Testing the initial conditions and dynamical evolution of star clusters using Gaia - I

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by  Richard J. Allison

Mon.Not.R.Astron.Soc.000,1–??(2010) Printed16January2012 (MNLATEXstylefilev2.2)
Testing the initial conditions and dynamical evolution of
star clusters using Gaia - I
Richard J. Allison1⋆
2
1 1 Institut fu¨r Theoretische Astrophysik, Zentrum fu¨rAstronomie der Universita¨t Heidelberg, Albert-Ueberle-Str.2, 69120 Heidelberg, Germany
0
2

n
a
J

3 ABSTRACT
1

We investigate how the properties of escaping stars are related to the initial con-
]
A ditions oftheir birthclusters.We findthat the number ofescapingstars,their spatial
distribution, and their kinematics show a dependence on the initial conditions of the
G
host cluster (substructure and virial ratio). Thus the properties of escaping stars can
. be used to inform us of the initial conditions of star formation, and also provide a
h
p window into the dynamical history of star clusters. The ESA Gaia mission will make
- observationsofthepositionsandpropermotionsoftheseescapingstarsattherequired
o accuracy to allow us to investigate the dynamical evolution of local star clusters and
r
t provide an important insight into the initial conditions of star formation.
s
a Key words: kinematics and dynamics - open clusters and associations: general
[

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v
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1 INTRODUCTION regation)canhaveaprimordialorigin.Escapingstars,how-
8
ever, offer a unique measure of the past dynamical state of
8 Howstarsformisoneoftheoutstandingquestionsinastron-
a star cluster. Clusters with a violent past could be distin-
2 omy. In particular, it is not clear if stars typically form in
. guishable from clusters with a relatively quiet history from
1 clusters (e.g., Lada & Lada 2003), in particular in a quasi-
theirescaper population.
0 staticway(Tan et al.2006),orforminahierarchicaldistri-
TheESAGaiamissionwillconductanall-skyastromet-
2 bution (Gutermuth et al. 2005; Bressert et al. 2010) which ric and spectrophotometric survey of ∼ 109 point sources
1 can then violently dynamically evolve into dense star clus- between 6th and 20th magnitude. The Gaia mission will be
: ters (Allison et al. 2009, 2010; Portegies Zwart et al. 2010).
v abletomeasurethedistance,positionandvelocitydistribu-
Xi Dynamical interactions rapidly erase initial substructure tionofthelocalGalacticstellarpopulationtohighaccuracy.
and any memory of the initial conditions, and so deduc-
Therefore Gaia will not only findco-moving groupsof stars
r ing the initial conditions from dynamically evolved clusters
a (dispersing clusters), but it will be able to identify escap-
and distinguishing thesepossibilities is difficult.
ing stars from clusters and measure their velocities at large
Constraining the initial conditions of star clusters
(angular and physical) distances from their birth cluster.
is important as their initial state directly influences
Inthisletterweshowhowescapingstarsmightbeused
the early dynamical evolution of the cluster, such as
toinvestigatethedynamicalhistoryofstarclustersandthe
its ability to remove any initial structure (Goodwin
conditions of their formation. In Section 2 we describe our
1998; Goodwin & Whitworth 2004; Smith et al. 2011),
initial conditions for relatively active and relatively quiet
dynamically mass segregate (Allison et al. 2009, 2010;
clusters. In Section 3 we show that theescaper populations
Moeckel & Bonnell 2009a; Moeckel & Bate 2010), and pro-
from the two clusters are very different and that these dif-
cessitsbinarypopulation(Kroupa1995;Parkeret al.2009,
ferences should be observable by Gaia. We discuss the im-
2011; Moeckel & Bate 2010). However, it is difficult to dis-
plications in Section 4, before concluding in Section 5.
tinguish between a cluster that has evolved dynamically to
be, for example, mass segregated, and one which was mass
segregated at its formation (primordial). Therefore it is im-
portant tobeable toestimate whetheracluster isdynami- 2 INITIAL CONDITIONS
cally evolvedornot;butthisisdifficulttomeasureasmost
WeperformN-bodysimulationsofyoungstarclusterswith
potential indicators of dynamical evolution (e.g. mass seg-
twodifferentsetsofinitialconditions.Theinitialconditions
are designed to produce one class of cluster that undergoes
rapid and violent dynamical evolution, and another class
⋆ E-mail:[e-mail] that has a relatively quiescent dynamical history. However,
(cid:13)c 2010RAS
2
both produce a central ‘cluster’ that appear very similar to Escaping stars are definedusing three criteria:
one another after a few Myr.
