Relativistic Outflow Formation by Magnetic Field around

Rapidly Rotating Black Hole

Shinji Koide 　（ Toyama University ）

Black Hole 2003, October 29 (Wed), 2003 @ Kyoto International Community House

General relativistic magnetohydrodynamic (MHD) simulation shows relativistic outflow is driven around rapidly rotating black hole by radial magnetic field . This is the first self-consistent solution of spontaneous relativistic outflow formation in black hole magnetosphere.

Relativistic Jets in the Universe

1) Active galactic nuclei, Quasars: γ>10

2) Microquasars: γ~ 3

3) Gamma-ray bursts: γ> 100

~

~

Relativistic Jet Formation Mechanism

Acceleration of plasma/gasCollimation of plasma/gas outflow

by Magnetic Field

Acceleration Mechanism of Magnetically-Driven Jet

(Blandford & Payne 1982)

Magnetic filed lines are twisted by rotating plasma around central object

Magnetic pressure Magnetic tension

Blow off plasma Rotate plasma

Centrifugal force

Plasma outflow

(Uchida & Shibata 1985)

,Keplerrot VV rotr

Steady-state theory & Numerical simulations (Kudoh & Shibata 1998) ：

Vjet ～ Vrot ～ VKepler

Rotatingplasma disk

BJet

Vrot rrot

Centralobject

Jet

Pinch effect of twisted magnetic flux tube

Any material, energy, and information rotatesthe same direction of the black hole rotation.The plasma in it behaves like heavy rapidly rotating disk!

Relativistic Jet Acceleration by Magnetic Field

RotatingPlasma Disk

B

Relativistic jet

RotatingBlack Hole

Frame Dragging Effect

Ergosphere

Relativistic jet ：

　　　 Vjet ～ c

VKepler ～ c

Direct Outflow from Ergosphere

・　 Central object:

Rapidly Rotating Black hole

　 　 (a=J/Jmax ～ 1)

・　 rrot ～ rH⇒　 (Black hole horizon)

Direct Outflow from Ergosphere

Initial Magnetic Field Configuration

• Uniform magnetic field case(Koide et al. 2002)

◆ Powerful energy emission⇒Negative energy region in ergospere No outflow◆

No centrifugal force along vertical magnetic field lines.

• Radial magnetic field case(split monopole field)

Significant centrifugal force alongoblique magnetic field lines.

Uniform, strongmagnetic field

rH

R

zUniform, thinplasma

r

Kerrblackhole

Ergosphere

Centrifugalforce

Radialmagnetic field

rH

R

z

r

Kerrblackhole

Ergosphere

Centrifugalforce

Thin plasma

Initial Condition: Simple system of rotating black hole, thin plasma, and radial magnetic field

• Black Hole: 　　　　　　　　　 (nearly maximally rotating blac

k hole)

• Magnetic Field ： Radial magnetic field

B0=B(R=rS,z=0)

• Plasma ：　 Magnetic field dominates: ρ0=0.018B02/c2

Gas falling near the horizon. Hydrostatic equilibrium far from black hole.

99995.0max

J

Ja

50 log3

18

log3 62

0

cp

0ˆ v

A

arrarrBB gg

r 2

22222

0

cos

A

arrBB g

2

2

0

cossin2

,

,

,

0ˆ v,86.6

H

ˆ

r

rcvr

General Relativistic MHD Equations in Kerr Space-Time

• General relativistic equation of conservation laws and Maxwell equations: 　　　

　　　　　　　∇ ( n U ) = 0 　　　 (conservation of particle number)

　　　　　　　∇ T = 0 (conservation of energy and momentum)

　　　　　　　∂ F∂F∂F = 0

　　　　　　　∇ F = - J

• Frozen-in condition: FU

= 0

• Kerr Metric ：　 ds2 = gdxdx;

g= - h02 ; gii= - hi

2 ;

　　　 g0i= - hi2i (i=1,2,3) ; gij= 0 (i≠j)

n: proper particle number density. p : proper pressure. c: speed of light.e : proper total energy density, e=mnc2 + p / ( -1). m : rest mass of particles. : specific heat ratio.U : velocity four vector. A : potential four vector. J : current density four vector.∇ : covariant derivative. g : metric.

