Mon.Not.R.Astron.Soc.000,1–15(2015) Printed27January2017 (MNLATEXstylefilev2.2) Infalling Young Clusters in the Galactic Centre: implications for IMBHs and young stellar populations J. A. Petts1?, A. Gualandris 7 1University of Surrey, Guildford, United Kingdom 1 0 2 SubmittedTEMP n a J ABSTRACT 5 2 The central parsec of the Milky Way hosts two puzzlingly young stellar popula- tions, a tight isotropic distribution of B stars around SgrA* (the S-stars) and a disk ] A of OB stars extending to ∼ 0.5pc. Using a modified version of Sverre Aarseth’s di- rect summation code NBODY6 we explore the scenario in which a young star cluster G migrates to the Galactic Centre within the lifetime of the OB disk population via h. dynamical friction. We find that star clusters massive and dense enough to reach the p central parsec form a very massive star via physical collisions on a mass segregation - timescale. We follow the evolution of the merger product using the most up to date, o yet conservative, mass loss recipes for very massive stars. Over a large range of initial r t conditions,wefindthattheverymassivestarexpelsmostofitsmassviaastrongstel- s lar wind, eventually collapsing to form a black hole of mass ∼20−400M , incapable a (cid:12) [ of bringing massive stars to the Galactic Centre. No massive intermediate mass black hole can form in this scenario. The presence of a star cluster in the central ∼ 10pc 1 within the last 15Myr would also leave a ∼ 2pc ring of massive stars, which is not v currently observed. Thus, we conclude that the star cluster migration model is highly 0 unlikely to be the origin of either young population, and in-situ formation models or 4 binary disruptions are favoured. 4 7 Key words: stars: winds – stars: evolution – stars: black holes – stars: massive 0 . 1 0 7 1 1 INTRODUCTION (Nagata et al. 1995; Cotera et al. 1996; Figer et al. 1999) : andQuintuplet(Okudaetal.1990;Nagataetal.1990;Glass v ThecentralparsecoftheMilkyWayhostsalmosttwodozen et al. 1990; Figer et al. 1999) clusters at ∼ 30pc. This led i He-1emission-linestars(Krabbeetal.1995;Paumardetal. X Gerhard(2001)topostulatethatefficientdynamicalfriction 2001) and a population of many other OB stars in a thin on star clusters forming a few parsecs from SgrA*, where r clockwisediskextendingfrom∼0.04−1.0pc(Eckartetal. a GMCs can more easily cool and fragment, could bring a 1999).Feldmeier-Krauseetal.(2015)recentlyextendedthe dense core of massive stars to the central parsec within the range of observations up to ∼ 4pc2 centred on SgrA*; the age of the He-1 population. radio source associated with the supermassive black hole (SMBH) at the centre of the Milky Way. The authors show Another model suggests that in-situ formation of the thattheOBpopulationisverycentrallyconcentrated,with clockwisediskispossibleifatidallydisruptedGMCspirals 90% projected within the central 0.5pc. The clockwise disk intoformasmallgaseousdisk,whichcanbedenseenough exhibits a top heavy mass function (α ∼ 1.7, Lu et al. to become Jeans unstable and fragment into stars (Bonnell (2013)). Krabbe et al. (1995) estimate the He-1 stars to be & Rice 2008; Mapelli et al. 2008; Alig et al. 2011; Mapelli only ∼ 3−7Myr old, which is puzzling as the tremendous etal.2012;Aligetal.2013).Theinfallingcloudneedstobe tidalforcesinthisregionmakeitdifficultforagiantmolec- ∼105M(cid:12) inordertoreproduceobservations(Mapellietal. ular cloud (GMC) to remain bound long enough for gas to 2012). Two large gas clouds of mass ∼5×105M(cid:12), M-0.02- coolandfragment(Phinney1989;Morris1993;Genzeletal. 0.07andM-0.13-0.08,areseenprojectedat∼7and∼13pc 2003; Levin & Beloborodov 2003). fromtheGalacticCentre,respectively(Solomonetal.1972). There appears to be very few He-1 stars farther than The top heavy mass function can be reproduced by the in- thecentralparsec,otherthaninside/neartheyoungArches situmodelsolongasthegashasatemperaturegreaterthan 100K, consistent with observations of the Galactic clouds. Therotationaxisoftheclockwisediskshowsastrongtran- ? E-mail:[email protected] sition from the inner to outer edge (Lu et al. 2009; Bartko (cid:13)c 2015RAS 2 J. A. Petts, A. Gualandris et al. 2009; Lu et al. 2013), suggesting that the disk is ei- etal.(2014)estimatethepresentdaybinaryfractionofthe ther strongly warped, or is comprised of a series of stellar disktobe0.3+0.34at95%confidence,withafractiongreater −0.21 streamers with significant variation in their orbital planes than0.85ruled outat 99% confidence.More recently,Gau- (Bartkoetal.2009).In-situformationiscurrentlyfavoured tam et al. (2016) predict that the binary fraction must be for the clockwise disk, as an infalling cluster would likely greater than 32% at 90% confidence. formadiskwithaconstantrotationaxis(Perets&Gualan- Anadditionalpopularscenarioisthetransportofstars dris 2010). A caveat of in-situ formation is that it requires fromyoungdensestarclustersthatmigratetotheGalactic near radial orbits incident upon SgrA*, perhaps requiring Centreviadynamicalfriction,withtheaidofanintermedi- cloud-cloud collisions (Wardle & Yusef-Zadeh 2008; Hobbs ate mass black hole (IMBH). Kim & Morris (2003) showed & Nayakshin 2009; Alig et al. 2011). thattosurvivetothecentralparsecfromadistance≥10pc, Interiortothediskliesamoreenigmaticpopulationof clusters either need to be very massive (∼106M ) or very (cid:12) B-stars in a spatially isotropic distribution around SgrA*, dense(centraldensity,ρ ∼108M pc−3).Kimetal.(2004) c (cid:12) with a distribution of eccentricities skewed slightly higher showed that including an IMBH in the cluster means the thanathermaldistribution(Gillessenetal.(2009),Mapelli the core density can be lowered, but only if the IMBH con- &Gualandris(2015)).These“S-stars”havesemi-majoraxes tains ∼ 10% of the mass of the entire cluster, far greater less than 0.04 pc, with S0-102 having the shortest period than is expected from runaway collisions (Portegies Zwart of just 11.5 ± 0.3yrs, and a pericentre approach of just & McMillan 2002). ∼260AU (Meyer et al. 2012). The S-star population could Fujii et al. (2009) (hereafter F09) revisited this prob- potentiallybeolderthanthediskpopulation,asthebright- lemusingthetree-directhybridcode,BRIDGE(Fujiietal. eststarinthispopulation,S0-2,isamainsequenceB0-B2.5 2007),allowingtheinternaldynamicsofthestarclustersto V star with an age less than 15Myr (Martins et al. 2008). beresolved.ThesmalltidallimitsimposedbySgrA*meant The other stars in this population have spectra consistent the clusters had core densities greater than 107M pc−3, (cid:12) with main sequence stars (Eisenhauer et al. 2005), and ob- leadingtorunawaycollisionsonamasssegregationtimescale servationallimitsrequirethemtobelessthan20Myroldin (Portegies Zwart & McMillan 2002; Portegies Zwart et al. order to be visible. 