to form GMCs more massive than the
Theresultant(dm/dt)self becomes: total mass that a single supernova remnant can sweep. The
(cid:18)dm(cid:19) m total mass initially contained within a sphere of 100 pc ra-
dt = T . (3) dius with HI density 10cm−3 is about 7.7×105M(cid:12). Thus
self f m =7.7×105M andthisgivesm =7.7×106M
crit (cid:12) trunc (cid:12)
Equation (3) with a constant T = 10 Myr indicates that
f giventhatweopttouseγ =10.
GMCs grow exponentially in mass. Given the minimum
The detailed modeling of m and γ change the rela-
trunc
mass for GMCs (m ) are ∼ 104M (see Appendix D),
min (cid:12) tiveimportanceofGMCself-growth/dispersalandCCC. In
GMCs require at least 100 Myr for the exponential growth
addition, rapid star formation triggered by CCC also needs
up to 106.5M (see also Appendix C). With additional 14
(cid:12) to be properly modeled if CCC becomes effective (see Sec-
Myr required for destroying GMCs due to star formation
tion4.3). However,thesedetailswouldnotlargelyimpactif
(see Section 3.2), we expect that observed massive GMCs
wefocusontheGMCMFslope(seeSection4.1). Therefore,
∼ 106.5M typically have their ages (cid:38) 114 Myr. 114 Myr
(cid:12) wewillreservethedetailedinvestigationforfutureworks.
is almost comparable or larger than a typical timescale for
Note that the second term on the left hand side of Equa-
the half galactic rotation ∼ 100 Myr. Therefore, this 114 tion(1)hasitsboundaryconditionatm=m =104M .
min (cid:12)
Myr indicates that massive GMCs ∼ 106.5M in inter-arm
(cid:12) The flux at this boundary in our formulation corresponds to
regions are not directly formed “in-situ” in inter-arm envi-
the minimum-mass GMC production rate. In later sections,
ronment; they may be remnants that were originally born
wewillexplainthatthisratediffersbetweensetups(seethe
in arm environment and survived the destructing processes
firstparagraphinSection4andthesecondparagraphinSec-
(e.g.,bystellarfeedbackandgalacticshear). Modelingsuch
tion5).
transitionfromarmenvironmentintointer-armenvironment
shouldbeinvestigatedfurthertostudytheobservedspurfea-
3.2. Dispersalterm
tures extended from spiral arms (e.g., Corder et al. 2008;
TIMEEVOLUTIONOFGMCMASSFUNCTIONS 5
Table1.Studiedsetups
Cases Parameters
T T Cloud-CloudCollisions ε Figures
f,fid d res
(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
Note.8casesthatwestudied.IndividualparametersaredescribedinSection3.Eachcolumnrepresentsasfollows;T showstheformation
f,fid
timescale.T givesthedispersaltimescale.Cloud-CloudCollisionsindicatewhetherweincludetheCCCterms(yes)ornot(no).ε
d res
representstheresurrectingfactorintroducedinSection5.1(i.e.thefractionalmassoutoftotaldispersedgasthatisconsumedtoformnewer
generationofGMCs),where“support”meansthatweartificiallykeepthepopulationofminimum-massGMCsfixed.Figuresindicatethe
correspondingfiguresinthisarticle.
The first term on the right hand side gives the GMC self- adopt Salpeter initial stellar mass function (Salpeter 1955).
dispersal rate. Here, self-dispersal means that massive stars In case of GMCs < 105M , the self-dispersal may be less
(cid:12)
born within GMCs destroy their parental GMCs by any effective (see Inutsuka et al. 2015) and more careful treat-
means (ionization, dissociation, heating, blowing-out, etc.). mentisdesired. Inaddition, T canbeoneorderofmagni-
∗
Thecharacteristicself-dispersaltimescale,T ,isgivenas: tudeshorterifCCCtriggersrapidformationofmassivestars
d
(c.f.Fukuietal.2016)(seealsoSection4.3). Forsimplicity,
T =T +T , (5)
d ∗ dest however, we do not consider any T and T variation in
∗ dest
thecurrentarticle(seealsoSection4.3).
