Mon. Not. R. Astron. Soc. 422, L28–L32 (2012) doi:10.1111/j.1745-3933.2012.01229.x

Intermediate-mass black holes in globular clusters Yu-Qing Lou1,2,3 and Yihong Wu1 1Department of Physics and Tsinghua Centre for Astrophysics (THCA), Tsinghua University, Beijing 100084, China 2Department of Astronomy and Astrophysics, the University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA 3National Astronomical Observatories, Chinese Academy of Sciences, A20, Datun Road, Beijing 100021, China

Accepted 2012 January 26. Received 2012 January 7; in original form 2011 September 8 Downloaded from https://academic.oup.com/mnrasl/article/422/1/L28/971090 by guest on 27 September 2021

ABSTRACT There have been reports of possible detections of intermediate-mass black holes (IMBHs) in globular clusters (GCs). Empirically, there exists a tight correlation between the central supermassive black hole (SMBH) mass and the mean velocity dispersion of elliptical galaxies, ‘pseudo-bulges’ and classical bulges of spiral galaxies. We explore such a possible correlation for IMBHs in spherical GCs. In our model of self-similar general polytropic quasi-static dynamic evolution of GCs, a heuristic criterion of forming an IMBH is proposed. The key 1/(1−n) result is MBH = Lσ , where MBH is the IMBH mass, σ is the GC mean stellar velocity dispersion, L is a coefficient and 2/3 < n < 1. Available observations are examined. Key words: accretion, accretion discs – black hole physics – hydrodynamics – instabilities – globular clusters: general – galaxies: bulges.

dispersion σ (Gebhardt et al. 2000; Zheng 2001; Ulvestad, Greene 1 INTRODUCTION & Ho 2007; Bash et al. 2008; Noyola, Gebhardt & Bergmann 2008; For star-forming molecular clouds, collapsed massive dense gas Zaharijas 2008; Zepf et al. 2007). These empirical correlations may cores eventually lead to luminous new-born stars burning nuclear shed light on the origin and evolution histories of IMBHs and their fuels. Analogously but on larger scales, we speculate that grossly host GCs. Among such observed relations, MBH and σ correlate spherical core collapses of globular clusters (GCs) should also cause tightly (e.g. Gebhardt, Rich & Ho 2002; Safonova & Shastri 2010). something singular around the dense centre; such massive central Self-similar solutions for general polytropic hydrodynamics of singularities may form intermediate-mass black holes (IMBHs). a self-gravitating fluid with spherical symmetry were constructed Observations of GC cores indicate that the central concentrations recently. Asymptotic behaviours of novel quasi-static solutions in a of non-luminous materials are likely due to IMBHs (Bahcall & single polytropic fluid have been revealed by Lou & Wang (2006, Ostriker 1975). Mass accretions on to IMBHs were proposed to 2007) and was used to model rebound (MHD) shocks in SNe. Such power GC ULX sources (e.g. Farrell et al. 2009). solutions were applied to clusters of galaxies (Lou et al. 2008) On even much larger scales, observations of galaxies reveal a for possible galaxy cluster winds. In this Letter, we invoke such strong correlation between the mass MBH of the central supermas- quasi-static solutions to model dynamic evolution of host GCs and sive black holes (SMBHs) and the mean velocity dispersion σ of the formation of IMBHs and to establish MBH–σ power laws. host stellar bulge in the form of log (MBH/M) =  + δlog (σ/σ0), Various aspects of GCs have been studied extensively (e.g. where  and δ are two coefficients and σ 0 is a velocity dispersion Benacquista & Downing 2011, and extensive references to excel- normalization (e.g. Tremaine et al. 2002). We naturally expect a lent reviews therein). We focus on the quasi-static self-similar GC similar MBH–σ power-law relation to exist for GCs. dynamic evolution (say, induced by the gravothermal instability) Evidence for central IMBHs in GCs has been debated extensively in the late phase of the pre-collapse regime; such asymptotic solu- and their possible existence bears important consequences for both tion for GC evolution leads to diverging mass density at the centre. the formation and evolution of GCs (e.g. Grindlay & Gursky 1976; Physically, as the inner enclosed core mass becomes sufficiently Maccarone et al. 2007, 2008; Zaharijas 2008). The central bright- high within a radius comparable to its Schwarzschild radius, an ness excesses observed in several GC cores (e.g. Djorgovski & King IMBH forms inevitably (e.g. through mergers of stellar mass black 1986) are compatible with the presence of central IMBHs. Proper- holes or runaway collisions and coalescence or merging of stars to ties of IMBHs correlate with various properties of GCs, including form supermassive stars and to trigger subsequent e± pair insta- stellar density profiles, central stellar dynamics, luminosities of bilities therein). After forming such a central IMBH in the core, GCs, mass-to-light ratios, surface brightness, rotation amplitudes, pertinent post-collapse mechanisms continue to operate: e.g. mass position angles, dark matter densities and the mean stellar velocity segregation maintains more massive stars around the central IMBH, while the ‘binary heating’ (e.g. Hurley, Aarseth & Shara 2007) from primordial stellar binaries that survived the IMBH formation tends E-mail: [email protected] to resist further core collapse or to drive post-collapse oscillations.

