Detection of Intermediate-Mass Black Holes in Globular Clusters Using Gravitational Lensing
aa r X i v : . [ a s t r o - ph . GA ] D ec Detection of Intermediate-Mass Black Holes in GlobularClusters Using Gravitational Lensing
Takayuki
Tatekawa
Department of Social Design Engineering, National Institute of Technology, Kochi College,200-1 Monobe Otsu, Nankoku, Kochi, 783-8508, JapanResearch Institute for Science and Engineering, Waseda University,3-4-1 Okubo, Shinjuku, Tokyo 169-8555, [email protected] andYuuki
Okamura
Department of Electrical Engineering and Information Science, National Institute of Technology,Kochi College,200-1 Monobe Otsu, Nankoku, Kochi, 783-8508, Japan (Received ; accepted )
Abstract
Recent observations suggest the presence of supermassive black holes at the centersof many galaxies. The existence of intermediate-mass black holes (IMBHs) in globu-lar clusters has also been predicted. We focus on gravitational lensing as a new wayto explore these entities. It is known that the mass distribution of a self-gravitatingsystem such as a globular cluster changes greatly depending on the presence or ab-sence of a central massive object. After considering possible mass distributions for aglobular cluster belonging to the Milky Way galaxy, we estimate that the effect on theseparation angle of gravitational lensing due to an IMBH would be of milliarcsecondorder.
Key words:
Gravitational lensing: IMBH: globular clusters
1. Introduction
The existence of supermassive black holes (SMBHs) at the centers of galaxies has beenmade evident by recent observations. For example, the shadow of the SMBH in the center1f M87 was directly observed by the Event Horizon Telescope (EHT) (The Event HorizonTelescope Collaboration 2019). Long-term observations of the movement of stars surroundingthe center of the Milky Way Galaxy suggest the existence of an invisible massive object, SgrA ∗ , considered to be an SMBH (Ghez et al. 2000; Gillessen et al. 2009). For other galaxies,the existence of SMBHs is indirectly suggested by the M − σ relation between the mass of theSMBH and the velocity dispersion of stars in a galaxy (Silk et al. 1998; Ferrarese and Merritt2000; Gebhardt et al. 2000; G¨ultekin et al. 2009).The question of how SMBHs form has not yet been definitively answered, althoughvarious scenarios have been considered over the years (Rees 1984). One of the scenariosinvolves an intermediate-mass black hole (IMBH) (Greene et al. 2020) that grows to becomean SMBH. On the basis of the M − σ relation, it seems possible that IMBHs may be located atthe centers of globular clusters. This has been discussed particularly in the case of M15 but hasnot been resolved (Gerssen et al. 2002; Baumgardt et al. 2003; McNamara et al. 2003; Kiselevet al. 2008; Murphy, Cohn, Lugger 2011; den Brok et al. 2014), although indirect verificationhas been performed using statistical properties of stars.In this paper, we consider a new method of detecting SMBHs and IMBHs. These celestialobjects’ immense mass bends the trajectory of light from background objects: the gravitationallensing (GL) effect. It can cause background objects to be observed multiple times, and singleimages to be brightened. Through the GL effect, it is possible to verify the existence of amassive object between the background light-source and the Earth.GL by globular clusters has been discussed in the past. Kains et al. (2016,2018) pro-posed a method for IMBH detection using gravitational microlensing. Bukhmastova (2003)has attempted to explain QSO-galaxy associations using the GL effect produced by globularclusters.GL is a small effect even for SMBHs; our treatment includes lensing by both the massiveobject and the surrounding stars. The density profile of the stars in the cluster may be obtainedfrom the equilibrium solution for a gravitational many-body system; it varies greatly dependingon the presence or absence of a massive central object. In this paper, we assume a sphericallysymmetric distribution of stars and analyze the GL effect for various characteristic densityprofiles.The paper is organized as follows. In Section 2, we discuss the mass distribution forglobular clusters. Two models of the distribution are described: 1) a model based on the staticstate of self-gravitating systems and 2) a phenomenological model. If there is an IMBH in aglobular cluster, the surrounding stars are thought to form a cusp. Therefore, the two modelsare also evaluated with such a cusp, so that in all we consider four types of models in this paper.In Section 3, the geodesic equation for light trajectories is discussed. Since the models assumespherical symmetry, the spacetime can be described by the Schwarzschild metric. We considerthe gravitational potential of each model in this metric. If the cusp is distributed across the2ntire area, the mass will diverge. Hence, it is necessary to connect the spacetime inside thecusp smoothly with that of the cuspless model. In Section 4, we compare the GL effect inmass distribution models with/without an IMBH, and estimate the maximum separation anglecaused by GL. The difference in the separation angle is found to be of sub-milliarcsecond orderwhen we assume that the lensing object is a globular cluster in the Milky Way Galaxy. InSection 5, the conclusions of this study are presented.
