ブラックホールと中性子星の周辺におけるプラズマ物理	當真賢二	  (TOMA	
  Kenji)	
  	
  東北大学学際科学フロンティア研究所	
  第27回理論懇シンポジウム 「理論天文学・宇宙物理学と境界領域」 2014年12月26日	

Outline	
1. 偏光：X線・γ線観測のフロンティア	  – BH時空の検証、強磁場中のQEDの検証	
  – γ線バーストの放射メカニズム	
  2. BHジェットの駆動メカニズム：一般相対論とプラズマ物理の境界にある問題	
  – パルサー風	  – Blandford-­‐Znajek	
  process	

1.	
  偏光：	
  X線・γ線観測のフロンティア	多くの衛星計画	
• GEMS,	  IXPE,	
  PolariS,	
  PolSTAR	  • POET,	
  POLAR,	
  SPHiNX,	
  GAP2	  • Astro-­‐H/SGD	  • GAP,	
  PoGOLite,	
  Tsubame	BH時空の観測的検証	

(SchniUman	
  &	
  Krolik	
  2009)	
• X線連星の明るい状態	
  • 標準円盤、散乱>>吸収→直線偏光	
  (Chandrasekhar	
  1960)	
  • 一般相対論的効果	
  (Stark	
  &	
  Connors	  1977;	
  Li+2009;	
  SchniUman	
  &	
  Krolik	
  2009)	
  

偏光角の波長依存性	強磁場中のQEDの検証	
• 強磁場中性子星の熱放射	
  
E<~

真空偏極	

eB
= 11.6B12 keVme c• ⇒ 基本モードは直線偏光	  • 磁場に垂直なモードは散乱されにくい	
  モード変換による偏光角のエネルギー依存	22

Physics of Strongly Magnetized Neutron Stars

強磁場の影響	
プラズマ偏極優勢	
 真空偏極優勢	中性子星
大気	

B = 1013 G
E = 5 keV

(Lai	
  &	
  Ho	
  2003)	

ガンマ線バーストの放射機構	
BH	
  +	
  accre^on	
  ﬂow	
バースト	

光球放射	

相対論的ジェット	

内部散逸放射	

準熱的（散乱>>吸収）	
   シンクロトロン放射	  
明るいほど低偏光	
揃った磁場なら高偏光	
•
•
•
	
  
•

GAPのγ線偏光観測	
  (Yonetoku+2011;2012)	
  明るい３例から高偏光(P>30%;	
  2σ)を検出	
  シンクロトロン放射、磁場駆動ジェットを示唆	
  (KT	  2013)	
  Tsubame（東工大）に期待	
  • バーストの可視偏光は未だ観測できていない	
  Reverse	  shock	残光	Forward	  shock	

variations in source brightness or observing conditions on timescales .1 sowing to the rapid rotation of the polaroid. There is no significant variation inatmospheric transparency or seeing (image point-spread function) over the
588-s exposure. b, Measured time-averaged polarization P of all objects versus

measurements in this Letter use this Monte Carlo estimator, although because
polarization in GRB 120308A is significantly non-zero, the derived errors(within ,1% absolute error) are comparable to standard error analyses for that
object (see Supplementary Information and Extended Data Figs 1, 2, 3, 4, 5, 6).

The theoretical maximum degree of linear polarization of synchrotron radiation emitted by electrons in a perfectly homogeneous magnetic field is P < 70%; the difference between the measured and thetheoretical maximum can therefore provide further constraints on thephysical properties of the emitting source. The measured net polarization can be less than the theoretical maximum because of (1) the dilution of polarized reverse-shock emission by unpolarized forward-shock
emission, (2) the combinationof ordered magnetic fields from thecentral
6
Takaki et al.

残光の可視偏光	(Mundell+2013	
  Nature)	

P (%)

30

(Takaki,	
  KT,	
  Kawabata+	
  in	
  prep.)	

