Molecular Cloud Evolution II. From cloud formation to the early stages of star formation in decaying conditions

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by  E. Vazquez-Semadeni

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PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
MOLECULAR CLOUD EVOLUTION II. FROM CLOUD FORMATION TO THE EARLY STAGES OF STAR
FORMATION IN DECAYING CONDITIONS
Enrique Va´zquez-Semadeni1, Gilberto C. Go´mez1, A. Katharina Jappsen2,3, Javier Ballesteros-Paredes1,
Ricardo F. Gonza´lez1 and Ralf S. Klessen4
Submittedto The Astrophysical Journal
ABSTRACT
We study the formation of giant dense cloud complexes and of stars within them by means of
SPH numerical simulations of the collision of gas streams (“inflows”) in the warm neutral medium
(WNM)atmoderatelysupersonicvelocities. Thecollisionscausecompression,coolingandturbulence
7
generationin the gas,forming a cloud that then becomes self-gravitatingandbegins to collapse glob-
0
ally. Simultaneously, the turbulent, nonlinear density fluctuations induce fast, local collapse events.
0
2 The simulations show that: a) The clouds are not in a state of equilibrium. Instead, they undergo
secular evolution. During the early stages of the evolution, their mass and gravitational energy Eg
n
increase steadily, while the turbulent energy E reaches a plateau. b) When E becomes compara-
a k g
ble to E , global collapse begins, causing a simultaneous increase in E and E that maintains a
J k | g| k
2 near-equipartition condition |Eg| ∼ 2Ek. c) Longer inflow durations delay the onset of global and
local collapse, by maintaining a higher turbulent velocity dispersion in the cloud over longer times.
2
d) The star formation rate is large from the beginning, without any period of slow and accelerating

star formation. e) The column densities of the local star-forming clumps are very similar to reported
3
values of the column density required for molecule formation, suggesting that locally molecular gas
v
5 and star formation occur nearly simultaneously. The MC formation mechanism discussed here natu-
7 rally explains the apparent “virialized” state of MCs and the ubiquitous presence of HI halos around
3 them. Also, within their assumptions, our simulations support the scenario of rapid star formation
8 after MCs are formed, although long (& 15 Myr) accumulation periods do occur during which the
0 clouds build up their gravitationalenergy, and which are expected to be spent in the atomic phase.
6
Subject headings: Instabilities—ISM:clouds—ISM:evolution—Shockwaves—Stars: formation—
0
Turbulence
/
h
p
- 1. INTRODUCTION 2003; Va´zquez-Semadeni et al. 2006;
o Ballesteros-Paredes& Hartmann 2006;
The evolution of molecular clouds (MCs) re-
r Ballesteros-Paredes 2006, see also the reviews by
t mains an unsolved problem to date. While tra-
s Mac Low & Klessen 2004 and Ballesteros-Paredes et
ditionally MCs have been thought of as virialized
a al. 2006). One approach that can shed light on this
: structures in the interstellar medium (ISM) (e.g.,
v de Jong, Boland & Dalgarno 1980; McKee et al. problem is to perform numerical simulations of the
i MC formation process by generic compressions in the
X 1993; Blitz & Williams 1999; McKee 1999), with
warm neutral medium (WNM). Numerous studies of
relatively long lifetimes (Blitz & Shu 1980) and a
r this kind exist, although self-gravity has in general
a significant delay before they begin forming stars (e.g.
not been included (Hennebelle & P´erault 1999, 2000;
Palla & Stahler 2000, 2002; Tassis & Mouschovias
Koyama & Inutsuka 2000, 2002; Inutsuka & Koyama
2004; Mouschovias,Tassis & Kunz 2006), re-
2004; Audit & Hennebelle 2005; Heitsch et al. 2005,
cent studies have suggested that MCs be-
2006; Va´zquez-Semadeni et al. 2006, hereafter Paper
gin forming stars shortly after they them-
I).
selves form, and are non-equilibrium entities
Thesestudieshaveshownthatasubstantialfractionof
(Ballesteros-Paredes,Va´zquez-Semadeni & Scalo 1999;
the internal turbulence, with velocity dispersions of up
Ballesteros-Paredes,Hartmann & Va´zquez-Semadeni
1999; Elmegreen 2000; toafew km s−1,canbe producedinshock-boundedlay-
Hartmann, Ballesteros-Paredes& Bergin 2001; ersbyacombineddynamical+thermalinstability,whose
Va´zquez-Semadeni, Ballesteros-Paredes& Klessen nature is still not well determined (Hunter et al. 1986;
Walder & Folini 1998, 2000;Koyama & Inutsuka 2002;
Electronicaddress: e.vazquez, g.gomez,j.ballesteros,[email protected]&x Koyama 2004; Heitsch et al. 2005, 2006;
1CentrodeRadioastronom´ıayAstrof´ısica(CRyA),Universidad Folini & Walder 2006). Paper I also showed that the
Nacional Auto´noma de M´exico, Apdo. Postal 72-3 (Xangari),
dense gas, there defined as the gas with densities larger
Morelia,Michoaca´n58089, M´exico
2Astrophysikalisches Institut Potsdam, An der Sternwarte 16, than100 cm−3,is atsystematicallyhigherthermalpres-
14482Potsdam,Germany suresthanthemeaninterstellarvaluebyfactorsof1.5–5,
3Canadian Institute for Theoretical Astrophysics (CITA), implying that MC formation by colliding streams of dif-
McLennan Physics Labs, 60 St. George Street, University of
fuse gas can account at least partially for the clouds’
Toronto,Toronto,ONM5S3H860,Canada
4Institut fu¨r Theoretische Astrophysik / Zentrum fu¨r As- excess pressure and turbulent nature.
tronomie der Universita¨t Heidelberg, Albert-U¨berle-Str. 2, 69120 Only a few works have included self-gravity in sim-
Heidelberg,Germany ulations of colliding flows. The pioneering work of
2 Va´zquez-Semadeni et al.
Hunter et al. (1986) considered the collision of su- gas (Koyama & Inutsuka 2002; Audit & Hennebelle
personic streams within MCs, finding that the frag- 2005; Heitsch et al. 2005, 2006, Paper I) suggest that
mentsproducedbythedynamicalinstability(whichthey the clumpiness is generated by the shock compression,
termed“Rayleigh-Taylor-like”)wereabletoaccretemass which nonlinearly triggers the thermal instability,
until they became gravitationally unstable and collapse. induces supersonic turbulence, and causes the formation
However, this study was designed to investigate flows of dense clumps in the compressed layers.
within MCs rather than the formation of the clouds It is thus essential to model the evolution of the cloud
themselves, and so it considered much smaller (sub- within the frame of its more diffuse environment includ-
parsec) scales and different density, temperature and ing both self-gravity and cooling. Not only this is indis-
cooling regimes than are relevant for the process of MC pensible in order to model the generation of turbulence
formation. and clumpiness, but also to model the cloud’s evolution
More recent numerical studies of MC formation as it exchanges mass and energy with its surroundings.
not including self-gravity (Koyama & Inutsuka 2002; Note that the generation of turbulence by the compres-
Heitsch et al. 2005) have estimated that the individual sion itself implies that the turbulence within the cloud
clumps formed in the compressed layer formed by the is driven while the large-scale converging motions per-
colliding streams are themselves gravitationally stable, sist, and decaying afterwards. This is a mixed regime
although the entire compressed-gas complex would be that also requires self-consistent modeling of the cloud
gravitationally unstable, had self-gravity been included. and its surroundings, rather than, for example, random
This raises the issue of whether the clumps can be Fourier driving in closed boxes.
