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.