where T is the typical timescale for star formation after
∗
GMC birth and Tdest is the typical timescale for GMC de- 3.3. Cloud-CloudCollisionterms
structiononcestarsbecomemain-sequencestars.Weassume
The second and third terms on the right hand side repre-
thetypicalstarformationtimescalewithinGMCsis∼T
f,fid sentthecollisionalcoagulationbetweenGMCs(i.e.theCCC
and thus employ T = 10 Myr. T can be estimated by
∗ dest process). The second term on the right hand side produces
line-radiationmagnetohydrodynamicssimulations. Inutsuka
a GMC with mass m(= m +m ) through CCC between
1 2
et al. (2015) reported one of such simulation results, which
GMCs with their mass m and m . Similarly, the last term
1 2
updatedtheworkinHosokawa&Inutsuka(2006)byinclud-
on the right hand side creates a GMC with mass m + m
2
ing magnetic fields. Their results indicate that, once a mas-
throughCCCbetweenGMCswiththeirmassmandm ,so
sive star with mass > 30M is formed in a GMC, the star 2
(cid:12) that the negative sign indicates that this collision decreases
destroysmorethan105M surroundingmolecularhydrogen
(cid:12) the number density of GMCs with mass m. K(m ,m ) is
within4Myr. Therefore,typicalT is10+4=14Myr. 1 2
d the product of the total GMC collisional cross section be-
T can be understood as the typical GMC “lifetime” av-
d tweenGMCswithmassm andm ,σ ,andtherelative
eraged over all the populations because T is the typical 1 2 col1,2
d velocitybetweenGMCs,V :
rel
timescale within which GMCs disperse in the system. This
m +m
isconsistentwiththeshortlifetimeindicatedbyobservations K(m ,m )=σ V =c 1 2V . (6)
1 2 col1,2 rel col Σ rel,0
(e.g., in LMC (Kawamura et al. 2009) and in M51 (Meidt mol
etal.2015)). Here,c isacorrectionparameter,Σ isaGMCcolumn
col mol
T is basically independent of GMC mass because, for density, and V is the fiducial relative velocity between
dest rel,0
example, 10 times massive GMCs generate 10 times more GMCs. c reflects various effects (e.g., geometrical struc-
col
> 30M stars, which in turn results in destroying 10 times ture, gravitational focusing, relative velocity variation) and
(cid:12)
moremoleculargas(seeInutsukaetal.2015). Therefore,if is expected to be on the order of unity (see discussions in
T isirrespectiveofGMCmass,T doesnotdependonGMC Appendix A). For simplicity, we use c = 1 throughout
∗ d col
mass. Thus,weopttouseT = 14Myrthroughoutthecur- this article. For the column density and the relative veloc-
d
rent article. Note that this argument of mass independence ity, we opt to employ an observed constant value Σ =
mol
isvalidonlyifparentalGMCsare> 105M becausethisis 2×1022µm cm−2 (seeSection3.1)andatypicalexpand-
(cid:12) H
theminimummassrequiredtogeneratestars>30M ifwe ingspeedofionization-dissociationfrontV =10kms−1
(cid:12) rel,0
6 M.I.N.KOBAYASHIETAL.
(seeAppendixB.3)respectively,whereµisthemeanmolec- 0
ularweightandm istheatomichydrogenmass. Tf,fid =10.0[Myr]
H ) Td =14.0[Myr]
Our calculation focuses only on collisional coagulation ]
(cid:12)-2
and ignores fragmentation at all. This assumption is based M
/
on the observational fact that cloud-to-cloud velocity dis- 3c
persion (∼ 10kms−1: e.g., Stark & Lee 2005) does not kp -4 Time[Myr]
/ 20
largely exceed the sound velocity (∼ 10kms−1) of inter- [ 40
cloudmedium(i.e.,WNM).However,thefragmentationmay n(cl -6 6800
have a severe impact under the CCC-dominated phase (i.e., og 120
l 160
the GMC number density is high) such as in the Galactic 200
-8
Center(Tsuboietal.2015). Thisisbeyondourcurrentscope
4 5 6 7
andwewillreportinourforthcomingpaper. log(GMCMass[M ])
(cid:12)
Note that T in principle should be the ensemble average
d
ofallthedestructiveprocessesincluding,forexample,shear Figure1. Case1: Differentialnumberdensityncl asafunctionof
GMCmass,basedonthefiducialcalculationwithCCC. Thecolor
(c.f.,Kodaetal.2009).However,shearmaybesub-dominant
correspondstothetimeevolution. Notethattheyellowlinecorre-
inthediskregionstypically> 5kpc(c.f.,Dobbs&Pringle
spondsto160Myrisalmostfullyoverlappedbytheblacklinefor
2013). Forsimplicity,weassumethattheinclusionofthese 200Myr,thusitisnotvisibleonthisplot. Asareference,weplot
processeswouldnotvaryT significantly. threegraystraightlines;dot-dashedlinefitsthecomputedGMCMF
d
slope (∼ −1.7), dashed line corresponds to the observed shallow
slopewhereasthedottedlineshowstheobservedsteepslope. The
4. RESULTS1: SLOPEOFGIANTMOLECULAR
calculatedGMCMFfitsintothemiddleoftheobservedsloperange.
