ブラックホール磁気圏での 磁気リコネクションの数値計算 熊本大学 小出眞路 RKKコンピュー 森野了悟 ターサービス(株) BHmag2012,名古屋大学,

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1 RKK ( ) BHmag2012,,

2 Outline Motivation and basis: Magnetic reconnection around astrophysical black holes Standard equations of resistive GRMHD Test calculations of resistive GRMHD A simulation of magnetic reconnection with initially uniform magnetic field around rotating black hole Summary and Future plans

3 Black Hole Magnetosphere Corona Plasma Disk Black Hole Ergosphere Plasma Black Hole Magnetosphere

4 ω φ β φ /h φ cβ φ < c m Kerr black hole cβ φ >c Total energy of particle m (energyat-infinity) Angular momentum L < 0 Horizon E! =!mc 2 " + # $ L Lapse function Ergosphere Frame-dragging potential (Extremely deep potential without Frame-dragging effect limitation)

5 Black Hole Magnetosphere Closed magnetic field lines between ergosphere and disk: Magnetic bridge Corona Plasma Disk Black Hole Ergosphere Plasma Black Hole Magnetosphere

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

7 Ideal GRMHD simulations: A phenomena caused by the magnetic bridge near the black hole Magnetic bridge Magnetic field Magnetic field Sub-relativistic jet Current loop Kerr black hole Accretion disk Ergosphere Kerr black hole Accretion disk Ergosphere Initial

8 Ideal GRMHD result With finite resistivity Intermittent Jet Kerr black hole Magnetic field Accretion disk Ergosphere Flare of X-ray Magnetic field Magnetic reconnection Kerr black hole Ergosphere Accretion disk heating Anti-parallel magnetic field is formed Magnetic reconnection must be important near black hole horizon! GRMHD with finite resistivity Near future important subject

9 Even with uniform magnetic field as its initial condition Uniform magnetic field around rotating black hole with No Accretion disk : Initial condition (1) Kerr black hole: maximally rotating rotation parameter, a=j/j max = (2) Magnetic field: Uniform around Kerr black hole (Wald solution) (3) Plasma: zero momentum, uniform, low density and pressure ρ 0 =0.1B 02 /c 2, p 0 =0.06ρ 0 c 2 (Koide, Shibata, Kudoh, Meier 2002) Ergosphere Uniform, strong magnetic field z Uniform, thin plasma θ Kerr black hole r H r R

10 A result of ideal GRMHD simulation Kerr black hole Ergosphere Magnetic field lines Power Radiation along Magnetic Field Lines cross Ergosphere, Plasma Propagation of Alfven wave: Electromagnetic energy transportation but No Outflow t = 7τ S Koide, Shibata, Kudoh, Meier 2002

11 Magnetic field lines Current sheet Longer term simulation of uniformly magnetized plasmas around Kerr BH ergosphere Frame-dragging effect Kerr black hole Kerr black hole Ω H Ideal GRMHD simulation of uniformly magnetized plasma around rotating black hole (Komissarov 2005) Frame-dragging potential, Magnetic reconnection? reconnection: Numerical calculation Angular momentum, L<0! " L < 0 Schwarzschild radius Initial: uniform magnetic field (Wald solution) sprit monopole-like magnetic field in ergosphere magnetic reconnection in ergosphere

12 Outline Motivation and basis: Magnetic reconnection around astrophysical black holes Standard equations of resistive GRMHD Test calculations of resistive GRMHD A simulation of magnetic reconnection with initially uniform magnetic field around rotating black hole Summary and Future plans

13 Covariant form of standard resistive GRMHD equations Maxwell equations: proper particle number density 4-velocity Energy-momentum tensor Field strength tensor Unit system c = 1 µ 0 = 1 4-current density F µ! U! = "( J µ + ( U! J! )U ) µ resistivity ( ) 2 Lapse function:! = h 2 0 +! h i " i Shift vector:! i = h i " i /# i! =! 1,! 2,! 3 (gravitational time delay) (velocity of dragged frame) ( )

14 !D!t 3+1 form of resistive GRMHD equations for numerical calculations, γρδ = "# $[%D(v + c &)]! general relativistic effect!p!t = "# $[%(T + c&p)] " ( D + ' + ) * c 2, - #(c 2 %) + % f curv " P :. Shear of framedragging special relativistic effect!"!t = #$ %[&(c2 P # Dc 2 v + ec')] # ($&) % c 2 P # T :(!B = "# $ [%(E " c& $ B) ]! ( J + " e c# ) + 1 $E!t c 2 $t = % & -! ' B + # ( ) c & E * 0 / +, 2. 1! " B = 0! e = " c 2 # $ E E + v! B = " - J $ # 2 ' % e $ 1 v & J # / ( ). c 2 ( ) * 0 +, v2 1 All of these equations are coupled! We have to consider Ampere s law and Gauss s law of electric charge. We use the numerical method of Watanabe & Yokoyama (2006).

