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