Intermediate Mass Black Holes near Galactic Center: Formation. and Evolution

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1 Intermediate Mass Black Holes near Galactic Center: Formation and Evolution Yuan Yuan Johns Hopkins University Abstract: As a missing link between stellar mass black holes and supermassive black holes (SMBH), black holes of intermediate mass (IMBH) could be formed in young dense stellar clusters. When the clusters are in an evolutionary phase in which stellar density in the core becomes so high that physical collisions between stars become frequent, the formation of a central subsystem of massive star will lead to collision away. An IMBH could be finally formed if the runaway grows beyond 300M. Great interest would be invoked when the IMBHs can be transported to the center of galaxies. Recent N-body simulations show that about 10% of the clusters born close to the center of the Galaxy could form IMBHs. After the parent clusters completely evaporate, these IMBHs will continue to sink toward the central SMBH due to dynamical friction and finally merge with SMBH via gravitational radiation, which may shed new light on the formation and evolution of SMBHs in the center of galaxies. I. INTRODUCTION Although there are rapidly growing evidences showing that many galaxies including the Milky Way host central supermassive black holes (SMBHs) with masses in the range of M, the formation is not well understood. In the last decade, the discovery of the missing link in BHs intermediate mass black holes (Matsumoto et al. 2001) of mass between M have opened up new possibilities for modeling SMBHs. Several scenarios have been proposed for the formation of these objects. Madau & Rees (2001) suggested that IMBHs could be the endproduct of the episode of pregalactic star formation. Many other 1

2 authors (Quinlan & Shapiro 1990; Portegies Zwart et al. 1999; Figer & Kim 2002; Miller & Hamilton 2002) have pointed out that they could also be formed in young dense stellar clusters via runaway stellar merging. The most famous IMBH candidate M82 X1 was detected in Using the Chandra data, Matsumoto et al. identified nine bright compact X-ray sources in the central region of starburst galaxy M82, of which the brightest source (M82 X1) had a luminosity of in 2000 January, corresponding to a BH with a minimum mass of 700 M (assuming the Eddington luminosity). T. Harashima et al.(2001) observed the same region in the infrared using Subaru telescope and found that at least four young compact star clusters coincide with these X- ray sources within the uncertainties of Chandra and Subaru, including the brightest M82 X-1, whose position is consistent with the star cluster MGG-11. Before that, many ultraluminous X-ray sources (ULXs), which could be good candidates for IMBHs, have also been observed to be associated with star clusters (Fabbiana, Schweizer & Mackie 1997). Based on these observations, which suggested that the X-ray sources were apparently formed in star clusters, Ebisuzaki et al. (2001) proposed a new possible scenario for SMBHs. This scenario of formation consists of several steps: First, IMBHs form in young compact star clusters through runaway merging of massive stars; second, when these IMBHs are forming, the host clusters sink toward the galactic center through dynamical friction and dissolve in the tidal field. The last step is the merging of IMBHs. Lots of numerical simulations are done for the first step (Holger & Makino 2003; Portegies Zwart et al, 2004). And the second step is supported by realistic mass-loss model, while the last step has not yet been investigated in detail. This paper gives a brief review about work done in this area with the influence from a central SMBH. The dynamical evolution of star clusters producing massive black holes and how the orbits of IMBHs evolve after the host clusters are disrupted are discussed. A short conclusion and discussion about the relation with the SMBH formation scenario is given in the last part. 2

3 II. IMBH FORMATION THROUGH RUNAWAY GROWTH A. Formation of the collision runaway Massive and young star clusters are always of greatest interest to astronomers, in which stellar dynamical time scales are short enough that the cluster can undergo significant structural change during the lifetimes of the most massive stars. In such clusters, dynamical evolution controls the early phases of the stars lives. As long as stellar evolution remains relatively unimportant the cluster s dynamical evolution is dominated by its half-mass relaxation time, i.e. the time scale on which two-body encounters transport energy around the system (Heggie & Hut 2003; Spitzer 1987) t rh 2Myr( r h [pc] )3/2 ( M c [M ] ) 1/2 N lnλ (1) Here, N is the number of bound objects in the system (a binary is a single object), Mc is the total mass of the cluster, rh is its half-mass radius, and lnλ ln(0.1n) 10. For realistic clusters there exists a broad range in stellar masses from 0.1M o dot to 100M o dot. During the early evolution of the cluster, massive stars sink toward the center via dynamical friction while lighter stars are expelled toward the periphery, leading to the well-known mass segregation effect. Pairs of stars bound in a tight binary will effectively behave like a single more-massive star, sinking to the core. Binney & Tremaine (1987) find that for a massive star of mass m at a distance r from the center the drifting rate inward is given by r dr dt = 0.43Gm σ lnλ (2) where is the cluster velocity dispersion. Thus, the in-spiral timescale for the star sinking from the half-mass radius rh to the center as dynamical friction reduces its velocity is t df < m > m 0.138N ln(λ <m> m r3 h )( GM c ) 1/2 (3) where Mc is the total mass of the cluster, and < m >= Mc/N is the mean stellar mass. Using < m > 0.5M, Portegies Zwart & McMillan (2002) find that empirically the most 3

