SAG: Stars and Galaxies Vol. 3, id. 3, 2020 December 28© 2020. Center for Astronomy, University of Hyogo.

Detection of intermediate-mass black holes in globular clusters
using gravitational lensing
Takayuki TATEKAWA 1,2 and Yuuki O KAMURA 3
1

Department of Social Design Engineering, National Institute of Technology, Kochi College,
200–1 Monobe Otsu, Nankoku, Kochi, 783–8508, Japan
2
Research Institute for Science and Engineering, Waseda University,
3–4–1 Okubo, Shinjuku, Tokyo 169–8555, Japan3
Department of Electrical Engineering and Information Science, National Institute of Technology, Kochi College,
200–1 Monobe Otsu, Nankoku, Kochi, 783–8508, Japan
tatekawa@kochi-ct.ac.jp(Received 2020 October 26; accepted 2020 December 23)

Abstract
Recent observations suggest the presence of supermassive black holes at the centers of many galaxies. The
existence of intermediate-mass black holes (IMBHs) in globular clusters has also been predicted. We focus on
gravitational lensing as a new way to explore these entities. It is known that the mass distribution of a self-gravitating
system such as a globular cluster changes greatly depending on the presence or absence of a central massive object.
After considering possible mass distributions for a globular cluster belonging to the Milky Way galaxy, we estimatethat the effect on the separation angle of gravitational lensing due to an IMBH would be of milliarcsecond order.
Key words: gravitational lensing — intermediate-mass black holes — globular clusters

1.

Introduction

The existence of supermassive black holes (SMBHs) at thecenters of galaxies have been made evident by recent observations. For example, the shadow of the SMBH in the center of
M 87 was directly observed by the Event Horizon Telescope(EHT: Event Horizon Telescope Collaboration et al. 2019).
Long-term observations of the movement of stars surrounding
the center of the Milky Way Galaxy suggest the existence of aninvisible massive object, Sgr A∗ , 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 –σrelation between the mass of the SMBH and the velocity dispersion of stars in a galaxy (Silk and Rees 1998; Ferrarese andMerritt 2000; Gebhardt et al. 2000; Gültekin et al. 2009).
The question of how SMBHs form has not yet been definitively answered, although various scenarios have been considered over the years (Rees 1984). One of the scenarios involves
an intermediate-mass black hole (IMBH: Greene et al. 2020)
that grows to become an SMBH. On the basis of the M –σ relation, it seems possible that IMBHs may be located at the centers of globular clusters. This has been discussed particularlyin the case of M 15 but has not been resolved (Gerssen et al.
2002; Baumgardt et al. 2003; McNamara et al. 2003; Kiselev
et al. 2008; Murphy et al. 2011; den Brok et al. 2014), althoughindirect verification has been performed using statistical properties of stars.
In this paper, we consider a new method of detecting SMBHs
and IMBHs. These celestial objects’ immense mass bends the

trajectory of light from background objects: the gravitational
lensing (GL) effect. It can cause background objects to beobserved multiple times, and single images to be brightened.
Through the GL effect, it is possible to verify the existence ofa massive object between the background light-source and theEarth.
GL by globular clusters has been discussed in the past. Kainset al. (2016, 2018) proposed a method for IMBH detection
using gravitational microlensing. Bukhmastova (2003) has attempted to explain QSO-galaxy associations using the GL effect produced by globular clusters.
GL is a small effect even for SMBHs; our treatment includes
lensing by both the massive object and the surrounding stars.
The density profile of the stars in the cluster may be obtained
from the equilibrium solution for a gravitational many-bodysystem; it varies greatly depending on the presence or absenceof a massive central object. In this paper, we assume a spherically symmetric distribution of stars and analyze the GL effectfor various characteristic density profiles.
The paper is organized as follows. In section 2, we discuss
the mass distribution for globular clusters. Two models of the
distribution are described: 1) a model based on the static stateof self-gravitating systems and 2) a phenomenological model.
If there is an IMBH in a globular cluster, the surrounding stars
are thought to form a cusp. Therefore, the two models arealso evaluated with such a cusp, so that in all we consider fourtypes of models in this paper. In section 3, the geodesic equation for light trajectories is discussed. Since the models assume spherical symmetry, the spacetime can be described by

