Wandering of the central black hole in a galactic nucleus and correlation of the black hole mass with the bulge mass
aa r X i v : . [ a s t r o - ph . H E ] F e b Wandering of the central black hole in a galacticnucleus and correlation of the black hole masswith the bulge mass
Hajime I
NOUE Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan ∗ E-mail: [email protected]
Received ; Accepted
Abstract
We investigate a mechanism for a super-massive black hole at the center of a galaxy to wanderin the nucleus region. A situation is supposed in which the central black hole tends to moveby the gravitational attractions from the nearby molecular clouds in a nuclear bulge but isbraked via the dynamical frictions by the ambient stars there. We estimate the approximatekinetic energy of the black hole in an equilibrium between the energy gain rate through thegravitational attractions and the energy loss rate through the dynamical frictions, in a nuclearbulge composed of a nuclear stellar disk and a nuclear stellar cluster as observed from ourGalaxy. The wandering distance of the black hole in the gravitational potential of the nuclearbulge is evaluated to get as large as several 10 pc, when the black hole mass is relatively small.The distance, however, shrinks as the black hole mass increases and the equilibrium solutionbetween the energy gain and loss disappears when the black hole mass exceeds an upperlimit. As a result, we can expect the following scenario for the evolution of the black hole mass:When the black hole mass is smaller than the upper limit, mass accretion of the interstellarmatter in the circum-nuclear region, causing the AGN activities, makes the black hole masslarger. However, when the mass gets to the upper limit, the black hole loses the balancingforce against the dynamical friction and starts spiraling downward to the gravity center. Fromsimple parameter scaling, the upper mass limit of the black hole is found to be proportional to he bulge mass and this could explain the observed correlation of the black hole mass with thebulge mass. Key words: accretion, accretion disk - galaxies: active - galaxies: nuclei
One of the most remarkable findings in astrophysics in the past a few decades should be the tightcorrelation of the central black hole mass to the velocity dispersion or mass of the bulge componentin the host galaxy (e.g. Ferrarese & Merritt 2000; Gebhardt et al. 2000; Merritt & Ferrarese 2001;Marconi & Hunt 2003). This discovery has motivated various theoretical and observational studies,mainly focusing on the co-evolution of the black hole with the ambient galactic bulge through severalfeedback processes (see Fabian 2012; Kormendy & Ho 2013; Heckman & Best 2014, for the review,and references therein).These studies are based on a general consensus that activities of active galactic nuclei (AGN)are results of mass accretion on to the central black hole and that the accreted matter is supplied fromthe outside of the nucleus, which is triggered by such activities as star bursts, galactic mergers and soon, on the host galaxy side.However, there still remains an important and difficult issue, how the accreted matter throwsaway its angular momentum and finally gets to the black hole at the center: The processes for matterto inflow from a region with a ∼ kpc distance to a region with a 10 ∼
100 pc distance have been studiedfairly well both observationally and theoretically, but the final path from the 10 - 100 pc region to anaccretion-disk region with a sub-pc distance to the black hole is still far being understood (see e.g.Jogee 2006; Alexander & Hickox 2012, for the review and references therein).Recently, Inoue (2020) argues that several observational properties of AGNs could well be ex-plained by considering a mass accretion caused by the passage of a black hole through the interstellarspace in a circum-nuclear region. If a black hole with the mass, M bh , passes with relative velocity, v ,through the interstellar space with number density, n is , the mass accretion rate to the black hole, ˙ M ,is approximately given as ˙ M ≃ π ( GM bh ) n is m p v , (1)where G and m p are the gravitational constant and the proton mass, respectively (e.g. Davidson &Ostriker 1973). Then, the X-ray luminosity, L X is estimated as2 X ≃ η ˙ M c ≃ × (cid:18) η . (cid:19) (cid:18) v . cm (cid:19) − (cid:18) n is cm − (cid:19) M bh M ⊙ ! erg s − , (2)where η is the energy conversion efficiency of the accretion onto the black hole. If we consider acase in which a black hole with the mass of 10 M ⊙ moves in such a circum-nuclear region as thenuclear stellar disk of our Galaxy (Launhardt, Zylka & Mezger, 2002), we can adopt n is ≃ cm − and v ≃ . cm s − as the typical values. We further assume η ≃ . taking account of the lowradiative efficiency of the accretion disks in AGNs (see e.g. Inoue 2020). L X estimated