Molecular Cloud Evolution IV: Magnetic Fields, Ambipolar Diffusion, and the Star Formation Efficiency

1.5 MB·English

by  Enrique Vazquez-Semadeni

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed19January2011 (MNLATEXstylefilev2.2)
Molecular Cloud Evolution IV: Magnetic Fields,
Ambipolar Diffusion, and the Star Formation Efficiency
1
1 Enrique V´azquez-Semadeni1⋆, Robi Banerjee2†, Gilberto C. Go´mez1‡,
0
2 Patrick Hennebelle3§, Dennis Duffin4¶, and Ralf S. Klessen2k
n 1Centro de Radioastronom´ıa y Astrof´ısica, Universidad Nacional Auto´noma de M´exico, Campus Morelia, Apdo. Postal 3-72,
a Morelia, 58089, M´exico
J 2Zentrumfu¨r Astronomie der Universita¨tHeidelberg, Institut fu¨r Theoretische Astrophysik, 69120 Heidelberg, Germany
3Laboratoire de radioastronomie millim´etrique (UMR 8112 CNRS), E´cole Normale Sup´erieure et Observatoire de Paris, 24 rue Lhomond, 75231 Paris
8
4Department of Physics and Astronomy, McMaster University,Hamilton, Ontario L8S4M1, Canada
1

]
A 19January2011
G
.
h ABSTRACT
p We investigate the formation and evolution of giant molecular clouds (GMCs) by
- the collision of convergent warm neutral medium (WNM) streams in the interstellar
o
medium, inthe presenceofmagnetic fields andambipolardiffusion(AD), focusing on
r
t the evolution of the star formation rate (SFR) and efficiency (SFE), as well as of the
s
a mass-to-magnetic-flux ratio (M2FR) in the forming clouds. We find that: 1) Clouds
[ formedbysupercriticalinflowstreamsproceeddirectlytocollapse,whilecloudsformed

bysubcriticalstreamsfirstcontractandthenre-expand,oscillatingonthescaleoftens
1
of Myr. 2) Our suite of simulations with initial magnetic field strength of 2, 3, and
v
4µG show that only supercritical or marginal critical streams lead to reasonable star
4
8 formingrates.Thisresultisnotalteredbytheinclusionofambipolardiffusion.3)The
3 GMC’sM2FRisagenerallyincreasingfunctionoftime,whosegrowthratedependson
3 thedetailsofhowmassisaddedtotheGMCfromtheWNM.4)TheM2FRisahighly
. fluctuatingfunctionofpositionintheclouds.Thisimpliesthatasignificantfractionof
1
acloud’smassmayremainmagneticallysupported,whileSFoccursinthesupercritical
0
1 regionsthatarenotsupported.5)Inoursimulations,theSFEapproachesstationarity,
1 becausemassisaddedtotheGMCatasimilarrateatwhichitconvertsmasstostars.
: Insuchanapproximatelystationaryregime,wefindthatthe SFEprovidesaproxyof
v
thesupercriticalmassfractioninthecloud.6)Weobservetheoccurrenceofbuoyancy
i
X of the low-M2FR regions within the gravitationally-contracting GMCs, so that the
latter naturally segregate into a high-density, high-M2FR “core” and a low-density,
r
a low-M2FR “envelope”, without the intervention of AD.
Key words: interstellar matter – magnetic fields – stars: formation – turbulence
1 INTRODUCTION MCs were considered to generally have strongly magnet-
ically subcritical mass-to-flux ratios (M2FRs), and super-
Magnetic fieldsinmolecular clouds(MCs),and thegradual
critical clouds were believed to be rare (e.g., Mouschovias
redistribution of magnetic flux within them by ambipolar
1991, sec. 2.1) so that the time for AD to allow their
diffusion (AD),havebeen thought to be crucial ingredients
cores to become supercritical turned out to be very long,
inregulatingstarformation(SF)anditsefficiency(SFE)for
of order 10-20 times larger than the clouds’ free-fall time
over two decades (see, e.g., the reviews by Shu et al. 1987;
(t )(e.g.,Ciolek & Mouschovias1994;Basu & Mouschovias
Mouschovias1991,andreferencestherein).Inearly studies, ff
1994).In this“standard model” of magnetically-supported,
AD-mediated subcritical clouds, the general mechanism for
theformationoflow-massstarswasthroughtheslowgravi-
⋆ E-mail:[email protected] tationalcontractionofisolatedcorescontainingaverysmall
† E-mail:[email protected]
fractionoftheclouds’mass,thusaccountingfortheverylow
‡ E-mail:[email protected]
observed global SFE of giant MCs (GMCs) (Myers et al.
§ [email protected]
¶ duffi[email protected] 1986; Evans et al. 2009).
k E-mail:[email protected]
2 V´azquez-Semadeni et al.
However, subsequent studies have suggested that both supersonicturbulence,V´azquez-Semadeniet al.(2008)have
MCs (McKee 1989) and their clumps (Bertoldi & McKee recentlynotedthatthenon-thermalmotions withinclumps
1992;Myers& Goodman1988;Crutcher1999;Bourke et al. must contain a significant compressive component, even if
2001; Crutcher et al. 2003; Troland & Crutcher 2008) are thedrivingispurelysolenoidal.Thiscompresivecomponent,
close to being magnetically critical, with a moderate pref- ratherthan opposinggravity,aidsit orisdrivenbyit.This
erence for being supercritical. This implies that, if a MC is is consistent with recent suggestions that, for example, the
subcritical, it is expected to be only moderately so as well, OrionAcloudiscollapsing,andproducingtheOrionNebula
in which case thetime for cores within it to become locally Cluster (ONC) in the process (Hartmann & Burkert 2007),
supercritical may be almost as short as the cores’ free-fall thatthenon-thermalmotionswithin theclumpNGC2264-
time (Ciolek & Basu 2001; V´azquez-Semadeniet al. 2005). C correspond mainly to gravitational contraction, rather
Moreover, MCs are generally believed to be supersonically than to isotropic turbulence (Peretto, Hennebelle & Andr´e
turbulent(see,e.g.,thereviewsbyVazquez-Semadeniet al. 2007), that massive star-forming regions may be immersed
2000; Mac Low & Klessen 2004; Elmegreen & Scalo 2004; in large-scale accretion flows (Galv´an-Madrid et al. 2009;
Ballesteros-Paredes et al. 2007; McKee & Ostriker 2007, Csengeri et al. 2010; Schneideret al. 2010), and that MCs
andreferencestherein),andinthiscase,ADmustbetreated in the Large Magellanic Cloud seem to follow an evolution-
nonlinearly. As a consequence, its characteristic timescale ary trend such that more evolved clouds are more mas-
is expected to decrease, suggesting again that it may be sive (Fukuiet al. 2009). Also, if the non-thermal motions
comparable to the free-fall time (Fatuzzo & Adams 2002; within clouds were homogeneous and isotropic turbulence,
Heitsch et al.2004).Thus,thegravitationalcontractionand itwould bedifficulttounderstandthecommon observation
collapse of a star-forming region must occur rapidly, essen- that SF occurs at localized spots within the clouds, rather
tially in the timescales corresponding to those of a mag- than scattered throughout their volumes (e.g., Kirk et al.
netically supercritical region, which is of the order of a 2006; Evans et al. 2009).
