2008 3/13 AIU08@KEK 1 Yousuke Takamori ( Osaka City Uni v. ) with Hideki Ishihara, Msashi Kimura, Ken-ichi Nakao,(Osaka City Uni v.) Masaaki Takahashi(Aichi Univ. of Edu.) ,Chul-Moon Yoo(YITP) Numerical Study of Stationary Black Hole Magnetosph eres -Toward Blandford-Znajek mechanism by fast rotating black holes-
Yousuke Takamori ( Osaka City Univ. ). Numerical Study of Stationary Black Hole Magnetospheres -Toward Blandford-Znajek mechanism by fast rotating black holes-. with Hideki Ishihara, Msashi Kimura, Ken-ichi Nakao,(Osaka City Univ.) - PowerPoint PPT Presentation
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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)
takamori
Pourpose of our study is understunding the origine of activity of AGN.This photo is one of AGN from Hubble Space Telescope.Astrophysists think there is a BH in the center of AGN and it works as an engine of AGN. The mechanism of energy generation is roughly divided into two ways.One is due to a release of the gravitational energy of acreting matter. The other is an extraction of the rotational energy of the accretion disk or the rotating BH.In this talk, we think the rotational energy of rotating BH as energy of AGN, and Blandford-Znajek mechanism as the extraction mechanism from rotating BH.
2008 3/13 AIU08@KEK 3
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
takamori
I'm going to explain B-Z mechanism.B-Z mechanism 提案された by Blandford and Znajek in 1977.They thought stationary axisymmetric force-freeelectromagnetic fields in a kerr background and showed there is a positive energy flux on the Kerr BH horizon in this condition.
2008 3/13 AIU08@KEK 4
・ 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.
takamori
What is global magnetic configuration E^{r}を大きくする?素朴には,in fast-rotating BH case E^{r} は大きくなるHowever in this case 先行研究では,in vaccum and extremal BH case magnetic filed line did'nt penetrate the horizon which mean ∂_{θ}A_{φ} |r=r+=0.This is known as マイスナー effect.And, ダイナミカルシミュレーションでは,,this is not vaccum,マイスナー effect did'nt appear.我々は定常解を求めこの問題を確かめようと考えています.
2008 3/13 AIU08@KEK 5
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
takamori
We consider stationary axisymmetric force-freeelectromagnetic fields in Kerr background.Useing stationary and axisymmetric, electromagnetic field is written by 3-scalarfunctions I, A_{\phi}, and A_{t}.A_{\phi} and A_{t} is components of the vectorpotential. And I is cuurent.Then assuming the force-free condition, electromagnetic fields is written by only1-scalar function A_{\phi}.
2008 3/13 AIU08@KEK 6
・ Force-free
・ Stationary axisymmetric electromagnetic field
Grad-Shafranov equation
Assumptions
Maxwell equations
Basic equation
・ Kerr background
takamori
結局,From Maxwell equations, the equationwhich determine the A_{\phi} is obtained.This equation is called Grad-Shafranov equation.
2008 3/13 AIU08@KEK 7
: vector potential
:Electric current
:Angular velocity of magnetic field
Grad-Shafranov(G-S) equation
takamori
具体的には,Grad-Shafranov equation はこれです.We obtain magnetic field configuration arrounda Kerr BH solving G-S equation.
2008 3/13 AIU08@KEK 8
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
takamori
Solving G-S equation, いくつか問題があります.Firstly, G-S equation is quasi-nonlinear secondorder partial differntial equation, so it is difficult to solveanalitically.Then this equation has two kind of singularsurfaces. One is event horizon defined by Δ=0and the other is the so-called light surfacedefined by D=0.At these surfaces, Ω_{F}の角速度で運動する観測者が光速になるところです.In general, there are two light surfaces inBH cases.Solving G-S equation numerically,due to the light surface singularities, it isdifficult to obtain a 連続滑らかな solution at the light surfaces.
2008 3/13 AIU08@KEK 9
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
takamori
light surface singularityがない場合は簡単にとけます.In this case, G-S equation is non-singularelliptic differential equation so imposing aboundary condition a smooth solution is obtained.
If second-order differential coefficients are finite at D=0,we obtain regularity condition N=0.Giving I and Ω_{F}, this condition is becoming the boundary condition at LS.したがって,G-S equation はLSのうちと外を別々に解くことになる.すると,このように求めた解 is discontinuous at the LS.
2008 3/13 AIU08@KEK 12
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
takamori
To obtain a slution which is contiuous at LS,this regularity condition is treated as an equation which determine IdI.Then, we solve G-S equation IdIを決めながらitarativeにThis method is successufl method 提案されたby Contopoulos in pulsar case.
2008 3/13 AIU08@KEK 13
・ 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
2008 3/13 AIU08@KEK 14
ILS
OLS
Blandford-Znajek Monopole Solution
Rigidly rotating
This is a solution under the slow-rotating BH approximation.
BH
for
2008 3/13 AIU08@KEK 15
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
2008 3/13 AIU08@KEK 16
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
2008 3/13 AIU08@KEK 17
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
takamori
Considering the B-Z mechanism, we are interesting a magnetic configurationwhich is smooth arround the horizon.So we use this condition at ILS.
2008 3/13 AIU08@KEK 19
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.
2008 3/13 AIU08@KEK 20
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
2008 3/13 AIU08@KEK 23
・ 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
2008 3/13 AIU08@KEK 24
Plan of this talk
・ Introduction
・ Grad-Shafranov equation
・ Test Simulation
Blandford-Znajek Monopole Solution
・ Future study
・ Summary
takamori
This is contents of my talk.First, I'm going to talk about our motivation for this study.Then, I'm going to introduce our numerical method to obtain a solution of stationary BH magnetosphere,and preliminary results.Finally, I'm going to summarize this talk.