CLOUDMASSFUNCTION
We perform the time integration of Equation (1) to deter-
0
minewhatcontrolstheobservedGMCMF. Table1liststhe
Tf,fid =10.0[Myr]
parametersweuse;especiallyCases1to4correspondtothe ) Td =14.0[Myr]
]
analysisinthissection. Throughoutthissection,weassume (cid:12)-2
M
thatminimum-massGMCsarealwayscontinuouslycreated /
3
so that the rate producing new-born minimum-mass GMCs pc Time[Myr]
k -4
compensates the decrease in the population of minimum- / 20
[ 40
mass GMCs due to self-growth, self-dispersal, and CCC. ncl 60
Thistreatmentmakesthenumberdensityofminimum-mass og( -6 81020
GMCs always constant. This steadiness of minimum-mass l 160
200
GMCsisalsopresumedfromrelativelyconstantstarforma-
-8
tionactivityobservedinnearbygalaxies(Madau&Dickin- 4 5 6 7
log(GMCMass[M ])
son 2014). Without any sufficient creation, minimum-mass
(cid:12)
GMCs are exhausted and the observed GMCMF cannot be
reproduced. InSection5,wewilldiscusspossiblevariation Figure2. Case2: Differentialnumberdensityncl asafunctionof
GMC mass, based on the fiducial calculation without CCC. The
intherateofproducingnew-bornminimum-massGMCsby color corresponds to the time evolution. As a reference, we plot
dispersedgas“resurrection”(i.e.,regenerationofGMCpop- threegraystraightlines;dot-dashedlinefitsthecomputedGMCMF
slope (∼ −1.7), dashed line corresponds to the observed shallow
ulationsfromdispersedgasproducedbymassivestars).
slopewhereasthedottedlineshowstheobservedsteepslope. The
Wesetthetimestepwidthas0.1%oftheshortesttimescale calculatedGMCMFfitsintothemiddleoftheobservedsloperange.
in each time step. We use the logarithmically spaced mass ThesimilaritywithFigure1indicatesthatCCCdoesnotimpacton
theGMCMFslopesignificantlybutmodifythemassiveend.
gridwherethei+1-thmassislargerbyafactor1.0125than
thei-thmass. TheinitialGMCMFhasadelta-functionlike
massdistributionwhereonlyminimum-massGMCsexist,by
which we can highlight how fast the GMCMF evolves. All certainGMCpairswhenatleastoneofGMCsinthepairhas
thefigurespresentedhereshowtheGMCMFuptothemass its cumulative number less than 1 because such GMCs are
wherethecumulativenumberofGMCs> 1,exceptthatthe less likely to exist in real galaxies so that this type of CCC
GMCMFs in Figs. 1 and 2 are shown beyond that mass to isalsoexpectedtoberare. Thespikesandkinksthatappear
show the CCC effect clearly, and also except that the GM- in the massive end (especially in Figs. 1 and 5) are due to
CMFinFig.6isshownbelowthatmassbecauseGMCsmore the numerical effect by this CCC turn-off procedure so that
massivethantheobservedonesarecreatedduetoourrather actualGMCMFsintheUniversewouldbesmoother. There-
artificialchoiceofanexcessivelyCCC-dominatedcondition sults of PAWS survey (c.f., Colombo et al. 2014a) are used
andwewouldliketofocusontheslopesintheobservedmass in the plots for comparing the GMCMF slopes between our
range. Note that we turn off the CCC calculation between calculationandobservations.