few free-fall times (Ostrikeret al. 1999; Heitsch et al. 2001; These findings all suggest that an important, and per-
V´azquez-Semadeniet al. 2005; Galv´an-Madrid et al. 2007), hapsevendominant,componentofthenonthermalmotions
althoughthesubcriticalenvelopemaystillbeheldupbythe observedinMCsandtheirsubstructureisactuallyconverg-
magnetic field, since, at the envelopes’ lower typical densi- ing flows, which may be driven by gravity or by external
ties, the AD timescale is indeed much longer than the dy- compressions. In fact, a model in which the nonthermal
namical timescales. motions in MCs are a gravitationally-driven mass cascade
Moreover,therealizationthatthemajorityofMCsmay hasbeenrecentlyproposedbyField et al.(2008).Moreover,
be supercritical, and that most stars, including low-mass the collision of converging flows has been shown to pro-
ones, form in cluster-forming regions (Lada & Lada 2003), duce turbulence in the compressed layers formed by them
have forced a reconsideration of the problem, to accomo- (Hunteret al. 1986; Vishniac 1994; Walder& Folini 2000;
date the fact that the standard model’s paradigm of low- Heitsch et al. 2005, 2006; V´azquez-Semadeniet al. 2006).
mass star formation in strongly subcritical clouds may be Klessen & Hennebelle (2010) have recently shown that the
theexceptionratherthantherule,evenfortheformationof turbulent kinetic energy observed in objects as diverse as
low-mass stars.1 Unfortunately,if most MCs are supercriti- galactic disks, MCs, and protostellar accretion disks is in
cal, one must once again face the old Zuckerman & Palmer general consistent with being driven by infall from the en-
(1974) conundrum that the Galactic star formation rate vironments of those objects. However, if the turbulence is
(SFR) should be roughly two orders of magnitude larger being driven by the gravitational contraction, it cannot be
than the one observed, of ∼ 3–4M⊙ yr−1 (see the supple- expectedtohaltthecontraction thatdrivesit.Thus,oneis
mentarymaterialofDiehlet al.2006).Thisisbecauseglob- faced with theZuckerman-Palmer conundrumagain.
ally supercritical clouds cannot be supported by the mag- OnewaytoavoidtheZuckerman-Palmerconundrumis
netic field,and thusshould be collapsing as a whole. iflargechunksofthemoleculargasintheGalaxyareindeed
Turbulence is often invoked as an additional source magneticallysubcriticalandthussupportedagainstgravity,
of support against the clouds’ self-gravity, as if it were while SF occurs precisely in those regions that are not, as
simply an extra source of pressure (Chandrasekhar 1951). recently suggested by Elmegreen (2007). Thus, the chaotic
Such a treatment, however, neglects the fundamental prop- spatialandstatisticaldistributionsofthephysicalvariables,
erty of turbulence that the largest velocities occur at the produced by the turbulence in the forming MCs, may play
largest scales, a property which is reflected in Kolmogorov a keyrole in thecontrol of theSFR and theSFE.
(1941)’s famous energy spectrum of the turbulence. Stud- In this paper we present numerical simulations of the
ies taking this into account have generally only considered formation and evolution of MCs, starting from their for-
it from the point of view of the energetics involved (e.g., mation out of generic compressions in the warm neutral
Bonazzola et al. 1987; V´azquez-Semadeni& Gazol 1995), medium (WNM), and reaching up to their star-forming
but have neglected the vector nature of the velocity field. epochs, in the presence of magnetic fields and AD, in or-
Having the largest velocity differences at the largest scales der to investigate the production of sub- and supercritical
impliesthattheeffectofclassicalvorticalturbulencewithin regions, and the rate at which clouds with various environ-
a cloud or clump should be primarily to distort it, rather mental conditions form stars. We focus on the effect of AD
thansupportit(Ballesteros-Paredes et al.1999a).Inhighly ontheSFEandtheglobalevolutionoftheclouds.Theplan
of the paperis as follows: in sec. 2 we present some general
considerationsontheevolutionoftheM2FRinMCs.Insec.
1 Oneinstanceofsuchaninfrequent,stronglymagnetizedcloud 3wepresentthenumericalmodelandtheparametersofthe
maybetheTaurusMC(Heyer etal.2008) simulations.Insec.4wethenpresentourresultsonthevari-
Molecular Cloud Evolution IV: Magnetic Fields 3
ability oftheM2FR,theevolution of theSFE,andtherole acloud,aclump,oracore.Althoughredistributionofmat-
ofADintheevolutionofbothsub-andsupercriticalclouds. ter along field lines does not in principle affect the total
Finally, sec. 5 presentsa summary and our conclusions. M2FR along the full “length” of a flux tube, this length
is a rather meaningless notion, since the flux tube may ex-
tend out to arbitrarily long distances. What is more mean-
ingful is the M2FR of the dense gas that makes up the
2 EVOLUTION OF THE M2FR IN
cloud, since the cloud is denser than its surroundings, and
MOLECULAR CLOUDS: FROM SUB- TO
thus it is the main source of the self-gravity that the field
SUPERCRITICAL
has to oppose. In fact, for the formation of a cloud out of
Although the most commonly considered mechanism for flow collisions in the WNM, the cloud’s density is ∼ 100
increasing the M2FR of a certain density enhancement is times larger than that of the WNM (Hennebelle& P´erault
the redistribution of magnetic flux among the central flux 1999; Koyama& Inutsuka 2002; Heitsch et al. 2005;
tubesofacloudbyAD(Mestel & Spitzer1956;Mouschovias Audit& Hennebelle 2005; V´azquez-Semadeniet al. 2006;
1977),anotherimportant,yetoften-neglectedmechanism is Hennebelleetal. 2008;Banerjee et al. 2009),andso thelat-
that, for a uniform medium permeated by a given mean ter’sself-gravityisnegligible.Thus,inthisproblem,natural
magnetic field strength B0, there is always a certain length boundariesforthesystemareprovidedbytheboundingsur-
alongthefield(termedthe“accumulationlength”byMestel face of the dense gas, allowing a clear working definition of
1985; see also Shu et al. 2007) such that flux tubes longer theM2FR.
thanthatcontain enoughmass perunitareatobemagnet- However, contrary to the very common assump-
ically supercritical. Thecriticality condition intermsof the tion of a constant cloud mass, the formation of
mass column density Σ = ρL and the field strength B0 for clouds by converging gas streams implies that the
a cylindrical geometry is (Nakano& Nakamura1978), cloud’s mass is a (generally increasing) function of time
(Ballesteros-Paredes et al. 1999b; V´azquez-Semadeniet al.
Σ/B0 ≈(4π2G)−1/2 ≈0.159G−1/2, (1) 2007; V´azquez-Semadeniet al. 2010; Banerjee et al. 2009;
Klessen & Hennebelle 2010), a result that has recently re-
whereρisthemassdensityandListhecylinderlength.This
ceivedobservationalsupport(Fukuiet al.2009).Thismeans
conditiongivestheaccumulationlength,intermsoffiducial
that, within the volume of the cloud, the M2FR is also an
values representativeof theISMin thesolar neighborhood,
increasing quantity, since the flux remains constant if the
as (Hartmann et al. 2001)
flow is along field lines, while the mass increases (see also
Lc ≈470(cid:18)5BµG0 (cid:19)(cid:16)1 cmn−3(cid:17)−1 pc, (2) Smhausse,ttahli.s2in00t7u)r.nIfimtphleiecslothuadtstthaertMs 2frFoRmoefsasencltoiauldlyiszeerxo-
pected to start out strongly subcritical (when the cloud is
where n = ρ/(µmH) is the number density of the medium, onlybeginningtoappear),andtoevolvetowardslargerval-
mH is the Hydrogen mass, and µ is mean particle weight, uesat latertimes.Rewritingeq.(2)for thecolumndensity,
taken as µ = 1.27. In principle, if the Galactic field is pri- we see that the cloud becomes supercritical when
marily azimuthal, then the Galactic ISM at large is mag-
netically supercritical in general, because field lines do not
end, and thus sufficiently long distances are always avail- Ncr =1.45×1021 B0 cm−2, (3)
able along them.2 Thus, the M2FR of a system is not a (cid:18)5µG(cid:19)
uniquely defined, absolute parameter, but rather depends on
wherethesystem’sboundariesaredrawn.Wealsostressthat where N ≡ Σ/µm is the number column density along
H
the M2FR depends on the local geometry of the consid- field lines.
