J. Astrophys. Astr. (2021) 42:12 Ó Indian Academy of Sciences https://doi.org/10.1007/s12036-021-09732-4Sadhana(0123456789().,-volV)FT3](0123456789().,-volV) LETTER Is the binding energy of galaxies related to their core black hole mass? C. SIVARAM1 and KENATH ARUN2,* 1Indian Institute of Astrophysics, Bangalore 560 034, India. 2Christ Junior College, Bangalore 560 029, India. E-mail: [email protected] MS received 28 February 2020; accepted 4 January 2021 Abstract. Most of the large galaxies host a supermassive black hole, but their origin is still not well understood. In this paper, we look at a possible connection between the gravitational binding energies of large galaxies, etc., and the masses of their central black holes. Using this relation (between the gravitational binding energy of the host structure and the black hole energy), we argue why globular clusters are unlikely to harbour large black holes and why dwarf galaxies, if they have to host black holes, should have observed mass-to-light ratios of *100. Keywords. Binding energy—black holes—dwarf galaxies. It is by now well established that most large galaxies correlations (Silk & Rees 1998). Also, there is no (spiral, elliptical, etc.) host a supermassive black hole understanding as to whether these relations evolve (SMBH) in their centre (Kaper et al. 2001). Again with the evolution of the galaxy (and the central BH) active galactic nuclei, quasars, etc., are powered by or are set at the time of their formation (Di Matteo the gravitational energy of matter accreting onto these et al. 2005; Volonteri & Natarajan 2009). Again, central SMBHs (Melia 2007). These black hole many other large stellar conglomerations such as (BH) masses are typically of several millions of globular clusters or dwarf galaxies also do not seem to solar masses and can be as large as a few billion solar host massive BHs in their core regions. masses, such as in the case of M87. The origin of these Here we propose a possible connection between the SMBHs at the galactic cores is still an enigma. Again gravitational binding energies of large galaxies, etc., the BH masses have been related to the galactic bulge and the masses of their central BHs. The idea is mass, suggesting a common origin (Zel’dovich & similar to what happens in the core collapse of mas- Novikov 1971; Shapiro & Teukolsky 1983). sive stars. The gravitational binding energy released in Various studies have pointed out the correlation stellar collapse is carried away mainly by neutrinos (as between the masses of the SMBHs at the centre of in the case of SN1987A). The total energy carried galaxies and the different characteristics of their host away is just the gravitational binding energy of the galaxies (Kormendy & Richstone 1995), such as the remnant neutron star (Burrows 1990). spheroid luminosity and spheroid mass (Magorrian In the case of SN1987A, the estimated 3 Â 1053 erg et al. 1998), the spheroid velocity dispersion (Gultekin carried away by neutrinos just corresponds to the et al. 2009), and the kinetic energy of random motions binding energy of a 1:4 M neutron star (Bahcall 1989). (Feoli & Mancini 2009). The Kormendy relation More massive stars would release more binding energy, connects the estimated mass of the BH to the K-band and this would correspond to the formation of a BH. The luminosity and the velocity dispersion of the bulge of gravitational binding energy released in the formation the host galaxy measured outside the BH sphere of of a BH would correspond to its rest energy, i.e., to the influence (Kormendy 1977). mass of the BH (for the same reason, a supernova would However, the underlying physics of these various not give rise to a white dwarf as a remnant, as the observations is not yet fully understood. It is thought binding energy of the white dwarf is two or more orders that feedback processes are responsible for such smaller than the energy released in the explosion). 