Table Of Content

AstrophysicsandSpaceScience DOI10.1007/s•••••-•••-••••-• How do binaries affect the derived dynamical mass of a star cluster? Kouwenhoven, M.B.N.1 • de Grijs, R.1,2 8 0 0 2 n a (cid:13)c Springer-Verlag•••• J 5 Abstract The dynamical mass of a star cluster can using the equation derived by Spitzer (1987): 2 be derivedfromthe virialtheorem,usingthe measured R σ2 half-mass radius andline-of-sightvelocity dispersionof M =η hm los , (1) ] dyn h the cluster. However, this dynamical mass may be a G p significantoverestimationoftheclustermassifthecon- where η 9.75 is a dimensionless proportionality con- - ≈ o tribution of the binary orbital motion is not taken into stant. The derivation of Mdyn using the expression r account. In these proceedings we describe the mass above is valid under the following assumptions: (1) t s overestimation as a function of cluster properties and the cluster dynamics are described by the Plummer a binary population properties, and briefly touch the is- model, (2) all stars are single and of equal mass, (3) [ sue of selection effects. We find that for clusters with the cluster is in virial equilibrium, and (4) no selec- 1 a measured velocity dispersion of σlos & 10 kms−1 the tion effects are present. Dynamical mass estimates v presenceofbinariesdoesnotaffectthedynamicalmass for numerous clusters have been obtained this way 1 4 significantly. For clusters with σlos . 1 kms−1 (i.e., (e.g., Mandushev et al. 1991; Smith & Gallagher 2001; 9 low-densityclusters),thecontributionofbinariestoσlos Maraston et al. 2004; Bastian et al. 2006; Larsen et al. 3 issignificant,andmayresultinamajordynamicalmass 2007;Moll et al.2007). Whenbinarystarsarepresent, . 1 overestimation. Thepresenceofbinariesmayintroduce however,Eq.(1)resultsinanoverestimationoftheclus- 0 adownwardshiftof∆log(LV/Mdyn)=0.05 0.4inthe ter mass. 8 log(LV/Mdyn) vs. age diagram. − Observations have shown that the majority of the 0 field stars are part of a binary or multiple system : v Keywords Star clusters: general— methods: numer- (Duquennoy & Mayor 1991; Fischer & Marcy 1992). i ical — binaries: general X It is believed that most (if not all) binary stars are r formed in binary systems, which is supported by a both observations (Mathieu 1994; Mason et al. 1998; 1 Introduction Kobulnicky & Fryer 2007; Kouwenhoven et al. 2005, An estimate for the mass of star clusters can be ob- 2007) and theory (Goodwin & Kroupa 2005). Stars tainedfromthevirialtheorem,usingtheprojectedhalf- inbinary systems exhibit notonly motionin the gravi- mass radius Rhm and line-of-sight velocity dispersion tationalpotentialofthe cluster (particle motion σpart), σlos. This dynamical mass estimate, Mdyn, is obtained but also orbital motion σorb in the binary system. Eq. (1) is applicable for σ (the centre-of-mass mo- part tionofthe binaries),but resultsin anoverestimationif Kouwenhoven, M.B.N. σ , the superposition of σ and σ , is measured. los part orb deGrijs,R. In this paper we therefore address the question: “How 1Department of Physics andAstronomy, Universityof Sheffield, dobinariesaffectthedynamicalmassofastarcluster?” HicksBuilding,HounsfieldRoad,SheffieldS37RH,UnitedKing- dom 2NationalAstronomicalObservatories,ChineseAcademyofSci- 2 Method and terminology ences, 20A Datun Road, Chaoyang District, Beijing 100012, P.R.China WeevaluatetheeffectofbinarityonM usingnumer- dyn ical simulations. We use the STARLAB package (e.g., 2 Whenchangingcertainpropertiesofthebinarypop- Table 1 The default properties of the model used in our ulationinastarcluster,suchastheincreasingtheaver- analysis, which we refer to as model R. In our analysis we vary the properties of modelled star clusters in order to agemassratio q orbinaryfractionFM,thetotalmass h i find the effect of these changes on the derived