The Formation and Evolution of Small Star Clusters

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by  Helen Kirk

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed21January2014 (MNLATEXstylefilev2.2)
The Formation and Evolution of Small Star Clusters
Helen Kirk,1,2⋆† Stella S. R. Offner,1,3‡ Kayla Redmond4
4 1Harvard-Smithsonian Centerfor Astrophysics, Cambridge, MA, 20138, USA
1 2 OriginsInstitute,McMaster University,Hamilton, ON, L8S 4M1, Canada; [email protected]
0 3Yale University, NewHaven, CT, 06511, USA; stella.off[email protected]
2 4Universityof North Carolina at Asheville, NC

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a
J 21January2014

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1
ABSTRACT

] Recent observations show that small, young, stellar groupings of ∼10 to 40 members
A
tend of have a centrally-located most massive member, reminiscent of mass segre-
G gation seen in large clustered systems. Here, we analyze hydrodynamic simulations
. which form small clusters and analyze their properties in a manner identical to the
h
observations.We find that the simulatedclusters possesssimilarproperties to the ob-
p
served clusters, including a tendency to exhibit mass segregation. In the simulations,
-
o the central location of the most massive member is not due to dynamical evolution,
r since there is little interaction between the cluster members. Instead, the most mas-
t
s siveclustermemberappearstoformatthecenter.Wealsofindthatthemoremassive
a stars in the cluster form at slightly earlier times.
[

1 INTRODUCTION Maschberger et al. 2010; Maschberger & Clarke 2011;
1
Kirk & Myers 2011; Parker & Meyer 2012). An MST is
v
Stars typically do not form in isolation but rather within
0 essentially a structure which connects all points together
groups of a hundred or more stars (Lada & Lada 2003).
1 through their minimum distances, much like the sketch of
Even star-forming regions that have low-stellar densities,
5 a constellation. We adopt the MST method for identifying
e.g., Taurus, contain clear groups of 10 or more stars (Kirk
4 stellar groups for easier comparison with observational
. & Myers 2011, hereafter KM11). The pervasiveness of stel- results. Ultimately, all methods work to identify regions
1
larclusteringfrom theveryearliest embeddedphaseofstar
0 of relatively high stellar density but are subject to some
formation (e.g., Gutermuth et al 2009, hereafter G09) to
4 subjective decision about the location of the cluster
old, dense globular clusters (e.g., Meylan & Heggie 1997)
1 boundary.
indicates that thenatureof stellar distributions holds clues
:
v about theformation and dynamical evolution of stars. Inspection of the highest stellar density regions indi-
i cates that the most massive stars tend to be located closer
X Inlocalstar-formingregions,Bressert et al.(2010)find
to cluster centers and in the highest stellar density regions
thatthedistributionofprotostellarseparationshasnochar-
r (Hillenbrand & Hartmann 1998a; Gouliermis et al. 2004).
a acteristicscale.Oneinterpretationofthisresultisthatthere
This appears to be true also for clusters where the most
is no obvious preferred scale to separate “clustered” stars
massive star is only a few solar masses (KM11). Since it is
from“non-clustered”stars.Simulationssuggestthatcluster-
only possible to assign stellar mass after an age of a couple
ingcanbedynamicallyerasedrapidlyormay,whenpresent,
million years, at which point most of the natal cloud gas
provide no distinctive signature in the separation distri-
hasbeenaccretedorexpelled,itisdifficulttodeterminethe
bution(Gieles et al.2012;Parker & Meyer2012).However,
primordial distribution of masses.
singling outhigherdensityconcentrationsisnonethelessin-
structivebecausethemostdenselydistributedstarsaremost Numerical simulations thus provide an important av-
likely togravitationally interact and evolveas an ensemble. enueforexploringearlyclusterproperties.Earlyworkmod-
A variety of definitions exist to dictate what is a eled thedynamical evolution of small clusters by beginning
stellar cluster. Lada & Lada (2003) propose a minimum with a set of stellar seeds in a gas potential (Bonnell et al.
of 35 “physically related” stars with total mass density 1997,2001;Delgado-Donate et al.2003).Inlaterwork,sim-
> 1.0M⊙ pc−3. This definition is derived by requiring ulationsalsofollowed theformation ofindividualstarsfrom
that a grouping survive evaporation for 108 years. To be the gravitational collapse of dense gas cores (e.g., Klessen
applied in practice, the Lada & Lada (2003) cluster defi- 2001; Bate et al. 2003; Bonnell et al. 2004; Offneret al.
nition and other similar approaches (e.g., Jørgensen et al. 2008; Girichidis et al. 2011). However, even with the aid
2008) use stellar surface density thresholds to determine of simulations it remains unclear whether observed mass
cluster membership. A number of authors have recently segregation is primordial or dynamical and depends par-
utilized a different method, the minimal spanning tree tially on the assumed model and initial conditions (e.g.,
(MST), to define stellar groups (Cartwright & Whitworth Bonnell & Davies 1998). For example, N-body simulations
2004; Gutermuth et al. 2009; Allison et al. 2009; ofclumpy,subvirial(i.e.,collapsing) clustersfindthatmass
2 Kirk et al.
segregation canoccurwithinaglobaldynamicaltimeofthe thenperturbthegas fortwotothreecrossing timesusinga
full system (Allison et al. 2009; Parker et al. 2013). In con- random velocity field. This field has a flat power spectrum
trast,Maschberger et al.(2010)findthatmarginally bound overwavenumbersk1 tok2 (seeTable 1),and we renormal-
hydrodynamicsimulationsproducemasssegregation within ize the perturbations to maintain a constant cloud velocity
0.5 Myr of formation. Girichidis et al. (2012) find that the dispersion. After the initial driving phase that produces a
steepness of the initial large-scale gas density profile from well-mixed turbulent distribution, we turn on gravity and
whichstarsareformedalsoaffectsthefinalspatialdistribu- allow collapse to proceed.