15 Standard resistive GRMHD equations =Relativistic one-fluid model with resistivity Conservation of particle number, momentum, and energy Maxwell equations Standard relativistic Ohm s law v g > c Resistivity Group velocity of electromagnetic wave is greater than light speed Causality broken? Dispersion relation of electromagnetic wave in resistive plasma Now numerical calculations are limited in special relativistic version.

16 v g > c v S' > c2 v g Electromagnetic wave packet

17 Generalized GRMHD equations ( ) = 0!! "U! 1 ne! $ µh! & ne U µ J! + J µ U! %& = 1 * 2ne!µ "µ # "p +, ( ) + "h ( ) + U µ # "µ Thermo-electromotive force ( ) 2 J µ J! 2 U µ U! # µ"h# ne ne J! Equation of continuity Equation of 3+1 formalism required for numerical calculation of the generalized GRMHD are much complicated so that no one wants to perform calculation with them. Conservation of momentum and energy ' ) () -. / F µ #" $ J µ! % + U! J! ( ) 1+ 0 Inertia and momentum of charge and current ( )U µ ' ( To clarify the limitation of the standard resistive GRMHD equations, we tried to perform acausal phenomena expected (not welcome) in the standard equations. Hall effect Standard Ohm s law Generalized general relativistic Ohm s law! < (2.5 ~ 3)" p!1 = (2.5 ~ 3) m e# µh v g! c

18 In the new coordinates with fast plasma flow, the wave packet has monotonic amplitude profile and then could not send a signal It seems that monotonic wave decreases. This shape of the wave is not wave packet anymore. The wave packet is not Lorentz invariant.

19 Wave-packet is not Lorentz invariant, and then cannot be accept as a relativistic object. The group velocity, which represents propagation velocity of wave packet, is not relativistic quantity. In fact, the definition of the group velocity, vg= ω k, is not scalar nor covariant variable. We cannot discuss causality using the group velocity. Instead of group velocity, head velocity vh=lim k ω k presents the propagation velocity of information. In resistive RMHD, the head velocity is light speed, and Lorentz invariant. Concerning causality, there is no limitation of standard resistive GRMHD! Koide & Morino, Phys. Rev. D, 84, , Jackson, Classical Electrodynamics (1962).

20 Outline Motivation and basis: Magnetic reconnection around astrophysical black holes Standard equations of resistive GRMHD Test calculations of resistive GRMHD A simulation of magnetic reconnection with initially uniform magnetic field around rotating black hole Summary and Future plans

21 Test calculations of resistive GRMHD Interaction between plasma and uniform magnetic field around black hole

22 Plasma fall in uniform magnetic field into a Schwarzschild black hole! = 10!2 r S c ( RMHD) Color: pressure a = 0 Unit of length r S = 2GM c 2 Unit of time! S = r S c

23 Plasma fall in uniform magnetic field into a Schwarzschild black hole! = 10 3 r S c ( ) Color: pressure a = 0

24 ! = 10!3 r S c!!!! a! J J max = 1 z / r S J max = GM 2 c Ergosphere Kerr Black Hole Twist by frame-dragging effect v = 0 HLL+MUSCL scheme r / r S

25 Outline Motivation and basis: Magnetic reconnection around astrophysical black holes Standard equations of resistive GRMHD Test calculations of resistive GRMHD A simulation of magnetic reconnection with initially uniform magnetic field around rotating black hole Resistive GRMHD simulation with initial uniform magnetic field around a rotating black hole (2 trial) Resistive GRMHD simulation with initial split monopole magnetic field around a rotating black hole Summary and Future plans

26 Current sheet Longer term simulation of uniformly magnetized plasmas around Kerr BH ergosphere Frame-dragging effect Kerr black hole Kerr black hole Ω H Ideal GRMHD simulation of uniformly magnetized plasma around rotating black hole (Komissarov 2005) Magnetic field lines magnetic reconnection (if resistivity is significant) Schwarzschild radius Initial: uniform magnetic field (Wald solution) sprit monopole-like magnetic field in ergosphere magnetic reconnection in ergosphere

27 Trial calculation with uniform magnetic field as initial field a = ! = Color: B 1/2 Lines: magnetic field Arrows: velocity

28 Trial calculation with uniform a = 0.94! = Color: B 1/2 Lines: magnetic field Arrows: velocity magnetic field (try #2)

29 Initial condition of magnetic field of a t = 0 simple case: Split monopole field z a = r Resistivity! = 0.01r S Relativistic Alfven velcoty v A = B!c 2 +!! "1 p + B2 Split-monopole magnetic field Relativistic magnetic reconnection c

30 Magnetic reconnection in split-monopole magnetic field around black holes! = 0.01r S Rapidly rotating (Kerr) black hole case: a = Color: B 1/2 Lines: magnetic field Arrows: velocity What mechanism causes the magnetic reconnection in the ergosphere?