Figure 2 | Evolution of optical polarization anda−910 brightness in GRB 120308A. a, b, Evolution of

偏光度	

20
10
0

T (°)

90 b

偏光角	

45
0

c

RINGO2RATCam

16

1.0
17

18

0.1

300

1,000
Mid-time since GRB trigger (s)時間	

1 2 0 | N AT U R E | VO L 5 0 4 | 0 5 D E C E M B E R 2 0 1 3

3,000

r ′ (mag)

Flux density (mJy)

光度	polarization degree P (a) and position angle h(b; degrees east of north) for GRB 120308A.
−10
10 Individual 0.125-s RINGO2 exposures at the eight
Polaroid angles are co-added over a desired timeinterval into eight images, on which absolute
−11
Model
10 aperture
photometry is performed and
P and h derived. Owing to the low read noise of thesystem, data from each rotation angle can be−12
10 stacked into temporal bins after data acquisition to
optimize signal-to-noise ratio versus time
resolution. Here the data were subdivided into four[mag]
16bins of duration ,84 s and one bin of ,252 s giving
roughly equal signal-to-noise ratio. The observed
polarization properties are robust to alternative
18
choices of temporal binning (see Supplementary
Information and Extended Data Figs 7, 8, 9). Error
20
bars, 61s, as described in Fig. 1b. c, Light curve of
GRB 120308A in red (555–690 nm) light using
RINGO2 and RATCam. Data have been cross30
calibrated to the SDSS r9 system via five objects incommon, with a possible systematic error of up to,6% between the two instruments due to colour
20
effects. Model fits using one peak (blue solid line)or two peaks (broken grey line for each component;
10
resultant combined light curve in solid grey) are
shown with an additional point26 constraining late0time
behaviour(see Supplementary
Information).
2
3
4
The 10
two-peak model is10
statistically slightly10
Time since GRB trigger [s]preferred. Error bars, 61s.

Swift XRT

Flux [erg cm−2 s−1]

[mjy]

1

Optical0.1

PD [%]
B caseB case
5

10

6

10

Fig. 4.— Model of the observed X-ray and optical LCs and PD (solid lines). These correspond to the B∥ case and the emission is ass
to come from the bright patch in the forward shock (see more details in the text). The PA changes by 90◦ at t ≃ 5.7 × 103 s. The dline represents the B⊥ case, for which the light curves are the same as those in the B∥ case.

広島大かなた望遠鏡による結果	

©2013 Macmillan Publishers Limited. All rights reserved
circumburst medium (Kumar & Barniol Duran 2010) andthe long-lived reverse shock emission (Uhm et al. 2012)
should also be examined, although we leave them as future work.
Swift satellite has revealed that many of X-ray afterglows in the early phase decay much shallower than predicted by the standard forward shock model, and the
X-ray LC observed at 5 × 102 s ! t ! 7 × 103 s in GRB
111228A corresponds to that phase. The origin of thisshallow decay is still under debate (Pennanen et al. 2014;

to the optical PA change, may not reproduce theserved polarimetric behavior either. The PA chang
the early phase optical afterglow would be a new
to understanding the origin of the X-ray shallow dphase.
Our findings demonstrate that the dense polarim
observations in addition to the multi-band LCs are q
powerful for constraining the theoretical models of G
afterglows. The optical polarizations of blazars also sinteresting behaviors (e.g., Marscher et al. 2008; A• 早期の可視残光がP>20%	
  ⇔ １日後ではP~1-­‐3%	
  (Covino+	
  2004)	
  • 後期可視残光から円偏光 P_c/P_l	  ≅	
  0.15	
  (Wiersema,	
  Covino,	
  KT+2013	
  Nature)	
  2.	
  BHジェットの駆動機構：一般相対論とプラズマ物理の境界にある問題	

パルサー風	6⇠ 10 ?

(Kennel	  &	
  Coroni^	  1984;	
  	
  
S.J.	
  Tanaka	
  &	
  Takahara	
  2010;	
  2013)	

単極誘導	
抵抗	

~
B

J~
~
V
＋	

導体円盤	

ー	
ー	
＋	

(Faraday	  1832)	

＋	

~ =
E

＋	

電位差	

=

~ ⇥B
~
V

Z

V Bdl

Goldreich	  &	  Julian	
  (1969)	
  model	• 定常軸対称	  • 星は伝導体で物質エネルギー優勢	  • 星外は磁場優勢だが粒子で満たされ	
  

⌦s

~ ·B
~ =0
E
~ =
E

J~
⇢<0

B' < 0パルサー風	⇢>0⌦F = ⌦s~ =E• 星の回転が磁気圏に電位差を作る	  • 粒子はE×Bドリフトで共回転か磁場に沿って運動	  • 光円柱の存在 ⇒ B_φ	
  ≠	
  0	
  
• 星の回転が電流を駆動	

~ · J~ =
E

~p
B

~
E

~
$⌦F~e' ⇥ B

~ ·S
~=
r

光円柱	

~ p) · V~'(J~ ⇥ B

1 ~
~'
~
E⇥B
S=
4⇡

~ ⇥B
~
V

~ ⇥ B/B~ 2
E×Bドリフト	 ~vd = E

Par^cle	
  In	
  Cell	  Simula^on	

trophysical Journal Letters, 795:L22 (5pp), 2014 November 1

Jr

6

ρ
(b)