dynamical of the cluster increases. The latter increase results in mass Mdyn. At the bottom of the table we list the line-of- larger particle velocity, which should not be fully at- sightvelocitydispersionσlos oftheindividualstars,σpart of tributed to the change in the distribution mass ratio thecentre-of-massmotionofthebinaries,andσ ofsolely orb distribution f(q) or in F . The increasein particle ve- M the orbital motion of the binary components. Each value locity is partially causedby the increasedcluster mass. represents thewidth of the best-fittingGaussian. The latter affect can be compensated for by adjusting Property Model R the number of particles N = S +B in the cluster. In Model Plummer thiswaythetotalclustermass“before”and“after”the Half-mass radius Rhm =5 pc change in the binary population is equal. The effect of Particles N =S+B N =18600 the changed binary property can now be studied accu- Total mass Mcl =104 M⊙ rately. Throughouttheseproceedingswekeepthetotal Mass segregation No mass for each comparison fixed to 104 M⊙, a typical Virial equilibrium Yes value for open cluster-like objects. Primary mass fKroupa;0.08−20 M⊙(M1) Binary fraction F =100% M Mass ratio fq(q)=1; 0<q 1 3 Dependence on cluster properties ≤ Eccentricity f (e)=2e; 0 e<1 e ≤ Orbital size fOpik(a); 10 R⊙ 0.02 pc In the sections below we describe how the derived dy- − Orientation Random namical mass depends on the structural parameters of σlos (measured) 1.20 kms−1 a cluster and on its stellar mass distribution. σ (centres-of-mass) 0.91 kms−1 part σorb (binaries) 0.14 kms−1 3.1 Mass distribution and number of particles N Eq. (1) assumes that all stars are single, equal-mass Portegies Zwart et al.2001)tomodelstarclusterswith stars. In reality, the stellar mass spectrum is defined different structural and binary population properties. by the mass function, such as the Kroupa (2001) IMF. The default properties of our cluster model are listed The effect of the mass distribution on the dynamical inTable1;werefertotheseas’modelR’.Wedetermine massdetermination,inparticularasafunctionoftime, thebest-fittingvelocitydispersionfromthevelocitydis- is studied in detail by Fleck et al. (2006), and will not tributions, and do not weigh by mass or luminosity. be discussed further here. In the introduction we described three velocity dis- The number ofparticlesN =S+B in a starcluster persions: the measured line-of-sight velocity dispersion isrelateditstotalmassviaM = M N,where M is cl σlos, the particle (centre-of-mass) velocity dispersion the averagemass of a particle (singhlesi/binaries),hwhiich σpart, and the binary orbital velocity dispersion σorb. isdefinedbythemassdistributionandthepairingprop- Whether binarity is important depends on which of erties of the binaries. For a cluster consisting of single these σpart or σorb dominate the measured σlos. We stars only, the dynamical mass given by Eq. (1); the discriminate between three types of clusters: determination of M is not dependent on the value dyn of N. This is not the case if binaries are present. In Particle-dominated clusters. The measured σ is los • thiscase,alargerN (orlargertotalmass)resultsinan dominated by σ ; Eq. (1) is a reasonable approxi- part mation and M /M .110%. increasedσpart,whilethe orbitalmotionofthe binaries dyn cl (reflected in σ ) remains unaffected. Clusters with a Intermediate-typeclusters. Thecontributionofσ orb part • largeN arethus lessaffected bybinarity,andallowfor and σ to σ is comparable; the dynamical mass orb los overestimation is 110%.M /M .200%. amoreaccuratedeterminationofMdynthanthosewith dyn cl small N. Binary-dominated clusters. The measured σ is los • dominated by σ . The dynamical mass overesti- orb mation is significant: M /M &200%. 3.2 The half-mass radius Rhm and stellar density ρhm dyn cl Typically, binary-dominated clusters have σ . 