2004). During collisions, the resulting very massive star The tidal forces in this region prohibit standard star (VMS) was rejuvenated using the formalism of Meurs & formation, so the S-stars must have formed farther out and van den Heuvel (1989), and collapsed to an IMBH at the migrated inwards. A possible formation mechanism of the endofitsmainsequencelifetime,extrapolatedfromthere- S-stars is from the tidal disruption of binaries scattered to sults of Belkus et al. (2007). The authors found that by low angular momentum orbits, producing an S-star and a allowingtheformationofa3−16×103M IMBH(seealso (cid:12) hyper-velocity star via the Hills mechanism (Hills 1991). Fujii et al. 2010), some stars could be carried very close to The captured stars would have initial eccentricities greater SgrA* via a 1:1 mean resonance with the infalling IMBH. than 0.97 (Miller et al. 2005; Bromley et al. 2006), but the The orbits of these “Trojan stars” were randomised by 3- presenceofacuspofstellarmassblackholesaroundSgrA* body interactions with the SMBH and IMBH, constructing could efficiently reduce the eccentricities of these orbits via aspatiallyisotropicS-starcluster.F09’ssimulation“LD64k” resonant relaxation within the lifetime of the stars (Perets transported23starstothecentral0.1pc,however,thereso- etal.2009).Additionally,Antoninietal.(2010)showthatif lutionofthesimulationis∼0.2pc,setbytheforcesoftening a binary is not tidally disrupted at first pericentre passage, ofSgrA*.Thesimulationalsobrought354starswithin0.5pc the Kozai-Lidov (KL) resonance (Kozai 1962; Lidov 1962) of SgrA*, 16 being more massive than 20M , analogous to (cid:12) cancausethebinarytocoalesceafterafeworbitalperiods, clockwise disk stars. The IMBH formed in LD64k is more producing an S-star and no hyper velocity star. massive than the observational upper limit of ∼ 104M , (cid:12) Alternatively, Chen & Amaro-Seoane (2014) show that derivedfromVLBAmeasurementsofSgrA*(Reid&Brun- stars from the clockwise disk can be brought very close to thaler2004).However,Fujiietal.(2010)statethatanIMBH SgrA*viaglobalKLlikeresonances,iftheclockwisediskof of 1500M(cid:12) is sufficient for the randomisation of stars (see gas originally extended down to ∼ 10−6pc (the lowest sta- also Merritt et al. 2009). blecircularorbitaroundSgrA*).Theauthorsalsoshowthat Despite the successes of the F09 model, IMBH forma- O/WRstarswouldbetidallydisruptedwithintheregionof tioninyoungdensestarclustersmaybeprohibited.VMSsof the observed S-star cluster due to their large stellar radii, the order 103M are expected to have luminosities greater (cid:12) whereas B-stars could survive, in agreement with observa- than107L (Kudritzki2002;Nadyozhin&Razinkova2005; (cid:12) tions.Recently,Šubr&Haas(2016)showedthataclockwise Belkus et al. 2007), driving strong stellar winds. F09 as- disk with 100% primordial binarity can produce ∼ 20 S- sumedthemasslossrateofstarsmoremassivethan300M (cid:12) stars in less than 4Myr. KL oscillations can efficiently drag tobelinearwithmass,however,recentworkonVMSwinds binaries close to SgrA*, producing an S-star and a hyper- showsteeperrelationsforstarsthatapproachtheEddington velocity star. This mechanism produces S-stars with eccen- limit (Kudritzki 2002; Vink 2006; Vink et al. 2011). F09’s tricitieslowerthanfromthedisruptionofbinariesoriginat- modelalsoneglectedtheeffectoftheevolvingchemicalcom- ing from outside the disk. However, in order to thermalize position on the luminosity, and hence the mass loss, of the the S-stars, ∼ 500M in dark remnants are still required VMS(Nadyozhin&Razinkova2005).Wenotethattheini- (cid:12) around SgrA* in order to match observations, consistent tial mass function (IMF) used in F09, although employed with Fokker-Planck models (Hopman & Alexander 2006). due to numerical constraints, meant there were ten times Three confirmed eclipsing binaries are observed within the more massive stars than expected from a full Kroupa IMF, clockwise disk, all being very massive O/WR binaries (Ott leadingtoanincreasedcollisionrateandbuildupoftheVMS et al. 1999; Martins et al. 2006; Pfuhl et al. 2014). Pfuhl mass. (cid:13)c 2015RAS,MNRAS000,1–15 InfallingYoungClustersintheGalacticCentre:implicationsforIMBHsandyoungstellarpopulations 3 No conclusive evidence for the existence of IMBHs in starclustershasyetbeenfound(SeeLützgendorfetal.2013, dv v 2015,foracomprehensivereviewonIMBHsinglobularclus- dtcl =−4πG2Mclρlog(Λ)f(v∗ <vcl)v3cl (1) ters). Sufficiently high mass loss could cause VMSs to end cl their lives as stellar mass black holes or pair-instability su- where vcl is the cluster velocity, Mcl is the cluster mass, pernovae at low metallicity (Heger & Woosley 2002). Pair- ρ is the local background density and f(v∗ < vcl) is the instability supernovae candidates have recently been found fraction of stars moving slower than the cluster; assuming at metallicities as high as ∼ 0.1Z (Gal-Yam et al. 2009; a Maxwellian distribution of velocities, valid in the cuspy (cid:12) Cookeetal.2012),withexpectedprogenitorsofseveralhun- models explored here. The Coulomb logarithm is given by: dred solar masses (Chen et al. 2015). The most massive star observed, R136a1, is a (cid:16)b (cid:17) (cid:18)min(ρ(R )/|∇ρ(R )|,R )(cid:19) 265+80M starinthe30DoradusregionoftheLargeMagel- log(Λ)=log max =log g g g , −35 (cid:12) b max(r ,GM /v2) lanic Cloud (LMC) (Crowther et al. 2010), with metallicity min hm cl cl (2) Z = 0.43Z . Crowther et al. (2010) suggest that it could (cid:12) where R is the galactocentric distance of the cluster and be a very rare main sequence star, with a zero age main g r is the half mass radius of the cluster. When coupled sequencemassof320+100M .However,itcouldbethecol- hm −40 (cid:12) withtheN-bodydynamics,r isthelivehalfmassradius, lision product of a few massive stars. R136a1 