TIMEEVOLUTIONOFGMCMASSFUNCTIONS 7
5 -1
Time[Myr] Tf,fid =10.0[Myr] Tf,fid =4.2[Myr]
4 4800 Td =14.0[Myr] ])(cid:12)-2 Td =14.0[Myr]
) 120 M
Myr] 3 126000 3/c -3
p Time[Myr]
T([col 2 n/[kcl -4 246000
g ( -5 80
lo og 120
1 l 160
-6 200
0 4 5 6 7
4 5 6 7 log(GMCMass[M ])
(cid:12)
log(GMCMass[M ])
(cid:12)
Figure4. Case3: Differentialnumberdensityncl asafunctionof
Figure3. ThethreetimescalesasafunctionofGMCmassinCase GMCmass,basedonthecalculationwithTf,fid =4.2Myrinclud-
1. ThegraysolidlinerepresentsT ,thegrayhorizontaldot-dashed ingCCC.Thecolorcorrespondstothetimeevolution.Notethatthe
f
linecorrespondstoT ,andthecoloredlinesshowthetimeevolu- twocoloredlines(i.e.,thepurplelinefor120Myrandtheyellow
d
tionofT .Notethatthethreecoloredlines(i.e.,thegreenlinefor linefor160Myr)areoverlappedbytheblacklinefor200Myrso
col
80Myr,theredlinefor120Myr,andthecyanlinefor160Myr)are thattheyarenotvisibleonthisplot.Asareference,weplottwogray
overlappedbythepurplelinefor200Myrsothattheyarenotvisible straightlines;dot-dashedlinecorrespondstotheobservedGMCMF
onthisplot. ThefigureindicatesthatGMCself-growthdominates slope(∼ −1.3)inthearmregions,whereasthedottedlineshows
inlow-massregime,butCCCdeformstheGMCMFatitshigh-mass the observed steep slope in inter-arm regions. The agreement be-
endwhereT isoneorderofmagnitudesmallerthanself-growth tweentheobservedslopeandthecomputedslopesuggeststhatthe
col
timescale.NotethatthegraylineofTf shownherehasafactordif- armregionstypicallyhaveashorterself-growthtimescale,Tf ∼ 4
ferencefromoriginalT definedasEquation(4)(seeAppendixE). Myr.Thisiswellcharacterizedbythesteadystatesolutionwithout
f
CCCshowninEquation(10). Again,theCCCeffectislimitedbut
modifiestheGMCMFmassive-end.
4.1. CCCcontributiontotheGMCMFslopesandmassive
ends determined by CCC at the high-mass end where Tcol (∼
O(1) Myr) becomes one order of magnitude smaller than
Figures1and2showourfiducialcalculations(i.e.,T =
f,fid T (∼ O(2) Myr). This indicates that the GMCMF slope
10 Myr, T = 14 Myr), which correspond to Case1 and f
d in m (cid:46) 105.5M is well characterized by the combination
Case2 respectively. The only difference between these two (cid:12)
of the GMC self-growth and dispersal. In the next section,
casesiswhetherthecalculationincludestheCCCterms(Fig-
we are going to focus only on the GMCMF slopes and will
ure 1) or not (Figure 2). Both computed GMCMFs demon-
discussthemassive-endbehaviorinSection4.3.
strate almost the same slope −α ∼ −1.7 (where n ∝
cl
m−α),whichsuccessfullyfitsintothemiddleoftheobserved 4.2. CharacteristicSlopeoftheGMCMF
range. This result indicates that CCC does not impact the
As shown in Section 4.1, CCC does not modify the GM-
GMCMF slope significantly whereas the shape of the mas-
CMFevolutionsignificantly. Especiallyinlowermassrange
siveendismodifiedbytherelativeimportancebetweenself-
(e.g., m < 105.5M ) CCC does not affect the mass func-
growth/dispersalandCCC. (cid:12)
tion,andhenceourformulationcanberewrittenwithoutthe
ToexaminetherelativecontributionofCCC,wecompute
CCC terms. Therefore, the evolution of differential number
thecollisiontimescale,T ,as:
col
densityofGMCswithmassm,n ,isnowsimplygivenas:
cl
n (m)
Tcol = (∂nclc/l∂t)CCC , (7) ∂∂ntcl + ∂∂m(cid:18)nclTm(cid:19)=−nTcl . (9)
f d
wherethedenominatoristhetimeevolutionduetoCCC:
Thisformulationhasbeenalready(andoriginally)proposed
(cid:18)∂n (cid:19) 1(cid:90) ∞(cid:90) ∞
cl = K(m ,m )n n by Inutsuka et al. (2015). One can obtain the steady state
∂t CCC 2 0 0 1 2 cl,1 cl,2 solution of this differential equation for m (cid:46) mcrit with a
×δ(m−m −m )dm dm constantT as:
1 2 1 2 f
−(cid:90) ∞K(m,m2)nclncl,2dm2. n (m)=n (cid:18) m (cid:19)−1−TTdf , (10)
0 cl 0 M
(8) (cid:12)
Figure 3 shows the time evolution of computed T , with where n is the differential number density normalized at
col 0
T and T overplotted together. The figure indicates that m = 1M . ThispredictsthatGMCMFshaveslopeswitha
f d (cid:12)
the GMC mass growth is determined by GMC self-growth singleexponent,whichiswellcharacterizedby−1−T /T .