2

T. Tatekawa and Y. Okamura
Vol. 3
Mtot r0
1
the Schwarzschild metric. We consider the gravitational potenρH (r) =.
(2)2π r (r + r0 )3
tial of each model in this metric. If the cusp is distributed acrossthe entire area, the mass will diverge. Hence, it is necessary to Note that the Hernquist model diverges at r = 0, whereas inconnect the spacetime inside the cusp smoothly with that of the Plummer model, the density distribution converges gently
the cuspless model. In section 4, we compare the GL effect in at the center.
mass distribution models with/without an IMBH, and estimate
For a globular cluster without an IMBH, we consider thethe maximum separation angle caused by GL. The difference Plummer model and the Hernquist model.
in the separation angle is found to be of sub-milliarcsecond order when we assume that the lensing object is a globular cluster 2.2. Mass distribution around an IMBH
in the Milky Way Galaxy. In section 5, the conclusions of thisIf a globular cluster includes an IMBH, the exchange of orstudy are presented.
bital energies causes the distribution of stars to take a characteristic form. Here we make several assumptions: 1) The
2. Mass distribution
distribution of stars is represented by a single-particle distribution function; 2) The mass of the IMBH is much smaller
2.1. Mass distribution for globular clusters
than the mass of the globular cluster core; 3) For simplicity, all
In this subsection, we explain mass distributions for glob- the stars around the IMBH have the same mass; 4) The distriular clusters. For simplicity, we assume a spherically sym- bution of stars is independent of angular momentum. Under
metric distribution and consider the equilibrium solution in a these assumptions, the static solution for the density distribuself-gravitating system under Newtonian gravity. Although the tion of stars by the Fokker-Planck equation obeys a specificdistribution function, in general depends on both energy and power law and is known as the Bahcall-Wolf cusp (Bahcall andangular momentum, we assume that the effect of angular mo- Wolf 1976; Merritt 2013).
mentum is small and only the energy dependence needs be conρB (r) ∝ r−7/4 .
(3)sidered. In this case, the velocity dispersion is isotropic at eachpoint in space, and the energy for matter is given by the kinetic The Bahcall-Wolf cusp has been verified by N -body simulaand potential energies.
tion for stellar systems around a massive object (Preto et al.
The distribution function tells us both the distribution of the 2004). In contrast to the Plummer models, in a Bahcall-Wolf
matter in the system and the velocity at which it moves. The cusp the mass density diverges at the center. To consider lensstars in the cluster may be thought of as a fluid obeying an ing by a Bahcall-Wolf cusp, it is necessary to consider the massequation of continuity, the collisionless Boltzmann equation, of the IMBH itself.
that contains a term depending on the gradient of a potential.
Suppose that the distribution function is a known function of 3. Geodesic equations
energy, and that the potential in the collisionless Boltzmann
equation is the gravitational potential solving the Poisson equa- 3.1. Geodesic equations for the spherically symmetric modeltion. The mass density is of course proportional to the gradientIn this paper, we consider the equilibrium state for globularof this potential. Therefore, it is possible in principle (if not al- clusters with spherical symmetry. Therefore, spacetime is deways in practice) to solve the collisionless Boltzmann equation scribed by the Schwarzschild metric (Misner et al. 1973; Hartle
and Poisson’s equation together to obtain the mass density. In 2003).
the case that the distribution function goes as a power of the