(c)

4

2

2

0

0

−2

−2

−4

−4

z/R∗

4

−6

Bφ

6

(a)

Chen & Belobor

−10

−5

−320 −240 −160 −80

0

5

x/R∗
0

80

160

10
240

320

−6
0
−300

2

4
−150

6

x/R∗
0

8

10
150

12
300

(Chen	  &	
  Bcylinder.
eloborodov	  2014)	
1. Magnetosphere of type I aligned rotator (poloidal cross section) at t = 100. Vertical dashed line shows
the light
Green curves
show the ma+	  +	
  eρ.-­‐	
   (c) Toroidal component of the magnetic field B .
aces. (a) Radialcurrent density Jr . (b) Net charge
density• component
PIC計算of electric＋ 電子の曲率放射とγ→eφ
version of thisis available in the online journal.)• figure
広い領域でe+e-­‐生成が可能とした場合、準定常的な電流回路が形成される	
  ~ ·B~ 6= 0• 電流シートに沿った電場がe+e-­‐生成	
   E• 高エネルギー粒子が磁力線を横切る	
ρp
ρe
6

(a)

but	
 |⇢|
6

|⇢GJ |
ρi(see	
  also	
  Yuki	
  &	
  Shibata	
  (c)2014)	
(b) 2012;	
  Philippov	
  &	
  Spitkovsky	
  BH	
  jets	
活動銀河核	

Lj . LEdd ' 1046 M8 ergs 1
= 10

100

ガンマ線バーストでは	

Lj

LEdd

> 100

中心エンジン	• パルサーと異なり、BHには降着流が付随するが・・・	
  • ジェットへの質量流入は適度に調節されているようだ	
  

ジェット	

Lj ⇠ Ṁj c2
BH	

nGJ ⇠ 10 2 B3 M8 cm 3

降着円盤	

10
⇠
10
max
= 10

• エネルギー源は何か？	
  • 質量源は何か？	
  • 加速機構は何か？	
  • 収束機構は何か？	
  100

有力視されているシナリオ	
低密度	

高密度	

ジェット	

円盤風	

• BH上空の低密度領域にエネルギーを注入	  • ブラックホール回転エネ
ルギーの定常的注入(Blandford	
  &	  Znajek	
  1977)→ 電磁場優勢ジェット	
  • 物質源は不明。非定常過程？中性子注入？(KT	  &	  Takahara	
  2012)	
  • ローレンツ力（磁気圧勾
配、磁気遠心力）による
物質加速	
  (cf.	
  KT	
  &	
  
Takahara	
  2013	
  PTEP)	
  • 外側のガス圧で絞る	  Blandford	
  &	  Znajek	
  (1977)	
無限遠での解	 H' =• Kerr時空、定常軸対称場	
  • 無限小回転BH	
  2⇡⌦F B rpsin ✓⌦H ⌧ 1• スプリットモノポール場	
  

B

rp

= const.

• Force-­‐free近似（電磁場優勢）	
  

H' = const.
H' = 2⇡(⌦F⌦F = ⌦H /2 + O(a3 )

⌦H )B

rp

sin ✓ 地平面での条件	

Force-­‐free	  /	
  MHD	
  simula^ons	
•
•
•
•
•

Koide,	
  Shibata,	
  Kudoh	
  (1999-­‐)	  Komissarov,	
  Barkov	  (1999-­‐)	  Gammie,	  McKinney,	
  Tchekhovskoy	
  (2003-­‐)	  De	
  Villiers,	
  Hawley,	
  Krolik	
  (2003-­‐)	  See	
  水田さん、高橋博之さんポスター	
  • Kerr時空は固定	  • 初期条件にB_pを設定	
  • 準定常的なポインティング流速生成	  Lj > Ṁacc c2も可能	

• MHD計算では、粒子を注入し続けなければならない	  • 負のエネルギー粒子なし(Komissarov	  2005)	
  and final (right) distribution of A! . Level surfaces coincide with magnetic field lines, and field line density corresponds to poloidal field
ate field lines follow density contours if "0 > 0:2"0; max .