has a large hm and M is well represented by the cluster mass enclosed inferred mass loss rate of (5.1+0.9)×10−5M yr−1, ∼ 0.1 cl −0.8 (cid:12) within its tidal radius, including stars with energies above dex larger than the theoretical predictions of Vink et al. the escape energy. (2001). Belkus et al. (2007) predict that the evolution of all stars more massive than 300M is dominated by stel- (cid:12) lar winds, with similar lifetimes of ∼2−3Myr. As such, it 2.2 Evolution of very massive stars is not surprising that R136a1 is the most massive star cur- rently observed, as more massive VMSs should be rare and Nadyozhin & Razinkova (2005) present similarity theory short lived. modelsofVMSs,forwhichthestellarpropertiescanbecal- WhilstitmaybeunlikelyforanIMBHtoformatsolar culatedbysolvingasetofdifferentialequations(Imshennik metallicity, a VMS could transport stars to SgrA* within &Nadezhin1968).VMSsarepredictedtohavelargeconvec- its lifetime. In this paper we test the feasibility of the star tivecorescontainingmorethan85%ofthemass,surrounded clustermigrationscenarioastheoriginofeitheryoungpop- by a thin extensive radiative envelope. In such stars the ulation in the Galactic Centre. opacity becomes larger than the electron scattering value, We evolve direct N-body models of star clus- and can be considered to come from Thomson scattering ters in the Galactic Centre, using the GPU-accelerated alone. Utilising such approximations, the authors provide code NBODY6df, a modified version of Sverre Aarseth’s simple formulae to calculate the core mass and luminosity, NBODY6 (Aarseth 1999; Nitadori & Aarseth 2012) which as functions of stellar mass and chemical composition. includes the effects of dynamical friction semi-analytically The luminosity of stars with µ2M ≥100 can be found (Pettsetal.2015,2016).Insection§2wedescribethetheory by substituting equation 36 of Nadyozhin & Razinkova behind our dynamical friction and stellar evolution models. (2005) into their equation 34: Insection§3wedescribethenumericalimplementation.Sec- tion §4 discusses prior constraints on the initial conditions (cid:16) p (cid:17) 64826M 1−4.5/ µ2M and describes the parameters of the simulations performed. L≈ , (3) In sections §5 and §6, we present our results and discuss 1+X their implications for the origin of the young populations. where L is the luminosity, M is the mass of the VMS, X Finally, we present our conclusions in section §7. is the core hydrogen abundance, and µ is the mean atomic mass of the core. Assuming a fully ionised plasma, µ takes the form: 4 µ= , (4) 2 THEORY 6X+Y +2 where Y is the core helium abundance. Equation 3 shows 2.1 Dynamical friction that at very large masses L∝M. However, unlike the F09 The dynamical friction model used in this paper is a semi- model,thisformulationoftheluminosityexplicitlyincludes analyticimplementationofChandrasekhar’sdynamicalfric- anL∝(1+X)−1dependence.Asthemasslossratedepends tion(Chandrasekhar1943),describedinPettsetal.(2015), on L, this leads to an increased mass loss rate in the late which provides an accurate description of the drag force on stages of evolution (see section §2.2.1). star clusters orbiting in analytic spherical background dis- Belkusetal.(2007)(hereafterB07)modelledtheevolu- tributions of asymptotic inner slope γ = 0.5,..,3 (see also tion of VMSs with zero age main sequence (ZAMS) masses Petts et al. 2016, for a generalised model accurate also for ofupto1000M ,assumedtohaveformedviarunawaycol- (cid:12) thecoredγ =0case).Thenoveltyofourmodelistheuseof lisions in a young dense star cluster. The authors numer- physically motivated, radially varying maximum and mini- ically evolve the chemical composition of the star through mumimpactparameters(b andb respectively),which theCoreHydrogenBurning(CHB)andCoreHeliumBurn- max min vary based on the local properties of the background. The ing(CHeB)phasesviaconservationofenergyandmassloss dynamical friction force is given by: from the stellar wind. In this section we briefly outline the (cid:13)c 2015RAS,MNRAS000,1–15 4 J. A. Petts, A. Gualandris model of Belkus et al. (2007) and describe how we include the empirical Vink et al. (2001) (here after V01) rates and stellar collisions and their effect on VMS evolution. the VMS rates follow a tight relation with Γ , almost inde- e As VMSs have large convective cores, one can reason- pendentofmass.Theauthorsfindthatthemasslossrateis ably approximate them as homogeneous (verified to be a proportional to: good approximation down to 120M , B07). Applying con- (cid:12) servationofenergy,thehydrogenfractioninthecoreduring (cid:26)Γ2.2, if 0.4<Γ <0.7 CHB evolves as equation 1 of B07: M˙ ∝ e e (9) Γ4.77, if 0.7<Γ <0.95. e e dX L(µ,M) During the CHB phase, we model the stellar wind of M (µ,M) =− , (5) cc dt (cid:15) the VMS using the formulae from V01, whilst correcting H for the proximity to the Eddington limit by fitting on the where M is the mass of the convective core and (cid:15) is the cc H datafromtable1ofV11.Inthiswayweobtainacoefficient hydrogenburningefficiency(i.e.theenergyreleasedbyfus- that allows us to convert the V01 rate to the Γ enhanced ing one mass unit of hydrogen to helium). e ratesofstarsapproachingtheEddingtonlimit(similarlyto Whenthecoreisdepletedofhydrogen,theVMSburns Chen et al. 2015). V11 modelled stars up to 300M , how- helium via equation 4 of B07 (see also Langer (1989a)): (cid:12) ever,asthelogarithmicdifferencebetweentheV11andV01 rates shows little dependence on mass, we extrapolate this dY L(µ,M) approach to higher masses. V11 state that their predicted M (µ,M) =− , (6a) cc dt (cid:15)ratio windvelocitiesareafactor2–4lessthanderivedempirically. (cid:15) =h(cid:16)BY − BO(cid:17)+(cid:16)BC − BO(cid:17)C0(Y)i, (6b) The effect of rotation is also neglected. It should be noted ratio AY AO AC AO that due to these two effects, and our extrapolation of the V11 models, we likely underestimate the mass loss of our where(cid:15) accountsforthefactthatCandOareproduced ratio VMSs. Therefore the masses of our VMSs and their result- inanon-constantratio,affectingtheenergyproductionper ing remnants should be taken as a conservative upper limit unit mass of helium burnt. Here, A and B are the atomic at solar metallicity. weights and binding energy of nuclei; with subscripts