Stellar Collisions and Blue Straggler Stars in Dense Globular Clusters

18 Pages·2013·2.6 MB·English

Draftversion September12,2013
PreprinttypesetusingLATEXstyleemulateapjv.6/22/04
STELLAR COLLISIONS AND BLUE STRAGGLER STARS IN DENSE GLOBULAR CLUSTERS
SouravChatterjee
DepartmentofAstronomy,UniversityofFlorida,Gainesville,FL32611.
Frederic A. Rasio
CenterforInterdisciplinaryExplorationandResearchinAstrophysics(CIERA)and
Dept. ofPhysics&Astronomy,NorthwesternUniversity,2145SheridanRd,Evanston,IL60208, USA.
3
1
0 Alison Sills
2 DepartmentofPhysicsandAstronomy,McMasterUniversity,1280MainStreetWest,Hamilton,ON,L8S4M1,CANADA.

p and
e Evert Glebbeek
S
DepartmentofAstrophysics/IMAPP,RadboudUniversityNijmegen,POBox9010,6500GL,Nijmegen,TheNetherlands.
1 Draft versionSeptember 12, 2013
1
ABSTRACT

] Blue straggler stars (BSS) are abundantly observed in all Galactic globular clusters (GGC) where
R
data exist. However, observations alone cannot reveal the relative importance of various formation
S channels or the typical formation times for this well studied population of anomalous stars. Using a
. state-of-the-artH´enon-type Monte Carlo code that includes all relevant physical processes, we create
h
p 128 models with properties typical of the observed GGCs. These models include realistic numbers
- of single and binary stars, use observationally motivated initial conditions, and span large ranges in
o central density, concentration, binary fraction, and mass. Their properties can be directly compared
r with those of observed GGCs. We can easily identify the BSSs in our models and determine their
t
s formation channels and birth times. We find that for central densities above ∼ 103M⊙pc−3 the
a
dominant formation channel is stellar collisions while for lower density clusters, mass transfer in
[
binaries provides a significant contribution (up to 60% in our models). The majority of these

3 collisionsare binary-mediated,occurringduring 3-body and4-body interactions. As a resulta strong
v correlation between the specific frequency of BSSs and the binary fraction in a cluster can be seen
4 in our models. We find that the number of BSSs in the core shows only a weak correlation with the
8 collision rate estimator Γ traditionally used by observers, in agreement with the latest Hubble Space
2
Telescope(ACS)data. Usinganidealized“fullmixing”prescriptionforcollisionproducts,ourmodels
7
indicate that the BSSs observed today may have formed several Gyrs ago. However, denser clusters
2. tend to have younger (∼1Gyr) BSSs.
0 Subject headings: methods: numerical — methods: statistical — blue stragglers — stars: kinematics
3 and dynamics — globular clusters: general
1
:
v
i 1. INTRODUCTION appropriate data exist, have shown this exotic stellar
X population (e.g., Piotto et al. 2002; Ferraro et al. 1995).
Blue straggler stars (BSSs) are H-burning stars bluer
r andbrighter(andthereforemoremassive)thanthemain- Many open clusters also show moderately large popula-
a tions(∼10s)ofBSSs(e.g.,Mathieu & Geller2009);they
sequence turnoff (of a star cluster; Sandage 1953). It
arealsofoundinsomedwarfgalaxies(e.g.,Mapelli et al.
is impossible for normal single star evolution to create
2007)andeveninthefield(e.g.,Preston & Sneden2000).
them. Instead, BSSs must be created when a main-
Stellar collisions can happen in many ways in a star
sequence (MS) star’s mass is increased either via mass
cluster. Two single stars can collide directly (s-s colli-
transfer in a binary (MTB) or by physical collision and
sion). If binaries are present, then binaries will interact
merger with another star. Both processes can also in-
with other single stars (b-s) or binaries (b-b). During
crease the residual H-burning life of the star (rejuvena-
b-s and b-b encounters, physical collisions can occur be-
tion) by bringing in new Hydrogen for burning in the
tween two or more MS stars taking part in the interac-
core (eg., Lombardi et al. 1995, 1996; Sills et al. 1997,
2001; Lombardi et al. 2002; Chen & Han 2009) 1. Since tions (Fregeau et al. 2004).
During MTB the details of the orbital evolution and
the BSSs are among the brighter members of a cluster,
mass transfer rate determine the final outcomes. Mass
they are relativelyeasier to detect in photometric obser-
transfer can be stable, such that a fraction of the com-
vations. All Galactic globular clusters (GGCs), where
panionmassistransferredtotheMSstarandthebinary
Electronicaddress: [email protected]fl.edu remains intact; or it can be unstable, leading to a more
1NotethatevenifnonewHydrogenissuppliedtothecore,the rapid dynamical evolution and the eventual merger of
starmaystillappearasaBSSduringitsownresidualMSlifeifits the two stars (Chen & Han 2009). Stellar mergers may
presentmassishigherthantheMSturn-off.
