Masaaki Takahashi (Aichi Univ. Edu.) Kevin Thoelecke (Montana State Univ.) Sachiko Tsuruta (Montana State Univ.)
Workshop "Challenges of AGN jets” @ NAOJ, Mitaka 2017/01/18-20
Blandford-Znajek mechanisum and Relativistic Jets
In my presentation, we assume a stationary and axisymmetric force-free magnetosphere in Kerr geometry and solve the structure of magnetic fields numerically.
We find three types of black hole magnetospheres, which depend on the rotations of the black hole and magnetic field lines. !We also discuss the efficiency of the electromagnetic energy extraction from the hole for the magnetospheres.
http://www.nationalgeographic.co.jp/news/news_article.php?file_id=2012120305
http://www.wired.com/wiredscience/2009/01/spectacular-new/
http://hubblesite.org/gallery/album/pr2000020a/
http://home.hiroshima-u.ac.jp/hasc/news/3c279/
RelativisticJets
Centaurus A (NGC5128)
Virgo A (M87)
Hercules A (3C348)
GRB models
Purpose and Goals
Acceleration of relativistic jet
Collimation of relativistic jet
BH-Accretion disk system
Disk wind and jet formation
Energy extraction from rotating BH
jet region
near BH region
near Black Hole region
inner Black Hole Magnetosphere
Basic Equations
The ideal MHD condition uF = 0The particle conservation law (nu); = 0Maxwell equations Fµ
; = 4jµ , F[µ;] = 0Polytropic relation (Tooper 1965) P = K
0
The equation of motion T; = 0
1. Number flux per unit magnetic flux () = nup
Bp
2. Angular velocity of the field lines F () = FtrFr
= FtF
3. Total energy of the magnetized flow E() = µut F4 B
4. Total angular momentum L() = µu 14 B
5. Entropy S()
Field-aligned ``conserved quantities'' flow’s parameters
GRMHD GeneralRelativisticMagneto-hydrodynamic
1.RelativisticBernoulliequation
!
!
2.totalEnergyofMHDFlow
E() = µut F
4B
energy in corotation frame gtavitational Lorentz factor
Alfven Mach number
poloidal magnetic field
toroidal magnetic field enthalpy!(rest mass energy+internal energy)
Poynting flux per the particle number fluxthe fluid part of energy
(E F L)2 = µ2 + M2(B2p + B2
)
M2 u2
p
u2AW
→ PLOT Solution
energy conversion KE <--> ME
GRMHD FlowsGeneralRelativisticMagneto-hydrodynamicFlows
3.Trans-fieldequation
7.3. #5&UV^–'I#?Q0%b *2"c 77
Force−balance
Poloidal flow
Magneticfield line
BH
The ideal MHD plasmas stream along magnetic field lines.
The configuration of field lines is determined by the force−balancebetween the field.
6. (GS )
(0% T µν;ν = 0 T_ 3 Ψ = Aφ(r, θ) = constant I$H0I?Q (∂Ψ
E+/>A,T)?Q )` Z]X[EJ HGINP E(Ψ), L(Ψ), ΩF (Ψ),
η(Ψ), M(Ψ) :19ORAF;I_ poloidal equation (2IBDJ2!/.T19
Q ) FLS@Dd Ψ(r, θ)e T< AMJ0%E7Q`
The force-balance equation is derived by Nitta et al.(1991) (=RK cold Y6W\ ^a
hot Y6W\ ^IC8DK-4 D.3 T)
α−M2
4π(∂r∂rΨ+ ∂θ∂θΨ) +
B2pρ
2w
4π√−g
!√−g
ρ2w(α−M2)
"′
+4πµ2ρ2wM2
ηη′ +2π
M2
#
gφφ(E2η2)′ + 2gtφ(ELη2)′ + gtt(L
2η2)′$
(7.47)
− 4πη2ρ2wM4
(µ2Gφ − eµuφ)Ω′F +
4π
M2[Gφ(Eη)′ +Gt(Lη)
′] (Eη)f = 0
where the prime (′) ≡ −(1/B2p)[(∂
rΨ)∂r + (∂θΨ)∂θ], Bp ≡ ρwBp and f ≡ ρwf .
7.3 –
T?Q=Fa
• Khanna — MNRAS
• Koide (2008)
References
[21] “Numerical 3+1 General Relativistic Magnetohydrodynamics: A Local Characteristic Ap-
proach”
Anton, L., et al. ApJ. 637, 296-317
7.3. #5&UV^–'I#?Q0%b *2"c 77
Force−balance
Poloidal flow
Magneticfield line
BH
The ideal MHD plasmas stream along magnetic field lines.