2. Mass Distribution
In this subsection, we explain mass distributions for globular clusters. For simplicity,we assume a spherically symmetric distribution and consider the equilibrium solution in a self-gravitating system under Newtonian gravity. Although the distribution function in generaldepends on both energy and angular momentum, we assume that the effect of angular momen-tum is small and only the energy dependence need to be considered. In this case, the velocitydispersion is isotropic at each point in space, and the energy for matter is given by the kineticand potential energies.The distribution function tells us both the distribution of the matter in the system andthe velocity at which it moves. The stars in the cluster may be thought of as a fluid obeying anequation of continuity, the collisionless Boltzmann equation, that contains a term dependingon the gradient of a potential. Suppose that the distribution function is a known functionof energy, and that the potential in the collisionless Boltzmann equation is the gravitationalpotential solving the Poisson equation. The mass density is of course proportional to thegradient of this potential. Therefore, it is possible in principle (if not always in practice) tosolve the collisionless Boltzmann equation and Poisson’s equation together to obtain the massdensity. In the case that the distribution function goes as a power of the energy, the combinedequation is called the Lane-Emden equation; the density is proportional to solution of thisequation raised to the power of the polytropic index n . Unfortunately, the coefficient of theproportionality is not a constant, but a generally complicated function. Only in the case of n = 5, the Plummer model (Plummer 1911; Binney and Tremaine 2008), does the Lane-Emdenequation yield a simple expression for the density. ρ P ( r ) = 34 π M tot r ( r + r ) / , (1)where r represents “Plummer length”. M tot is the total mass of the cluster.Another well-known model is that of Hernquist, which realizes de Vaucouleurs’ 1 / ρ H ( r ) = M tot π r r r + r ) . (2)Note that the Hernquist model diverges at r = 0, whereas in the Plummer model, the densitydistribution converges gently at the center.For a globular cluster without an IMBH, we consider the Plummer model and theHernquist model. If a globular cluster includes an IMBH, the exchange of orbital energies causes thedistribution of stars to take a characteristic form. Here we make several assumptions: 1) Thedistribution of stars is represented by a single-particle distribution function; 2) The mass of theIMBH is much smaller than the mass of the globular cluster core; 3) For simplicity, all the starsaround the IMBH have the same mass; 4) The distribution of stars is independent of angularmomentum. Under these assumptions, the static solution for the density distribution of starsby the Fokker-Planck equation obeys a specific power law and is known as the Bahcall-Wolfcusp (Bahcall and Wolf 1976; Merritt 2013). ρ B ( r ) ∝ r − / . (3)The Bahcall-Wolf cusp has been verified by N -body simulation for stellar systems around amassive object (Preto et al. 2004). In contrast to the Plummer models, in a Bahcall-Wolfcusp the mass density diverges at the center. To consider lensing by a Bahcall-Wolf cusp, it isnecessary to consider the mass of the IMBH itself.