also happen through interactions involving higher-order
2 Chatterjee et al.
hierarchical systems. For example, a hierarchical triple, numerical data.
forcertainconfigurationscanincreasethe eccentricityof For many years, this kind of realistic modeling of
the inner orbit via the Kozai effect (Kozai 1962). If the GGCs (particularly using high enough N and f ) re-
b
eccentricitybecomessufficientlyhigh,thetwoinnerstars mained very challenging because of the extreme com-
couldthenbedriventomerge(Perets & Fabrycky2009). putational cost of direct N-body simulations. Hence,
Although this may be a significant channel for BSS for- much more simplified models were used to study the
mation in less dense open clusters (Perets & Fabrycky BSS formation channels and other properties such as
2009), it is not clear whether these triples can survive their radial distribution in the cluster (Mapelli et al.
long enough in denser GGCs to be effective. 2006). Recently, modernH´enon-typeMonte Carlocodes
The interactions that produce BSSs depend on the (H´enon 1971), modified to include the relevant physi-
cluster global properties, including the central density cal processes (in particular, strong interactions and sin-
(ρ ), binary fraction (f ), distribution of initial binary gle and binary stellar evolution), have opened new pos-
c b
orbitalproperties,andmass-segregationtimescale. Thus sibilities (e.g., Chatterjee et al. 2010; Giersz & Heggie
the BSSsina clustercan,inprinciple,providea window 2011; Hypki & Giersz 2013; Chatterjee et al. 2013;
onto the system’s dynamical history. In this respect the Giersz et al. 2013).
formationchannel(s)oftheBSSsandtheirtypicalforma- The current version of our Cluster Monte Carlo code
tion times are of greatinterest. Unfortunately, although (CMC)iswellsuitedto themodeling ofGGCs including
observationsinvariouswavelengthbandshavetaughtus allrelevant physicalprocesses,with largeN ∼105−106
a lot about the stellar properties of the BSSs, it is hard and realistic f values. CMC has been developed
b
to determine how a particular BSS formed or how long and rigorously tested for over a decade (Joshi et al.
agotheinteractionoccurred. Theoreticalmodelingisthe 2000, 2001; Fregeau et al. 2003; Fregeau & Rasio
only way to approach these questions. 2007; Chatterjee et al. 2010; Umbreit et al. 2012;
A popular approach so far to learn about BSS forma- Pattabiramanet al.2013; Chatterjee et al. 2013). Using
tion has been to carefully study the correlation between this code it is now possible to produce large numbers
thenumberofBSSs(normalizedinsomewaybyanother of highly detailed cluster models covering the full range
characteristicstellarpopulationinthesameregionofthe of observationally motivated initial parameters. For
cluster) and some dynamically important cluster prop- example, recent studies with CMC have explored a
erty,suchasthecollisionrateestimatorΓ(Pooley & Hut large range in parameter space and successfully created
2006), f (Sollima et al. 2008), or the total cluster mass a library of cluster models that have properties very
b
(most recently Leigh et al. 2013). While these observed similar to the observed GGCs (Chatterjee et al. 2010,
correlationscaninprincipleteachusalotaboutthepos- 2013).
sibleformationchannelsofBSSs,directunderstandingof Here we reportresults fromour analysisof 128cluster
these processes is impossible to achieve without careful models produced with CMC with finely sampled output
modeling. For example, a BSS created via MTB could focusing on BSS formation and evolution. These models
actually be the result of some past dynamical encounter include all relevant physical processes, and include real-
that changed the binary orbit in such a way that mass isticN,andf ,andhencecanbedirectlycomparedwith
b
transfer occurred. On the other hand, a collision may observedGGCs without any need for rescaling. In addi-
be mediated by resonant dynamical interactions involv- tiontoalldynamicalproperties,singleandbinarystellar
ing binaries, blurring the distinction between the simple evolutionis alsofollowedindetailinthesemodels. Thus
“collisional” and “binary” origins often discussed in ob- our simulations provide an unprecedented opportunity
servational studies (e.g., Knigge et al. 2009; Leigh et al. to understand the dominant dynamical and stellar evo-
2011a). Inadditiontothesecomplications,itisalsohard lution processes responsible for BSS production in clus-
to determine observationallywhat the binary fractionin ters, spanning a large range in cluster parameters, and
the cluster is at present, and it is impossible to know allow us to test for various correlations between BSSs
what it was in the past. Even in the complete absence and cluster properties.