ered system. For instance, a system with spherical symme- The critical column density for magnetic criticality
try has a slightly lower critical value of µcrit ≈ 0.13G−1/2 given by eq. (3) turns out to be very similar, at least for
(Mouschovias & Spitzer 1976). Measuring the criticality of solar neighbourhood conditions, to thecritical column den-
thestreamswithrespecttothisvaluewouldleadtoasuper- sity of hydrogen atoms necessary for cold atomic gas to be-
criticalconfigurationforourrunsB3ratherthansub-critical come molecular, NH ∼1–2×1021 cm−2 (e.g., Franco & Cox
configurations. 1986; van Dishoeck & Black 1988; vanDishoeck & Blake
Now consider a cloud or clump that is formed by the 1998; Hartmann et al. 2001; Glover & Mac Low 2007a,b;
accumulation ofgas along fieldlines in general.3 Intherest Glover et al. 2010). Thus, the evolution of a cloud is such
of this discussion, we will generically refer to the result- that it starts out as an atomic and subrcritical diffuse cloud
ing density enhancement as a “cloud”, referring to either (Va´zquez-Semadeni et al. 2006) and, as it continues to ac-
crete mass from the warm atomic medium, it later be-
comes molecular and supercritical, roughly at the same time
2 Note,however,thatsupercriticalitydoesnotnecessarilyimply (Hartmann et al.2001).Thisisfullyconsistentwiththeob-
collapse,sincethegasmaybethermallyorotherwisesupported, servation that diffuse atomic clouds are in general strongly
as is likely the case for the diffuse warm medium at scales of
subcritical (Heiles & Troland 2005),whileMCs areapprox-
hundredsofparsecs.
imately critical or moderately supercritical (Crutcher1999;
3 Sincecompressionsperpendiculartothemagneticfieldcannot
Bourkeet al. 2001; Troland & Crutcher2008).
inducecollapse,andcompressionsobliquetothefieldcanproduce
collapsebyreorientingthedirectionsoftheflowandthefieldlines Moreover, the critical column density given by eq. (3)
(Hennebelle&P´erault2000),ourassumedconfigurationinvolves is also very similar to that required for rendering cold gas
nolossofgenerality. gravitationally unstable, which is estimated to be
4 V´azquez-Semadeni et al.
P/k 1/2 3 THE NUMERICAL MODEL
Ngrav ≈0.7×1021(cid:18)3000 K cm−3(cid:19) cm−2 (4)
3.1 The numerical code and setup
(Franco & Cox 1986; Hartmann et al. 2001). Thus, at so- We use the adaptive mesh refinement (AMR) code FLASH
lar neighborhood conditions, a forming cloud is expected (Fryxellet al. 2000) with MHD, modified to include the
to become molecular, magnetically supercritical, and self- ADmoduledevelopedbyDuffin& Pudritz(2008).TheAD
gravitating at roughly thesame time. treatment takes the single-fluid approximation, and uses a
It is important to note that the mass accretion onto simpleprescriptiontoavoidtheneedtotracktheiondensity
a cloud due to gas stream collisions is likely to start inthisapproximation.Thisprescriptionessentiallyturnsoff
along essentially just one dimension. This mode of mass AD at low (n . 103 cm−3) densities. For more details, we
accretion may be slow, and it has been argued that it refer thereader toDuffin & Pudritz (2008).
may involveexcessively long times (e.g., McKee & Ostriker The simulations also use a sink particle prescription
2007). However, numerical simulations of the process show (Bate et al. 1995; Jappsen et al. 2005; Banerjee et al. 2009;
that, once the gas has transitioned to the cold, dense Federrath et al. 2010). A sink particle is created in a cell
phase, it soon becomes gravitationally unstable, even if the density there reaches a threshold density n =
sink
though it may remain mainly in the atomic phase, and 2×105 cm−3,and thecell is a local minimum of thegravi-
three-dimensional gravitational contraction can then en- tationalpotential.Whenacellformsasink,thelattertakes
sue, providing a much faster mode for increasing the allthemassinexcessofn intheregionwherethedensity
sink
column density (V´azquez-Semadeniet al. 2007; Elmegreen n satisfies n > n . The sink particles have an accretion
sink
2007; Heitsch & Hartmann 2008; Hennebelle etal. 2008; radiusof 0.065 pc,corresponding to roughly 1 Jeans length
Banerjee et al. 2009). The same is true if the global con- at n and T ∼20 K.
sink
vergence of the flow is driven by larger-scale gravita- Concerning the heating and cooling, we use the same
tional instabilities (e.g., Kim et al. 2003; Li et al. 2005; prescription used in V´azquez-Semadeniet al. (2007) and
Kim & Ostriker 2007). Of course, this increase of the col- Banerjee et al. (2009), which is derived from the fit by
umn density due to gravitational contraction of the dense Koyama& Inutsuka (2002) to the results of the chemistry
gas is only relevant to molecule formation. During such a andcoolingcalculationsofKoyama & Inutsuka(2000).This
process, theM2FR remains constant if thecloud’s mass re- prescriptionimpliesthatthesimulatedISMisthermallyun-
mains fixed or varies on timescales much longer than the stable in the density range 1.n.10 cm−3, which, under
contraction, and the latter occurs under ideal MHD condi- thermal balance between heating and cooling, corresponds
tions.Thegravitationalcontractioncanonlycontributetoa to thetemperature range 5000&T &500 K.
furtherincreaseoftheM2FRifthegravitationalpotentialof We model the convergence of WNM flows as the colli-
thecloudcausesittoaccretefurtheramountsofdiffusegas, sionoftwolarge-scalecylindricalstreams.Oursetupissim-
which transitions to the dense phase as it is incorporated ilar(thoughnotidentical)tothenon-magneticSPHsimula-
into thebulk of thecloud. tionofV´azquez-Semadeniet al.(2007) labeled L256∆v0.17
In all of the processes discussed so far, AD has not (see their Fig. 1). Each stream is 112 pc long and has a ra-
played a role. This is of course due to the well known fact dius of 32 pc. The streams collide at the plane x = 0 pc,
that AD is not dynamically relevant until densities as high and are embedded in a (256 pc)3 simulation box, in which
as nAD ∼ 105 cm−3 are reached (Mouschovias et al. 1985). the coordinates range from −128 to 128 pc. The numerical
Such densities are only reached in the dense cores of MCs box is periodic, and the streams are completely contained
and therefore AD is not expected to cause any important within it, ending at a distance of 16 pc from the x bound-
effectsontheglobalevolutionofMCs.Suchcores,however, aries.Theresultingcloudoccupiesarelativelysmallvolume
may be magnetically subcritical if they form by turbulent far from the boundaries, and so it can interact freely with
compressions within the cloud before AD becomes locally itsdiffuseenvironment,with relatively little effect from the
important,evenifthecloudisgloballysupercritical.Thisis boundaries. Most importantly, the cloud is free to grow by
because,underidealMHD,acoreformedwithinaninitially accretion from theWNM.
uniform cloud must have a smaller M2FR than that of the The cylindrical streams are given an initial, moder-
cloud (V´azquez-Semadeniet al. 2005). In a sense, the core ately supersonic inflow velocity v so that they collide at
inf
repeatsthepatternfollowedbyitsparentcloud,initiallybe- the centre of the numerical box. The inflow speed of each
ing strongly subcritical and evolving towards higher values stream is measured with respect to the isothermal sound
oftheM2FR,beinglimitedbytheM2FRofitsparentcloud, speed of 5.7 km s−1 that corresponds to theinitial temper-
until AD becomes important and allows its M2FR to over- ature of 5000 K. This isothermal inflow Mach number is
take that of the parent cloud, perhaps becoming supercrit- denoted M . We also add 10% random velocity pertur-
inf
ical and allowing the core to collapse. However, this notion bations to the bulk stream speeds, in order to trigger the
hasnotbeentestedinthecontextoftheglobalevolutionof instabilities that generate turbulence in the forming cloud
aGMC,in particulartakingintoaccountthepropertythat (Vishniac 1994; Heitsch et al. 2005; Pittard et al. 2005;
thecloud’s M2FR should evolve(generally increasing) with V´azquez-Semadeniet al. 2006). The box is initially filled
time. In the remainder of thepaper we investigate thissce- withWNMatauniformdensityofn=1 cm−3 (ρ=2.12×
nario, by means of numerical simulations of the formation 10−24 g cm−3, using a mean atomic weight of 1.27). At the
andevolutionofaGMC,includingAD,andfocusinginpar- temperatureofT =5000Kforthewarmphase,thisimplies
ticular on the resulting SFE.Weare particularly interested thatthecoldphasecomesintohydrostaticthermalpressure
inthestar-formingpropertiesofthecloudasittransitsfrom balancewiththeWNMatadensityn≈100 cm−3 (seeFig.