C 2012 The Authors Monthly Notices of the Royal Astronomical Society C 2012 RAS Forming IMBHs in globular clusters L29

The N-body GC simulations by Hurley et al. (2007) of up to 105 stars we derive two coupled non-linear ordinary differential equations and initial 5 per cent binaries may eventually reach a GC core bi- (ODEs) for first derivatives α and v in terms of x: nary frequency as high as 40 per cent at the end of the core-collapse D(x,α,v)α = N (x,α,v),D(x,α,v)v = N (x,α,v), (2) phase. Shown in their fig. 3, this core binary frequency actually 1 2 fluctuates between ∼10 and ∼40 per cent. We would expect that where the three functionals D, N1 and N2 are defined explicitly an IMBH forms at the GC centre rapidly during the core-collapse in Hu & Lou (2009) with zero magnetic field. An exact global phase, but we do not know exactly when. Physically, this IMBH static solution of equation (2) in physical dimensions, known as the would engulf the central stars and binaries at the epoch of IMBH singular general polytropic sphere, has u = 0: formation. Such an explosive event may give rise to a powerful A nAK1/n ρ = K1/nr−2/n,M= r(3n−2)/n, (3) gamma-ray burst and a shock wave surrounding the GC centre. The 4πG (3n − 2)G slower relaxation, evolution and accretion then persist on a much −q A ≡ n2 −1/(n−3nq/2) longer time-scale. It is crucial to search for such evidence. where the coefficient [ 2(2−n)(3n−2) ] . This solution GCs are close to spherical; in their dynamic evolution, lumpiness serves as an asymptotic ‘quasi-static’ solution for small x,i.e.they Downloaded from https://academic.oup.com/mnrasl/article/422/1/L28/971090 by guest on 27 September 2021 observed could either result from merging disturbances and tidal are leading order terms of v(x)andα(x) and there exist higher order disruptions or provide source of fluctuations that may be classified terms for a self-similar asymptotic hydrodynamic evolution. into acoustic modes, gravity modes and vortical modes on larger For such quasi-static self-similar hydrodynamic asymptotic so- scales (Lou & Lian 2012). These perturbations may be unstable to lutions at small x, we consistently presume two relations trigger gravothermal instability. Analogous to nuclear burnings in a α = Ax−2/n + JxS−1−2/n and v = LxS , (3) star to resist stellar core collapse, the ‘binary heating’ from primor- dial stellar binaries may delay core collapse in GCs (e.g. Meylan & in two coupled non-linear ODEs (2), and derive two non-linear J S L Heggie 1997). As the source of ‘binary heating’ is exhausted during algebraic equations for three coefficients , and : a GC evolution, the inner core collapse is inevitable with possible n(S − 1)J = (S + 2 − 2/n)AL, (4) gamma-ray bursts. This ultimately leads to the formation of IMBH   which may accrete materials from immediate environs.   n2 (3n − 2) (3n − 4) + W S2 + S 2(3n − 2) 2 n 2 GCMODELFORAM –σ POWER LAW (5) BH n2 + (3n − 2)2(1 − 4/n)W + = 0, We adopt the same hydrodynamic perspective of Lou & Jiang (2008; (3n − 2) hereafter LJ) for the large-scale spherical GC dynamic evolution. W ≡ n2q/ − n n − GCs are smaller and less massive than typical galactic bulges. The where parameter combination [2(2 )(3 2)]. Once random stellar velocity dispersion and the ‘binary heating’ from the proper roots of S are known, two coefficients J and L are related primordial stellar binaries in the core provide an effective pressure by equation (4); only one is free to choose. The existence of SPS  > against the GC self-gravity. This may justify our fluid formalism for solution (3) and the requirement of (S) 1 constrain the parameter GC cores. In contrast to LJ, we emphatically focus on the reported regime of such quasi-static solution. Oscillatory behaviours can also tentative candidates of IMBHs and the properties of the host GCs. emerge (Lou & Wang 2006; Wang & Lou 2007, 2008), which may Our main goal is to extend the theory of LJ to GCs and examine be relevant to the interesting controversy of gravothermal and post- W = whether IMBHs and properties of GCs can be sensibly fitted with collapse oscillations in GCs. For a sufficiently small 0, we can > data. By data comparisons, our results appear encouraging for such have two roots S 1 from quadratic equation (5). It also happens > < a physical perspective. Our physical model predictions for IMBHs for one root S 1 and the other root S 1. and host GCs can be tested by further observations. Meanwhile, As a physical requirement for sensible similarity solutions of a v α we also compare with SMBHs in their host galactic pseudo-bulges. general polytropic flow, both (x)and (x) approach zero at large → + →+∞ v → This may hint at a general validity of such a self-similar dynamic x. Thus for either x 0 or x , the reduced velocity 0, → + →+∞ evolution in spherical self-gravitating systems. which means at time t, for either r 0 or r the flow → In spherical polar coordinates (r, θ, φ), the non-linear general speed u 0, or at a radius r,whent is either short or long enough, → polytropic hydrodynamic partial differential equations (PDEs) of the radial flow speed u 0. This model describes a self-similar spherical symmetry are PDEs (1)–(4) of LJ with the same notations. dynamic GC evolution towards a quasi-static configuration a long The Poisson equation is automatically satisfied. As the bulk flow of time later and may grossly fit relaxed spherical GCs. “stellar fluid” is slow in GCs, we invoke here the quasi-static self- From the general polytropic EoS (LJ), the effective pressure is similar dynamic solutions of Lou & Wang (2006). We introduce the P = (4π)γ −1Gq+γ −1(3n − 2)q K1−3q/2ργ Mq , transformation in the dimensionless independent variable x: 1/2 and σ L(r, t) = (γ P/ρ) is the local stellar velocity dispersion in a / n / n− α(x) GC. Asymptotically as t →+∞, σ (r, t) eventually becomes r ≡ K1 2t x, u ≡ K1 2t 1v(x),ρ≡ , L πGt 2 / / n q/ q+γ − / n− /n 4 σ (r) = γ 1 2K1 (2 )n 2A( 1) 2r( 1) . (6) Kt2n−4β x K3/2t 3n−2m x (1) L P ≡ ( ) ,M≡ ( ) , 4πG (3n − 2)G To check against the data, we derive the spatial average of velocity dispersion σ in a GC. The GC outer boundary is here taken as with K and n being two constant scaling parameters; here, u, ρ, either radius rc where mass density ρc is indistinguishable from P,andM are radial velocity, mass density, effective pressure and the surrounding or the tidal radius. Within rc ofaGC,thespatial enclosed mass, respectively, while four functions v(x), α(x), β(x), average of stellar velocity dispersion σ L(r)isgivenby and m(x) are, respectively, dimensionless reduced radial velocity,  r 3 c mass density, pressure and enclosed mass of x only. σ = σ (r)4πr2dr = QK1/2 πr3 L Substituting self-similar transformation (1) into non-linear PDEs 4 c 0 ≡ + γ − / − 1+q/2 1/2 (1−n)/2 3nq/4 1/2 (1)–(4) of LJ and defining parameter q 2(n 2) (3n 2), ≡ [3n γ /(4n − 1)](4πGρc) A K .

