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Mon.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 segregation 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 segregation 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 profile. 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 masssegregation 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 configuration 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 cluster 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 massive 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 REFERENCES be very useful in locating binary and multiple systems by looking for short links within the MST. 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