Self-consistent simulations of Nuclear Cluster formation through Globular Cluster orbital decay and merging
aa r X i v : . [ a s t r o - ph ] A p r Mon. Not. R. Astron. Soc. , 1–7 (????) Printed 14 June 2018 (MN L A TEX style file v2.2)
Self-consistent simulations of Nuclear Cluster formationthrough Globular Cluster orbital decay and merging
R. Capuzzo-Dolcetta ⋆ P. Miocchi † Dipartimento di Fisica, “Sapienza” Universit´a di Roma , P.le A. Moro, 5, Roma I-00185, Italy.
Accepted ????. Received ????; in original form ????
ABSTRACT
We present results of fully self-consistent N -body simulations of the motion offour globular clusters moving in the inner region of their parent galaxy. Withregard to previous simplified simulations, we confirm merging and formation ofan almost steady nuclear cluster, in a slightly shorter time. The projected surfacedensity profile shows strong similarity to that of resolved galactic nuclei. Thissimilarity reflects also in the velocity dispersion profile which exhibits a centralcolder component as observed in many nucleated galaxies. Key words: stellar dynamics – methods: numerical – galaxies: kinematics anddynamics – globular clusters: general
The motion of massive globular clusters (GCs) in a galaxyis determined by both the smooth general potential andthe graininess of the stellar field. This latter, fluctuat-ing, component acts as a decelerating mechanism: the socalled dynamical friction (hereafter DFR). The role ofDFR, which dissipates cluster orbital energy and angularmomentum, has been found to be crucial in determiningthe time evolution of globular cluster orbits in a galaxy,especially for orbits plunging in the inner, high densitygalactic regions (e.g. Capuzzo-Dolcetta & Vicari 2005).The decay time may, indeed, be short enough for mas-sive clusters to limit their motion to the inner galacticregions. However, any general astrophysical considera-tion regarding dynamical friction cannot be based juston approximated evaluations based on Chandrasekhar-like formulas, even in their generalizations apt to treataxisymmetric or triaxial cases (Pesce, Capuzzo-Dolcetta& Vietri 1992; Capuzzo-Dolcetta 1993; Capuzzo-Dolcetta& Vicari 2005).Several effects are indeed neglected by the analyticalapproach. For example, it is not considered the changein magnitude of the DFR due to the increase of the spa-tial extension of the cluster caused by the formation, andsubsequent ‘expansion’, of its tidal tails, and, moreover,how the DFR acts on the different parts of the cluster(e.g., it could be stronger on its core than on the tails).Recent numerical experiments have tried to shed light onthis aspect finding that stars stripped from the cluster by ⋆ E-mail: [email protected] † E-mail: [email protected] the field, but still close enough to the system, continue tocontribute to the mass of the decelerating system (Fell-hauer & Lin 2007), and that, in general, the real DFReffect can be stronger than that estimated by the usualChandrasekhar formula (probably because of the furtherfriction due to tidal effects, see Fujii et al. 2007). An-other unexplored question is how the gravitational feed-back on the very inner part of the galaxy influences theDFR strength; this may be important when, during thefinal stages of orbital decay, the GC orbit gets very nearto the centre so to enclose a galactic mass comparablewith that of the cluster itself.Clarifying the role of the above-mentioned dynamicaleffects is important also in the attempt to understand themechanisms leading to the formation of Nuclear Clusters(NCs). In fact, it can help in discriminating the variousscenarios proposed, especially in supporting the validityof multiple merging of smaller sub-systems as a forma-tion mechanism (see, e.g. Oh & Lin 2000; Fellhauer etal. 2002; Bekki et al. 2004; Capuzzo-Dolcetta & Miocchi2008). At this regard, we cite the recent observational ev-idence of the existence of a very young and massive starcluster in NGC 2139, located 2 ′′ offset from the kine-matical centre of the host galaxy, suggesting a formationthat was independent and non coeval with that of thegalactic bulge and, moreover, the observed environmentis such that the system can decay to the centre in a timeso short to keep intact its structure and become what isnormally called NC (Andersen et al. 2008, see also B¨okeret al. 2002; Walcher et al. 2006).Reliable indications would follow by straightforward,direct N -body integrations, which can be enormouslytime consuming when treating self-consistently the mo- c (cid:13) ???? RAS R. Capuzzo-Dolcetta, P. Miocchi tion of a GC in a dense galactic environment. In thisLetter we present the preliminary results of a fully self-consistent N -body simulation concerning the close inter-action of a sample of four massive GCs in the central re-gion of a galaxy. Both the clusters and the galaxy are rep-resented by mutually interacting particles, thus includingself-consistently both DFR and tidal interactions. Thisstudy can give depeer insight into the decay and merg-ing of GCs in galactic nuclear regions, a scenario firsttackled by semi-analitical approaches (Tremaine et al.1975; Capuzzo-Dolcetta 1993) and then pursued by onefull N -body experiment though with a resolution muchlower than