SFRff=0), by counting all
assuming a mean stellar mass
Myr for Class II protostars.
versus αvir,e for these nearbyr,e values for the four clouds
αvir,e > 10). We also plot thelog with nH,min=400 cm−3
ouds with SFRff=0, SFRff
hough not as rapidly as in the
Mach number of these clouds
e of 7.7 from our numericalmean temperature in both
ion is expected to be a bit
uds than for the numericalre 15 shows the prediction of4, while the dashed line is the
pplied to the clouds fromgures. The number of clouds
low to deﬁne a clear relation
cloud SFRs are approximately

2. On average, SFRff in the simulation decreases withhave been carefully derived by Federrath et al. (2016) and thanks
increasing αvir,e.
to the well-deﬁned timescale of star formation determined through3.
The SFRff–αvir,e relation from the simulation has a large
the orbital model of CMZ clouds by Kruijssen et al. (2015), thescatter that is not explained by cloud to cloud variations
values of SFRff and αvir,e are relatively well determined for theof the other non-dimensional parameters b, β,and %.
CMZ (Barnes et al. 2017). The SFR value derived by Barnes et al.
This scatter is most likely due to a combination of time(2017), SFRff=0.01–0.04, associated with the value αvir,e=variations in the high-density tail of the gas density PDF,
Paolo
Padoan,Troels et
Haugbølle,ke Nordlund,and Søren Frimann
4.3±2.3
from Federrathal. (2016), areÅshownby the shaded
random ﬂuctuations of SFRff in clouds with a low number
area
in
Figure15.
Thesevalues
for
the
CMZ
are
clearly
consistent
ApJ 840, 48 (2017)of stars (sink particles), and a lack of statisticalwith our numerical results and with the prediction of our revised
equilibrium of the MC turbulence, due to the transient
PN11 model.
nature of the MCs.
4.
The
PN11 model has been revised, with the most
7. Summary and Conclusions
important modiﬁcation being the choice of the timescale.
We have studied the SFR as a function of cloud parameters
Whilein PN11we chose the free-fall time of the criticalby generating a large sample of realistic MCs, formed ab initio
density, tff,cr, in the revised model we assume it is thein a simulation of SN-driven turbulence in an ISM region oftimescale of formation of the characteristic postshock
250 pc size. This simulation is a continuation of our previous
structures responsible for the high-density tail of the gas
SN-driven experiment presented in Papers I, II and III, where
density PDF, tPDF=2 θ tdyn, because tff,cr<tPDF. This
we Astrophysicalhad already Journal,
demonstrated
that wechoice results in an SFR a few times lower
The
840:48 (16pp),
2017 could
May 1 recover MC
Padoanthan
et al.in theproperties consistent with the observations. In this work, thanks
PN11 model, for characteristic parameters of MCs.
to a signiﬁcant increase of the spatial resolution and theab initio 5.
in Applied
the simulation,
to an ensemble
to the each
MCscontributing
from the simulation,
the ofrevised14 introduction of sink particles, we can further test if the SFR in
realistic initial
andboundary
conditions
withprobability
model results in an SFRff–αvir,e relation with a rather
distributionssmallthat mayclosely
matchis thoseof realbyMCs.
the clouds is consistent with the observations as well. Although
scatter,which
ﬁt wellthe Therelation
1a 2given cloud cannot beofinstantaneous
value
of
the
SFR
the global SFR in the simulation will be addressed elsewhere,ff
SFRff, a = 0.4 exp (-1.6 avir ). This relation is consistentaccurately predictedbased onthe incloud
physicalparameters,
we have anticipated that it corresponds to a depletion time ofwith our previous
resultPadoanet al. (2012),
where weand can evendeviatesigniﬁcantly
fromthepredictionof the ontheorder of 1 Gyr, in agreement with global galactic values,
had already concluded that SFRff depends primarily
theoretical model,
becauseof the ever-changing nature of MCs
which is also an important test for the simulation. The main
the virial
parameter.
embeddedinthecomplexﬁlamentary
structure of the cold
results of this work are summarized in the following.
6. The values of SFR
ff in the MCs selected from theISM, and simulation,
the continuous
driving
by
SNe,
with
occasional
averaged in logarithmic
intervalsof α ,
1. The SFR per free-fall time in the MCs selected from the
explosions in close proximity or even within the cloud. Despite vir,efollow the prediction of our revised PN11 model, SFRff,αsimulation follows a broad probability distribution, with athis chaotic nature of the SFR in MCs, we have found that an(at least for α 8). The model predictions are also
peak at SFRff≈0.025, and a maximum value of approxiensemble average obtainedvir,eby selecting a very large number offollowed closely by individual clouds with Nsink>100,
mately 0.2.
clouds from many snapshots of the simulation yields valuesoffor which the role of random ﬂuctuations of SFR is
2. On average, SFRff in the simulation decreases with
SFRff within logarithmic intervals of αvir,e that nicely ﬁt the ff
expected to be small.
increasing αvir,e.
model prediction. We conclude that our simulation provides
7.
The
SFR measured in well studied nearby MCs from
3. The SFRff–αvir,e relation from the simulation has a large
support for the theoretical framework of turbulent fragmentadirect
counts of protostars is consistent with the SFR
scatter that is not explained by cloud to cloud variations
tion, while exposing the chaotic and unpredictable nature of thevaluesof the clouds in the simulation and with our modelof the other non-dimensional parameters b, β,and %.
star-formation process.
predictions. As in the simulation, the scatter is large alsoThis scatter is most likely due to a combination of timein theof SFR
increases
towardvariations in the high-density tail of the gas density PDF,
We thank
theobservational
anonymousvalues
referee
forff, andseveralusefullargervaluesofα.
vir,ehelped us improve the paper.
random ﬂuctuations of SFRff in clouds with a low number
comments and corrections that
8. TheSFR in theestimated
the aid byof a thestudy ofof stars (sink particles), and a lack of statisticalComputingresources
for CMZ,
this work
werewithprovidedthephysicalproperties
ofthe“Brick”
cloud
andequilibrium of the MC turbulence, due to the transientNASA High-End Computing (HEC) Program through the of anorbital modelof the clouds
in the Division
CMZ, is also
consistentnature of the MCs.
NASA Advanced
Supercomputing(NAS)
at Ameswith
ournumerical
results
andourtheoreticalpredictions.
4. The PN11 model has been revised, with the most
Research Center. P.P. acknowledges support by the Spanish
important modiﬁcation being the choice of the timescale.
MINECO
under
project
T.H.investigation
is supported is theOneof themost AYA2014-57134-P.
valuable ﬁndings of thisWhilein PN11we chose the free-fall time of the criticalby a Sapere
Aude Startingfrom The
Danishfor from
large scatterof SFRffGrant, including
bothlargeCouncil
variations
density, tff,cr, in the revised model we assume it is the
Independent
Researchat Centrevalues
for Star
Planetcloud toResearch.
cloud, evenat comparableof andαvir,e, and time
timescale of formation of the characteristic postshock
Formation
is funded
the Danish
Research
variations
within by
individualclouds.National
This result
has been
structures responsible for the high-density tail of the gas
Foundation.
achieved thanks to the very large number of clouds formed
density PDF, tPDF=2 θ tdyn, because tff,cr<tPDF. This
choice results in an SFR a few times lower than in the15
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We thank the anonymouscomments and corrections that
Computing resources for thisNASA High-End Computing
NASA Advanced Supercomput
Research Center. P.P. acknowle
MINECO under project AYA20
by a Sapere Aude Starting Grant
Independent Research. Research
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y1olution and theest if the SFR ins well. Althoughessed elsewhere,
depletion time ofgalactic values,
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lowing.