tion of stellar masses. The simulations each have a 2563 base grid and four
Inprinciple,thedetailsofclusteringandmasssegrega- levelsofAMRrefinement,whereweautomatically addnew
tion can be used to differentiate between theoretical mod- gridstosatisfytheJeanscriterionandadoptaJeansnumber
els. In practice, however, it is uncertain whether observed of 0.125 (Truelove et al. 1997). We introduce a sink parti-
early mass segregation favors the turbulent core model cle when the Jeans condition is violated on the finest level
(Krumholzet al. 2007) or competitive accretion model (Krumholzet al. 2004). These particles approximately rep-
(Bonnell et al. 2001). In the former scenario, massive stars resent young stellar objects, and henceforth, we shall refer
formfromhigh-columndensitygas,whichisincidentallyof- tothemas“stars.”Sincewedonotincludemasslossdueto
tencentrallylocated.Inthelatterscenario,starsformingat protostellar outflows, the simulated particle masses should
the cloud center are deep within the gravitational well and be considered upper limits. We also merge particles if they
thushavethelargest reservoirofavailablegas,whichallows approachwithinfourfinecellsandifonehasm∗ 0.1M⊙.
≤
them to grow to higher masses. The sink particle-gas interaction is smoothed on scales of
Mostsimulationanalyseshavefocussedonsystemsrep- one fine cell, such that the dynamics of closely approach-
resenting larger-cluster formation, including the formation ing particles, especially when their mass is comparable to
of massive stars and high stellar density environments. We the cell gas mass, is not well modeled. In any event,we ex-
focus here on the less-explored regime of the formation of pecttheformationofsmallfragmentstobesuppressedwith
small (N=10-40), sparse (1-10’s pc−2) intermediate mass the addition of either radiative transfer or magnetic fields
(M < 4 M⊙) stellar groups comparable to those in KM11. (e.g, Offneret al. 2009; Commer¸con et al. 2011). Thus, our
In this regime, stellar dynamics might be expected to play stars may also sometimes represent binary or multiple star
asmaller role, although KM11 doobservemass segregation systems ratherthan individualstars.
atanearlyage.Wealsoinvestigatetheeffectofglobalcloud Table 1 displays the simulation parameters. The pa-
properties, such as Mach number, temperature, and turbu- rameters of the fiducial Rm6 simulation are set such that
lent driving scale, on cluster properties. thecloud is virialized:
Ourworkextendspreviousstudiesinseveralimportant 5σ2(L/2)
ways. First, we quantitatively define subclusters within the α= =1, (1)
GM
simulation and then follow the evolution of these groups
where L is the cloud size, M is the cloud mass, and σ =
andtheirpropertiesoverseveralclusterdynamicaltimesfor
c /√3 is the one-dimensional velocity dispersion, where
arange of cloud properties. Second,we applyobservational M s
c isthesoundspeed.Thefiducialsimulationalsoobeysthe
constraints and quantitatively compare the simulated clus- s
observed linewidth-size relation (e.g., Solomon et al. 1987;
ter properties with observations of similar clusters. Finally,
McKee & Ostriker 2007):
weanalyzesimulationswithcontinuousturbulentenergyin-
jection rather than simulations of isolated clouds wherein R 0.5
turbulence is allowed to decay. The former picture is more σ=0.72 km s−1. (2)
(cid:18)1pc(cid:19)
similartothe“distributed”starformationinnearlyregions,
whichhaverelativelylowstellarsurfacedensitiesandarenot The other simulations vary in temperature, Mach number,
strongly centrally condensed. drivingscale,andphysics.Table1illustratesthatthediffer-
In Section 2 we describe our simulation parameter encesleadtosignificantstatisticaldifferencesinthenumber
study. In Section 3 we discuss the identification of stellar densityofstars.Forreference,simulation Rm6shasparam-
groups using minimal spanning trees (MST). We analyze etersidenticaltoRm6butusesadifferentturbulentrandom
the cluster properties in Section 4, including the details of seed and produces a similar surface density of stars, n∗, to
the mass distributions and member separations. In Section theRm6 simulation.
5 we compare with observations of young clusters. Finally,
we summarize ourconclusions in Section 6.
