1. 2008 3/13AIU08@KEK1 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-

2. 2008 3/13AIU08@KEK2 Introduction Possible origin of energy 1.Gravitational energy 2.Rotational energy ・ Accretion Disk ・ Rotating BH Blandford-Znajek mechanism (Blandford ＆ Znajek 1977)

3. 2008 3/13AIU08@KEK3 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

4. 2008 3/13AIU08@KEK4 ・ 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.

5. 2008 3/13AIU08@KEK5 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

6. 2008 3/13AIU08@KEK6 ・ Force-free ・ Stationary axisymmetric electromagnetic field Grad-Shafranov equation Assumptions Maxwell equations Basic equation ・ Kerr background

7. 2008 3/13AIU08@KEK7 ： vector potential :Electric current :Angular velocity of magnetic field Grad-Shafranov(G-S) equation

8. 2008 3/13AIU08@KEK8 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

9. 2008 3/13AIU08@KEK9 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

10. 2008 3/13AIU08@KEK10 For rotating BH and rotating magnetic field Numerical boundary Numerical 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

11. 2008 3/13AIU08@KEK11 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.

12. 2008 3/13AIU08@KEK12 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

13. 2008 3/13AIU08@KEK13 ・ 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. OLS ILS ＢＨ Numerical boundary Numerical domain

14. 2008 3/13AIU08@KEK14 ILS OLS Blandford-Znajek Monopole Solution Rigidly rotating This is a solution under the slow-rotating BH approximation. ＢＨ for

15. 2008 3/13AIU08@KEK15 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

16. 2008 3/13AIU08@KEK16 Results :B-Z monopole solution :Numerical solution OLS :Red line :Green line

17. 2008 3/13AIU08@KEK17 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. OLS

18. 2008 3/13AIU08@KEK18 Future Study Numerical boundary Numerical 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

19. 2008 3/13AIU08@KEK19 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.

20. 2008 3/13AIU08@KEK20 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.

21. 2008 3/13AIU08@KEK21 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

22. 2008 3/13AIU08@KEK22 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

23. 2008 3/13AIU08@KEK23 ・ 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

24. 2008 3/13AIU08@KEK24 Plan of this talk ・ Introduction ・ Grad-Shafranov equation ・ Test Simulation Blandford-Znajek Monopole Solution ・ Future study ・ Summary