sub- to supercritical. 2of V´azquez-Semadeniet al. 2007).However, in oursimu-
Molecular Cloud Evolution IV: Magnetic Fields 5
lations the density of thecold phase is higher, because it is sion will again be toallow SFto occur more readily than if
in balance with the sum of the thermal and the ram pres- mediatedbyADaloneinsubcriticalcases,andsoourSFRs
sureofthecollidingstreams(V´azquez-Semadeniet al.2006; mustagainbeconsideredupperlimitstotheonescausedby
Hennebelle etal. 2008; Banerjee et al. 2009). AD alone.
3.2 Resolution issues
3.3 The simulations
We start our simulations at a base resolution of 5123, i.e.
We consider five numerical simulations with three reason-
∆x = 0.5pc, at the convergence point of the flows. Addi-
ably realistic values of the initial, uniform magnetic field
tonally,weallowthecodetorefineupto4additionallevels,
B0, of 2, 3, and 4 µG, respectively. These values span the
the highest of which corresponds to a maximum resolution
observed range of values of the uniform component of the
of 8192 grid points, or a grid spacing of ∆x = 0.03 pc in
Galacticmagneticfield(Beck2001).Theinitialfieldisalong
eachdirection.Forthedynamicalmeshrefinementweusea
thex-direction.
Jeans-typecriterion(Truelove et al.1997,see,however,Fed-
With respect to the cylindrical criticality criterion, eq.
errathetal.(2011,submitted)foramorestringentcriterion
(1), these cases respectively correspond to µ/µcrit ≈ 1.36,
inthepresenceofmagneticfields),requiringthelocalJeans
0.91 and 0.68, so that the first case is magnetically super-
lengthtoberesolvedwithatleast10gridcellswhilerefining
critical while the other two are subcritical. Note, however,
isactive. Beyondthelast refinementlevel,theJeanslength
that the subcritical cases are only so because of the finite
beginstobemorepoorlyresolved,untilamaximumallowed
density of n =2×105 cm−3 is reached, at which a sink extent(256 pc) of thenumerical box. For B0 =3 and 4µG,
sink lengths of 280 and 380 pc, respectively, would be required
particle is formed. We refer the reader to Banerjee et al.
torenderthesystemmagneticallycritical.Also,becausethe
(2009) for a discussion of the justification and possible lim-
criticalvalueofthemass-to-fluxratiodependsonthegeom-
itations of this choice with regards to thermal issues. Here
etry of the considered configuration, individual (molecular)
we discuss issues related with gravity and AD.
clumps could be supercritical if compared to the slightly
Thevalueofn weusewaschoseninordertoreason-
sink lower µcrit of Mouschovias & Spitzer (1976) (see discussion
ablyensurethatthematerialgoingintosinkparticlesisac-
in Sec. 4.2).
tually gravitationally bound. Indeed, Galv´an-Madrid et al.
The supercritical case is considered only in the AD
(2007) found that, when cores are defined by means of
regime, as we do not expect the absence of AD to make
a density threshold, most cores defined by thresholds &
105 cm−3 proceeed to collapse. Instead, when cores are de- a significant difference in this case. The B0 = 3 and 4µG
casesareconsideredbothinthe“ideal”andADregimes,to
fined by lower thresholds (say, ∼ 104 cm−3), a significant
investigatetheeffectofADonthestar-formingpropertiesof
fraction of them is transient,reboundinginstead of collaps-
magnetically subcritical clouds. Note that we have written
ing (see also Ballesteros-Paredes et al. 2003; Klessen et al.
theword“ideal”withinquotationmarksbecausewecannot
2005; V´azquez-Semadeniet al. 2005). Now, at n = n
sink avoid the effect of numerical diffusion, even if we turn off
and T = 20 K, the Jeans length is LJ = 0.066 pc, and theAD.Exceptfor thevalueof themagnetic field strength
sowemarginally fail tofulfill theminimum Jeanscriterion,
and whether AD is on or off, the simulations are otherwise
ofresolvingLJwithatleast4cells.However,thisshouldnot identical, all having an inflow speed of 13.9 km s−1, corre-
introduceanysignificanterrors,aswearenotconcernedhere
sponding to an isothermal Mach number M =2.44 with
inf
with the fragmentation of the core into multiple stars, nor
respect to theunperturbed,initial medium at T =5000 K.
with their mass distribution, but only with the total mass
Therunsarelabeledmnemonically,sothatthefirsttwo
going intostars.
characters of the run’s name indicate the field strength in
Another issue is that numerical diffusion can have an
µG(e.g.,“B3” denotesB0 =3µG),andthelast twodenote
effect similar to that of AD, as discussed by Klessen et al.
whether the simulation is in the ideal MHD case (“MH”)
(2000). Specifically, since the scale of the densest cores is a
orincludesAD(“AD”).Table1summarizestheparameters
few grid cells, numerical diffusion can cause the magnetic
used in each of the fiveruns. The last column in this Table
flux to diffuse out of them, in a similar manner to AD. In-
gives themaximum time reached by each simulation.
deed, we do occasionally observe the occurrence of gravita-
tionalcollapseinmagneticallysubcritical,idealMHDsimu-
lations, in which theoretically this should not occur. More-
3.4 Considerations on measuring the mass-to-flux
over,sincen effectivelyconstitutesanupperlimittothe
sink ratio
density that can be reached by any cell in the simulation,
andsincewehavechosenavalueofn thatisofthesame In what follows, we will be presenting measurements of the
sink
order as nAD, the highest densities in the code will be of M2FRinvariousregionsofthesimulations.However,thisis
theorderofnAD,andnumericaldiffusionandADwill have not an unambiguoustask in general, and in fact the M2FR
comparable effects. This limitation could be avoided by us- canbemeasuredusingdifferentprocedures.Inprinciple,the
ing an even larger number of refinement levels but, since M2FRshouldbemeasuredalongfluxtubes,inorderforthe
the simulations are already very numerically expensive (∼ measurementtobedirectlyrepresentativeofthedynamical
200,000 CPU hours per run), this option is not presently effectofthefieldonthegas.Thus,themeasurementshould
feasible. Alternatively, we could give up on satisfying the be performed by tagging a bundle of field lines, and inte-
Jeans criterion, simply raising n without increasing the grating the density along the path defined by them. Unfor-
sink
allowed numberofrefinementlevels,anoptionthatwemay tunately, such a measurement is extremely difficult to per-
attemptelsewhere.Inanycase,theeffectofnumericaldiffu- form,eveninthesimulations.Amagneticfluxtubemaylose
6 V´azquez-Semadeni et al.
Table 1.Run parameters Inordertoappreciatetheamountofdistortionthatmay
be present in the field within the clouds, in Fig. 1 we show
Run B0 AD µ/µcrit Finaltime cross sections through the centre of runs B3-MH and B4-
name [µG] [Myr] MHalongthe(x,y)plane,showingthedensityfieldandthe
component of the magnetic field on this plane. We observe
B2-AD 2.0 on 1.36 31.4
that in run B3-MH the field lines are not strongly bent,
B3-MH 3.0 off 0.91 26.1
except at the sites of local collapse. This suggests that the
B3-AD 3.0 on 0.91 35.6
M2FRs we measure by the projection method should not
B4-MH 4.0 off 0.68 48.4
B4-AD 4.0 on 0.68 59.2 exceedingly overestimate theactual flux-tubevalue.