3 CLUSTER IDENTIFICATION
3.1 Simulated Stellar Catalogs
2 NUMERICAL SIMULATIONS
Ourmethod of cluster identification was designed tomimic
We analyze six simulations of molecular clouds forming thatin KM11.Thelocation ofstars formed within thesim-
stars.WeperformallsimulationswiththeORIONadaptive ulations was first projected onto three planes (xy, xz, and
mesh refinement (AMR) code (Truelove et al. 1998; Klein yz), as an observer would view the simulated region from
1999). The simulations do not include magnetic fields and a large distance away along the z, y, and x axes. Since the
fiveassume asimple isothermal equation of state. Thesim- simulations are performed within a periodic box, we repli-
ulation procedure is described in Offneret al. (2009) and catethesimulated stellar distribution toa totalof threeby
Offneret al. (2013),so we give only a brief summary here. three boxes, to ensure any clusters with members across a
We initialize the simulations with uniform density and box edge are properly identified. In our final analysis, only
The Formation and Evolution of Small Star Clusters 3
Table 1.SimulationProperties
Runa L(pac) M(Ma⊙) T(Ka) Ma k1..k2a n(p∗cb−2) t(ceMnydr) N∗c S(%F)Ec
Rm6d 2 600 10 6.6 1..2 18.2 0.95 88 17.6
Rm6s 2 600 10 6.6 1..2 18.8 0.95 100 18.1
Rm9d 2 600 10 8.9 1..2 3.2 0.95 17 3.5
Rm4d 2 600 10 4.2 1..2 13.8 0.89 80 10.4
Rk34 2 600 10 6.6 3..4 3.2 0.91 14 2.6
Rt20 2 600 20 6.6 1..2 1.5 0.95 17 3.8
Rrte 0.65 185 10 6 1..2 30.6 0.32 18 7.2
aSimulation ID, box length, total initial gas mass, initial gas temperature, and Mach number,
respectively.
bThe projected number density of stars after one freefall time including only those stars with
m∗>0.03M⊙.
cThe timeelapsed inthe simulation after gravity was turned on, and the total number of stars,
andstarformationefficiency(stellarmassdividedbytotalmass)atthisfinaltime.
dPropertiesofthissimulationwerepreviouslyanalysedinOffneretal.(2013).
eThis simulation from Offneretal. (2009) includes radiative transfer; it models heating due to
accretionandnuclearburning.
clusters identified which have centres within the inner box Barrow et al.1985);eachconnectinglinesegmentisreferred
are included, to ignore duplicate clusters. toasabranch.OncethefullMSTstructureiscalculated,we
In real molecular clouds, dense, star-forming cores are follow theprocedure of G09 and KM11 to identify clusters.
estimatedtoloseabouttwo-thirdsoftheirmasstoprotostel- Figure 1 shows an example of thisfull procedure.
laroutflowsandwindsbeforethefinalmassofthestarisset, In the left panel of Figure 1, the full MST structure is
assuming that there is a one-to-one mapping between the shown for the xy projection of the Rm6 simulation at its
densecoreandinitialstellarmassfunction(e.g.,Alves et al. final time step (0.95 Myr). Since the simulation has been
2007). Sinceoutflows are not included in these simulations, replicated three times in either direction to allow us to ac-
we reduce the masses of the sink particles formed by two countfortheeffectofperiodicboundaries,severalstructures
thirds;theexact factor usedhaslittle effect on ourfinalre- canbeseenmorethanonce.Forclarity,onlyasmallportion
sults, the majority of which involve relative stellar masses. of theouter replicated simulation boxes are displayed.
Weeliminateanysinkparticleswhichhavere-scaledmasses WithinthefullMSTstructure,clustersarevisuallyap-
of < 0.03 M⊙. Observationally, YSOs at very low masses parent as regions of small star-star separations, i.e., stars
become difficult to detect; the surveys analyzed by KM11 connected by short branches. Clusters can therefore be de-
sufferfromincompletenessaround0.01to0.03M⊙.Finally, finedasstarsallinterconnectedbybranchlengthslessthan
in each of the three projected views, we remove the lower somevalue,thecriticalbranchlength,L .G09foundthat
crit
mass memberof any pairs separated by less than 1000 AU. nearby clustered regions have MSTs with a characteristic
This separation corresponds to the approximate resolution cumulative distribution of branch lengths: a steep rise at
limit of the stellar catalogs analyzed by KM11. Since the small branch lengths followed by a turn-over to a shallow
observed stellar masses were derived from spectral types, slope,illustrated inFigureA1.Giventhischaracteristic cu-
any low-mass companions closer than the minimum reso- mulative branch length distribution, the branch length of
lution would not have been counted in the observational the turn-over is an obvious choice for L , and that is ef-
crit
mass estimate. Note that our results are not strongly sen- fectivelywhatG09andKM11adopt.AppendixAdiscusses
sitive to the precise choice in cutoff values; the main effect the determination of L in more detail. With such a pro-
crit
is to slightly change the value of L used to define clus- cedure, regions which have different mean clustering prop-
crit
ters(discussedinthefollowing section).InAppendixA,we ertieshavedifferentL values,which morenaturallycap-
crit
demonstrate that our final results are robust to changes in turesthelocalstellaroverdensitiesthanadoptingaconstant
L . We note that the simulations without radiative feed- value would do. In KM11, there is a factor of a few differ-
crit
back likely over-estimate the amount of fragmentation on ence between L found for the sparse stellar distribution
crit
small scales (Offneret al. 2009; Bate 2009). By excluding inTauruscomparedwiththemuchdenserclusteredIC348.
verysmallstarsandthesmallermemberofclosepairs,which For our analysis of the simulations, if we followed an
wedoinordertoadheretoobservationallimits,wealsocor- identicalmethod,wewouldpotentiallyhaveadifferentL
crit
rect for this effect. measured for each simulation, at each time step, in each of
thethreeviewingangles. Withtheanalysisin AppendixA,
wedemonstratethatitissufficientforourpurposestoadopt
a single constant value of L for each simulation (i.e., no
3.2 MST Identification crit
variationwithtimeorviewingangle).Wedonotfindstrong
With this final set of simulated stellar catalogs, we identify evidence for real variations in L within each simulation,
crit
small stellar clusters for each of the three projections us- andmaintainingasingleL allowsforeasierintercompar-
crit
ing minimum spanning trees. An MST represents the mini- isons within each simulation. Allowing for a different L
crit
mumtotallengthbywhichallpointscanbeconnected(e.g. between the various simulations is important, as different
4 Kirk et al.
Figure 1.Theminimalspanningtreestructurecomputed fortheRm6simulationatthefinaltimestep(0.95Myr)whenviewedalong
the z axis (xy projection). The original simulation is contained within the dashed square, while the outer borders show part of the
replicated boxes. Theblue circles show the locations of the sink particles,with their sizescaling withmass (see legend inright panel),
whilethe red lines show the MST structure. Note that additional clustered structure occurs across the original simulation boundaries.