C 2012 The Authors, MNRAS 422, L28–L32 Monthly Notices of the Royal Astronomical Society C 2012 RAS L30 Y.-Q. Lou and Y. Wu

We invoke a heuristic criterion of forming an IMBH in a GC (LJ). 2 An IMBH mass MBH is given by MBH = rsc /(2G)wherers is its Schwarzschild radius and c is the speed of light. By solution (3) and when the enclosed mass at radius r approaches the condition nAK1/n r(3n−2)/n = rc2/(2G), (3n − 2)G an IMBH forms with the Schwarzschild radius 2 1/n n/(2n−2) r = rs = [(3n − 2)c /(2nAK )] . Only those asymptotic quasi-static GCs with n < 1 can thus form central IMBHs (see fig. 1 of LJ). Consequently, M = c2/ G n − c2/ nA n/(2n−2)K1/(2−2n), BH [ (2 )][(3 2) (2 )] (7) Downloaded from https://academic.oup.com/mnrasl/article/422/1/L28/971090 by guest on 27 September 2021 or equivalently, the explicitly MBH–σ power law     n/(2−2n) 1/(1−n) c2 2nA σ M = ≡ Lσ 1/(1−n), BH 2G (3n − 2)c2 Q Figure 1. Mass of IMBH M / M versus the mean velocity dispersion where the exponent 1/(1 − n) > 3 since 2/3 < n < 1 (LJ). BH σ/(200 kms−1) for the nine GCs (Tables 1 and 2). The solid line is the To validate our quasi-static model for the nine GCs with available 7 −1 3.6251 least-squares fit: MBH = 4.1020 × 10 M(σ/200 km s ) , with three observational data and references summarized in Tables 1 and2,we −3 parameters {n, γ , ρc} being (0.7241, 1.99, 4.40 M pc ). fit a MBH–σ power law in Fig. 1. Applying the least-squares crite- rion to the data of Table 1 (e.g. Meylan & Mayor 1991; Safonova & and rc = 13, 16, 32, 14, 17, 20, 5, 16, 28 pc, respectively, in rough = × 7 σ/ Shastri 2010), we obtain MBH 4.1020 10 M( agreement with the estimated data in orders of magnitude. −1 3.6251 200 km s ) , with parameters {n, γ , ρc} being (0.7241, 1.99, Harris (1996) reported rc of 47 Tuc, NGC 2808, ω Cen, M62, −3 ρ 4.40 M pc ). This c value appears somewhat larger but grossly NGC 6388, NGC 6397 and M15 to be 56, 43, 88, 18, 18, 11 and consistent with the data in rough orders of magnitude (e.g. Meylan 64 pc, while Bahcall & Hausman (1977) reported rc of M80 to be −3 1987). Specifically, ρc ∼ 0.87 M pc for GC 47 Tuc ∼18 pc and Ma et al. (2007) estimated rc of G1 to be ∼81 pc. ρ ∼ −3 (Meylan 1988), c 0.091 M pc for GC NGC 6397 (Meylan We now consider a sample of SMBHs in galactic pseudo-bulges ρ ∼ −3 & Mayor 1991) and c 0.42 M pc for GC G1 (Meylan et al. (e.g. Kormendy & Kennicutt 2004) summarized in Table 3. For the ρ 2001). No published c values for NGC 2808, M80, M62, NGC reasons in Hu (2008), we do not regard galaxy NGC 3227 as con- 6388 and M15 are available. The nine GCs 47 Tuc, NGC 2808, ω taining a pseudo-bulge. In Table 3, σ e is defined by Gebhardt et al. Cen, M80, M62, NGC 6388, NGC 6397, M15 