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arXiv:astro-ph/0507383v1 16 Jul 2005
Magnetohydrodynamics in full general relativity: Formulation and tests
Masaru Shibata and Yu-ichiou Sekiguchi
Graduate School of Arts and Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153-8902, Japan
A new implementation for magnetohydrodynamics (MHD) simulations in full general relativity
(involving dynamical spacetimes) is presented. In our implementation, Einstein’s evolution equations
are evolved by a BSSN formalism, MHD equations by a high-resolution central scheme, and induction
equation by a constraint transport method. We perform numerical simulations for standard test
problems in relativistic MHD, including special relativistic magnetized shocks, general relativistic
magnetized Bondi flow in stationary spacetime, and a longterm evolution for self-gravitating system
composed of a neutron star and a magnetized disk in full general relativity. In the final test, we
illustrate that our implementation can follow winding-up of the magnetic field lines of magnetized
and differentially rotating accretion disks around a compact object until saturation, after which
magnetically driven wind and angular momentum transport inside the disk turn on.
04.25.Dm, 04.40.Nr, 47.75.+f, 95.30.Qd
I. INTRODUCTION
Hydrodynamics simulation in general relativity is probably the best theoretical approach for investigating dynamical
phenomena in relativistic astrophysics such as stellar core collapse to a neutron star and a black hole, and the
merger of binary neutron stars. In the past several years, this field has been extensively developed (e.g., [1–7]) and,
as a result, now it is feasible to perform accurate simulations of such general relativistic phenomena for yielding
scientific results (e.g., [6–9] for our latest results). For example, with the current implementation, radiation reaction
of gravitational waves in the merger of binary neutron stars can be taken into account within ∼ 1% error in an
appropriate computational setting [6,7]. This fact illustrates that the numerical relativity is a robust approach for
detailed theoretical study of astrophysical phenomena and gravitational waves emitted.
However, so far, most of the scientific simulations in full general relativity have been performed without taking into
account detailed effects except for general relativistic gravity and pure hydrodynamics. For example, simplified ideal
equations of state have been adopted instead of realistic ones (but see [7]). Also, the effect of magnetic fields has
been neglected although it could often play an important role in the astrophysical phenomena (but see [10]). In the
next stage of numerical relativity, it is necessary to incorporate these effects for more realistic simulations. As a step
toward a more realistic simulation, we have incorporated an implementation for ideal magnetohydrodynamics (MHD)
equations in fully general relativistic manner. In this paper, we describe our approach for these equations and then
present numerical results for test problems computed by the new implementation.
Magnetic fields indeed play an important role in determining the evolution of a number of relativistic objects. In
the astrophysical context, the plasma is usually highly conducting, and hence, the magnetic fields are frozen in the
matter. This implies that a small seed field can wind up and grow in the complex motion of the matter, resulting
in a significant effect in the dynamics of the matter such as magnetically driven wind or jet and angular momentum
redistribution. Specifically, in the context of the general relativistic astrophysics, the magnetic fields will play a role in
the following phenomena and objects: Stellar core collapse of magnetized massive stars to a protoneutron star [11] or a
black hole, stability of accretion disks (which are either non-self-gravitating or self-gravitating) around black holes and
neutron stars, magnetic braking of differentially rotating neutron stars [10] which are formed after merger of binary
neutron stars [6,7] and stellar core collapse [14–16,8,9], and magnetically induced jet around the compact objects
(e.g., [17]). To clarify these phenomena, fully general relativistic MHD (GRMHD) simulation (involving dynamical
spacetimes) is probably the best theoretical approach.
In the past decade, numerical implementations for GRMHD simulation in the fixed gravitational field have been
extensively developed (e.g., [18,17,19–24]). In particular, it is worth to mention that Refs. [19–23] have recently
presented implementations for which detailed tests have been carried out for confirmation of the reliability of their
computation, in contrast with the attitude in an early work [17]. They are applied for simulating magnetorotational
instability (MRI) of accretion disks and subsequently induced winds and jets around black holes and neutron stars.