f d
in low-mass regime where T (∼ O(1) Myr) is one or- Indeed, the computed GMCMF shows the slope consistent
f
der of magnitude shorter than T (∼ O(2) Myr), and is withthisexponent−1−T /T withoutCCC(seeFigure2),
col f d
8 M.I.N.KOBAYASHIETAL.
1 isbecauseT isbasicallyindependentofGMCmassasex-
d
Tf,fid =22.4[Myr] plainedinSection3.2andalsobecauseobservationssuggest
) Td =14.0[Myr]
](cid:12) thatthestarformationefficiencyisalmostthesamethrough-
M -1
outthegalacticdisksinnearbygalaxies(e.g.,Schrubaetal.
/
3
c 2011) and in the Galaxy (Izumi et al. in prep.). However,
p Time[Myr]
/k -3 20 Td mightbelongerininter-armregionscomparedwitharm
[ 40
ncl 60 regionsduetoitslessstarformingactivitywhereTd canbe
og( -5 81020 upto∼ 30Myr(theupperlimitmeasuredinM51byMeidt
l 160 et al. 2015), or might be a factor longer in smaller clouds
200
(cid:46) 105M whereitisinvalidtoapplyourmass-independent
(cid:12)
4 5 6
assumptionontheclouddestructionrateduetomassivestars.
log(GMCMass[M ])
(cid:12) ThesignificanceofTd variationshouldbeinvestigatedmore
andwewillleavethisforfuturework.
Figure5. Case4: Differentialnumberdensityncl asafunctionof
GMC mass, based on the calculation with T = 22.4 Myr in- Our results suggest that the variation of the GMCMF
f,fid
cluding CCC. The color corresponds to the time evolution. Note slopesaregovernedbyT /T diversityindifferentenviron-
f d
thattheyellowlinefor160Myrisoverlappedbytheblacklinefor
ment on galactic scales. We predict that future large radio
200Myr,thusitisnotvisibleonthisplot. Asareference,weplot
twograystraightlines;thedot-dashedlinerepresentstheobserved surveys reveal that the GMCMFs have a single power-law
steepslope(∼ −2.6)intheinter-armregions,whereasthedashed exponent with which the ratio T /T may be uniquely con-
f d
lineshowstheobservedshallowslopeinthearmregions.Although
strained.
theGMCMFshowstheslope∼−2.2relativelyshallowerthanthe
observed one ∼ −2.6 due to the coagulation by CCC, the basic
4.3. PossibleModificationintheMassive-end.
correspondencesuggeststhattheinter-armregionstypicallyhavea
longer self-growth timescale Tf ∼ 22 Myr. This is well charac- The shape of the GMCMF massive-end, typically >
terized by the steady state solution without CCC shown in Equa- 106M , is determined by the relative importance of GMC
tion(10). (cid:12)
self-growth and CCC as shown in previous sections. The
quantitativediscussioninvolvesmanyuncertaintiesfromour
andinthelowermass(m < 105.5M )parteveninthecase modeling; for example, more detailed prescription for γ is
(cid:12)
withCCC(seeFigure1). necessary for GMC self-growth (i.e., Tf), and it is also re-
Toexaminethevalidityofthe−1−Tf/Td prediction,we quiredtomodelT∗variationduetothedrasticstarformation
conductthecalculationagainbyemployingvariousT . Fig- invoked by CCC, which is inferred by observations in the
f
ures4and5arethetworepresentativeresultsandbothcases Galaxy(c.f.,Fukuietal.(2014);alsoseeSection6.1). These
exhibit the slope well characterized by −1 − T /T . Fig- arebeyondourcurrentscopeandtobediscussedinourforth-
f d
ure 4 employs a shorter formation timescale, T = 4.2 comingpaper.
f,fid
Myr, and reproduces the observed shallow slope in arm re- Despite these limitation, our results indicate that, if CCC
gions. Similarly,Figure5showstheresultwithT =22.4 becomes effective, another structure may appear in the
f,fid
Myr,whichreproducestheobservedsteepslopeininter-arm massive-end.Thispossiblyexplainstheextrapower-lawfea-
regions.Therefore,thearmregionspresumablyhavealarger ture observed in some regions (e.g., Material Arms) from
numberofmassivestarsformingHII regionsandsupernova PAWSsurvey(Colomboetal.2014a).