of binaries, the calculation of the Γ parameter involves Previous works have pointed out that some dynami-
many properties of the cluster that are hard to measure cal properties such as core radius (r ) or ρ for a clus-
c c
accurately. ter can have very different values according to differ-
The only way to explore and probe these complicated ent definitions used by theorists and observers (most re-
interactions in an evolving cluster is by numerical mod- centlyChatterjee et al.2013). Hence,toavoidconfusion,
eling of the full N-body system with stellar evolution, we restrict our analysis in this paper to the standard
using a realistically large N, the full stellar mass func- theoretical definitions used in most N-body simulations
tion, and treating binary stars in detail. One must then (Casertano & Hut 1985), unless otherwise specified. A
also include all relevant physical processes, such as two- differentand independent analysisof the same models is
body relaxation, strong encounters including physical presentedinSills et al.(2013),whereallrelevantquanti-
collisions,b-bandb-sinteractions,MTB,tidalstripping tiesaredeterminedusingthestandardobservational def-
in the Galactic potential, etc. In these models, if all in- initions. In this paper we focus on two basic questions:
teractionsarerecorded,inadditiontothe overallcluster (1) What is the dominant physical process that creates
properties and individual stellar properties at frequent the BSSs observed in GGCs? (2) With our adopted re-
enoughoutput times, then all necessaryinformationcan juvenation prescription (Hurley et al. 2000, 2002), what
be obtainedaboutBSSformationandevolution. Inpar- is the typical age (t ) of the observed BSSs?
age
ticular, after BSSs have been identified in any snapshot The paper is organized as follows. In Section 2 we
of the system, the detailed past history of each BSS go- describe briefly our simulation setup, explain how the
ing all the way back to t = 0 can be extracted from the BSSsareidentifiedinthemodels,andprovidedefinitions
Collisions and Blue Straggler Stars 3
for various categories of BSSs based on their formation “mockobserved”values(r ,andρ )toremaincon-
c,obs c,obs
histories. We also show comparisons between modeled sistent. Most recently Chatterjee et al. (2013) discussed
andobservedBSSsintermsofobservableproperties. We in detail various definitions for the cluster parameters
discuss the relative importance of the various formation and the relationships between theoretical and observed
channels in Section 3. The time elapsed since formation estimates ofthese quantities. Here,we estimate r as
c,obs
for the BSSs in our models is explored in Section 4. In theradiuswherethesurfaceluminositydensitydropsby
Section 5 we investigate correlations of the number of afactorof2fromthecentraldensity(e.g.,Spitzer1987).
BSSsinthecluster(N )withf andΓ. Wesummarize Thevaluesforρ areestimatedfromthe peakcentral
BSS b c,obs
and conclude in Section 6. surface luminosity density and r using the prescrip-
c,obs
tion givenin Djorgovski(1993, their Equation4) follow-
2. NUMERICALMETHOD
ing Harris (1996, 2010 edition). We assume M/L=2 to
We use CMC and adopt a large grid of initial condi- convert luminosity to mass when required to determine
tions over the range of values typical of the observed the mock observed values.
young massive clusters (e.g., Scheepmaker et al. 2007, Our grid of simulations in this study covers star clus-
2009)to create128detailedstar-by-starmodels. All our ters att ≈12Gyrwith r between0.6 and2.8pc. This
cl c
simulatedclustershaveinitialvirialradiibetweenrv =3 range in rc corresponds to rc,obs in the range between
and4pc. InitialN isvariedbetween4and10×105stars. ≈ 0.06 and 5.07pc. Similarly, this grid spans clusters
The positions and velocities of the stars (and center of withρc between≈7×102 and105M⊙pc−3 whichcorre-
mass for the binaries) are assigned according to King sponds to ρc,obs between≈3×102 and4×106M⊙pc−3.
profiles with W0 between 4 and8. We vary the initial fb Final M arebetween1×105 and4×105M⊙, andcorre-
between 0.05 and 0.3. The masses of the single stars, or sponding M values are similar. Among the 128 mod-
obs
primaries in case of a binary, are chosen from the IMF els, 19 have reached the binary-burning stage (thought
presented in Kroupa (2001, their Equations 1 and 2) in to be equivalent to the post core-collapsed GGCs) be-
the stellar mass range 0.1−100M⊙. Secondary binary fore t = 12Gyr, while the rest are still contracting
cl
companion masses are sampled from a uniform distribu- (i.e.,theyarenon-core-collapsedGGCs,Chatterjee et al.
tion of mass ratios in the range 0.1M⊙−mp, where mp 2013). Relevant model properties are listed in Table
is the mass of the primary. The semi-major axes, a, for 1. For more detailed comparison between model and
stellarbinariesarechosenfromadistributionflatinloga GGC properties see Chatterjee et al. (2013). For mock
within physical limits, namely between 5× the physical observed BSS values for these models and further com-
contactofthecomponentsandthelocalhard-softbound- parisons with GGC properties see Sills et al. (2013).
ary (Heggie & Hut 2003). Although initially all binaries
inourmodelsarehardattheirrespectivepositions,some 2.1. Definition of blue straggler stars in our models
of these hard binaries can become soft during the evolu- We define BSSs asstarsthatarestill inthe H-burning
tionofthecluster. Theclustercontractsundertwo-body stage but with masses > 1.1M , where, M is the
TO TO
relaxation and the velocity dispersion increases making MS turn-off mass of the cluster. Note that in real clus-
initiallyhardbinariessoft. Moreover,binariessinktothe tersBSSsarechosenfromacolor-magnitudediagramby
core due to mass segregation where the velocity disper- adefiningboxfortheBSSsthatdependon,forexample,
sion is higher than that at the initial binary positions. the quality of the data, the filters used for the observa-
We include these soft binaries in our simulations until tion, and the scatter width of the MS (e.g., Leigh et al.