and G1 correspond (2000) and Tremaine et al. (2002) as the luminosity-weighted rms to K = {1.2, 2.0, 7.7, 1.5, 2.1, 3.1, 0.22, 1.9, 5.8} × 1019 cgs unit velocity dispersion within a slit aperture of length 2Re,whereRe is the effective or half-light radius of a galactic bulge. Parameter σ is Table 1. Nine GCs with their reported central IMBHs and mean 8 the rms velocity dispersion within a circular aperture of radius R /8. stellar velocity dispersions σ are tabulated here. e Ferrarese & Merritt (2000) used ‘central stellar velocity dispersion’ 1  0.04 3 −1 σ c. There is a relation σ = σc(8Rap/Re) (Jørgensen, Franx & GC name Other name MBH ( × 10 M) σ (km s ) 8 Kjaergaard 1995), where Rap 2 arcsec (e.g. Davies et al. 1987). . +0.5 ± σ σ NGC 104 47 Tuc 1 0−0.5 11.6 0.8 Actually, the difference between e and c are much smaller than NGC 2808 – 2.7 13.4 their errors (e.g. Hu 2008; 2009). ω ± NGC 5139 Cen 30 4 22.8 A pseudo-bulge border is at radius rc where the mass den- NGC 6093 M80 1.6 12.4 ± 2.5 sity ρ reaches a value ρc indistinguishable from the environs. ± NGC 6266 M62 3.0 14.3 0.4 With the least-squares fit to Table 3 data, we obtain M = +5.7 BH NGC 6388 – 5.7 18.9 ± 3.6 7 −1 3.7680 −2.85 2.4350 × 10 M(σ/200 km s ) , with parameters {n, γ , ρ } ± c NGC 6397 – 0.05 4.5 0.6 being (0.7346, 1.999, 1.8 M pc−3), respectively. This estimated . +0.7 ± NGC 7078 M15 2 5−0.8 14.1 3.2 ρ . +5.0 ± c appears grossly consistent with the data in the order of mag- G1 (M31) Mayal II 18 0−5.0 25.1 1.7 −3 nitudes, e.g. ρc ∼ 0.1 M pc for the pseudo-bulge in the Milky Way (Lopez-Corredoira et al. 2005). No published ρc Table 2. Nine GCs with the reported total GC masses MGC. values for other pseudo-bulges are available. The eight pseudo- bulges NGC 1068, NGC 2787, NGC 3079, NGC 3384, NGC × 6 GC Name MGC ( 10 M) Major relevant references 3393, Circinus, IC 2560 and Milky Way correspond to values of K = {3.4, 5.8, 1.3, 3.5, 5.0, 0.85, 1.4, 2.6} × 1021 cgs unit and 47 Tuc 1.26 Pryor & Meylan (1993) r = 1.3, 1.7, 0.79, 1.3, 1.5, 0.64, 0.82, 1.1 kpc, respectively, in rough NGC 2808 1.46 Servillat et al. (2008) c ω Cen 3.1 Miocchi (2010) agreement with the data in orders of magnitudes, e.g. Veilleux ∼ M80 1.0 Pryor & Meylan (1993) et al. (1999) reported rc of NGC 3079 to be 3.6 kpc, while M62 0.63 Pryor & Meylan (1993) Busarello et al. (1996) estimated rc of NGC 3384 to be ∼1 kpc NGC 6388 2.6 Lanzoni et al. (2007) NGC 6397 0.062 Heggie & Giersz (2009) 1 M15 0.44 van den Bosch et al. (2006) The prime distinguishes this approximation for σ 8 from the actual value of σ σ  /σ G1 (M31) 7.37 ± 2.15 Ma et al. (2009) 8 and the ratio 8 8 may depend systematically on the velocity dispersion of a galaxy.

