E E T Ti F R T Ti Energy Extraction from Rotating Black Hole by Magnetic Reconnection

EEttifRttiEnergy Extraction from Rotating Black Hole by Magnetic Reconnection

Shinji Koide Kumamoto University

Collaborators: Kenzo Arai (Kumamoto University)

Koide & Arai, ApJ 682, 1124-1133 (2008)

SCSAMM workshop 2009.8.28(Fri) Outline

• Motivation • Energy extraction mechanism from black hole: Penrose process • A mechanism of energy extraction from black h o le throug h magne tic reconnec tion • Summary and future aspects One of Motivation: Magggpnetic reconnection in the ergosphere

ΩH Ideal GRMHD simulation of uniformly magnetized plasma around rotating black hole (Komissarov 2005) Kerr x Schwarzschild radius black hole Magnetic rS field lines t  60 ergosphere 2c magnetic Frame-dragging reconnection x (if resistivity is effect significant) Kerr black hole a  0.9

Initial: uniform magnetic field (Wald solution) sprit monopole magnetic field in ergosphere ֜ magnetic reconnection in ergosphere ֜ Notation of space-time and Ergosphere Space-time around rotating black hole (Kerr metric)   x 0 , x1 , x 2 , x3  t,r,, metric ds2  g dx dx 3 g 2 (ittild(gravitational red-shift,    g  i0 Lapse function 00  gravitational potential) i1 gii g Anggyular velocity of    i0 i g frame-dragging  ii   g  Shift vector  (velocity of frame dragging)  ZAMO coordinates xˆ 0 , xˆ1 , xˆ 2 , xˆ 3  (local Minkowski frame) (zero angular momentum observer)

Ergosphere:   1 Any matter, energy, and information never propagate against the rotation of black hole rotation in the ergosphere.

Unit system: c  1,  0  1,  0  1 Conservative quantity of a particle aroundiblkhld rotating black hole • Energy-at-infinity (total energy): Rest mass energy + kinetic energy + gravitational potential E   Eˆ   L   Eˆ    P  “Frame-dragging potential” Energy observed b y ZAMO Eˆ  m  • Angular momentum Lorentz factor ˆ   L  g  P  Momentum Pˆ  m v  When L  Eˆ /   in ergosphere, the energy-at-infinity becomes negative, E   0. Penrose Process Extraction of energy of rotating black hole through particle fission in ergosphere

   Conservation of angular momentum LA  LB  LC    Conservation of energy EA  EB  EC A→B+C (fission ֜ redistribution of angular Larger energy-at-infinity momentum) C than injected particle A E   E  ˆ   C A When LB  EB /  in ergosphere, EB  0. B Fission Negative If particle B is swallowed by black energy-at- infinity Kerr Horizon hole, total mass of black hole E   0 B Black decreases. The particle C obtains

 HlHole energy larger than that of injected  1     Ergosphere particles: EC  EA  EB  EA   1  M 'BH  M BH  EB  M BH Frame Dragging Effect Energy extraction from black hole Mechanism of extraction of black hole rotational energy • Penrose process Through nega tive • MHD Penrose process energy-at-infinity • Blandford-Znajek mechanism To realize negative energy-at-infinity, we require redistribution of angular momentum.

Mechanisms Carrier of negative energy Force for redistribution Penrose Particle Particle fission MHD Penrose Plasma Magnetic tension Blandford-Znajek Electromagnetic field ー Magnetic tension How about magnetic reconnection? Magnetic reconnection： Plasma acceltibleration by reconnec tiftion of magne tifildtic field lines Ideal MHD condition =frozen-in condition Plasma acceleration B0

0 Electric J 0 resistivity Plasma outflow Magnetic field lines Breakdown region of ideal MHD condition Magnetic sling-shot

Plasma

acceleration Terminal velocity vterm Magnetic tension Terminal velocityyp of plasma outflow from ma gnetic reconnection when magnetic energy is converted to kinetic energy of plasma flow (Non-relativistic case) 2 2 0 2 B0 B0 vterm  vterm vA (Alfven vel ocit y) 2 2 0 In relativistic case, we have to consider inertia of pressure. Magnetic reconnection and particle fission

Particle fission Penrose process PtilParticle

Magnetic reconnection Magnetic field lines NhiNew mechanism Plasma of energy extraction outflow from rotating BH? ergosphere Frame-dragging effect