theyarenaturallydisruptedviastrongencountersinthe 2007;Sills et al.2013). Themass-basedcriterionismore
cluster. theoretically motivated and is much simpler to use for
The numerical setup and initial properties for these allmodels (although, inaccessible to observers). We find
simulations are similar to the work presented in that in our models the BSSs identified using the mass-
Chatterjee et al. (2013) which shows that similar initial basedcriterionpopulatetheexpectedregionofsynthetic
conditions result in models at the end of the simulation Hertzsprung-Russell diagram (HRD) created using our
(at cluster age tcl ≈ 12Gyr) with properties including models (Figure 1 for an example). Observationally mo-
rc, half-mass radius (rh), ρc, total mass (M), and relax- tivatedselectionboxesareadoptedtoextractBSSsfrom
ation timescale at half-mass (trh) very similar to those these same models in Sills et al. (2013) and we find that
observedintheGGCs. Thedifferencesintheinitialcon- thenumbersobtainedfromeithercriteriaaretightlycor-
ditionsbetweenthesesimulationsandthosepresentedin related. Furthermore, the selected BSSs using either
Chatterjee et al. (2013) are as follows. Models with zero criteria do not show any systematic differences in their
initial fb are discarded in this study since observations formation channels, formation ages, or radial positions.
suggest that this is not the case for the GGCs in reality Sinceinthisstudywerestrictourselvestothetheoretical
(Davis et al.2008;Milone et al.2012)2. Inaddition,the understanding of the BSSs, in particular to understand
models presented in this study explore a larger range in the dominant formation channels and typical ages since
initial fb, and W0. formation, we simply adopt the much simpler theoreti-
While in most cases we present theoretical estimates cally motivated mass-based definition for BSSs for this
ofcentralquantitiessuchasrc andρc (Casertano & Hut study.
1985), whenever we directly compare model quantities
withobservedGGCquantities,weusethecorresponding 2.2. Classification of blue-straggler stars based on their
formation channels
2 In fact, comparison between the observed BSS numbers in At the last snapshot of the model (t ∼ 12Gyr) we
the GGCs and the numbers obtained via mock observations of cl
identifytheBSSsfollowingthecriteriondescribedinSec-
these models indicates that even initial fb =0.05 maybe too low
(Sillsetal.2013). tion 2.1. The full dynamical history for each of these
4 Chatterjee et al.
3
2
1
)
⊙
L
/
Leff
( 0
g
o
l
−1
−2
−3
4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4
log(T /K)
eff
Fig. 1.—ExampleofasyntheticHertzsprung-Russell(HRD)diagramforarepresentativemodelwithmanybluestragglerstars(run110,
Table 1). Dots denote all stars in the cluster. Red squares and blue triangles denote blue-straggler stars in single and binary systems,
respectively at the integration stopping time of tcl = 12Gyr. Blue straggler stars are defined as H-burning stars with stellar masses
≥1.1MTO, where MTO is the main-sequence turn-off mass. BSSs in this model populate the expected region in the synthetic HRD. In
addition,thesingleandbinarymain-sequences,giantbranch,white-dwarfcoolingsequence areclearlyvisible.
BSSs is retrieved. We identify the interaction (such as way that led to the mass transfer event that ultimately
s-s,b-s,orb-bcollisions,orMTB)thathadincreasedthe createdtheBSS.TheseperturbedMTBorSEchannelsare
mass of this star for the first time such that it would be duly noted and discussed. However, interactions of this
identified as a BSS at t . We classify each BSS into one type do not change the formation classification.
cl
of 5 channels according to this first interaction. The 5
channels are: stable mass transfer in a binary denoted 2.3. Comparison of Model Blue Stragglers with
as MTB, binary-stellar-evolution-driven merger denoted Observed Blue Straggler Properties
as SE, single-single collisions denoted as s-s, collisions
Here we briefly investigate whether the modeled BSS
mediated by binary-single interactions denoted as b-s,
properties are consistent with some of the well studied
andcollisionsmediatedbybinary-binaryinteractionsde-
observed BSS properties in the GGCs. Note that these
noted as b-b. Note that some BSSs may have a compli-
are motivated to add further validation of the model
catedhistory. Forexample,afteramasstransferepisode
BSSs. More detailed analysis of these properties are po-
in a binary, a MS star may have attained a BSS mass;
tentially interesting but beyond the scope of this study
lateronthisstarcouldstillcollidewithanotherMSstar,
and we differ that to Sills et al. (2013) and future work.