considered in isolation. The question of which gas Finally, such a unified (or “holistic”) description of
parcels are involved in the collapse of a particular molecular cloud formation and evolution, as outlined
object is a delicate one, and lies, for example, at the by Va´zquez-Semadeni, Ballesteros-Paredes& Klessen
heart of the debate of whether competitive accretion (2003), including the star formation epoch, allows
is relevant or not in the process of star formation also the investigation of a number of key problems,
within molecular clouds: A model in which clumps are such as the evolutionary and star-formation timescales,
isolated dense objects immersed in a more tenuous, the evolution of the cloud’s gravitational and kinetic
unbound medium leads to the conclusion that competi- energies,and whether it settles into virialequilibrium or
tive accretion is irrelevant (Krumholz, McKee & Klein not, and the effect of self-gravityon the cloud’s physical
2005). Instead, competitive accretion plays a central conditions.
role (Bonnell et al. 1997; Klessen, Bate & Burkert In this paper we investigate numerically the evolution
1998; Bonnell et al. 2001; Klessen & Burkert 2000, of interstellar gas as it collects from the WNM, shocks,
2001; Klessen 2001a,b; Bonnell & Bate 2006) in suffers a phase transition to a cold and dense state, and
a scenario in which the local density peaks are finallybeginstoformstars,bymeansofnumericalsimu-
the “tips of the iceberg” of the density distribu- lationsofcollidinggasstreamsintheWNM,inthepres-
tion, the flows are organized on much larger scales ence of self-gravityand coolingleading to thermalinsta-
(Ballesteros-Paredes,Va´zquez-Semadeni & Scalo 1999; bility. Sincethisisaproblemthatnaturallycollectslarge
Ballesteros-Paredes,Hartmann & Va´zquez-Semadeni amounts of gas in small volumes, first by compression-
1999; Bate, Bonnell & Bromm 2003), and the evolution triggered thermal instability and then by gravitational
andenergeticsofthedensitypeakscannotbeconsidered instability, it is convenient to use some Lagrangian nu-
in isolation, but as part of the global flow instead merical scheme, and we opt for a smoothed particle hy-
(Ballesteros-Paredes,Va´zquez-Semadeni & Scalo drodynamicscode,sacrificingthepossibilityofincluding
1999; Klessen, Heitsch & Mac Low 2000; magnetic fields. We also omit for now a self-consistent
Heitsch, Mac Low & Klessen 2001; description of the stellar feedback on the gas and of the
Shadmehri, Va´zquez-Semadeni & Ballesteros-Paredes chemistry. Thus,oursimulationswillnotadequatelyde-
2002; Tilley & Pudritz 2004, 2005; Ballesteros-Paredes scribethe lateststagesofMC evolution(in whichstellar
2006; Dib et al 2006). In a similar fashion, then, energy feedback may significantly influence the dynam-
accuratedeterminationofwhethergravitationalcollapse ics of the cloud) nor are they capable of predicting the
can be triggered by moderate-Mach number stream transition from atomic to molecular gas accurately. We
collisions in the WNM, requires to self-consistently defer these tasks to subsequent papers.
include self-gravity in the numerical simulations. Theplanofthepaperisasfollows. In 2wefirstgivea
§
Simulations of molecular cloud formation by qualitative description of the physical system, the main
the passage of a self-gravitating, clumpy medium phenomenainvolved,andourexpectationsfortheevolu-
through Galactic spiral shocks (Bonnell et al. 2006; tion. In 3 we describe the numerical code, the physical
§
Dobbs, Bonnell & Pringle 2006)haveshownthatrealis- physical setting, and the choice of parameters. Then, in
tic velocitydispersionsanddensitiescanbe generatedin 4 we describe the results concerning the evolutionof the
the resulting post-shock clouds, which are furthermore cloud’s mass and its kinetic and gravitational energies,
driven to produce local collapse events leading to star and the onset of star formation. In 5 we discuss the
§
formation, although a large fraction of the mass in implications and limitations of the recent work. Finally,
the clouds remains gravitationally unbound. However, in 6 we presenta summary and draw some conclusions.
§
because these simulations were isothermal, the gas
had to be assumed to already be clumpy and at very 2. QUALITATIVEDISCUSSION
low temperatures ( 100 K) previous to the passage It is convenientto firstpresent an overviewof the sys-
∼
through the shock, while recent studies including tem, its expected evolution, and the physical processes
cooling leading to thermal bistability of the atomic on which it relies (see also the description in Paper I).
Molecular Cloud Evolution II 3
Since we are interested in studying the formation of a cloud begins collapsing as a whole. Because it contains
dense cloud out of the diffuse medium, we take as initial many Jeans masses, local collapse events begin to occur
conditionauniform-densityregionwithintheWNM,ofa at the nonlinear density peaks produced by the turbu-
fewhundredparsecsacross,withaninitiallycompressive lence, if gravity locally overcomes all available forms of
velocityfieldconsistingoftwooppositely-directedinflows support. Because the density enhancements are nonlin-
with speed v and Mach number , with respect to ear,the local collapse events occur on shorter timescales
inf inf
M
the WNM temperature. This compressive field is gener- than the global contraction.
ically representative of either the general turbulence in The magnetic field should also be considered.
the WNM, or of motions triggered by some large scale Hennebelle & P´erault (2000) have shownthat the tran-
agent, such as gravitational or Parker instabilities, or sition from diffuse to dense gas can occur even in
shellcollisions. SincethevelocitydispersionintheWNM the presence of magnetic fields aligned with or oblique
is transonic(Kulkarni & Heiles 1987; Heiles & Troland to the direction of compression. Moreover, most evi-
2003), we consider values of 1. dence,boththeoreticalandobservational,pointstowards
inf
M ∼
As is well known, the atomic ISM is thermally MCs and giant molecular complexes being magnetically
bistable, with two stable phases being able to coex- supercritical or marginally critical at least (Crutcher
ist at the same pressure but different densities and 1999; Hartmann, Ballesteros-Paredes& Bergin 2001;
temperatures, mediated by a thermally unstable range Bourke et al. 2001; Crutcher, Heiles & Troland 2003).
(Field 1965; Field, Goldsmith & Habing 1969). The Since the magnetically supercritical case is not qualita-
collision of transonic WNM streams produces a thick tively different from the non-magnetic case, the above
shock-bounded slab, in which the gas is out of ther- discussion is applicable to supercritical clouds as well.
mal equilibrium between radiative heating and cool- Hartmann, Ballesteros-Paredes& Bergin (2001) have
ing. This shocked slab is nonlinearly thermally un- approximately quantified the above scenario, estimating
stable (Koyama & Inutsuka 2000; Kritsuk & Norman thatthe accumulationtime fromthe WNM shouldbe of
2002), and as it flows towards the collision plane, the orderof 10–20Myr,the accumulatedgasmust come
it undergoes a phase transition to the cold neu- from distances up to a few hundred parsecs,and that at
tral medium (CNM), causing the formation of a the end of the accumulation period, the gas should be
thin cold layer, (Hennebelle & P´erault 1999, 2000; becoming molecular and self-gravitating at roughly the
Koyama & Inutsuka 2000, 2002; Inutsuka & Koyama same time, because the column density for self-shielding
2004; Audit & Hennebelle 2005; Heitsch et al. 2005, and formation of molecular gas,
2006, Paper I). Subsequently, the boundary of this
N 1 2 1021 cm−2, (1)
layer is destabilized, probably by a combination of ss ∼ − ×
thermal instability and nonlinear thin-shell instability is very similar to that required for rendering the dense
(Vishniac 1994), producing fully-developed turbulence gas gravitationally unstable,
in the layer for sufficiently large (Hunter et al.
inf
1986; Koyama & Inutsuka 2002; IMnutsuka & Koyama N 1.5 1021P1/2 cm−2, (2)
gi ∼ × 4
2004; Heitsch et al. 2005, 2006, Paper I). As the layer
where P is the pressure expressed in units of 104
becomes turbulent, it thickens, and becomes a fully 4
K cm−3 (see also Franco & Cox 1986). That
three-dimensional structure (Paper I), to which we sim-
is, the gas is converted to the molecular phase
ply refer as “the cloud”.