ω Koide & Arai 2008 Escape and fall of a pair of plasma elements ejected from magnetic reconnection region (Schematic picture) Magnetic Backward reconnection plasma E   0 ejection B E   E  C C A B Forward plasma Kerr ejection black Magnetic field lines Interchange hole of plasma ergosphere

Very complex phenomena: numerical simulation with general relativistic MHD with resistivity (resistive GRMHD) Simulation of magnetic reconnectiiion in ergosp here •We require numerical simulation to understand global mechanism of the magnetic reconnection in ergosphere. • HhHowever, we have no numer ilhifical technique of resistive GRMHD. (This is reasonable because of standard formulation has causality problem. ) • To estimate the energy extraction of black hole due to magnetic reconnection, we use simplified model with combination of slab model of magnetic reconnection and general relativistic potential. Analytic model of magnetic reconnection in ergosphere

Rest frame of Local slab bulk plasma approximation out C B Energy-at-ifiitinfinityoflf plasma Bulk plasma A element ejected Plasma element  v forwardly/backwardly with the vˆKepler c0 velocity vˆK K ejected forwardly ejection velocity v’ = v’   ± out  B Plasma element  E  ˆ ˆ  C ejected backwardly E  E   P  E  (‘+’ = ‘C’, ‘ー’ = ‘B’) B  C ˆ  1 E  H ˆ  U int ˆ ˆ ˆ ˆ P  H  v Thermal Enthalpy energy v  v' Velocityyp of plasma vˆ  K out  1 v v' element ejected  K0 out observed by ZAMO θ=π/2 1/ 2  2 ˆ  1 vˆ Lorentz factor Slab model of relativistic magnetic reconnection Anti-parallel magnetic field Magnetic x' (bulk plasma reconnection x' P'  0 rest frame) point

Forward Backward ejection Current sheet ejection v’out v’ out y' y' Py'0 Py'0   0 Pair-plasma flow: Redistribution of Resistive (angular) momentum! region   0 (()gLocal approximation) We neglect • Tidal force and Coriolis’ force • Global magnetic field reaction Terminal velocity of plasma outflow from relativistic magnetic reconnection: inertia effect of pressure. Complete energy conversion of magnetic energy to kinetic energy of plasma elements (assumption):

2 Magnetic and thermal  1U B0 Particle energy  'out H    H   1 U energy per particle   ' 2n after reconnecti on out 0 before reconnection

Then, we get maximum of 4-velocity of the plasma element due to magnetic reconnection:

2 1/ 2 1  u 2  1  u 2   u'  ' v'  1 A    1  1 A   D out out out       2  2  2  2   2  u 2   where A D  1    4  2  1/ 2 1    p  p    p  u  0  0 ,   0 0  0 A  2 2  2  2 2  B0  1 B0  B0 B0  1 B0  Relativistic magnetic reconnection ＜Including inertial effect of (thermal) energy＞ 

 Analytical model (Koide & Arai 2007) 2 1/ 2 1  u 2  1  u 2   u'  ' v'  1 A    1  1 A   D out out out       2  2  2  2  

1/ 2 uout '    p  u  0  0 A B 2  1B 2    out 'vout '  0 0  B2 Alfven four-velocity, 0 h0 1 p    p    0 0  0 2  2 2  B0 B0  1 B0  Pressure per enthalpy p p0 / h0 P  2 B0 / 2 Γ: Polytoropic index Simulation of Relativistic Magnetic Reconnection Watanabe & Yokoyama, ApJ. 2006

Momentum density 2 ：  B 2  ~ c  ~ c   0  Possibility of redistribution of significant momentum Comparison between analytic and numerical solutions of relativistic magnetic reconnection 

 Analytical model (Koide & Arai 2007) 2 1/ 2 1  u 2  1  u 2   u'  ' v'  1 A    1  1 A   D out out out       2  2  2  2   uout ' Numerical simulation   out 'vout ' (Watanabe & Yokoyama 2006)

p P  2 B0 / 2 Conditions of energy extraction through magnetic reconnection

C   E B (1) Formation of plasma backward flow E E  BH wihith negat ive energy-at-ifiiinfinity :      0  H (2) Escape of forward plasma flow to infinity: Mass of plasma element   E  M            1    0 H   1  Sppgyecific energy-at-infinityyp of forward and backward plasma ejection estimated by slab model  E   ' vˆ  ' 2 1       ˆ 1  vˆ  ' vˆ     ' 2 1  out  K out    H K  3 K out K 3 out ' 2 ˆ 2vˆ 2   out K K  Energy extraction condition through magnetic reconnec tion in ergosp here θ=π/2 Zero vˆ  vˆ pressure 0 Kepler EnergyNegative extraction Bulk plasma case fromenergy-at-infinity black hole in zeroftiforma pressuretion case rotates with  Keplerian velocity B0  U  out  0 h A 0    0 Relativistic out reconnection Alfven Escape to infinity 2 2 4-velocity B0  h0  c  Horizon