Left: The fullMST structure. Middle: The MST structure remainingafter branches withlengths larger than Lcrit (here 0.07 pc) have
beenremoved.Groupingsofmorethantensinkparticleswhichremainconnectedareclassifiedasclusters.Right:Aclose-upviewofthe
singleclusterfoundinthisexample.Here,non-membersareshowninblack,andtheclustercentreisindicatedbytheredplussign.
Table 2.MSTstatistics tobedetermined,however,asmallLcrit wouldbeexpected,
given the high stellar densities in thesimulations.
Sim Lcrit a Medianb Meanb Minb Maxb Lratc
(pc) (pc) (pc) (pc) (pc)
A second statistic associated with L is the ratio
crit
Rk34 0.12 0.21 0.34 0.007 1.2 0.027
of L to the maximum separation between stars. In ob-
crit
Rm4 0.06 0.05 0.13 0.004 1.1 0.013
servedclusters,regionswithsparserclusterssuchasTaurus
Rm6 0.07 0.04 0.10 0.005 0.95 0.016
tend to have a larger L values as well as a larger maxi-
Rm6s 0.07 0.05 0.09 0.005 0.6 0.013 crit
Rm9 0.1d 0.40 0.37 0.005 0.75 0.022 mumseparationbetweenstars.Indenserclusters,bothLcrit
Rrt 0.1d 0.12 0.12 0.016 0.3 0.061 and the maximum separation between stars are smaller,
Rt20 0.1 0.14 0.25 0.005 1.2 0.025 and Lcrit (and by definition, also the maximum separa-
tion)hasalsobeenfoundtoincreasewhenamoreextended,
aThe critical branch length used for identifying clusters in the
sparser, region around a cluster is included in the analysis
simulation(allprojections,alltimesteps).
(Masiunas et al.2012).WefindthattheratioofL tothe
bThe median, mean, minimum, and maximum branch lengths crit
maximumseparation betweenstars is relatively constant in
inthe final time step of each simulation, averaged over all three
projections. theobservations, ranging from 1% to 3% for theclusters in
cThe ratio of Lcrit to the maximum projected separation be- KM11. Performing a comparable measurement on the sim-
tween sourcesinthefinal timestep ofeachsimulation,averaged ulations reveals that they are generally consistent with the
overallthreeprojections. observations,asshowninTable2.Allofthesimulationsex-
dLcrit ispoorlydeterminedduetothesmallnumberofsources. cept for Rrt have ratios between 1% and 3%, while in Rrt,
the ratio is 6%. Rrt was one of two simulations where the
initialconditionscanresultindifferentclusteringproperties. valueof L measured was highlyuncertain, so thepoorer
crit
The Lcrit adopted for each simulation is given in Table 2. agreement with observations is perhapsunsurprising.
As can be seen in Table 2, L varies by a factor of
crit
about two between the different simulations, ranging from
0.06 pc to 0.12 pc. In KM11, the relatively sparse Taurus OnceL isdetermined,andtheinitialfullMSTstruc-
crit
stars have an L of 0.52 pc, while the tightly clustered ture is “pruned” of all branches longer than L (middle
crit crit
stars in IC348 have an L of 0.083 pc; G09 found L panel of Figure 1); stars which remain connected through
crit crit
values between 0.086 pc and 0.7 pc in their sample. The short branches are potential clusters. KM11 impose an ad-
values of L measured in the simulations span a smaller ditionalcriterionthatclustershavemorethantenmembers
crit
range than these two observational surveys, and tend to be to allow for meaningful analysis of cluster properties. The
small. The L values are, however, larger than that used right panel of Figure 1 shows a zoomed-in view of theclus-
crit
inMaschberger et al.(2010)(of0.025pc)fortheanalysisof teridentifiedin themiddlepanelwith thered plusshowing
a competitive accretion simulations showing the formation thecluster’scentre.AsinKM11,wedefinethecentreasthe
ofalarger,moredenselyclusteredregion.TheMSTanalysis median position ofall clustermembers; thisisless proneto
inGirichidis et al.(2012)isrestricted totheΛ method bias than, e.g., the mass-weighted mean position for mea-
MST
ofAllison et al.(2009),which,inthecaseoflarge-N clusters surements of mass segregation. In Appendix B we demon-
withoutsubstructure,canbeaneffectivemethodtomeasure stratethat usinginstead themean clustermemberposition
masssegregation. TheΛ methoddoesnotrequireL as thecluster’s centrehas little impact on ourresults.