On the other hand, little effort has been paid to numerical implementations of fully GRMHD (in the dynamical
gravitational field). About 30 years ago, Wilson performed a simulation for collapse of a magnetized star in the
presence of poloidal magnetic fields in general relativity. However, he assumes that the three-metric is conformally
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flat [25], and hence, the simulation is not fully general relativistic, although recent works have indicated that the
conformally flat approximation works well in the axisymmetric collapse (e.g., compare results among [15], [26], and
[27]). The first fully GRMHD simulation for stellar collapse was performed by Nakamura about 20 years ago [28]. He
simulated collapse of nonrotating stars with poloidal magnetic fields to investigate the criteria for formation of black
holes and naked singularities. Very recently, Duez et al. have presented a new implementation capable of evolution
for the Einstein-Maxwell-MHD equations for general cases [10]. They report successful results for test simulations.
Valencia group has also developed a GRMHD implementation very recently [29].
In this paper, we present our new implementation for fully GRMHD which is similar to but in part different from
that in [10]∗. As a first step toward scientific simulations, we have performed simulations in standard test problems
including special relativistic magnetized shocks, general relativistic Bondi flow in stationary spacetime, and long term
evolution of fully general relativistic stars with magnetic fields. We here report the successful results for these test
problems.
Before proceeding, we emphasize that it is important to develop new GRMHD implementations. In the presence of
magnetic fields, matter motion often becomes turbulence-like due to MRIs in which a small scale structure often grows
most effectively [30]. Furthermore in the presence of general relativistic self-gravity which has a nonlinear nature,
the matter motion may be even complicated. Perhaps, the outputs from the simulations will contain results which
have not been well understood yet, and thus, are rich in new physics. Obviously high accuracy is required for such
frontier simulation to confirm novel numerical results. However, because of the restriction of computational resources,
it is often very difficult to get a well-resolved and completely convergent numerical result in fully general relativistic
simulation. In such case, comparison among various results obtained by different numerical implementations is crucial
for checking the reliability of the numerical results. From this point of view, it is important to develop several numerical
implementations in the community of numerical relativity.
implementations, reliability of the numerical results will be improved each other. Our implementation presented
here will be useful not only for finding new physics but also for checking numerical results by other implementations
such as that very recently presented in [10,29].
In Sec. II, we present formulations for Einstein, Maxwell, and GRMHD equations. In Sec. III, numerical methods
for solving GRMHD equations are described. In Sec. IV, methods for a solution of initial value problem in general
relativity is presented. In Secs. V and VI, numerical results for special and general relativistic test simulations are
shown. In the final subsection of Sec. VI, we illustrate that our implementation can follow growth of magnetic fields of
accretion disks in fully general relativistic simulation. Sec. VII is devoted to a summary and a discussion. Throughout
this paper, we adopt the geometrical units in which G = c = 1 where G and c are the gravitational constant and the
speed of light. Latin and Greek indices denote spatial components and spacetime components, respectively. ηµν and
δij(= δij) denote the flat spacetime metric and the Kronecker delta, respectively.
By comparing several results computed by different
II. BASIC EQUATIONS
A. Definition of variables
Basic equations consist of the Einstein equations, general relativistic hydrodynamic equations, and Maxwell equa-
tions. In this subsection, we define the variables used in these equations. The fundamental variables for geometry are
α: lapse function, βk: shift vector, γij: metric in three-dimensional spatial hypersurface, and Kij: extrinsic curvature.
The spacetime metric gµνis written as
gµν= γµν− nµnν, (1)
where nµis a unit normal to a spacelike spatial hypersurface Σ and is written as
nµ=
?1
α,−βi
α
?
,ornµ= (−α,0). (2)
In the BSSN formalism [35], one defines γ ≡ ηe12φ= det(γij): determinant of γij, ˜ γij= e−4φγij: conformal three-
metric, K = Kk
k: trace of the extrinsic curvature, and˜Aij ≡ e−4φ(Kij− Kγij/3): a tracefree part of the extrinsic
∗For instance, our formulation for Einstein’s evolution equations, gauge conditions, and our numerical scheme for GRMHD
equations are different from those in [10] as mentioned in Secs. II and III.