with thoseparameters roughly agree to the observed values. Then, Inoue (2020) further points out a possibilitythat the central black hole might wander around the galactic center and sometimes get close to thecircum-nuclear, 10 - 100 pc region in which dense interstellar matter could exist, referring to the recentobservational report by Combes et al. (2019) that the AGN positions were frequently off-centered byseveral tens pc in the circumnuclear structures.In this paper, we study a mechanism for the central black hole to move around the nuclearcenter and investigate how the black hole mass evolves there. We propose a mechanism for the blackhole to move and estimate how far it can go out from the center in section 2. As a result, it is shownthat the central black hole could be possible to come to the 10 - 100 pc region. Finally, we discuss ifthe observed correlation between the black hole mass and the bulge mass could be explained in thescope of this scenario, in section 3. Wandering of the central massive object in a large gravitationally bound system has already beenstudied for a case of a cD galaxy in a cluster of galaxies by Inoue (2014), where a situation is consid-ered in which member galaxies come closest to the cluster center in turn and cause the central objectto be pulled to random directions one after another. The encounter between the central object andthe innermost galaxy induces a movement of the central object, trying to establish the equipartitionof their kinetic energies between them. If the central object starts moving, however, the ambient darkmatter particles brake the moving through the dynamical friction. Then, Inoue (2014) approximatelycalculated the energy gain rate of the central object from the innermost galaxy passing by the centralobject and the energy loss rate through the dynamical friction to the diffuse dark matter, and obtainedthe wandering velocity of the central object by balancing the energy gain rate and the energy loss rate.Adopting a different situation to that of Inoue (2014), we now consider the case that a super-massive black hole stays at the center of a galactic bulge which consists of a large number of stars,loading most of the bulge mass, and interstellar gas including a certain number of molecular clouds.3he study by Inoue (2014) can be applied to the present case, by replacing the central cD galaxy,member galaxies and diffuse dark matter, in a cluster of galaxy, with a central black hole, molecularclouds and field stars, in a bulge, respectively.
Following the case of our Galaxy studied by Launhardt, Zylka & Mezger (2002), a nuclear bulge issupposed to surround a super-massive black hole with a mass, M bh , and to consist of a nuclear stellardisk (NSD) and a nuclear stellar cluster (NSC). Although the NSD should have a spheroidal shape, weapproximate its structure to be spherically symmetric with a radius, R d and the total mass, M d , t , forsimplicity. The density in the NSD, ρ d , is assumed to be constant, according to the phenomenologicalresult on the density distribution of the inner part of the Galactic NSD (Launhardt, Zylka & Mezger,2002). Thus, we have the relation as M d , t = 4 π ρ d R . (3)The NSC is also assumed to have a spherical structure with radius, R c , and total mass, M c , t . Wefurther assume M d , t ≃ M c , t , (4)and R d ≃ R c , (5)following the case of the Galaxy again.Several percent of the NSD mass is the interstellar matter and its significant fraction could bein dense, clumpy regions called as molecular clouds. The molecular clouds in the NSD are assumedto have a mass spectrum as dNdM mc = N u M mc , u M mc M mc , u ! − γ for M mc ≤ M mc , u (6)where M mc is the mass of a molecular cloud, N ( M mc ) is the number of the molecular clouds with themass ≤ M mc in the NSD, M mc , u is the upper mass limit and N u is the normalization number. Fromthis equation, we get the total mass of the molecular clouds, M mc , t , in the NSD as M mc , t = 12 − γ N u M mc , u , (7)when γ < . The mass spectral index, γ , of the several nearby galaxies including the galactic centerregion of our Galaxy are observed to be ∼ ∼ Attractions of the molecular clouds to the black hole are approximately estimated in appendix 1. Fromthe result in equation (A5), the energy flow rate to the black hole with wandering velocity, u , from thenearest molecular clouds with a mass spectrum in equation (6) and velocity, v , can be expressed as dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) mc − bh ≃ αM bh G R v Z M mc , u M mc , e M dNdM mc dM mc , ≃ α − γ − γ M bh G M mc , t M mc , u R v , (8)Here, r ( i )1 in equation (A5) has been replaced with the average radius, r , of the spherical volumein which only one molecular cloud in a minute mass range between M mc and M mc + ∆ M exists onaverage and it has been approximated by an equation as r ≃ R ( dN/dM mc )∆ M . (9) α represents the term of β / (2 β ) in equation (A5). As mentioned in appendix 1, β is considered tobe less than unity and hence it makes α larger than unity. α should also include the deviation factorof the r value in the spherically symmetric case from the real spheroidal case, and it could be largerthan unity too. Taking account of these factors, α is likely to be significantly larger than unity. Thelower limit mass, M mc , e , in the integration is the mass of the molecular cloud in the equipartition ofthe kinetic energy with the black hole, and is defined with the following equation as, M mc , e v M bh u . (10)Molecular clouds with the mass larger than M mc , e only contribute to the energy transfer from themolecular clouds to the black hole. Considering v ≫ u in the present case, we practically set5 mc , e ∼ If the black hole moves with velocity, u , against the field stars in the NSD, it should get the dynamicalfriction force, F df , calculated as F df = 4 π ( GM bh ) ρ d ln Λ u A ( X ) , (11)where A ( X ) = erf(X) − √ π e − X (12)(Binney & Tremaine, 2008). Here, X is defined as X = u/ ( √ σ ) , (13)and σ is the velocity dispersion of the stars. According to Binney & Tremaine (2008), the factor, Λ ,in the Coulomb logarithm is roughly given to be ( M d , t /M bh )( r bh /R d ) , where r bh is the orbital radiusof the black hole around the NSD center. When M bh ≃ − M d , t and r bh is of the order of . R d , ln Λ is around 3. From this equation, the energy transfer rate from the black hole to the ambient starsthrough the dynamical friction, dE/dt | bh − st is dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bh − st ≃ F df u ≃ π ( GM bh ) ρ d ln Λ √ σ A ( X ) X . (14)
The equation for the balance between the energy gain rate and the energy loss rate of the central blackhole is given as dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) mc − bh = dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bh − st , (15)and this yields the following relation with helps of equations (8), (14) and (3) as √ α σv − γ − γ M mc , u M mc , t M mc , t M d , t ! ≃ M bh M d , t A ( X ) X . (16)The value of X is calculated in appendix 2 as functions of the normalized distance of the blackhole to the nucleus center, x , defined in equation (A7) and the relative mass of the black hole to thetotal nuclear bulge mass, m bh , defined as m bh = M bh M n , t ≃ M bh M d , t , (17)6here M n , t is the total nuclear bulge mass given by M n , t = M c , t + M d t . We can now see the rightside of equation (16) as functions of x and m bh and rewrite it as B = C ( x, m bh ) , (18)where B = √ α σv ! − γ − γ M mc , u M mc , t ! M mc , t M d , t ! , (19) C ( x, m bh ) = m bh A ( X ) X , (20)and X = X ( x, m bh ) . If m bh is given, we can determine the equilibrium position of the black holefrom this equation.Figure 1 shows the C-values as a function of x for six given m bh values from − . to − .The line of B = 1 × − is drawn in the figure to show a case in which the C curve for m bh = 10 − is tangential to the B line at the position with the open circle. We see from this figure that when m bh is − . , − and − . , the equilibrium position of the black hole gets ∼ . , ∼ . and ∼ . respectively. Since the typical size of the NSD, R d , could be ∼ pc, we can say that the black holecould wander in the NSD region at a distance of several 10 pc from the nucleus center in these cases. If the above scenario really takes place, we can expect that a black hole with an appropriate masswanders in the NSD region. Then, since the NSD region contains a significant amount of interstellarmatter, matter accretion on to the black hole is expected to happen during the passage of the black holethrough the interstellar space and this makes the black hole mass larger. As seen from figure 1, theposition of the black hole moves inward as the black hole mass increases, and eventually gets to theposition where the curve C touches to the line B. Since mass accretion continues even at this position,the black hole mass further increases and the black hole goes into a situation in which the B and Clines do not cross each other any more. Now that the dynamical friction is always dominant to theattractions from the molecular clouds, it continuously forces the black hole in-fall. Since the rotationalvelocity of the black hole increases as the radius decreases as seen in figure 2, the dynamical frictiongets stronger and stronger, associated with the in-fall in this situation. Furthermore, the stellar densityof the NSC is much higher than that of the NSD and thus the dynamical friction could be muchstrengthened after the black hole enters the NSC region. As a result, the black hole is expected tospiral downward to the nucleus center in a fairly short time after it reaches to the tangential pointbetween the B and C lines. 7 ig. 1.