Y, C During CHeB VMSs are depleted of hydrogen and are andOrepresentinghelium,carbonandoxygenrespectively. C0(Y)isthederivativeoftheC(Y)fitfromLanger(1989b) expected to show Wolf-Rayet like features. We follow the approach of Belkus et al. (2007) and extrapolate the mass with respect to Y (see B07 for the derivation of Equation loss formula of Nugis & Lamers (2000): 6b).DuringCHeB,µisdefinedas(Nadyozhin&Razinkova 2005): log(M˙)=−11+1.29log(L)+1.7log(Y)+0.5log(Z). (10) 48 B07 explored models with Wolf-Rayet like mass loss µ= , (7a) 36Y +28C+27O rates (arbitrarily) up to 4 times weaker, which only left a remnant twice as massive. The uncertainty arisingfromex- whichbyassumingY +C+O=1andusingthefittoC(Y) trapolation of this formula should be of little significance by Langer (1989b), can be rewritten solely as a function of to the transport of young stars to the central parsec, as Y as: post main sequence VMSs are not massive enough to ex- 48 perience substantial dynamical friction after the cluster is µ= . (7b) 19Y +C(Y)+27 disrupted (B07). However, if sufficiently chemically rejuve- nated, a CHB VMS may be capable of bringing stars to Subsequent stages of evolution are rapid and explosive. We the central parsec before losing most of its mass. Thus the assumethataftercoreheliumburningtheremnantcollapses evolution during the CHB stage is of most interest. to a black hole with no significant mass loss. Wemakesurethatinbothburningphasesthepredicted mass loss never exceeds the photon tiring limit, the maxi- mummasslossratethatcantheoreticallybeachievedusing 2.2.1 Mass loss 100%ofthestarsluminositytodrivethewind(Owockietal. The chemical evolution of the VMS is coupled to the 2004): mass evolution, as the luminosity of the star sets the wind (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)−1 L R M strength. Vink et al. (2011) (hereafter V11) show that the M˙ =0.032 . (11) wind strength is heavily dependent on the proximity to tir 106L(cid:12) R(cid:12) M(cid:12) theEddingtonlimit,whengravityiscompletelycounterbal- Here, the radii, R, of stars are taken from the mass-radius ancedbytheradiativeforces,i.e.grad/ggrav =1,wheregrad relation of Yungelson et al. (2008), which is in excellent and ggrav are the radiative and gravitational forces, respec- agreement with Nadyozhin & Razinkova (2005)’s similarity tively.Forafullyionisedplasma,theEddingtonparameter, theorymodelsofVMSs,butrequireslesscomputationalre- Γe,isdominatedbyfreeelectronsandisapproximatelycon- sources to calculate. The OB disk population is less than stant throughout the star (V11): 7Myr old. Hence, we assume approximately solar abun- dances such that X = 0.7, Y = 0.28 and Z = 0.2 (Pols 0 0 0 (cid:18) (cid:19)(cid:18) (cid:19)−1 et al. 1998). g L M Γ = rad =10−4.813(1+X ) , (8) e g s L M grav (cid:12) (cid:12) 2.2.2 Rejuvenation following collisions where X is the surface hydrogen abundance of the star. s V11’s fig. 2 shows that the logarithmic difference between Nadyozhin&Razinkova(2005)showthatVMSshavenearly (cid:13)c 2015RAS,MNRAS000,1–15 InfallingYoungClustersintheGalacticCentre:implicationsforIMBHsandyoungstellarpopulations 5 all of their mass in their large convective cores. Repeated Name Mcl rhm W0 N mlow collisions can efficiently mix the core and the halo, keeping (105M(cid:12)) (pc) (M(cid:12)) thestarrelativelyhomogeneous.ThewindoftheVMSalso 1lo 1.06 0.200 6 32k 1.0 ensureshomogeneity,asthelooseradiativeenvelopeisshed- 1hi 1.06 0.200 6 128k 0.16 dedbythestellarwind,leavingthesurfacewithcomposition 1kr 1.06 0.200 6 186k 0.08 similar to the core. We chemically rejuvenate a VMS following a collision Table 1. Initial conditions of the isolated simulations. Column with another star. We assume that stars colliding with the 1liststhenameofthesimulation.Thenamingconventionisde- VMS efficiently mix with the convective core such that: scribed in section §4. Columns 2,3 show the mass and half mass radius of the cluster. Column 4 shows the dimensionless central X M +X M potentialoftheKingmodel.Column5showsthenumberofpar- Xnew = star sMtar VMS VMS (12) ticles, and column 6 gives the lower mass limit of the IMF. The VMS+star uppermasslimitis100M(cid:12) forallmodels. SimilarlyforYandZ.WeapproximateX (t)andY (t) star star formainsequencestarsbyinterpolatingthedetailedstellar modelsofSchalleretal.(1992).IfaCHeBVMScollideswith (2003) for the central region of the Galaxy, yet has simple ahydrogenrichmainsequencestar,weassumethatCHBis analytic properties. We place a central fixed point mass of reignited. When two VMSs collide their composition is also 4.3×106M to represent SgrA* (Gillessen et al. 2009). assumed to be well mixed. (cid:12) 3 NUMERICAL METHOD 4.1 Physical and numerical constraints on the initial conditions Tomodeltheeffectsofdynamicalfrictiononself-consistent starclustermodels weusetheGPU-enableddirectN-body There are two constraints on the initial conditions of the code NBODY6df (Petts et al. 2015), which is a modified clusters.Firstly,theymustreachtheGalacticCentrewithin versionofAarseth’sdirectN-bodycodeNBODY6(Aarseth the age of the young populations. We therefore wish to 1999; Nitadori & Aarseth 2012). In this paper we model model clusters that can potentially reach the Galactic Cen- the background as an analytic stellar distribution with a tre in less than 7Myr, so that we may test the migration central black hole (see section §4). In Petts et al. (2015) we model for both the clockwise disk and the S-stars. We ob- only tested our dynamical friction model for cases without tain tight constraints on the initial orbital parameters by a central black hole, however we discuss how to trivially integratingtheorbitsofpointmassesintheGalacticCentre add a black hole to the model in Appendix A. A validation potentialincludingdynamicalfriction.Fig.1showscontours of this approach via comparison with full N-body models ofequalinspiraltimefordifferentinitialmasses,apocentres computedwithGADGET(Springeletal.2001)isalsogiven and initial velocities. Initial conditions to the right of each in the appendix. linearesuchthattheclusterscanreachwithin0.5pcinless We introduced an additional modification to the code than7Myr.Archeslikeclusters(initialmass4−6×104M , (cid:12) to properly model the evolution of a VMS, as described in Harfst et al. 2010)) could reach the