selected from thestribution, with avalue of approxidecreases withation has a large
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Padoan et al.

One of the most valuable ﬁndings of this investigation is thelarge scatter of SFRff, including both large variations fromcloud to cloud, even at comparable values of αvir,e, and time
variations within individual clouds. This result Padoan
has been
et al.
achieved thanks to the very large number of clouds formed
ab initio in the simulation, each contributing to an ensemble ofrealistic initial and boundary conditions with probability 15
distributions that may closely match those of real MCs. The
instantaneous value of the SFRff of a given cloud cannot be
accurately predicted based on the cloud physical parameters,
and can even deviate signiﬁcantly from the prediction of thetheoretical model, because of the ever-changing nature of MCs
embedded in the complex ﬁlamentary structure of the cold
ISM, and the continuous driving by SNe, with occasional
explosions in close proximity or even within the cloud. Despite
this chaotic nature of the SFR in MCs, we have found that an
ensemble average obtained by selecting a very large number ofclouds from many snapshots of the simulation yields values of
SFRff within logarithmic intervals of αvir,e that nicely ﬁt themodel prediction. We conclude that our simulation provides
support for the theoretical framework of turbulent fragmentation, while exposing the chaotic and unpredictable nature of thestar-formation process.
We thank the anonymous referee for several usefulcomments and corrections that helped us improve the paper.
Computing resources for this work were provided by theNASA High-End Computing (HEC) Program through theNASA Advanced Supercomputing (NAS) Division at Ames
Research Center. P.P. acknowledges support by the Spanish
MINECO under project AYA2014-57134-P. T.H. is supported
by a Sapere Aude Starting Grant from The Danish Council for
Independent Research. Research at Centre for Star and Planet
Formation is funded by the Danish National Research
Foundation.
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Figure 1. Square root of the projected density of the whole 250 pc
computational domain, integrated along the direction of the mean magnetic
ﬁeld and computed at t=70.0 Myr (upper panel) and t=74.7 Myr (lower
panel), corresponding to 14.6 Myr and 19.3 Myr after the inclusion of selfgravity, respectively. The data has been extracted at a resolution of 0.24 pc.
The grayscale intensity starts at a column density of approximately
1.0 Me pc−2 and saturates at a maximum column density of 149 Me pc−2.
The projected density is computed using only cells with a number density
above the threshold nH,min=200 cm−3, to illustrate the mass distribution
corresponding to our lower-density MC catalog.

particles. Here, it is worth mentioning that, while producing
realistic SFR values within MCs, this simulation currentlyhas
a global SFR corresponding to a gas depletion time of the orderof 1 Gyr, consistent with global galactic values (Bigiel
et al. 2011). Figure 3 shows the global star-formation
efﬁciency, SFE, versus time (solid line), where
SFE (t ) º M (t ) M(1)

Figure 2. Same projected-density ﬁelds as in Figure 1, but with a compressedgrayscale intensity range, making the saturation value of 149 Me pc−2 darker,
thus allowing the sink particles to stand out as dots of the highest intensity
value. All sink particles with masseslarger than 1 Me are shown (the largest
mass is 117.1 Me), which amount to 1408 sinks in the upper panel and 2888
sinks in the lower panel. One can clearly see (young) sinks inside the densest
ﬁlaments, as well as many (older) sinks that have already left their parent
clouds, including clusters that have cleared their surrounding gas thanks to SN
explosions of their most massive members.

and Ms the total mass in sink particles. The depletion time, tdep,
is deﬁned as
tdep º Mbox (dMs (t ) dt ) = 1 (d SFE (t ) dt12
).