furthergrowinginmassandgettingrejuvenatedforasec-
ond time. For BSSs that are formed via such multiple
2.3.1. Radial Distribution
channels, only the first one is considered for the classi-
fication to avoid ambiguity. In this particular example One of the observationally accessible properties of
theformationchannelforthisBSSwouldbe classifiedas the BSSs in the GGCs is their radial distribution. The
MTB.Nevertheless,suchcomplicatedmultiple episodes of radial distribution of the BSSs in a cluster depends on
rejuvenation of the same MS star is not frequent. There a collection of the host cluster’s properties including
isanothersourceofcomplication. ABSSclassifiedasMTB ρ , relaxation time, and f . Thus analysis of the radial
c b
mayhavehadinteracteddynamicallypriortotheRoche- distribution of the BSSs has been proposed as a tool to
lobe overflow that had not changed its mass. Such an observationally constrain the dominant formation chan-
interactionwouldchangetheorbitalpropertiesofthebi- nel,andvariousdynamicalproperties ofthe hostGGCs.
nary,sendingitalongaverydifferentevolutionarypath- As a result, the radial distribution of the BSSs has
Collisions and Blue Straggler Stars 5
been studied with interest for many GGCs via observa- and the core radii both are consistent with the observed
tions (e.g., Fusi Pecci et al. 1992; Dalessandro et al. Milky Way populations.
2008b; Ferraro et al. 1995; Beccari et al. 2008; In addition to these caveats about different identifica-
Dalessandro et al. 2008a; Moretti et al. 2008a,b; tion criteria for BSSs, one should also note that, within
Ferraro et al. 2012) as well as via theoretical modeling BSE, and the current version of CMC, the rejuvenation
(e.g., Mapelli et al. 2006). Three qualitatively different prescriptionusedforallmergers,whetherfromcollisions
types of radial distributions are observed among the orbinaryevolution,andMTBassumes“fullmixing”(for
GGCs,namely,singleandbimodal,andflatdistributions details see Hurley et al. 2000, 2002). This is an oversim-
(for a compilation see Ferraro et al. 2012). We check plification. Inreality,the amountofHydrogencoremix-
whether the radial distributions of our model BSSs are ing and thus the degree of rejuvenation of the MS star
consistent with those observed in the GGCs. depends on the kinematic details of the encounter (e.g.,
To find the relative frequency of the BSSs in the Sills et al.1997,2001). Becauseofthissimplificationthe
cluster as a function of their radial positions we define lifetimes of the BSSs after rejuvenation in our models
a double-normalized ratio R following Ferraro et al. areactuallyupper limits (atleastfornon-rotatingstars).
BSS
(2012) given by This may also lead to an over-productionof BSSs in our
models compared to real clusters.
3. RELATIVEIMPORTANCEOFBLUESTRAGGLER
δN /N
R = BSS BSS, (1) FORMATIONCHANNELS
BSS
δL/L
cl In this section we investigate the relative contribu-
tions from various BSS formation channels. Figure 4
where δN denotes the number of BSSs within some
BSS shows an example of branching ratios for the BSSs in
radial bin, N denotes the total number of BSSs, δL
BSS the whole cluster, and in the core, for model run32 (Ta-
denotes the stellar luminosity within that bin, and L
cl ble 1). This is a model with moderate central density
denotes the total luminosity of the cluster. R is a
measure of whether the BSSs follow the luminosBitSySden- ρc = 7×103M⊙pc−3. Collisions produce most of the
BSSsinthismodel. AmongthecollisionalBSSs,thes-s
sityprofileoftheclusterortheyareover/underabundant
channel contributes the least. The binary mediated col-
in some parts.
lisional channels, namely b-s and b-b contribute about
Figure2showsR asafunctionoftheradialposition
BSS 36% and 50%, respectively. BSSs produced via the MTB
in units of the core radius. In our models we find exam-
channel contributes about 11% and SE contributes only
ples of bimodal distributions (bottom panel, observed,
3.2%. As expected collisional channels dominate even
e.g., in M55) as well as centrally peaked distributions
more among the core BSSs. Interestingly, although the
(top panel, observed, e.g., in M80, Ferraro et al. 2012,
b-sandb-b contributionsincrease if only coreBSSs are
and references therein). In our models we do not find
considered instead of all BSSs in the cluster, s-s contri-
flat radial distributions observed, for example, in NGC
bution decreases. This trend is seen often in our models
2419 (Dalessandro et al. 2008b). This is expected since
because of the low number density of single MS stars
all our models are relaxed and hence a non-segregated
within the core at late times, due to mass segregation.
radial distribution for the BSSs is not expected in our
About 60% of all BSSs formed via MTB or SE in this
models. Ferraro et al. (2012) suggest that the minimum
example model have had at least one prior strong en-
distance r from the center where the central peak of
min counter. Thus for this particular model a significant
R ends in units of r is anti-correlated with the core
BSS c fractionofeventhe MTBandSEBSSs arenotcreatedvia
relaxation timescale due to mass-segregation effects of
purebinarystellarevolution,andhavebeendynamically
the relatively high-mass BSSs. Results from our models
modifiedbeforebinarystellarevolutioncouldrejuvenate
are consistent with this finding. However, we find that
them.
the determination of r can have large errors and de-
min Branching ratios for various BSS production channels
pendsstronglyontheradialbinsusedtocalculateR .