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curvature. Here, η denotes the determinant of flat metric; in the Cartesian coordinates, η = 1, and in the cylindrical
coordinates (̟,ϕ,z), η = ̟2. In the following, ∇µ, Di, and˜Didenote the covariant derivatives with respect to gµν,
γij, and ˜ γij, respectively. ∆ and˜∆ denote the Laplacians with respect to γij and ˜ γij. Rij and˜Rij denote the Ricci
tensors with respect to γij and ˜ γij, respectively.
The fundamental variables in hydrodynamics are ρ: rest-mass density, ε : specific internal energy, P : pressure, and
uµ: four velocity. From these variables, we define the following variables which often appear in the basic equations:
ρ∗≡ ρwe6φ,
vi≡dxi
dt
h ≡ 1 + ε +P
w ≡ αut.
(3)
=ui
ut= −βi+ γijuj
ut,(4)
ρ,
(5)
(6)
Here, ρ∗is a weighted baryon rest mass density from which the conserved baryon rest mass can be computed as
M∗=
?
ρ∗η1/2d3x.(7)
The fundamental variable in the ideal MHD is only bµ: magnetic field. The electric field Eµin the comoving frame
Fµνuνis assumed to be zero, and electric current jµis not explicitly necessary for evolving the field variables. Using
the electromagnetic tensor Fµν, bµis defined by [31]
bµ≡ −1
2ǫµναβuνFαβ,(8)
where ǫµναβis the Levi-Civita tensor with ǫt123=√−g and ǫt123= −1/√−g. Equation (8) implies
bµuµ= 0. (9)
Using Eq. (8), Fµνin the ideal MHD is written as
Fµν= ǫµναβuαbβ, (10)
and thus, it satisfies the ideal MHD condition
Fµνuν= 0. (11)
The dual tensor of Fµν is defined by
F∗
µν≡1
2ǫµναβFαβ= bµuν− bνuµ.(12)
For rewriting the induction equation for the magnetic fields into a simple form (see Sec. II D), we define the three-
magnetic field as
Bi≡ −e6φγi
jF∗jµnµ= e6φ(wbi− αbtui). (13)
Here, we note that Bt= 0 (i.e., Bµnµ= 0), and thus, Bi= γijBj. Equations (13) and (9) lead to
bt=Bµuµ
αe6φ
and bi=
1
we6φ
?
Bi+ Bjujui
?
(14)
Using the hydrodynamic and electromagnetic variables, energy-momentum tensor is written as
Tµν= TFluid
µν
+ TEM
µν. (15)
TFluid
µν
and TEM
µν
denote the fluid and electromagnetic parts defined by
TFluid
µν
= (ρ + ρε + P)uµuν+ Pgµν= ρhuµuν+ Pgµν,(16)
TEM
µν = FµσFσ
ν −1
4gµνFαβFαβ=
?1
2gµν+ uµuν
?
b2− bµbν,(17)
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where
b2= bµbµ=B2+ (Biui)2
w2e12φ
.(18)
Thus, Tµνis written as
Tµν= (ρh + b2)uµuν+
?
P +1
2b2?
gµν− bµbν. (19)
For the following, we define magnetic pressure and total pressure as Pmag= b2/2 and Ptot≡ P + b2/2, respectively.
The (3+1) decomposition of Tµνis
ρH≡ Tµνnµnν= (ρh + b2)w2− Ptot− (αbt)2,
Ji≡ −Tµνnµγν
Sij≡ Tµνγµ
(20)
i= (ρh + b2)wui− αbtbi,
j= (ρh + b2)uiuj+ Ptotγij− bibj.
(21)
iγν
(22)
Using these, the energy-momentum tensor is rewritten in the form
Tµν= ρHnµnν+ Jiγi
µnν+ Jiγi
νnµ+ Sijγi
µγj
ν. (23)
This form of the energy-momentum tensor is useful for deriving the basic equations for GRMHD presented in Sec. II
C. For the following, we define
S0≡ e6φρH,
Si≡ e6φJi.
(24)
(25)
These variables together with ρ∗and Biare evolved explicitly in the numerical simulation of the ideal MHD (see Sec.
II C).