−1
2
2
energy, the combined equation is called the Lane-Emden equads2 = − 1 + 2 Ψ(r) d (ct)2 + 1 + 2 Ψ(r)dr2cction; the density is proportional to the solution of this equation
raised to the power of the polytropic index n. Unfortunately,+ r2 dθ2 + sin2 θdφ2 ,(4)
the coefficient of the proportionality is not a constant, but agenerally complicated function. Only in the case of n = 5, the where Ψ(r) corresponds to the Newtonian gravitational potenPlummer model (Plummer 1911; Binney and Tremaine 2008), tial.
From the Schwarzschild metric, geodesic equations may bedoes the Lane-Emden equation yield a simple expression forderived. Hereafter we define the time component as
the density.
ρP (r) =3
Mtot r02,
2
4π (r + r02 )5/2

(1)

where r0 represents “Plummer length”. Mtot is the total massof the cluster.
Another well-known model is that of Hernquist, which realizes de Vaucouleurs 1/4 law, the relationship between surface brightness and distance from the center for elliptic galaxies (de Vaucouleurs 1948). Although the Hernquist model isphenomenological, it has been widely applied, and with muchsuccess (Hernquist 1990). The density in the Hernquist model
is given by:w ≡ ct .
The geodesic equations are as follows:


d2 w
2
2 dΨ
dw dr
1
+
=
−
Ψ
,
2
2
2
dλ
c dr
c
dλ dλ

2

2
2 dΨ
d2 r
dw
1+ 2Ψ
=− 2
dλ2
c dr
c
dλ

−1  2
2
dr
2 dΨ
1+ 2Ψ
+ 2
c dr
c
dλ

  22dϑ+r 1+ 2Ψ
c
dλ

(5)

(6)

id. 3Detection of IMBHs in GCs using GL




2

dφ
,
(7)
dλ
 2
d2 ϑ
2 dr dϑ
dφ
,
(8)
=
−
+
sin
ϑ
cos
ϑ
2
dλ
r dλ dλdλ
2 dr dφ 2 cos ϑ dϑ dφ
d2 φ
=−
−
.
(9)
2
dλ
r dλ dλsin ϑ dλ dλBecause we consider a null geodesic, we have introduced theparameter λ instead of the world interval s. Moreover, because
the trajectory of light can be modeled in a plane, when we notice one trajectory of light, we can ignore the φ components.
Hereafter we set ϑ = π/2.
+ r sin2 ϑ 1 +

3.2.

2
Ψ
c2

We consider four models for globular clusters. When there
is no IMBH, the densities given by equations (1) and (2) correspond to the potentials
GMtot,(r2 + r02 )1/2
GMtot
ΨH (r) = −,r + r0ΨP (r) = −

(10)
(11)

where ΨP and ΨH refer to the potentials in the Plummer andHernquist models, respectively. When the globular cluster includes an IMBH, we set a Bahcall-Wolf cusp at the center.
Outside the cusp, the effect of the IMBH is tiny. At the boundary of the cusp r = R, we connect smoothly with the Plummer
or Hernquist model. In connecting the models, we first focus
on the continuity of the terms of the geodesic equation. Then
we set the first derivative of the potential to be continuous.
For the Plummer model, in the inner region of the model(r < R), the potential is then
GMBH
GMtot
− 2
r
(R + r02 )1/2
 

r 1/4
64
+ πGR2 ρ0
−1 ,
(12)
5
R
 r −3/4dΨP+B(in) (r) GMtot 16.
(13)=+ πGRρ02
drr
5
RIn the outer region of the model (r > R), the potential isΨP+B(in) (r) =−

GMBH
GMtot
ΨP+B(out) (r) = −,− 2
r(r + r02 )1/2
dΨP+B(out) (r) GMtot
GMtot r
.
+ 2
=
dr
r2
(r + r02 )3/2

(14)
(15)

(16)

Similarly, for the Hernquist model, in the inner region(r <R), the potential is
GMBH GMtot−r
R + r0
 

r 1/4
64
−1 ,
+ πGR2 ρ0
5R

GMBH GMtot−,r
r + r0dΨH+B(out) (r) GMtot
GMtot r=+.
drr2
(r + r0 )2
ΨH+B(out) (r) = −

(19)
(20)

In the Hernquist model, the density parameter ρ0 , which ensures that the boundary conditions match at the point wherer = R, turns out to be

3.3.

