Table Of ContentMon.Not.R.Astron.Soc.000,1–7(2008) Printed21January2009 (MNLATEXstylefilev2.2)
Using the minimum spanning tree to trace mass
segregation
9
0 Richard J. Allison1⋆, Simon P. Goodwin1, Richard J. Parker1,
0 Simon F. Portegies Zwart2,3, Richard de Grijs1,4 and M. B. N. Kouwenhoven1
2
n 1 Department of Physics and Astronomy, Universityof Sheffield, Sheffield, S3 7RH, UK
a 2 SectionComputational Science, Universityof Amsterdam, Amsterdam, The Netherlands
J 3 Astronomical Institute “Anton Pannekoek”, University of Amsterdam, Amsterdam, The Netherlands
4 NAOC-CAS,Beijing, China
4
1
]
A
G ABSTRACT
. We present a new method to detect and quantify mass segregation in star clusters.
h
It compares the minimum spanning tree (MST) of massive stars with that of random
p
stars. If mass segregation is present, the MST length of the most massive stars will
-
o be shorter than that of random stars. This difference can be quantified (with an
r associatedsignificance)tomeasurethedegreeofmasssegregation.Wetestthemethod
t
s onsimulatedclustersinboth2Dand3Dandshowthatthemethodworksasexpected.
a We apply the method to the OrionNebula Cluster (ONC) and showthat the method
[
isabletodetectthemasssegregationintheTrapeziumwitha‘masssegregationratio’
1 ΛMSR =8.0±3.5(whereΛMSR =1isno masssegregation)downto16M⊙,andalso
v that the ONC is mass segregatedat a lower level (∼2.0±0.5) down to 5 M⊙. Below
7 5 M⊙we find no evidence for any further mass segregation in the ONC.
4
0 Key words: methods: data analysis
2
.
1
0
9 1 INTRODUCTION regation. InSections 3 and 4we introduceour newmethod
0
and test it against a number of artificial clusters and the
: Itisthoughtthatmanyyoungstarclustersshow‘massseg-
v ONC. In Section 5 we discuss the method’s limitations and
regation’ – where the massive stars are more centrally con-
i somewaysinwhichitmightbeimproved,andwesummarise
X centrated than lower-mass stars (see Section 2 for a review
ourresults in Section 6.
of current methods for detecting mass segregation).
r
a There are two possible origins of mass segregation in
star clusters. It may be dynamical, in that mass segrega-
tion evolves within an initially non-mass segregated clus- 2 MASS SEGREGATION
ter due to dynamical effects (Chandrasekhar 1942; Spitzer
2.1 What is mass segregation?
1969).Oritmaybeprimordial,wheremasssegregationisan
outcomeofthestarformation process(Murray & Lin1996; In this paper we will take as a working definition of mass
Bonnell & Bate 2006). Information about the presence and segregation that the most massive stars are not distributed
degree of mass segregation thusprovides strong constraints inthesamewayasotherstars.Inparticular,thattheyhave
on star cluster formation and evolution. However, thereare a more concentrated distribution.
currently no good methods by which mass segregation can This is not the only possible definition of mass segre-
bedeterminedandquantifiedabsolutely.Thisseriouslylim- gation. It would be possible to define mass segregation as
its the interpretation of observations of mass segregation, a significant difference in the mass functions1 (MFs) at dif-
and especially our ability to compare different clusters. ferent positions in a cluster. Indeed, this is often the way
In this paper we present a new method, based on the masssegregationiscurrentlydefined(seeSection2.2).How-
minimumspanningtree,whichallowsustoquantifythede- ever,wefeel suchadefinition isproblematic asit raises the
greeofmasssegregationinacluster,andexaminehowmass question of how many stars need to be included in order to
segregationchangeswithstellarmass.InSection2wereview properly sample the MF and therefore tell if two MFs are
thelimitationsofcurrentmethodsofdeterminingmassseg-
1 Wedeliberatelydistinguishbetweenthepresent-daymassfunc-
⋆ E-mail:r.allison@sheffield.ac.uk tionandtheinitialmassfunction.