Galactic centre in less section§2.2.Whenaphysicalcollisioncreatesastargreater than 7Myr if they formed at ∼ 5pc, or from 7−10pc if than100M(cid:12)weflagitasaVMSandtreatitsevolutionsep- largeinitialeccentricitieswereassumed.Moremassiveclus- aratelyfromthestandardSSEpackageinNBODY6(Hurley terscaneasilymigrate∼10pcin7Myr.Wenotethatthese etal.2000)viathemethoddescribedinsection§2.2.Asthe inspiral times are lower limits, as real clusters would lose masslosscanbeverylargeforVMSs,finetimeresolutionis massfromstellarwindsandtides.Wechoosetomodelonly neededtopreventoverestimationofthemassloss.Weintro- those clusters for which a point mass object of the same duce a new routine which integrates the mass and compo- mass can reach the Galactic Centre within ∼7Myr. sition of the star between the dynamical time steps using a Secondly,thesizeoftheclustersislimitedbytheirsmall timestepof0.1years,sufficientlyaccuratetoresolvetheevo- tidallimitswhensoclosetoSgrA*.Approximatingtheclus- lution.V11predictterminalwindspeedsofafewthousand ter as a point mass, the tidal radius is given by (Binney & kms−1 for VMSs, as such we assume that the stellar wind Tremaine 2008): escapes the cluster and simply remove this mass from the GM VMS. An arbitrary number of VMSs can potentially form r3 = cl , (13) and evolve simultaneously in the simulation. t ωp2+(cid:0)ddR2Φ2(cid:1)p (cid:16) (cid:17) where ω and d2Φ are the angular velocity of a circu- p dR2 4 INITIAL CONDITIONS p lar orbit and the second derivative of the potential at peri- In NBODY6df the background potential is assumed static centre, respectively. The high mass requirement for fast in- and analytic; an assumption valid over the short timescales spiral, coupled with the small tidal limits, means that all considered here (less than 7Myr). We adopt a Dehnen models are inherently very dense and runaway mergers are model(Dehnen1993),representingthecentralregionofthe expected. Although it is unknown whether such dense clus- Galaxy.Weuseaslopeγ =1.5,scaleradiusa=8.625pcand ters are likely to form in the Galactic Centre, we explore totalmassM =5.9×107M ,whichcloselyreproducesthe these initial conditions in order to test the feasibility of the g (cid:12) observedbrokenpower-lawprofileobtainedbyGenzeletal. inspiral model. (cid:13)c 2015RAS,MNRAS000,1–15 6 J. A. Petts, A. Gualandris Name Mcl rhm W0 ρ¯c N mlow fbin Ra v (105M(cid:12)) (pc) (M(cid:12)pc−3) (M(cid:12)) (%) (pc) (vc) 4lo15_W4 4.24 0.589 4 6.78×105 217 1.0 0 15 1.0 4lo15_W4v75 4.24 0.360 4 2.86×106 217 1.0 0 15 0.75 2lo10 2.12 0.220 6 1.32×107 216 1.0 0 10 1.0 2lo10_W4 2.12 0.318 4 2.09×106 216 1.0 0 10 1.0 2lo10_v75 2.12 0.141 6 5.35×107 216 1.0 0 10 0.75 2lo10_v5* 2.12 0.220 6 1.32×107 216 1.0 0 10 0.5 2lo10_v2* 2.12 0.220 6 1.32×107 216 1.0 0 10 0.2 2lo5 2.12 0.135 6 5.75×107 216 1.0 0 5 1.0 2lu5 2.12 0.135 6 5.75×107 29k 1.0 0 5 1.0 1lo10 1.06 0.175 6 1.26×107 215 1.0 0 10 1.0 1hi10 1.06 0.175 6 1.26×107 217 0.16 0 10 1.0 1hi10_b 1.06 0.175 6 1.26×107 217 0.16 5 10 1.0 1lo10_W4 1.06 0.273 4 1.69×106 215 1.0 0 10 1.0 1lo10_v75 1.06 0.112 6 4.94×107 215 1.0 0 10 0.75 1lo10_v5* 1.06 0.175 6 1.26×107 215 1.0 0 10 0.5 1hi10_v5* 1.06 0.175 6 1.26×107 217 0.16 0 10 0.5 1lo10_v2* 1.06 0.175 6 1.26×107 215 1.0 0 10 0.2 1hi10_v2* 1.06 0.175 6 1.26×107 217 0.16 0 10 0.2 1hi10_v2b* 1.06 0.175 6 1.26×107 217 0.16 5 10 0.2 1lo5 1.06 0.107 6 6.42×107 215 1.0 0 5 1.0 1hi5 1.06 0.107 6 6.42×107 217 0.16 0 5 1.0 1hi5_b 1.06 0.107 6 6.42×107 217 0.16 5 5 1.0 1hi5_ms 1.06 0.107 6 6.42×107 217 0.16 0 5 1.0 1hi5_W4d 1.06 0.055 4 2.03×108 217 0.16 0 5 1.0 1lu5 1.06 0.107 6 6.42×107 14.7k 1.0 0 5 1.0 Table 2. Initial conditions of the simulations. Column 1 lists the name of the simulation. The naming convention is described in section§4.Columns2and3givethemassandhalfmassradiusofthecluster.Column4givesthedimensionlesscentralpotential of the King model. Column 5 shows the average core density. Column 6 shows the number of particles, and column 7 the lower masslimitoftheIMF.Theuppermasslimitis100M(cid:12) forallmodels.Column8showstheinitialbinaryfraction.Columns9and 10showtheinitialpositionandvelocity,inunitsof pcandthecircularvelocity,respectively.ModelsaresetuptobeRoche-filling atfirstpericentrepassage;unlesstheyaremarkedbyanasterisk,inwhichcasetheyareRoche-fillingattheirinitialpositions. 10 9 v 0 = 0.2 v c =0.5vc 75vc 1200 11lhoi v0 =0. 1000 1kr ) 8 v0 =vc M v0 ( 800 c) 7 ss p a (a,0 6 S M 600 R M 400 5 V 4 200 3 0 0 1 2 3 4 5 4 5 t (Myr) 10 10 Cluster Mass (M ) Figure 2. Time evolution of the VMS mass formed in simu- lations with different lower mass cutoffs in the IMF. The solid Figure1.ContoursofTdf =7Myrasafunctionofclustermass, blue,dashedredanddot-dashedgreenlinesshowthemassofthe initial distance, Ra,0, and initial velocity, v0, given in units of VMSinsimulations1lo,1hiand1kr,respectively.Thediagonal the local circular velocity, vc. Models to the right of each line crossesshowtheendofCHB,theverticalcrossesshowre-ignition approachwithin0.5pcofSgrA*inlessthan7Myr.Thehalfmass of CHB, and the solid circles show where the remnants collapse radiusoftheclusterisassumedtobe0.1pc. toblackholes. 4.2 Initial Mass function limit of 0.08M would yield the most physically realistic (cid:12) WesamplestarsfromaKroupainitialmassfunction(IMF) results, but at a computational cost unfeasible for a pa- withanupperlimitof100M (Kroupa2001).Alowermass rameter study of such massive clusters at the current time (cid:12) (cid:13)c 2015RAS,MNRAS000,1–15 InfallingYoungClustersintheGalacticCentre:implicationsforIMBHsandyoungstellarpopulations 7 (365k-730kparticlesforthemostmassivemodelsexplored). inSana&Evans(2011),whichshowthatshortperiodsand However, truncating the low end of the IMF means that loweccentricitiesarepreferredinmassivebinaries.Forlower onesamplestoomanymassivestarsascomparedwithafull mass stars the periods are drawn from the Kroupa (1995) Kroupa IMF. To quantify the difference this has on VMS period distribution and are assigned thermal eccentricities. formation, we ran three test simulations at different mass The mass of a binary and its initial position in the cluster resolutions, in the absence of a tidal field. Simulations 1lo, are assumed to be independent. 