The initial conditions vary the initial amount of struc- (i) Themagnitudeofthevelocityisgreaterthanthestars
ture and the initial virial ratio. The initial spatial distribu- current escape speed;
tion of stars is created by using a fractal distribution, as (ii) Themagnitudeof theradial component of theveloc-
described in detail in Goodwin & Whitworth (2004). Using ity vectoris larger than thetangential component;
a fractal distribution allows a simple procedure with which (iii) Thestarisoutsidesome’escaper’radius;whichisset
structureddistributionscanbecreatedandreproduced.The at two times thehalf mass radius.
method also benefits from having only a single parameter
to control the amount of substructure – the fractal dimen-
sion(D).Thefractaldimensionparamaterisesthestructure
3.1 Spatial distribution of ejected stars
suchthatalowervalueimpliesmoresubstructure(i.e.more
clumpy), and D =3.0 is a random distribution. The initial Our two clusters evolve in very different ways for the first
velocitiesaresetasdescribedinAllison et al.(2010)andare 2 Myr. The clumpy and cool cluster (D1.6Q0.3) undergoes
scaled to a global virial ratio (Q). We define the viral ratio a dense collapse phase and re-expansion before reaching
as Q=T/|Ω| (where T and Ω are thekinetic and potential a Plummer-like equilibrium state by 4 Myr as described
energiesofstars,respectively),suchthatavirialised cluster in (Allison et al. 2010). The smoother, virialised cluster
has Q=0.5. (D2.6Q0.5) gentlyrelaxes to aPlummer-likeequilibrium in
In this study we investigate the effect of initial struc- roughly a crossing time (see also, Allison et al. 2010). The
ture and virial ratio on the properties of escaping stars. final states of the clusters are illustrated in Figure 1. The
We present two different initialisations: 1) an initially cool left-handpanelsshowtheinner3pcradiusregionoftypical
and highly structured cluster (D = 1.6, Q = 0.3); and 2) examplesofthetwoclustersafter4Myr,ascanbeseenthe
an initially virialised and less structured cluster (D = 2.6, spatial distribution of the two clusters is very similar and
Q = 0.5), labelled D1.6Q0.3 and D2.6Q0.5, respectively. memory of their initial conditions has been erased. How-
Each simulation contains 1000 stars, has an initial maxi- ever,theright-handpanelsshow thespatial distribution on
mum radius of 1 pc, includes no primordial binaries or gas, a20pcradiusscaleinwhichthereisclearlyadifference:the
andathree-partpowerlawisusedforthestellarinitialmass D1.6Q0.3clusterhavingmore,andmorewidelydistributed,
function (IMF, Kroupa2002), escaping stars.
M−0.3 m06M/ M⊙ <m1, Clumpy and cool clusters undergo a much more dy-
N(M)∝ MM−−12..33 mm1266MM// MM⊙⊙ <<mm23,, (1) nouams iwcaolrkevoonluttihoeneaffnedctsmofoointhitiaanldspraetlaiaxleddicstlurisbtuertsio.nPsrehvais-
 shownthatclusterswhichareinitiallyclumpiertendtohave
with m0 = 0.08 M⊙, m1 = 0.1 M⊙, m2 = 0.5 M⊙ and amoreviolent,dynamicalevolution,undergoingmasssegre-
m3 = 50 M⊙. No stellar evolution is included because of gation(Allison et al.2009,2010;Moeckel & Bonnell2009b),
the short duration of the simulations (∼ 4 Myr). We use forming trapezium-like systems (high-order multiples con-
thestarlabN-bodyintegratorkiratorunoursimulations taininghigh-massstars;Allison & Goodwin2011),andeven
(Portegies Zwart et al.2001).Weanalyse50realisationsfor influencing the binary properties (Parkeret al. 2011). In
each initial condition. smoother, relaxed clusters these effects are much less pro-
Forthisinitial studywehavesimplified theinitial con- nounced, and it is likely that features such as trapezium
ditions. Complications such as initial binaries, tidal fields, systemsandmasssegregation cannothaveadynamicalori-
natalgas,etchavebeenignoredtoallowasimplenumerical gin(Bonnell & Davies1998;Allison et al.2010)andsomust
experiment to be performed. We also ignore various obser- havea‘primordial’origin inthenatureofstarformation it-
vational selection effects and assume perfect knowledge of self.