(2 )

In Figure 3, the short-dashed and long-dashed lines show SFE
versus time for depletion times tdep=1.5 and 0.8 Gyr,respectively. One can see that the global SFR corresponds
approximately to these values of tdep in the approximate time

The Astrophysical Journal, 840:48 (16pp), 2017 May 1

Figure 3. Star-formation efﬁciency vs. time for the whole computational
volume (black thick solid line), and for individual MCs from the simulation,
selected with nH,min=200 cm−3 (blue thin lines) and nH,min=400 cm−3 (redthin lines). The thin line of each cloud covers 1.68 Myr, the time interval
during which the clouds are followed (notice that the SFE for ﬁve MCs with
Mcl > 2 × 104 Me (nH,min = 200 cm−3) and for six MCs with Mcl>104 Me(nH,min = 400 cm−3) has been plotted with thicker lines). The vertical dottedlines mark the times of the 10 simulation snapshots where the MCs wereselected, at intervals of 1.5 Myr. The short-dashed and long-dashed lines show
SFE vs. time for depletion times tdep=1.5 Gyr starting at t=61 Myr andtdep=0.8 Gyr starting at t=67 Myr, respectively. The two green shaded
areas show two examples of SFE vs. time for tdep=0.05 Gyr, characteristic of
the MCs from the simulation, starting at the ﬁrst and fourth cloud selection
times. This cloud depletion time is evaluated as 1 á (SFRff t ff ) ñ averaged overall clouds with non-zero SFR.

intervals 61–67 Myr and 67–74 Myr. At the same time, theclouds selected in the simulation as described in the following
section exhibits a much steeper time dependence of their local
SFE, as shown by the short thin lines in Figure 3, and by theshaded areas illustrating the average cloud depletion time of

The Astrophysical Journal, 840:48 (16pp), 2017 May 1

Padoan et al.

Padoan et al.

search is carried out at a uniform resolution of 5123 cells, that is, aspatial resolution of 0.49 pc, but the cloud properties are computed
using all the tracer particles identiﬁed within each cloud. Because
the tracers record all the hydrodynamical variables interpolated attheir position, and due to the very large number of tracers inhighdensity regions, MC properties are derived with the hydrodynamical variables sampled at the highest local spatial resolution of
the AMR grid, up to the highest resolution of 0.0076 pc in thedensest regions.
MCs are selected from 10 snapshots, at equal intervals of
1.5 Myr, with the ﬁrst snapshot at 4.0 Myr after the inclusion ofself-gravity in the simulation. Only clouds satisfying the
Thefollowing
AstrophysicalJournal,
840:48 (16pp),
2017 May(1)1 the cloud mass isthreeconditions
are retained:
Mcl>1000 Me, (2) the rms velocity is σv<4 km s−1(toavoid MCs too strongly affected by recent nearby SNe5), (3)the cloud does not disperse during the next 1.5 Myr, meaningthat it is not doubling its effective size in that time interval (toavoid MCs whose properties are evolving too rapidly). With
these conditions, the number of MCs is reduced from 391 to
313, 203 clouds with nH,min=200 cm−3 and 110 clouds with
nH,min=400 cm−3. The conditions allow for a better comparison with the SFR model that does not account for transient
processes like SN feedback or cloud dispersal. It may yield asample more suitable for the comparison with the observations
as well,because MCs strongly affected by SNe, or in the
Figure 4. Properties of MCs selected from the simulation with density thresholds nH,min=200 cm−3 (empty circles) and nH,min=400 cm−3 (ﬁlled circles). The solid
processandof
being
alsovalues
sufferfroma strong
σv averaged
within logarithmic intervals of Re (upper panels), and the values of αvir,e and βdashedlines aredispersed,
the power-law ﬁtsmay
to the meanof Mcl andAstrophysical Journal, 840:48 (16pp), 2017 May 1
averaged inside logarithmic intervals of Mcl (lower left panel) and σv (lower rightinpanel).
thefeedbackby H II regions, which is not modeled The
simulation.
In thefollowing,
we by
will
for each andcloud
formedand dispersed
thecompute
SN-driven turbulence,theythe threeis then deﬁned as the following non-dimensional SFR:
, and β,thenon-dimensionalparameters,ﬁlamentary
αvir, %structure
are part of an interconnectedthat expressing
extends
SFRff º SFR [Mtr (t j ) tff ] ,(15)
the outer scaleof the turbulence,
approximatelymagnetic
ratios upof to turbulent,
gravitational,
thermal,and
70–100pc (deAvillez& Breitschwerdt
Joung etestimates
theal. ratiowhereof tff is the free-fall time at the mean density of the cloud.
energies.
The
virialparameter,
α , 2007;
2009; Padoan et al. 2016b), so they canvir
hardly be considered as
To evaluate the ﬂuctuations of SFRff, we also measure it within
turbulent
andgravitationalenergiesinaspherical
cloudofwell-deﬁned and long-lived isolated entities. This complex
14 shorter time intervals of length δt=Δt/14=0.12 Myr.
one-dimensional
velocity
dispersion
radius ever-changing
Rcl, mass Mnatureof MCsis not a major concern
for thisFigure 5 shows SFRff versus αvir,e for our two cloud samples
cl, andwork, as&ourgoal is tocorrelate the SFR with the physical(empty circles). There is a clear trend of decreasing SFR with
McKee1992):σv (Bertoldi

Padoan et al.