1hi and 1kr have lower cutoffs of 1.0, 0.16 and 0.08M , (cid:12) respectively. We model star clusters as King models with dimensionless central potential, W = 6, and with no pri- 4.4 Simulations 0 mordial mass segregation. The parameters of the isolated The initial conditions are described in Table 2 and simulations are displayed in Table 1. Fig. 2 shows the VMS are referred to by the following naming convention: massasafunctionoftimeforsimulations1lo,1hiand1kr, <M><mf><R >, where < M > is the cluster mass in a showing that better sampling of the low end of the IMF in- units of ∼ 105M , < mf > is the mass resolution of the (cid:12) hibitsthegrowthoftheVMS.Thisoccursbecauseprimarily simulation,and<R >istheinitialgalactocentricdistance a highmassstarsbuilduptheVMS,duetotheirshortdynam- in pc. For most simulations we sample from a Kroupa IMF ical friction timescales and large cross sections for collision. with an upper mass cut off, m = 100M . The “lo” reso- up (cid:12) In simulation 1kr, although half the cluster mass is com- lution models have a lower mass cut off, m = 1M and low (cid:12) prised of stars less massive than 0.58M , only 37 stars less (cid:12) mean mass, h m i = 3.26M . The “hi” resolution models ∗ (cid:12) massive than 0.58M are consumed throughout the entire (cid:12) havem =0.16M andh m i=0.81M .Forsimulations low (cid:12) ∗ (cid:12) lifetime of the VMS. The VMS initially grows very rapidly. with<mf >=luweuseanIMFidenticaltothemassfunc- However, the late main sequence evolution is dominated by tion of the clockwise disk (Lu et al. 2013). The simulation thestrongstellarwindoftheheliumrichVMS.Throughout name is followed by a suffix describing additional informa- its lifetime, the VMS in simulation 1kr removes 2244M (cid:12) tion about the simulation. The suffix W4 denotes that the of material from the cluster through its stellar wind, ∼2% dimensionless central potential, W , is initially 4 instead of 0 oftheclustermass.DuringCHeB,simulations1loand1hi 6. The suffix vX indicates an eccentric orbit with initial ve- reignite CHB via collision with a massive main sequence locity, 0.Xv (where v is the circular velocity at the initial c c star,resultinginalowerremnantmassatcollapse.Thelate position). The Suffix “ms” indicates that the cluster is pri- evolution is very stochastic, however this is not important mordially mass segregated. Finally, the suffix “b” denotes for the migration of young stars to the Galactic Centre, as the inclusion of primordial binaries (see section §4.3). Most theVMSonlyprovidesgravitationalbindingenergycompa- models are Roche-filling at first pericentre passage, apart rable to normal cluster stars during its CHeB phase. from runs marked with an asterisk, which are Roche-filling Fig.2showsthatalowerlimitof0.16M issufficientto (cid:12) at their initial positions. The model with the suffix “d” is resolve the mass evolution of the VMS, and as we are only extremely Roche under-filling at its initial position. interested in the final distribution of OB stars, this IMF is sufficient for our simulations. We cannot evolve the most massive clusters at high mass resolution, as these models 5 RESULTS become too computationally expensive. As a compromise, wetestalargerangeofinitialconditionswithalowerlimitof In all models, the clusters are completely tidally disrupted 1M(cid:12),andre-runaselectionofinitialconditionswithalower in less than 7Myr. Massive clusters migrate farther in than limit of 0.16M(cid:12) to obtain more realistic results. We can lowermassclustersonthesameinitialorbits,duetoshorter simultaneouslyusethelowresolutionsimulationstoexplore dynamical friction timescales and less efficient tidal strip- the possibility of an initially top heavy mass function for ping.Howeveranyclusterthatreaches∼3pcisrapidlydis- clusters forming close to SgrA*. A very top heavy function solved by its shrinking tidal limit as it approaches SgrA*. isobservedfortheclockwisedisk(Luetal.2013),however, Clusters on eccentric orbits inspiral faster, as they pass it is unknown whether a top heavy IMF is expected from through denser regions of the cusp periodically. However, the collapse of GMCs at ∼5−10pc from SgrA*. clusters on very eccentric orbits (e.g. 2lo10_v2∗, e ∼ 0.9) disruptonthefirstfewpericentrepassages,depositingstars at large distances along the initial cluster trajectory. 4.3 Binary fraction Most simulations naturally form a VMS in less than Some simulations include a population of primordial bina- 1Myr due to their high initial densities. However, the ini- ries.Binariesareinitialisedasfollows.Firstlyallstarsmore tial rapid mass accretion soon loses to the increasing mass massive than 5M are ordered by mass. The most mas- loss rate and relaxation of the cluster, causing the VMS to (cid:12) sive star is then paired with the second most massive star, collapse to a black hole of ∼ 20−250M after 2−5Myr (cid:12) and so on. This choice is motivated by observational data (300−400M formodelswithaLuetal.(2013)IMF),typ- (cid:12) showing that massive OB stars are more likely to form in ically before their parent clusters completely disrupt. Table binarysystemswithmass-ratiosoforderunity(Kobulnicky 3 shows the maximum mass, remnant mass and lifetime of &Fryer2007;Sana&Evans2011).Onceallstarsmoremas- the VMS formed in each simulation. The clusters become sive than 5M are in binaries, lower mass stars are paired completely unbound at ∼2−3pc, and the IMBHs formed (cid:12) atrandomuntilthespecifiedbinaryfraction(thefractionof are not massive enough to experience significant dynami- starsinitiallyinabinarysystem)isreached(Kroupa2008). cal friction and drag stars close to SgrA* (dynamical fric- For stars more massive than 5M , the periods and eccen- tion timescales for even a 400M IMBH are longer than (cid:12) (cid:12) tricities are drawn from the empirical distributions derived 100Myr). Conversely, the evolution of the VMS does not (cid:13)c 2015RAS,MNRAS000,1–15 8 J. A. Petts, A. Gualandris RunName MVMS,max MVMS,rem tVMS 2lo10 2lo10 M(cid:12) M(cid:12) Myr 104 Semi-Major Projected 2 4lo15_W4 - - - 3 Axes Distances 10 10 4lo15_W4v75 139.1 19.8 6.16 all 7 Myr 2lo10 3472.9 85.7 3.26 2 >8M 15 Myr 2lo10_W4 703.2 53.0 4.86 10 101 2lo10_v75 3858.4 79.5 2.98 1 2lo10_v5* 2069.5 119.7 2.92 10 2lo10_v2* - - - 2lo5 2804.1 169.0 2.78 100 100 2lu5 4086.6 378.5 2.54 2lo10_v2* 2lo10_v2* 4 1lo10 1541.8 73.5 3.11 10 2 