4 massive stars (m > 20M ) will segregate rapidly to the cluster center, forming a dense stellar subcore on a time scale t cc 0.2t rh. Though for a static cluster core there is usually too little initial mass in the high-density region to form a 1000M object, if taken into account the dynamical evolution, a central density increase of 2 3 orders of magnitude is typical in a core-collapse time t cc, boosting even a relatively low-density core into the range where collisions between massive stars are very common, and may eventually form a collision runaway. FIG. 1. ( Portegies Zwart et al., 2004) The area of parameter space for which runaway collision can occur, and where the process is prevented. t df is the dynamical friction time scale for 100M stars, and c the cluster concentration parameter. Solid circles indicate simulations resulting in runaway merging, while open circles correspond to those in which no runaway merging occurred. They found that the requirement for the formation of an IMBH in MGG-11 is that the cluster was born with c 2 and t df 4Myr. If the mass of this runway grows beyond 300M, it will collapse to an IMBH without losing significant mass in a supernova explosion (Heger et al. 2003). Portegies Zwart et al. (2004) successfully applied this model to explain the ULX source M82 X-1 associated 4

5 with the star cluster MGG-11. They found that for a 100M star in MGG-11, t df is around 3Myr, comparable to the main-sequence lifetime of the most massive stars. On the other hand, for another cluster MGG-9, t df is as long as 15Myr, which shows no evidence of an IMBH. They conclude that one important cause for the formation of the IMBH in MGG-11 is that the massive stars bound in it can reach the center of the cluster before exploding as supernovae, whereas those in MGG-9 can not. Their numerical results are shown in figure 1, where the concentration parameter is defined as the ratio of the core radius to the tidal radius:. In their simulation, both high initial concentration and short dynamical friction time scales are required to lead to the final collision runaway. B. Influence from a central tidal field Another factor that can not be neglected is the mass loss of cluster due to many complicating effects including stellar evolution and the evaporation of the cluster that fills its Jacobi surface in an external potential driven by tidal stripping. Portegies Zwart et al. (2001) have studied the evolution of young compact star clusters within 200 pc of the Galactic center. Their calculations included the effects of both stellar and binary evolution and the external influence of the potential of galaxies. They found that the mass of a typical model cluster decreased almost linearly with time according to M c = M c0 (1 τ t dis ) (4) Here Mco is the initial mass, is the cluster age in terms of the instantaneous two-body relaxation time (t rj ) within the Jacobi redius, and t dis is the cluster s disruption time. They found that their model clusters dissolved within 0.3 of t rj. So the early dissolution of the cluster reduces the runaway mass by prematurely terminating the collision process. Generally the main requirement for a successful collision runaway is that the star cluster must experience core collapse before massive stars explode as supernovae and before the cluster dissolves in the Galactic tidal field. Once the runaway collapses to an IMBH, the 5

6 collision growth rate slows down dramatically, and the host cluster loses its mass even faster due to the tidal disruption of stars by the IMBH. For small clusters, the mass of the runaway is effectively limited by the total number of high-mass stars in the system. For sufficiently dense larger clusters, the runaway mass is determined by the fraction of stars that can mass segregate to the cluster core while still on the main sequence. A convenient fitting formula combining these two situations is (Portegies Zwart & McMillan 2002; McMillan & Portegies Zwart 2004 ) m r 0.01M c (1 + t rh 100Myr ) 1/2 (5) FIG. 2. [ Portegies Zwart et al. (2005)] Orbital evolution of a star cluster with 45000M and a Salpeter initial mass function, a lower mass limit of 0.2M and a Wo=9 King model initial density profile. It spirals into the Galactic center from an distance of 2pc (lines from top left to bottom right), while producing an runaway merger product/imbh mr (bottom left to top right). Solid lines show the result of the semi-analytic model (with lnλ = 8), the dotted lines represent high-precision N-body calculations. Figure 2 shows both the N-body simulation and the results of semi-analytic model based on equations above for a star cluster born with stars in a circular orbit at a distance 6