MST crit
The Formation and Evolution of Small Star Clusters 5
4.2 Mass Evolution
We next examine the stellar mass properties within each
cluster identified. Figure 3 shows the evolution of the mass
of the most massive cluster member for the Rm6 simula-
tion, for each of the three projections, every 0.01 Myr (top
panel).Notethat only onecluster with morethan10mem-
berswasidentifiedineachprojection.Themostmassivestar
formed in this simulation is located within a bona-fide 3D
clusteringofsources,andhencethesamepointisplottedfor
the clusters identified in all three projected views. At early
times, the cluster has fewer members, and due to the small
motionsofthestarsnearthecluster’speriphery,thegroup-
ingtendstoalternatebetweenmeetingandnotmeetingthe
cluster definition (more than 10 members all connected by
MST branch lengths of less than L ). This leads to some
crit
apparent gaps in thetime-sampling plotted.
Figure 3 shows that the most massive cluster member
tends to grow with time due to accretion, in this case, fol-
lowing a roughly linear trend. Accretion also occurs in the
other stars in the simulation, however, this is less apparent
whenexaminingthemedianclustermembermassasafunc-
tion of time (middle panel of Figure 3). At earlier times in
particular, this accretion signature is masked by the addi-
tion of new cluster members. The majority of new cluster
members are new stars which formed in the simulation (or
stars which have now accreted sufficient material to sur-
pass theobservational threshold), and their addition to the
cluster tends to decrease the median cluster mass. This ef-
Figure2.Thedistributionofstellarmassesatthefinaltimestep fect islargest at theearlier times when thetotal numberof
ofeachsimulation.ThetworunsofRm6withdifferinginitialtur-
cluster members is small. Other simulations have also been
bulentseedsareplottedinthesamepanel,withRm6sshownby
shown to produce time-independent IMFs that agree with
thedashedblueline.Thedottedredlineineachfiguresshowsthe
the observed IMF (Krumholzet al. 2012; Bate 2012). This
the Kroupa IMF distribution (as given in Weidneretal. 2010),
is generally attributed to either stellar feedback or dynam-
scaledtothesametotalnumberofstars,andgiventhesamemin-
imummasscutoff.Thenumbersinthetoprightcornershowthe ical interactions regulating stellar masses. Since the IMF is
probabilitythatthemassdistributioninthesimulationisdrawn observed to be robust to variation across a broad range of
fromthesameparentsampleastheKroupaIMF. initialconditions andenvironments(Bastian et al. 2010),it
is unlikely to havea preferred characteristic timescale.
The ratio of the mass of the most massive and median
cluster members (the “mass ratio” shown in the bottom
4 RESULTS
panelofFigure3)givesaroughsenseofhowgravitationally
4.1 Mass Distribution dominantthemostmassiveclustermemberisexpectedtobe
in terms of the cluster’s dynamical evolution. We find that
Figure 2 shows the stellar masses at the final time step in
themass ratio tendsto increase slowly with time, spanning
each simulation, with the masses reduced by a factor of 3
aslightlysmallerrangeofvaluesthanthemassratiosfound
from the sink cell masses, and the observational limit of
in KM11. This is discussed in more detail in Section 5.
0.03 M⊙ applied. Earlier time steps generally follow the
same trend but with fewer sources. Observed young stellar
massfunctionstendtobeconsistentwiththestandardIMF,
4.3 Separation Distribution
with the possible exception of Taurus, which has a relative
excess of sub-solar, K7-M1 type stars (e.g., Luhman et al. We similarly analyze the evolution of the spatial positions
2009). Figure 2 shows the stellar mass distributions (solid oftheclustermembers.ThetoppanelofFigure4showsthe
blacklines)comparedtotheKroupaIMF(reddottedline). evolutionoftheseparationofthemostmassiveclustermem-
The simulation mass distributions are consistent with a ber in the Rm6 simulation from their cluster centres. From
Kroupa IMF; a two-sided Kolmogorov-Smirnov test (e.g., thisfigure, it is clear that theseparation neithermonotoni-
Conover 1999) gives probabilities between 10 and 40% that callyincreasesnordecreases–rather,itisvariable.Muchof
the two are drawn from the same parent sample. We note this variability, and all instances of large changes between
that earlier time steps in the simulations give a similar re- adjacenttimesteps,arecausedbyvariationsinclustermem-
sult, and the two runs of Rm6 are statistically similar. For bership,ratherthansubstantialmotionofthemostmassive
reference, the Jeans mass of Rm6 is MJ = 8.0M⊙, where cluster member. Particularly at earlier times, when the to-
M = 4/3πρ¯(L /2.0)3 and the Jeans length is defined as talnumberofclustermembersissmall,theadditionorsub-
J J
L = c [π/(Gρ¯)]1/2. This is slightly higher than the mean traction of a cluster member on the outskirts (with a sepa-
J s
stellar mass we findin thesimulations. ration of roughly L to its nearest neighbour), can cause
crit
6 Kirk et al.
Figure 3. The variation in maximum mass, median mass, and Figure 4.Thevariationinoffset fromtheclustercentre forthe
mass ratio in the Rm6 simulation as a function of time. Red mostmassivemember,themedianclustervalue,andtheratioof
circles, green diamonds, and blue asterisks show the results for the two (the ‘offset ratio’) in the Rm6 simulation as a function
the three projections, and earlier times are shown with lighter of time, sampled every 0.01 Myr. Red circles, green diamonds,
colours.Thesimulationsweresampledevery0.01Myrtoprovide andblueasterisksshowtheclustersidentifiedineachofthethree
morereadableplots. projections,andlightershadesindicateearliertimesteps.