1hi10 886.903 48.7 3.77 10 3 1hi10_b 1232.4 17.3 3.94 10 1lo10_W4 721.1 20.6 5.05 1lo10_v75 3413.8 199.8 2.84 N102 1 10 1lo10_v5* 1462.2 82.2 2.77 1hi10_v5* 927.5 56.8 2.92 101 1lo10_v2* 125.3 16.4 4.37 1hi10_v2* 234.9 22.7 3.07 100 100 1hi10_v2b* 481.3 32.6 2.58 2lo5 2lo5 1lo5 3310.7 245.6 2.60 104 1hi5 1415.7 62.0 2.44 2 10 1hi5_b 1747.1 80.1 2.52 3 10 1hi5_ms 1338.8 65.8 2.99 1hi5_W4d 2044.4 77.8 2.91 2 10 1 1lu5 3950.6 355.1 2.42 10 1 10 Table 3.PropertiesofVMSsformedinthesimulations.Column 1liststhenameofthesimulation.Column2showsthemaximum 100 100 massviastellarcollisions,column3showstheresultingremnant -1 0 1 -1 0 1 10 10 10 10 10 10 mass after CHeB, and column 4 shows the epoch in the cluster a(pc) RG(pc) evolutionwhentheVMScollapses. Figure3.Left:Finaldistributionsofsemi-majoraxesofstarsin simulations2lo10,2lo10_v2∗and2lo5atT =7Myr.Thesolid appear to significantly inhibit the inspiral of the cluster, as green histogram shows all the stars. The dashed blue histogram only(cid:46)2%oftheinitialclustermassistypicallyexpelledby showsmainsequencestarsmoremassivethan8M(cid:12)atT =7Myr. the VMS throughout its lifetime. Right:Finaldistributionsoftheprojecteddistancesofstarsfrom For each model, Table 4 shows the final distribution SgrA*inthesamesimulations.Thedashedblueanddot-dashed of stars after complete cluster dissolution and death of any redhistogramsshowthedistributionsofmainsequencestarsmore VMSs.Weshowthedistributionsofsemi-majoraxesforall massive than 8M(cid:12) at T =7Myr and T =15Myr, respectively, stars and main sequence stars more massive than 8M at projectedtorotateclockwiseonthesky.They-valuesofthepro- (cid:12) jecteddistributionsarere-normalisedtotheexpectednumberof 7Myr, as well as how many of these stars have final semi- starshadthemodelbeensimulatedwithafullKroupaIMF. major axes smaller than 1pc. We use a 8M cut-off as (cid:12) these are the faintest main sequence stars spectroscopically observable in the Galactic Centre with current telescopes, Modelswithveryeccentricorbits(e.g.2lo10_v2∗)can K ≥ 15.5 (Do et al. 2009, 2013; Lu et al. 2013; Feldmeier- bringstarsclosetoSgrA*,however,veryfewstarshavefinal Krause et al. 2015). Although photometric studies can see semi-major axes smaller than 1pc. No stars more massive objects down to magnitudes of K <19−18 (∼2M main (cid:12) than 8M are scattered to semi-major axes smaller than sequence stars, Genzel et al. 2003), it is impossible to de- (cid:12) 1pc in either 1lo10_v2∗ or 2lo10_v2∗. This is likely due terminewhetherthesestarsareyoungorold.Wealsoshow tothepreferentiallossoflowmassstars,whereashighmass theprojecteddistributionsofvisiblemainsequencestarsat starsremaininsidetheclusterforlonger,andenduptracing 7Myr and 15Myr (as the S-star population may be older the final cluster orbit. than the disk population, see section §1). Simulation 2lo5 is the only non-radial model to bring stars to the central parsec, and the only model that brings a significant number of massive stars. However, one would 5.1 Low Resolution Models expecttoalsosee∼3000massivestarsintherange1−10pc, Fig. 3 shows the final distributions of the semi-major axes about10timesmorethanreachthecentralparsec.Theright andprojectedpositionsofstarsforarepresentativeselection side of Fig. 3 shows the distributions of projected distances of the models with a lower mass cutoff of 1M (< mf >= of stars that are spectroscopically visible at 7 and 15Myr. (cid:12) lo). In all models the final distributions are broad, with a The amplitudes of the distributions are normalised to the standarddeviationof∼2pc.Othersimulationsshowsimilar expectednumberofstarshadthesimulationbeenrunwith distributions, with less massive and less eccentric models a Kroupa IMF. The stars are projected to rotate clockwise dissolving farther out (see Table 4). inthesky.Itcanbeseenthatforallsimulations,morethan (cid:13)c 2015RAS,MNRAS000,1–15 InfallingYoungClustersintheGalacticCentre:implicationsforIMBHsandyoungstellarpopulations 9 RunName haiall hai>8M(cid:12) N(<1pc) N(<1pc,>8M(cid:12)) hD2Di7Myr hD2Di15Myr (pc) (pc) (pc) (pc) 4lo15_W4 9.20±4.15 8.86±3.94 0 0 9.22±4.90 9.31±4.95 4lo15_W4v75 5.23±3.27 4.89±3.11 0 0 5.17±3.88 5.34±4.00 2lo10 6.90±2.92 6.19±2.76 0 0 6.24±3.00 6.38±3.03 2lo10_W4 6.43±2.70 6.00±2.62 0 0 6.05±2.86 6.13±2.85 2lo10_v75 4.61±2.30 4.20±2.09 0 0 4.36±2.56 4.44±2.55 2lo10_v5* 5.51±2.86 5.37±2.83 0 0 6.31±4.50 6.40±4.62 2lo10_v2* 5.52±3.17 5.44±3.14 15 0 6.95±5.48 6.85±5.44 2lo5 2.53±1.33 2.23±1.23 3970 270 2.26±1.36 2.32±1.31 2lu5 3.33±1.54 3.24±1.50 3 0 3.29±1.64 3.32±1.68 1lo10 8.79±2.49 8.24±2.35 0 0 8.29±2.57 8.47±2.67 1hi10 8.02±2.49 7.18±2.25 0 0 7.22±2.40 7.47±2.54 1hi10_b 8.76±2.49 8.71±3.38 0 0 8.07±2.58 8.08±2.76 1lo10_W4 8.88±2.53 8.40±2.29 0 0 8.43±2.48 8.49±2.53 1lo10_v75 6.42±2.09 6.19±1.87 0 0 6.42±2.51 6.54±2.60 1lo10_v5* 6.09±2.36 5.90±2.20 0 0 6.93±3.68 6.94±3.74 1hi10_v5* 5.87±2.32 5.77±2.20 0 0 6.67±3.71 6.66±3.67 1lo10_v2* 5.64±2.87 5.56±2.89 9 0 7.56±5.38 7.44±5.19 1hi10_v2* 5.62±2.88 5.58±2.79 10 0 7.63±5.43 7.82±5.76 1hi10_v2b* 5.70±2.90 6.56±3.38 8 1 8.13±5.51 8.00±5.35 1lo5 4.04±1.36 3.81±1.28 0 0 3.85±1.51 3.90±1.55 1hi5 3.60±1.33 3.18±1.25 0 0 3.23±1.39 3.24±1.31 1hi5_b 3.98±1.36 4.62±2.69 0 0 3.62±1.24 3.71±1.27 1hi5_ms 4.24±1.47 3.20±1.11 0 0 3.25±1.37 3.30±1.48 1hi5_W4d 3.08±1.27 2.56±1.13 0 0 2.63±1.53 2.64±1.49 1lu5 4.36±1.38 4.27±1.35 0 0 4.30±1.44 4.34±1.41 Table4.Finaldistributionsofstarsoriginallyfromthecluster.Simulationsarerununtiltheclusteriscompletelyunboundandany VMSscollapse,uptoamaximumof7Myr.Column1showsthenameofeachsimulation.Column2showsthemeansemi-major axis of all stars remaining in the simulation and the standard deviation. Column 3 shows the same for stars more massive than 8M(cid:12) thatarestillonthemainsequenceat7Myr.Column4showsthetotalnumberofstarswithfinalsemi-majoraxeslessthan 1pc, and column 5 shows only those which are are still on the main sequence and more massive than 8M(cid:12) at 7Myr. Columns 6 and7showthedistributionsofprojecteddistancesfromSgrA*formainsequencestarsvisibleat7and15Myr,respectively.The dissolvedclustersareprojectedsothattheresultingdiskofstarsrotatesclockwiseinthesky. 