7 of 2pc form the Galactic center( Portegies Zwart et al., 2005). The orbit decay of the cluster due to dynamical friction with stars in the host galaxy can be similarly described by equation (2), which is valid when the cluster is in orbit radius larger than 0.1 pc. In figure 2, the solid lines (semi-analytic model) and dotted lines (N-body calculation) match well with each other, indicating that the simple analytic model produces satisfactory results. Figure 3 gives a snapshot of the simulation of the same cluster projected in three different planes at an age of 0.35Myr, which has been significantly flattened by the Galactic tidal field, especially along the direction of angular momentum. By the time, a 1100M collision runaway star has formed in the center and 30% of the cluster has already dispersed. The total cluster is almost disrupted in about 0.7 Myr. FIG. 3. Snapshot of the same dissolving cluster, showing the projection on various plane of the N-body simulation. Many bound stars in the cluster now have already been dispersed and the rest stars have spread out into the shape of a disk spanning the inward-spiraling orbit. The merger products are generally out of thermal equilibrium and often rapidly rotating. Generally their thermal time scale significantly exceeds the mean time between collisions even if the runaway grows to 1000M. So the accreting object will be unable to reach 7

8 thermal equilibrium before the next merger. So far we do not know clearly about the detailed evolution of the stars hundreds or thousands of mass of sun. Van Beveren pointed out that if the wind mass loss rate exceeds the accretion rate due to mergers, then the entire runaway process may fail. But mass loss rates are relatively low while most of the accretion is occurring. The collision rate during the period of rapid growth typically exceeds, comparable to the maximum loss rates derived for massive stars. Thus the stellar winds are unable to prevent the occurrence of repeated collisions only if the mass loss rate is very high (more than ) and sustained over the lifetime of the massive stars. Based on detailed supernova calculations, we can also assume that stars having masses larger than 260M collapse to black holes without significant mass loss in supernova explosions. In the simulation of the cluster in figure 2 and 3 by Portegies Zwart et al. (2005) shows that the runaway star loses about 200M in the stellar wind and collapses to a 1000M IMBH at about 2.4 Myr. III. EVOLUTION AFTER THE FORMATION OF IMBH With an IMBH, the cluster loses its mass even faster due to tidal field because the IMBH heats the whole system and stars flow over the tidal boundary much faster. Mass loss slows down considerably when the number of stars has dropped to less than a few hundred, since by then most mass is in the central BH and the relaxation time increases with decreasing cluster mass. The numerical results from H. Baumgardt et al. (2004) shows a system of about 100 stars, composed mainly out of main sequence stars and white dwarfs, is still bound to the IMBH after a Hubble time. IRS13E (Maillard et al. 2004) looks like such a remnant cluster. IRS13E is located at about 0.16 pc from the central black hole in the Galaxy Sgr A*. It appears as composed of at least six hot, massive stars within a projected diameter of 0.02 pc. The common westward direction and similar amplitude of the proper motions with a mean value of 280 km/s for the main components indicate that the stars are physical 8

9 bound. Several members in it have already reached the WR stage, which means it may be a young cluster of a few Myr old. A natural hypothesis is that there is a central black hole of mass M keeping these stars bound. Constraints on the BH mass are obtained from the radial velocities of the sources. Assuming circular orbits, the 3rd Kepler s law gives directly a total mass of 1300M. If we only assume the system is bound and release the circular orbits hypothesis, we get a lower limit to the mass: M. A. Orbit evolution with a central SMBH Great interest would be invoked when the IMBHs can reach the inner pc of galaxies. From the formula of the dynamical friction ( Binney & Tremaine 1987), we find that the time scale on which clusters sink to the galactic center is (taking lnλ = 8) T df r v c ( 1kpc )2 ( 250kms )(5 106 M Θ )yr (6) 1 M c So clusters initially within 1kpc of the center of the galactic center can reach the center within several Gyr. If the cluster dissolve significantly before they reach the central parsec, the time scale for the IMBH to sink into the galactic bulge would increases greatly, may even be larger than the lifetime of the galaxy. In another word, only IMBHs born in clusters near the galactic center or those having dissolve time scale (time scale for losing most of its stars) not less than the dynamical time scale T d f could reach the very central area and participate in merging processes. After the host clusters have almost been disrupted, the orbital evolution of IMBHs can also been described by the dynamical friction formula we used before (equation 2), or in detail, by Chandrasekhar dynamical friction formula in the case of Maxwellian distribution (Binney & Tremaine 1987) dv BH dt = 4πlnΛG2 ρm BH G(X) (7) VBH 2 where G(X) = erf(x) 2X π exp( X 2 ), and X V BH /( 2σ). Here σ is the velocity dispersion. Using the circular velocity 9