The configuration of field lines is determined by the force−balancebetween the field.
6. (GS )
(0% T µν;ν = 0 T_ 3 Ψ = Aφ(r, θ) = constant I$H0I?Q (∂Ψ
E+/>A,T)?Q )` Z]X[EJ HGINP E(Ψ), L(Ψ), ΩF (Ψ),
η(Ψ), M(Ψ) :19ORAF;I_ poloidal equation (2IBDJ2!/.T19
Q ) FLS@Dd Ψ(r, θ)e T< AMJ0%E7Q`
The force-balance equation is derived by Nitta et al.(1991) (=RK cold Y6W\ ^a
hot Y6W\ ^IC8DK-4 D.3 T)
α−M2
4π(∂r∂rΨ+ ∂θ∂θΨ) +
B2pρ
2w
4π√−g
!√−g
ρ2w(α−M2)
"′
+4πµ2ρ2wM2
ηη′ +2π
M2
#
gφφ(E2η2)′ + 2gtφ(ELη2)′ + gtt(L
2η2)′$
(7.47)
− 4πη2ρ2wM4
(µ2Gφ − eµuφ)Ω′F +
4π
M2[Gφ(Eη)′ +Gt(Lη)
′] (Eη)f = 0
where the prime (′) ≡ −(1/B2p)[(∂
rΨ)∂r + (∂θΨ)∂θ], Bp ≡ ρwBp and f ≡ ρwf .
7.3 –
T?Q=Fa
• Khanna — MNRAS
• Koide (2008)
References
[21] “Numerical 3+1 General Relativistic Magnetohydrodynamics: A Local Characteristic Ap-
proach”
Anton, L., et al. ApJ. 637, 296-317
GRMHD
Force-free BH Magnetosphere
Slowly-rotating case : Analytical study
Blandford & Znajek 1977
Pan & Yu 2015
!
Numerical Simulation
Komissarov 2004
O(a3)
O(a2)
a/m 1 F /!H O(1)
initially split-monopole
near BH (inner region)
previous works
The Magneto-frictional Method
r ·B = 0 B = r↵r B ·r↵ = 0B ·r = 0
J B = 0
J ·r = 0
J ·r↵ = 0 [r (r↵r)] ·r↵ = 0
[r (r↵r)] ·r = 0
1. the Clebsch variables
3. given initial-configuration
4. iteration for self-consistent solution
5. … until no-Fictional force state
2. Force-free (Lorentz force) + Frictional force
Stationary solution of Force-free BH Magnetosphere
Rotating Black-Hole Magnetosphere
Stationary and Axisymmetric
Magnetically-dominated plasma
Initially split-monopole magnetic field
Separation Surface
Inner Light Surface
Outer Light Surface
Event Horizon
Kerr BH
singular surface
singular surface
fast point
Alfven point
magnetic field lines
< GR-version > Kerr Black Hole
a = 0.1m
a = 0.3m
a = 0.6m
F = 0.1!H F = 0.5!H F = 0.8!H
Z
R
Initially split-monopole
Self-consistent solutions
Black Hole magnetosphere
conical jet/wind
!
collimated jet-like structure
!
toward the equatorial disk
slowly rotating black hole or
rapidly rotating black hole slowly rotating magnetosphere
rapidly rotating black hole rapidly rotating magnetopshere
F (1/2)!H
Thoelecke et al. , submitted to PRD
Extracted BH-energy
Blandford-Znajek (1977) power
Direction Efficiently
Black Hole
Ergosphere
1
2
0
wave front
1 2 3
a = 0.99 m , r = 1.7m0
(r/m
) sinq
(r/m) cos q
SPACETIME DRAGGING
Space-time dragging effect
by rotating black hole
Negative potential region → “Penrose process”
Magnetic field lines are also dragged near the horizon. !Energy extraction due to magnetic torque: !“Blandford-Znajek process “
We see , and the energy flux streams outward.
T =14
BpB ( F )4g
(A,)2
Lr = g T
B < 0
Er = FLr
BHCLOUD
ENERGY EXTRACTION FROM BH
TORQUETORQUE
Angular Mom./ Energy Flux
Lorentz force to act on BH from the outside
When there is a global magnetic field around BH, ・・・?