that presented here (Oh & Lin 2000, who used N = 500 ‘particles’ for each GC and N ∼ to representthe galaxy). Simulations presented here are useful to testand validate previous recent results (Capuzzo-Dolcetta &Miocchi 2008, hereafter CM08) obtained in a simplifiedscheme in which the clusters are N -body objects movingin a fixed analytical galactic potential and subjected to adynamical friction braking given analitically by the Pesceet al. (1992) formula. Our N -body simulation has been carried out employ-ing a parallel tree-code with a leap-frog time integra-tor using individual and variable time steps (Miocchi &Capuzzo-Dolcetta 2002). The simulation is set in a waysimilar to that described in CM08, but for the role ofthe external galactic field which was, there, representedas a triaxial, analytical potential while, here, it is self-consistently ( N -body) represented starting from initialconditions sampling a given equilibrium density profile.We resume briefly the main characteristics of our mod-elization. We choose to follow four massive GCs as tem-plates of a multi-merger among stellar systems decayed,conserving their individuality, in the inner galaxy region.A quick orbital decay induced by DFR is due to initiallarge values for the total mass of the clusters, whose re-sistance to external tidal disturbance was guaranteed bytheir sufficient initial compactness. The GC sample cor-responds to the four densest clusters dealt with in CM08and, moreover, every GC is composed by N = 2 . × equal mass stars. See Tab. 1 for the initial values of theirstructural parameters.The stellar bulge where GCs move is here repre-sented as another N -body system, with N b = 512 , ∝ [ − E ] / with E being the particle total energyper unit mass). The Plummer density profile ρ b ( r ) = ρ b0 (cid:2) r/r bc ) (cid:3) − / , (1)where ρ b0 = 3 √ M b / πr is the central density, has twoscale parameters, the core mass ( M b ) and the core radius( r bc ) which are used, in the following, as mass and lengthunits, respectively. Consequently, time, velocity and den-sity will be measured in units of the galactic crossingtime t b = ( r /GM b ) / , of v b = ( GM b /r bc ) / and of the central density ρ b0 , respectively. Due to the infiniteextension of the Plummer sphere, it has to be truncatedat a distance r cut large enough to guarantee the stabilityof the sampling N -body representation. This is done bythe choice of r cut = 12 . r bc so as to contain 99% of the to-tal mass. In order to have nearly the same galactic meangravitational field of that in the CM08 model, we set the M b and r bc values as equal to those of the correspondingparameters of the galactic density distribution used inCM08, even though this gives a slightly larger ρ b0 . Thefour clusters [(a), (b), (c) and (d)] start their evolutionfrom the same initial conditions chosen in CM08. As clus-ters reference centres, we chose their centre-of-densities(CDs) as defined in Casertano & Hut (1985). Finally,note that the simulation results are scale-invariant: anyquantity can be re-scaled by fixing arbitrarily the valuesof M b and r bc in physical units. The rapidity of the merger is evident in Fig. 1, as wellas in the upper panel of Fig. 3: the merger process iscompleted at about t m ≃ t b , when the Lagrangian radiiof the four GCs, seen as a whole system, flatten to a quasi-stable configuration (see Fig. 2, upper panel). In physicalunits: t m ≃ . (cid:18) r bc
100 pc (cid:19) / (cid:18) M b M ⊙ (cid:19) − / Myr . (2)Of course this ‘merging’ time depends much on the or-bital initial conditions of the progenitor clusters and mea-sures the time of the merger since the time in whichthe GCs are already confined within the galactic coreregion. Nevertheless, given that DFR has been convinc-ingly shown to be an efficient mechanism to drag GCsin the very inner regions of a galaxy within a time muchshorter than a Hubble time (see Capuzzo-Dolcetta 1993,Capuzzo-Dolcetta & Vicari 2005 and the discussion inMiocchi et al. 2006), this result supports the hypothesisstating that a GCs ‘sedimentation’ can take place at thegalactic centre well within the galaxy life-time.It is interesting noting (see the upper panel of Fig. 3),that the CD of the merger remnant oscillates within0 . r bc from the galactic CD. If, e.g., r bc ∼
200 pc thisdisplacement corresponds to ∼ ∼ . ′′ at the Virgo cluster distance), so the NC would appearsubstantially centered. The off-centered NC position isdue to the gravitational feedback between the very cen-tral part of the galaxy and the clusters, which naturallyderives from the self-consistent nature of the model. Suchan interaction reflects into the ‘perturbed’ behaviour ofthe 1% Lagrangian radius (below ∼ . r bc ) of the galaxythat is evident from Fig. 2 (lower panel).The lower panel of Fig. 3, indicates that a first merg-ing event between 2 clusters [(b) and (d)] occurs ratherearly (at t < ∼ . r bc far from the galactic centre. The fact that such a bi-nary coalescence takes place before the complete orbitaldecay, suggests that these events can occur even offsetfrom the galactic centre, in spite of the strong tidal field. c (cid:13) ???? RAS, MNRAS , 1–7 uclear Cluster formation through Globular Cluster decay and merging Figure 1.