the four dimensions of phase space that Gaia will best de- The spatial distribution of stars in the central regions
termine,viz.positionsonthesky,andpropermotions.Gaia of these initially very different clusters (left-hand plots in
itself, as well as follow-up observations will also determine Figure 1) show few differences. But if we move out beyond
the distance and radial velocities of stars, but to a lower the pc-scale into the 10s pc-scale (right-hand plots) we can
accuracy. immediatelyobservethattheseclustershavehadaverydif-
ferent dynamical history. Essentially, the differences in the
numbers,distanceandvelocitiesofescapersisawindowinto
3 THE VARIANCE OF PROPERTIES the dynamical history of the cluster. More and faster esca-
pers means that a cluster is dynamically older and has had
MEASURABLE BY GAIA
more two-body encounters within it to eject stars.
Hereweinvestigatetwopropertiesofescapingstarsthatare The D1.6Q0.3 cluster shown in Figure 1 (top panels)
measurable byGaia: has 80 escaping stars as projected on the sky (in three di-
mensions it has actually 149 escaping stars). This is com-
(i) The numberof escaping stars and theirspatial distri-
pared to 15 escapers in projection (30 in three dimensions)
bution around theirparent cluster;
fortheD2.6Q0.5 cluster(bottompanels).These resultsare
(ii) Thekinematics of theescaping stars.
typical throughout our ensembles, with clumpy, cool clus-
Wewillshowthatthesepropertiesaredependentonthe ters producing an average of 57 escapers, and smooth, re-
initial conditions, and therefore contain much information laxed clusters producing only 12 in projection (115 and 20
on theinitial conditions of star formation. in three dimensions, respectively).
(cid:13)c 2010RAS,MNRAS000,1–??
Testing the initial conditions and dynamical evolution of star clusters using Gaia 3
(a)
(b)
Figure1.Thetwo-dimensionalprojectionof(a)aninitiallycoolandclumpycluster(Q=0.3&D=1.6),and(b)aninitiallyvirialised
andsmoother distribution(Q=0.5&D=2.6) at4Myrs.Theplots onthe leftshow the central regions on2pcscale, thedifferences
betweenthespatialdistributionsarenotobvious.However,therightsideplotsshowazoomedoutview(20pc),herethedifferencesare
clear.Theinitiallyclumpier cluster has amuchmorepronounced population ofescaping stars,caused byits moreenergetic dynamical
evolution.
3.2 Kinematics
The differences between the spatial distributions of the es-
caping stars from the two clusters is also seen in the kine-
maticsoftheirescapingstars.Figure2showsthecumulative
distribution of the escaping stars from the D1.6Q0.3 clus-
ter (black solid line) and the D2.6Q0.5 (red dashed line).
There is a clear difference between the kinematics of the
two sets of escaping stars. The D2.6Q0.5 cluster shows a
small spread of escaper velocities, with 90 per cent of the
ejectedstarswithina0.5 km s−1spreadaround1.5 km s−1,
withamedianescapervelocityof1.2 km s−1.Themaximum
escapervelocityfromthisclusteris<10 km s−1.Whilethe
D1.6Q0.3 cluster, has a larger 2 km s−1spread for 90 per
cent of escapers (with a median velocity of 2 km s−1), and
hasamuchmore pronouncedhigh velocity tail. 10 percent
ofescapingstarshavevelocitiesover5 km s−1andthemax- Figure 2.Thecumulativedistributionfunctionofthevelocities
of escaping stars for the cool and clumpy (black, solidline) and
imum escaper velocity is 15 km s−1.
virialisedandsmoother(red,dot-dashedline)clustersat4Myrs.
It should be noted that there is some degeneracy be- The differences between the velocity distributions are caused by
tween thedistance of an escaping star from thecluster and the dynamical evolution of the clusters. The cool and clumpy
its ejection velocity which Gaia can break. An ejected star clusterisabletoproducefastermovingescapingstarsthroughits
at 10 pc from thecluster may havebeen ejected 4 Myrago dynamicearlyevolutionandejectstarsathighvelocitiesthrough
at 2.5 km s−1, or 1 Myr ago at 10 km s−1. The difference binarystarformation.Thevirialisedsmootherclusterhasamore
betweenthesetwosituationsisimportantasa10kms−1 re- quiescent early evolution, and does not produce a population of
fastmovingescapingstars.
quiresamuchstrongerinteraction(probablywithamassive
binary) than a 2.5 km s−1 ejection event (see below).
