1. General Relativistic MHD Simulations with Finite Conductivity Shinji Koide (Kumamoto University) Kazunari Shibata (Kyoto University) Takahiro Kudoh (NAOJ) EANAM2006 @KASI, Daejeon, Korea, 2006.11.3(Fri)

2. Outline Numerical results of Ideal general relativistic MHD (ideal GRMHD) simulation: Jet formation by magnetic bridges between the ergosphere and disk around a rapidly rotating black hole. Anti-parallel magnetic field is formed along the jet ⇒ Magnetic reconnection Numerical method of GRMHD with finite conductivity (σGRMHD): Numerical algorithm and simple tests –Essential role of implicit method

3. Motivation: Relativistic Jets in the Universe Mirabel, Rodriguez 1998 γ>100 AGN γ>10 γ～ 3γ～ 3 ~ ~ Several M lys Several lys Gamma-ray burst Forming Spinning Black hole (?) ~ 1 km ~ 1AU  -rays X-ray, optical, radio emission ~ Light years Relativistic jet Gravitational collapse

4. Active galactic nuclei, Quasars: γ>10, L jet ~ several M pc Stellar mass black hole binaries (Microquasars): γ~ 3, L jet ~ several pc Gamma-ray bursts: γ> 100, L jet ~ 1AU- several pc ~ ~ Acceleration of plasma/gas Collimation of plasma/gas outflow 1) Magnetic field 2) Radiation pressure 3) Gas pressure Models: Relativistic Jets The jet formation mechanism may be common. These relativistic jets are formed by drastic phenomena around black holes. However, the confirmed model has not yet shown. Points of models

5. Black Hole Magnetosphere （ Corona ） Plasma Disk Black Hole Ergosphere Magnetic Field Lines Plasma Closed magnetic field lines between ergosphere and disk

6. Plasma Magnetic Field induced by Current Loop around Black Hole Plasma Disk Black Hole Ergosphere Magnetic Field Lines Current loop Magnetic bridges R0R0

7. Hayashi, Shibata, and Matsumoto (1996) Nonrelativistic MHD Simulation with Dipole-Magnetic Field and Disk Magnetic bridge Anomalous resistivity: Magnetic island (Plasmoid)

8. Frame-dragging effect Twist of magnetic bridge by ergosphere Plasma Ergosphere Magnetic bridge Disk rotation Twist by frame-dragging effect Rapidly Rotating Black Hole Current loop

9. Ideal General Relativistic Magnetohydrodynamics To investigate dynamics of the magnetic bridge between the ergosphere and the disk, we have to consider the interaction of the plasma and magnetic field near the black hole. Simplest approximation for it is given by ideal GRMHD where electric conductivity σ is infinite (σ→∞). (Ideal GRMHD)

10. general relativistic effect Special relativistic total energy density 3+1 Formalism of Ideal GRMHD Equation where (conservation of particle number) (equation of motion) (equation of energy) (Maxwell equations) (ideal MHD condition) : (Lapse function), : (shift vector) special relativistic effect Special relativistic mass density,  Special relativistic total momentum density No coupling with other Eqs. ～ similar to nonrelativistic ideal MHD (conservative form) (equation of state)

11. Numerical Method The ideal GRMHD equations are similar to those of nonrelativistic ideal MHD. Therefore, we can use the numerical techniques developed for nonrelativistic MHD calculations. In this study, we use simplified TVD method. Simplified TVD method This method is developed by Davis (1984) as a simplest shock capturing scheme for hydrodynamics. Merit: We don’t need eigen-vector of Jacobian matrix of equations like primary TVD scheme. Just maximum of eigen-value of the Jacobian is used. It is easily applied for complex equations like GRMHD equations.

12. Results of Ideal GRMHD Simulations Physical Review D 74, 044005 (Aug., 2006) http://link.aps.org/abstract/PRD/v74/e044005

13. Solid white line: Magnetic field line Color: Initial condition of Ideal GRMHD simulation t =0 Almost maximally rotating Black hole Magnetic bridge Disk: Kepler rotation -4 -2 0 2 4 Corona: hydrostatic +background pressure (Specific-heat ratio: ) Ergosphere -6

14. Condition of Ideal GRMHD simulation -4 -2 0 2 4 -6 Axisymmetry Mirror symmetry ( 210 × 70 mesh 2 ) Solid white line: Magnetic field line Color: t =0 Calculation region:

15. Time evolution: Mass density, magnetic configuration Solid white line: Magnetic field surface Color: Arrow: velocity

16. Solid line: Magnetic field surface Color: Arrow: Velocity

17. Solid line: Magnetic field line, Arrow: Velocity v max : 0.4c - 0.6c Mass density, velocity, magnetic pressure at Magnetic pressure, -4 -2 0 2 4 -5 1 0 -2 -3 -4

18. Solid line: Magnetic field line Color: Arrow: Velocity v max : 0.4c - 0.6c Final stage of calculation ： Density, velocity, magnetic configuration -4 -2 0 2 4