C 2012 The Authors, MNRAS 422, L28–L32 Monthly Notices of the Royal Astronomical Society C 2012 RAS Forming IMBHs in globular clusters L31

Table 3. A sample of SMBHs in pseudo-bulges of galaxies.

8 −1 −1 Galaxy MBH ( × 10 M) σ e (km s ) σ c (km s )

NGC 1068a 0.15 165 ± 17 165 ± 17 b . +0.04 NGC 2787 0 41−0.05 210 210 . +0.025 ± ± NGC 3079 0 025−0.013 146 15 146 15 c . +0.01 NGC 3384 0 16−0.02 160 160 NGC 3393 0.31 ± 0.02 184 ± 18 184 ± 18 Circinus 0.011 ± 0.002 75 ± 20 75 ± 20 IC 2560 0.029 ± 0.006 137 ± 14 137 ± 14 Milky Wayd 0.036 ± 0.003 132.5 132.5 Note. As in Hu (2008), NGC 3227 may not possess a pseudo-bulge.

aThe SMBH mass has been updated by Das et al. (2007). Downloaded from https://academic.oup.com/mnrasl/article/422/1/L28/971090 by guest on 27 September 2021 bThe stellar velocity dispersion is taken from Sarzi et al. (2001). cThe stellar velocity dispersion is from Busarello et al. (1996). dThe SMBH mass has been updated by Falcke, Markoff & Bower (2009) and the stellar velocity dispersions are from Dehnen et al. (2006). Figure 3. Masses of central IMBH MBH (in M) versus GC masses (in M) of the nine GCs. The straight line is the least-squares fit to the data in −4 1.2128 a log–log display by MBH/M = 1.25 × 10 (MGC/M) , with n = 0.7252.

Table 4. A sample of SMBHs in galactic pseudo-bulges.