BSS are expected to depend on ρ and f in a cluster. We
c b
divide our full set of models into subsets based on the
2.3.2. Core Blue Straggler Number and Core Radius
initialf andpresentthebranchingratiosforeachsubset
b
Are the number of BSSs in the core (N ) in our separately as a function of ρ .
BSS,c c
models consistent with those in the GGCs? Figure 3 Figure 5 shows the branching ratios for s-s, b-s, b-b,
shows N as a function of r both for our mod- MTB,and SE formation channels as a function of ρ . The
BSS,c c,obs c
els and for observed GGCs. Note that since we are di- differentpanelsshowsubsetsofmodelswithdifferentini-
rectly comparing with observed BSSs, we use rc,obs for tial fb. For models with ρc > few ×103M⊙pc−3 colli-
this comparison. The BSSs are, however, chosen using sional channels clearly dominate. In lower density clus-
the same mass-based criteria. The observed BSS counts ters contribution from MTB increases. Throughout the
areobtainedfromtheACSsurveydata(Sarajedini et al. full range in f in our models (f = 0.05 – 0.30), the
b b
2007) as selected by Leigh et al. (2011b). Our models same trend is observed. This indicates, that even if f
b
clearly have very similar core radii compared to the ob- is increased (within the studied range in ρ ) the relative
c
servedGGCs. WeidentifyaslightlylargerN . More contribution from MTB does not increase proportionally.
BSS,c
observationally motivated selection criteria presented in While increasing f increasesthe probability that a BSS
b
Sills et al. (2013) for the same models show that if these can be created via MTB (simply because of the increased
systems were observed,the number of BSSs actually de- number of binaries), the increased number of binaries
tected would have been smaller by about a factor of 2. alsoincreasestheb-sandb-bBSSformationrates. Typ-
ThusweconcludethatN asproducedinourmodels icalρ valuesintheGGCsareordersofmagnitudehigher
BSS,c c
6 Chatterjee et al.
8
7
6
5
S
BS 4
R
3
2
1
0
0 2 4 6 8 10
5
4
3
S
S
B
R
2
1
0
0 5 10 15 20 25 30
r/r
c
Fig. 2.—ExamplesoftwoqualitativelydifferentradialdistributionsforourmodelBSSs. Verticalandhorizontalaxesdenotethedouble-
normalizedratioRBSS (Equation1)andradialpositioninunitsofrc,respectively. Toppanelshowsanexampleofcentrallypeakedradial
distribution(run108;Table1). Bottompanelshowsanexampleofabimodaldistribution(run110),onepeaknearthecenterr/rc<5and
asecondpeakatlarger/rc≈22. Azoneofavoidanceisobservedatr/rc betweenabout5and17.
than ∼ 103M⊙pc−3 (e.g., Harris 1996; Chatterjee et al. fer process has occurred (with respect to all BSSs pro-
2010, 2013). Hence, we conclude that the collisional duced via MTB and SE). These interactions can poten-
channels dominate production for the BSS populations tially alter the orbital properties of these binaries ini-
observed in the majority of the GGCs. If only the core tiating very different evolutionary pathways for them.
BSSs are considered, the importance of the collisional BSSsformedfromthesedisturbedbinariesclearlywould
channels increases as expected. We find that through- not have formed the BSSs with the exact same proper-
outthe fullrangeofρ inourmodelscollisionalchannels ties and at the same times without these encounters. As
c
dominate the formation of core BSSs (Figure 6). expected, for the models with higher ρ values MTB and
c
Although binary stellar evolution is not the dominant SE BSSs have a higher chance of being perturbed be-
channel for a large range of central densities typical of fore production. For a given ρ a range of these values
c
GGCs, binaries themselves are nevertheless crucial for are also seen. This is due to the different concentrations
significant BSS production. Figure 7 and 8 show the for these clusters. Models with initially lowerconcentra-
relative contributions from various collisional formation tionsproduceasmallerfractionofperturbedMTBandSE
channels for the BSSs in the whole cluster and in the BSSs. Themajorityofthemodelswithρc &104M⊙pc−3
core, respectively, as a function of ρ . For the full range produce more perturbed MTB and SE BSSs than unper-
c
in f and ρ in our models BSSs produced via s-s are turbed. Low density (ρ . 103) models on the other
b c c
rare. Binary-mediated collisions (b-s or b-b) dominate. handwillhavearelativelylargerfractionofBSSsviathe
As f increases, so does the relative contribution from MTB and SE channels created from the primordial undis-
b
b-b. For example, for models with initial f = 0.05, turbed binaries. Similar results were also found previ-
b
the b-s channelcontributes moreto the collisionalBSSs ously by Sollima et al. (2008). Our results support their
compared to the b-b channel for all ρc values. In con- conclusion that for ρc . 103M⊙pc−3 stellar evolution
trast, for initial f = 0.3 the contribution from the b-b channels contribute &50%ofthe BSSs, and a high frac-
b
channel dominates for all ρ values we have considered. tion of them are produced from dynamically unmodified
c
Figure 9 shows the fraction of BSSs produced via binaries.