1 los For a cluster with a given total mass and virial ra- kms−1, while particle-dominated clusters have σ & los tio,thevelocitydispersiondependsstronglyonitssize: 10 kms−1 (see Sect. 6). σ R−1/2; see Eq. (1). Particles in a large (i.e., part ∝ hm Howdobinariesaffectthedynamicalmassmeasurementofastarcluster? 3 ρhm (stars pc-3) for model R ρhm (stars pc-3) for model R 106 104 102 100 106 104 102 100 10 1000 14 10000 a=a0 8 12 σ-1(km s)los 246 M/M (%)dyncl 246800000000 σ-1(km s)los 102468 a=a0 f(a) M/M (%)dyncl 1010000 f(a) 0 1 0 0.1 1 10 100 1000 0.1 1 10 100 1000 0.1 1.0 10.0 0.1 1.0 10.0 Semi-major axis (AU) Semi-major axis (AU) Projected half-mass radius (pc) Projected half-mass radius (pc) Fig. 2 The effect of the binary semi-major axis a on the Fig. 1 Theeffectofthehalf-massradiusR anddensity hm measuredline-of-sightvelocitydispersion σ andtheover- los ρ on the measured line-of-sight velocity dispersion σ hm los estimation of the dynamical mass of the cluster. All clus- andtheoverestimationofthedynamicalmassofthecluster. ter properties are listed in Table 1. The solid curves show The dashed lines indicate the results calculated from the the results for clusters in which all binaries have an iden- centre-of-mass motion, and the solid curve the results for themeasured line-of-sight velocities. tical a = a0, with a0 along the horizontal axis. The dot- tedcurveindicatetheresultsfortworealistic distributions: f(a) ∝ a−1 (O¨pik’s law) and the Duquennoy& Mayor (1991) period distribution. The results for both distribu- sparse) cluster move slower in the potential than in a tions are indistinguishable in this figure. The dashed lines tight, high-density cluster. The orbital motion of bi- indicate the results calculated from the centre-of-mass mo- nary systems is independent of the cluster properties. tion. In a low-density cluster the binary orbital motion is thus expected to dominate σ , while the presence of los 4 Dependence on binary population properties binaries is negligible in high-density clusters. The dependence on R and the average density within the hm half-mass radius ρ is illustrated in Fig. 1. Note that The reliability of a dynamical mass determination de- hm thedynamicalmassofadistantmassiveOBassociation pends not only on the properties of the cluster (which (Mcl 104 M⊙; Rhm 20 pc) can be overestimated are reflected in σpart), but also the properties of the ≈ ≈ by almostan orderofmagnitude, if its binaries are not binary population (which are reflected in σorb), in par- properly taken into account. ticular on the size of the binary orbits and the binary fraction. The effect of each of these binary population 3.3 The virial ratio Q properties is discussed in the sections below. After star formation the remaining gas from which the 4.1 The semi-major axis distribution f(a) starshaveformedisejectedbystellarwindsandsuper- nova explosions. During this phase a star cluster may The distribution over semi-major axes (or periods) is loose a significant fraction of its mass, resulting in a one of the most important parameters that affect the reduced gravitationalpotential. The cluster is now out interpretation of the observed σ , as the orbital ve- los ofvirialequilibrium,possiblyevenunbound,andstarts locity of a component is a binary system is propor- expanding. Although Eq. (1) assumes that the cluster tional to a−1/2. Clusters containing wide binaries are is in virial equilibrium, a correction for expanding or less affected by binarity than those containing tight contracting clusters is easily made. Let Q E /E K P binaries. This effect is clearly shown in Fig. 2. In ≡− bethevirialratio,withE thetotalkineticenergyand K binary-dominated clusters, which have σ σ , the los orb EP the total potential energy of the