(McKinney	
  &	
  Gammie	
  2004)	

• 最近では輻射輸送計算が盛ん	  非定常シナリオ	
(Perfrey,	
  Giannios	
  &	
  Beloborodov	  L632015)	

Black hole jets without large-scale flux

L64

K. Parfrey, D. Giannios and A. M. Beloborodov
4 DISCUSSION AND CONCLUSIONS• 反対巻きのB_pループを順に降着させる	
  • BHとdiskが逆回転の場合に活動的	
  We have argued that small-scale magnetic fields, sourced in an accretion disc where they are amplified by MRI turbulence, can launch
powerful relativistic jets when coupled to a rapidly rotating black
hole. In this scenario, the existence and transport of large-scale
net flux are unnecessary. For prograde discs, there is a minimum
poloidal length-scale lcrit below which the magnetic loops can remain closed when connecting the black hole to the disc, preventing
jet production. There is no such minimum loop scale for retrograde
accretion, allowing these flows to power jets even for lower discthicknesses (if l ∝ H). Retrograde discs also naturally contain loops
with larger flux "l near the disc’s inner edge, and so can produce
more powerful jets since ⟨Lj ⟩ ∝ "2l .
Prograde discs can, however, create jets via this mechanism provided field loops of sufficient size are present. Several effects are
expected to mitigate in favour of prograde jet launching. A coronal• 回転エネルギー → Poyn^ng	
  ﬂux →
軸付近での大規模磁気リコネクションで
熱化	

	

D

時間	

Blandford-­‐Znajek過程についての論点	
• パルサーでは	
  
~ = V
~ ⇥B
~
E

⌦F = ⌦s~ ·S~=
r~ · J~
E

1. 至る所で電磁場優勢の状況において、電位差はどのように生じ、維持されるのか？	
  2. 電流はどこで駆動され、どう閉じるのか？	
  3. BH回転エネルギーはどうPoyn^ng流速に転換するのか？Penrose過程との関係は？	

Kerr	
  space-­‐^me	

3.1 The 3+1 decomposition of space–time

The space–time metric can be generally written as
ds 2 = gµν dx µ dx ν = −α 2 dt 2 + γij (β i dt + dx i )(β j dt + dx j ),(12)

• Boyer-­‐Lindquist	  座標	
  • 地平面で特異	
  the shift vector and γ ij thewhere α is called the lapse function, •β i 空間座標は直交	  " >"three-dimensional 1metric 0tensor of thespace-like hypersurfaces.
• これまでほとんどの解析的
The hypersurfaces are regarded as the研究はこれを使っている	
  absolute space at differ-ent instants of time t (cf. Thorne
et al. 1986).
For Kerr space–•
Kerr-­‐Schild	
  
座
標	
  
"0
to the existences oftime, ∂t gµν = ∂ϕ gµν = 0. These correspond• 地平面で正則	
  • 空間座標は直交していない	
  the coordinates (t, ϕ, r, θ),
the Killing vector fields ξ µ and χ µ . In
• MHD計算に用いられている	0) and χ µ = (0, 1, 0, 0).
ξ µ = (1, 0, 0,エルゴ領域	
The local fiducial
observer (FIDO; Bardeen et al. 1972; Thorne⇠t2 = gtt = ↵2 + 2 > 0ε<0粒子を落としてエet al. 1986), whose world line is perpendicular to theabsolute space,
ネルギーを取り出す
BL > 0is described by the粒子は必ず	
coordinated'four-velocity
ことが可能	  !
"u · ⇠t < 0 がありえる	 （Penrose過程）	
i " =
1 −β

ng vectors 䛸 FIDOs

ଶ

3.1 The 3+1 decomposition of space–time

時空の3+1分解	௝space–timewritten as᫬✵䛾3+1ศゎ+ ߛ௝௞ (ߚThe݀‫ ݐ‬+݀‫ ݔ‬௝ )(ߚ௞ ݀‫ݐ‬metric+ ݀‫ ݔ‬௞can) be generally

ds 2 = gµν dx µ dx ν = −α 2 dt 2 + γij (β i dt + dx i )(β j dt + dx j ),⮬↛䛺┤஺ᇶᗏ䜢ᙇ䜛് 0䠅

Time-like Killing vector:
௜
݀‫ݔ‬
=0
Time-­‐like	
  Killing	
  

(12)