Examples of a constant B line and C curves, which determine the equilibrium position of the black hole between the gravitational attractions fromthe nearby molecular clouds and the dynamical friction from the ambient field stars in the nuclear stellar disk. The straight line corresponds to the case of B = 1 × − , and the curves represent the C -values as functions of x in six cases of m bh . The cross point between the B -line and the C -curve exhibits theequilibrium position of the black hole for given B and m bh values. The arrow shows the movement of the equilibrium point associated with the mass increaseof the black hole. The open circle at the position where the B line and the C -curve touches with each other indicates the upper limit mass of the black holefor the given B value. The above arguments indicate that a black hole with a relatively small mass increases its massthrough mass accretion caused by wandering in the NSD, but that no significant mass increase occursafter it reaches the position where the B and C lines touches with each other. Thus, the presence ofan upper mass limit of the black hole is predicted from the present scenario.It should be noted here that the key parameters introduced in the above arguments are severalratios of pairs of masses or radii. They are M d , t - M c , t and R d - R c for the nuclear bulge, M mc , u - M mc , t for the mass spectrum of the molecular clouds, M mc , t - M d , t for the mass fraction of themolecular clouds in the NSD, and M bh - M n , t for the mass ratio of the black hole to the nuclearbulge. Hence, if these ratios are universal to every galaxies, the observed proportionality between theblack hole mass and the bulge mass which is probably proportional to the nuclear bulge mass can beexplained with this scenario.In order for the tangential point between the B and C lines to appear, the shape of the C curveshould be convex downward and it needs the presence of the NSC having the appropriate mass andradius relative to those of the NSD. As seen in figure 2, the profile of u has the minimum where x isclose to 0.1 and this makes the C-curve convex downward. The gravity by the NSD matter within x is8ominant to that of the NSC in the region far from the NSC and u is proportional to x . However, thegravity from the NSC becomes dominant in the region near the NSC and u tends to be proportionalto x − . This situation causes the appearance of the minimum in u . Although the ratios in equations(4) and (5) are based on the observations of the Galaxy (Launhardt, Zylka & Mezger, 2002) and thoseinformations from other galaxies are poor yet, Ferrarese et al. (2006) reports that the centers of mostgalaxies are occupied by the NSCs, and suggests the tight correlation of the NSC masses with themasses of the host galaxies.The ratio between M mc , u and M mc , t appears, together with the index, γ , of the mass spectrumof the molecular clouds, in equation (19) to determine the B value. M mc , u and N u are estimated tobe roughly . M ⊙ and . from the mass spectrum of the molecular clouds in the Galactic centerregion obtained by Miyazaki & Tsuboi (2002). From these values and equation (7) adopting γ ∼ . ,we get M mc , t ∼ . M ⊙ and an estimation for the second parenthesis in equation (19) of − γ − γ M mc , u M mc , t ∼ − . . (21)The ratio of M mc , t to M d , t is also included in the estimation of B . Since M d , t is reported to be ∼ M ⊙ in the case of the Galaxy (Launhardt, Zylka & Mezger, 2002), we have M mc , t M d , t ∼ − . . (22)The molecular gas content per unit stellar mass is shown to be fairly constant for all late-type galaxiesof type Sa to Im and is about 10% (Boselli et al. 2014). Here, it is set to be about 2%, consideringthat our concern is on the nuclear bulge and the molecular clouds. We further need to set the valueof α in the right side of equation (19) to identify the value of B ; it could be as large as ten or so asdiscussed in subsection 2.2. Hence, we assume α ∼ and then have roughly √ α σv ∼ . , (23)considering σ ∼ v and ln Λ ∼ . Inserting the parameter values evaluated above, B is approximatelyestimated as B ∼ − . . (24)As seen from figure 1, the upper limit of m bh is slightly lower than ∼ − . for B ≃ − . .Thus, the present scenario predicts that the ratio of the upper mass limit of the black hole to thebulge mass could be a little less than − . , if the mass of the nuclear bulge is about 10% of that ofthe galactic bulge. This is smaller than the observed mass ratio ∼ − (Merritt & Ferrarese 2001;Maeconi & Hunt 2003) by about a factor of 5 or so, but the difference could be within the ambiguitiesin the order estimations of the present study. 9 Summary