Galaxies log MBH ( +, −)logMs,B−V

NGC 1068 7.18 10.36 NGC 3079 6.40 (0.30, 0.30) 10.19 NGC 3393 7.49 (0.03, 0.03) 10.57 Circinus 6.04 (0.07, 0.09) 9.57 IC 2560 6.46 (0.08, 0.10) 10.26 Milky Way 6.56 10.11

Note. Here, log MBH ( +, −) is the logarithm of the mass and 1σ error of the SMBH and log Ms,B−V is the logarithm of the host pseudo-bulge stellar mass inferred from the K-band mass-to-light ratio M/L derived from B − V colours. The SMBH mass and the bulge mass of NGC 1068 are from Das / σ/σ σ = Figure 2. Mass MBH M versus the mean velocity dispersion 0 ( 0 et al. (2007). The SMBH mass of Milky Way is from Falcke −1 200 km s ) for galactic pseudo-bulges and nine GCs together in a log–log et al. (2009). plot. The line is the least-squares fit to the combined data of galactic pseudo- bulges and nine GCs by log (MBH/M) = 7.3529 + 3.4125log (σ/σ0), with and MGC are related by the power law n = 0.7070. This fit may reveal the underlying mechanism(s)    1/3 (3n−2)/(2−2n) 4πρ n 2G M = c M 1/(3−3n). BH n − c2 GC and Cavichia et al. (2011) estimated rc of Milky Way to be 3 2 ∼1.4 kpc. In Fig. 3, we show the IMBH mass M versus the GC mass Fig. 2 shows the correlation between MBH and the mean velocity BH dispersion σ of the host for samples of galactic pseudo-bulges and MGC. By the least-squares fit to these data, we obtain −4 1.2128 of nine GCs. The joint least-squares fit is MBH/ M = 1.25 × 10 (MGC/M) , −1 = ρ −3 log(MBH/M) = 7.3529 + 3.4125 log(σ/200 km s ), with n 0.7252. The c value is 4.93 M pc , grossly consistent with the data in orders of magnitudes. Note that the n value of with n = 0.7070. The n value of M –σ power law for nine GCs BH MBH–σ power law for the nine GCs is 0.7241. These two n values alone is 0.7241. These two n values are close. It seems thus that the are fairly close, indicating that our model may consistently explain M –σ power law may actually extend down to GCs. BH both the MBH–σ and MBH–MGC power laws for the nine GCs. InTable 4, we list the central SMBH masses and the stellar masses of galactic pseudo-bulges (Hu 2009). As noted, NGC 3227 is not 3 RELATION FOR MBH–MGC POWER LAW taken as a galaxy having a pseudo-bulge. Moreover, NGC 2787 and

By our physical model analysis, the total mass MGC of a GC is NGC 3384 are two galaxies with composite structures consisting   of both pseudo-bulges and small inner classical bulges (e.g. Erwin 3n/2 n πρ (2−3n)/2 A 2008); we do not treat them as pseudo-bulges. We include the Milky (4 c) 3/2 MGC = K . = ± × 6 (3n − 2) G Way as having a pseudo-bulge with MBH (3.6 0.3) 10 M (e.g. Falcke et al. 2009) and stellar bulge mass Ms = (1.3 ± 0.5) × 10 By equation (3), a smaller ρc corresponds to a larger rc and thus a 10 M (e.g. Dwek et al. 1995). In Table 4, Ms,B−V is the pseudo- larger MGC with 2/3 < n < 1. By relation (7), the two masses MBH bulge stellar mass estimated by K-band mass-to-light ratio M/L