cluster. Clusters velocity dispersion scales as σ a−1/2,≈and there- los with Q < 0.5, Q = 0.5 and Q > 0.5 are contract- fore M /M a−1. In the par∝ticle-dominated case dyn cl ing, in virial equilibrium, and expanding, respectively. ∝ (which occurs if most binaries are wide), the dynam- For any value of Q, the relationbetween the true mass ical mass is a good representation of the true cluster and the dynamical mass of a star cluster is given by mass. The intermediate case occurs approximately at M =(2Q)−1M . Goodwin & Bastian (2006) define cl dyn log(a/R ) 4.7,where R is the half-mass radius hm hm theeffectivestarformingefficiency(eSFE)ǫasthestar- ≈− of the cluster. Note that the two most commonly used formingefficiencythatonewouldderivefromthevirial orbital size distributions, f(a) a−1 (O¨pik’s law; e.g. ratiounderthe assumptionthatthe star-formingcloud ∝ Poveda & Allen2004)andthelog-normalperioddistri- wasoriginallyinvirialequilibrium: Q=(2ǫ)−1. Under bution(Duquennoy & Mayor1991)practicallygivethe this assumption, the dynamical mass overestimation is same results. given by M /M =ǫ−1. dyn cl 4 the massive primary star barely moves, while the low- 1.20 160 mass companion orbits at high velocity. Depending on 1.15 150 whichstarismeasured,theorbitalvelocitycontribution σ-1(km s)los 111...001050 M/M (%)dyncl 111234000 tliosocmiσtelyoasosucfartenhdeb.beArisghmhigatehllsetor,raavlnaedrrgategh.eerMmefaoossrsterfmaretoqisoutetmhnutalssyslietvhaedesstvtaeor- 110 0.95 a larger dynamical mass overestimation. However, in 100 0.90 practice the effect of an uncertainty in the mass ratio 0 20 40 60 80 100 0 20 40 60 80 100 distribution is much smaller than for an uncertainty in Binary fraction (%) Binary fraction (%) f(a), F and the selection effects. M Fig. 3 The effect of the binary fraction FM on the measured line-of-sight velocity dispersion σ and the overesti- los mationofthedynamicalmassofthecluster. Modelparam- 5 Selection effects eters are listed in Table 1. The dashed lines indicate the results calculated from the centre-of-mass motion, and the Selectioneffectsplayanimportantrolefortheinterpre- solid curve the results for the measured line-of-sight veloci- tation of the measured σ , in particular (i) the pro- los ties. jected radiusfromthe cluster centre atwhich the mea- surement is performed and (ii) the stellar mass range 4.2 The binary fraction FM that is included in the observations. The line-of-sight velocity dispersion of the centres-of-mass at a certain The binary fraction is an important parameter, as it projected distance ρ from the cluster centre is determines the relative weight that is given by σ part (the single stars) and σorb (the binaries) to the mea- σ2 (ρ)= 3πGMcl 1+ ρ2 −1/2 (2) sured value of σlos. The dynamical mass of a star clus- part 64 Rhm (cid:18) Rh2m(cid:19) ter with a low binary fraction is expected to be only barely overestimated. As the binary fraction increases, (Heggie & Hut 2003). Anexpressionforthe dynamical the overestimation gradually becomes larger. In the mass overestimation as a function of ρ is obtained by intermediate-case, the dynamical mass overestimation substituting Eq. (2) into Eq. (1): scalesmoreorlesslinearlywithF ,whichisillustrated M in Fig. 3. Mdyn √2 1+ ρ2 −1/2 . (3) M ≈ (cid:18) R2 (cid:19) cl hm 4.3 The eccentricity distribution f(e) For velocity dispersions measured in the cluster centre Starsinaneccentricorbitspendmostoftheirtimenear this results in a mass overestimation by 40%. Mea- ∼ apastron,wheretheirvelocityissmall,andasmallfrac- surements at the half-mass radius provide the correct tion of their time near periastron, where their velocity Mdyn, while measurements in the cluster outskirts re- is large. The average velocity of a set of eccentric bi- sultinanunderestimationofthemass. Apossiblymore naries