roughly simultaneously with the onset of gravita-
Moreover,thethermalpressureinthedensecloudisin
tional instability, explaining why non-star-forming
pressurebalancewiththetotal(thermal+ram)pressure
MCs are rare, at least in the solar neighborhood
intheinflows,whichissignificantlylargerthanthemean
(Hartmann, Ballesteros-Paredes& Bergin 2001;
thermalpressureintheISM(PaperI).Consequently,the
Hartmann 2003; Ballesteros-Paredes& Hartmann
density in the cold layercan reach densities significantly
2006). Note, however, that this near equality is a
larger than those typical of the CNM. Thus, this cloud
coincidence (e.g., Elmegreen 1985, 1993a), since N
is turbulent, dense, and overpressured with respect to gi
depends on the ambient pressure, and therefore non-
the mean ISM pressure; i.e., it has properties typical of
star-forming molecular clouds may be more common
molecularclouds(PaperI).Themainmissinglinkinthis
in higher-pressure environments, possibly explaining
scenario is whether the atomic gas can be readily con-
observations that up to 30% of the GMCs appear
vertedtomolecular,althoughseveralstudiessuggestitis
to be starless in some external galaxies (Blitz et al.
feasible (Bergin et al. 2004; Glover & Mac Low 2006).
2006). On the other hand, this result may be due in
Since we do not incorporate any chemistry nor radiative
part to incompleteness effects. Indeed, recent Spitzer
transferinourcode,wecannotfollowthistransitionself-
observations have uncovered young stellar objects in
consistently, and simply assume that once the dense gas
Galactic molecular cores previously believed to be
has a sufficiently large column density (see below), it is
starless (e.g., Craspi et al. 2005; Rho et al. 2006).
rapidly converted to the molecular phase.
Inthe remainderofthe paper,we proceedtoquantify
In summary, the compression produces a turbulent
thisscenariobymeansofnumericalsimulationsdesigned
cloud of dense and cold gas. As it becomes denser and
for this purpose.
colder,its Jeansmassdecreasessubstantially. Moreover,
its own mass is increasing, so it rapidly becomes much 3. THEMODEL
moremassivethanitsownJeansmass. Also,asthecloud
3.1. Parameters and numerical scheme
becomes more massive, its total gravitationalenergy (in
absolute value) |Eg| increases substantially, and eventu- We consider a cubic box of length Lbox on a side, ini-
ally overcomesevenits turbulentkinetic energy,andthe tially filled with WNM at a uniform density of n =
0
4 Va´zquez-Semadeni et al.
1 cm−3, with a mean molecular weight of 1.27, so that n . Due to the limited mass resolution we can afford,
thr
themeanmassdensityisρ =2.12 10−24gcm−3,which the sinks should in general be considered star clusters
0
implies total masses of 6.58 104M×⊙ and 5.26 105M⊙ rather than isolated stars, although some of them may
× ×
for the two box sizes used, L = 128 pc and 256 pc, actually correspond to single stars. Nevertheless, with
box
respectively. The gas has a temperature T = 5000 this caveat in mind, we refer to the sinks as “stars” in a
0
K, the thermal-equilibrium temperature at that density generic way.
(cf. 3.3). Within the box, we set up two cylindrical
§
and oppositely-directed inflows, each of length ℓinf, ra- 3.3. Heating and cooling
dius r = 32 pc, and speed v = 9.20 km s−1,
inf inf ± The gas is subject to heating and cooling functions of
corresponding to a Mach number = 1.22 with
Minf the form
respect to the sound speed of the undisturbed WNM,
c =7.536 km s−1 (see fig. 1). For the two values of
WNM
ℓinf weconsiderhere(48pcand112pc),themassesofthe Γ=2.0 10−26 erg s−1 (3)
cylindricalinflowsare1.13×104 M⊙ and2.64×104M⊙, Λ(T) × 1.184 105
respectively. To triggerthe dynamicalinstability of the =107exp − ×
compressed slab, a fluctuating velocity field (computed Γ (cid:18) T +1000 (cid:19)
inFourierspace)isinitiallyimposedthroughoutthe box 92
at wavenumbers 4 kL 8, and having amplitude +1.4 10−2√T exp − cm3. (4)
≤ box ≤ × (cid:18) T (cid:19)
v . The streams are directed along the x direction
rms,i
and collide at half the x extension of the box. Table 1 These functions are fits to the various heating (Γ) and
gives the values of these parameters for the simulations cooling(Λ)processesconsideredbyKoyama & Inutsuka
presented here. The runs are named mnemonically as (2000), as given by equation (4) of Koyama & Inutsuka
Lxxx∆v0.yy,wherexxxindicatesthephysicalboxsizein (2002).5 The resulting thermal-equilibrium pressure
parsecs and 0.yy indicates the value of vrms,i in km s−1. Peq,definedbytheconditionn2Λ=nΓisshowninfig.2
We use the N-body+smoothed particle hydrodynam- as a function of density. We have abandoned the piece-
ics (SPH) code GADGET (Springel, Yoshida & White wise power-law fit we have used in all of our previous
2001), as modified by Jappsen et al. (2005) to include papers (e.g., Va´zquez-Semadeni, Passot & Pouquet
sink particles, a prescription to describe collapsed ob- 1995; Passot,Va´zquez-Semadeni & Pouquet
jects without following their internal structure. We use 1995; Va´zquez-Semadeni, Passot & Pouquet 1996;
1.64 106 particlesforthe runswith Lbox =128pc,and Va´zquez-Semadeni, Gazol & Scalo 2000; Gazol et al.
3.24×106particlesfortherunwithLbox =256pc. Since 2001; Gazol, Va´zquez-Semadeni & Kim 2005), includ-
×
the larger-box run contains 8 times more mass than the ing Paper I, mainly because the present fit is applicable
smaller-box ones but uses only twice as many particles, at densities typical of molecular gas.
the minimum resolvedmass in the larger-boxrun is four The usual procedure to apply cooling to the hydrody-
times largerthan in the smaller-boxrun. Unfortunately, namicevolutioninvolvesthe considerationofthe cooling
the computing resources available in our cluster did not rate in the Courant condition to restrict the simulation
allow us to further increase the number of particles. time step. For the problem at hand, the high densi-
We take asunits for the code a massof 1M⊙, a length ties reached behind the shocks, for example, would im-
ℓ = 1 pc and a velocity v = 7.362 km s−1, implying a ply an exceedingly small time step and the simulation
0 0
time unit t =l /v =0.133 Myr. would become computationally unfeasible. But this ap-
0 0 0
pears completely unnecessary, because all this means is
3.2. Sink particles that the thermal evolution happens faster than the dy-
namical one, and therefore, as far as the hydrodynamics
Sink particles are created when a group of SPH parti-
is concerned, it is instantaneous. Therefore, it is more
cles become involved in a local collapse event. At that
convenient to use an approximation to the thermal evo-
point, the SPH particles are replaced by a sink particle
lution of the gas, which constitutes an extension of that
whose internal structure is not resolved anymore. The
used in Paper I, and which allows us to simply correct
sink particle inherits the total angular momentum and
theinternalenergyafterthehydrodynamicstephasbeen
mass of the SPH particles it replaces, and henceforth
performed, with no need to adjust the timestep.
movesonlyinresponsetoitsinertiaandthegravitational
Considerthe thermalequilibriumtemperature,T , as
eq
force, although it can continue to accrete SPH particles,
a function of the gas density, and the corresponding in-
should they come sufficiently close to the sink particle.
ternal energy density e (T is a well defined function
eq eq
The creation of a sink particle requires a number of
of the density n as long as Λ(T) is monotonic. This is
conditions to be satisfied. First, the localdensity should
true within the range of temperatures occurring in the
exceedsomeuser-definedthresholdvaluen . Ifthisoc-
thr simulations). Consider also the time required to radiate
curs,thenthecodecomputesthetotalmassandangular
thethermalenergyexcess(ifT >T ;acquiretheenergy
eq
momentum ofthe groupof particlesaboven to deter-
thr deficit, if T <T )
eq
minewhethertheyaregravitationallybound. Iftheyare,
then they are collectively replaced by the sink. At sub-
e e
sequent times, further SPH particles can be accreted by τ = − eq . (5)
thesinkiftheycomewithinan“accretionradius”racc of Λ (cid:12)(cid:12)n2Λ−nΓ(cid:12)(cid:12)
the sink and are gravitationally bound to it. In all cases (cid:12) (cid:12)
we use n = 105 cm−3 and r = 0.04 pc. This value 5 Notethateq.(4)inKoya(cid:12)ma&Inutsu(cid:12)ka (2002)contains two
thr acc typographicalerrors. Theformusedhereincorporatesthenecesary
of racc amounts to roughly 1/3 of the Jeans length at corrections,kindlyprovidedbyH.Koyama.