most massivemember’soffset tothemedianoffset, the‘off-
a significant change in the cluster centre position. On rare
setratio’,asafunctionoftime.Thisratiogivesanindication
occasions, a star near the cluster boundary may itself be
ofhowcentrally-locatedthemostmassiveclustermemberis:
separated from one or two additional non-cluster members
ratiosmuchlessthanoneimplythatthemostmassiveclus-
bylessthan L ,in which case, when itbecomes less than
crit ter member lies near the cluster centre, while ratios much
L from an existing cluster member, either through mo-
crit largerthanonewouldimplytheopposite;randomlylocated
tion or the appearance of a new star inbetween, the cluster
mostmassivememberstendtohaveoffsetratiosaroundone
gains several new members, and the cluster centre is more
(KM11).TheoffsetratioshowninFigure4showssignificant
strongly influenced. The reverse situation can also occur –
scatter, although slightly less than either the most massive
oneorseveralclustermembersneartheoutskirtswhich are
star’sormedianstars’offsetsindividually,sincebothcanbe
separated by nearly L from the next cluster members
crit affected by changes to the location of the cluster centre in
may move slightly farther away, reducing the overall clus-
asimilar manner.Itisnotablethattheoffset ratiotendsto
termembershipandchangingtheclustercentreaccordingly.
be less than one at all times in the simulation, for all pro-
Themotion ofthemost massive clustermemberitself, con-
jections,andthereisnoobviousevolutionwithtime.These
tributesrelativelylittletothechangesobservedinFigure4,
points will bediscussed in more detail in Section 5.
and we discuss thisin more detail in thefollowing section.
ThemiddlepanelofFigure4showsthevariationinthe
medianvalueofallclustermember’soffsetsfrom thecentre
4.4 Dynamical Evolution of Cluster Members
with time. This variation is primarily due to the change in
the cluster centre, induced by cluster membership changes. Thepreviousresultsshowsignificantmasssegregation from
As with the most massive member’s offset, the large, sud- theearliesttimesclustersareidentified.Aclusterbyourdef-
den variation tend to be when multiple stars became mem- inition, however, must havea minimum of eleven members,
bers(non-members)duetosmallmotionstakingthembelow soit ispossible thatsome dynamical evolution occurs prior
(above) the L separation from the next nearest cluster to the times analyzed above. To assess the amount of dy-
crit
member. Several of the large-scale variations in themedian namical evolution,in thissection weinvestigate themotion
offset can also beseen in themassive star offset. of the most massive cluster members beginning from their
The bottom panel of Figure 4 shows the ratio of the timeofformation.Ifthemostmassivemembermovessignif-
The Formation and Evolution of Small Star Clusters 7
icantlyandinteractsdynamicallywithotherstars,thenthe timesinthesimulation–thelargestmotionseenisthebulk
masssegregation displayedbytheMSTclustersmaynotin motion of theentiregroup. Themost massive clustermem-
fact be primordial. ber moves very little with respect to the surrounding stars:
Therearetwotimescalesthatarerelevanttotheprocess only several lower mass stars have significant relative mo-
of dynamical mass segregation. An estimate for the mass tions as they join (or are expelled from) thecluster.
segregation timescale is given bythetime it takes themost The lower panel of Figure 5 shows a similar view for a
massivemembertomigrate totheclustercenter.Forastar largercluster,identifiedinthexzprojection,whichsatisfied
of mass M in a cluster with N stars this can be expressed: the MST cluster definition for a large fraction of the simu-
lation in multiple views. These views are ‘messier’, showing
M M N R
tseg(m) h itrelax h i , (3) greater interactions between cluster members. Despite this,
≃ M ≃ M 8 ln(N) v
h i the most massive cluster member usually appears near the
where v is the average velocity of a star and R is the middle of the stars, rather than moving there due to early
h i
cluster radius. (Spitzer 1969; Allison et al. 2009). For stars dynamical interactions. Finally, note that this figure illus-
initially moving with v cs, N = 15, M = 0.6M⊙ and tratesthat(larger)clustersmayincludememberswhichare
h i ≃ h i
R=0.2pc,astarof3M⊙ willmigratetotheclustercenter only associated along the line of sight: in the bottom left
in tseg 0.14 Myr. panelofFigure5,atearliertimes(lowerrightofplot),there
∼
Thisestimateimplicitly assumesthatthesystemisgas are clearly two highly separated groupings of stars (similar
freeandthatthegravitationalpotentialisdominatedbythe positioninxbutnoty),whichappearasasinglegroupingin
stars. However, the simulated systems here are gas domi- themiddleplot(similarxandzatthelowerright).Figure5
nated, which acts to damp the effect of two-body interac- also demonstrates that thecluster evolution proceeds with-
tions. Overall, less than 20% of the total gas in the sim- out the merging of small clusters (e.g., Maschberger et al.
ulations turns into stars by 1 tff, although the volume re- 2010), although individual stars may join. The stellar den-
strictedtotheclusteritselflikelyhasahighermassfraction sities in thesimulations are sufficientlylow that only1 or2
in stars. In a gas dominated system, the characteristic dy- clusters are present at once, so it is unlikely for clusters to
namicaltimescalecanbeapproximatedbythefreefall time: interact and merge.