Time sequence of the four GCs merger. Field stars are not shown.
Table 1.
Parameters list for the initial cluster models, expressed in galactic units. Reportedare: the GC mass ( M ), the limiting radius ( r t ), the King radius ( r c ), the King concentrationparameter ( c ), the half-mass radius ( r h ), the central density ( ρ ), the half-mass crossing time, t ch ≡ [ r / ( GM )] / , and the King velocity parameter ( σ K ).cluster M r t r c c r h ρ t ch σ K (a) . × − .
16 1 . × − . . × −
510 3 . × − . (b) . × − .
16 1 . × − . . × −
270 4 . × − . (c) . × − .
14 1 . × − .
99 2 . × −
400 3 . × − . (d) . × − .
14 1 . × − .
89 2 . × −
160 4 . × − . (cid:13) ???? RAS, MNRAS , 1–7 R. Capuzzo-Dolcetta, P. Miocchi
Figure 2.
Evolution of the Lagrangian radii of the four GCs asa whole (upper panel) and of the galaxy (lower panel) averagedover a time window = 3 t b . They correspond to the radii ofthe sphere centered at the system CD and enclosing a givenfraction of the total mass (as indicated). This ‘pre-decay’ coalescence was also found in the low-resolution self-consistent simulation by Oh & Lin (2000)(their model 1b), though in their case this phenomenontakes place much farther from the galactic centre.By comparing the merger time with that of theCM08 simulation, it is found that here the orbital decayis ∼ . t <
20, the clusters CDs are always
Figure 3.
Upper panel: time evolution of the distance of thecluster CD from the galactic centre ( r CD ). Lower panel: timebehaviour of the distance between each cluster CD and itsfirst-neighbour. All the behaviours have been averaged over atime window = 3 t b . Solid lines refer to cluster (a); dotted: (b);short-dashed: (c); long-dashed: (d). closer to the centre of the galaxy respect to the case ofthe CM08 simulation. For later times the comparison isno longer reliable, because of the very small values of r CD .In Fig. 5 the total surface density profile is plottedfor the last configuration and the typical appearance of anucleated galaxy central profile comes clearly out. As anexample, the similarity with the VCC 1871 profile in theVirgo cluster (Cˆot´e et al. 2006) is particularly evident.The central ‘over-density’ associated to the NC presenceis, however, less pronounced with respect to that obtainedin CM08 and this is due to the lower concentration of theNC in comparison with the last configuration in CM08 c (cid:13) ???? RAS, MNRAS , 1–7 uclear Cluster formation through Globular Cluster decay and merging Figure 4.
Time evolution of the ratio between the distance ofthe cluster CD from the galactic centre and the same quantityevaluated in the CM08 simulation ( r CD , ), averaged over atime window = 3 t b . Line symbols are the same as in Fig. 3. Figure 5.
Projected surface density profile of the whole sys-tem (galaxy plus NC) in the final configuration (solid line) and,for the sake of comparison, of the galaxy only (long dashedline). case. Perhaps, the lower NC central density may be a con-sequence of the too large collisional effect generated bythe interaction of the clusters with the ‘particles’ repre-senting the very inner region of the galaxy that, becauseof the limited resolution, are probably too massive. Onlyhigher resolution simulations of the very last stages of theorbital decay and merging would allow to establish theactual role of this effect (Capuzzo-Dolcetta & Miocchi, inpreparation).