increasing virial parameter, though with a very large scatter. The
parameter of MCs at a given time, rather than to follow theirscatter increases with increasing αvir,e and must have primarily a2long-term evolution.
2
⎞
⎛
5
R
s
40time tinterval
2Ek Myr
random origin due to variations from cloud to cloud at ﬁxed virial
ff
To deﬁne the SFR
Δt=1.68
v clover the⎟⎜,(3Figure
) and
a
º
=
~
vir
parameter
time variations
each
cloud, because we doafter the cloud identiﬁcation,we
2 tonly the totalEmass in the
6. Timevariation withinof SFRff in each cloud, with clouds ordered byGM
3ptrack⎝ virialdyn ⎠parameterg for all theβ at black
constant
not increasing
detect any value
strongofdependenceof SFRff on%ff orFigure 5. SFR
free-fallassociatedtimecl vs. witheffective(solidline). Each error bar
their time-averaged
SFR
tracerperparticles
the initialcloud deﬁnition. We(and thescatterthe
is muchsmallerthe case SFRof theofSFR
αvir,eclouds selectedfromthe simulation.
The arrows
theisvaluesof αextends
betweenmaximum
andinminimum
a cloud, among the
do notaccount
for the possibility
that show
new gasaccreted
tovir,etheof
predicted
our modelusingparameters,
as shown Myr
in size. Values ofThefromerror
the solidThelineSFR
are isthethe where
MCs withSFR
thedynamical
time
is connecteddeﬁnedasinterval.
values bymeasured
within14 thetimecloudintervalsof Δt/14=0.12
cloud
orff=0.
removeditbarsduringthat timeby
FigureThe originof thisscatter isSFR
discussed
Section that
5.4, never form stars
values of SFR
ff averaged within logarithmic intervals of αvir,e. The dashedindicatethat theminimum
is zero.in Clouds
10−511).
thus
where
our model
predictions.
illustrate
the shownamountin the ﬁgure (35ourºrevisedcomputed with theline is SFRff,α, the analytical ﬁt to
the wholeintervalΔt=1.68ToMyrare nottdyn
R cl PN11
sv,3Dmodel.
(4during) we presentof clouds
time variations,
showwithin Figurephysical parameters of the MCs fromsimulation, given by Equation (28).
200 maximum
cm−3 and and6 clouds for nH,
in the wecatalognH,min 6= theMtr (tthej ) - Mtr (t j + Dt )minimum
values−3of SFRff measured in the 14 time intervals
The dotted line is a function
SFR ºlike αvir,α, but with a ,smaller exponential(14)
min = 400 cm .)
D
t
The
last
equality
in
(3)
is
exact
in
the
case
of
an
idealized
δt=0.12 Myr in each cloud. The characteristic scatter of SFRff
coefﬁcient, to trace roughly an upper envelope of the plot. The ﬁlled circles
due to time variations is over one order of magnitude.
>2×104density.
Me (upperFor
panel)more
and therealistic
six MCs cloudindicate
the ﬁve MCs
withofMcluniform
spherical
cloud
4
where,
M
(t)
is
the
total
mass
of
tracers
in
the
cloud
that
have
not
tr
The mean values of SFRff in logarithmic bins
with Mcl>10 Me (lower panel).
13 of αvir,e aremass distributions,virial parameteris tjonly
an approxbeen accreted ontothe
sink particles
at the time t, andis the timeshown by the error-bar symbols in Figure 5, where the size of
to of
the kinetic
jth snapshot
which the cloudistheerror bars is equal to the standard error of the mean (the rms Figure 6. Time variatio
imationcorrespondingof the ratio
and ingravitationalenergies.
The
5. SFRbyperthesquarefree-fall timevs.ofeffective
parameter
for allthe in increasing value of their
the bin). Despite
scatter,timethesemean&values
identiﬁed.theThe sizable
SFR per free-fall(Krumholz
McKee follow2005) Figure
dividedroot
the thevirial
numberof datapointsrms Mach number is the ratio of the three-dimensional rms

clouds selected from the simulation. The arrows show the values of αvir,e ofvery closely the analytical ﬁt to our model (see Section 5.3) the
SFRff=0. The error bars connected by the solid line are thevelocity
and
the
sound
speed,c:7 MCsof withs
SFRff averaged within logarithmic intervals of αvir,e. The dashedapplied to all the MCs selected from the simulation, values

extends between the mavalues measured within 1
10−5 indicate that the mMay 1

physical parameters of the MCs from the simulation, given by Equation (28).
clouds in the catalog with nH,min = 200 cm−3 and 6 clouds for nH,−3
The dotted line is a function like αvir,α, but with a smaller exponential
min = 400 cm .)coefﬁcient, to trace roughly an upper envelope of the plot. The ﬁlled circles
etTheal. Astrophysical Journal, 840:48 (16pp), 2017 May 1
six MCs
indicate the ﬁve MCs with Mcl>2×104 Me (upper panel) and thePadoan
with Mcl>104 Me (lower panel).