• t=一定面の法線：ﬁducial	  ఓiobserver	  (FIDO)	
vectorand γ ij thewhere αఓ is called the lapse function,ߦ௧β the shift
✓
◆
݊
i
1
three-dimensional metric tensor of the space-likeµ hypersurfaces.
n =
,
௜
↵
↵
ߚ ݀‫ݐ‬The hypersurfaces
are ௜ regarded as the absolute space at differ)ent instants of time ݀‫ݔ‬t (cf. Thorne et al. 1986).n For
Kerr
space–
~
௜
=
(
↵,
0)
µ
௜ + ݀‫ ݔ‬௜ )(‫ ݐ‬+ ݀‫ݐ‬, ‫) ݔ‬
(‫ݐ‬
+
݀‫ݐ‬,
‫ݔ‬
time, ∂t gµν = ∂ϕ gµν = 0. These correspond to the existences of
µ
µ
and
χ
. In the coordinates(t, ϕ, r, θ),theߙ݀‫ݐ‬
Killing vector fields
ξ• FIDOは自然な正規直交݀‫ݏ‬0
基底を張る	  
ξ µ = (1, 0, 0, 0) and χ µ = (0, 1, 0, 0).
t=一定面	• FIDOは角運動量ゼロ	  The local fiducial
observer (FIDO; Bardeen
et al. 1972; Thorne௜)(‫ݐ‬, ‫ݔ‬
(ZAMOともいう)	
et al. 1986), whose world line is perpendicular to then ·absolute⇠' = 0space,is described by the coordinate four-velocity
!
"
i
1 −β
µ
ν
dxi=0	

3+1	  Electrodynamics	

respectively (see Appendix A for more details). The current J is

http://mnras.oxfordjournals.org/ by guest on June 27, 2014

fordjournals.org/ by guest on June 27, 2014m http://mnras.oxfordjournals.org/ by guest on June 27, 2014

mentumofabsolute
this observerisn · χ of=zerogµνelectricn χ and= magnetic
he coordinate
angularvelocityofthespace.
The
condition
constitutive
equations:of the absolute space. The condition of zero electric and magnetic the ergosphere (outer thick
FIDO
is also a forzero
angularmomentumobserversusceptibilities
forgeneralgeneralfully
ionizedplasmasleads
to following
fully
ionizedplasmasleads
to following
H = αsusceptibilities
B3.2− β The× D, 3+1(18) ) − α/√γϕϕ = 0.2, 0.1, −(14)
electrodynamics
E
=
α
D
+
β
×
B,
(
constitutive
equations:e et al.
1986).equations:
Note that the FIDO frame is notconstitutive
ij kin the order of increasing r.
CF.
Atinfinity,
α
=
1
and
β
=
0,
where
×
F
denotesejkanthe
beEECBH.
used
asaconvenientorthonormal
basis
to
as
The
BL
coordinates= ααorderB,=
DD++ββ×(17) (17)In
to×B,study
the testHelectromagnetic
field
inouter
Kerrthickspace–time,=αB −ρ βare× D,((場の古典論;	
  Komissarov	  2004)	
line.
The
spin
p
oingularity
that
E
=
D
and
H
=
B.
Here,
D,
B
and
the
electric
=
∞)
at
the
event
(g
rr (Thorne
cal physics
al. 1986;
Punsly & Coroniti
√ et
we
adopt
the
3+1
electrodynamics
ofbythewhich
wasdeijversion
k2
magneticfield
andchargedensityasmeasured
FIDOs,
=
1
+
1
−
a
.
teld,
horizon
is
r
C
F
.
At
infinity,α=
1 and β =
where
C
×
F
denotes
e
µ
µ⌫
µ
⇤
µ⌫
H
j
k
H
ααBBF−−ββ×D,(18) (18)
H=
==
×t,⌫
D,
E
⇠
,
H
=
F
⇠
,
08).
座標基底に関する電磁場	  t,⌫by (Komissarov
seealsoH Landau
Lifshitz
espectively
Appendix
A for moredetails).
current
is Here,-like in velopedthe(seeergosphere,
where
so2004a,that E =The
D and
=J B.
D,
B
and ρ1975;
arespace.
the elecof&theabsolute
T
ij
k
µ
µ⌫
µ
⇤
µ⌫
~0) asFIDO	  kj Fk . At(t,–Lindquist(BL)
coordinatesϕ,r,θ field)1=and(see
infinity,α =αβ2==chargeC××For	
  ZforAMO	
n(= covariant
↵,
C
. At
infinity,
1 and
β0,0,susceptibilities
whereC=
denotes
eijCanddius
ofwhereouterboundarythemagnetic
densitymeasuredbygenerFID
elatedtothe
thecurrentasmeasuredby
FIDOs,
j,n⌫asD
FFdenotes
n
, eB
=
F
µand
j Freferences
k field,
The
Maxwell
Komissarov
2009,
therein).
⌫of
√
that
==D
and
Hµν
D, D,
B and
ρ(see
are
the µ
electric
2=
= 1E+
1D−
a∗2 coordinatecos
θ.B.B.Here,
mit)
is so
rrotate
DOs
with
the
angularrespectively
A for more details). The current J
so
E
and
H=
Here,
Bµνvelocityandρ Appendix
aretheelectrices that
F
=
0
and
∇
F
=
4πI
are
reducedto equations:
equations∇constitutiveνfield,
magneticfield
andchargedensityasmeasuredby
FIDOs,
Jptotes= α field,j −theρβ.
(19)ν shapestoflat one. The
themeasured
current asby
measuredmagnetic
field
and chargerelated
densitytoas
FIDOs,by FIDOs, j , as