12 Page 2 of 4 J. Astrophys. Astr. (2021) 42:12 By analogy, we argue that whatever processes led The binding energy of the cluster is as follows: to the formation of the BHs at the galaxy cores was GM2 connected with the formation of the galaxy and should ðÞBE ¼ cluster 1051 erg, ð7Þ cluster R thus be related to the gravitational binding energy of cluster the final structure which forms. which is lesser than the binding energy of even neu- We note that in the case of galactic structures, the tron stars. binding energy of the parent galaxy is comparable In the case of some globular clusters such as the with that of the BH hosted by the galaxy. The binding Mayall II (G1) in the Andromeda galaxy, there is 4 energy of a typical galaxy such as the Milky Way is evidence of a 2 Â 10 M BH. There are also studies given by (Meylan et al. 2001) that have suggested that the 2 globular cluster G1 could in fact be the core of a dwarf GMspiral ðÞBE ¼ 1061 erg elliptical galaxy. The BH in G1 could have been spiral R ÀÁspiral formed at an earlier epoch when the cluster was more 12 Mspiral 10 M ; Rspiral 10 kpc : ð1Þ closely packed or due to the merger of few clusters. X-ray and radio emissions from Mayall II appear to be The galaxy harbours a 106 M BH. The energy consistent with an intermediate-mass BH. The energy (mass) of the BH is as follows: 3 2 57 of the 10 M BH is E ¼ MBHc 10 erg. In order 2 2 GMBH 61 for the cluster to have a binding energy of this order, ðÞBE BH¼ MBHc ¼ 10 erg: ð2Þ RBH its size should be The equality of Equations (1) and (2) is suggestive GM2 of what happens in the release of gravitational binding R ¼ cluster 10À4 pc, ð8Þ M c2 energy in stellar collapses leading to the formation of BH 6 a remnant compact object. where Mcluster 10 M . Again for large elliptical galaxies, mass is few As the potential decreases, the kinetic energy of the 14 orders higher, that is Melliptical 10 M , so the cluster increases, hence spreading it out over time. binding energy is as follows: One of the possible models for the formation of an ðÞBE 1063 erg: ð3Þ IMBH is the collapse of a cluster of stars (Miller & elliptical Hamilton 2002). The collapsed core can accrete Thus, galaxies, such as M87 harbour billion solar matter ejected during the formation. If a collection of mass BH, whose binding energy is as follows: a thousand 10 solar mass stars in a volume of a parsec 63 cube collapses, ejecting 30% of its mass, then the total ðÞBE BH 10 erg: ð4Þ 33 mass of the ambient gas, nmP 6 Â 10 kg. If *30% Therefore, we have, in the case of galaxies, the of the mass of each star is ejected, then the accretion 2 2 3 7 binding energy of the galaxy being comparable with rate given by m_ ¼ 16pnmPG M =c 4 Â 10 the BH energy: kg sÀ1. For a constant density, the dynamical friction GM2 (Chandrasekhar 1942) is given as follows: M c2 ¼ galaxy : ð5Þ BH R G2M2q Hence, the fraction of BH mass to the galactic mass Fd ¼ C ; ð9Þ v2 is given by m M GM where G is the gravitational constant, M the mass of BH ¼ galaxy 10À5 À 10À6: ð6Þ M Rc2 the moving object, q the density, and vm the velocity galaxy of the object in the frame in which the surrounding 8 This implies that smaller galaxies of 10 M could matter was initially at rest. C is not a constant but harbour BHs of 100 M . depends on how vm compares with the velocity dis- This is in agreement with the observations of host persion of the surrounding matter. galaxy mass and its central BH mass, as shown in The dynamical friction comes into effect in the Table 1 (Bandara et al. 2009). evolution of the cluster due to the interaction between The result from Equation (6) could also provide a the ambient gas and dust, and the central Intermediate reason as to why globular clusters do not host a BH. Mass Black Hole (IMBH). The initial BH is subjected J. Astrophys. Astr. (2021) 42:12 Page 3 of 4 12 Table 1. Correlation between SMBH mass (MBH) and mass of its host galaxy (Mgalaxy). Galaxy system MBHðÞM MgalaxyðÞM MBH=Mgalaxy SDSS J0029-0055 2:65 Â 108 1:04 Â 1013 2:5 Â 10À5 SDSS J0252?0039 6:48 Â 107 1:27 Â 1013 5:1 Â 10À6 SDSS J0330-0020 1:88 Â 108 1:51 Â 1013 1:2 Â 10À5 SDSS J0728?3835 1:90 Â 108 1:73 Â 1013 1:1 Â 10À5 SDSS J0737?3216 1:28 Â 109 2:41 Â 1013 5:3 Â 10À5 SDSS J0822?2652 4:67 Â 108 2:00 Â 1013 2:3 Â 10À5 SDSS J0955?0101 1:14 Â 108 1:21 Â 1013 9:4 Â 10À6 SDSS J0959?0410 2:25 Â 108 1:08 Â 1013 2:1 Â 10À5 SDSS J1142?1001 2:81 Â 108 1:68 Â 1013 1:7 Â 10À5 SDSS J1403?0006 1:74 Â 108 1:18 Â 1013 1:5 Â 10À5 SDSS J1451-0239 2:21 Â 108 1:17 Â 1013 1:9 Â 10À5 SDSS J1538?5817 1:06 Â 108 1:17 Â 1013 9:1 Â 10À6 SDSS J2238-0754 1:47 Â 108 1:44 Â 1013 1:0 Â 10À5 SDSS J2303?1422 3:83 Â 108 2:57 Â 1013 1:5 Â 10À5 SDSS J2341?0000 1:44 Â 108 1:87 Â 1013 7:7 Â 10À6 to a gravitational force due to the star cluster, which is by about 1 kilo-parsec from the centre.