T : energy momentum tensor, T = pg + (e+p)UU+FF -gF

F/4.F : field-strength tensor, F =∂ A -∂A

(Maxwell equations)

Vector Form of General Relativistic MHD Equation(3+1 Formalism) 　

σPfPTP

:)()]([ curv2

2

c

cDc

t

)]([ Hvv

Dt

D

σTPvvP :)()]([ 222H

ceDcct

)( BβEB

ct

Eβ

BE

βJctc

c 2e

1

0B E2e c

0BvE

whereβv cH

3

1curv

j

jjji

ijij

i TGTGf ji

jiij x

h

hhG

1

(conservation of particle number)

(equation of motion)

(equation of energy)

(Maxwell equations)

(ideal MHD condition)

: (Lapse function)

ji

j

iij xh

h

c

h iii : (shift vector)23

1

20

i

ii

c

hh

general relativistic effect

special relativistic effect

: (shift velocity)

Special relativistic mass density,

Special relativistic total momentum density

Special relativistic total energy density

c

Lines: Magnetic fieldsurfacesArrows: Velocity ofplasma

z/rS

R/rS

Kerr blackhole

Ergosphere

Numerical Result: Initial Condition

2

2

logc

B

rS=2GMBH/c2

R/rS

z/rS

c

Kerr blackhole

Lines: Magnetic fieldsurfacesArrows: Velocity ofplasma

Ergosphere

2

2

logc

B

Vmax=0.86c(Lorentz factor 2.0)

τS=rS/c (Unit of time)

Relativistic Outflow driven by Magnetic Field from Ergosphere

Magnetic field lines

Kerr black hole

Ergosphere

t = 10.7S

Plasma

Magnetic field flux tube acts as propeller screw!

Time Evolution

Lines: Magnetic fieldsurfaces

Arrows: Velocity ofplasma

0

Kerr blackhole

rS=2GMBH/c2

Color: AzimuthalComponent ofMagnetic Field, Bφ

2

Vmax=0.86c(Lorentz factor 2.0)

τS=rS/c (Unit of time)

Plasma Acceleration Force)(EM BJEv W

)(ip pW v)(EM BJEv

W

)(polpolEM BJEv W

EMWMagnetic tension⇒　 Centrifugal force

Magnetic pressure/ten-sion blow off plasma⇒

SummaryGeneral Relativistic MHD simulation shows that magnetic fi

eld configuration dominates relativistic outflow formation near rapidly rotating black hole ( ).

□ Uniform magnetic field case (Koide et al. 2002): Powerful magnetic energy emission. No outflow.

■ Radial magnetic field case (present result): Relativistic plasma outflow from ergosphere. Lorentz factor, 2.0 (Vmax=0.86c).

• The plasma is accelerated by magnetic force (Lorentz force). Contribution of magnetic pressure/tension ( blow o⇒ff plasma) and magnetic tension ( centrifugal force) are ⇒almost comparable.

• The outflow is not pinched by magnetic field significantly and no collimated jet is found.

After the long term simulation, it is expected that the magnetic tension of the twisted magnetic flux tube pinches the outflow to form a relativistic jet.

99995.0a

A Model of Relativistic Jet Formation of Gamma-ray Bursts: Collapsor/Hypernova model

Fe

C+O

HHe

collapse

RelativisticJet from Ergosphere

Jet fromDisk

Accretion DiskKerrBH

Ergosphere

Magnetic Field Lines

Rotating C+O star(M~30MSUN) Central region of relativistic jet

formation of gamma-ray burst

MagneticField Lines

Magnetic reconnection?

Falling Plasma

22

2c

B