3. Geodesic Equations
In this paper, we consider the equilibrium state for globular clusters with sphericalsymmetry. Therefore, spacetime is described by the Schwarzschild metric (Misner, Thorne,Wheeler 1973; Hartle 2003).d s = − (cid:18) c Ψ( r ) (cid:19) d( ct ) + (cid:18) c Ψ( r ) (cid:19) − d r + r (cid:16) d θ + sin θ d φ (cid:17) , (4)where Ψ( r ) corresponds to the Newtonian gravitational potential.From the Schwarzschild metric, geodesic equations may be derived. Hereafter we definethe time component as w ≡ ct . (5)The geodesic equations are as follows:d w d λ = − c dΨd r (cid:18) c Ψ (cid:19) d w d λ d r d λ , (6)4 r d λ = − c dΨd r (cid:18) c Ψ (cid:19) d w d λ ! + 2 c dΨd r (cid:18) c Ψ (cid:19) − d r d λ ! + r (cid:18) c Ψ (cid:19) d ϑ d λ ! + r sin ϑ (cid:18) c Ψ (cid:19) d φ d λ ! , (7)d ϑ d λ = − r d r d λ d ϑ d λ + sin ϑ cos ϑ d φ d λ ! , (8)d φ d λ = − r d r d λ d φ d λ − ϑ sin ϑ d ϑ d λ d φ d λ . (9)Because we consider a null geodesic, we have introduced the parameter λ instead of the worldinterval s . Moreover, because the trajectory of light can be modeled in a plane, when we noticeone trajectory of light, we can ignore the φ components. Hereafter we set ϑ = π/ We consider four models for globular clusters. When there is no IMBH, the densitiesgiven by Eqs. (1) and (2)correspond to the potentialsΨ P ( r ) = − GM tot ( r + r ) / , (10)Ψ H ( r ) = − GM tot r + r , (11)where Ψ P and Ψ H refer to the potentials in the Plummer and Hernquist models, respectively.When the globular cluster includes an IMBH, we set a Bahcall-Wolf cusp at the center. Outsidethe cusp, the effect of the IMBH is tiny. At the boundary of the cusp r = R , we connectsmoothly with the Plummer or Hernquist model. In connecting the models, we first focus onthe continuity of the terms of the geodesic equation. Then we set the first derivative of thepotential to be continuous.For the Plummer model, in the inner region of the model ( r < R ), the potential is thenΨ P+B(in) ( r ) = − GM BH r − GM tot ( R + r ) / + 645 πGR ρ "(cid:18) rR (cid:19) / − , (12)dΨ P+B(in) ( r )d r = GM tot r + 165 πGRρ (cid:18) rR (cid:19) − / . (13)In the outer region of the model ( r > R ), the potential isΨ P+B(out) ( r ) = − GM BH r − GM tot ( r + r ) / , (14)dΨ P+B(out) ( r )d r = GM tot r + GM tot r ( r + r ) / . (15)To satisfy the condition of connection at the boundary ( r = R ), the density parameter ρ must be ρ = 5 M tot π ( R + r ) / . (16)5imilarly, for the Hernquist model, in the inner region( r < R ), the potential isΨ H+B(in) ( r ) = − GM BH r − GM tot R + r + 645 πGR ρ "(cid:18) rR (cid:19) / − , (17)dΨ H+B(in) ( r )d r = GM tot r + 165 πGRρ (cid:18) rR (cid:19) − / . (18)In the outer region of the model ( r > R ), the potential isΨ H+B(out) ( r ) = − GM BH r − GM tot r + r , (19)dΨ H+B(out) ( r )d r = GM tot r + GM tot r ( r + r ) . (20)In the Hernquist model, the density parameter ρ , which ensures that the boundary conditionsmatch at the point where r = R , turns out to be ρ = 5 M tot πR ( R + r ) . (21) For consideration of GL by globular clusters, we must fix the values of some parameters.The mass of the IMBH M BH , radius of the Bahcall-Wolf cusp R , total mass of the lensingobject M tot , and radius of the lensing object r lens should all be considered. Because, in boththe Plummer and Hernquist models, a scale length r is included, we must decide on its valueas well. For the null geodesic, we must consider the distance between the center of the globularcluster and the background object r star ; we must also consider the distance x e from the Earthto the center of the globular cluster.Both the Plummer model and the Hernquist model include a length parameter r (Eqs.(1) and (2)), a characteristic quantity that determines the shape of the density distribution. Inorder to compare the two models, it is desirable to make contained within radius r approxi-mately the same in both. To do this, we use a relation between r and r lens . For the Plummermodel, we set r = r lens / M P ( r lens ) = Z r lens ρ P ( r ) · πr d r = M tot r ( r + ( r lens / ) / ≃ . M tot . (22)For the Hernquist model, we set r = r lens / M H ( r lens ) = Z r lens ρ H ( r ) · πr d r = M tot r ( r lens + r lens / ≃ . M tot . (23)For both models, about 70 percent of total mass is thus within the radius of the lensing object.In this paper, the mass of the IMBH is fixed at M BH = 10 M ⊙ . The upper boundof the distance to the background star is fixed at r star < [lyr]. The globular cluster isassumed to belong to the Milky Way; catalogs of Milky Way globular clusters have beenpublished (Harris 1996; Hilker et al. 2020). The parameters (total mass, calculated radius6rom apparent dimension, and distance from the Earth) of the clusters in the present paper arechosen to be typical of relatively large Milky Way globular clusters according to the catalogs: • × ≤ M tot ≤ × [ M ⊙ ] • ≤ r lens ≤
100 [lyr] • × ≤ x e ≤ × [lyr]Here x e means the distance from the Earth to center of globular cluster. The boundary ofBahcall-Wolf cusp is fixed at R = 1[lyr].