Nevertheless, a more definite way to estimate the
amount of overestimation of the M2FR incurred in by the
Table2.Initialconditionsandthefinalsimulatationtimeofthe projection method is to use a different estimator. One such
runs presented inthis work. To calculate the magnetic critically estimatoriswhatwecallthe“localmethod”,whichconsists
oftheentiresystemweusethecriticalmass-to-fluxratio,µcrit≈ in measuring the M2FR for individual cells in the simula-
0.16G−1/2, for a cylindrical geometry of Nakano&Nakamura tions, using the total magnetic field strength. Specifically,
(1978) wemeasurethegasmassM andreadoffthetotalmagnetic
fieldstrengthB inacell,inordertocalculatetheM2FR as
M
coherenceifthefieldlinesthatcomposeitdivergefromeach µ≈ , (6)
other at long distances. Also, the field near and within the Bdx2
cloud can be significantly distorted, dueboth to the turbu- where dx is the cell’s side length. This estimator is gives a
lenceinthecloud,andtoitsgravitationalcontraction,even lower bound to the M2FR in a magnetic flux tube, since it
if the initial flow direction is along thefield lines. only countsthe mass in a single cell within that tube.This
Observationally, the M2FR is often estimated by mea- methodhasnoobservationalanaloguebut,byusingthetwo
suringtheratioofcolumndensitytomagneticfieldstrength, methods,weexpecttobracketthetruedistributionofvalues
N/B, along lines of sight (LOSs) through the cloud of in- of theM2FR, at least in a statistical sense.
terest (e.g. Crutcher et al. 2003). However, this procedure
actually intersectsmanydifferentfluxtubes,andthusgives
only an approximation to the actual M2FR of a single flux
4 RESULTS
tube. As discussed by Crutcher (1999), if a cloud is flat-
tened,itsplaneis perpendiculartothemagnetic fieldlines, 4.1 Global evolution and star formation
andthesystemisobservedatanangleθ,thentheobserved
We first direct our attention to the global evolu-
M2FR,orequivalently,theN/B ratio,willberelatedtothe
tion of the clouds. The supercritical run B2-AD
actual one by (N/B) =N/(Bcos2θ). Alternatively,as is
obs evolves very similarly to the non-magnetic runs
the case for the measurements we present below, if the ob-
presented by V´azquez-Semadeniet al. (2007) and
servation is performed along an LOS that is perpendicular
V´azquez-Semadeniet al. (2010) and the strongly su-
to the cloud, but the magnetic field is at an angle θ with
percritical runs presented by Hennebelleetal. (2008) and
respecttotheLOSandtothenormaltotheplane,thenwe
Banerjee et al. (2009). The cloud starts out as a thin
have that (N/B) = N/(Bcosθ), with |θ| 6 π/2. In ei-
obs cylindrical sheet that fragments and thickens as time
ther case, (N/B) tends to overestimate the actual value,
obs increases, until it becomes gravitationally unstable and
and on some occasions very large values may be artificially
begins a global radial contraction at t ∼ 9 Myr. Shortly
measured. This implies that a map of (N/B) is actually
obs afterthat(t∼12Myr)starformation beginsinthedensest
amapofupper limitstotheactualN/B.Thisled Crutcher
fragments (“clumps”), while the fragments continue to fall
(1999) to introduce statistical correction factors of 1/2–1/3
towards the global centre of mass, and by t ∼ 24 Myr a
to the set of M2FR values obatined in the observations he
densecloud of radius∼10 pchasformed there,which does
considered.
not appear to contract further. This lack of contraction,
The equivalent procedure for the simulation data is to
however, is only apparent, because in fact gas is being
measure M2FR along LOSs through the clouds in our sim-
consumedwithinthecloudbySF,andgasfrom theoutside
ulations. We choose the LOSs to lie along the x-direction,
continues to fall onto the cloud. Figure 2 shows this run at
since this is the direction of the mean magnetic field and
t=10, 20 and 30 Myr, illustrating its evolution.
of the colliding WNM streams, and thus it is the direc-
On theotherhand,thesubcritical runsB3 and B4un-
tionalongwhichthecolumndensityisdynamicallyrelevant.
dergo a period of initial contraction followed by a rebound,
Specifically, we then measure the M2FR as
eventuallysettlingintoanoscillatoryregime,whichconsists
Σ ρdx of alternating periods of contraction and expansion around
µ= hBxi ≡ L−1RLLBxdx, (5) bthyeLmia&gnNetaoksatmatuicraeq(2u0il0i4b)r.iuTmhessteatoes,cailslaptrioevnisouasrleyboebsstersveeend
R
where L is the stretch of the cloud along the x direction. in animations of the simulations (not shown), but they can
The path L is chosen so as to contain the full extent of also be observed in Figs. 3 and 4, which show snapshots of
thecloud’s thickness.Thisisdoneforeveryposition on the thedensityfieldofrunsAD-B3andAD-B4atvarioustimes,
plane of the cloud, to obtain maps of the M2FR over the respectively.InFig.3,itcanbeseenthatthecentraldensity
cloud’ssurface.Werefertothisas“theprojection method” islargerattheintermediatetimeshowninthemiddlepanel
of measuring theM2FR in thesimulations. than at the final time shown at the right panel. A similar
Molecular Cloud Evolution IV: Magnetic Fields 7
behavior is seen in Fig. 4, where the central density is seen increase, the dense gas mass decreases, these effects being
tobelargerattimest=20.5andt=48Myrthanatt=34 relatively more noticeable for the strong-field runs B4. In
Myr.TheB3runs,havingaweakermeanfield,contract for general, as can beseen from thebottom leftpanel of Fig. 5,
a longer time (up to t ∼ 25 Myr) and reach a smaller size the total number of sinks in the B4 and B3 cases is larger
(radiusR∼20pc)thantheB4runs(maximumcontraction only by a few sinks when AD is included. This reinforces
at t∼20Myr,with radiusR∼25 pc).TheB4 runs,which ourconclusionfromSec.3.2thatnumericaldiffusionhasan
werefollowed tolongertimes,clearly exhibittheoscillatory effect of comparable strength to that of AD in our simula-
regime,withaperiodof∼30Myr.Theoscillationscanalso tions,sinceADisabletoinducethecollapseofafewclumps
beseen in Fig. 5, which we now discuss. in addition to those that collapse because of numerical dif-
Figure 5 shows, for all the runs, the evolution of the fusion. In the case of the B3 runs, the relative effect of AD
total dense gas mass and total sink mass (top left panel), is smaller because the same difference of a few extra sink
the time derivative of the total sink mass M˙ (top right particles is a smaller fraction of the total number of sinks
sinks
panel),thetotalnumberofsinks(bottomleftpanel),andthe produced.