1000 young stars are observed out to ∼ 10pc. Considering axes for all the stars and main sequence stars more mas- currentobservationallimitations,ifaclusterwerepresentin sivethan8M at7Myr.Thedistributionsareverysimilar, (cid:12) thecentral∼10pcwithinthelast∼15Myr,alargenumber however 1hi10_v5∗ has a smaller ratio of spectroscopically of stars would be observable up to ∼10pc, suggesting it is visible stars to all stars due to differences in the IMF. The unlikely that any clusters have inhabited this region in the panelsontherightshowtheprojecteddistributionsofmain last ∼15Myr. sequence stars visible at 7 and 15Myr. For the projected Simulations 2lo10 and 2lo10_W4 have the same initial distributions, the number of stars is re-normalised to the orbitandmass,yet2lo10_W4islessconcentrated.Thelower expectednumberofstarshadthesimulationbeenrunwith density and longer relaxation timescale cause 2lo10_W4 to aKroupaIMFfrom0.08−100M .Althoughmassivestars (cid:12) formalessmassiveVMSthan2lo10.However,theVMSin are consumed to construct the VMS, this is a small frac- 2lo10_W4 lives longer as not all the most massive stars are tion of the population. The distributions look very similar consumed within ∼1Myr. The models end up with similar in shape and magnitude, indicating that models run with final distributions of the resulting disk (see Table 4). The a lower limit of 1M produce similar final distributions to (cid:12) same trend is seen for the less massive analogues, 1lo10 simulationsthatbettersampletheIMF.Thisverifiestheva- and1lo10_W4.Thetwomostmassivesimulations4lo10_W4 lidity of the normalisation approach used on the projected and 4lo10_W4v75, are massive enough to reach the central visible distributions in Fig. 3. parsecfrom15pcin∼7Myr,butwithcentraldensitieslow Fig. 5 demonstrates how simulations 1hi5, 1hi10, enough to suppress the formation of VMSs. However, these 1hi10_v5∗ and 1hi10_v2∗ evolve with time. The top two simulations are more susceptible to tides, and are tidally panels show the evolution of the Galactocentric distance of disrupted at large radii. the cluster and the mass enclosed within the tidal radius. ThebottomtwopanelsshowtheevolutionoftheVMSmass and the half mass radius of the cluster. Simulations 1hi5, 5.2 Higher resolution models 1hi10and1hi10_v5∗quicklyformaVMSandexpanddue Fig. 4 shows a comparison between simulations 1lo10_v5∗ to rapid two body relaxation in the dense core. The expan- and1hi10_v5∗,whichhavethesameinitialconditions,ex- sionlowersthecoredensityandthusthecollisionrate.The cept1hi10_v5∗bettersamplesthelowmassendoftheIMF. reducedcollisionrateallowstheVMSstorapidlyburntheir The panels on the left show the distributions of semi-major fuelandcollapsewithoutsignificanthydrogenrejuvenation. (cid:13)c 2015RAS,MNRAS000,1–15 10 J. A. Petts, A. Gualandris 1lo10_v5* 1lo10_v5* 10 1 104 Semia-lMl ajor Axes PDrisotjaenctceeds 102 8 0.8cM 103 >8M 7 Myr pc) 6 0.6l (1 102 15 Myr 101 R (g 4 0.40M5 2 0.2) 1 10 0 0 100 100 1600 N 1hi5 1hi10_v5* 1hi10_v5* 1400 1hi10 0.25 104 102 M)11020000 11hhii1100__vv52** 0.20hmr 103 (MS 800 0.15 (p MV 600 0.10c) 102 101 400 0.05 200 1 0 0.00 10 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 T (Myr) 100 100 -1 0 1 -1 0 1 10 10 10 10 10 10 Figure5.Theevolutionofgalactocentricdistance,clustermass, a(pc) RG(pc) VMS mass and cluster half mass radius as a function of time, for simulations 1hi5 (solid blue lines), 1hi10 (dashed red lines), Figure 4. Comparison between simulations 1lo10_v5∗ and 1hi10_v5∗(dot-dashedgreenlines)and1hi10_v2∗(dottedma- 1hi10_v5∗, which have almost identical initial conditions, the gentalines). latter sampling better the low end of the IMF. The left pan- els show the distribution of semi-major axes of all stars (solid green line) and stars more massive than 8M(cid:12) at T = 7Myr 1hi5 1hi5 (jedcatsehdeddibstluaneclienseo).fTmhaeinrisgehqtupeannceelsstsahroswmtohreedmisatrsisbivuetitohnasnof8pMro(cid:12)- 104 Semi-Major Axes PDrisotjaenctceeds 102 all at T = 7Myr (dashed blue) and T = 15Myr (dot-dashed red). 3 >8M 7 Myr The stars are projected so that the disk rotates clockwise in the 10 15 Myr sky.They-valuesoftheprojecteddistributionsarere-normalised 2 1 to the expected number of stars had the model been simulated 10 10 withafullKroupaIMF. 1 10 Simulation1hi5formsamoremassiveVMSthantheother 100 100 simulations as it is initially ∼ 10 times as dense, however N 1hi5_ms 1hi5_ms the resulting increased luminosity decreases its lifetime. In simulation1hi10_v2∗,theclusterbecomesunboundbefore 104 102 themassivestarscanreachthecentreofthecluster,however 3 10 theinitialdensityishighenoughthata235M VMSforms (cid:12) by the first pericentre passage. The self-limiting nature of 2 1 10 10 the VMS formation is discussed in section §6. 1 10 5.3 Models with extreme initial conditions 100 100 -1 0 1 -1 0 1 10 10 10 10 10 10 The young clockwise disk population exhibits a top heavy a(pc) R (pc) mass function, with power law index α ∼ 1.7 (Lu et al. G 2013).Inthecontextoftheclusterinspiralscenariothishas Figure 6. Comparison between simulations 1hi5 and 1hi5_ms, been explained by mass segregation inside the cluster, with whichhavealmostidenticalinitialconditions,howeverthelatter themost massive starsreachingthe central parsec,and low is primordially mass segregated. The left panels show the dis- mass stars being preferentially lost due to tides during in- tribution of semi-major axes of all stars (solid green line) and spiral. However, as we have shown in section §5.1, clusters stars more massive than 8M(cid:12) at T =7Myr (dashed blue line). lose massive stars as well as low mass stars throughout in- Therightpanelsshowthedistributionsofprojecteddistancesof spiral,viadynamicalejectionsandtheshrinkingtidallimits spectroscopically visible stars at T = 7Myr (dashed blue) and as the clusters approach SgrA*. In order to test the effect T =15Myr(dot-dashedred).Thestarsareprojectedsothatthe disk rotates clockwise in the sky. The y-values of the projected of mass segregation, we ran simulation 1hi5_ms, which we distributions are re-normalised to the expected number of stars primordiallymasssegregatedusingthemethoddescribedin hadthemodelbeensimulatedwithafullKroupaIMF. Baumgardtetal.(2008).Forsimulations1hi5and1hi5_ms, Fig. 6 shows the semi-major axes of all stars and main se- quencestarsmoremassivethan8M atT =7Myr,aswell (cid:12) as the distributions of projected distances of spectroscopi- (cid:13)c 2015RAS,MNRAS000,1–15