at a certain point in time is therefore relatively important selection effect is introduced by the large small. For a cluster with highly eccentric orbits, the brightness difference between stars of different masses. contributionofσorb toσlos is thereforesmallerthanfor In reality, the determination of σlos is dominated by a similar cluster with circular binaries. The dynami- the properties of stars in a certain mass range (more cal mass overestimationthus decreases with increasing specifically, a certain brightness range). The measured average eccentricity. However, the effect of an incor- σlos may therefore not be representative for the clus- rectly adopted eccentricity distribution on the calcu- ter as a whole. This may result in a further dynami- lated M is small, as compared to that of the un- calmassoverestimation(e.g.,Kouwenhoven& de Grijs dyn certainty in f(a), F , and the selection effects. For 2007). A detailed analysis of the selection effects is M model R our simulations show M /M = 80% for a necessary to properly take these selection effects into dyn cl cluster with circular binaries, while M /M = 50% account. dyn cl if all binaries have e=0.95. 4.4 The mass ratio distribution f(q) 6 When can binaries be ignored? Equal-mass stars in a binary system orbit each other Whenstudyingastarclusterindetail,itisimportantto at equal velocities. For a very low mass ratio binary, find out how reliable the measured dynamical mass is. Howdobinariesaffectthedynamicalmassmeasurementofastarcluster? 5 σ =10kms-1 107 part 5.0 0.7 Particle dominated 0.5 M (M)clsun 110056 η ~ 9.75 σpart=1 km s-1 σσ-1 - (km s)lospart 00000.....01234 175200505%0%%%%M/Mdyncl 112234......250500 000000......123456∆ log (L/M)Vdyn ass Intermediate 1.0 0.0 er m 104 η > 9.75 0 2σlos4 (km 6s-1) 8 10 0 2σlos4 (km 6s-1) 8 10 st u Cl 103 Fig.5 Giventhemeasuredline-of-sightvelocitydispersion σ , what is the effect of binarity? Can the effect of bina- los Binary dominated rity on the dynamical mass derivation be ignored? And if η >> 9.75 102 not,howsevereistheoverestimationoftheinferreddynam- 0.01 0.10 1.00 10.00 100.00 ical mass? For a given σ , the curves indicate the differ- los Half-mass radius Rhm (pc) enceσlos−σpart (left-handpanel), andthedynamicalmass overestimation M /M (right-hand panel). Results are dyn cl Fig. 4 The effect of binarity on the dynamical mass de- shownformodelswithFM =0%(dottedcurves),FM =25% termination for a cluster with a mass M and a half-mass (dashedcurves),andfor50%,75%and100%(solidcurves). cl radius Rhm. For young massive clusters, binarity generally We additionally indicate the shift ∆log(LV/Mdyn) in the has a small effect on Mdyn, while for distant OB associa- (LV/Mdyn)vs. age diagram thatisintroducedbythepres- tions the overestimation of M can be up to an order of enceof binaries. dyn magnitude. potential. Thiseffectmayresultinasignificantdynam- Fig.4 showsfor clustersofdifferent size R andmass icalmassoverestimation. Dependingonthe magnitude hm M whethertheyareparticle-dominated,intermediate- of the dynamical mass overestimation, we distinguish cl case, or binary-dominated. The figure shows that between three types of clusters: particle-dominated the low-density OB associations are generally binary- (σorb σpart), intermediate-case (σorb σpart), and ≪ ≈ dominated;theirdynamicalmassasobtainedfrominte- binary-dominated clusters (σorb σpart). ≫ grated spectral lines would significantly overestimated Theorbitalvelocityofbinariesisindependentofthe theirtruemass. Youngmassiveclusters,withmassesof clusterproperties(size,mass,etc.). Whetherornotbi- 105−6 M⊙ and typical half-mass radii of a few parsec, nary motion affects σlos is thus depends on the cluster are of the intermediate case. Their dynamical masses properties. For clusters with a high stellar density (i.e, are