19. Solid line: Magnetic flux surface Color: Arrow: Velocity Magnetic configuration of final stage: Numerical magnetic island -4 -2 0 2 4 Magnetic island (Plasmoid) Ideal GRMHD: No magnetic reconnection Magnetic Island: Numerical Anti-parallel magnetic field : Numerical

20. Schematic picture of phenomena caused by the magnetic bridge near the black hole Magnetic surface Accretion disk Ergosphere Kerr black hole Kerr black hole Current loop Magnetic surface Accretion disk Ergosphere Magnetic bridge Sub-relativistic jet Initial Ideal GRMHD result Summary of Results of Ideal GRMHD and Expected Phenomena beyond Ideal case

21. Kerr black hole Magnetic surface Kerr black hole Flare of X-ray Magnetic reconnection Accretion disk Ergosphere Accretion disk heating Intermittent Jet Magnetic surface Ideal GRMHD result GRMHD with finite conductivity Anti-parallel magnetic field is formed Mixture of hot and cool plasma: Constant polytropic index EoS is not good approximation

22. Development of Numerical Method for GRMHD Simulation with Finite Conductivity Fairly new topic. But no new results of physics. Only explanation of new required method and preliminary tests.

23. Previous GRMHD Simulations = ideal GRMHD with polytropic EoS Koide, Shibata, Kudoh 1999 Gammie 2003 DeVillier & Hawley 2003 Komissarov 2004 McKinney 2005 σ=∞ ， Γ=5/3, 4/3 This assumption neglect astrophysically important effects But no GRMHD simulation with finite conductivity and more appropriate EoS.

24. general relativistic effect Special relativistic total energy density GRMHD Equations with Finite Conductivity (σGRMHD) (conservation of particle number) (equation of motion) (equation of energy) (Maxwell equations) (Ohm’s law with finite conductivity) special relativistic effect Special relativistic mass density,  Special relativistic total momentum density conductivity no correspondence to non-relativistic MHD

25. general relativistic effect Special relativistic total energy density GRMHD Equations with Finite Conductivity (σGRMHD) (conservation of particle number) (equation of motion) (equation of energy) (Maxwell equations) (Ohm’s law with finite conductivity) special relativistic effect Special relativistic mass density,  Special relativistic total momentum density ～ N. Watanabe & T. Yokoyama, ApJ 647, pp. L123-L126 (astro-ph/0607285)

26. Electric conductivity → finite: Explicit (before improved EoS (Equation of State)) Recalculation of dynamics of magnetic bridge with large conductivity (σ=100c 2 /τ) Electric conductivity → finite: Implicit (before improved EoS (Equation of State)) Recalculation of dynamics of magnetic bridge with large conductivity (σ=10,000c 2 /τ ) Improved EoS (Electric conductivity: finite) Recalculation of dynamics of magnetic bridge with large conductivity (σ=100c 2 /τ) Numerical method ofσGRMHD: Tests

27. Solid line: Magnetic field line, Arrow: Velocity Explicit method: Comparison of results of ideal and finite  GRMHD simulations at (no anti-parallel magnetic field) -4 -2 0 2 4 Color: Ideal GRMHD: Finite  :

28. Solid line: Magnetic field line, Arrow: Velocity Explicit method: Comparison of results of ideal and finite  GRMHD simulations at (no anti-parallel magnetic field) -4 -2 0 2 4 Ideal GRMHD: Finite  : Stop due to numerical instability

29. Solid line: Magnetic field line, Arrow: Velocity Implicit method: Comparison of results of explicit and implicit methods with very large conductivity at -4 -2 0 2 4 Color: σ=10 4 /cr S Explicit (ideal) Implicit (simplified)

30. ― Comparison between different EoS’s ― Ryu, Chattopadhyay, & Choi 2006 RP : Exact (Synge 1957) Γ=4/3, 5/3: Constant polytropic index TM : Mignone et al (2005) RC : Ryu et al (2006) Equation of State (EoS) Exact improved

31. Solid line: Magnetic field line, Arrow: Velocity Comparison of results of finite  GRMHD simulations before/after improved EoS at (explicit) -4 -2 0 2 4 Color: Γ ＝５／３ (before improvement) Improved EoS (TM)

32. Summary Ideal GRMHD: –The magnetic bridges between the ergosphere and disk around rapidly rotating black hole can not be stationary and expand explosively to form a jet. –The anti-parallel magnetic field is formed along the jet where the magnetic reconnection will take place, which may influence the jet propagation. GRMHD with finite conductivity (σGRMHD) is required to investigate the magnetic reconnection. We showed the new numerical method of σGRMHD and test calculations for it. –Implicit method is essential.

33. Near future plan Development of correct implicit σGRMHD code σGRMHD simulations of magnetic bridge between the ergosphere and disk around rapidly rotating black hole; Importance of magnetic reconnection in the mechanism of relativistic jet formation.