2 R. J. Allison et al.
different.Also,howmuchdothemass rangessampled need that the cluster centre must be determined; it is generally
tooverlapinordertodifferentiateMFslopes?Bothofthese chosen as where the luminosity is highest - i.e. where the
questions are difficult to answer without assuming a MF a massive stars are.
priori.
3 MINIMUM SPANNING TREE METHOD
2.2 Current methods for detecting mass
segregation In this section we detail our new method for defining and
quantifying mass segregation using a minimum spanning
The current methods can be loosely separated into two
tree (MST).
groups,thosethatinvolvefittingadensityprofileandchar-
acteristic radii tovarious mass ranges; and thosethat trace
thevariation of theMF with radius. 3.1 The MST
Noneof thecurrentmethodsto examinemass segrega-
tion give a model independent,quantitativemeasure of the The MST of a sample of points is the path connecting all
amount of mass segregation in a cluster (and most are not points in a sample with the shortest possible pathlength,
quantitativeatall).Thecurrentmethodsofdeterminingthe which contains no closed loops (see, e.g. Prim 1957). The
presenceofmasssegregation alsorelyonadeterminationof lengthoftheMSTisunique,buttheexactpathmightdiffer
thecentreofthecluster,whichinitselfmaybeverydifficult in some special circumstances. For example, an equilateral
to define. triangleofstarsnamed1,2,and3hastwopossibleMSTs–
Density profiles are usually fitted with a model profile one connecting stars 1 & 2 and 2 & 3, and one connecting
(e.g. a King (1966) model) or as a cumulative distribution stars 1 & 2 and 1 & 3. But the lengths of both MSTs are
functionfordifferentmass‘bins’.Thisallowsthevariationin identical.
distribution to be easily seen, and the ‘degree’ of mass seg- We have used the algorithm of Prim (1957) (see also
regation can be measured by the change of the slope of the Cartwright & Whitworth (2004)) to construct our MSTs.
distribution function. From these distributions characteris- We construct an ordered list of the separations between all
tic radii for each mass bin (e.g. core radius) can be found. possible pairs of stars. Starsare thenconnected together in
This method has been used in many studies (Hillenbrand ‘nodes’ starting with the shortest separations and proceed-
1997;Pinfield et al.1998;Raboud1999;Adamset al. 2001; ing through the list in order of increasing separation, form-
Littlefair et al. 2003; Sharma et al. 2008). However, theac- ingnewnodesaslongastheformationofthenodedoesnot
curacy of this method has been brought into question by create a closed loop.
Gouliermis et al. (2004), who show that this method is
highly dependent on the number of mass bins, the size of
3.2 Quantifying mass segregation
the bins, and on the models used to fit the density pro-
file. Converse & Stahler (2008) use a method which is sim- Followingourdefinitionofmasssegregation(seeSection2.1)
ilar to comparing cumulative distributions. They study the we can detect and quantify mass segregation by comparing
amountofmassinsideaprojectedradiusandobtainaquan- the typical MST of cluster stars with the MST of the most
tifiable measure of the degree of mass segregation from the massive stars.
area between this curve and a curve for a cluster with no ThemethodisbasedonfindingtheMSToftheN most
masssegregation.However,thisreliesonhavingamodelfor massive stars and comparing this to the MST of sets of N
a non-mass segregated cluster. Gouliermis et al. (2008) use randomstarsinthecluster.IfthelengthoftheMSTof the
the variation of the ‘Spitzer radius’ (the Spitzer radius is most massive stars is significantly shorter than the average
the r.m.s. distance of stars in a cluster around the centre length of the MSTs of the random stars, then the massive
of mass) with luminosity todeterminethepresenceofmass starshaveadifferent,andmoreconcentrated,distribution–
segregation. hencetheclusterismasssegregated.Theratiooftheaverage
A variation of the MF with radius will show the pres- randomMSTlengthtothemassivestarMSTlengthgivesa
ence of mass segregation as a shallowing of the slope of the quantitativemeasureofthedegreeofmasssegregation with
MF with radius (de Grijs et al. 2002a,b,c; Gouliermis et al. anassociatederror(i.e.howlikelyitisthatthemassivestar
2004;Sabbiet al.2008;Kumaret al.2008;Harayama et al. MST is thesame as that of a random set of stars).