Molecular Cloud Evolution II 5
Then, we compute the new internal energy density e′, the paper, these figures are also available as full-length
after a time step dt, as animations. The timesindicatedinthe framesareinthe
code’s internal unit, t =0.133 Myr.
0
e′ =e +(e e )exp( dt/τ ). (6) The animation shows that the gas at the collision site
eq eq Λ
− −
begins to undergo a phase transition to the cold phase
Note that, if the gas is cooling down (or heating up)
(theCNM)atroughlyonecoolingtimeafterthecollision
rapidly, τ dt, exp( dt/τ ) 0 and the gas immedi-
Λ Λ (Hennebelle & P´erault 1999; Va´zquez-Semadeni et al.
≪ − →
ately reaches its equilibrium temperature, without ever
2006), becoming much denser and colder. Because this
overshootingbeyondthatvalue. Conversely,ifthe gasis
gasis in equilibrium with the total(thermal+ram)pres-
atverylowdensityorisclosetotheequilibriumtemper-
sureoftheinflow,thedensityovershootsfarbeyondthat
ature, τ dt, and equation (6) becomes,
Λ of standard CNM, well into the realm of molecular gas,
≫
with mean densities of several hundred cm−3 and tem-
e′ =e dt(n2Λ nΓ), (7)
peratures of a few tens of Kelvins. This implies a much
− −
where we, again, use the fact that Λ(T), as given by lower Jeans mass than that in the conditions of the ini-
equation(4),isamonotonicfunctionofthetemperature. tialWNMandquicklythecloud’smassexceedsitsJeans
mass.
3.4. Gravitational instability in cooling media Defining the “cloud” as the material with n >
50 cm−3, thecloud’smassbecomeslargerthanitsmean
The standard Jeans instability analysis is modified in
Jeans mass at t 2.2 Myr in runs with L = 128
box
a medium in which the balance between heating and ∼
pc and at t 3 Myr in the run with L = 256 pc,
instantaneous cooling produces a net polytropic behav- ∼ box
at which times the mean density in the dense gas is
ior,characterizedbyaneffective polytropicexponentγe, n 65 cm−3, the mean temperature is T 50 K,
which is the slope of the thermal-equilibrium pressure h i ∼ h i ∼
versus density curve (fig. 2). In this case, the Jeans the Jeans mass is MJ ∼1200 M⊙, and the free-fall time
isτ L /c 11Myr,wherecisthe localsoundspeed.
ff J
length is found to be given by (e.g., Elmegreen 1991; ≡ ∼
For comparison, the total mass in the two cylinders,
Va´zquez-Semadeni, Passot& Pouquet 1996)
which contain the material that initially builds up the
LJ ≡(cid:20)γγeGπρc02(cid:21)−1/2, (8) atchlnoedudc2l,.o2u6isd×1e.11v03e4n×Mtu1a⊙0l4liynMbt⊙heceionrmutnehsewmrituuhncsLhwbmoixtoh=reL2mb5o6axsp=sciv.1e2T8thhpuacsn,,
its mean Jeans mass. For example, in run L256∆v0.17
where c is the adiabatic sound speed and γ is the heat
at t = 17.26 Myr, the (mass-weighted) mean density of
capacity ratio of the gas. Equivalently, this instability
the dense gas is n 600 cm−3, the mean temperature
criterioncan be described as if the effective sound speed h i∼
is T 37 K, and γ 0.75. These values imply a
iisn Bthoeltzsymsatenmn’swceorensgtiavnetnabnydcµeffis=thpeγmekeTa/nµ,mwolheecruelakr mtimehaeniteh∼ffeecctloivuedJceoanntsamineass∼s M2J.3∼5 301004MM⊙,⊙w.hiNleotaet tthhaatt
weight. Note that, when γe = 0, the thermal pressure the cloud+sink mass mig∼ht exce×ed the mass in the in-
doesnotreacttodensitychangesandthereforeisunable
flows, since ambient material surrounding the cylinder
to oppose the collapse, rendering the medium gravita-
is dragged along with the flow, effectively increasing the
tionally unstable atallscales (L =0). In a more realis-
J amount of gas in the inflow (see below).
tic situation where the cooling is not instantaneous, the
The animations show that global collapse does indeed
Jeanslengthwillbelimitedbythecoolinglength. Inthis
occur, although at significantly later times than when
work, the cooling has a finite characteristic timescale,
the cloud’s mass becomes larger than the Jeans mass,
given by eq. (5).
probablyreflectingthe roleofturbulence asanopposing
At the initial uniform density n = 1 cm−3 and tem-
0 agent to the collapse. Indeed, the clouds in the 128-pc
peratureT =5000K,γ 0andtheJeanslengthisalso
0 e ∼ boxes begin global contraction at t 8 Myr, while the
near zero. This can be seen in fig. 2, where the vertical ∼
cloudinthe256-pcbox,withmorethantwicetheinflow
dotted line shows the initial uniform density, at which
duration, begins contracting at t 12.5 Myr.
the localtangentto the Peq vs. ρ curve is seen to have a Figure5(top panel)showsthe e∼volutionofthecloud’s
nearly zero slope. However, when the compression pro-
mass in run L256∆v0.17, together with the mass in col-
duces a dense cloud at the midplane of the box, the rest
lapsed objects (sinks), to be discussed further in 4.5.
ofthegasdecreasesitsdensityandentersthefullystable §
It is clearly seen that the cloud’s mass is not constant,
warm phase. For reference, at a density n = 0.5 cm−3
but rather evolves in time. Gas at cloud densities first
andacorrespondingequilibriumtemperatureT =6300
eq appears after 2 Myr of evolution, and the cloud’s mass
K, γ =0.834 and L =1324 pc, implying a Jeans mass
e J increases monotonically until t 19.5 Myr, at which
MwiJth=L3b.o4x7=×112087Man⊙d.LUbonxd=er2t5h6especcwonoduiltdioinnsit,iaolulyr rcuonns- tAimfteeritthriseatcimhees,iatbmeagxinimsutomdevcarleua∼eseMbcelc∼aus2e.7of×th1e04rMap⊙id.
tain 10−3 and 10−2 Jeans masses, respectively.
∼ ∼ conversionof gas to stars.
4. RESULTS By the end of the simulation, the total mass in stars
plus dense gas is almost three times the initial mass in
4.1. Mass evolution
the inflows. In practice, the inflows last longer than the
In general, the simulations evolve as described in Pa- simple estimate τinf ℓinf/vinf would indicate, and in-
≡
per I and in sec. 2. As an illustration, figs. 3 and 4 volvemoremassthanthemassinitiallywithinthecylin-
showselectedsnap§shotsofrunL256∆v0.17viewededge- ders. This is due to the fact that, as the cylinders be-
on and face-on, respectively. In the electronic version of gin advancing, they leave large voids behind them that
6 Va´zquez-Semadeni et al.
have the double effect of slowing down the tails of the of stellar masses in the simulation at that time.6 We
inflows and of dragging the surrounding gas behind the thenuseTable1ofD´ıaz-Miller, Franco & Shore (1998),
cylinders. For v = 9.20 km s−1, τ = 5.22 Myr for whichgivestheionizingfluxasafunctionofstellarmass,
inf inf
runs with ℓ = 48 pc (L128∆v0.24 and L128∆v0.66), toestimatethefluxassociatedtoeachmassbin,andthen
inf
and τinf = 12.17 Myr for the run with ℓinf = 112 pc integrateoverallrelevantmassestoobtainthetotalF48.