B. Einstein’s equation
Our formulation for Einstein’s equations is the same as in [6] in three spatial dimensions and in [34] in axial
symmetry. Here, we briefly review the basic equations in our formulation. Einstein’s equations are split into constraint
and evolution equations. The Hamiltonian and momentum constraint equations are written as
Rk
k−˜Aij˜Aij+2
Di˜Ai
3K2= 16πρH, (26)
j−2
3DjK = 8πJj, (27)
or, equivalently
˜∆ψ =ψ
8
˜Rk
k− 2πρHψ5−ψ5
j) −2
8
?˜Aij˜Aij−2
3K2?
, (28)
˜Di(ψ6˜Ai
3ψ6˜DjK = 8πJjψ6, (29)
where ψ ≡ eφ. These constraint equations are solved to set initial conditions. A method in the case of GRMHD is
presented in Sec. IV.
In the following of this subsection, we assume that Einstein’s equations are solved in the Cartesian coordinates
(x,y,z) for simplicity. Although we apply the implementation described here to axisymmetric issues as well as
nonaxisymmetric ones, this causes no problem since Einstein’s equations in axial symmetry can be solved using the
so-called Cartoon method in which an axisymmetric boundary condition is appropriately imposed in the Cartesian
coordinates [32–34]: In the Cartoon method, the field equations are solved only in the y = 0 plane, and grid points
of y = ±∆x (∆x denotes the grid spacing in the uniform grid) are used for imposing the axisymmetric boundary
conditions.
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We solve Einstein’s evolution equations in our latest BSSN formalism [35,6]. In this formalism, a set of variables
(˜ γij,φ,˜Aij,K,Fi) are evolved. Here, we adopt an auxiliary variable Fi≡ δjl∂l˜ γij that is the one originally proposed
and different from the variable adopted in [10] in which ∂i˜ γijis used. Evolution equations for ˜ γij, φ,˜Aij, and K are
(∂t− βl∂l)˜ γij= −2α˜Aij+ ˜ γikβk
,j+ ˜ γjkβk
,i−2
?
3˜ γijβk
?
,k, (30)
(∂t− βl∂l)˜Aij= e−4φ
?
α
?
Rij−1
3e4φ˜ γijRk
k
−DiDjα −1
,j˜Aki−2
3e4φ˜ γij∆α
??
+α(K˜Aij− 2˜Aik˜Ak
?
?
?˜Aij˜Aij+1
j) + βk
,i˜Akj+ βk
3βk
,k˜Aij
−8παe−4φSij−1
3˜ γijSk
k
?
, (31)
(∂t− βl∂l)φ =1
6
−αK + βk
,k
?
, (32)
(∂t− βl∂l)K = α
3K2?
− ∆α + 4πα(ρH+ Sk
k). (33)
For a solution of φ, the following conservative form may be adopted [6]:
∂te6φ− ∂i(βie6φ) = −αKe6φ. (34)
For computation of Rijin the evolution equation of˜Aij, we decompose
Rij=˜Rij+ Rφ
ij, (35)
where
Rφ
ij= −2˜Di˜Djφ − 2˜ γij˜∆φ + 4˜Diφ˜Djφ − 4˜ γij˜Dkφ˜Dkφ,
˜Rij=1
2
(36)
?
δkl(−hij,kl+ hik,lj+ hjk,li) + 2∂k(fkl˜Γl,ij) − 2˜Γl
kj˜Γk
il
?
. (37)
In Eq. (37), we split ˜ γijand ˜ γijas δij+ hijand δij+ fij, respectively.˜Γk
˜ γij, and˜Γk,ij= ˜ γkl˜Γl
In addition to a flat Laplacian of hij,˜Rijinvolves terms linear in hijas δklhik,lj+ δklhjk,li. To perform numerical
simulation stably, we replace these terms by Fi,j+ Fj,i. This is the most important part in the BSSN formalism,
pointed out originally by Nakamura [28]. The evolution equation of Fiis derived by substituting Eq. (30) into the
momentum constraint as
ijis the Christoffel symbol with respect to
ij. Because of the definition det(˜ γij) = 1 (in the Cartesian coordinates), we use˜Γk
ki= 0.
(∂t− βl∂l)Fi= −16παJi+ 2α
+ δjk?
?
fkj˜Aik,j+ fkj
,j˜Aik−1
2
˜Ajlhjl,i+ 6φ,k˜Ak
i−2
?
