2008 3/13AIU08@KEK1 Yousuke Takamori （ Osaka City Univ. ） with Hideki Ishihara, Msashi Kimura,...

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.

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

takamori

light surface singularityがある場合は境界条件を課すと，一般的に滑らかな解が得られません．その理由を次のスライドで説明します．

2008 3/13 AIU08@KEK 11

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.

takamori

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


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


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


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


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.

2008 3/13AIU08@KEK1 Yousuke Takamori （ Osaka City Univ. ） with Hideki Ishihara, Msashi Kimura,...

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