the bin). Despite the sizable scatter, these mean values follow
very closely the analytical ﬁt to our model (see Section 5.3)
applied to all the MCs selected from the simulation,
2
SFRff, a = 0.4 exp (-1.6 a1vir,e). This agreement is nearly
equally good for both cloud samples, with the lower-density
one (nH,min = 200 cm−3) yielding only slightly smaller meanvalues (though always within approximately one standard errorof the mean from the analytical ﬁt of the model). The
agreement with the model breaks down only at large values of
αvir,e, where the statistical signiﬁcance of the mean values of
SFRff is low, due to the very small number of data points perbin (the standard error of the mean can still be small, if mostpoints are close to each other by chance).
The arrows in Figure 5 indicate the values of αvir,e of theclouds,where SFRff=0. The upper panel shows that asigniﬁcant fraction of clouds in the sample with nH,min=
Figure 8. Probability distributions of SFRff of our two samples of MCs
200 cm−3 (approximately 18%) have SFRff=0, even atextracted from the simulation.
relatively low values of αvir,e. The fraction drops to 5% in
the case of nH,min= 400 cm−3 (lower panel). We ﬁnd that onlyin Figures 5 and 7) are SFRff= 0.04±relatively small clouds have SFRff=0. Figure 7 shows that(shown as ﬁlled circles
−3
3
0.01
(n
=
200
cm
) and SFRff=0.05±0.02 (nH,min =H,minSFRff=0 only in clouds with Mcl<5×10 Me or−3
Mcl<2×103 Me in the lower and higher nH,min samples400 cm ).
Figure 6.TheTimelower
variationof SFRff ofin eachcloud, inwithFigure
clouds 7
orderedrespectively.
envelope
the plots
rises by The probability distributions of the SFR are shown inal parameter for all the(solidblackline). Eacherror Figurebarincreasing
value of theircloudtime-averaged
SFR
8. Although they both peak at SFRff≈0.025, theffsharplywithincreasing
mass,
so
that
the
scatterinSFRw the values of αvir,e ofextends between the maximum and minimum SFR of a cloud, amongffthe
densercloudshave,peronfree-fallaverage,larger
SFRfffrom=
rathersmall
for themostmassiveclouds.
The
averageby the solid line are
Figure
7. SFR
vs. cloudmassvalues,
for the withsame MCs
valuesmeasured
within14 timeintervals2017
of Δt/14=0.12
Myr size.values
Values ofThetheis
AstrophysicalJournal,
840:48(16pp),
May
1
Padoan et theal.
−5
ls of αvir,e. The dashedfor the10ﬁveand0.03±0.03
nH,min=200ﬁgures.
cm−3 and SFRff= 0.04±0.04
simulationfor
as in
the previous
indicate sixmostthat the minimum
SFR clouds
is zero. Clouds
neverform stars
massivein thethat
twosamples−3
model computed with theduring the whole interval Δt=1.68 Myr are not shown in the ﬁgure (35bins of α catalog
closely
follow the revised model (see for nH,min=400 cm . These values are comparable togiven by Equationlogarithmic(28).
clouds in the vir,ewith nH,min = 200 cm−3 and 6 clouds for nH,−3
Figure 5). Observational
efforts
shouldalso aim at compiling very observational estimates of the SFR in MCs, as discussed
h a smaller exponential
min = 400 cm .)e plot. The ﬁlled circles
8 in Section 6.
large samples of MCs with estimated values of SFRff and cloud below
r panel) and the six MCs
In Figure 9, we plot the mass fraction turned into stars Padoan
in the et al.
parameters.
The Astrophysical Journal, 840:48 (16pp), 2017 May 1
time interval Δt=1.68 Myr after the cloud is identiﬁed, which
is given by SFR×Δt, versus the initial star-formation
mean values follow
6. Comparison with Observations
efﬁciency at the time the cloud is identiﬁed, SFE0, given byel (see Section 5.3)
The average SFR in MCs can be deduced from the global
m the simulation,
SFR in the Galaxy. For example, Krumholz & Tan (2007)
SFE 0 º Mtr,s (t j ) (Mtr,s (t j ) + Mtr (t j )) ,(16)
eement is nearly
derived a value of SFRff=0.02, and Murray (2011) a value
h the lower-density
approximately three times smaller, SFRff=0.006, due pri- where Mtr,s(tj) is the total mass of the tracer particles inside theghtly smaller mean
y one standard marilyerror to a different IMF assumption. To move beyond global
cloud and already accreted onto sinks at the time tj when thethe model). values
The and derive the dependence of the SFR on physicalcloud is identiﬁed, and Mtr(tj) is the total mass in tracers foundly at large valuesofparameters
it is necessary to study the properties of individualinthe same cloud and still in the gas phase.
the mean valuesof
MCs and directly measure their SFR. Focusing on the mostr of data points per
We can express the initial SFE as SFE0=SFR0×tcl, where
activestar-forming
regions
intheGalaxy,selectedbytheirll be small, if most
average SFR up to the time the cloud is identiﬁed
0 is the15. SFR per free-fall time vs. effective virial parameter for the MC
free–free emission, Murray (2011) found a rather large mean SFRFigureis thecloud
age.
Althoughsome
the MCs(t < sampletj), andbytclEvanset al.
(2014;
ﬁlled
circles) andonly
for theMCs ofselectedfrom our=0.14–0.24,
and,
more
importantly,
a
very
wide
SFR,
SFR
ff
lues of αvir,e of the−3
cmthe(emptycircles). The
simulation
with density constantthreshold nSFR
have
an
approximately
duringtime
intervalH,min=400
range
anel shows thata of values, from 0.001 to 0.59. Vutisalchavakul et al.
dashedis the sameﬁt to ourSFR
modelpriorprediction,
ff,α, as inif we line
assumedthatanalytical
the averageto theSFRcloud(2016)
= selected star-forming regions based on a catalog of H II Δt, previous
mple with nH,minﬁgures. The dotted line is our model prediction for a single set of
identiﬁcation
were
equal
to
the
average
SFR
after
the
cloud
FRff=0, evenat Figureregions(Anderson
et al. 2014),
but ofderived
much
lower
SFRs,
8. Probability
distributions
SFRff ofour twosamplesof MCs parameters, using the average Mach number value in the observational sample,
SFR0=SFR,
could
of a asextracted
the simulation.
on drops to 5%
in on
the observedwe
cloudshave derive
the same themeanagetemperature% = 9.4 (assuming
based
the from
22 μm
band of the WISE satellite (Wright identiﬁcation,
the tclouds
the simulation),
the same b as
andtclβ≡SFE
as the averagevaluesthe time
when it andis selected,). We ﬁnd thatetonly
cl, at from
0/SFR.
al. 2010), with an average value of SFRff=0.007, of the cloud,from thevalues
simulation,b=0.48
and β=0.37(because
theydashedare unknown
for theFigure 7 showsorderthat ofConstant
of
t
are
shown
in
Figure9bylines.
(shownasﬁlled
circlesin
Figures
5
and
7)
are
SFR
=
0.04±
cl
ff
the
mean
Galactic
value.
The
mid-infrared(MIR)
observed clouds). The green shaded area shows the estimated SFRff and αvir,e<5×103 Memethod
or 0.01
envelopeof Federrath
the plotet al.
in (2016)
Figureis verywell
200systematicallycm−3) and SFR= forupper
H,min =
ff=0.05±0.02
H,min Thethey(nused
mayunderestimate
the (nSFR
the CMZ
clouds, fromand 9Barnes
et al. (2017).
her nH,min samples 400 cm−3).
described by a line of constant cloud age, tcl≈1.5 Myr. Thus,byafactoroftwotothree,
sotheircorrected
averagevalue
may
ots in Figure 7 rises
Theas probability
distributions
SFR byareMurray
shown ifin the SFR prior to the time of the cloud identiﬁcation werebe as large
0.02. Although
the listsofof theobjects
distributed around the model prediction shown by the dotted
the scatter in SFR
ff
Figure8. Although they
both
peak
at SFRff≈0.025,
constant
and equal to the average SFR measured after the cloud(2011)
andVutisalchavakul
etal.
(2016)
couldinprinciple
be the
The average values
Figureclouds
7. SFRhave,
per free-fall
vs. cloud larger
mass forthe same
MCsSFR
from =the line. Furthermore, the values of SFRff of the clouds from thedenseron
average,values,withff
used to comparewith thecloud properties, there is a identiﬁcation, none of the cloudsselected from the simulation
simulationthe
as inSFR
the previous
ﬁgures.
in the two samples

Padoan et al.