C 2012 The Authors, MNRAS 422, L28–L32 Monthly Notices of the Royal Astronomical Society C 2012 RAS L32 Y.-Q. Lou and Y. Wu derived from B − V colour. We do not use the pseudo-bulge stellar Das V., Crenshaw D. M., Kraemer S. B., 2007, ApJ, 656, 699 mass calculated by K-band mass-to-light ratio M/L derived from Davies R. L., Burstein D., Dressler A., Faber S. M., Lynden-Bell D., r − i colour, because IC 2560, Circinus and NGC 3393 lack such Terlevich R. J., Wegner G., 1987, ApJS, 64, 581 data. Adopting the least-squares criterion to the data in Tables 4, we Dehnen W., McLaughlin D. E., Sachania J., 2006, MNRAS, 369, 1688 obtain Djorgovski S. G., King I. R., 1986, ApJ, 305, L61 Dwek E. et al., 1995, ApJ, 445, 716 −8 1.3813 MBH/M = 4.2825 × 10 (Mbulge/M) , Erwin P., 2008, in Bureau M., Athanassoula A., Barbuy B., eds, Proc. IAU Symp. 245, Formation and Evolution of Galaxy Bulges. Cambridge = ρ −3 ρ with n 0.7587. The c value is 0.08 M pc . This estimated c Univ. Press, Cambridge, p. 113 value appears grossly consistent with the data in orders of magni- Falcke H., Markoff S., Bower G. C., 2009, A&A, 496, 77 tudes (see Section 2). Farrell S. A., Webb N. A., Barret D., Godet O., Rodrigues J. M., 2009, Nat, 460, 73 Ferrarese L., Merritt D., 2000, ApJ, 539, L9 4 CONCLUSIONS AND DISCUSSION Gebhardt K., Pryor C., O’Connell R. D., Williams T. B., Hesser J. E., 2000, Downloaded from https://academic.oup.com/mnrasl/article/422/1/L28/971090 by guest on 27 September 2021 In summary, for a self-similar quasi-static spherical hydro- AJ, 119, 1268 dynamic evolution of a general polytropic GC (LJ), our Gebhardt K., Rich R. M., Ho L. C., 2002, ApJ, 578, L41 Grindlay J., Gursky H., 1976, ApJ, 205, L131 model analysis shows M = Lσ 1/(1−n) and M = BH BH Harris W. E., 1996, AJ, 112, 1487 n − / πρ n (3n−2)/(6n−6) c2/ G (3n−2)/(2n−2)M 1/(3−3n) [(3 2) (4 c )] [ (2 )] GC , Heggie D. C., Giersz M., 2009, MNRAS, 397, L46 / < < with 2 3 n 1. They agree grossly with the available data. Hu J., 2008, MNRAS, 386, 2242 First, we have the exponent 1/(1 − n) > 3. Secondly, n is inde- Hu J., 2009, MNRAS, arXiv:0908.2028 pendent of uncertainties in estimates of rc and ρc. Thirdly, the tight Hu R. Y., Lou Y.-Q., 2009, MNRAS, 396, 878 correlation supports a causal connection between the formation and Hurley J. R., Aarseth S. J., Shara M. M., 2007, ApJ, 665, 707 evolution of an IMBH and the dynamics of host GC. Finally, while Jørgensen I., Franx M., Kjaergaard P., 1995, MNRAS, 276, 1341 forming an IMBH at the GC centre, the spherical general polytropic Kormendy J., Kennicutt R. C., 2004, ARA&A, 42, 603 GC relaxes in a self-similar quasi-static phase for a fairly long lapse. Lanzoni B., Dalessandro E., Ferraro F. R., Miocchi P., Valenti E., Rood Index n holds the key in our self-similar quasi-static GC model. R. T., 2007, ApJ, 668, L139 Lopez-Corredoira M., Cabrera-Lavers A., Gerhard O. E., 2005, A&A, 439, As ρ = AK1/nr−2/n/(4πG)andM = Lσ 1/(1−n), the smaller the BH 107 value of n is, the steeper the density profile becomes and the smaller Lou Y.-Q., Jiang Y. F., 2008, MNRAS, 391, L44 (LJ) / − > σ the exponent 1 (1 n) 3oftheMBH– relation is. When density Lou Y.-Q., Lian B., 2012, MNRAS, 420, 2147 profile is less steep, there are more materials around the centre and Lou Y.-Q., Wang W. G., 2006, MNRAS, 372, 885 the accretion is more effective. Then it would be more effective Lou Y.-Q., Wang W. G., 2007, MNRAS, 378, L54 to form IMBHs given that IMBHs are formed by the collapse of Lou Y.-Q., Jiang Y. F., Jin C. C., 2008, MNRAS, 386, 835 mostly stars towards the centre. The outcome is that for a certain σ , Ma J. et al., 2007, MNRAS, 376, 1621 the smaller the mass of an initially formed IMBH is, the steeper the Ma J. et al., 2009, RAA, 9, 641 density profile is and the smaller the value of n is . Maccarone T. J., Kundu A., Zepf S. E., Rhode K. 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