2008 3/13 AIU08@KEK 1
Yousuke Takamori ( Osaka City Univ. )
with Hideki Ishihara, Msashi Kimura, Ken-ichi Nakao,(Osaka City Univ.)
Masaaki Takahashi(Aichi Univ. of Edu.) ,Chul-Moon Yoo(YITP)
Numerical Study of
Stationary Black Hole Magnetospheres
-Toward Blandford-Znajek mechanism by fast rotating black holes-
2008 3/13 AIU08@KEK 2
Introduction
Possible origin of energy
1.Gravitational energy
2.Rotational energy
・ Accretion Disk
・ Rotating BH
Blandford-Znajek mechanism
(Blandford & Znajek 1977)
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Angular-Velocity of BH
Angular Velocity of
Magnetic Field
Energy flux at the event horizon
Blandford-Znajek(B-Z) Mechanism
If
there is a positive energy flux
outward at the even horizon.
BH
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・ Non electro vacuum and dynamical case
Numerical simulation suggests “Meissner effect” is not seen in
maximally rotating Kerr BH case (Komissarov & McKinney 2007).
: energy flux:Angular velocity of BH
:Angular velocity of Magnetic Field
:Magnetic field
・ Electro vacuum and stationary case
at maximally rotating Kerr BH horizon (Bicak 1976).
“Meissner effect”
It is important to clarify the angular velocity of Kerr BH
and the magnetic field configuration for maximal energy extraction.
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Assumptions
・ Stationary axisymetric
・ Kerr background
・ Force-free
Electric filed and Magnetic filed is written by
:Electric current
:Vector potential
:Current density vector
:field strength tensor
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・ Force-free
・ Stationary axisymmetric electromagnetic field
Grad-Shafranov equation
Assumptions
Maxwell equations
Basic equation
・ Kerr background
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: vector potential
:Electric current
:Angular velocity of magnetic field
Grad-Shafranov(G-S) equation
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Property of G-S equation
・ G-S equation is quasi-nonlinear second order partial
differential equation.
・ G-S equation has two kind of singular surfaces.
: Event horizon
: Light surfaces
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For non-rotating BH and non-rotating
magnetic field
Numerical boundary
Numerical domain
Impose a boundary condition.
Dirichlet, Neumann etc.
A smooth solution in the numerical domain is obtained.
G-S equation is non-singular elliptic
differential equation.
BH
equatorial plane
rotational axis
2008 3/13 AIU08@KEK 10
For rotating BH and
rotating magnetic field
Numerical boundaryNumerical domain
impose a boundary condition.
Dirichlet, Neumann etc.
A smooth solution in the numerical
domain will be not obtained.
There are two light surfaces in
G-S equation.
Inner light surface
(ILS)
Outer light surface
(OLS)
BH
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If and are given functions,
At
We can solve G-S equation in both
sides of a light surface, independently.
is Neumann boundary
condition at the light surfaces.
A solution will be discontinuous
at the light surfaces.
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This equation is treated as the equation which determines .
Treatment of Light Surface
(Contopoulos et al, 1999)
G-S equation can be solved by using iterative method .
Then a solution is smooth and continuous at the light surface.
・ G-S equation at the light surface
・ Regularity condition at the light surface
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・ As a first step of our study, we constructed
numerical code in the domain including
the outer light surface.
Test simulation
・ We tried to obtain a Blandford-Znajek
monopole solution as a test simulation.
OLSILS
BH
Numerical boundaryNumerical domain
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ILS
OLS
Blandford-Znajek Monopole Solution
Rigidly rotating
This is a solution under the slow-rotating BH approximation.
BH
for
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Computational domain and Set Up
We solved G-S equation in the domain including
the outer light surface.
We solve numerically.
We put as
BH
We factorize as
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Results
:B-Z monopole solution
:Numerical solution
0
5e-005
0.0001
0.00015
0.0002
0.00025
0 5 10 15 20 25 30 35 40 45 50
"pr01.dat"
"bz01.dat"
OLS
:Red line
:Green line
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Near the Outer Light Surface
about 20% discrepancy
Slow-rotating BH approximation is not guaranteed far from BH
(Tanabe & Nagataki 2008). Then this result is consistent.
3e-006
4e-006
5e-006
6e-006
7e-006
8e-006
9e-006
1e-005
1.1e-005
1.2e-005
20 25 30 35 40 45 50
"pr01.dat"
"bz01.dat"
OLS
2008 3/13 AIU08@KEK 18
Future Study
Numerical boundaryNumerical domain
ILS OLS
・ We should construct a numerical code
to study the domain including the ergo
region.
・ We have to determine at the
inner light surface.
・ The outer light surface is treated as
a numerical boundary.
Ergo region
We are constructing a numerical code which determines
at the inner light surface.
BH
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BH
・ We know and its derivative at the outer
light surface. Then we can construct a solution
for G-S equation beyond the outer light surface
as a Cauchy problem.
Beyond the Outer Light Surface
integration direction
If we solve G-S equation as a Cauchy problem,
we can not impose a boundary condition here.
・ However, numerical simulation is not stable
because G-S equation is elliptic equation.
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Summary・ We constructed the numerical code in the domain
including the outer light surface.
As a test simulation, we obtained numerical solutions
with the boundary condition similar to B-Z monopole
solution.
・ Slow-rotating approximation is not so good near and
beyond the outer light surface.
・ We are constructing a numerical code which determines
at the inner light surface.
2008 3/13 AIU08@KEK 21
Numerical procedure
を解く
初期 A_{φ} と境界条件を与える.
D=0 となる場所を探す.
D=0 で N=0 から電流を決める.
LS 以外
LS 上
2008 3/13 AIU08@KEK 22
Treatment of Two Light Surfaces
If we determine IdI from ILS(OLS) regularity condition
OLS(ILS) regularity condition become boundary condition
at the OLS(ILS)
given
determined
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・ There is the regularity condition at
the event horizon (Znajek 1977).
We are constructing a numerical code which determine
at the inner light surface.
Our approach
・ The physical environment far from BH
is complicated.
・ Because we study B-Z mechanism, we want to
treat the event horizon as the numerical boundary.
BH
OLS
ILS
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Plan of this talk
・ Introduction
・ Grad-Shafranov equation
・ Test Simulation
Blandford-Znajek Monopole Solution
・ Future study
・ Summary