Rather than relying solely on projected views, we can
3π 1/2
t = , (4) alsoconsiderthefull3Dpictureatonce.Figure6showsthe
ff
(cid:18)32Gρ(cid:19)
root-mean-squaredthree-dimensionaltrajectory oftheclus-
where ρ = µpn is the mass density and µp = 2.33mH termembersasafunctionoftimeforthesame twoclusters
is the mean particle mass. For a mean clump number den- asshowninFigure5.Inthisfigure,themostmassivecluster
sity of n = 104 cm−3, t = 0.34 Myr, which is slightly member’s3Dpositionisshownasalargecircleateachtime
ff
longer than the N-body dynamical times. Protostars form step, while other cluster members are indicated by small
from thedensegas (n>105 cm−3),which has asmall rela- colouredplusses.Attimestepswhenthestarsmeetourcri-
tiveoffset velocity from thelower densitycore envelopegas teriaforacluster,aproxyforthe3Dclustercenterisshown
(e.g., Kirk et al. 2007; Offneret al. 2009; Kirk et al. 2010), by the square. In the upper panel, the stars only meet the
so a single star shouldn’t have a significantly different ve- MSTclusterdefinitionforashorttime,whileinthebottom
locity than the gas it forms from. The core itself, however, panel,theclusterdefinitionissatisfiedforalongerperiodof
may have some advection velocity, t , that is comparable time,although still less than halfof thetime inwhich stars
adv
to the sound speed or slightly higher. Protostars retaining haveformed.
theirnatalsonicvelocitymightmigrate 0.2pcin0.5Myr. Several important points are quickly apparent from
∼
Thesetimesaresignificantlyshorterthanthetimeover these figures. First, the most massive member typically
whichthesimulationsformstars( 0.6-1.0Myr),soinprin- formsearly;seeSection4.5.Additionalprotostarswhichare
∼
ciple, the stars have sufficient time to become mass segre- eventually included in the MST-defined cluster form later,
gatedduetodynamicalinteractions.Weexaminethispoint sometimes in close proximity to the most massive member
in both theobserved projected view of clusters and the full and sometimes further away, migrating towards the posi-
3D view. tionofthemostmassivemember.Second,themotionofthe
Figure 5 shows the projected positions of two clusters most massive member is .√3cs and is relatively constant.
as a function of time for views along the x, y, and z axes This is significant because dynamical interactions between
(yz,xz,andxyprojections).Inthesefigures,wetakethelist cluster members may “heat” up the velocities as the clus-
of cluster members at a given time and projection and plot ter becomes virialized (Proszkow & Adams 2009). Finally,
the2Dviewsoftheclustermemberpositionssampledevery although cluster members do appear to be dynamically in-
0.01 Myr. Cluster member positions are indicated by the teracting(e.g.,toppanelFigure6),themost massivemem-
coloured plussigns, whilethemost massiveclustermember beris already close totheMST-definedcenteroften several
is shown with the circle, whose size scales with time. The 0.1 Myr before a cluster is identified and before many close
direction of motion of the most massive member can be in- interactions occur.
ferred byfollowing thecirclefrom small tolarger sizes.The Some 2D-defined clusters contain members not associ-
toppanelshowsasmallclusterwhichonlysatisfiedallMST ated in 3D space. In Figure 6, this results in an offset of
cluster criteria for several time steps in the xz projection. the three-dimensional center position relative to the most-
Several points can be taken from this plot: first, the three massivemember.Interestingly,thesemoredistantmembers
projected views have a fairly similar appearance, implying donot effect theconclusion that theprotostars are primor-
thatmanyoftheclustermembersareindeedclusteredin3D dially mass segregated. If anything, the random projected
space. Second, many of the stars remain clustered at early placement of these MST members would make the clusters
8 Kirk et al.
Figure5.Theevolutionoftheprojectedmemberpositionsalongthex,y,andz axesforeachclustermemberasafunctionoftimefor
two clusters in Rm6. The most-massive member is indicated with a black circle that grows as a function of time. The top cluster was
identified fromthe y projection (middlepanel) andthe bottom cluster was identified fromthe z projection (rightpanel). Positions are
shownonlyevery0.01Myrforclarity.
appearlessmasssegregatedthantheyactuallyare,byshift- What is immediately apparent from this figure is that the
ing the apparent cluster centreaway from its truelocation. most massive stars all started forming early in the simula-
Observedmass-segregatedclustersarethuslikelytobemore tions – less than one third of the time since the first star
mass segregated than they appear due to the inclusion of formed. Some low mass stars also formed early, but other
unrelated stars dueto chance alignments. low mass stars continueto form throughout thesimulation,
as shown by the blue and green circles. In Rm6 and Rm6s,
whereMSTclusters ofmorethantenmemberswereidenti-
4.5 Stellar Formation Times fiedusingthebestfitLcrit,themost massivestarformed is
amemberof aclusterin at least oneprojection. This trend
As noted in Section 4.4, the most massive member of each doesnotextendmuchfurtherinthemass-ranking,however,
clustertendstoform early inthesimulation. Figure7illus- in several instances somewhat massive stars appear to be
trates the relative formation times of thestars in the simu- associated with smaller (N 10 member) groups for at least
lation,showingthefinaltimestepofeachsimulationexcept some time steps. We emph≤asize that although the massive
Rrt(excludedsinceitformsthefewest stars),viewedinthe starsmaystartformingearly,thisdoesnotimplythatthey
xy projection. The stars are shown as the coloured circles, also stop forming early. Since there is no stellar feedback
withthecirclesizescalingwithmass,andtheMSTstructure included in the simulation, which could reduce or halt ac-
is shown with the black lines. The colours assigned to each cretion, it is not surprising that stars continue to accrete.
starrepresenthowearlyinthesimulation theyfirstformed: A longer formation time for higher mass stars is consistent
red and purple indicate earlier times, while green and light withcertaintypesofanalyticaccretion modelsinwhichac-
blue indicate later times, all calculated as the fraction of cretion rates depend weakly or are independent of stellar
time elapsed since the first star formed in the simulation.