5Mtot
.
16πR(R + r0 )2(21)

Choice of parameters

For consideration of GL by globular clusters, we must fix thevalues of some parameters. The mass of the IMBH MBH , theradius of the Bahcall-Wolf cusp R, the total mass of the lensing
object Mtot , and radius of the lensing object rlens should all beconsidered. Because, in both the Plummer and Hernquist models, a scale length r0 is included, we must decide on its value
as well. For the null geodesic, we must consider the distance
between the center of the globular cluster and the backgroundobject rstar ; we must also consider the distance xe from the
Earth to the center of the globular cluster.
Both the Plummer model and the Hernquist model include
a length parameter r0 (equations (1) and (2)), a characteristic quantity that determines the shape of the density distribution. In order to compare the two models, we wish to makethe mass contained within radius r0 approximately the same inboth. To do this, we use a relation between r0 and rlens . For
the Plummer model, we set r0 = rlens /2:
Z rlens
MP (rlens ) =ρP (r) · 4πr2 dr0

3
Mtot rlens=

2(rlens+ (rlens /2)2 )≃ 0.72 Mtot .

3/2

(22)

For the Hernquist model, we set r0 = rlens /5:
Z rlensρH (r) · 4πr2 dr
MH (rlens ) =0

To satisfy the condition of connection at the boundary (r =R), the density parameter ρ0 must be
5Mtot
ρ0 =
.
16π(R2 + r02 )3/2

dΨH+B(in) (r) GMtot 16.
(18)=+ πGRρ0drr2
5RIn the outer region of the model (r > R), the potential isρ0 =Derivation of potential term for models3 r −3/4

ΨH+B(in) (r) =−(17)

2
Mtot rlens(rlens + rlens /5)2
≃ 0.69 Mtot .

=

(23)

For both models, about 70% of the total mass is thus within theradius of the lensing object.
In this paper, the mass of the IMBH is fixed at MBH =104 M⊙ . The upper bound of the distance to the backgroundstar is fixed at rstar < 106 lyr. The globular cluster is assumed
to belong to the Milky Way; catalogs of Milky Way globularclusters have been published (Harris 1996; Hilker et al. 2020).
The parameters (total mass, calculated radius from apparent
dimension, and distance from the Earth) of the clusters in thepresent paper are chosen to be typical of relatively large Milky
Way globular clusters according to the catalogs:

4T. Tatekawa and Y. Okamura• 1 × 106 ≤ Mtot ≤ 5 × 106 M⊙• 10 ≤ rlens ≤ 100 lyr• 1 × 104 ≤ xe ≤ 10 × 104 lyr

Here xe means the distance from the Earth to the center of theglobular cluster. The boundary of Bahcall-Wolf cusp is fixed at
R = 1 lyr.
4.
4.1.

Effect of gravitational lensing
Model settings

Various physical and geometrical quantities required for analyzing the GL effect of globular clusters will now be defined.
Figure 1 shows the positional relationship between the Earth,
globular cluster, and background star. Even if spacetime iscurved by gravity, the trajectory of light can be analyzed ina plane. The center of the globular cluster is defined as the origin. Then the Earth is set at (xe ,0). This position is denoted as
point E. A background star is positioned at point S. Becauseof the effect of GL, however, this star appears to observers on
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. The intersection point of the y-axis and line segment ES ′ is definedas the “impact parameter” ylens . The relationship between theimpact parameter and θ is
ylens
.
(24)
tan θ =
xe
The distance between the center of the globular cluster and thebackground star is given by rstar .
4.2.