F
H
outgoing plasma
torqueThe square of magnetic field
(1)
(2)
Znajek condition (1977)
TORQUE ACTS ON THE MAGNETIC FIELD
If BH is dragging the magnetosphere
The rate of black hole Energy extraction
for high spin, Peak shifts 0.5 → 0.58
P = 5.2 1019r4B
2
G2
M2
M2
erg/s
1
2
Q(1Q)
a2
r2+ + a2
Z
0
A2, sin
r2+ + a2 cos2 d
P =
Z
r+
T rtpgdd
=1
2
Z
r+
FA,pgF rd
Poynting flux from EH. (BZ power 1977)
angular velocity of MFL
Q F /!H Q =
Relativistic Jet powered by Black Hole
conical jet/wind
!
collimated jet-like structure
!
toward the equatorial disk
slowly rotating black hole or
rapidly rotating black hole slowly rotating magnetosphere
rapidly rotating black hole rapidly rotating magnetopshere
F (1/2)!H
inner region
a/m
F /!H
0.50.1 0.8
0.1
0.3
0.6
F (1/2)!H
Collimations of Extracted Poynting flux
for high-spin, EM-Flux converges toward Rotational axis
R
50%80%
95%Z
BH
BZ-power
F = 0.1!H
Jet region
outer magnetosphere
ApproximatedGSeq.&asolution
trans-magnetosonic region ( by β-model )
highly relativistic outflows
!
narrow opening angle
Assumptions:
RL R < RLE
BR/BZ 1
(E mc2)
Jet region (SR)
log x~0 2 3 4 5ï1 1
0
2
3
4
5
6
ï1
1
0.80.60.4
0.2
^ ^
=0/1.0
x =
x~~
L
x
x~~
F~
loge 0
z~
→ jet bending
I. Ek Em
II. Ek Em
III. Ek Em
MHD GS sol.
(equipatition)
I
III
II
→ radial
log x
log 0z
MHD outflowsTomimatsu & Takahashi 2003
0
outer region
log x~0 2 3 4 5ï1 1
0
2
3
4
5
6
ï1
1
0.80.60.4
0.2
^ ^
=0/1.0
x =
x~~
L
x
x~~
F~
loge 0
z~
IIII IIz~
x~1
Black Hole
Disk
^ =
0
E E~k m
I .
III .II .
E Ek m>>
E Ek m<<
x~L x~F
e0Tomimatsu & Takahashi 2003
Ek
Em
Radial → jet bending
collimate
I. Ek Em
II. Ek Em
III. Ek Em
MHD GS sol.
Light surface Fast mag. surface
collimate
radial
Analytical models of BH magnetosphere
(equipatition)
trans-magnetosonicsolutions
radial
poloidal velocity
IIII IIz~
x~1
Black Hole
Disk
^ =
0
E E~k m
I .
III .II .
E Ek m>>
E Ek m<<
x~L x~F
e0
E/mc2 10
light surface
fast point
~ equipartition
BH
Jet convergence profile
image
Does this reflect a magnetic field shape?
radial ?
paraboloidal
Does this reflect a magnetic field shape?
B
Jet convergence profile
radial ?
paraboloidal
image
IIII IIz~
x~1
Black Hole
Disk
^ =
0
E E~k m
I .
III .II .
E Ek m>>
E Ek m<<
x~L x~F
e0
Collimation (Radial → Paraboloidal) came to be observed !?
Tomimatsu & MT 2003
Stationary sol.
of SR-GS
A Self-consistent Jet model was obtained :
!
!
We have obtained self-consistent solutions of force-free black hole magnetospheres.
Summary
Depending on the BH-spin and the angular velocity of magnetosphere , the BZ-power transports to the axial direction or to the equator.
a (or !H)
F
Radial : in EM-dominated region
Pallaboloidal: in KE-dominated region
for Jet (SR)
near BH (GR)
Appendixs
MAGNETIC INTERACTION
磁気的現象を介して、BH情報を得られるか?
The dragging of space-time trails the magnetic field lines.
Energy Flux
MagneticFieldLines
F
H
0 < < HF
BH
Accretion Disk
このとき、外向きの電磁場のエネルギーフラックスが発生! = 「回転エネルギー引き抜き」
ENERGY FLUX
磁気的現象を介して、BH情報を得られるか?
T↵ =
+
P
c2
u↵u g↵P +
1
4
F↵
F +
1
4g↵FF
T 00em =
1
8
E2 +B2 uem
cT 0iemei =
c
4E B S
エネルギー密度
エネルギー密度流速 (ポインティング・フラックス)
眞榮田さんへ (Ver. 2)高橋真聡 です (2013.08.23)
BH地平面における各物理量の θ依存性について、シミュレーション結果を考察する参考として、簡単な磁場形状モデルを適応させてみました。複雑に見えるグラフをシンプルに理解できるかもしれません。
(1) Poynting-flux (r-compornant):
Er(r, θ) ≡ T rt = ϵ0
EθBφ√−g
where√−g = Σ sin θ , Σ = r2 + a2 cos2 θ and Eθ = ΩF Fθφ = ΩF Aφ,θ .