(L256∆v0.17). Instead, in the first two runs, the inflow Now, from eqs. (9) or (10) one can solve for Mc,4 as a
actually lasts 11 Myr, while in the latter run it lasts function of NOB, to find the minimum cloud mass that
22.5 Myr. T∼his can be observed in the animation of survivestheionizingradiationfromtheexistingOBstars
fi∼g. 3, in which the velocity field arrowsclearly show the in the simulation. When this minimum surviving mass
gradual decay of the inflows, and their longer duration is larger than the actual cloud mass, we expect that the
compared to the simple linear estimate. cloud will be disrupted. Since we define the “cloud” as
thegaswithdensitiesabove50 cm−3,wetaken =0.05.
3
4.2. Cloud disruption Also, for simplicity we assume cs,15 tMS,7 1.
∼
For run L256∆v0.17, fig. 6 shows the evolution of the
Our simulations do not include any feedback
cloud’smasstogetherwiththeminimumdisruptedcloud
from the stellar objects, while this process is prob-
mass under the two estimates, eqs. (9) and (10). We see
ably essential for the energy balance and pos-
that the cloud is expected to be disrupted at t = 20.3
sibly the destruction of the cloud (e.g., Cox
with either estimate.
1983, 1985; Franco, Shore & Tenorio-Tagle 1994;
HMaarttzmnearnn2,0B0a2l;leNstaekraoms-uPraare&deLsi&2B0e0r5g;inLi & Nakam20u0ra1; anAdttthheiscltoimude’sthmeamssasissiMncslin=ks2.i6s7M×si1n0ks4M=⊙5,.2s5o×th1a0t3Mth⊙e
star formation efficiency (SFE) up to this time has been
2006; Mellema et al. 2006). Thus, the simulations
cannot be safely considered realistic after their stellar M∗
SFE= 16%. (11)
content would likely disrupt the cloud. We now present
Mcl+M∗ ≈
an a posteriori estimation of the time at which this
occurs. This number is still larger than observational estimates
Franco, Shore & Tenorio-Tagle (1994)havesuggested for cloud complexes (Myers et al. 1986), and suggests
thatthemaximumnumberofOBstarsthatthecloudcan that additional physical processes, such as longer inflow
supportatanytimeisgivenbythenumberofHIIregions durations ( 4.6) or magnetic fields ( 5.2) may be neces-
§ §
required to completely ionize the cloud. In particular, sary to further reduce the SFE.
they found that the maximum number of massive stars We conclude from this section that after t 20.3 Myr
≈
that can be formed within a molecular cloud is runL256∆v0.17isprobablynotrealisticanymore,asfar
as the evolution of a real MC is concerned, and we thus
16M n3/7 restrict most of our subsequent discussions to times ear-
N c,4 3 , (9) lierthanthat,exceptwhenlatertimesmaybeillustrative
OB,inside ≃ F458/7(cs,15 tMS,7)6/7 ofsomephysicalprocess. Self-consistentinclusionofstel-
lar feedback in the simulations, similarly to the studies
where Mc,4 is the mass of the cloud, in units of 104 of Nakamura & Li (2005) and Li & Nakamura (2006),
M⊙, n3 is the number density in units of 103 cm−3, toinvestigatethefinalstagesofMCevolutionwillbethe
F48 is the photoionizing flux from the massive stars, in subject of future work.
units of 1048 s−1, c is the isothermal sound speed
s,15
in units of 15 km s−1, and t is a characteristic OB 4.3. Turbulence evolution
MS,7
star main sequence lifetime, in units of 107 yr. How- The collision also generates turbulence in the dense
ever, as Franco, Shore & Tenorio-Tagle (1994) pointed gas,andsothe cloudcanbe consideredto be inadriven
out, clouds are more efficiently destroyed by stars at turbulentregimewhiletheinflowspersist,andinadecay-
the cloud edge because the lower external pressure en- ing regime after the inflows subside, although the tran-
sures that the ionized gas expands rapidly away from sition from driven to decaying is smooth, rather than
the cloud, driving a fast ionization front into the dense abrupt, since the inflows subside gradually, as explained
material. Thus, in this case, these authors find that the above. Figure 5 (bottom panel) shows the evolution of
maximum number of OB stars that the molecular cloud the velocity dispersion σ for the dense gas along each of
can support is given by: the three coordinate axes for run L256∆v0.17. During
thefirst10Myrofthecloud’sexistence(2.t.12Myr)
NOB,edge ≃ F3/53(Mc c,4tn13/5 )6/5. (10) σthya≈t tσhez t∼ur4buklmensc−e1a,paplethaorsugtho biteisaninistoetrreosptiinc,gwtoithnoσtxe
48 s,15 MS,7 first increasing and then decreasing. It reaches a maxi-
In order to determine the time at which the MC in mumof morethan twice as large( 8.5 km s−1) as that
∼
oursimulationhasformedasufficientnumberofstarsto of σ and σ at t 6 Myr, reflecting the fact that the
y z
≈
disrupt it, we proceed as follows. Since the individual inflows are directed along this direction. This suggests
sinkparticlesinthe simulationsingeneralcorrespondto thatthegenerationoftransverseturbulentmotionsisnot
clusters rather than to single stars, we cannot use the
sink particle masses directly. Instead, for each temporal 6 Note that this estimate for the number of massive stars as a
output of our simualtion, we fit a standard initial mass function of time is probably a lower limit, as recent simulations
suggestthatmassivestarsinclusterstendtoformfirst,whilestars
function (Kroupa 2001) to the total mass in sink par-
formedlatertendtobelessmassivebecauseofthecompetitionfor
ticles at that time, to obtain the the “real” distribution accretionwiththeexistingmassivestars(e.g.Bonnell 2005).
Molecular Cloud Evolution II 7
100% efficient. Of course, this effect is probably exag- pointofthenumericalbox,togetherwiththeevolutionof
gerated by our choice of perfectly anti-parallel colliding the(absolutevalueofthe)gravitationalenergy(E )for
g
| |
streams. Clouds formed by obliquely colliding streams the entire simulation box. This cylinder contains most
are likely to have more isotropic levels of turbulence. of the cloud’s mass thoughout the simulation, although
After t 6 Myr, σ begins to decrease, reflecting it also contains sizable amounts of interspersed WNM.
x
the weaken∼ing of the inflows, settling at σ 6 km s−1 Figure 8 (bottom panel), on the other hand, shows the
x
by t 12 Myr. At this time, σy and σz b∼egin to in- evolution of the same energies but with Ek and Eth cal-
crease∼, reflecting the global collapse of the cloud on the culated for the dense gas only.