2008).Massfunctionsarecalculatedforvariousannuli,such The algorithm proceeds in thefollowing way:
thatinamasssegregatedclustertheradiiclosesttothecen- 1.Determine the lengthof the MSTof theNMST most mas-
tre of the cluster will contain the most massive stars, and sive stars; lmassive. The MST of a subset of the NMST most
sotheywillhaveshallower slopesthanthoseatlargerradii. massive (or what ever subset is of interest) stars is con-
This method also suffers from a strong dependence on the structed.This MST has length lmassive.
choice of annuli and mass bins for the MF (Ascenso et al. 2.DeterminetheaveragelengthoftheMSTofsetsofNMST
2008).Gouliermis et al. (2004) findthatauniquesetof an- random stars; hlnormi. Sets of NMST random stars are con-
nulitouseforcomparisonisnotsimpletochoosebecauseof structed, and the average length, hlnormi of their MSTs is
thevariationoftheshapeandslopeoftheMFasthechoice determined. There is an dispersion associated with the av-
ofbinsizechangesforindividualclusters.Aparticularprob- eragelengthoftherandomMSTswhichisroughlygaussian
lemisthatlow-massstarsareoftendifficulttodetectinthe and so can be quantified by the standard deviation of the
centralregions leading torelatively small regions ofoverlap lengths hlnormi±σnorm.
in the MFs at different radii (which are the slopes to be Numericalexperimentshaveshownthat50randomsets
compared).Afurthercomplicationisintroducedbythefact issufficienttoobtainagoodestimateoftheerrors.However
Using the MST to trace mass segregation 3
using hundreds of random sets tends to result in smoother
N
MST
trends. We suggest using 500 – 1000 random sets for low
5 10 100 200
NMST and at least 50 for high NMST.
Note that theerror σnorm on hlnormi is always large for Inverse-MS 14.1% 15.5 17.1 15.2
smallNMST duetostochasticeffectswhenrandomlychoos- MS 12.6% 14.1 17.0 17.2
NoMS 73.3% 70.4 65.9 67.6
ing a small numberof stars (see Section 4.2).
3.Determinewithwhatstatisticalsignificancelmassive differs
fromhlnormi.Wedefinethe‘masssegregationratio’(ΛMSR) Table 1. The percentage of Plummer spheres which show evi-
denceofinverse-masssegregation(Inverse-MS),masssegregation
as the ratio between the average random path length and
(MS), and no mass segregation (No MS) for various numbers of
that of themassive stars: membersintheMST,NMST,in3dimensions.
hlnormi σnorm
ΛMSR = ± (1)
lmassive lmassive
where an ΛMSR of ∼ 1 shows that the massive stars are NMST
distributed in the same way as all other stars, an ΛMSR 5 10 100 200
significantly > 1 indicates mass segregation, and an ΛMSR Inverse-MS 13.1% 14.9 17.2 17.1
significantly <1indicates inverse-masssegregation (i.e. the MS 10.4% 12.2 17.0 19.1
massivestarsaremorewidelyspacedthanotherstars).The NoMS 76.5% 72.9 65.8 63.8
more (significantly) different the ΛMSR is from unity, the
more extreme is thedegree of (inverse-)mass segregation. Table2.AsTable1butforclustersprojectedinto2dimensions
4. Repeat the above steps for different values of NMST to aswouldbeobserved.
determine at what masses the cluster is segregated, and to
what degree at each mass. Clearly, the choice of NMST is
arbitrary, so the ΛMSR needs to be determined for many
values of NMST to gain information on how the degree of
mass segregation changes with mass. 4.2 Mass segregated clusters
This method has three major advantages over current
We then test the method with mass segregated clusters.