4. Effect of Gravitational Lensing
Various physical and geometrical quantities required for analyzing the GL effect of glob-ular clusters will now be defined. Fig. 1 shows the positional relationship between the Earth,globular cluster, and background star. Even if spacetime is curved by gravity, the trajectoryof light can be analyzed in a plane. The center of the globular cluster is defined as the origin.Then the Earth is set at ( x e , E . A background star ispositioned at point S . Because of the effect of GL, however, this star appears to observerson Earth to be at point S ′ . The angle between the x-axis and line segment ES ′ is defined as θ . Then the deflection angle (angle between line segments ES and ES ′ ) is defined as Θ. Theintersection point of the y-axis and line segment ES ′ is defined as the “impact parameter” y lens .The relationship between the impact parameter and θ istan θ = y lens x e . (24)The distance between the center of the globular cluster and the background star is given by r star . As an example for our study, one well-known globular is selected and verified. OmegaCentauri is the most massive globular cluster of the Milky Way. The existence of an IMBHin Omega Centauri has been discussed from both theoretical and observational perspectives(Noyola et al. 2010; Haggard et al. 2013; Baumgardt et al. 2019). The effect of the GL will becalculated using the known parameters of Omega Centauri: M tot = 4 × M ⊙ ,r lens = 82 [lyr] ,x e =1 . × [lyr]. For this calculation, r star is fixed at 5 × [lyr] (van de Ven et al. 2006; D’Souzaet al. 2013).When the cluster excludes the IMBH, the mass distribution is given by the Plummermodel or the Hernquist model. The deflection angle is shown in Fig. 2. For the case of theHernquist model (hereafter, model H), the deflection angle increases sharply as the impactparameter decreases. Conversely, for the case of the Plummer model (hereafter, model P),7 arth (E)y y lens x e x Apparent position of star (S’ )Background star (S)
Globular cluster θ r star Fig. 1.
Schematic picture of the positional relationship between the Earth, globular cluster, and back-ground star. The center of the globular cluster is defined as the origin, and the trajectory of light fromthe background star to the Earth (solid line) is analyzed. lense [lyr] Model PModel H D e f l e c t i on ang l e Θ [ a r cs e c ] Fig. 2.
The deflection angle by the Plummer model (Model P) and the Hernquist model (Model H) as afunction of impact parameter. the deflection angle approaches a constant value at y lens = 0. This is due to the difference ingravitational potential between models: for model P, the potential flattens at the center, butfor model H, the potential sharpens there.When the cluster includes an IMBH, the deflection angle dramatically changes at smallimpact parameters. For the case of the Plummer model with an IMBH (hereafter, model P-BH),the difference of angle becomes significant at y lens ≤ r ≃
0. For the case of the Hernquist modelwith an IMBH (hereafter, model H-BH), although the effect of the IMBH (or cusp) appears atsmall impact parameters, it is not as clear as in the case of model P-BH (Fig. 4). Perhaps thisresult is caused by H-BH’s less extreme potential slope near the center.8 lense [lyr] Model PModel P-BH D e f l e c t i on ang l e Θ [ a r cs e c ] lense [lyr] Model PModel P-BH D e f l e c t i on ang l e Θ [ a r cs e c ] Fig. 3.
Comparison of deflection angle with and without IMBH in the Plummer model. The figure onthe right is an enlarged view of y lens from the figure on the left. lense [lyr] Model HModel H-BH D e f l e c t i on ang l e Θ [ a r cs e c ] lense [lyr] Model HModel H-BH D e f l e c t i on ang l e Θ [ a r cs e c ] Fig. 4.