3K,i
?
−2α,k˜Aij+ βl
,khij,l+
?
˜ γilβl
,j+ ˜ γjlβl
,i−2
3˜ γijβl
,l
?
,k
.(38)
We also have two additional notes for handling the evolution equation of˜Aij. One is on the method for evaluation
of Rk
kfor which there are two options, use of the Hamiltonian constraint and direct calculation by
Rijγij= e−4φ(˜Rk
k+ Rφ
ij˜ γij).(39)
We always adopt the latter one since with this, the conservation of the relation˜Aij˜ γij= 0 is much better preserved.
The other is on the handling of a term of ˜ γijδklhij,klwhich appears in˜Rk
k. This term is written by
˜ γijδklhij,kl= −δklhij,kfij
,l,(40)
where we use det(˜ γij) = 1 (in the Cartesian coordinates).
As the time slicing condition, an approximate maximal slice condition K ≈ 0 is adopted following previous papers
(e.g., [36]). As the spatial gauge condition, we adopt a hyperbolic gauge condition as in [37,6]. Successful numerical
results for merger of binary neutron stars and stellar core collapse in these gauge conditions are presented in [6,7,26,8].
We note that these are also different from those in [10].
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FIG. 14. Snapshots of the density profile of a neutron and a disk in x−z plane at t = 0, 9.78, and 19.40 ms for RB = 2×10−4.
For ρ > 1010g/cm3, the density is denoted by the same color (red), and for 1010g/cm3≥ ρ ≥ 107g/cm3, the color is changed
(from red to green).
FIG. 15. Evolution of Mdisk and Jdisk for RB = 0 (dotted curves), 1.3 × 10−5(long-dashed curves), 5 × 10−5(solid curves),
and 2 × 10−4(dashed curves).
In this simulation, magnetic energy decreases monotonically due to a small expansion of the disk induced by the
magnetic pressure. In this case, no instability sets in. This is a natural consequence since the field lines are parallel
to the rotational motion, and hence, they are not wound by the differential rotation. Obviously, the assumption of
the axial symmetry prohibits deformation of the magnetic field lines and plays a crucial role for stabilization. If a
nonaxisymmetric simulation is performed, MRI could set in [64,65,23].
VII. SUMMARY AND DISCUSSION
In this paper, we describe our new implementation for ideal GRMHD simulations. In this implementation, Einstein’s
evolution equations are evolved by a latest version of BSSN formalism, the MHD equations by a HRC scheme, and
the induction equation by a constraint transport method. We performed a number of simulations for standard test
problems in relativistic MHD including special relativistic magnetized shocks, general relativistic magnetized Bondi
flow in the stationary spacetime, and fully general relativistic simulation for a self-gravitating system composed of a
neutron star and a disk. Our implementation yields accurate and convergent results for all these test problems. In
addition, we performed simulations for a magnetized accretion disk around a neutron star in full general relativity. It
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is shown that magnetic fields in the differentially rotating disk are wound, and as a result, the magnetic field strength
increases monotonically until a saturation is achieved. This illustrates that our implementation can be applied for
investigation of growth of magnetic fields in self-gravitating systems.
In the future, we will perform a wide variety of simulations including magnetized stellar core collapse, MRI for
self-gravitating neutron stars and disks, magnetic braking of differentially rotating neutron stars, and merger of
binary magnetized neutron stars. Currently, we consider that the primary target is stellar core collapse of a strongly
magnetized star to a black hole and a neutron star which could be a central engine of gamma-ray bursts. Recently,
simulations aiming at clarifying these high energy phenomena have been performed [66,67]. In such simulations,
however, one neglects self-gravity and also assumes the configuration of the disks around the central compact object
and magnetic fields without physical reasons. On the other hand, Newtonian MHD simulations including self-gravity
consistently have recently performed in [68]. However, stellar core collapse to a black hole and gamma-ray bursts are
relativistic phenomena. For a self consistent study, it is obviously necessary to perform a general relativistic simulation
from the onset of stellar core collapse throughout formation of a neutron star or a black hole with surrounding disks.