Typical example – ω Centauri

As an example for our study, one well-known globularcluster is selected and verified. ω Centauri is the most massive globular cluster of the Milky Way. The existence of an
IMBH in ω Centauri has been discussed from both theoreticaland observational perspectives (Noyola et al. 2010; Haggard
et al. 2013; Baumgardt et al. 2019). The effect of the GL
will be calculated using the known parameters of ω Centauri:
Mtot = 4 × 106 M⊙ , rlens = 82 lyr, xe = 1.56 × 104 lyr. For
this calculation, rstar is fixed at 5 × 105 lyr (van de Ven et al.
2006; D’Souza and Rix 2013).
When the cluster excludes the IMBH, the mass distribution is given by the Plummer model or the Hernquist model.
The deflection angle is shown in figure 2. For the case of the
Hernquist model (hereafter, model H), the deflection angle increases sharply as the impact parameter decreases. Conversely,
for the case of the Plummer model (hereafter, model P), the deflection angle approaches a constant value at ylens = 0. This isdue to the difference in gravitational potential between models:for model P, the potential flattens at the center, but for modelH, the potential sharpens there.
When the cluster includes an IMBH, the deflection angle
dramatically changes at small impact parameters. For the caseof the Plummer model with an IMBH (hereafter, model P-BH),
the difference of angle becomes significant at ylens ≤ 20 lyr(figure 3). Although the boundary of Bahcall-Wolf cusp isfixed at 1 lyr, the trajectory of light is affected well outside thisradius. Due to the effect of the IMBH, the angle diverges atVol. 3r ≃ 0. For the case of the Hernquist model with an IMBH
(hereafter, model H-BH), although the effect of the IMBH (orcusp) appears at small impact parameters, it is not as clear as in
the case of model P-BH (figure 4). Perhaps this result is causedby H-BH’s less extreme potential slope near the center.
4.3.

Dependence on parameters

We investigate the effect of changing the parameters on thedeflection angle. The standard values of the parameters are as
follows:• Total mass of lensing object: Mtot = 106 M⊙• Radius of lensing object: rlens = 50 lyr• Distance from the Earth: xe = 5 × 104 lyr• Distance between the center of the globular cluster andthe background object:rstar = 5 × 105 lyr
The above four parameters are then changed one by one, withthe following results:
Figure 5 shows the dependence of the deflection angle on
total mass Mtot . As Mtot increases, the angle increases because gravity strengthens. Figure 6 shows the dependence ofthe deflection angle on the radius of the lensing object rlens .
The lensing effect increases as the mass is concentrated in anarrower area.
We also show the dependence on the distance between theglobular cluster and the Earth. The angle θ changes as the
distance between the Earth and the globular cluster changes.
Figure 7 shows the θ dependence of the deflection angle. Since
we are considering globular clusters belonging to the MilkyWay, the upper limit of the distance is set to 105 lyr. The distance dependence of the angle, therefore, seems from the figure
to be insignificant.
Finally, we consider the dependence on the distance to thebackground object (figure 8). As the distance to the background object increases, the angle approaches a limiting curve
that, except in model P, rises steeply at low impact parameters.
4.4.

Observability of gravitational lensing effects

Finding the trajectories of two light beams reaching the
Earth from the same astronomical object generally requires
solving a boundary value problem. The trajectory that connects the Earth and astronomical object is determined. As asimple case, we consider a situation where the Earth, lensing
object, and background object are aligned (figure 9). In thissituation, the separation angle of the two trajectories due to GL
can be calculated easily: it becomes 2Θ, or twice the refraction
angle. If the separation angle is large enough, multiple celestial
images will be observed. Even if the separation angle is small,the effect of GL will increase the brightness of the backgroundobject.
The Earth, lensing object, and background object are aligned
when the initial emission angle and the refraction angle are
equal. Therefore, for each model, the situation where the initial emission angle and the refraction angle coincide has been
investigated. For analysis, we consider the parameters given by
the case of ω Centauri. The results are shown in table 1.
From table 1, the presence of a black hole affects the separation angle by about 10 × 10−2 arcsec.

id. 3Detection of IMBHs in GCs using GL

5yApparent position of star (S’ )Background star (S)

ylens

rstar
Earth (E)θxxeGlobular cluster

Fig. 1. Schematic of the positional relationship between the Earth, globular cluster, and background star. The center of the globular cluster is defined as
the origin, and the trajectory of light from the background star to the Earth (solid line) is analyzed.
0.025