The starting point of the study in this paper is the recent argument by Inoue (2020) that a passage ofthe super-massive black hole through interstellar matter in a circum-nuclear region could be an originof AGN activities. Even if this hypothesis is correct, however, a serious question arises: How can theinterstellar matte in the circum-nuclear region get close to the black hole at the nucleus center? Nodirect link has yet been found observationally or theoretically between the nuclear region with the 10 ∼
100 pc distance and the accretion-disk region with a sub-pc distance.Reversing the way of thinking, here, we have investigated a possibility that the black holewanders around the nucleus center and comes close to the outer nuclear region, rather than the matterin the circum-nuclear region going down to the center. A situation has been considered in whicha certain number of molecular clouds move around in a nuclear bulge and the central black hole ispulled by nearby molecular clouds. We have estimated the balance between the energy transfer ratefrom the molecular clouds to the black hole through the gravitational acceleration from the nearbymolecular clouds and that from the black hole to the ambient stars through the dynamical frictionfrom the ambient stars in the nuclear bulge. Then, it is found that the black hole can have enoughkinetic energy to wander in a region with a radius as large as several 10 pc, when the black hole massis relatively small. At the same time, it is predicted that a black hole with the relatively small massincreases its mass through mass accretion but loses its stable orbit when its mass exceeds the criticalmass. The studies carried out in this paper include several bold assumptions, approximations andsimplifications that should be checked more precisely. In spite of this, the proposed scenario of awandering black hole in the circum-nuclear region at a distance of several 10 pc is very attractive inthe following two ways.One is that it is possible to give an answer to the difficult issue, which has long been studied invarious ways but is yet unresolved, of how the matter in the 10 - 100 pc region can reach the vicinityof the black hole and cause AGN activities. Studies have currently been based on the thought that thematter in the 10 - 100 pc region should be flowing further inward to the sub-pc region of the accretiondisk and have been faced with the difficult problems of the barriers of the angular momentum or thestar formation. The present scenario does not need to break those hard barriers.The other is that it is possible to provide the simple interpretation of another difficult issue;the correlation of the black hole mass with the ambient bulge mass. With this interpretation, thecorrelation between the two masses can be understood as a mere result of the whole accretion historiesof the black hole, irrespectively of what has been happening in interactions between the black hole10nd the host galaxy, such as co-evolution or feedback.The present scenario, which provides possible and fairly simple interpretations of these diffi-cult issues, should be worthy of further study both observationally and theoretically.
Acknowledgments
The previous version of this manuscript was submitted to PASJ once but withdrawn since the author could not respond to the critical comment fromthe referee in a timely manner. The present version is the result of reconsidering the point from the referee deeply. The author is very grateful to theprevious referee for the useful comment. The referee to the present version is also appreciated for the helpful comments.