Figure 9. Cloud mass fraction converted into stars in the time interval during
which we follow the MCs, Δt=1.68 Myr, vs. the cloud initial SFE, SFE0.
Assuming that the SFR prior to the time of the cloud selection, SFR0, wasconstant and equal to the average one measured during Δt after the cloud
selection, SFR0=SFR, we derive an estimate of the MC age at the time of
selection, tcl≡SFE0/SFR, shown by the dashed lines for ages of 1.5, 6.0, and
24 Myr. As in previous ﬁgures, the ﬁlled circles are the ﬁve MCs with
Mcl>2×104 Me (upper panel) and the six MCs with Mcl>104 Me (lower
panel). The arrows indicate the values of SFE0 for clouds with SFR=0.

The MC lifetime was studied in Paper I, using our previous
SN-driven simulation, where we found that it was approximately
four dynamical times on average, tlife/tdyn=4.5±1.7,
resulting in cloud lifetimes consistent with those estimated for
MCs in the Large Magellanic Cloud (Kawamura et al. 2009).
Using the average cloud dynamical times in our two cloud
samples, we would estimate an average cloud lifetime of
tlife=14.6± 4.5 Myr for nH,min=200 cm−3 and tlife=11.1±3.5 Myr for nH,min=200 cm−3. Most clouds in thetwo samples have smaller estimated ages than such lifetimes,suggesting that most of the ages derived from the assumption of
constant SFR are reasonable.
5. The Turbulent Fragmentation Model

14

The SFR can be modeled as the result of the gas fragmentationby the supersonic turbulence (Padoan 1995; Krumholz &McKee 2005; Padoan & Nordlund 2011; Federrath & Klessen
2012; Hopkins 2013). Given the probability density function(PDF) of the gas density in supersonic MHD turbulence and a292 Low-mass young stellar population and star formation history of the cluster IC#40 1805 in the W4 H II region
Neelam Panwar, M. R. Samal,
Jose, W. P. Chen, D. K. Ojha, K. Ogura, H. P. Singh and R. K. Yadav
2688A. K.
N.Pandey,
Panwar etJ.al.
MNRAS 468, 2684 (2017)
背景：• 小質量星もそのおよそ半数はmassive young clusters or
OB associationsの中で誕生する• 大質量星の影響はあるか？
例 質量関数、星形成効率、若い星のまわりの円盤の進化• IC 1805（距離~2.0kpc）: HII領域W4の中心にある若い散開
星団 W4の中はOB星の星風により空洞化 bright-rimmed
cloudsが見られる
観測データ：• CFHT: deep V- and R-band images (検出限界0.2 Msun）• Spitzer: IRAC (3.6, 4.5, 5.8, 8.0 micron) and MIPS (24micron)• 2MASS: JHKs bands• Chandra: 0.5-8.0 keV
結果：
1. YSO候補384星を同定（Class I/II 101星＋Class III 283星）
2. 平均年齢~2.5 Myr、質量0.3-2.5 Msun、cluster半径~5pc
2688
3. 質量関数 power index -1.23+-0.23
Salpeter (-1.35) や他の星形成領域とconsistent
4. 円盤進化• 円盤を持たない星は円盤を持つ星より古い• 円盤を持つ星の割合が大質量星の近くで下がるという証拠は（統計の範囲内で）ない
5. IC 1805/W4は巨大なフィラメントから生まれたかもしれない

2688

N. Panwar et al.

N. Panwar et al.

Figure 2. (a) IRAC ([3.6]–[4.5])/([5.8]–[8.0], (b) IRAC/2MASS and (c) IRAC/MIPS colour–colour diagrams for the sources (black dots) in IC
error bars represent the 1σ uncertainty in respective colours. Class I and Class II sources are shown with red square and blue diamond symbols, reFigure 2. (a) IRAC ([3.6]–[4.5])/([5.8]–[8.0], (b) IRAC/2MASS and (c) IRAC/MIPS colour–colour
for with
themagenta
sources(black dots) in IC 1805. The
Transition discdiagrams
candidates are shown
triangles.
error bars represent the 1σ uncertainty in respective colours. Class I and Class II sources are shown with red square and blue diamond symbols, respectively.
or [4.5] − [24] > 4. Out of four Class I sources identified inproperties of 647 stars in the IC 1805 using Chandra/ACTransition disc candidates are shown with magenta triangles.

or [4.5] − [24] > 4. Out of four Class I sources identified in
Section 3.1.1, two are reclassified as Class II, while the remaining two do not possess 24 µm measurements. Fig. 2(c) shows theSection 3.1.1, two are reclassified as Class II, while the remaining two do not possess 24 µm measurements. Fig. 2(c) shows the
IRAC/MIPS colour–colour diagram for the sources in the region.
In Fig. 2, transition disc candidates based on the MIPS photometry
are marked as magenta triangles.

vations. ACIS-I consists of four 1024 × 1024 pixel2 CCD
cover a 17 arcmin × 17 arcmin FOV, and covers the 0.5
energy band with a spectral resolution of ∼150 eV at 6
PSF radius of 0.5 arcsec within ∼2 arcmin of the on-axis
degrading to ∼6 arcsec at a 10 arcmin off-axis angle. W
these Chandra X-ray sources for the counterparts of odata set with a less stringent matching radius of 3 arcse
et al. 2013a) because of the large off-axis beam. The a
between X-ray and optical positions is excellent in majocases (∼80 per cent), with offsets <1 arcsec. In a few cas