remnantsthusexperiencetherecurrentsupersoniccompres-
5. RESULTS2: FATEOFDISPERSEDGAS
sion twice more frequently than the fiducial case, and the
5.1. Resurrectingfactor
inter-armregionstypicallyhaveasmallernumberofmassive
stars thus experience less frequent supersonic compression, ForallthetimeevolutionsoftheGMCMFwehaveshown
whichresultsinaboutfactortwolongerformationtimescale in the previous sections, we assume that minimum-mass
than the fiducial case. This may also explain the observed molecular clouds are continuously provided and the dis-
steep slope at outskirts of galaxies (e.g., the observed steep persed gas is removed from the system and never restored
slope in the CO luminosity function in Galaxy M33; see intothesystem.
Gratieretal.2012)wherestarformationislessactivecom- However,inreality,dispersedgasshouldreturntotheISM
pared with normal disk regions. Note that the above argu- and become either seeds of newer generation of molecular
mentmightbemodified,iftheeffectoflarge-scaledynamics cloudsormassaccretingontothepre-existingGMCs. Toes-
(e.g.,interactionwithshockwavesorstrongshearflows)may tablishacompletepictureofgasresurrectingprocessesinthe
play an important role in the destruction of GMCs than the ISM,weneedtoevaluatethefateofsuchdispersedgas. This
stellar feedback, especially in the outskirts of galaxies with aspiration requires detailed numerical simulations, ideally,
prominentspiralstructures. threedimensionalradiationmagnetohydrodynamicssimula-
NotethatwehereassumethattheT variationbetweendif- tions, which are however computationally too expensive to
d
ferentregionsislimitedcomparedwiththeT variation.This conduct. Instead in this article, we quantify the amount of
f
TIMEEVOLUTIONOFGMCMASSFUNCTIONS 9
dispersed gas by introducing the “resurrecting factor”, ε , 2
res
which is the mass fraction out of the total dispersed gas Tf,fid =10.0[Myr]
that are transformed to newer generation of minimum-mass ])(cid:12)0 Tεrdes==114..00[Myr]
M
GMCs. InSection4,weartificiallysettheproductionrateof
/
3 -2
minimum-mass GMCs to keep its number density constant. c
p Time[Myr]
k
Hereinstead,weevaluatethisproductionrateduetogasres- / 20
[ -4 40
urrectionas: ncl 60
∂(n∂ctlm)(cid:12)(cid:12)(cid:12)(cid:12) =εresM˙total,dispδ(m−mmin), (11) log( -6 81120260000
res
-8
where M˙ is the total dispersed gas mass produced 4 5 6 7
total,disp log(GMCMass[M ])
from the system per unit time, mmin is the minimum-mass (cid:12)
of GMCs (i.e. 104M in this study), and δ is the Dirac
(cid:12) Figure6. Case5: Differentialnumberdensityncl asafunctionof
delta function. By this definition, ε can be considered as GMC mass including CCC with the fiducial timescales (T =
res f,fid
the probability that dispersed gas becomes minimum-mass 10Myr and Td = 14Myr) and with εres = 1. The color corre-
sponds to the time evolution. As a reference, we plot three gray
GMCs before they accrete onto the pre-existing GMCs and
straightlines;thedot-dashedlineistheslope∼−1.7predictedby
thusthe1−εres fractionofdispersedgasareconsumedfor Equation (10), the dotted line represents the observed steep slope
the self-growth of pre-existing GMCs if a GMCMF is in a (∼ −2.6)intheinter-armregions,whereasthedashedlineshows
the observed shallow slope in the arm regions. The slope of the
steadystate.
GMCMF is initially between −1.7 and −2.6 but becomes steep-
Fromnow,wearegoingtofocusonlyonsteadystateGM- ened and the total mass in the system keeps growing so that this
CMFsforsimplicity. WemayassumethattheGMCMFsare does not reach any steady state (c.f., Figure 8). The kink in the
quasi-steadyintheGalaxyandnearbymassivespiralgalax- massive-end at 40 Myr indicates the onset for the CCC-dominant
phase. After40Myr,CCCbecomesdominantandmassiveGMCs
ies because they presumably have already undergone active > 107M are continuously created, which is not consistent with
(cid:12)
starformationphaseatredshift∼ 2andhaverelativelycon- observations, either. Thesteepenedslopeatthelow-massendap-
stant star formation activity at the present day (e.g. Madau pearsduetooverloadingresurrection.