The velocity cumulative distribution functions shown
(cid:13)c 2010RAS,MNRAS000,1–??
4
in Figure 2 can be separated into two populations, high- using their current relaxation times (Bastian et al. 2008;
(& 5 km s−1) and low-velocity (. 5 km s−1). The high ve- Portegies Zwart & Chen 2008; Allison et al. 2010).
locity tail is populated by stars ejected by high-mass bi- Escaping stars are therefore a very useful population
nary systems. This population is mostly missing from the to investigate, as they are dynamical in origin - there is no
D2.6Q0.5 clusters because they are not able to easily form obviousprimordialprocessthatwouldproduceasignificant
thesebinarysystems(theclustersare initially single stars). population of escaping stars from a cluster. Thus, measur-
While the more dynamic early evolution of the D1.6Q0.3 ingthepopulationofescapingstarsfromastarcluster(and
clusters allows the formation of high-mass binary systems their properties, e.g. velocities) tells us about the dynam-
(e.g., Allison & Goodwin 2011). In clusters with a primor- ical history of the cluster. This work is the first step into
dial binary population the formation of high-mass binary looking at how the escaping population is related to initial
systemsiseasier,asformingbinariesthroughdynamicsisa conditions of star clusters.
difficult process (Kroupa 1995). Starsthatescapefromclustersretaininformationabout
If primordial massive binaries are present then a high- thedynamical stateof thecluster at thetimethat theyare
velocitytailshouldalsobepresenteveninquiescentclusters, liberated.Thus,thekinematicsofescapingstarscanprovide
althoughwewouldstillexpectasmallernumberofejections us with information about the state of their host cluster at
asthenumberofencounterswiththemassivebinaryshould thetimeoftheirescape.Forexample,high-velocityescapers
belower(duetotheyoungerdynamicalageofthequiescent are an indication of either the formation of a binary or an
cluster). We will return to this in a later paper when we interaction with one. The kinematics of a star ejected by a
examine theeffect of primordial binaries. binary can inform us of the properties of the binary, and
the velocity and distance of the escaping star allow us to
constrainatimeatwhichtheinteractionoccurred(seee.g.,
Tan 2004; Zapata et al. 2009). In particular, examining the
4 DISCUSSION
high-velocity escapers produced by interactions with mas-
We have investigated the influence of the initial conditions sivebinariesandTrapezium-likesystemsmayconstrain the
of star cluster on the properties of escaping stars. We have formation time of those systems and distinguish between
shown that after 4 Myr of evolution clusters with very dif- primordial and dynamical models for their formation.
ferent initial conditions in both substructure and virial ra- The Gaia mission provides the first opportunity to ex-
tio)appeartohavesimilarspatialdistributionsonpc-scales. amine the dispersed population around star clusters. As it
However,on larger scales (∼10pc) thespatial distributions is all-sky it will findstars at great (angular) distances from
of the clusters are very different (Figure 1). Clusters which clustersandtheadditionalkinematicanddistanceinforma-
are initially cool and clumpy havesignificantly more escap- tion provided will allow stars to be associated with their
ingstars, andtheseescapershaveahigheraveragevelocity, birth cluster even if they have travelled a large distance on
and a pronounced high-velocity tail (Figure 2). Therefore, thesky.
onecanlearnaboutthedynamicalevolutionandinitialcon- What we have not addressed in this letter are the ob-
ditions of star clusters, by observing the population of es- servational limitations of the Gaia data. Sky position and
caping stars around star clusters. propermotionswill beobtainedtoanaccuracy ofpositions
Disentangling the effects of dynamical evolution and and proper motions are expected to be ∼ 200µarcseconds
birth properties in star clusters is a complex problem. For and∼150µarcseconds, respectively,for20thmagnitudeob-
example, can one use young star clusters (<1 Myrs) to in- jects (Lindegren 2010). This means that Gaia will have ac-
vestigatetheprimordialbinarypopulation?Dynamicalsim- curate positions and proper motions for G-dwarfs at 1 kpc,
ulationshaveshown thatstarclustersprocesstheirprimor- but for M-dwarfs only out to a few 100 pc. Therefore the
dialbinarypopulationastheydynamicallyevolve,therefore massesofescapersfrommoredistantclusterswillbeanim-
itisimportantthatweareabletoestimatehowdynamically portant observational limitation. We will return to this in
evolved a cluster is if we are to infer information about its later work.