methods.Firstly,itgivesaquantitativemeasureofmassseg-
These are created by first producing Plummer spheres of
regation with an associated significance, allowing different
1000 stars in the same way as before. The positions of the
clusters to be directly compared. Secondly, it does not rely
most massive xpercentofthestarsarethenswapped with
ondefiningaclustercentreoranyspeciallocationinaclus-
randomlychosenstarspositionedinsidethexpercentnum-
ter(allowingittobeappliedtohighlysubstructuredregions
ber radius, so only the most massive stars are inside this
as well). Thirdly, it is applicable not just to the most mas-
innerradius.Theroutinecreatesclustersinwhichthemost
sivestars,buttoanysubsetofstars(orbrowndwarfs) that
massive stars are centrally concentrated, and so are mass
one might think has a different distribution to the‘norm’.
segregated. This is not necessarily a realistic configuration
It should be noted that whilst we are applying our
asNaturemightproduce,butitisoneinwhichwecaneasily
method to simulated data and therefore know the mass of
control thedegree of mass segregation.
each star, an absolute determination of the mass is not re-
Eachclustercontains1000stars.Therefore,a5percent
quired, just a measure of the relative masses (i.e. luminos-
MS places the50 most massive stars within theinner 5per
ity).
cent numberradius. For 10 per cent MS, it is the 100 most
massivestars,andsoon.Wevarytheradiiinwhichthemost
massive stars are placed from the5per cent numberradius
4 THE METHOD IN ACTION
tothe80 percent numberradius. Notethat at large values
4.1 A non-mass segregated cluster theclusterismorelikeoneinwhichthelow-massstarshave
beenconcentratedattheoutskirtsratherthanoneinwhich
WetestedthemethodonalargesampleofPlummerspheres
thehigh-mass stars are centrally concentrated.
(Plummer 1911) consisting of 1000 single stars with masses
Tables 3 and 4 show the percentage of clusters which
sampledfromaKroupainitialmassfunction(Kroupa2002).
the method finds to have mass segregation at > 1σ signifi-
Themassesareassigned randomlytostarssotheseclusters
cancefordifferent numbersofstars in theMST(NMST).In
should not be mass segregated (except by chance).
bracketsafterthepercentageisthetypicalsignificanceofthe
Tables 1 and 2 show the fractions of random clusters
mass segregation. As before, Table 3 uses the 3D positions
that are found at 1σ significance to be either inverse-mass
of thestars, while in Table 4 a 2D projection is used.
segregated (Inverse-MS: ΛMSR <1), mass segregated (MS:
It is clear from Tables 3 and 4 that the ability of the
ΛMSR > 1) or that showed no mass segregation (No MS:
method to detect mass segregation depends on a combina-
ΛMSR ∼ 1) for various numbers of members in the MST
tionofthelevelofmasssegregationandthenumberofstars
(NMST).Table 1 uses the3D positions of thestars, Table 2
intheMST.Inthefirstcolumn,whenNMST =5themethod
isfora2Dprojection aswouldbeobserved.Inboththe2D
isabletodetectmasssegregation atroughly2σ significance
and 3D clusters we find that false positives and negatives
when the %MS is low. As it only uses the 5 most massive
(i.e. ΛMSR > 1σ from unity) occur with equal frequency
stars to look for mass segregation, it is not good at finding
around1/6th ofthetime–exactlywhatwouldbeexpected
masssegregation whenthe%MSinvolveshundredsofstars.
if the error is Gaussian.
In contrast, when NMST = 500 (i.e. half of the stars
in the cluster), the method is poor at spotting when the
4 R. J. Allison et al.
%MS NMST
5 10 100 200 500
5 100(2) 100(3) 100(3) 90(2) 40(1)
10 100(2) 100(3) 100(7) 100(4) 70(1)
20 100(2) 100(3) 100(7) 100(10) 100(3)
50 80(2) 100(2) 100(5) 100(8) 100(15)
80 30(1) 80(1) 100(4) 100(6) 100(10)
Table 3.Thefrequency of apositivedetection of masssegrega-
tion for various NMST and various degrees of mass segregation
(%MS) in a 3D cluster. The numbers in parentheses denote the
typical significance (in sigma) at which mass segregation is de-
tected.
