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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