binary stellar evolution (MTB and SE collectively) that
havebeen dynamically perturbed before the mass trans- 4. TIMESINCEFORMATIONFORTHEBLUE
STRAGGLER STARS
Collisions and Blue Straggler Stars 7
103
models
observed
bs
o
n rc, 102
hi
t
wi
s
S
S
B
f
o
r
e
mb 101
u
n
100
10-2 10-1 100 101
r (pc)
c,obs
Fig. 3.—NumberofBSSsinthecore(NBSS,c)vsrc,obs. BlackcirclesandredsquaresdenotemodelandobservedBSSs,respectively. The
observed NBSS,c values are obtained from the HST/ACS survey (Sarajedinietal. 2007) as selected by Leighetal. (2011b). Our models
populate very similar rc,obs values compared to the observed GGCs. However, our models show slightly higher NBSS,c numbers due to
themass-basedselectioncriteria. IfthesemodelscouldbeobservedusingtheHST/ACSfiltersandtheBSSswereidentifiedusingamore
observationallymotivated selectionboxNBSS,c woulddecreasebyaboutafactorof2(Sillsetal.2013).
all BSSs core BSSs
SS coll BS coll SS coll BS coll
0.3 35.9 0.0
35.7
0.5 SE
3.2 SE
5.9
MTB
10.6
50.0 MTB
57.8
BB coll
BB coll
Fig. 4.— Example of branching ratios for the various formation channels of BSSs at tcl =12Gyr (run32; Table 1). This model has a
moderatecentraldensity(ρc ≈7×103M⊙pc−3;Table1). Thisparticularmodelwaschosenasanexampleduetoitshighnumber(312)
of BSSs. The dominant formation channel is stellar collisions in this model. Among the collisional BSS production channels collisions
mediatedbyb-binteractionsdominatefollowedbyb-smediatedcollisions. s-scollisionsarerare. MTBisthedominantchannelamongthe
binary-stellar-evolutiondrivenBSSproductionchannels. SEcontributeonlytoasmalldegree.
8 Chatterjee et al.
1.0
f =0.05
0.5 b
0.0
1.0
f =0.1
o 0.5 b
ti
a
r
g 0.0
n
hi 1.0
c
n
a
r f =0.2
b 0.5 b
0.0
1.0
f =0.3
0.5 b
0.0
103 104 105
ρc(M pc−3)
⊙
Fig. 5.—Branching ratios of various formationchannels for all BSSs inour models vs the respective cluster’s ρc. The different panels
from top to bottom show subsets of models with different initial fb in an increasing order. The respective initial fb values are shown in
each panel. The circles, squares, and triangles denote collisional, MTB, and SE channels, respectively. Clearly, for ρc > few ×103M⊙pc−3
collisionalformationschannels dominateforallinitialfb. BSSsfromSEarerarethroughoutthefullrangeinρc andfb inourmodels.
Collisions and Blue Straggler Stars 9
1.0
f =0.05
0.5 b
0.0
1.0
rc
n
hi 0.5 fb=0.1
t
wi
o
ti 0.0
a
r 1.0
g
n
hi f =0.2
c 0.5 b
n
a
r
b
0.0
1.0
f =0.3
0.5 b
0.0
103 104 105
ρc(M pc−3)
⊙
Fig. 6.— Same as Figure 5 but including only the BSSs within the core. Collisional channels dominate the core BSSs throughout the
fullrangeinρc andfb. SEmergersarerarethroughouttherangeofρc inourmodels. MTBchannelonlybecomescomparabletocollisional
channelsifρc<103M⊙pc−3.
One of the biggest uncertainties in any star-by-star of t are also interesting in light of the current under-
age
N-body model using analytical formulae (Hurley et al. standing that most GGCs (those observationally cate-
2000) for hydrodynamic mergers and stellar evolution is gorized as non-core-collapsed) are still contracting (e.g.,
the degree of rejuvenation. Unfortunately, the degree Chatterjee et al.2013). As a result,relevantglobalclus-
of rejuvenation depends on the detailed kinematics of ter properties such as ρ , r , and r have not attained
c c h
the encounter (e.g., Sills et al. 1997, 2001) as well as the steady-state values typical of core-collapsed, “binary-
stellar properties such as rotation, and the ages of the burning” clusters (Chatterjee et al. 2013). Hence, for
parent stars (Chen & Han 2009). Incorporating these these contracting clusters it is not sufficient to know the
details is beyond the scope of this study and can only present-day observed cluster properties to fully under-
be done via live stellar and binary evolution codes that standthe formationefficiencyofBSSs. Instead,the past
canrigorouslyincorporatethesephysicaleffectswhilethe dynamical history over at least the last ∼ t should
age
whole cluster is dynamically evolving. This is a hard be taken into account. Of course the results (based on
task and has yet to be achieved fully self-consistently the simple full mixing assumption) presented here can
by any group. Hence, although the analytical formulae also be directly compared with future results from sim-
andprescriptionsusedinBSE(Hurley et al.2000,2002) ulations incorporating more sophisticated rejuvenation
are known to over-estimate the degree of rejuvenation treatments when they become available.