phere
are
shown inFig. Appendix
1.
ϕ respectively
(see
A for
more details).
current J isThe
covariantenergy–momentum
equation
the Theelectromag−β
>
0,
(14)
B
+
∇
×
∇
·
B
=
0,
∂
E = α D + β (15)× B, (
relatedtothecurrentasmeasured
by
j0,
,ρβ.
as
J FIDOs,
=Eα =
j −of
t
ates haveno coordinate
singularrelatedtothecurrentasmeasuredby
FIDOs,
j,as
etic field ∇ν Tµν = −Fµν I ν gives us the energy equation as
=αj−
ρβ.
(19)
the KSJspatial
coordinates
are noThe covariant energy–momentum
equation of the electrom
$
!
"
#
J
=
α
j
−
ρβ.
(19)
same
direction
as
the
BH.
The
BL
coordinatesννppendix
A),· andthen4πρ,
one should
=4π
−FJ,
us =
the α
energyequation
as
netic
field
∇
1
1 ∇
ν Ttheµν I givesH
B
−
β
×
D,
µ
D
=
−∂
D
+
∇
×
H
=
(16)
The
covariantenergy–momentum
equationofelectromagt(E · D of+ the
B · singularity
H) + ∇ ν· (grrE=
×
H at
= −E
· J, $(20) !
t
the
own
coordinate
atial
structureelectromag"# ∞)The
covariantequationof event
theaselectromag√−Fµν I gives
energyequation
netic
field
∇ν Tµν = energy–momentum
8π
4πus the1
1√√
2 . as+ ∇ ·i whereννnates.
k × =F−E
=
1
+
1
−
a
dius netic
ofwherehorizonisrdenotes
=
−F
I
gives
us
the
energyequationfieldevent
∇
(E
·
D
+
B
·
H)
E e×ijCH· J, (
∂
$
!
"
# the
ν T·µC
µν
t
H
∇
and
∇
×
C
denote
(1/
γ
)∂
(
γ
C
)
and
∂
C
,
rei
j
k
i
Poyn^ng	
  
ﬂ
ux	
8π
4π
1
1
where C µ·#F denotes C Fi , andij$theangularmomentumequation as√!
k ∇theij =k"−E(Espace-like
· D+B
·
H)
+
·
E
×
H
· J,Levi-Civita(20) so∂ξt spectively,isinergosphere,
where
ctor
that
E =1974)では、D and Hγ)ϵisthepseudo-tensor
真空解(Wald	
  $ and e # = (1/
#
1
1
i
8π
4π
F denotes
C·FJ,
the angular momentum equation a
i , and(20)
12∂>
1 the
· Dradius+ B · H)+ ∇outer· whereEC×· H=of−E
t 0. (E+βTheofboundarythe
H_φ=0.	
  magnetic
No	
  Poyn^ng	
  