10 V BH = GM s /a (8) we can get the time evolution of the semi-major axis a. Matsubayashi et al. (2005) adopt Tremaine s -model with central black hole and outer slope of -5 for the stellar distribution in the N-body simulation and got the change of the value of a according to time (left figure in figure 4). They use unit length of 0.86pc, unit time of yrs and unit mass of M. Theoretical curve calculated by equation 7 and 8 is also shown in the figure. Matsubayashi et al. found out that the difference between the theoretical prediction and numerical result is good for the early period. In the later phase, the numerical simulation shows the slowing down of the evolution. Defining the hardening rate as, the N-body simulation gives hardening rate smaller than the theoretical prediction for a > A natural explanation is the loss cone depletion, similar to what happens in the case of SMBH binaries. When the IMBH sinks to the center, the Lagrangrian radius corresponding to the position of the IMBH starts to expand due to the back reaction of dynamical friction to the black hole. When the stellar mass inside the IMBH semi-major axis becomes comparable to the IMBH itself, the effect of back reaction becomes significant, which remarkably reduce the number density of field stars. In another word, IMBH kicked out the neighboring stars. In Matsubayashi et al. s simulation, the evolution of the semi-major axis finally got stuck when the semi-major axis reached a (about pc) after 5Myr beginning from a 0.1(0.086pc). This is also in agreement with the result gained by Portegies Zwart et al. (2005), who approached it in another way by assuming that the region within a mpc of the central SMBH in the Galaxy is depleted of stars, based on the argument that the total Galactic mass inside that radius, excluding the SMBH, is probably less than 10 3 M (Ghez et al, 1998, Ghez et al, 2003). 10

11 FIG. 4. Matsubayashi et al. (2005). Evolution of the semi-major axis and eccentricity of the IMBH. The left figure shows results for semi-major axis with an initial value Dotted, dash dotted, dashed, and solid curves are the results of models with different masses and different numbers of field stars. Thin curve shows the theoretical result. The right figure shows evolution of the eccentricity in the same four models. They use unit length of 0.86pc, unit time of yrs and unit mass of M. The right one in figure 4 shows the evolution of eccentricities of the IMBH in four runs. In all runs eccentricity shows systematic increase, in some runs there are even phases when e approaches to very close to unity, which has not been observed in previous studies of massive black hole binary. However, the configuration of the IMBH-SMBH here is quite different, so the evolution can be different. In the case of SMBH binary, two massive companions have similar mass and field stars which interact with the binary are not strongly bound to one of them. In the case of IMBH-SMBH binary, almost all field stars are strongly bound to the central SMBH, and have almost Keplarian orbits. The celestial mechanical effects such as the mean-motion resonance and Kozai mechanism can play important roles in the evolution, working as the transport mechanism for angular momentum from rapidly-rotating objects 11