yz plane, while σ remains nearly stationary, unaffected The gravitational energy is shown for the entire box
x
by the globalplanarcollapse,until t 18 Myr,atwhich because of the practical difficulty to evaluate it only for
time it also begins to increase. This∼coincides with the the cloud’s mass,althoughwe expectthe latter to domi-
onset of star formation, and probably reflects the local, nate the global gravitational energy when the cloud has
small-scale, isotropic collapse events forming individual become very massive. Indeed, we observe that Eg at
| |
collapsed objects. late times is very large and dominates the other forms
of energy. A brief period of positive values of E is ob-
g
| |
4.4. Evolution of the density, pressure and temperature served for 0 . t . 9 Myr, which can be understood as
distributions follows: Because the boundary conditions are periodic,
Poisson’s equation for the gravitational potential is ac-
Animportantconsiderationforunderstandingthepro-
tuallymodifiedto havethe densityfluctuationρ ρ as
duction of local collapse events is the distribution of the −h i
its source. This is standard fare in cosmological simula-
density and temperature as the cloud evolves. This is
tions(see,e.g., Alecian & L´eorat 1988;Kolb & Turner
showninfig.7,wherethenormalizedmass-weightedhis-
1990). Moreover, gravity is a long-range force, and so
togramsofdensity(left panel)andoftemperature(right
large-scale features tend to dominate E . Thus, during
panel) are shown at times 4, 8, 12, 14, 16, and 20 Myr g
the earlyepochs ofthe simulation,when the cloudis be-
for run L256∆v0.17. It can be seen that a cold phase
ginning to be assembled, the dominant density features
already exists at t = 4 Myr, although at this time the
are the voids left behind by the cylinders because they
cloud’sconditionsdonotgreatlyexceedthoseofthestan-
are large, while the cloud is small and not very massive.
dard CNM, meaning that turbulence is only building up
However, once the cloud becomes sufficiently massive,
in the cloud at this stage, at which the simulation re-
and the voids have been smoothed out, the cloud domi-
sembles a two-phasemedium. At t=8 Myr, the density
natesE . TheperiodofpositiveE isomittedfromboth
maxima and temperature minima have shifted to more g g
panels of fig. 8.
extreme values, which persist up to t = 12 Myr, indi-
Comparison of the energy plots (figs. 8 top and bot-
catingthatthe density andtemperature fluctuationsare
tom) with the evolution of the gas and sink masses (fig.
predominantly created by the supersonic turbulence in
5 top) and the animations shows some very important
the cloud(Va´zquez-Semadeni et al. 2006). However,by
points. First and foremost, the cloud is never in virial
t = 14 Myr, the density and temperature extrema are
equilibrium over its entire evolution. Instead, we can
seen to be again moving towards more extreme values,
identify three main stages of evolution. First, over the
and this trend persists throughout the rest of the sim-
period 8.5 . t . 14 Myr, E increases monotonically,
ulation, indicating that gravitational collapse has taken | g|
transitingfrombeingnegligiblecomparedtoE andE
over the evolution of the density fluctuations (see also k th
to becoming larger than either one of them. The exact
4.5).
§ time atwhichthis occursdepends onwhatsystemis be-
It thus appears that in this particular simulation, the
ing considered. It occurs at t 10 Myr when only the
turbulent density fluctuations act as simply as seeds ∼
energies in the dense gas are considered, while it occurs
for the subsequent growth of the fluctuations by grav-
at t 18 Myr when the entire cylinder, which includes
itational instability, as proposed by Clark & Bonnell ∼
substantial amounts of warm gas, is considered. Since
(2005). However, both in that paper as in the present
global gravitational contraction starts at t 12 Myr, it
study, the global turbulence is already decaying, and by ∼
appears that the true balance is bracketed by the esti-
definition it cannot then continue competing with self-
matesbasedonthefullcylindervolumeandonthedense
gravity. Minimally, a fundamental role of turbulence,
gas, though closer to the latter. Thus, over the interval
even in this decaying state, must be to create nonlin-
8.5 . t . 12 Myr, the increase in E is driven mostly
ear density fluctuations, ofmuchlargeramplitudes than | g|
by the cooling and compression of the gas.
wouldbe createdbythermalinstabilityalone,whichcan
Second,fromt 12Myrtot 24Myr, E continues
collapse in shorter times than the whole cloud (cf. 4.5 ∼ ∼ | g|
and 5.1). In any case, simulations in which the in§fl§ow to increase (Eg becomes more negative),but now driven
by the global collapse of the cloud. Over this period, we
lastsbeyondtheonsetofcollapsewouldbedesirable,but
seethatE forthedensegas(fig.8,bottom panel)closely
unfortunately, as mentioned in 4.6, we have not found k
it feasible to attempt them wit§h the present code and follows |Eg|, approximately satisfying the condition
computational resources,and such a study must await a E =2E . (12)
g k
| |
different numerical scheme and physical setup.
We see that in this case, this condition is a signature of
collapse, not of gravitational equilibrium, even though it
4.5. Energy evolution and star formation
lastsfornearly12Myr. Themaintenanceofthisequipar-
Figure 8 (top panel) shows the evolution of the ki- tition arises from the fact that the cloud is converting
netic (E ) and thermal (E ) energies within a cylinder gravitationalpotential energy into kinetic energy, but is
k th
of length 16 pc and radius 32 pc centered at the middle replenishing the former by the collapse itself.
8 Va´zquez-Semadeni et al.
Note that we have cautioned in 4.2 that our simula- increases from zero to 15% of the cloud’s mass in 3
§ ∼ ∼
tions may not be realistic after t 20.3 Myr, because Myr (from t=17 to t=20 Myr).
≈
by that time the stellar energy feedback may be suffi- The mainconclusionsfromthis sectionarethat a)the
cienttoreverttheglobalcollapseanddispersethecloud, cloud evolves far from equilibrium all the way from its
but even in that case the cloud will have evolved out of inception through its final collapse; b) as soon as E
g
equilibrium up to that point. dominates the dynamics, the cloud develops equiparti-
Finally,aftert 24Myr,the gravitationalandkinetic tionindicative ofthe collapse,notofequilibrium, and c)
∼
turbulent energies saturateand begin to vary in approx- star formationis rapid in comparisonwith the evolution
imate synchronicity. Although this late stage may not and collapse of the whole cloud.
be representative of actual MCs, it is important to un-
derstand what is happening in the simulation. The near 4.6. Effect of the inflow duration and initial velocity
constancy of E and E could naively be interpreted
g k dispersion
| |
as a final state of near virial equilibrium. However, in-
spection of the animations shows otherwise. At t 24 Inpreviouspapers(Va´zquez-Semadeni, Kim & Ballesteros-Paredes
Myr,the mainbody ofthe cloudis completingits g∼lobal 2005; Va´zquez-Semadeni et al. 2006), it has been ar-
collapse,andthemassinstarsisbeginningtoexceedthe gued that the star formation efficiency is a sensitive
mass in dense gas, which is itself decreasing. So, at this function of whether the turbulence in the cloud is in a
point, the gravitational energy is beginning to be domi- driven or in a decaying regime. Our system is driven
natedbythestars,ratherthanbythegas. Moreover,the at first, and gradually transits to a decaying regime, as
face-onanimation(fig.4)showsthatatthistime(t=180 pointed out in 4.3. In all of the runs in this paper,
§
incodeunits,shownintheanimations)thestarclusteris global cloud contraction and the subsequent star forma-
beginning to re-expand, after having reached maximum tion occur after the inflows have weakened substantially
compression. This leads to the decrease of Eg between (t>τinf). The main motivation behind runL256∆v0.17
t 24 and t 28 Myr. Meanwhile, E , wh|ich| is domi- was precisely to model an inflow of as long a duration
k
na∼ted by the∼dense gas, decreases because the dense gas as possible, to approximate the case of a driven cloud,
is being exhausted. Nevertheless, the outer “chaff” of although this goal was not completely achieved. For ex-
the dense gas, mostly in the form of radial filaments, is ample, Hartmann, Ballesteros-Paredes& Bergin (2001)
continuingto fallontothe collapsedcloud. This leadsto suggest accumulation lengths of up to 400 pc. Runs
a new increase in both energies, as this is a secondary with even larger boxes (e.g., 512 pc) would be desirable,
collapse. This situation repeats itself at t 32 Myr but they are either prohibitely expensive, or have an
(t=244in code units) after the secondarycol∼lapse ends excessively poor mass resolution. In addition, vertical
and its second-generation star cluster begins to expand stratification effects would have to be considered. This
away. The end of each collapse and the re-expansion of will require a transition to the code Gadget2, which
the clusters is marked by kinks in the graph of M allows for non-cubic boundary conditions and/or inflow
sinks
versus time in fig. 5 (top), indicating a decrease in the boundary conditions, a task we defer to a subsequent
star formation rate. paper.