ﬂux.	
an
field
8πCB)
4π
(D ×
+ ∇Ci·Fi , and−(E· m)Dmomentum− (H · m)B$#√#field,t
where· F· mdenotes
the angularequationas
1 respectively (see Appen
4π # 1 + 1 1 − a 2 cos2 θ.
the4πstationaryisagnetic
field
in Kerrlimit)
space–time,$#i res =
(
D
×
B)
·
m
+
∇
·
∂
t
where
momentum equationas −(E · m) D − (H · m)B
1 C · F denotes C Fi ,$and
1 the angular4π
4π
cs of thede( Dasymptotes
×whichB)·MacDonald
mwas+∇∂t 2#version
−(E
m)
D−
(H ·shapesm)Bspace–timeto· 4πthe(1982)
flat
The
relatedto the1$# ·one.
Thorne&andThorneet al. (1986)
developed
thecurrent
3+1 as4π$+ B · H)m(21)
+ 1(E ·&DLifshitz
ee also Landau
1975; =1−(ρ E + J ×1 B) · m,× B) · m of
+are
∇the
· shown∂t 2electrodynamics
−(E
·
m)
D
−
(H
·
m)B
zon
and
in
Fig.
1
.
2the( Dergosphereversionwithout
introducing
E=or−(ρ
H,E andshowed
$
(E
·
D
+
B
·
H)m
+
J
× B) · m,(+
Maxwelltherein). The
4π covariant
4π
1
J
=
α
j
−
ρβ.
2
D
+ haveBequations,· H)m=−(ρcanEpaper,+find
J ×singularB) energy
· m,
(21)
+of (E
the·tothese
expressions
this
such
as equations(22) and (29).
ild4πI(KS)
coordinates
noin
=
are
reducedwheremµ some=∂one
theden$coordinate
ϕ . From2
1
where 角運動量ﬂux	
m = ∂ . From these equations, one can find the energy d

(13)

χν =
erver
s not
sis toroniti(see
ityutiliz(14)l equaD field
nates
×
E=
y, oneevent
s−atathe2.
lution.
where= 0.
ϕutilizofequathe
mentum
2 θ.
Dos
field
pulsar×E=
hapes
y, oneat thepulsar case.
In terms of the vector potential, on
Figure 1. The event horizon (inner thick line) and the outer boundary of(Komissarov
one finds
the ergosphere (outer thick line) of Kerr
space–time.2004a).
The thinThus,
lines represent

Kerr	
  BH	
  磁気圏	√
) − α/ γϕϕ = 0.2, 0.1, −0.1, −0.14,r −0.17,−0.2 in theθBL coordinates
1√−1∂γθ ϕϕ(,= 0Bis identical= √ ∂to• 外部電流で作られたB_pがエルゴ領域を貫いている	  r (,in the order of increasing r. The line Bof )=−√α/theγγ• 磁気圏プラズマは低密度でcollisionlessであるが、次式を満たしてはいる	
  outer thick line. The spin parameter is set to be a = 0.9.

~ ·B
~ =whereD
0 we have defined ( ≡ Aϕ . It is easiof the absolute space. The condition
zero electric
magnetic
whichofmeansthat ( isandconstant
along easusceptibilities
for force
generalionizedplasmas2859
leads• 重力はローレンツ力に比べて無視できる（地平面ごく近傍を除いて）	
The
condition
D · Bto=following
0 and equatio
Electromotive
infully
the BZ
process
constitutive equations:
Taking account of Eϕ = 0, one can write

In the steady, axisymmetric state, the angular momentum equaE (21)= αisDreduced+ β ×to B,
E = −ω × B, ω = %F m. (17)
tion
!
"
!
"
Hϕ
H
ϕ
this equation
into ∇ × E =
∇· −
(25)
B p = B i ∂i 電位差パラメータはパルサーの場合と同様に定義される	
−= −( JSubstituting
p × B p ) · m.
H = 4πα B − β × D, 4π
(18)
tains
~p
定常軸対称として	
S
Electromotive
force
in
the
BZ
process2859
The energy equation (20) is reduced
ij k to
CjﬂF"
where
C
×
F
denotesePoyn^ng	  ux	
k . i At infinity, α = 1 and β = 0,!
"!
B ∂i %F = 0.
H
H
ϕ
ϕ
i
that
E=
D=and
= FB.theHere,ρ(26)are the electric
∇so
· the
−%steady,
B ∂i H−%
Baxisymmetric= −ED,· J pB, and equaInstate,
angularmomentum
F
p
4π
4π
H
< 0each B'FIDOs,field,
magnetic
densitymeasured
constant by
along
is, %astion(21) isreduced tofield and chargeThat
F is! can be"deduced! equation
" A
which
by
(25)
togetherwithequations~ pThealso
described
byEi =respectively(seei Appendix
for
more
details).
current~ −%F ∂iJ(,iswhich
Hϕ
Hϕ
B
E
by(25)
the(23)
equations
imply
HϕBisp )generated∇· and− (24).
· m.
B p These=B ∂= −(that
Jp ×
i −
line
is
equipotential,
and %F corresponds
relatedtothecurrentasmeasuredby
FIDOs,
j,aspoloidal4π
currents which have4πthe component perpendicular to the~p propertieJ
betweenthefieldlines.
These
The
energyis reduced
to component of the Poyntpoloidal
B equation
field, and(20)thenthe poloidal

Coronitiθ) (seeocity(14)

dinatese event
1 − a2.
where
y of thecos2 θ.
shapes.
ngulars are noshouldromag-

where we have defined ( ≡ Aϕ . It is easilyof the absolute space. The conditionof zeroand magnetic
which means
thatelectric( is constantalong eachsusceptibilities for general fullyTheionizedplasmasfollowingcondition
D · leads
B = 0toand
equationconstitutive equations:
Taking account of Eϕ = 0, one can write

起電力の起源	E = α D + β × B,

E = −ω × B,

ω = %F m.