(e.g., Glebbeek & Pols 2008), for example, for collision Figure 10 shows examples of the distributions of t
age
productsofnon-rotatingstars,thisremainscurrentlythe fortwoofourmodels. Thesemodelsarechosensuchthat
state of the art. For the purpose of this study, how- atleast21BSSsarecreated(toensureenoughstatistics)
ever,we expect thatthe relative contributions fromMTB, and the relative contributions from collisional channels
SE,andvariouscollisionalchannelsshouldnotchangeby f have extreme values 0.5 (run60),and 1 (run17;Ta-
coll
much given that all channels are treated using maximal ble 1). The central densities differ by a little over an
mixingwithinBSE,hence rejuvenationmaybe similarly order of magnitude.
over-estimatedin all cases. The BSSs in the high-f (hence high-ρ ) model are
coll c
Despite these caveats,itis importantto study the dis- typically younger compared to those in the low-f
coll
tributions of t for our model BSSs since they provide model. The broadly peaked distributions seen here are
age
us with useful upper limits. The typical distributions qualitatively similar in all our models. However, the ex-
10 Chatterjee et al.
1.0
f =0.05
b
0.5
0.0
S
S 1.0
B f =0.1
al b
n
o 0.5
si
olli
c 0.0
:
o 1.0
ati fb=0.2
r
g
n 0.5
hi
c
n
a 0.0
r
b 1.0
f =0.3
b
0.5
0.0
103 104 105
ρc(M pc−3)
⊙
Fig. 7.—BranchingratiosbetweenvariouscollisionalBSSformationchannelsvsρc. Squares,triangles,andcirclesdenoteb-s,b-b,and
s-scollisions,respectively. Eachpanel showsasubsetofmodelswithagivenvalueofinitialfb as mentionedoneachpanel. Throughout
the studied range of ρc and fb contribution from s-s channel is minor. Collisions from b-s and b-b dominate. For high fb =0.3 values
b-bcollisionsdominateoverb-scollisionsincreationofBSSs.
act positions of these peaks depend on the cluster prop- between the median t values for BSSs in the whole
age
erties. For the low-density model the bulk of the BSSs cluster and in the core decreaseswith decreasing f . We
b
have ages between t ≈ 0 and 7Gyr. In contrast, the find that the oldestBSSs in some clusters can be almost
age
BSSs in the high-density model typically have t be- as old as the respective clusters.
age
tween 0 and 3Gyr. This is consistent with previous re-
sults suggested for the observed BSS population in 47 5. NUMBEROFCOREBLUE-STRAGGLERSTARSVS
Tuc(Sills et al.2000). HalfofallBSSsinthelow-density BINARYFRACTIONANDΓ
modeliscontainedwithinanageof≈3Gyr,whereas,for After the large-scale surveys using the Hubble Space
the high-density model this value is ≈0.9Gyr. Telescope (e.g., the ACS survey; Sarajedini et al. 2007;
Thedifferenceinthepeaksoftheagedistributionsbe- Dotter et al.2007;Milone et al.2008)providedhighres-
tween the high and low-density models is expected. In a olution, homogeneously observed and analyzed data for
high-density model the dominant formation channel for a large number of GGCs, a number of studies have in-
BSSs is stellar collisions (s-s, b-s, and b-b; Figures 4, vestigated various correlations between the number of
5). ContributionfromMTBincreasesasρc decreases(Fig- BSSs in these clusters and their global properties (e.g.,
ure5). StellarcollisionstypicallyformhighermassBSSs Leigh et al.2007;Sollima et al.2008;Knigge et al.2009;
thanthoseformedviaMTB.SincealowermassBSShasa Leigh et al. 2013). In particular, the correlations or
longer residual lifetime than a higher mass counterpart, lack thereof between N and two cluster properties
BSS,c
the typical tage becomes shorter as ρc increases due to namely the binary fraction in the core (fb,c) and the
highercontributionfromcollisionalBSS-formationchan- collisional Γ generated a lot of interest. Both of these
nels. Indeed we find that the median for the t values quantities are of greatdynamical importance. The pres-
age
for the BSSs in our models negatively correlate with ρc ence of primordial binaries increases the BSS formation
t(ocotrhreelastaimonecroeaeffisocnietnhtermρce,tdaigaen=v−al0u.e44o;fFtigurfoer1t1h)e. cDouree froartmesatthiornouvgihaasl-lschiasndnierelsctelxycerepltaste-ds.toIntshteeacdo,lloisniloynBalSΓS
age
BSSs is typically slightly lower compared to the median given by
value of t for all BSSs in a cluster. This effect is more
age
prominent for higher fb models. The number of BSSs Γ= ρc 2 rc 3 vc,σ −1 (2)
outsiderc islowforlowfb models. Hence,the difference (cid:18)M⊙pc−3(cid:19) (cid:18)pc(cid:19) (cid:16)kms−1(cid:17)