%MS NMST
5 10 100 200 500
5 100(2) 100(2) 100(3) 70(2) 30(2)
10 100(2) 100(2) 100(6) 100(2) 50(2)
20 100(2) 100(2) 100(6) 100(9) 80(2)
50 70(1) 100(2) 100(5) 100(7) 100(12)
80 0(-) 60(1) 100(3) 100(6) 100(10)
Figure 1. The evolution of ΛMSR with NMST for a Plummer
Table 4. As Table 3, but for a 2D projection as would be ob- sphere with an initial 5 percent MS. The dashed line indicates
served. ΛMSR=1i.e.NoMS.
mass segregation only involves a few stars. However, it be-
comesextremelygoodwhenfindingmasssegregationinvolv-
ing many hundredsof stars.
LookingacrossTables3and4,thehighestsignificances
for findingmass segregation (upto10 or15σ results) occur
when NMST is equal to the number of stars that have been
masssegregated.Thebestresultsareforlarge NMST asthe
variance in hlnormi is smallest when NMST is large. Note
that,as mentionedabove,whenNMST issmall thevariance
in hlnormi is large dueto stochastic effects.
Therefore, it is vitalto varyNMST in order toexamine
mass segregation. However, this is an important diagnostic
tool, as variations of ΛMSR and its significance with NMST
contain information about how, and down to which mass
mass segregation is found.
InFig.1weshowthevariationoftheΛMSR withNMST
for a 5 per cent level of mass segregation (i.e. the 50 most
massive stars are segregated). The ΛMSR is around 9 for
NMST < 50, and then drops sharply toward unity when
NMST > 50. This shows two features of mass segregation
that can be extracted using this method.
Firstly,itshowsthatonlythe50mostmassivestarsare
masssegregated(asdescribedabove).However,italsoshows Figure 2. The evolution of ΛMSR with NMST for a Plummer
that the50 most massive stars areequally mass segregated. spherewithaninitial1percentand7percentMS.Thedashed
Thatis,themethodplacesthe50mostmassivestarsatthe lineindicatesΛMSR=1i.e.NoMS.
centreoftheclusterbutdoesnotordertheminanyspecific
way within thecentre.
4.3 A complex mass segregated cluster
To illustrate thiswe showin Fig. 2 thevariation of the
ΛMSR withNMST foraclusterinwhichthe10mostmassive We also apply the method to a more realistic and complex
starsareplacedintheinner1percentradius,andthe11th cluster.InFig.3weshowacomplex1000-bodycluster2.The
to70thmostmassivestarsinthe1–7percentradius.The
method clearly detects that there are two levels of mass
segregation – one involving the 10 most massive stars, and 2 The cluster has been evolved from a cold fractal distribution
one which involves the70 most massive stars. (see, Goodwin&Whitworth 2004); we will discuss simulations
Using the MST to trace mass segregation 5
Figure3.Thedistributionofstarsinaclusterevolvedfromcold,
fractalinitialconditions. Thetrianglesshow thepositionsofthe
five most massive stars, the squares the positions of the 6th to
12thmostmassivestars. Figure 4. The evolution of the 2D ΛMSR with NMST for the
cluster illustrated in fig. 3. The dashed line indicates an ΛMSR
ofunity,i.e.nomasssegregation.
trianglesshowthepositionsofthe5mostmassivestarsand
the squares the positions of the 6th to 12th most massive
stars (thereason for these choices will beexplained below).
Interestingly, the cluster has collapsed into a config-
uration in which the 5 most massive stars form a dense
Trapezium-likeclustertooneside,andthe6thto12thmost
massive stars have formed a separate ‘clump’ on the other
side. For this reason the cluster has no well-defined centre
which serves to illustrate the strength of the MST method
in dealing with unusualand clumpy clusters.
In Fig. 4 we show the change of the 2D ΛMSR with
NMST fortheclusterillustrated inFig. 3.Fig. 4shows that
inthisclusterthereare3‘levels’ofmasssegregation.Firstly,
there is the presence of a Trapezium-like system formed
from the five most massive stars (with an ΛMSR of around
12).Secondly,thereisanothersystem,containingthe6thto
12thmostmassivestars,whichshowssignificantlylessmass
segregation than the Trapezium-like system (an ΛMSR of
around 4). Thirdly, the 50 most massive stars in the clus-
ter show a steadily decreasing degree of mass segregation.