15

properties of 647 stars in the IC 1805 using Chandra/ACIS observations. ACIS-I consists of four 1024 × 1024 pixel2 CCD chips that
3.1.4 YSOs from Chandra X-ray data
cover a 17YSOs
arcmin× 17 arcmin FOV, and covers the 0.5–8.0 keV
are often strong X-ray emitters compared to their main-61.429279
22.10 ± 0.02
20.66 ± 0.01
5 based on the YSO sample 1(left-hand 37.888229
panel) and 2MASS√ data (right-hand panel). The continuous curve
37.895050
21.28 ± 0.01
19.79 ± 0.01
o the observed data points. 2The error bars
represent ± N61.506271
errors.
3
37.919338
61.502380
19.87 ± 0.01
18.61 ± 0.01
...
...
...
...
...

ter. Hence, we also
sc-bearing sourcesn the 2MASS and
Myr for the cluster,using all the YSOs.
is probably due torcumstellar extinconary stages (Josenearly parallel to
es not have muchmstellar extinction
ass 0/I sources, but
PMS sources such
umstellar discs rebservations, which
mention that Joseet of optically visages are largely in
l CMDs. A binary
onsequently CMD
ase of equal-massuence in the CMD
ect of binarity caners (∼10 Myr) forNGC 7160; Siciliaoung clusters likethe isochrones arege estimation will
he stars in clusters
Similarly, though
optical bands (e.g.
z-Ledesma, Mundt
lity tends to moven little effect on the5). Despite low difumber of disc-lessad due to the comsince the observed

3.01 ± 0.65
1.20 ± 0.30
1.57 ± 0.57

0.31 ± 0.02
0.30 ± 0.02
0.57 ± 0.08⊙

Figure 8. (a) Histograms of the age distribution of the Class II and Class III candidates, (b) cumulative distributions of CTTSs and WTTSs in the cluster
region as a function of stellar age.

by assuming a normal error distribution and using Monte Carlosimulations (see e.g. Chauhan et al. 2009). Since we have used the T-association and of Chauhan et al. (2009) for the BRC regions,evolutionary model by Siess et al. (2000) for all the PMS stars, the that WTTSs are relatively older than CTTSs, which supports theage and mass estimations given in Table 4 should not be affected notion that CTTSs evolve into WTTSs. Here again, we would likeby systematic errors. A sample of Table 4 is given here and the to remind readers that these results are affected by the uncertainty
complete table is available in the electronic version.
associated with the membership of the stars (see Section 3.1.4),
4 DISCUSSION
4.1 MF of the YSOs

particularly the WTTSs (spectroscopic information is needed toconfirm the true nature of WTTSs and CTTSs).
4.3 Disc fraction variation in the cluster