SFE (bottom right panel), defined as Indeed, run B3-AD forms stars at a much higher rate
thanrunB4-AD,eventhoughbotharesubcritical.Thetime
SFE(t)= Msinks(t) , (7) derivativeofthesinkmass,M˙sinks,inrunB3-ADisroughly
M (t)+M (t)
dense sinks one order of magnitude larger than that of run B4-AD, as
where M is the mass of the gas with density n > seeninthetoprightpanelofFig.5.Infact,theformationof
dense
100 cm−3, and M is the total mass in sink particles. sinkscompletelystopsinrunB4-ADaftert≈27,ascanbe
sinks
Wetake M˙ as a proxyfor theSFR. seen in the bottom left panel of this figure. The very slight
sinks
The evolution of the runs is seen to depend sensitively increase in the sink mass observed during this period (top
on the magnetic field strength. For t & 7 Myr, the dense left panel, dash-dotted red line) is dueto accretion onto the
gas mass oscillates byfactors of 2–4 forthesubcritical runs existing sink particles, rather than to the formation of new
B3andB4,inboththeMHDandtheADcases,evidencing particles. Run B2-AD, on the other hand, is seen to have a
againtheoscillatoryregimeinwhichtheserunsengage.The larger SFR than run B3-AD, although only by factors of a
maximaofthemassinthesubcriticalrunscoincidewiththe few. Also, it can be seen from all panels of Fig. 5 that the
times of maximum compression. This may be partially an onset of SFis delayed as B0 is increases, but that thepres-
artifact of the threshold density we have chosen for defin- enceof ADshortens thisdelay.It is important tonotethat
ing the cold gas. Because the global magnetic support for allSFintherunsoccursafterthecloudhasbeenassembled
the cloud prevents it from contracting much, the gravita- anditsM2FRhasreachedanearlystationaryvalue(t>10
tional binding of the cloud is generally weak. This in turn Myr).
means that the during periods of maximum expansion, a Toconcludethissection,thebottomrightpanelofFig.5
significant fraction of the cloud, although still in the cold showstheevolutionoftheSFE,definedbyeq.(7),inallsim-
phase, may be below the density threshold of 100 cm−3 we ulations. Again, a continuous trend of increasing SFE with
have used for defining the cloud. Nevertheless, if the mean decreasingB0 isobservedthroughoutoursetofsimulations.
density of the cloud varies, it is likely that the molecular The supercritical run B2-AD reaches an SFE of ∼ 35% at
fraction should actually vary as well, since molecular gas t = 30 Myr. For comparison, at this time, run B3-AD has
may be dissociated if the cloud’s column density decreases reachedanSFEof∼25%,whilerunB4-ADhasreachedonly
sufficiently(Glover et al.2010).Thus,duringperiodsofex- SFE∼3%.Althoughatfacevaluethesenumberswouldsug-
pansion, the cloud may contain a lower molecular fraction. gest that run B4-AD compares best to the observed SFEs
Inaddition,significantamountsofgasmaybeinatransient ofGMCs(Myerset al.1986;Evanset al.2009),thisconclu-
statebetweenthewarmandcoldphasesoftheISM(seethe sion may bepremature,sincetheadditional effectof stellar
reviews by V´azquez-Semadeniet al. 2003; Hennebelle et al. feedbackinreducingtheSFE(e.g.,V´azquez-Semadeniet al.
2009; Vazquez-Semadeni2009, and references therein), be- 2010) is not taken into account in the present simulations.
ing thuslost from thecold, densephase. We conclude that the SFR and the SFE can depend sensi-
The supercritical run B2, on the other hand, does not tively on the mean field strength, even for globally subcrit-
exhibitsuchstrongoscillationsinitsdensegasmasscontent. ical cases, and that the marginally subcritical run has an
Instead,thedensegasmassincreasesrapidlyatfirst.Thisis SFEcomparable to that of thesupercritical case.
because this run does not engage in any radial oscillations,
butsimplyproceedsdirectlytocollapse. Interestingly,how-
4.2 Spatial and probability distribution of the
ever, the dense gas mass later becomes roughly stationary,
mass-to-flux ratio
although this stabilization is not due to the cloud being in
anysortofequilibrium,butrathertothefactthatitisform- Akeypieceofinformationneededtounderstandthebehav-
ingstarsatroughlythesamerateitaccretesmassfrom the ior of simulations is the spatial distribution of the M2FR,
WNM. Indeed, the red lines in the top left panel, as well as as well as the evolution of its global average value. In what
thetoprightpanelofFig.5showthatthetotalsinkmassin follows,wediscusstheM2FRestimatedusingtheprojection
thesimulationincreasesatasteadypace,ofroughly400M⊙ method (cf. eq. [5]) along LOSs parallel to the x axis (i.e.,
Myr−1 duringthe time interval20<t<30 Myr. perpendicular to the plane of the cloud), taking L as the
All runs, including those that are subcritical, form path−10<x<10pcalongthedirectionoftheinflows.We
“stars” (i.e., sink particles), since, as discussed in Sec. 3.2, consider the M2FR normalized to the critical value given
numerical diffusion acts in a similar manner to AD. Never- by eq. (1). Given the flattened geometry of our clouds, we
theless,whenADisincludedthetotalsinknumberandmass consider that this is a more realistic value of the critical
8 V´azquez-Semadeni et al.
M2FR than the other commonly encountered critical value tube. Thus, local clumps may become magnetically super-
of (6π2G)−1/2 ≈ 0.13G−1/2, which holds for spherical ge- critical even within the globally subcritical simulations, as
ometry (e.g., Shu 1992). Figure 6 shows snapshots of the in the low-mass mode of the “standard model” of magneti-
normalized M2FR for runs B2-AD, B3-AD and B4-AD in callyregulatedSF(Shuet al.1987;Mouschovias1991),and
the top row, and of runs B3-MH and B4-MH in the bottom turbulent extensions of it (Nakamura & Li 2005). Accord-
row, all at t=20 Myr. In all cases, the spatial distribution ingtothetop rightandand bottom leftpanelsofFig.8,the
of the M2FR is seen to fluctuatestrongly. fraction of the volume (resp. mass) that is in locally super-
Comparing the MH and AD cases, it is interesting to critical regions increases from ∼ 3% (resp. ∼ 10%) in run
note that the spatial structure of the M2FR is similar in B4-ADto∼17%(resp.∼58%)inrunB2-AD.Insummary,
thelarge-scale features, but differsin theshapeand precise there exist plenty of mechanisms that contribute to the de-
locationofthefine,small-scaleones.Thisisamanifestation velopmentofahighlyinhomogeneousspatialdistributionof
of the system being chaotic, so that the subtle variations theM2FR,bothintheidealMHDandinthediffusivecases.
in the magnetic forces at the densest structures induced by This is consistent with recent observational determinations
ADaresufficienttochangethedetailsinthetopologyofthe suggestingthatthemagneticfieldstrengthisrandomlydis-
gas. Presumably, for sufficiently long times the differences tributedinMCs,withonlyitsmaximumvaluesscalingasa
in structure will reach eventhe largest scales. On theother power law of thedensity (Crutcher et al. 2010).
hand, simple visual inspection of the images is not enough Second,wenotefromthetopleftpanelofFig.8thatthe
to discern any trend of systematically larger values of the M2FR histogram for the supercritical run B2-AD is wider
M2FR in the presence of AD. To quantify this, we show thanthehistogramsofeitherofthesubcriticalruns,having
in Fig. 7 the histograms of the M2FR. The histograms are a larger fraction of both sub- and supercritical LOSs. This
computed for all LOSs within a circular region centered at can be understood as a consequence of the combined ac-
(y,z)=(0,0), with a 20-pc radius. From these, we see that tion of AD enhanced by turbulence and mass conservation.
theinclusion of ADcauses theproduction of asmall excess For weaker magnetic fields, the Alfv´enic Mach number is
of high-M2FR cells in comparison with the non-AD cases, larger, implying larger density fluctuations, in which AD
and that this effect is most noticeable in the strong-field is enhanced (Fatuzzo & Adams 2002; Heitsch et al. 2004;
(B4)case,inwhichthemaximumvalueoftheM2FRisover Li & Nakamura 2004), thusallowing the formation of more
a factor of 2 larger than in the non-AD case. Instead, in strongly supercritical clumps,which aredenserandcontain
the B3 case, the excess is marginal, suggesting that when moremass.Inturn,thisimpliesastrongerevacuationofthe
the system is very close to being supercritical, numerical remainingregions,whicharethoseleftwithsmallermasses,
diffusion dominates over AD. This result suggests that the and therefore with lower values of the M2FR.