Comparison of deflection angle with and without IMBH in the Hernquist model. The figure onthe right is an enlarged view of y lens from the figure on the left. We investigate the effect of changing the parameters on the deflection angle. The stan-dard values of the parameters are as follows: • Total mass of lensing object: M tot = 10 M ⊙ • Radius of lensing object: r lens = 50[lyr] • Distance from the Earth: x e = 5 × [lyr] • Distance between the center of the globular cluster and the background object: r star = 5 × [lyr]The above four parameters are then changed one by one, with the following results:Fig. 5 shows the dependence of the deflection angle on total mass M tot . As M tot increases,the angle increases because gravity strengthens. Fig. 6 shows the dependence of the deflectionangle on the radius of the lensing object r lens . The lensing effect increases as the mass isconcentrated in a narrower area.We also show the dependence on the distance between the globular cluster and the Earth.The angle θ changes as the distance between the Earth and the globular cluster changes. Fig. 7shows the θ dependence of the deflection angle. Since we are considering globular clustersbelonging to the Milky Way, the upper limit of the distance is set to 10 [lyr]. The distance9 lense [lyr]M tot =1 x 10 M tot =3 x 10 M tot =5 x 10 lense [lyr]M tot =1 x 10 M tot =3 x 10 M tot =5 x 10 lense [lyr]M tot =1 x 10 M tot =3 x 10 M tot =5 x 10 lense [lyr]M tot =1 x 10 M tot =3 x 10 M tot =5 x 10 (a) (b)(c) (d) D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] MMM MMMMMMMMM
Fig. 5.
Dependence of the deflection angle on the total mass M tot for (a) Plummer model, (b) Plummermodel with IMBH, (c) Hernquist model, and (d) Hernquist model with IMBH. dependence of the angle therefore seems to be insignificant from the figure.Finally, we consider the dependence on distance to the background object (Fig. 8). Asthe distance to the background object increases, the angle approaches a limiting curve that,except in Model P, rises steeply at low impact parameters. Finding the trajectories of two light beams reaching the Earth from the same astronom-ical object generally requires solving a boundary value problem. The trajectory that connectsthe Earth and astronomical object is determined. As a simple case, we consider a situationwhere the Earth, lensing object, and background object are aligned (Fig. 9). In this situation,the separation angle of the two trajectories due to GL can be calculated easily: it becomes2Θ, or twice the refraction angle. If the separation angle is large enough, multiple celestialimages will be observed. Even if the separation angle is small, the effect of GL will increasethe brightness of the background object.The Earth, lensing object, and background object are aligned when the initial emissionangle and the refraction angle are equal. Therefore, for each model, the situation where theinitial emission angle and the refraction angle coincide has been investigated. For analysis,we consider the parameters given by the case of Omega Centauri. The results are shown inTable 1. 10 a) (b)(c) (d) lense [lyr]r lense = 10 [lyr]r lense = 50 [lyr]r lense = 100 [lyr] 0 0.005 0.01 0.015 0.02 0.025 0 10 20 30 40 50 60 70 80 90 100Impact parameter y lense [lyr]r lense = 10 [lyr]r lense = 50 [lyr]r lense = 100 [lyr] 0 0.01 0.02 0.03 0.04 0.05 0 10 20 30 40 50 60 70 80 90 100Impact parameter y lense [lyr]r lense = 10 [lyr]r lense = 50 [lyr]r lense = 100 [lyr] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] lense [lyr]r lense = 10 [lyr]r lense = 50 [lyr]r lense = 100 [lyr] Fig. 6.
Dependence of the deflection angle on the radius of the lensing object r lens for (a) Plummermodel, (b) Plummer model with IMBH, (c) Hernquist model, and (d) Hernquist model with IMBH. Table 1.
Impact factor y lens when initial emission angle equals refraction angle Θ Models y lens [ × lyr] Θ[ × − arcsec]Model P 0 .
724 0 . . . .
81 2 . . . a) (b)(c) (d) lense [lyr]x e = 1 x 10 [lyr]x e = 5 x 10 [lyr]x e = 10 x 10 [lyr] 0 0.001 0.002 0.003 0.004 0.005 0 10 20 30 40 50 60 70 80 90 100Impact parameter y lense [lyr]x e = 1 x 10 [lyr]x e = 5 x 10 [lyr]x e = 10 x 10 [lyr] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0 10 20 30 40 50 60 70 80 90 100Impact parameter y lense [lyr]x e = 1 x 10 [lyr]x e = 5 x 10 [lyr]x e = 10 x 10 [lyr] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] lense [lyr]x e = 1 x 10 [lyr]x e = 5 x 10 [lyr]x e = 10 x 10 [lyr] Fig. 7.