Appendix 1 Energy transfer rate from the molecular clouds to the black hole
We consider a situation that molecular clouds in a nuclear bulge consist of I species having differentmasses, M ( i )mc , and number in the nuclear bulge, N ( i ) , ( i = 1 , ..., I ), from one another.Defining r ( i )1 as the radius of a sphere in which only one i -th the molecular cloud exists onaverage, we further introduce an assumption that all the gravitational forces from the i -th molecularclouds other than that from the nearest one are completely canceled by one another and that the blackhole receives only a force from the nearest one. The force per mass, f ( i ) , can be approximated as f ( i ) ≃ GM ( i )mc β ( r ( i )1 ) , (A1)where β is the averaging factor of the distance between the nearest molecular cloud and the blackhole during the passage of one particular nearest cloud and should be less than unity. One among themolecular clouds in the i -th specie enters the spherical region with the radius r ( i )1 one after another,and the average passage time, ∆ t ( i ) , can be expressed as ∆ t ( i ) ≃ β r ( i )1 v , (A2)where β is another averaging factor over various passing orbits, of the order of 1.Let us introduce a three-dimensional diagonal-coordinate ( x , y , z ), and express the three axiscomponents of the specific gravitational force which the black hole gets from the i -th molecular cloudsas f ( i ) k ( k = x , y , z ). Then, the velocity of the black hole, u , at time, t , can be calculated as u k = Z t I X i =1 f ( i ) k dt ′ ≃ I X i =1 J ( i ) X j =1 ( f ( i ) k ∆ t ( i ) ) j , (A3)for k = x , y , z . Here, j represents the sequence number of the successive replacements of the closestmolecular cloud to the black hole, and its total number in t is J ( i ) = int [ t/ ∆ t ( i ) ] . Since the direction11f the velocity increment during each ∆ t ( i ) should be random, the average of f ( i ) k ∆ t ( i ) over a largenumber of J ( i ) should be zero, and thus the average of u k over a sufficiently long time of t shouldbecome zero for all x , y and z in k . The average of u k , u k , is, on the other hand, calculated as u k ≃ I X i =1 J ( i ) X j =1 ( f ( i ) k ∆ t ( i ) ) j ≃ I X i =1 ( f ( i ) k ∆ t ( i ) ) , (A4)since all the cross terms should become zero on average. From equation (A4) and with the help ofequations (A1) and (A2), we get the energy transfer rate from all the molecular clouds to the blackhole, dE/dt | mc − bh , as dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) mc − bh ≃ M bh du dt + du dt + du dt ! ≃ M bh I X i =1 ( f ( i ) ∆ t ( i ) ) ∆ t ( i ) ≃ M bh I X i =1 β ( GM ( i )mc ) β ( r ( i )1 ) v . (A5) Appendix 2 Kinematics of the black hole and the ambient stars in the NSD
A.2.1 Black hole velocity
We assume that the black hole circularly rotates around the gravity center of the NSD and black holesystem. Under the environmental conditions introduced in subsection 2.1, the mass M n of the NSD +NSC matter within a distance, r ( > R c ), from the NSD center is given as M n = 4 πρ d r − R ) + M c , t = M d , t ( x − x ) + M c , t , (A6)where x = rR d , (A7)and x c = R c R d . (A8)Then, the Keplerian circular velocity of the black hole, u k , is calculated as a function of x by thefollowing equation as u ≃ D r GM n r D GM d , t R d " x + ( M c , t /M d , t ) − x x , (A9)where D is defined as D = M n M n + M bh . (A10)Since M n is a function of x , D is also a function of x . This factor, D , comes from a simple expectationthat the black hole tends to rotate around the gravity center between the NSD within r and the blackhole, although more detailed considerations on the situation of the whole NSD + black hole system isnecessary in practice. A.2.2 Velocity dispersion of the stellar motions
Applying the thermo-dynamics to the stellar system in the NSD, we introduce the pseudo-pressure ofthe stellar motions, P , as P = ρ d w , (A11)where w is the pseudo sound velocity of the stellar system. The dynamical structure could be obtainedby an equation as dPdr = − ρ d GM n r , (A12)and this equation yields the following equation, on the assumption of the constant ρ d , as dw dr = − GM n r . (A13)The solution of this differential equation is, with the help of equation (A6), w = w − GM d , t R d " x − x M c , t M d , t − x ! (cid:18) x c − x (cid:19) , (A14)where w c is w at r = R c . If we set the outer boundary condition as w = 0 at r = R d , w c is calculatedas w = GM d , t R d " − x M c , t M d , t − x ! (cid:18) x c − (cid:19) . (A15)From the above two equations, we get w = GM d , t R d " − x M c , t M d , t − x ! (cid:18) x − (cid:19) . (A16)From the analogy to the thermo-dynamics, w could relate to σ as w = σ . (A17)13 ig. 2. (Top) The square of the expected rotational velocity, u k , of the black hole normalized by GM d , t /R d as a function of x , (middle) the square of thepseudo-sound velocity of the stellar system, w , normalized by GM d , t /R d as a function of x and (bottom) X as a function of x , in a case of M c , t /M d , t =10 − and x c = 10 − . The solid and dashed curves in the top panel respectively correspond to the cases with and without the correction for the movementof the black hole around the gravity center between the black hole and the mass of the nuclear bulge within the black hole distance. A.2.3 X - x relation