Subsequent phenomena such as ejection of jets and onset of MRI of disks should be investigated using the output of
the collapse simulation. In previous papers [26,9], we performed fully general relativistic simulations of stellar core
collapse to formation of a neutron star and a black hole in the absence of magnetic fields. As an extension of the
previous work, simulation for stellar core collapse with a strongly magnetized massive star should be a natural next
target.
It is also important and interesting to clarify how MRI sets in and how long the time scale for the angular momentum
transport after the onset of the MRI is in differentially rotating neutron stars. Recent numerical simulations for
merger of binary neutron stars in full general relativity [6,7] have clarified that if total mass of the system is smaller
than a critical value, the outcome after the merger will be a hypermassive neutron star for which the self-gravity is
supported by strong centrifugal force generated by rapid and differential rotation. Furthermore, the latest simulations
have clarified that the hypermassive neutron star is likely to have an ellipsoidal shape with a large ellipticity [7],
implying that it can be a strong emitter of high-frequency gravitational waves which may be detected by advanced
laser interferometric gravitational wave detectors [69]. In our estimation of amplitude of gravitational waves [69], we
assume that there is no magnetic field in the neutron stars. However, the neutron stars in nature are magnetized, and
hence, the hypermassive neutron stars should also be. If the differential rotation of the hypermassive neutron stars
amplifies the seed magnetic field via winding-up of magnetic fields or MRI very rapidly, the angular momentum may
be redistributed and hence the structure of the hypermassive neutron stars may be significantly changed. In [7], we
evaluate the emission time scale of gravitational waves for the hypermassive neutron stars is typically ∼ 50–100 ms
for the mass M ∼ 2.4–2.7M⊙assuming the absence of the magnetic effects. Here, the time scale of ∼ 50–100 ms is
an approximate dissipation time scale of angular momentum via gravitational radiation, and hence in this case, after
∼ 50–100 ms, the hypermassive neutron stars collapse to a black hole because the centrifugal force is weaken. Thus, it
is interesting to ask if the dissipation and/or transport time scale of angular momentum by magnetic fields is shorter
than ∼ 50–100 ms so that they can turn on before collapsing to a black hole. Rotational periods of the hypermassive
neutron stars are 0.5–1 ms. Thus, if the magnetic fields grow in the dynamical time scale associated with the rotational
motion via MRI, the amplitude and frequency of gravitational waves may be significantly affected. According to a
theory of MRI [30], the wavelength of the fastest growing mode is ∼ 10(B/1012gauss)(ρ/1015g/cm3)−1/2(P/1 ms)
cm where B, ρ, and P denotes a typical magnetic field strength, density, and rotational period, respectively. This
indicates that a turbulence composed of small eddies (for which the typical scale is much smaller than the stellar
radius) will set in. Subsequently, it will contribute to a secular angular momentum transport for which the time scale
is likely to be longer than the growth time scale of MRI ∼ a few ms although it is not clear if it is longer than ∼ 100
ms. On the other hand, if the transport time scale is not as short as ∼ 100 ms, other effects associated with magnetic
fields will not affect the evolution of the hypermassive neutron stars. Indeed, Ref. [12] indicates that the typical time
scale associated with magnetic braking (winding-up of magnetic field lines) depends on the initial strength of the
magnetic fields, and it is much longer than the dynamical time scale as ∼ 100 (1012gauss/B) s. In this case, the
hypermassive neutron stars can be strong emitters of gravitational waves as indicated in [69]. As is clear from this
discussion, it is important to clarify the growth time scale of magnetic fields in differentially rotating neutron stars.
This is also the subject in our subsequent papers [63].
ACKNOWLEDGMENTS
We are grateful to Stu Shapiro for many valuable discussions and to Yuk-Tung Liu for providing solutions for
Alfv´ en wave tests presented in Sec. V and valuable discussions. We also thank Miguel Aloy, Matt Duez, Toni Font,
S. Inutsuka, A. Mizuta, Branson Stephens, and R. Takahashi for helpful discussions. Numerical computations were
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performed on the FACOM VPP5000 machines at the data processing center of NAOJ and on the NEC SX6 machine
in the data processing center of ISAS in JAXA. This work was in part supported by Monbukagakusho Grant (Nos.
15037204, 15740142, 16029202, 17030004, and 17540232).
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