Dependence of the deflection angle on distance between globular cluster and the Earth x e for (a)Plummer model, (b) Plummer model with IMBH, (c) Hernquist model, and (d) Hernquist model withIMBH. From Table 1, the presence of a black hole affects the separation angle by about 10 × − [arcsec].We have investigated the effects of other parameters also. The value that maximizes theseparation angle is selected within the range of the parameters treated in Section 4.3: • M tot = 5 × M ⊙ , • r lens = 10 [lyr], • x e = 1 × [lyr], • r star = 10 × [lyr].With this choice of parameters, the separation angle of the H-BH model takes its maximumvalue, about 0 .
76 [arcsec]. The smallness of this angle makes it difficult to observe the lensingphenomenon from the ground due to complications with the Earth’s atmosphere.
As we mentioned in section 3, globular clusters in the Milky Way have been cataloged(Harris 1996; Hilker et al. 2020). Our calculations indicate that the GL effect is small even inthe case of Omega Centauri. It is unlikely that a new globular cluster will be found near theEarth. It takes enough time for known globular clusters to be close to Earth. Here we considera different situation. 12 lense [lyr]r star = 1 x 10 [lyr]r star = 5 x 10 [lyr]r star = 10 x 10 [lyr] 0 0.001 0.002 0.003 0.004 0.005 0 10 20 30 40 50 60 70 80 90 100Impact parameter y lense [lyr]r star = 1 x 10 [lyr]r star = 5 x 10 [lyr]r star = 10 x 10 [lyr] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0 10 20 30 40 50 60 70 80 90 100Impact parameter y lense [lyr]r star = 1 x 10 [lyr]r star = 5 x 10 [lyr]r star = 10 x 10 [lyr] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] D e f l e c t i on ang l e Θ [ a r cs e c ] lense [lyr]r star = 1 x 10 [lyr]r star = 5 x 10 [lyr]r star = 10 x 10 [lyr] (a) (b)(c) (d) Fig. 8.
Dependence of the deflection angle on distance to background object r star for (a) Plummer model,(b) Plummer model with IMBH, (c) Hernquist model, and (d) Hernquist model with IMBH. Earthx e xGlobular clusterBackground star Fig. 9.
Schematic of the ideal case in which the Earth, globular cluster, and background star are aligned.Dashed tangential lines indicate the trajectory at the Earth and background star, respectively. Theseparation angle becomes twice the refraction angle. D e f l e c t i on ang l e θ [ a r cs e c ] Impact parameter y lense [lyr]r lens =10 [lyr]r lens =50 [lyr]r lens =100 [lyr]
Fig. 10.
Comparison of deflection angle when radius of lensing object is changed. D e f l e c t i on ang l e Θ [ a r cs e c ] Impact factor y lense [lyr]M tot =1 x 10 M solar M tot =3 x 10 M solar M tot =5 x 10 M solar Fig. 11.
Comparison of deflection angle when total mass of globular cluster is changed.
Suppose that cold dark matter can condense into something like a globular cluster (Carrand Lacey 1987). Such an object would not emit light by itself and would not be included inthe existing catalogs of globular cluster. If such a “dark globular cluster” exists and is closerto the Earth than other globular clusters, how much of a GL effect would it produce?Here we consider dark globular clusters that can be treated by the Hernquist model,with and without IMBHs. We fix two parameters: • x e = 5 × [lyr] • r star = 5 × [lyr]Fig. 10 shows the deflection angle when the radius of the lensing object is changed. Inthis figure, the total mass of the globular cluster is fixed at M tot = 10 M ⊙ . Fig. 11 shows thedeflection angle when the total mass of the lensing object is changed. In this figure, the radiusof the globular cluster is fixed at r lens = 50 [lyr],In these models, the deflection angle can be up to about 2 [arcsec]. If a dark globularcluster containing an IMBH is within a few thousand light years from the Earth, it may bedetected by the GL effect in future observations.14 . Conclusion eferences Alcock, C. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al, he Event Horizon Telescope Collaboration. 2019, ApJ, 875, L1Tyson, J. A. et al. et al.2006, A&A, 445, 513