0.025
Model P
Model P-BH

Model P
Model H
0.02
Deflection angle Θ [arcsec]

Deflection angle Θ [arcsec]

0.02

0.015

0.01

0.005

0

0.015

0.01

0.005

0
0

10

20

30
40
50
60
70
Impact parameter y lense [lyr]

80

90

100

0

10

20

30
40
50
60
70
Impact parameter y lense [lyr]

80

90

100

0.025
Model P
Model P-BH

Fig. 2. The deflection angle by the Plummer model (Model P) and the
Hernquist model (Model H) as a function of impact parameter.
Deflection angle Θ [arcsec]

0.02

Table 1. Impact factor ylens when initial emission angle equals refraction angle ΘModelsylens [×103 lyr]Θ [×10−2 arcsec]

Model P

0.724

0.957

Model P-BH

10.1

13.1

Model H

1.81

2.39

Model H-BH

10.8

14.2

We have investigated the effects of other parameters also.
The value that maximizes the separation angle is selected
within the range of the parameters treated in section 4.3:
• Mtot = 5 × 106 M⊙ ,• rlens = 10 lyr,• xe = 1 × 104 lyr,• rstar = 10 × 105 lyr.
With this choice of parameters, the separation angle of the HBH model takes its maximum value, about 0.′′ 76. The smallness of this angle makes it difficult to observe the lensing phe-

0.015

0.01

0.005

0
0

1

2
3
Impact parameter y lense [lyr]45Fig. 3. Comparison of deflection angle with and without IMBH in the
Plummer model. The figure on the right is an enlarged view of ylens
from the figure on the left.

nomenon from the ground due to complications with the Earths atmosphere.
4.5.

Case of dark globular cluster

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 in the caseof ω Centauri. It is unlikely that a new globular cluster will befound near the Earth. It takes enough time for known globular clusters to be close to Earth. Here we consider a different
situation.

6

T. Tatekawa and Y. Okamura
0.04
Model H
Model H-BH

Deflection angle Θ [arcsec]

0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0

10

20

30
40
50
60
70
Impact parameter y lense [lyr]

80

90

100

0.04
Model H
Model H-BH

Deflection angle Θ [arcsec]

0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0

1

2
3
Impact parameter y lense [lyr]45Fig. 4. Comparison of deflection angle with and without IMBH in the
Hernquist model. The figure on the right is an enlarged view of ylens
from the figure on the left.

Suppose that cold dark matter can condense into something
like a globular cluster (Carr and Lacey 1987). Such an object
would not emit light by itself and would not be included in theexisting catalogs of globular clusters. If such a “dark globularcluster” exists and is closer to the Earth than other globularclusters, how much of a GL effect would it produce?
Here we consider dark globular clusters that can be treatedby the Hernquist model, with and without IMBHs. We fix two
parameters:• xe = 5 × 103 lyr• rstar = 5 × 105 lyr
Figure 10 shows the deflection angle when the radius of thelensing object is changed. In this figure, the total mass of theglobular cluster is fixed at Mtot = 106 M⊙ . Figure 11 shows
the deflection angle when the total mass of the lensing object
is changed. In this figure, the radius of the globular cluster isfixed at rlens = 50 lyr.
In these models, the deflection angle can be up to about 2′′ .
If a dark globular cluster containing an IMBH is within a few
thousand light years from the Earth, it may be detected by theGL effect in future observations.
5.

Conclusion

We considered a method to determine the presence of an
IMBH in a globular cluster by examining the GL effect. Under
the assumption of spherical symmetry, we considered the mass
distribution of globular clusters with and without IMBHs, andthen calculated the separation angle of light trajectories basedon this. (Note that, in this study, it was assumed that the trajectories could pass through the interior of the globular cluster, butsuch a pass-through might be impossible if the globular cluster