It is important to recall as well that the condition Nevertheless,comparisonofthe runs with differentin-
given by eq. (12) is not sufficient for virial equilibrium. flowlengthsdoesshedlightontheeffectofalongerdriv-
The necessary and sufficient condition for this is that ingduration. ThecloudinrunL256∆v0.17,whoseinflow
the second time derivative of the moment of inertia of hasℓinf =112pc andlasts 22Myr(althoughitbegins
∼
the cloud be zero, and eq. (12) cannot guarantee this, weakening at T τinf = 12.2 Myr), begins contracting
∼
as many other terms enter the full virial balance of the at t 12.5 Myr and begins forming stars at t 17 Myr,
∼ ∼
cloud (Ballesteros-Paredes,Va´zquez-Semadeni & Scalo while both runs with ℓinf = 48 pc, for which the inflow
1999; Ballesteros-Paredes 2006; Dib et al 2006). Thus, lasts 11 Myr (begins weakening at t τinf = 5.2 Myr),
∼
the observed closeness of E and E in actual molecu- start to contract at t 8 Myr (recall that the inflows
lar clouds must exclusivel|y gb|e considkered an indication have the same density∼and velocity in all runs). Run
of near equipartition and probably of collapse, but not L128∆v0.24 begins forming stars at t =15.75 Myr, and
of virial equilibrium (see also Klessen et al. 2005). run L128∆v0.66 does so at t=13.77 Myr.
Thus, rather than evolving in near virial equilibrium, These results clearly suggest that a longer inflow du-
the cloud evolves far from equilibrium. E starts out ration delays the onset of both global collapse and star
g
fromessentiallyzeroandincreasesmonoto|nic|allyuntilit formation, in spite of the fact that the cloud formed by
catches up with the thermal and kinetic energies. From it is more massive. This is attributable to the fact that
thattimeon,gravitydominatestheenergybalance,lead- theturbulence inthe cloudis continuouslydrivenby the
ingtocollapse,whichinturncausesanearequipartition inflow. This is verified by comparing the three compo-
between E andE ,althoughbothenergiescontinueto nentsofthevelocitydispersionforrunsL256∆v0.17and
g k
vary syst|ema|tically. L128∆v0.24, respectively shown in fig. 5 (bottom) and
Finally, figure 5 (top) also shows that star formation fig. 9. Both runs show an initial transient peak in σx
begins at t 17.2 Myr, roughly 5 Myr after global con- ending at t 4 Myr. However, after this transient, σx
traction of ∼the cloud has begun. Yet, the star-forming inrunL256∆≈v0.17increasesagain,reachingamaximum
local collapse events proceed much more rapidly than of σx 8.5 km s−1, and decreasing afterwards, until it
the global collapse of the cloud. This indicates that the nearly∼stabilizes at a value σ 6 km s−1 at t 11.5
x
turbulence in the cloud has created nonlinear density Myr. During this time interval∼, σ σ 4 k∼m s−1.
y z
fluctuations, whose local free-fall time is shorter than Instead, in run L128∆v0.24, σ does≈not in∼crease again
x
that of the whole cloud. In particular, the mass in stars after the initial transient, and remains at a much more
Molecular Cloud Evolution II 9
moderate level of σ 5 km s−1, while σ σ cess than the bulk of the cloud. In turn, this suggests
x y z
3 km s−1. Clearly, the∼turbulence level is sign≈ificantl∼y that our model cloud would be observed as a collec-
higher in the longer-inflow run. tion of predominantly molecular clumps immersed in a
Note that the delay in the onsetof globalcollapseand large, predominantly atomic, substrate. This sugges-
oflocalstarformationcannotbeattributedtotheampli- tion is consistent with recent observations that substan-
tudeoftheinitialvelocityfluctuationsintheinflow,since tial amounts of atomic gas coexist with the molecular
runsL128∆v0.24andL256∆v0.17,whichhavecompara- phase in MCs (Li & Goldsmith 2003; Goldsmith & Li
ble amplitudes, differ substantially in these times, while 2005; Klaassen et al. 2005), with perhaps even warm
runs L128∆v0.24 and L128∆v0.66 begin global contrac- gas mixed in (Hennebelle & Inutsuka 2006). Of course,
tion at almost the same time and differ by only 15% a more conclusive confirmation of these estimates must
in the time at which they begin forming stars, in∼spite await simulations in which the chemistry and radiative
of one having more than twice the velocity fluctuation transfer are properly taken care of.
amplitude of the other. This leaves the inflow duration
as the sole cause of delay of both the global and local 5. DISCUSSION
collapses. 5.1. Implications
4.7. Local column density and star formation 5.1.1. Dynamic evolution and global versus local collapse
The initial conditions in our simulations are assumed The results presented in 4 provide general support
§
to consist exclusively of atomic gas. However, the final to the scenario outlined in Elmegreen (1993b);
densityandtemperatureconditionsaretypicalofmolec- Ballesteros-Paredes,Va´zquez-Semadeni & Scalo
ularclouds,somoleculeformationmustoccursomewhere (1999);Ballesteros-Paredes,Hartmann & Va´zquez-Semadeni
alongthewayintheevolutionoftheclouds. Oneimpor- (1999); Hartmann, Ballesteros-Paredes& Bergin
tant shortcoming of our simulations is that they can- (2001);Va´zquez-Semadeni, Ballesteros-Paredes& Klessen
notdistinguishbetweenatomicandmoleculargas,asno (2003); Hartmann (2003); Mac Low & Klessen (2004);
chemistryisincluded. Thus,wecannotdirectlytellfrom Heitsch et al. (2005, 2006); Va´zquez-Semadeni et al.
the simulations how long does it take for star formation (2006); Ballesteros-Paredeset al. (2006). The clouds in
to begin after molecular gas forms. We can, neverthe- our simulations are never in a state of virial equilibrium
less, measure the column density of the dense gas in before they convert most of their mass into stars. In-
star-forming regions, and compare it to the estimates stead, they are in a continuously evolving state, initially
for self-shielding given by Franco & Cox (1986) and obtaining their mass and turbulence simultaneously as
Hartmann, Ballesteros-Paredes& Bergin (2001), under they form out of the compression and cooling of diffuse
solar Galactocentric conditions of the background UV warm gas (see also Hennebelle & P´erault 1999, 2000;
field. The former authors find a threshold column Koyama & Inutsuka 2002; Inutsuka & Koyama 2004).
desity for self-shielding of N 5 1020 cm−2, The gas initially has negligible self-gravity compared to
ss
while the latter authors quote ∼values×N 1– its thermal support, but it quickly becomes super-Jeans
ss
2 1021 cm−2 from van Dishoeck & Black (1988∼) and because of the compression, the cooling, and the mass
va×n Dishoeck & Blake (1998). So, we take a reference increase of the dense gas, until its self-gravitating
value of N =1021 cm−2. energy eventually becomes comparable with the sum
ss
For comparison,in Table 2 we report the column den- of its thermal and turbulent energies, at which point
sities of the dense gas in the first four regions of star it begins to undergo global collapse. After this time,
formation in run L256∆v0.17 at the time immediately gravity becomes the main driver of the large-scale
beforethey beginformingstars. We seethatthe column motions in these (semi-) decaying simulations, causing
densities fall in the range 0.5–2 1021 cm−2, suggesting a near-equipartition, Eg 2Ek, which however is
that star formation locally occur×s nearly simultaneously indicative of collapse, n|ot|eq∼uilibrium.
withthe conversionofthe gasfromatomic tomolecular. The nonlineardensity fluctuations inducedby the tur-
Moreover,wecancomparethetimeatwhichweexpect bulencecollapsefasterthanthewholecloud,astheyhave
the bulk ofthe cloudto become molecular(i.e., the time shorter free-fall times, and star formation proceeds vig-
atwhichthemeancolumndensityinthecloudisreaching orously before global collapse is completed. This result
N ) with the time atwhichstar formationbegins in the is in contrast with the standard notion that linear den-
ss
cloud. A lower limit for the time at which the mean sity fluctuations cannot lead to fragmentation because
column density in the cloud equals N is given by the fastest growingmode of gravitationalinstability ina
ss
nearly uniform medium is an overall contraction of the
N N
t & ss = ss Myr, (13) whole medium (Larson 1985). Thus, a crucial role of
ss 2n0vinf 5.8 1019 cm−2 theturbulenceinthemediumistocreatenonlinear den-
×
where n and v have been defined in 3.1. This es- sityfluctuationsthathaveshorterfree-falltimesthanthe
0 inf
§
timate is a lower limit because, as mentioned in 4.1, entire cloud.