primordialstate.However,canwedistinguishbetweenapri-
mordialbinarypopulationthatlooksdynamicallyprocessed
andapopulationthatactuallyhasbeen?Likewisewithmass
5 CONCLUSIONS
segregation -howdoesonedistinguishbetween‘primordial’
and dynamical mass segregation? Clearly, dynamical mass We have investigated how the properties of escaping stars
segregation mustonlyoccurindynamicallyevolvedclusters arerelatedtotheinitialconditionsoftheirhostclusters.We
(asthisoccursontherelaxationtimescale).Therefore,mass findthatthenumberofescapingstarsandtheirspatialdis-
segregatedclustersthatarenotdynamicallyevolvedbutare tribution/kinematics show a dependenceon the initial con-
mass segregated are primordially mass segregated. ditionsofthehostcluster(substructureandvirialratio).Es-
Measuringthetruedynamicalageofastarclusterisdif- capingstarsfrominitially coolerandclumpierclusterstend
ficult. Clusters such as the Orion Nebula Cluster have cur- tobemorenumerous,andhavealargerspreadinvelocities.
rentrelaxation timesof∼50Myrs,whichimpliesthatthey While virialised and smoother clusters are less efficient at
are dynamically very young, having an age ≈ 1/10 its re- producing escaping stars. These simulations have been car-
laxation time. However, dynamical simulations have shown ried out with simple initial conditions (i.e. no binaries, gas,
that current cluster timescales can be misleading. Clusters tidal fields), although there is no reason to think that the
that undergo an initial collapse and re-expansion have had addition of more complex initial conditions/processes will
much shorter relaxation times in their past, and can there- significantly alter the results as the results depend on the
fore be dynamically more evolved than they appear when dynamical age of thecluster. But following work will inves-
(cid:13)c 2010RAS,MNRAS000,1–??
Testing the initial conditions and dynamical evolution of star clusters using Gaia 5
tigate the influence of more complicated initial conditions. Portegies Zwart S. F., Chen H.-C., 2008, in A. de Koter,
Thepropertiesofescapingstarscanbeusedtoinformusof L. J. Smith, & L. B. F. M. Waters ed., Mass Loss from
theinitial conditionsof star andcluster formation, andcan StarsandtheEvolutionofStellarClustersVol.388ofAs-
provideuswithawindowintothedynamicalhistoryofstar tronomical Society of the Pacific Conference Series, Re-
clusters. constructing the Initial Relaxation Time of Young Star
TheESAGaiamissionwillprovidetheall-skycoverage ClustersintheLargeMagellanicCloud:TheEvolutionof
and dynamical information required toinvestigate escaping StarClusters. p. 329
populationsfromstarclustersandprovideanimportantin- Portegies Zwart S.F., McMillan S.L.W., HutP., Makino
sight into theinitial conditions of star formation. J., 2001, MNRAS,321, 199
Smith R., Slater R., Fellhauer M., Goodwin S., Assmann
P., 2011, MNRAS,416, 383
Tan J. C., 2004, ApJL, 607, L47
6 ACKNOWLEDGMENTS
TanJ.C.,KrumholzM.R.,McKeeC.F.,2006,ApJL,641,
RJA would like to thank Simon Goodwin, Simon Portegies L121
Zwart, RichardParker andAndreasKoch for usefuldiscus- Zapata L. A., Schmid-Burgk J., Ho P. T. P., Rodr´ıguez
sionsthatimprovedthismanuscript.RJAalsoacknowledges L.F., Menten K. M., 2009, ApJL, 704, L45
support from the Alexander von Humboldt Foundation in
theformofaresearchfellowship.Thisworkhasmadeuseof
theIcebergcomputingfacility,partoftheWhiteRoseGrid
computing facilities at theUniversity of Sheffield.The sim-
ulations reported in this paper also made use of the Kolob
cluster at the University of Heidelberg, which is funded in
part by theDFG via Emmy-Noethergrant BA 3706.
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