(17)

Substituting this equation into ∇ × E = 0
H = α B − β × D,
(18)
【命題】
⌦F = 0, H' = tains
0 の定常状態は維持できない。 	
where C × F denotes eij k CjiFk . At infinity, α = 1 and β = 0,B ∂i %F = 0.
(KT	
  &	
  Takahara	
  2014)	
B=
0
BL座標で考えると	
so that E = D' and H = B. Here, D, B and ρ are the electric
2 alongfield, magnetic field1andchargedensityasmeasuredby eachFIDOs,
isconstant
B fie
That
is,
%
F
2
2
~
~
~
~
E = 0, D(see= Appendix⇥B
Dby =Bcurrent=−%
also
described
E
respectively
A
for
more
details).
The
J is m2i
F ∂i (, which↵↵line is byequipotential,
and %F corresponds torelated to the current as measured
FIDOs, j , as
エルゴ領域において	 ↵2 < 2 between（KS座標でも同じ結論）	
These properties a! D2 the> field
B 2 lines.
J = α j − ρβ.
the pulsar case, discussed in Section(19)2.
B_pを横切って電流が駆動され、	
 H' 6= 0

The covariant energy–momentum equation of the electromagν
ν
T
=
−F
I
gives>
us0,
the E
energyequation as
netic
field
∇
~
ν
µν
µ
荷電粒子の流れは電場を弱くする	
⌦
=
6
0
F
$
!
"
#
1
1
(E · D + B · H) + ∇ ·E × H = −E · J, (20)∂t 起電力の起源はエルゴ領域である。地平面は本質的でない。	
8π
4π

光円柱	
⌦

p

↵

~p
B
>0

''

f =0
⌦ = ⌦F• 光円柱の位置	

f (⌦F , r, ✓) ⌘ (⇠t + ⌦F ⇠' )2

=

↵2 +

⌦)2 = 0'' (⌦F（BL座標）	
• ２つの光円柱が存在	
  
• 外側	
  

⌦F

f =0
• 内側	
  

⌦F

⌦= p
⌦=

↵
''

p

↵
''

内側の光円柱は必ずエルゴ領域内にある	
Downloaded fro

e 1. The event horizon (inner thick line) and the outer boundary ofgosphere (outer thick line) of Kerr space–time. The thin lines represent√
粒子はΩ〜Ω_Fの領域(f<0)から両側に流れる
α/ γϕϕ = 0.2, 0.1, −0.1, −0.14, −0.17, −0.2 in the BL coordinates√⇒the Ω〜Ω_Fの領域に粒子注入が必要	
  order of increasing r. The line of ) − α/ γϕϕ = 0 is identical to
thick line. The spin parameter is set to be a = 0.9.

赤道面を貫く磁力線	
~
E
~p
S

• エルゴ領域と外側の光円柱を貫く
磁力線：	
   ⌦F > 0, H' 6= 0
• 電流を駆動するには、どこかでD2	
  >	
  B2が維持され	
  force-­‐free/MHDが破れている必要がある 	  	
  • 赤道面近傍で	
  

(B 2

J~p

D2 )↵2 =B ✓ B✓ f

• f	
  >	
  0	
  が必要	
  K. Toma and F. Takahara
• B	
  ≈	
  DとしてΩ_Fの値が決まるだろう	
  

2862

⌦F,max /⌦H~ · J~p < 0
E
(KT	
  &	
  Takahara	
  2014)	

f (⌦F , r) > 0

エルゴ領域内の赤道面の r	まとめ	• 偏光：X線・γ線観測のフロンティア	  – BH時空の検証、強磁場中のQEDの検証	  – ガンマ線バーストの放射機構	
  • BHジェットの駆動機構	
  – Kerr時空による単極誘導	  – 起電力の起源はエルゴ領域	
  – 電流は赤道面で駆動されうる	  – 地平面を貫く磁力線についてはforce-­‐free/MHDが成立し、Poyn^ng	
  ﬂuxは粒子を介さず地平面から直接放射される