The Formation and Evolution of Small Star Clusters 9
hereclearlymatchtheobservationswell–nearlyallhaveoff-
set ratios less than one, and similar, though smaller range
of mass ratios. There is no trend in the offset ratio vary-
ing with the mass ratio or most massive cluster member’s
mass.Thehistogram inthetoppanelofFigure8showsthe
distribution of offset ratios for both the simulation (black
line) and observations (grey line) - both are similar. A KS
testshowstheprobabilitythatbothobservedandsimulated
offset distributions are drawn from thesame parent sample
is 25% (top right corner); the two distributions are clearly
consistent.
Figure 9 shows the comparable results for the Rm6s
simulation,showingthesamebroadcharacteristics:thema-
jority of offset ratios are much less than 1, and mass ratios
around 10 to 20. Rm6s appears to have generally slightly
higher mass ratios, and a broader distribution of offset ra-
tios. The small differences between the two thus reflect the
differencebetween thetwo random turbulentseeds.
None of the other simulations had any clusters with
morethan10membersusingtheirbest-fitL .Rm4would
crit
havehadaclusteridentifiedatmanytimestepsifasmaller
minimum number of members were adopted for the clus-
ter definition (e.g., > 5 members). All of the other simula-
tions require both a smaller minimum number of members
and a larger L in order for a cluster to be identified in
crit
more than one or two time steps and a single projection.
We explore the effect of a more relaxed cluster definition
in AppendixA3, and find that theRm6 and Rm6s clusters
follow the same general behaviour as seen in Figure 8 and
Figure6.Thethree-dimensionalposition(x =px2+y2+z2) 9,and‘clusters’identifiedintheremainingsimulations also
of all cluster members for the same two clu|s|ters as in Figure 5. behave similarly (see Figure A4). This implies that neither
Themostmassivememberisindicated bythe blackcircles.The thespecific cluster definitionused northeinitial conditions
boxesshowtheclustercentercomputedfromtheMST.Theline hasastrong influenceon ourresults. At least in theregime
givesthetrajectoryforaparticlemovingwithanrmsvelocityof ofsmall-clusterformation,ourresultssuggestaninteresting
√3cs=0.35kms−1. conclusion.Changingtheinitialconditionsinthesimulation
fromthefiducialvalueswhichbestmatchobservations,toa
largerorsmallerMachnumber,highertemperature,smaller
mass (Bate & Bonnell 2005; Myers 2009; McKee & Offner
turbulent driving scale, or including radiative effects has a
2010). A tendency for more massive stars to begin form-
significant impact onthenumberof stars formed, andtheir
ing early has also been noted in some previous simulations
general clustering properties (stellar density,visual appear-
(e.g., Klessen 2001; Bate et al. 2003; Bonnell et al. 2004;
ance,etc).Despitethis,themostmassivestarstendtoform
Maschberger et al. 2010). Furthermore, Smith et al. (2009)
in what will become the centre of the cluster, suggesting
findthatthedensecoresinwhichmassivestarsform outof
that themechanism for this is broadly applicable.
tend form first as well.
We note that the degree of mass segregation in local
star-forming regions appears to be somewhat sensitive to
how mass segregation is defined.For example, Parker et al.
5 COMPARISON WITH OBSERVATIONS
(2011) and Parker et al. (2012) do not find mass segre-
KM11 found in their observational survey of young, small, gation in Taurus and ρ Ophiuchi, respectively. In fact,
−
and nearby clusters, that the most massive cluster member Parkeret al.(2011)claimtofindinversemasssegregationin
tended to be centrally located (offset ratios of <1), with Taurus. These results initially appear to be in conflict with
no dependence on other cluster properties, such as num- ourMSTanalysisandtheresultsofKM11,whoalsoanalyze
ber of members, mass, or mass ratio. Figure 8 shows the Taurus,butthedifferencecanbeunderstood bycomparing
mass and offset ratios found in the Rm6 clusters compared the two samples. Parkeret al. (2011) include all the stars
with those in the observational survey. In the lower panel intheiranalysis, essentially assumingthattheentireregion
of Figure 8, circles show the simulated clusters, with the behaves as one cluster, while KM11 make a point of iden-
circle size varying with the mass of the most massive clus- tifying groupings within the sample, i.e., small-N analogs
ter member, colour varying with projected view, and shad- of higher mass clusters. We assert that subdivision is natu-
ing varying with the time, sampled every 0.01 Myr. The ral in regions such as Taurus, where star formation is dis-
greylettersindicatetheobservedclustervaluesinKM11for tributed and stars in different areas appear to evolve with
Taurus, Lupus3, ChaI, and IC348. The simulated clusters little knowledge of other subgroups.
10 Kirk et al.
Figure 7.Therelativeagesofstarsformedineachofthesimulations.Foreachsimulationthexy projectionisshownatthefinaltime
step,withthesameplottingconvention forthestarsandMSTstructureaspreviousfigures,e.g.,Figure1.Thecolourassignedtoeach
starindicateswhenitformedinthesimulationasafractionofthetimeelapsedsincethefirststarinthesimulationformed(seelegend
onbottom right);largerpercentages thus indicate younger (morerecentlyformed)stars.Thesimulationsshownare,fromlefttoright,
toprow:Rk34andRm4,middlerow:Rm6andRm6s,bottom row:Rm9,Rt20.Rrtformedthefeweststarsandisomittedforbrevity.