Beyond the 50th most massive star the cluster shows no
evidencefor mass segregation at all.
We notethat thesecond clump was not obvious in our
initial‘eyeballing’ofthesimulation.Itonlybecameobvious
whenweplotted thepositions ofthe6th to12th most mas-
sive stars after the MST method alerted us to there being
something special about these stars. Figure5.TheevolutionofΛMSRwithNMSTfortheONC.The
dashedlineindicatesanΛMSR ofunity,i.e.nomasssegregation.
4.4 The Orion Nebula Cluster
masses. We note that this is not an ideal dataset as many
Finally, we apply the method to real data, specifically the
(presumablylow-mass) starslack masses. Howeverit serves
Orion Nebula Cluster (ONC) data of Hillenbrand (1997).
to illustrate the method.
We use the 900 stars for which Hillenbrand (1997) provide
Hillenbrand & Hartmann (1998) show, using cumula-
tivedistribution functionsand mean stellar mass as afunc-
like this in detail in a later paper, for now the cluster is merely tion of radius, that the ONC is mass segregated. They find
usedasanillustrationofthemethod. evidence for mass segregation in the ONC for stars more
6 R. J. Allison et al.
massive than 5 M⊙within 1 pc, with less compelling evi- 4. Repeat the above steps for different values of NMST
dencebeyond this radius. to determine at what masses the cluster is segregated, and
Figure 5 shows the change of ΛMSR with NMST for to what degree at each mass.
the ONC. The MST method clearly picks out the mass ByexaminingthedifferencebetweentheMSTofasub-
segregation of the Trapezium system at NMST = 4 with setof(massive)starsandanequalnumberofrandomstarsit
ΛMSR = 8.0±3.5. However, the MST method also shows ispossibletoquantifythelevelof(inverse-)masssegregation
that there appears to be a secondary level of mass segrega- inacluster.Testsonartificialclustersshowthatthemethod
tioninvolvingthe9mostmassivestars(>5M⊙)intheONC behavesas expected and can identify mass segregation.
in agreement with Hillenbrand & Hartmann (1998). Whilst We apply the method to the Orion Nebula Cluster
thesignificanceoftheΛMSR islow(around2.0±0.5),there (ONC)andfindthattheTrapeziumismasssegregatedwith
is a clear trend in that each of the 5th to 9th stars show anΛMSRof8.0±3.5.Wealsofindthatthe5thto12thmost
the same increased level of mass segregation. However, un- massive stars down to around 5 M⊙also show evidence of
likeHillenbrand & Hartmann(1998) wefindnoevidenceof being mass segregated with an ΛMSR of 2.0±0.5. Below 5
mass segregatation below 5 M⊙. M⊙we find no evidencefor mass segregation.
ACKNOWLEDGEMENTS
5 FURTHER APPLICATIONS AND FUTURE
WORK We thank Stuart Littlefair for useful discussions. RJA and
RJP acknowledge financial support from STFC. MBNK
In future papers we will apply our new method to more was supported by PPARC/STFC under grant number
observational and theoretical data to look for mass segre- PP/D002036/1. SPZ is grateful for the support of the
gation, and also to examine if low-mass stars havedifferent NetherlandsAdvancedSchoolinAstrophysics(NOVA),the
distributions to brown dwarfs and/or high mass stars. LKBF and the Netherlands Organization for Scientific Re-
Binaries.Inourtestswehavenotincludedbinarystars, search(NWO).Weacknowledgethesupportandhospitality
and they will have an effect on the lengths of MSTs. If a oftheInternationalSpaceScienceInstituteinBern,Switzer-
binary is resolved, then it is very likely that the two com- land where part of thiswork was doneas part of aInterna-
ponents will be linked as a node (if this is not the case it tional Team Programme.
would be unclear that the system really was a binary). In-
deed, triples and quadruples will also be linked as a subset
withintheMST.ThisraisesthepossibilitythatMSTscould
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