31 A expected mechanism of magnetic reconnection around Kerr black hole Magnetic pressure gradient drives the magnetic reconnection? Ω H Black Hole framedragging effect B φ Initial magnetic field magnetic pressure - B 2 φ Current sheet - B 2 φ Magnetic reconnection? magnetic pressure B φ

32 Magnetic reconnection in split-monopole magnetic field around black holes Schwarzschild black hole case: a = 0! = 0.01r S Color: pressure Lines: magnetic field Arrows: velocity

33 Magnetic reconnection in split-monopole magnetic field around black holes Schwarzschild black hole case: a = 0! = 0.01r S Color: pressure Lines: magnetic field Arrows: velocity

34 Magnetic reconnection in split-monopole magnetic field around black holes Schwarzschild black hole case: z arrow: velocity, color: density z v p r v p, max =0.62c 5 r

35 Initial condition of magnetic field t = 0 z a = 0 Relativistic Alfven velcoty v A = B!c 2 +!! "1 p + B2 Relativistic magnetic reconnection c r Split-monopole magnetic field

36 Magnetic reconnection rate Lapse function General relativistic a = 0!E " =!#J " Faraday s law !B!t [ ] = "# $ %(E " c& $ B) z r

37 Evaluation of tearing instability Maximum growth rate of tearing instability:! max! " A "1 R "1/2!! " A "1 a CS k! A = a CS v A! = 0.01r S, a CS = 0.1, v A = 1! A = 0.1! S, R = 10! max = 3" S!1 ( ) "2/5 R "3/5, ( ka CS ) max! R "1/4 (Alfven transit time) (Non-relativistic)! max = 2" k max! r S R = a CS v A! Thickness of current sheet (Magnetic Reynolds number) However, the magnetic reconnection does not dependent on the perturbation of the initial velocity at the current sheet. This indicates that the tearing instability has no essential role in the magnetic reconnection.

38 Analysis of numerical results of magnetic reconnection around Schwarzschild black hole 1.5 t=0 1.5 t=10 z z r r Pressure

39 Preliminary model of the magnetic reconnection in ergoshpere: Mechanism of Daruma-Otoshi MHD equilibrium Break-down High pressure gas Daruma-otoshi: Japanese traditional toy. When one piece is hit to shit out, the pieces upper and lower are connected.

40 Expected flow from ergosphere driven by magnetic reconnection in initial uniform magnetic field case flow flow flow flow Even in the initial uniform magnetic field case, around the rotating black hole, magnetic reconnection is caused spontaneously and hollow flow from the ergosphere is expected to appear. Ideal GRMHD simulation of uniformly magnetized plasma around rotating black hole (Komissarov 2005)

41 Kerr black hole case z a = r Color: azimuthal component of magnetic field, B Solid lines: magnetic field lines Arrows: velocity of plasma In Kerr black hole case: v p,max =0.71c (at t=10 s ) In Schwarzschild black hole case: v p,max =0.63c (at t=10 s ) More explosive magnetic reconnection is occurred in Kerr black hole case compared with the case of Schwarzschild black hole case.

42 Summary I present the basis of resistive GRMHD, with respect to plasma dynamics around black holes, the standard equations, and its causality. I performed simulations of resistive GRMHD with splitmonopole magnetic field around black holes. The primary results showed spontaneous magnetic reconnection. I proposed a model, daruma-otoshi mechanism of magnetic reconnection around black holes. The numerical results suggest even in initial uniform magnetic field case, the magnetic reconnection is caused spontaneously around the rotating black hole. In conclusion, in black hole magnetosphere, it is full of magnetic reconnection!

43 Future plans Further calculations and analysis of magnetic reconnection around black holes are needed to confirm the mechanism of the magnetic reconnection near the black hole. To perform numerical simulations with longer-term, more realistic magnetic field configuration and clarify importance of magnetic reconnection around astrophysical black holes, numerical code with more precise numerical methods, such as HLLD, are required. Radiation should be included in GRMHD simulations. I consider test EM wave calculation with resistive GRMHD. Comparison between magnetic field configurations suggested by our future precise numerical simulations and forthcoming radio observations with high spatial resolution.