Vol. 3is extremely dense. In that case, background objects could not
be seen.)A catalog of globular clusters in the Milky Way Galaxy has
been published (Harris 1996; Hilker et al. 2020). Typical parameters, based on those listed in this catalog, served as input to see how strong a GL effect a cluster can produce. For
ω Centauri, a dense globular cluster that is the most massive
belonging to the Milky Way, the separation angle of the GL effect caused by the globular cluster with an IMBH was found to
be on the order of sub-milliarcseconds. For other globular clusters belonging to the Milky Way, the separation angle would beeven smaller.
Observing multiple images may be difficult due to the small
separation angle of the GL. The brightening effect caused by
GL may be more easily observed. Light that travels throughmultiple trajectories from background objects can be concentrated onto the Earth by the GL effect, which causes sources
to appear brighter than they are in reality. Systematic observations of the GL effect on galaxies have been made in
the past (Tyson et al. 1990; Dahle et al. 2002; Kubo et al.
2009; Jaelani et al. 2020). We are considering similar surveys
for globular clusters.
This method has been applied to MACHO observations withsmall separation angles (Alcock et al. 2000). If a single observation reveals that the brightening effect of the globular cluster
due to the GL effect is quite strong, the density distribution of
the globular cluster may be high, suggesting the presence ofan IMBH. By observing the change in the brightening rate ofa globular cluster as it passes between background objects andthe Earth, it may be possible to detect the presence or absenceof an IMBH, but such continuous observations would require amore extensive observation program than the MACHO Project.
Quantitative evaluation of brightening due to the GL effect will
be derived in future work.
The authors would like to thank Shigeyuki Karino, TakahikoMatsubara, Kouji Nakamura, Hiroyuki Nakano, Hisaaki
Shinkai, and Kousuke Sumiyoshi for their useful discussion at
32nd RIRONKON symposium (Dec. 25-27, 2019, National
Astronomical Observatory, Japan). We would like to thank
Editage (www.editage.com) for English language editing.
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id. 3Detection of IMBHs in GCs using GL
0.025

6

Mtot=1 x 10 M
Mtot=3 x 106 M
6
Mtot=5 x 10 M

0.02
Deflection angle Θ [arcsec]

Deflection angle Θ [arcsec]

0.025

6

Mtot=1 x 106 M
Mtot=3 x 10 M
6
Mtot=5 x 10 M

0.02

0.015

0.01

0.005

0.015

0.01

0.005

0

0
0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

90

100

0

10

20

0.05

0.05

6

Mtot=1 x 10 M
Mtot=3 x 106 M
6
Mtot=5 x 10 M

80

90

100

90

100

6

Mtot=1 x 10 M
Mtot=3 x 106 M
6
Mtot=5 x 10 M

0.04
Deflection angle Θ [arcsec]

0.04

30
40
50
60
70
Impact parameter ylense [lyr]

(b)

(a)

Deflection angle Θ [arcsec]

7

0.03

0.02

0.01

0

0.03

0.02

0.01

0
0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

90

100

0

10

20

(c)

30
40
50
60
70
Impact parameter ylense [lyr]

80

(d)

Fig. 5. Dependence of the deflection angle on the total mass Mtot for (a) Plummer model, (b) Plummer model with IMBH, (c) Hernquist model, and (d)Hernquist model with IMBH.

0.025

rlense = 10 [lyr]rlense = 50 [lyr]rlense = 100 [lyr]0.02
Deflection angle Θ [arcsec]

0.02
Deflection angle Θ [arcsec]

0.025

rlense = 10 [lyr]rlense = 50 [lyr]rlense = 100 [lyr]

0.015

0.01

0.005

0.015

0.01

0.005

0

0
0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

90

100

0

10

20

(a)

0.05

0.05

90

100

90

100

rlense = 10 [lyr]rlense = 50 [lyr]rlense = 100 [lyr]0.04
Deflection angle Θ [arcsec]

Deflection angle Θ [arcsec]

80

(b)

rlense = 10 [lyr]rlense = 50 [lyr]rlense = 100 [lyr]

0.04

30
40
50
60
70
Impact parameter ylense [lyr]

0.03

0.02

0.01

0

0.03

0.02

0.01

0
0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

(c)

80

90

100

0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

(d)

Fig. 6. Dependence of the deflection angle on the radius of the lensing object rlens for (a) Plummer model, (b) Plummer model with IMBH, (c) Hernquist
model, and (d) Hernquist model with IMBH.