§
v is not constant, but actually decreases in time. For These results bring back the scenario of global
inf
N = 1021 cm−2, we obtain t & 17.3 Myr, suggesting cloud collapse proposed by Goldreich & Kwan (1974),
ss ss
thatthe bulkofthe cloudisstillexpectedtobepredom- but with a twist that avoids the criticism of
inantly atomic by the time star formation is beginning. Zuckerman& Palmer (1974), namely that MCs could
Together, these simple estimates suggest that stars notbe in globalcollapse because the star formationrate
form roughly simultaneously with the conversion of gas wouldbeexceedinglyhigh. Oursimulationssuggestthat
from atomic to molecular, with the local star form- MCsmaybeundergoingglobalgravitationalcontraction,
ing regions being more advanced in the conversion pro- buttheefficiencyisreducedbecauselocalcollapseevents,
10 Va´zquez-Semadeni et al.
whichinvolveonlyafractionofthetotalcloudmass,pro- formation, as well as to the rate of the star formation
ceed faster than the global collapse. Once a sufficiently process itself.
largenumberofstarshaveformed,theirenergyfeedback Note that the latency period is in fact by definition of
maypartiallyorcompletelyhaltthecollapse. Furtherre- the order of the crossing time of the entire large-scale
ductionmaybeprovidedbysupercriticalmagneticfields compressive wave, in agreement with observational evi-
(Va´zquez-Semadeni, Kim & Ballesteros-Paredes 2005; dence (Elmegreen 2000).
Nakamura & Li 2005). This scenario is consistent with
therecentproposalofHartmann & Burkert (2006)that 5.1.5. Implications for HI envelopes and magnetic criticality
the Orion A cloud may be undergoing gravitational col-
The formation of GMcs by an accumulation process
lapse on large scales.
such as the one modeled here has two important ad-
5.1.2. Cloud’s mass variation and cloud boundaries ditional implications. First, because the process starts
with lower-density gas (the WNM in our simulations)
Thefactthatthecloud’smassisnotconstantisequiv-
that is compressed by some external agent, GMCs,
alent to the property that the locus of a Lagrangian
which are the “tip of the iceberg” of the density distri-
boundary of the cloud defined (by means of a threshold
bution, are expected to have in general HI envelopes,
density)atanygiventimedoesnotremainatthecloud’s
which would constitute the corresponding “body of the
boundary as time progresses, but instead it is later in-
iceberg”. This gas would include transition material
corporated into the interior of the cloud. This implies
traversing the unstable phase between the warm to the
that the cloud does not always consist of the same set
cold phases, which would appear to have a nearly iso-
of fluid parcels, and that in a virial balance analysis of
baric behavior (Gazol, Va´zquez-Semadeni & Kim 2005;
the cloud, there is non-zeroflux ofthe physicalvariables
Audit & Hennebelle 2005; Va´zquez-Semadeni et al.
through Eulerian boundaries defined at any given time
2006), in agreement with the conclusion reached by
(Ballesteros-Paredes,Va´zquez-Semadeni & Scalo 1999;
Andersson & Wannier (1993) from observations of
Shadmehri, Va´zquez-Semadeni & Ballesteros-Paredes
HI envelopes of MCs. Such envelopes are routinely
2002; Dib et al 2006; Ballesteros-Paredes 2006).
observed (e.g., Blitz & Thaddeus 1980; Wannier
5.1.3. Absence of slow, accelerating star formation phase Lichten & Morris; Andersson, Wannier & Morris
1991; Andersson & Wannier 1993;
It is important to note that the star formation rate
Ballesteros-Paredes,Hartmann & Va´zquez-Semadeni
(given by the slope of the mass in sinks versus time in
1999; Brunt 2003; Blitz et al. 2006, see also the
fig.5[top])islargefromthestart,andweobservenolong
combined HI and CO maps of Blitz & Rosolowsky
period of slow, accelerating star formation, contrary to
2004). Weaker compressions than we have modeled
the suggestionofPalla & Stahler (2000,2002) that star
may produce mostly thin CNM sheets, with little or no
formation accelerates in time. Problems with this sug-
molecular gas, as discussed in Va´zquez-Semadeni et al.
gestion have been discussed by Hartmann (2003), and,
(2006).
within the framework of their assumptions and limita-
Second,this scenarioofmolecularcloudformationim-
tions, our simulations do not confirm it. An important
plies that the mass-to-flux ratio of the cloud is a vari-
question is whether this result will persist when mag-
able quantity as the cloud evolves. This ratio is equiv-
netic fields are included. It is possible that during the
alent to the ratio of column density to magnetic field
early stages of the cloud’s evolution, it may behave un-
strength (Nakano & Nakamura 1978), with the critical
der magnetically subcritical conditions, giving very low
colum density given by Σ 1.5 1021[B/5µG] cm−2.
star formation rates (SFRs), and then transit into a su- ∼ ×
Although in principle under ideal MHD conditions the
percritical regime, with higher SFRs ( 5.1.5).
§ criticality of a magnetic flux tube involves all of the
5.1.4. Latency period mass contained within it, in practice it is only the
mass in the dense gas phase that matters, because
The clouds in our simulations do spend a long la-
the diffuse gas is not significantly self-gravitating at
tency period between the beginning of the compres-
the size scales of MC complexes. As pointed out
sive motion that creates the cloud and the time at
byHartmann, Ballesteros-Paredes& Bergin (2001),the
which they begin forming stars ( 14–17 Myr in the
∼ abovevalueofthedensegas’columndensityformagnetic
three runs we have considered). However, this long
criticality is very close to that required for molecule for-
period is most probably spent in the atomic phase,
mation (eq. [1]) and for gravitational binding (eq. [2]),
since the column densities of the star-forming re-
and therefore, the cloud is expected to become magneti-
gions in the simulations are comparable to those re-
callysupercriticalnearlyatthesametimeitisbecoming
quired for molecule formation (Franco & Cox 1986;
molecular and self-gravitating.
Hartmann, Ballesteros-Paredes& Bergin 2001). More
specifically, Bergin et al. (2004) have found that the
5.2. Limitations and future work
timescale for CO molecule formation is essentially that
required for reaching a dust extinction of A Our simulations are limited in a number of aspects.
V
∼
0.7. Note also that during this time, the for- First and foremost, as already mentioned in 3.2, one
§
mation of H can be fast within density peaks major shortcoming of our simulations is the absence of
2
(cores), even if most of the mass of the cloud is feedback from the stellar objects onto the cloud. This is
still in the low-density regime (Pavlovskiet al. 2002; certainly an unrealistic feature. Second, our clouds are
Glover & Mac Low 2006). Thus,the conceptof“rapid” in a regime of turbulence decay after the inflows have
starformationreferstothetimeelapsedbetweentheap- subsided. Thismayormaynotbeanunrealisticfeature,
pearance of a (CO) molecular cloud and the onset star and rather it may represent a fraction of the population