Yousuke Takamori （ Osaka City Univ. ）

1. 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.) Masaaki Takahashi(Aichi Univ. of Edu.) ,Chul-Moon Yoo(YITP) AIU08@KEK

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

3. Blandford-Znajek(B-Z) Mechanism Energy flux at the event horizon Angular-Velocity of BH ＢＨ If there is a positive energy flux outward at the even horizon. Angular Velocity of Magnetic Field AIU08@KEK

4. : 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” ・Non electro vacuum and dynamical case Numerical simulation suggests “Meissner effect” is not seen in maximally rotating Kerr BH case (Komissarov＆ McKinney 2007). It is important to clarify the angular velocity of Kerr BH and the magnetic field configuration for maximal energy extraction. AIU08@KEK

5. Assumptions ・Kerr background ・Stationary axisymetric Electric filed and Magnetic filed is written by :Electric current :Vector potential ・Force-free :Current density vector :field strength tensor AIU08@KEK

6. Basic equation Maxwell equations Assumptions ・Kerr background ・Stationary axisymmetric electromagnetic field ・Force-free Grad-Shafranov equation AIU08@KEK

7. Grad-Shafranov(G-S) equation ：vector potential :Electric current :Angular velocity of magnetic field 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 AIU08@KEK

9. For non-rotating BH and non-rotating magnetic field rotational axis Numerical domain G-S equation is non-singular elliptic differential equation. Numerical boundary Impose a boundary condition. Dirichlet, Neumann etc. ＢＨ A smooth solution in the numerical domain is obtained. equatorial plane AIU08@KEK

10. For rotating BH and rotating magnetic field Numerical domain Numerical boundary There are two light surfaces in G-S equation. impose a boundary condition. Dirichlet, Neumann etc. ＢＨ A smooth solution in the numerical domain will be not obtained. Outer light surface (OLS) Inner light surface (ILS) AIU08@KEK

11. At If and are given functions, is Neumann boundary condition at the light surfaces. We can solve G-S equation in both sides of a light surface, independently. A solution will be discontinuous at the light surfaces. AIU08@KEK

12. Treatment of Light Surface ・Regularity condition at the light surface This equation is treated as the equation which determines . (Contopoulos et al, 1999) ・G-S equation at the light surface G-S equation can be solved by using iterative method . Then a solution is smooth and continuous at the light surface. AIU08@KEK

13. Test simulation Numerical domain Numerical boundary ・As a first step of our study, we constructed numerical code in the domain including the outer light surface. ・We tried to obtain a Blandford-Znajek monopole solution as a test simulation. ＢＨ ILS OLS AIU08@KEK

14. Blandford-Znajek Monopole Solution This is a solution under the slow-rotating BH approximation. OLS Rigidly rotating for ＢＨ ILS AIU08@KEK

15. Computational domain and Set Up We put as We factorize as We solve numerically. ＢＨ We solved G-S equation in the domain including the outer light surface. AIU08@KEK

16. Results OLS :Red line :Green line :B-Z monopole solution :Numerical solution AIU08@KEK

17. Near the Outer Light Surface about 20% discrepancy OLS Slow-rotating BH approximation is not guaranteed far from BH (Tanabe＆Nagataki 2008). Then this result is consistent. AIU08@KEK

18. Future Study Numerical domain Numerical boundary ・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. ＢＨ Ergo region OLS ILS We are constructing a numerical code which determines at the inner light surface. AIU08@KEK

19. Beyond the Outer Light Surface ・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. ・However, numerical simulation is not stable because G-S equation is elliptic equation. integration direction ＢＨ If we solve G-S equation as a Cauchy problem, we can not impose a boundary condition here. 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. 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 AIU08@KEK

22. Treatment of Two Light Surfaces If we determine IdI from ILS(OLS) regularity condition determined OLS(ILS) regularity condition become boundary condition at the OLS(ILS) given AIU08@KEK

23. Our approach OLS ・There is the regularity condition at the event horizon (Znajek 1977). ・Because we study B-Z mechanism, we want to treat the event horizon as the numerical boundary. ・The physical environment far from BH is complicated. ILS ＢＨ We are constructing a numerical code which determine at the inner light surface. AIU08@KEK

24. Plan of this talk ・Introduction ・Grad-Shafranov equation ・Test Simulation Blandford-Znajek Monopole Solution ・Future study ・Summary AIU08@KEK