& Dickinson 2014). Note that, although we have already
proved that the CCC effect is limited, we fully solve Equa- weneedtoreduceε . Figure7showsaresultwithε =
res res
tion (1) including the CCC terms to compute the time evo- 0.09 where we successfully reproduce a slope −α = −1.7
lutioninallthefollowingresults. WemainlyanalyzeCases throughoutallthemassrangeandachieveasteadystateGM-
5 to 8 in Table 1. The spikes and kinks that appear in the CMF. Tocheckthesteadiness,wecomputethetotalmassin
massiveend(especiallyinFigs.6, 9,and11)areduetothe the system as a function of time with various ε and Fig-
res
numericaleffectbytheCCCturn-offprocedureasexplained ure 8 shows the result. ε = 0.09 indicates that the GM-
res
inthesecondparagraphinSection4. CMFsettledownonthesteadystatewithaslightdecrement
initstotalmassbeforethecalculationendsat200Myr. Con-
5.2. NotAlltheDispersedGasareConsumedfor trarily, excessive input ε (cid:29) 0.09 leads to growth of the
res
Minimum-MassGMCCreation totalmass anddo notbecome steadyby 200Myras, forex-
Here, we examine the fiducial timescales Tf,fid = 10 ample, already seen in Figure 6. Less input εres (cid:28) 0.09
Myr and T = 14 Myr. First, we employ an extreme case leadstotoomuchgasdispersalintotheISMandeventually
d
where ε = 1; all the dispersed gas is consumed to form decreasesthetotalmassintheGMCphase,whichwouldnot
res
minimum-mass GMCs (Case 6) and Figure 6 shows the re- reach any steady state before 200Myr. In Section 5.5, we
sult. Initially the GMCMF exhibits slopes slightly steeper will investigate an analytical justification why εres ∼ 0.09
than −1.7 predicted by Equation (10). However the slope reachesasteadystateandpresentthepossiblerangeofεres
becomesmoresteepenedinthelow-massregimeandtheto- thatprovidesasteadyGMCMF.
talmassinthesystemkeepsincreasingsothattheGMCMF
5.3. TheObservedGMCMFSlopesinArmRegions
does not show any steady state (see Figure 8). Therefore,
theslopesareprovisionalandtransitional. Indeed,theCCC WenowfocusontheshorterformationtimescaleT =
f,fid
becomes significantly effective after 60 Myr so that GMCs 4.2Myr(Case7)toexaminewhatε wouldreproducethe
res
more massive than observed ones are instantly created after slopes observed in arm regions. Figure 9 shows the GM-
60Myr,whichdonotreproducetheobservedGMCMFs. CMFtimeevolutionwithε = 0.0123,whichsuccessfully
res
Thesteepenedslopeobservedinthelow-massendinFig- reproduces the observed shallow slope ∼ −1.3. Again, to
ure 6 appears due to overloading ε , because pre-existing check the steadiness, we compute the total mass in the sys-
res
GMCs cannot acquire the excessive amount of resurrecting temandFigure10showstheresultwithε from5.8×10−3
res
gas faster than a given self-growth timescale (in this case to 0.021. The figure also confirms that ε = 0.0123 pro-
res
10Myr). To achieve a steady state with a shallower slope, duces a steady state GMCMF. This factor 0.0123 indicates