Young clusters are the best objects to study the initial MFs. Since
The disc fraction of YSOs in clusters decreases with ages (e.g.
young low-mass stars have limited time to segregate from the cluster
and/or loose mass through evolutionary processes, the variation Haisch, Lada & Lada 2001; Hernández et al. 2008). Haisch et al.
of the MF gives clues about the physical conditions of the star (2001) quantitatively concluded that in young clusters an initiallyformation process. The MF is defined as the number of stars per very high (80–90 per cent) disc fraction decreases to 50 per centunit logarithmic mass interval, and is generally represented by a in ∼3 Myr and reaches almost 10 per cent at ∼5 Myr. Generally,
massive stars seem to lose their discs earlier than lower mass stars.
power law with a slope,For example, most low-mass stars (spectral type K5 and later) lose
their discs in 5–7 Myr (e.g. Sicilia-Aguilar et al. 2005), whereas
where N(log m) is the distribution of the number of stars per unit
for the intermediate-mass Herbig Ae/Be stars the corresponding
logarithmic
thecandidates
star clusters
the region.
solar The Figure 7. The MF of the candidate YSOs in the mass range (0.3 ≤ M/M⊙
Figure 6.mass
V /(Vinterval.
− R) CMDMostfor theofYSO
in the incluster√time-scale is <3 Myr (e.g. Hernández et al. 2005).
neighbourhood
havefrom
MFSiess
slopesnear
the Salpeter
MFGirardi
slope, etthat
PMS isochrones
et al.
(2000)and MS fromal. is,(2002), ≤ 2.5), derived from the optical data. The error bars represent ± N errors.
In
a
young
cluster,thediscfraction
dependsalso
onthe
spatialline shows the least-squares fit to the mass ranges described! = corrected−1.35. for the adopted distance and reddening are also overplotted. The The continuous
calculations
external
radiationand in theTheoreticalarrow is a reddening vector corresponding to Av = 3 mag. Diamonds position.
text. The valueof the slope predict
obtainedthat
is givenin the UV
figure.
Since
ourrepresent
YSO sample
is complete
down
to 0.3respectively.
M⊙ , for theMFof high-mass stars can photoevaporate outer discs only within 0.3–
squaresthe Class
II and ClassI sources,Blue
crossesstudyrepresent
we havethe
taken
those
YSOs
that haveasmasses
in therange 3.1.4
X-rayonly
sourcesthatare considered
YSOs (seeSections
0.7 pc (Johnstone, Hollenbach & Bally 1998; Adams et al. 2004).
0.3 ≤
2.5.
The mass distribution of our YSO sample A decrease
andM/M
3.2.2)
study.
(e.g. Hartmann
et al. 2005;
et al. 2013a).
‘standard⊙in≤thisin disc frequencies
in Pandey
the immediate
vicinityTheof massive
has a best-fitting slope, ! = −1.23 ± 0.14 (see Fig. 7). Ninkov stars model’by
Kenyon&Hartmann(1995)
postulates
that
the
CTTSs
has
been
found
in
severalmassiveclusters,
e.g.
NGC
2244,
2694 N. Panwar et al.
et al. (1995) studied the cluster IC 1805 and found a slope of NGCevolve
WTTSs
theiretcircumstellar
discs or
theirand
Pismisby24losing(Balogal. 2007; Mercer
etatal.least2009;
spread in colour due to member YSOs is larger than their mean 6611 to! = −1.38 ± 0.19 for masses between 2.5 and 30 M⊙ . A similar
innerparts.
In
the
Taurusregion,
the
WTTSs
are
systematically
older
Fang
et
al.
2012),
suggesting
rapid
destructionofcircumstellar
discs
uncertainties due to photometric colours (see Fig. 6), we expect thatvalue of ! (= −1.3 ± 0.2) has been reported by Massey et al. (1995)
than
the CTTSs (Bertout,
Cabrit2007),(agebut ∼2.5
the statisticalin suchenvironments.
IC 1805Siessis a &
young
clusterMyr)the YSOs are at different evolutionary stages.
for the massive stars of the W4 complex. Our YSO MF is consistentsignificanceis
low(Kenyon&Hartmann
1995;
Hartmannwith several low-mass (<2.5 M⊙ ) stars that have discs, thus 2001;
theWe considered that the errors associated with the determination
with those reported for other active star-forming regions. Erickson
Clarke & Palla 2003). Contrary to the above results,of age and mass are mainly of two kinds; random errors in photom- Armitage,
represents a good site for studying the influence of massive
et al. (2011) derived the MF of an extinction-limited YSO sample cluster
are studies which favour the assumption that CTTS and WTTS
etry and systematic errors due to different theoretical evolutionary there
on the evolution of discs.
(with masses >0.2 M⊙ ) consistent with the field star MF. Kang stars arecoeval and possess indistinguishable stellar properties (Lawson,
tracks. We estimated the effect of random errors by propagatingIn IC 1805,
it is argued that the region is mainly powered by three
et al. (2009) did not find any difference in the YSO MF slopes of& Huenemoerder 1996; Gras-Velázquez & Ray 2005).
them to the observed estimations of V , (V − R) and E(V −O-type
R) Feigelsonstars(Lefloch
et al. and1997).
BD in+60502
and BD+60501the star-forming regions W51 A (! = −1.17 ± 0.26) and W51 B
The coevality of CTTSs
WTTSs
a star-forming
regioncan bethe spectral
type
O4.5III
(Sota
et
al.
2014)
and
O7V
(! = −1.32 ± 0.26). We note that a few older Class III sources are ofexplainedby assuming that YSOs have intrinsically a wide(Sota
rangerespectively,
andaccretionare locatednearand/or
the assumed
clus-ofmight have been excluded as contaminants due to our conservative et al.of2011),
disc masses
and their
activity
the dispersal
(see
Fig.
9), inwhereasHD 15570wideis an
O4Iofstar(Sota
selection criteria (see Section 3.1.4). Even then, we believe our ter centre
the disc
takes
place
a correspondingly
range
time-scales2011), etlocated∼4.7
arcminof the cluster centre. The
result should represent the IMF for the cluster, for example, and et al.(Furlan
al. 2006;
Bertout
et al.east2007).
low-mass isochrones are very close to each other.
radial velocitiesof BD+60502andtheBD+60501
aresample−42 kms−1
In this work,
we havederivedages
of a large
of YSOs−1(Kharchenko
al. 2007)
andhence,−52 km(Kharchenko
2007),thein the ICet1805
cluster,
it issworth
to attempt et
to al.
address
problem consistentof the evolutionaryof CTTSs
and
WTTSs.(−55Fig. 8(a)respectively,
with thestatus
molecular
cloud
velocityto! = d log N(log m)/d log m,

4.2 Evolution of T-Tauri stars in IC 1805

Figure 9. The upper panel shows the surface distribution of WTTSs (bluecircles) and CTTSs (red circles) superposed on the Spitzer 4.5 µm image of the cluster. The concentric annuli have a width of 1.8 arcmin. The
massive stars are shown with diamond symbols and cluster centre with cross.
The lower panel shows the CTTS fraction as a function of the radial distancefrom the cluster centre. The two filled squares represent the locations of thetwo massive stars near the cluster centre.

shows the histograms of the ages of the CTTS and WTTS canTTSs are low-mass stars (< 3 M⊙ ) that are contracting towards thedidates within the cluster region in the mass range 0.3–2.5 M⊙ .
2684–2698(2017)
MS. They are generally classified into weak-line TTSs (WTTSs) MNRASThis 468,
manifests
that theCTTSs are relatively younger (mean age
and classical TTSs (CTTSs) on the basis of the strength of the Hα∼2.3 Myr) than the WTTSs (mean age ∼2.6 Myr). The cumulative
Figure8. (Strom(a) Histogramsof the WTTSs
age distribution
Class
II and Class distribution
III candidates,
distributionsof CTTSs
and WTTSs
the cluster
emissionline
et al. 1989).
exhibitofa theweak,narrowof(b)thecumulative
ages of CTTSs
and WTTSs
is shown
in Fig.in8(b).
region≤as 10
a functionof stellar age.
Hα (EWÅ) in emission
with no or little IR excess, while
We obtained the Kolmogorov–Smirnov test value of 11 per cent for
CTTSs display a strong Hα emission line (EW ≥ 10 Å) and large
the assumption that the two samples drawn are from the same popT-association
andof Chauhan
et al. considered(2009) fortothe
BRC regions, ulation. Although the statistical significance is less, our result is in
IR excesses.
Class II
sourcesare generally
correspond
that
WTTSs
are relatively
than CTTSs,
which
supports the agreement with those of Bertout et al. (2007) for the Taurus–Auriga
in theopticalcategoryto CTTSsolder
and Class
III sources
to WTTSs
notion that CTTSs evolve into WTTSs. Here again, we would like16