relative importance of AD and numerical diffusion depends Third,fromtheimagesinFig.6,inwhichthedotsindi-
on the mean field strength, an issue that deserves further catethepositionsofthesinkparticles,weseethatnotallof
exploration, but which is out of the scope of the present the regions that appear supercritical according to the pro-
paper. jectionmethodproceedtoformstars.Thismaybeeitherbe-
Returning to Fig. 6, and focusing on the top row of causetheyaretrulysupercriticalalbeitlocallyJeans-stable,
images, which show the variation of the M2FR’s spatial or because they are actually magnetically subcritical, and
distribution as a function of the magnetic field strength, onlyappearsupercriticalduetotheprojectioneffect.There-
several points are worth noting. First, as mentioned above, fore, it is important to determine the degree to which the
the M2FR is seen to be highly inhomogeneous in all three M2FRisoverestimatedbytheprojection method.Wedefer
runs. This is also illustrated in Fig. 8, whose top left panel adetailed energy-balancestudyoftheclumpsandcoresfor
shows histograms of the M2FR in the three runs at the a future paper, but here we can take a first step towards
same time as that shown in Fig. 6. The top right and bot- addressing this problem by comparing the maps and his-
tom left panels of Fig. 8 show the corresponding cumu- tograms of theM2FR obtained with theprojection method
lative distributions, respectively weighted by volume and tothoseobtainedwiththe“localmethod”(cf.Sec.3.4).Fig-
by mass. Finally, the bottom right panel shows the mass- ure 9 shows histograms of the M2FR using this method for
weighted cumulative distribution for high-column density themiddleplaneof eachsimulation at t=20Myr.Herewe
(N > 1021 cm−2) LOSs only. From these figures, it is seen do not integrate over any LOS in order to show the largest
that µ fluctuates by over one order of magnitude in the excursionsthattheM2FRcanexhibitinlocalcells.Asmen-
subcritical runs, and by two in the supercritical one. Part tionedinSec.3.4,thelocalmethodgiveslowerlimitstothe
of this variability, especially the highest values of µ, may actualM2FRinthefluxtubetowhichthelocalcellbelongs.
be an artifact of the measurement procedure, as discussed Theµ-histogramsusingthelocalmethodexhibitanum-
in Sec. 3.4. Nevertheless, significant actual fluctuations of ber of interesting features. First, it is seen that no super-
the M2FR on the plane are expected, since the cloud is critical cells are seen in both of the subcritical runs. At
turbulent and clumpy, due to the combined action of the least for run B3-AD this is necessarily an underestimation
thermal, Kelvin-Helmholz and nonlinear thin-shell (NTSI, of the actual M2FR, since sink formation has already oc-
Vishniac 1994) instabilities (V´azquez-Semadeniet al. 2006; curred in this run at the time at which the histograms are
Heitsch et al. 2006). In the ideal MHD case, segments of made(t=20Myr).Second,thesupercriticalrunB2-ADex-
magnetic flux tubes must have M2FRs smaller than that hibits a small but finite fraction of supercritical cells, even
of the whole tube at all times. However, in the presence of with this M2FR-underestimating method, indicative of the
diffusion(numericaland/orambipolar),Lagrangian regions abundance of supercritical regions in this case. Third, the
where the density is large enough can lose magnetic flux histograms are seen to peak at µ/µcrit ∼ 10−2, while those
and reach M2FR values larger than theinitial valuefor the obtained with theprojection method peak at µ/µcrit ∼0.6.
Molecular Cloud Evolution IV: Magnetic Fields 9
Thus, the difference between the two methods is so large 4.3 Evolution of the global M2FR
that the true distribution remains relatively unconstrained
WeproceednowtodiscusstheevolutionofthemeanM2FR
between them.
of the clouds in the simulations, and its range of variabil-
A final criterion that can be used to determine the
ity. Figure 11 shows the evolution of the mean and the 3σ
accuracy of the observational-like projection method is to
rangeofthenormalizedM2FR,µ/µcrit,forrunsB2-AD(top
compare the supercritical mass fraction obtained with this
panels), B3-AD (middle panels), and B4-AD (bottom pan-
methodwiththeSFE,whichisinfactameasureofthemass
els). The computation of the M2FR is performed using the
fractionthathasbecomesimultaneouslyJeans-unstableand
projection method with the same path length and over the
magnetically supercriticaloverthecloud’shistory.Unfortu-
samecircularregionasthoseusedforthehistogramsofFig.
nately, in principle this relationship is not trivial, because
8. The left panels of Fig. 11 show the mean and 3σ range
thesupercritical mass fraction is an instantaneous quantity
computed for the set of all lines of sight contained in the
in the simulation, while the stellar mass in the cloud is a
circular region, while theright panels show thesequantities
time-integrated quantity.However,ifan approximatelysta-
computed only for the set of lines of sight for which the
tionarystateisestablishedinwhichfreshgasiscontinuously column density is larger than 1021 cm−2.
addedtothecloudsbyaccretion fromtheWNM,replenish-
These plots illustrate thefact, discussed in Sec. 2,that
ing thegas used up to form stars (V´azquez-Semadeniet al.
the M2FR of a cloud is an evolving quantity, which first
2010), then the (stationary) SFE may be considered as a
increases as the cloud gathers material from the WNM in-
lower limit to the (stationary) supercritical mass fraction. flows that assemble it. At the inflow speed of 13.9 km s−1,
The notion that the SFE is a lower limit of the supercriti-
the 112-pc-long inflows are entirely incorporated into the
cal mass fraction allows for thepossibility that efficiency of
cloudinapproximately8Myr,whichindeedcorrespondsto
conversionofgas tostarsisstilllessthanunityevenwithin
thetimescaleatwhichtheM2FRofallthreecloudsisseen
thesupercritical, Jeans-unstable gas.
to have reached a roughly stationary value. This value is
Withthisinmind,weplotinFig.10theSFEversusthe seen to be larger for weaker mean field, as expected. How-
supercriticalmassfractioninthethreerunsB2-AD,B3-AD ever, a second increase in the M2FR is seen to occur after
and B4-AD, as read off from Figs. 5 (bottom right panel) this first stabilization. This can be attributed to the fact
and 8 (bottom left panel). The plotted value of the SFE is thattheinflowsdragpartofthesurrounding,initiallystatic
the mean between the extremes taken by the SFE over the WNM along with them, as they leave a partial vacuum be-
timeintervalafterwhichtheinitialrapidgrowthhasended, hindthem.Thisdraggedmaterialflowsatlowerspeeds,and
and the error bars denote these extremes. The dotted line reaches the cloud at later times (for further discussion, see
indicates a least squares fit to thedata points, given by V´azquez-Semadeniet al. 2007).
It is interesting that hµ/µcriti for the set of all lines of
sight is smaller than unity at all times for all three runs,
SFE≈−0.034+0.54 Msup . (8) even the supercritical one, B2-AD. This is likely a conse-
M quence of mass conservation again (cf. Sec. 4.2). Because
(cid:16) (cid:17)
these statistics are weighted by area over the circular re-
gion, the supercritical regions, which are denser, occupy a
Of course, this fit is totally empirical, and is provided only smaller fraction of the surface area of the cloud, and there-
asaguidelineforthetrendoftheSFEwiththesupercritical fore the area-weighted average M2FR is smaller than unity
massfraction.Inparticular,itismeaninglessbelowthevalue inallcases.However,theaveragesforthehigh-columnden-
of the supercritical mass fraction that produces an SFE of sity LOSs, shown in the right panels of Fig. 11 are larger
zero. than unity for all times in run B2-AD, and for nearly 20
MyrinrunB3-AD.Instead,forrunB4-AD,eventhishigh-
Weobservethat,in all threecases, theSFEis afew to
N average barelyreachesvalueslargerthanunity,andonly
several times smaller than the supercritical mass fraction.
overlessthan10Myr.Finally,inallcases,thehigh-N aver-
Undertheassumptionofstationarity,thisthensuggeststhat
age M2FR decreases at late times, a phenomenon that can
thesupercriticalmassfractionobtainedthroughtheprojec-
be attributed to the high-M2FR gas consumption by star
tion methoddoes notdifferfrom thereal onebymorethan
formation.
factors of a few, on average.