Figure 6.
Projected radial behaviour of the velocity disper-sion in the last system configuration. Line symbols are as inFig. 5.
An important result concerns the peculiar behaviourof the global velocity dispersion profile. Confirming thefinding in CM08, the velocity dispersion of the whole sys-tem (galaxy plus NC) shows a clear decrease towards thecentre (see Fig. 6). As discussed in CM08, this is a clearsign of the presence of two kinematically distinct sys-tems that are relaxed into a different dynamical status.Such a peculiar feature has been actually found in mostof the Virgo cluster nucleated dwarf ellipticals observedby Geha, Guhathakurta & van der Marel (2002, see theirFig. 5). The decreasing trend to the centre of the veloc-ity dispersion is also consistent with the solution of theJeans equations for a sample of NCs observed in late-typespirals (Walcher et al. 2005). Finally, either a flatteningor a slight central decrease are found and discussed in Oh& Lin (2000), too.
Massive globular clusters suffer of dynamical frictionbraking such to be limited to move in the inner part ofthe host galaxy. The interaction of decayed globular clus-ters among themselves and with the external tidal fieldmay lead to subsequent merger events, whose modes arenot easily understood without detailed N -body simula-tions. In a previous paper (CM08) we followed numeri-cally the interaction and merging of four globular clusterseach represented as an N = 250 ,
000 stars system mov-ing in an analitical triaxial potential, taking dynamicalfriction into account by means of the Pesce et al. (1992)formula.The present paper has the aim to check the CM08results via a full self-consistent N -body modelization, i.e.sampling also the galactic environment by a large num-ber (= 512 , c (cid:13) ???? RAS, MNRAS , 1–7 R. Capuzzo-Dolcetta, P. Miocchi simulations which are also necessary to study reliably thelong-term evolution and stability of the formed NC.
ACKNOWLEDGMENTS
The simulation was conducted at the CINECA super-computing centre, under the INAF-CINECA agreement(grant cne0in07 ). REFERENCES
Andersen D.R., Walcher C.J., B¨oker T., Ho L.C., vander Marel R.P., Rix H.-W., Shields J.C., 2008, submit-ted to ApJ.Bekki K., Couch W.J., Drinkwater M.J., Shioya Y.,2004, ApJ, 610, L13B¨oker T., Laine S., van der Marel R.P., Sarzi M., RixH.-W., Ho L.C., Shields J.C., 2002, AJ, 123, 1389Capuzzo-Dolcetta R., 1993, ApJ, 415, 616Capuzzo-Dolcetta R., Miocchi P., 2008, accepted on ApJ(astro-ph/0801.1072) (CM08)Capuzzo-Dolcetta R., Vicari A. 2005, MNRAS, 356, 899Casertano S., Hut P., 1985, ApJ, 298, 80Cˆot´e P. et al., 2006, ApJS, 165, 57Fellhauer M., Baumgardt H., Kroupa P., Spurzem R.,2002, Cel. Mech. Dyn. Astron., 82, 113Fellhauer M., Lin, D.N.C., 2007, MNRAS, 375, 604Fujii M., Iwasawa M., Funato Y., Makino J., 2007, sub-mitted to ApJ (astro-ph/0708.3719)Geha M., Guhathakurta P., van der Marel R.P., 2002,AJ, 124, 3073Miocchi P., Capuzzo-Dolcetta R., 2002, A&A, 382, 758Miocchi P., Capuzzo-Dolcetta R., Di Matteo P., VicariA., 2006, ApJ, 644, 940Oh K.S., Lin D.N.C., 2000, ApJ, 543, 620Pesce E., Capuzzo-Dolcetta R., Vietri M., 1992, MN-RAS, 254, 466Tremaine S., Ostriker J.P., Spitzer L. Jr., 1975, ApJ,196, 407Walcher C. J. et al., 2005, ApJ, 618, 237Walcher C.J., B¨oker T., Charlot S., Ho L.C., Rix H.-W.,Rossa J., Shields J.C., van der Marel R.P., 2006, ApJ,649, 692 c (cid:13) ???? RAS, MNRAS , 1–7 uclear Cluster formation through Globular Cluster decay and merging This paper has been typeset from a TEX/ L A TEX file prepared by the author. c (cid:13) ???? RAS, MNRAS000