12 to slowly-rotating objects, which can account for the increase of eccentricity. Actually, Matsubayashi et al. found out that the slow increase of the eccentricity directly corresponds to the decrease of the total angular momentum. After the IMBH reached the inner part and kicked out the neighboring stars, a hole in the distribution of stars has been created. After that the IMBH can strongly interact with stars with semi-major axis larger or comparable to its apocenter distance only when it is at the apocenter of its orbit. If it loses kinetic energy at its apocenter, it becomes highly eccentric. This situation is very different from the SMBH binary. Since the mass of IMBH is much lower than the central black hole, it has to come enough close to the field stars to interact effectively with them. So the IMBH has more chance to lose energy at apocenter than at percenter, and the eccentricity increases naturally. IV. THE FINAL FATE OF IMBH AND MERGING RATE The major characteristic of figure 4 is that the eccentricity of IMBH, after the evolution of semi-major axis stopped, can reach close to 1. From the viewpoint of the evolution of SMBH, we want to ask what the final fate of the IMBH is. If we take into account the gravitational radiation effects, the time scale of merging for the IMBH-SMBH binary through gravitational wave radiation is given by (Peters 1964) M s t gr a 3F (e)( 0.01pc )4 ( ) 2 m BH ( ) 1 yr (9) M M where F (e) depends on e as F (e) = (1 e 2 ) 7/2 ( e e4 ) 1 Although the time scale for a 1mpc orbit to decay by gravitational radiation exceeds the age of the Galaxy for circular motion, it will be significantly reduced to about several Myr when the eccentricity reaches about Thus we expect that the majority of IMBHs which arrive in the galactic center eventually merge with the SMBH on a time scale of a few Myr. Portegies Zwart et al. did not take into account the effect of the increase of eccentricity, a steady population of several IMBHs was needed in their model to drive the final merging. 12

13 However, the merging time scales got by both approaches are similar. In the simulation of Portegies Zwart et al. (2005), about 10% of the clusters born close to the center of the Galaxy could form IMBHs and a time average IMBH in-fall rate of roughly one per 7Myr was obtained. A total of IMBHs have reached the Galactic center over the age of the Galaxy ( 10Gyr), carrying a total mass of M, sufficient to build the accumulate majority of mass of the SMBH. Matsubayashi et al. (2004) estimated the rate of BH merger events by assuming that every galaxy has SMBH and each galaxy experience 1000 IMBH mergers. They found that the merging rate, may including both hierarchical (IMBH-IMBH) merger and monopolistic (IMBH-SMBH) merger, as per year. This number is quite attractive for the gravitational radiation observation. The frequency of the gravitational wave in the final merging phase is 0.1 to 102 Hz, within the target range of LISA and DECIGO. V. DISCUSSION AND SMBH FORMATION SCENARIO Based on currently growing evidences supporting that several ULXs associated with star clusters might be IMBH candidates, many authors have proposed that IMBH could be formed in young dense stellar clusters via runaway stellar merging, which has been successfully used to explain the most famous IMBH candidate ULX M82 X1 whose position is consistent with the star cluster MGG-11. In this scenario, the core collapse is driven by mass segregation in the dense cluster. And due to the high density in the central region a runaway growth of a huge star are expected via frequent mergers which may be the progenitor of an IMBH. The two main requirements for a successful collision runaway are that the star cluster must experience core collapse before massive stars explode as supernovae and before the cluster dissolves in the Galactic tidal field. Specifically, this requires a dense young cluster with high concentration. If the cluster is not dense enough for mass segregation to occur within 10 Myr, most massive stars will evolve into neutron stars or stellar mass black holes 13

14 before they spiral into the center, and eventually form a long-lived core with lots of compact objects, as some globular clusters are observed to be in this stage. Another effect to stop the core collapse is formation of large portion of binaries in dense cores. Binary stars are important sources of energy. Encounters between single stars and a binary will often cause energy transfer from the binary to the other by forcing the two companions closer. Numerical simulations of globular clusters have demonstrated that binaries can hinder the process of core collapse in globular clusters. In some extreme cases, large portion of binaries can make the core to re-expand. By taking account of realistic simulations, just as IMBHs can form through runaway collision in dense young star clusters, people expect that SMBHs in galactic center could be formed via lots of IMBH merger events. Ebisuzaki et al. (2001) and Matsubayashi et al. (2004) considered the following possible processes: In the early stage, IMBHs formed in clusters sink to the center by dynamic friction, constructing a multiple IMBHs system. BH binaries are formed and become harder and harder by three-body interactions with other IMBHs. Once one BH (product of hierarchical merger) has become more massive than others, it becomes extremely unlikely to be ejected from the central region. The seed BHs continue to swallow other IMBHs monopolistically and the central BH starts to grow. Matsubayashi et al. (2004) also calculated the frequency and amplitude of two limiting cases for the growth of SMBH- hierarchical and monopolistic growth. For the hierarchical growth model, the amplitude of gravitational radiation is larger than that for monopolistic model. The signals from equal-mass BH binaries with m 1 = m 2 > 10 4 M will be detectable by both LISA and DECIGO. While for the monopolistic growth, the signals will only be detected by DECIGO with event rate around 50. So we expect that LISA and DECIGO will establish the actual merging history from the statistics of gravitational radiation signals. 14

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