8

T. Tatekawa and Y. Okamura
0.005

4

xe = 1 x 104 [lyr]xe = 5 x 10 [lyr]4xe = 10 x 10 [lyr]0.004
Deflection angle Θ [arcsec]

Deflection angle Θ [arcsec]

0.005

4

xe = 1 x 104 [lyr]xe = 5 x 10 [lyr]4xe = 10 x 10 [lyr]0.004

Vol. 3

0.003

0.002

0.001

0

0.003

0.002

0.001

0
0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

90

100

0

10

20

(a)

90

100

90

100

xe = 1 x 104 [lyr]xe = 5 x 104 [lyr]4xe = 10 x 10 [lyr]

0.014
0.012
Deflection angle Θ [arcsec]

Deflection angle Θ [arcsec]

0.012

80

(b)

xe = 1 x 104 [lyr]xe = 5 x 104 [lyr]4xe = 10 x 10 [lyr]

0.014

30
40
50
60
70
Impact parameter ylense [lyr]

0.01
0.008
0.006
0.004
0.002
0

0.01
0.008
0.006
0.004
0.002
0

0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

90

100

0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

(d)

(c)

Fig. 7. Dependence of the deflection angle on the distance between a globular cluster and the Earth xe for (a) Plummer model, (b) Plummer model with
IMBH, (c) Hernquist model, and (d) Hernquist model with IMBH.

0.005

5

rstar = 1 x 105 [lyr]rstar = 5 x 10 [lyr]5
rstar = 10 x 10 [lyr]0.004
Deflection angle Θ [arcsec]

0.004
Deflection angle Θ [arcsec]

0.005

5

rstar = 1 x 105 [lyr]rstar = 5 x 10 [lyr]5
rstar = 10 x 10 [lyr]

0.003

0.002

0.001

0

0.003

0.002

0.001

0
0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

90

100

0

10

20

(a)

90

100

90

100

rstar = 1 x 105 [lyr]rstar = 5 x 105 [lyr]5
rstar = 10 x 10 [lyr]

0.014
0.012
Deflection angle Θ [arcsec]

Deflection angle Θ [arcsec]

0.012

80

(b)

rstar = 1 x 105 [lyr]rstar = 5 x 105 [lyr]5
rstar = 10 x 10 [lyr]

0.014

30
40
50
60
70
Impact parameter ylense [lyr]

0.01
0.008
0.006
0.004
0.002
0

0.01
0.008
0.006
0.004
0.002
0

0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

(c)

80

90

100

0

10

20

30
40
50
60
70
Impact parameter ylense [lyr]

80

(d)

Fig. 8. Dependence of the deflection angle on the distance to a background object rstar for (a) Plummer model, (b) Plummer model with IMBH, (c)Hernquist model, and (d) Hernquist model with IMBH.

id. 3Detection of IMBHs in GCs using GL

92

Mtot=1 x 106 Msolar
Mtot=3 x 106 Msolar
Mtot=5 x 106 Msolar

xe
Earth

Background star
Globular clusterx

Fig. 9. Schematic of the ideal case in which the Earth, globular cluster,and background star are aligned. Dashed tangential lines indicate thetrajectory at the Earth and a background star, respectively. The separation angle becomes twice the refraction angle.
2

Deflection angle θ [arcsec]

1.8
1.6

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.01

0.1

1
Impact factor y lense [lyr]

10

100

Fig. 11. Comparison of deflection angle when the total mass of theglobular cluster is changed.
rlens=10 [lyr]rlens=50 [lyr]rlens=100 [lyr]

1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.01

Deflection angle Θ [arcsec]

1.8

0.1
1
Impact parameter y lense [lyr]

10

100

Fig. 10. Comparison of deflection angle when the radius of the lensing
object is changed.

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