In general, we conclude from Sections 4.2 and 4.3 that
A finalpoint worth notingis that,asshown in thebot-
theM2FRisatime-dependentfunctionoftimeasacloudis
tom right panel of Fig. 8, the supercritical mass fraction
builtupbyconvergingWNMstreams,whoseaveragegener-
in the high-column density gas is significantly larger in all
allyincreaseswithtime,althoughinthehigh-N regionsthe
threerunsthanthesupercriticalmassfraction forgasatall
average later decays due to consumption by SF. Spatially,
columndensities(bottom leftpanel).Thisreinforcesthesce-
theM2FRexhibitslargefluctuations,whose3-σrangespans
nario that the M2FR is determined mainly by the column
overoneorderofmagnitudeeveninthestrong-fieldcases.It
density acquired by the individual regions in the clouds by
isthehigh-M2FRtailofthedistributionthatisresponsible
accretionofgasalongfieldlines,andlessimportantlybythe
for star formation.
local effect of ambipolar (and/or numerical) diffusion.
We conclude from this section that the projection
method gives estimates of the M2FR that are within less
4.4 Buoyancy of subcritical regions
thanafactorofafewfromtheactualdistribution,andthat
the dominant mechanism that determines the local M2FR An unexpected feature we have observed in these simula-
is theaccumulation of gas along field lines. tionsisthatthesubcriticalandsupercritical regions donot
10 V´azquez-Semadeni et al.
maintaintheirrelativepositionsfixedthroughouttheevolu- estimatestheM2FRasgivenbyeq.(6),givingalowerlimit
tion of the simulations. Instead, the subcritical regions ex- to theM2FR.
hibit “buoyancy”, so that they tend to separate themselves
We next studied the evolution of the M2FR and the
from the global contraction of the clouds, even in the glob-
star-forming properties of clouds formed by both sub- and
ally subcritical cases that rebound at later times. This is
supercritical inflows. Weconcluded thatin oursimulations,
most clearly seen in animations of the simulations, but can
the effect of numerical diffusion is at a comparable level to
beseeninFig.12,whereweshowthat,astimeproceeds,the
thatofAD.Wefoundthatthesubcriticalcases doundergo
subcriticalregionsbecomesegregatedfromthesupercritical
an initial phase of contraction, followed by a re-expansion,
ones, moving outwards, and developing cometary shapes,
settlingintoanoscillatoryregime.Thesupercriticalcase,on
with their headspointing outwards as well.
the other hand, proceeds directly to collapse, as expected.
This behavior can be described as a macroscopic-scale
All cases form stars, although at greatly different rates,
analogue of the very process of AD. In the latter, the neu-
producing what appears more a continuum of star-forming
trals sink into the gravitational potential well, percolating
regimesasthemeanmagneticfieldstrengthisvaried,rather
through the ions, which remain attached to the magnetic
thanabimodalregimeofhighSFRinsupercriticalcasesand
fieldlines.Inourcase,thesupercriticalregionssinkintothe
low SFR in subcritical ones, as was the case in the “stan-
potentialwell,whilethesubcriticalonesremainintheouter
dard” model of magnetically regulated SF (Shuet al. 1987;
partsofthewell,beingheldupbythemagnetictension.The
Mouschovias 1991). In particular, the marginally subcriti-
process is also reminiscent of the interchange mode of the
calcaseB3-AD,throughtheactionofdiffusion,reachedSF
Parker instability (Hughes& Cattaneo 1987).
efficiencies (SFEs) comparable to those of the supercritical
case B2-AD. The onset of SF is delayed by up to 15 Myr
inthemost stronglymagnetized cases westudied,although
5 SUMMARY AND CONCLUSIONS this delay is reduced by a few to several Myr when AD is
included (when it is not,all SFactivity isdueto numerical
In this paper we have presented a study of the forma-
diffusion).TheSFEsobservedinoursimulationsrangefrom
tion and evolution of GMCs by the convergence of WNM
∼ 35% for run B2-AD to ∼ 3% for run B4-AD. However,
streams, or “inflows”, in the presence of magnetic fields
since stellar feedback, which would reduce the SFE even
and AD. As described by many groups (see the review by
further,isnotincludedinthesesimulations, itislikelythat
Vazquez-Semadeni 2010, and references therein), this pro-
theefficiencyofrunB4-ADisactuallytoolowincomparison
cessinvolvesthetransitionoftheatomicgasfromthewarm,
with observed values.
diffusephase to thecold, denseone, allowing the fresh cold
gastoquicklybecomeself-gravitatingandbegintocontract. We then investigated the spatial and statistical distri-
We first reviewed the general evolution of the gas and butionoftheM2FR,findingthatthisisahighlyfluctuating
the M2FR expected in this type of systems, noting that, quantity. The fragmentation of the cloud by the combined
as originally pointed out by Hartmann et al. (2001), the actionofthermal,nonlinearthin-shell,andgravitationalin-
mass-to-flux ratio (M2FR) of the cloud is expected to in- stabilities leads to the formation of clumps of high density
crease in time, so that the cloud becomes molecular, self- andhighM2FR,andoflow-density,low-M2FRpatches.The
gravitating, and magnetically supercritical at roughly the fluctuations in the M2FR we observed span between one-
same time, provided that there is enough mass in the con- and-a-half and two orders of magnitude, the distribution
vergingstreamstorenderthemsupercritical.Thiscondition being wider for weaker magnetizations. These results are
requiresthat,forsolarneighborhoodconditions,theinflows qualitatively consistent with recent observational determi-
extend beyond the accumulation length given by eq. (2). nations suggesting that the magnetic field strength in MCs
Flows that do not extend to such distances are expected is strongly fluctuating(Crutcher et al. 2010).
toform subcriticalcloudswhich,however,maybepredomi-
Next,wediscussedtheevolutionofthemeanM2FRand
nantlyatomic.Thissuggeststhat,inparticular,theconverg-
its3-σ rangeinthevarioussimulations,findingthatingen-
ing flows induced by the spiral-arm potential wells, which
eralitinitiallyincreasesasexpectedbytheassemblyprocess
have typical size scales ∼ 1 kpc, will in general induce the
ofthecloud,tolaterreacharoughlystationaryregime.The
formationofmagneticallysupercriticalmolecularclouds.On
M2FRofthehigh-column-densityLOSs,ontheotherhand,
theotherhand,convergingflowsinducedbysmaller-scalein-
tendsto decrease at later times, dueto theconsumption of
flows,suchassupernovashocks,orsimplyturbulentrandom
gas in this regime by SF.
motionsinthegas,mayleadtotheformationofsubcritical,
partially atomic clouds. Finally, we reported the occurrence of an unexpected
Wethendiscussedthedifficultiesinherenttomeasuring effect: the buoyancy of the low-M2FR regions with respect
the M2FR, even under controlled conditions such as those to the high-M2FR ones, through a process that appears as
of the simulations. Other than measuring the M2FR along the macroscopic-scale analogue of AD: the high-M2FR re-
magnetic flux tubes, which is impossible to perform obser- gions sink deeper into thepotential well of thecloud, while
vationally,andperhapsevennumerically,weconsideredtwo the low-M2FR ones remain supported by the field in the
methodsformeasuringtheM2FR:the“projection”method, outerpartsofthecloud,sothatthecloudsevolvetowardsa
whichmimickstheobservationalprocedureofmeasuringthe segregatedstatewithlowM2FRintheirperipheryandhigh-
ratio of column density to field strength along the line of M2FRtowardstheircentre,evenonscalesmuchlarger,and
sight (LOS), and which gives an upper limit to the actual densities much lower than, those directly affected by AD.
M2FR,andthe“local” method,whichsimply measuresthe The process also bears resemblance with the intercheange
massandmagneticfieldinagridcellinthesimulation,and mode of theParker instability.