overestimated by a few per cent. From a practical large Mcl or small Rhm), Mdyn is generally unaffected. point-of-view, Fig. 5 shows how the measured velocity In the latter case σpart > 10 kms−1. The dynamical dispersion σ should be interpreted, and how the re- massoverestimationincreasesstronglywith(i)ahigher los sults depend on the intrinsic binary fraction. In the binary fraction and (ii) a smaller average orbital size. right-hand panel of Fig. 5 we additionally indicate the The dependence on the mass ratio distribution and ec- downwards shift ∆log(LV/Mdyn) = log(Mdyn/Mcl) in centricity distribution is small: ∆(Mdyn/Mcl) . 5%. the luminosity-to-mass vs. age diagram as a results of We additionally show that observing a certain subset binarity. of the cluster introduces a selection effects, which may result in a further mass overestimation by up to 40%. Analysisofthebrightest(generallymassive)starsinthe 7 Conclusions cluster may further overestimate the dynamical mass. The dynamical mass overestimation can be negligible The total mass of a star cluster is often inferred from forthemostmassiveclusters,whileitmayoverestimate its line-of-sight velocity dispersion σ and half-mass thetruemassbyuptoanorderofmagnitudeforsparse los radius R , assuming virial equilibrium. The latter OB associations. The full results of this study have hm approach includes the assumption that no binaries are been presented in Kouwenhoven& de Grijs (2007). A present. However, most stars are known to be in bi- follow-uppaper,whichtreatstheselectioneffectsprop- nary systems, and their orbital motions provide an ad- erly, is in preparation. ditional contribution to the measured σ . The latter los value is now no longer representative for the (centre- Acknowledgements M.B.N.Kouwenhovenwassup- of-mass)motion ofthe starsand binaries in the cluster ported by PPARC/STFC (grant PP/D002036/1). 6 References Bastian, N., Saglia, R. P., Goudfrooij, P., et al. 2006, As- tron. Astrophys.,448, 881 Duquennoy,A.&Mayor,M.1991,Astron.Astrophys.,248, 485 Fischer, D. A. & Marcy, G. W. 1992, Astrophys. J., 396, 178 Fleck, J.-J., Boily, C. M., Lan¸con, A., & Deiters, S. 2006, Mon. Not. R. Astron.Soc., 369, 1392 Goodwin, S. P. & Bastian, N. 2006, Mon. Not. R. Astron. Soc., 373, 752 Goodwin,S.P.&Kroupa,P.2005,Astron.Astrophys.,439, 565 Heggie,D.&Hut,P.2003,TheGravitationalMillion-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics (Cambridge University Press) Kobulnicky, H. A. & Fryer, C. L. 2007, Astrophys. J., 670, 747 Kouwenhoven,M.B.N.,Brown,A.G.A.,PortegiesZwart, S., & Kaper, L. 2007, Astron.Astrophys., 474, 77 Kouwenhoven, M. B. N., Brown, A. G. A., Zinnecker, H., Kaper, L., & Portegies Zwart, S. F. 2005, Astron. Astro- phys.,430, 137 Kouwenhoven, M. B. N. & de Grijs, R. 2007, Astron. As- trophys., in press (ArXiv:0712.1748) Kroupa, P. 2001, Mon. Not.R. Astron. Soc., 322, 231 Larsen, S. S., Origlia, L., Brodie, J. P., & Gallagher, III, J. S. 2007, Mon. Not. R. Astron. Soc., in press (ArXiv:0710.0547) Mandushev, G., Spasova, N., & Staneva, A. 1991, Astron. Astrophys.,252, 94 Maraston,C.,Bastian,N.,Saglia,R.P.,etal.2004,Astron. Astrophys.,416, 467 Mason, B. D., Gies, D. R., Hartkopf, W. I., et al. 1998, Astron. J., 115, 821 Mathieu, R. D. 1994, Annu. Rev. Astron. Astrophys., 32, 465 Moll,S.L.,Mengel,S.,deGrijs,R.,Smith,L.,&Crowther, P. 2007, Mon. Not.R. Astron. Soc., 382, 1877 Portegies Zwart,S.F.,McMillan, S.,Hut,P.,&Makino, J. 2001, Mon. Not. R.Astron. Soc., 321, 199 Poveda, A. & Allen, C. 2004, in Revista Mexicana de As- tronomia y Astrofisica Conference Series, ed. C. Allen & C. Scarfe, 49 Smith,L. J. & Gallagher, J. S.2001, Mon. Not.R.Astron. Soc., 326, 1027 Spitzer, L. 1987, Dynamical evolution of globular clusters (Princeton University Press) ThismanuscriptwaspreparedwiththeAASLATEXmacrosv5.2.