10 M.I.N.KOBAYASHIETAL.
0 -1
Tf,fid =10.0[Myr] Tf,fid =4.2[Myr]
])(cid:12)-1 Tεrdes==104..009[Myr] ])(cid:12)-2 Tεrdes==104..001[2M3yr]
M M
/ -2 /
3c 3c -3
p Time[Myr] p Time[Myr]
k -3 k
/ 20 / 20
[ 40 [ -4 40
ncl -4 60 ncl 60
( 80 ( 80
og 120 og -5 120
l -5 160 l 160
200 200
4 5 6 4 5 6
log(GMCMass[M ]) log(GMCMass[M ])
(cid:12) (cid:12)
Figure7. Case6: Differentialnumberdensityncl asafunctionof Figure9. Case 7: Differential number density ncl as a function
GMC mass including CCC with the fiducial timescales (T = of GMC mass including CCC with T = 4.2 Myr and ε =
f,fid f,fid res
10Myr and T = 14Myr) and with ε = 0.09. The color 0.0123. Thecolorcorrespondstothetimeevolution. Notethatthe
d res
corresponds to the time evolution. As a reference, we plot three twocoloredlines(i.e.,thepurplelinefor120Myrandtheyellow
gray straight lines; the dot-dashed line is the slope predicted by linefor160Myr)areoverlappedbytheblacklinefor200Myrso
Equation (10), the dotted line represents the observed steep slope thattheyarenotvisibleonthisplot.Asareference,weplottwogray
(∼ −2.6)intheinter-armregions,whereasthedashedlineshows straightlines;thedot-dashedlineistheshallowslope(∼−1.3)ob-
theobservedshallowslopeinthearmregions. ThecomputedGM- servedinthearmregions, andthedottedlinerepresentsthesteep
CMFsuccessfullyreproducesthefiducialslope∼ −1.7predicted slope(∼ −2.6)observedintheinter-armregions. Thecomputed
byEquation(10)andlandsonasteadystateasshowninFigure8. GMCMF successfully reproduces the shallow slope ∼ −1.3 pre-
dictedbyEquation(10)andreachesasteadystatestudiedinFig-
ure10.
7
Tf,fid =10.0[Myr]
Td =14.0[Myr]
)
](cid:12)6
M ) 6
]
[ (cid:12)
al M
g(Mtot 5 εres0.2 M[total 5
lo 4 00..00917 og( εres
l 0.021
0 50 100 150 200 00..00102538 TTfd,fi=d 1=4.40.2[M[Myry]r]
Time[Myr] 4
0 50 100 150 200
Time[Myr]
Figure8. Thetotalmassdependenceonεres asafunctionoftime
withthefiducialsetup(T = 10MyrandT = 14Myr). The
differentcolorrepresentstfh,fieddifferentε fromd0.0017to0.2.The Figure10. Thetotalmassdependenceonεres asafunctionoftime
res withT = 4.2MyrThedifferentcolorrepresentsthedifferent
GMCMF with ε = 0.09 reaches a steady state after 130 Myr f,fid
res ε from5.8×10−3 to0.021. TheGMCMFwithε = 0.0123
elapse. ε = 0.09reproducesthefiducialslope−α = −1.7as res res
res reaches a steady state, M , which is ∼ 4×105M . ε =
showninFigure7.Resurrectingmorethanε =0.09(e.g.,0.2in total (cid:12) res
res 0.0123reproducestheshallowslope−α=−1.3asshowninFig-
thisfigure)increasethetotalmassinthesystemandtheGMCMF
ure 9. Resurrecting more than ε = 0.0123 (e.g., 0.021 in this
wouldnotreachanysteadystate. Similarly,resurrectinglessthan res
figure)increasethetotalmassinthesystem.Similarly,resurrecting
ε =0.09(e.g.,0.0017inthisfigure)eventuallydecreasethetotal
res lessthanε =0.0123(e.g.,5.8×10−3inthisfigure)eventually
massinthesystemafter50Myr.Thesteadyresurrectingfactor(∼9 res
decreasethetotalmassinthesystemafter60Myr. Thesteadyres-
percent)ispredictable,whichwewillexplaininSection5.5.
urrectingfactor(∼1percent)ispredictable,whichwewillexplain
inSection5.5.
that almost 99 per cent of dispersed gas are accreting onto
andfuelingpre-existingGMCsduetothemultipleepisodes Section5.5.
ofcompressionandthatonly1percentofdispersedgasare
5.4. TheobservedGMCMFSlopesinInter-ArmRegions
turned to form newer generation of GMCs. This extreme
fraction is expected due to massive GMCs; massive GMCs We now focus on the longer formation timescale T =
f,fid
havelargersurfaceareathanlessmassiveGMCsandcollect 22.4 Myr (Case 8) to examine what ε would reproduce
res
large amount of diffuse ISM gas to grow. Therefore, small the slopes observed in inter-arm regions. Figure 11 shows
ε ,orlarge1−ε ,islikelytoberealizedinarmregions, the GMCMF time evolution with ε = 0.45. This GM-
res res res
whichhavemanymassiveGMCs. Wewilldiscusstherange CMFhasitsslope∼−2.4,shallowerthantheobservedslope
ofε around0.0123thatalsokeepstheGMCMFsteadyin ∼ −2.6 due to the coagulation by CCC, which is also ob-
res