(2) The boundary condition at the horizon : toroidal compornent of magnetic fields
BHφ [Ψ(θ)] = (ωH − ΩF )
(r2H + a2) sin θ
ΣH(Aφ,θ)H
(3) Blandford-Znajek Power (Poynting-flux at the horizon):
ErH [Ψ(θ)] = ϵ0ΩF (ωH − ΩF )
(r2H + a2)Σ2
H
(Aφ,θ)2H
—————
(4) BZ Power の θ依存性について:
BH spin a は given として、以下の2つの分布が数値シミュレーションによって得られればよい (GRMHD定常解と比較する際に必要となる情報!);
* ΩF (θ) at the horizon ?* Aφ(θ) at the horizon ?
これらの分布を決めるのは、初期磁場分布、磁場強度、降着率、降着率の θ 分布、、、どのように決まってくるのか?
以下では、これらの関数形を手で与えてみて、BZ power の θ分布の様子を調べてみる ( ⇔ 数値計算との比較する)。
磁力線(磁場の向き)は、南半球よりブラックホールに入り、北半球から出るものとする [ Br と Bφ (or Bφ )は北半球と南半球では、それぞれ逆向きとなる]。Bφ と Bφ では、符号が逆向きになるので注意(gµν の符号を (+,−,−,−) に選んでいるため)。
北半球:Br > 0 (at Gt > 0 region), Bφ > 0 (Bφ < 0)南半球:Br < 0 (at Gt > 0 region), Bφ < 0 (Bφ > 0)
1
3.1. Basic equations and some definitions 43
2. 一般化されたオームの法則The electric current is the sum of the two terms coressponding respectivery to the convec-tion current and to the conduction current;
Jµ = cεuµ + σFµνuν , (3.5)
where ε is the proper density of electric charge. 1
We assume the ideal MHD conditions (σ → ∞)
Eν = uµFµν = O(1/σ) = 0 . (3.7)
where Eν is the comoving electric field.
3. 粒子数保存の式(nuµ);µ =
1√−g
(√−g nuµ),µ = 0 (3.8)
4. 運動方程式重力場中のプラズマの運動は、運動方程式
Tµν;ν = 0 (3.9)
によって記述される。ここで、エネルギー運動量テンソル Tµν ( energy momentum tensor) は以下のように与えられる [メトリックの符号を (+,−,−,−) と選んだ場合] :
Tµν = (ρ + P )uµuν − Pgµν +14π
!FµδF ν
δ +14gµνF 2
"(3.10)
以下の議論では、熱伝導や粘性による散逸効果は無視する。2 ここで、ρ, P , n は、全エネルギー密度 (the total energy density), プラズマの圧力 (the pressure), 粒子数密度 (the properparticle density)。 電磁場テンソル (the electromagnetic field tensor) Fµν は、Maxwell 方程式を満たなければならない。また、uµ は流体の四元速度 (the four-velocity of plasma)である。3 4
1The conductivity of the fluid, σ, is
σ =nee2τ
mp[1 + (eτB/mp)], (3.6)
where ne is the electron density, τ is the collision time, e and mp are the electoron’s charge and mass, andB2 ≡ −BαBα (Bekenstein and Oron 1978)[?].
2The corresponding relativistic expressions for the stress-energy tensor (the relativistic Navier-Stokes equation)are discussed in [?].
3特殊相対論における完全流体については、例えば、シュッツ「相対論入門 (上)」江里口、二間瀬 訳 を参考にされたし。
4 エネルギー運動量保存則 について (「一般相対論」佐々木節 著 など参照)(i) uµ に平行な成分 · · · 流体のエネルギー保存則
−uαT αβ;β = (ρc2uβ);β + Puβ
;β = 0 (3.11)
(ii) uµ に垂直な成分 · · · 速度場に対するオイラー方程式の共変形
γαµT αβ
;β =#
ρ +P
c2
$uα
;βuβ + γαβP,β = 0 (3.12)
ここで、γαβ ≡ −gαβ+(1/c2)uαuβ は、uα に垂直な空間的3次元方向への射影テンソルである。(γαβuβ = 0 , γα
µγµβ =
γαβ , γµ
µ = 3 .)
BLANDFORD&ZNAJEKPROCESS(1977)
BZ flux 角速度の差 と 磁場強度 に依存
Znajek condition (1977)
ideal MHD condition