2  h0  0c  p0  1 r / r Enthalpy density S Energy extraction condition through magnetic reconnec tion in ergosp here θ=π/2 Finite p0 / h0  0 pressure p / h  0.3 EnergyNegativeEnergy extraction extraction 0 0 case fromenergy-at-infinityfrom black black hole hole in in p / h  0.5 pftiforma=030.3tihoncase 0 0 0p0=0.50h0 case  B0   0  U A out h0 Relativistic   out  0 reconnection  B2  Escape to infinity  0 2    h0  c   2 

Horizon p0 / h0  0

p0 / h0  0.3 2  h   c  p p0 / h0  0.5 0 0  1 0 Enthalpy density r / rS Slower rotating black hole case

p / h  0 Negative 0 0 energy-at-infinity Finite  p0 / h0  0.3  out  0 formation pressure p0 / h0  0.5 case

B 0  U  A  out  0 h0 Escape to infinity

Note that there is no circular orbit in ergo- p / h  0 sphere when 0 0 p0 / h0  0.3 a 1/ 2 r / rS Possibility of relativistic magnetic reconnection in individual objects Relativistic magnetic reconnection is required to extract black hole rotational energy by magnetic reconnection:

B0  Bcrit  0 0 c U A 1, p0  0  B idiidlindividual 0 crit possibility (g cm-3) (G)

AGN M87 2×10-17 500 YES (Collapsar GRB 4 1011 2 1015 Marginal model) × × Micro- GRS1915 6 10-5 8 108 NO QSO +105 × ×

Typical density near the central (SI unit system) black hole, which is assumed. Energy extraction from black hole through magnetic reconnection in situation of magnetorotational instability

Ergosph er e Frame-dragging effect Disk Rotating black hole Closed magnetic flux tube Frame- Magnetic dragging flux tube Closed effect magnetic flux tube

Koide & Arai, submitted Numerical simulations with resistive GRMHD is demanded for further investigation. Summary We show one of a peculiar phenomena of magnetic reconnection around rotating black hole, extraction of black hole rotational energy through it. We clarify criterion condition of the mechanism in a rather simple situation. • When we consider the magnetic reconnection in plasma rotating with Keplerian velocity around a rapidly rotating black hole (a=0.95), we need relativistic reconnection to extract the black hole energy. • When the ro ta tion o f black h o le becomes s lower (a= 0. 9), the region of magnetic reconnection which extracts the black hole rotational energy becomes thinner and stronger magnetic reconnec tion is requ ire d. On the case o f a 1/ 2 there is no circular orbit in the ergosphere and no possibility of the energy extraction in the simple situation. Future Aspects • To confirm the energy extraction mechanism of magnetic reconnection without artificial simplification, we have to perform global numerical simulation of generalized GRMHD with resistivity (Resistive GRMHD). Then we give consideration of the global maggynetic field and complex plasma dynamics. • However, standard resistive RMHD/GRMHD equations have causality problem, while ideal RMHD/GRMHD has no problem of causality because no electromagnetic wave can propagate in the ideal RMHD plasma. • We have to reconsider the basic equations of resistive GRMHD/RMHD on the base of relativistic two-fluid equations (Khanna 1998, Koide 2008, 2009) or Boltzmann-Vlasov equations (Meier 2004, next talk) when we perform the resistive GRMHD simulations. For example, :Two-fluid model ֜ ggqeneralized RMHD equations – (e.g. S. Koide, Phy. Rev. D 78, 125026 (2008), S. Koide, ApJ, 696, 2220-2233 (2009)) When we consider inertia of current (electron) properly, we confirmed the group velocity is less than light speed in plasmas whose plasma parameter is much greater than unity. – Numerical simulation with generalized GRMHD eqqquations is required but it is difficult to perform because we have to take into account of displacement current term and inertia of electric charge and current . Especially , Ohm’ s law becomes very complex. This will be our next work.