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 keV
me 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 s
owing to the rapid rotation of the polaroid. There is no significant variation in
atmospheric 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 the
theoretical maximum can therefore provide further constraints on the
physical 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 combination
of ordered magnetic fields from the
central
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 and
a
−9
10 brightness in GRB 120308A. a, b, Evolution of
偏光度
20
10
0
(°)
90 b
偏光角
45
0
c
RINGO2
RATCam
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 time
interval 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 the
system, 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]
16
bins 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 in
common, 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 late
0
time
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
Optical
0.1
PD [%]
B case
B 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 d
line 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) and
the 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 this
shallow decay is still under debate (Pennanen et al. 2014;
to the optical PA change, may not reproduce the
served polarimetric behavior either. The PA chang
the early phase optical afterglow would be a new
to understanding the origin of the X-ray shallow d
phase.
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 s
interesting 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
2
−300
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) Radial
of electric current density Jr . (b) Net charge
density
•  component
PIC計算　＋　電子の曲率放射と
γ→e
φ
version of this
is 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 ~
•  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)
~ 6= 0
·B
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
= 10
1
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 ⇠ M˙ j c2
BH
nGJ ⇠ 10
max
降着円盤
• 
• 
• 
• 
2
B3 M8 cm
⇠ 1010
= 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？ 3
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 r
p
sin ✓
⌦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 > M˙ acc 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 L63
2015)
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 disc
thicknesses (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 the
where α is called the lapse function, • β i 空間座標は直交 " >"
three-dimensional 1metric 0tensor of the
space-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 of
time, ∂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 > 0
ε<0粒子を落としてエ
et al. 1986), whose world line is perpendicular to the
absolute space,
ネルギーを取り出す
BL > 0
is 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
FIDOs
3.1 The 3+1 decomposition of space–time
時空の3+1分解
+
space–time
metric
3+1written as
( The +
)(
+ 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 ),
0
Time-like Killing vector:
=0
Time-‐like Killing dxi=0
(12)
•  t=一定面の法線：ﬁducial observer shift
vector(FIDO)
and γ ij the
where α is called the lapse function, β i the
✓
◆
i
1
three-dimensional metric tensor of the space-like
µ hypersurfaces.
n =
,
↵
↵
The hypersurfaces are regarded as the absolute space at differ)
ent instants of time t (cf. Thorne et al. 1986).n For
Kerr
space–
~
=
(
↵,
0)
µ
( + , )
(
+
,
+
)
time, ∂t gµν = ∂ϕ gµν = 0. These correspond to the existences of
(t, ϕ, r, θ),
the Killing vector fields ξ µ and χ µ . In the coordinates
•  FIDOは自然な正規直交
0
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
( , )
(ZAMOともいう)
et al. 1986), whose world line is perpendicular to the
n ·absolute
⇠' = 0space,
is described by the coordinate four-velocity
!
"
i
1 −β
µ
ν
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, 2014
m http://mnras.oxfordjournals.org/ by guest on June 27, 2014
mentum
ofabsolute
this observer
is
n · χ of=zero
gµνelectric
n χ and
= magnetic
he coordinate
angular
velocity
of
the
space.
The
condition
constitutive
equations:
of the absolute space. The condition of zero electric and magnetic the ergosphere (outer thick
FIDO
is also a for
zero
angular
momentum
observer
susceptibilities
forgeneral
general
fully
ionized
plasmas
leads
to following
fully
ionized
plasmas
leads
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 not
constitutive
ij k
in the order of increasing r.
C
F
.
At
infinity,
α
=
1
and
β
=
0,
where
×
F
denotes
e
j
k
anthe
beEECBH.
used
as
a
convenient
orthonormal
basis
to
as
The
BL
coordinates
= ααorder
B,
=
DD++ββ×
(17) (17)
In
to×B,study
the testHelectromagnetic
field
inouter
Kerrthick
space–time,
=α
B −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
ofbythe
which
was
deijversion
k
2
magnetic
field
and
charge
density
as
measured
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
see
alsoH Landau
Lifshitz
espectively
Appendix
A for more
details).
current
is Here,
-like in veloped
the(see
ergosphere,
where
so2004a,
that E =The
D and
=J B.
D,
B
and ρ1975;
arespace.
the elec
of&the
absolute
T
ij
k
µ
µ⌫
µ
⇤
µ⌫
~0) asFIDO kj Fk . At(t,
–Lindquist
(BL)
coordinates
ϕ,
r,
θ field
)1=and
(see
infinity,
α =α
β2=
=charge
C
××
F
or Zfor
AMO
n
(= covariant
↵,
C
. At
infinity,
1 and
β0,
0,susceptibilities
where
C=
denotes
eijCand
dius
ofwhere
outer
boundary
the
magnetic
density
measured
bygener
FID
elated
tothe
the
current
as
measured
by
FIDOs,
j,n⌫as
D
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 coordinate
cos
θ.B.B.Here,
mit)
is so
rrotate
DOs
with
the
angular
respectively
A for more details). The current J
so
E
and
H=
Here,
Bµνvelocity
and
ρ Appendix
are
the
electric
es that
F
=
0
and
∇
F
=
4πI
are
reduced
to equations:
equations
∇
constitutive
ν
field,
magnetic
field
and
charge
density
as
measured
by
FIDOs,
Jptotes
= α field,
j −theρβ.
(19)
ν shapes
to
flat one. The
themeasured
current asby
measured
magnetic
field
and chargerelated
densitytoas
FIDOs,by FIDOs, j , as
phere
are
shown in
Fig. Appendix
1.
ϕ respectively
(see
A for
more details).
current J is
The
covariant
energy–momentum
equation
the The
electromag−β
>
0,
(14)
B
+
∇
×
∇
·
B
=
0,
∂
E = α D + β (15)
× B, (
related
to
the
current
as
measured
by
j0,
,ρβ.
as
J FIDOs,
=Eα =
j −of
t
ates have
no coordinate
singularrelated
to
the
current
as
measured
by
FIDOs,
j
,
as
etic field ∇ν Tµν = −Fµν I ν gives us the energy equation as
=αj−
ρβ.
(19)
the KSJspatial
coordinates
are no
The covariant energy–momentum
equation of the electrom
$
!
"
#
J
=
α
j
−
ρβ.
(19)
same
direction
as
the
BH.
The
BL
coordinates
ν
ν
ppendix
A),· and
then4πρ,
one should
=4π
−FJ,
us =
the α
energy
equation
as
netic
field
∇
1
1 ∇
ν Tthe
µν I givesH
B
−
β
×
D,
µ
D
=
−∂
D
+
∇
×
H
=
(16)
The
covariant
energy–momentum
equation
of
electromagt
(E · D of
+ the
B · singularity
H) + ∇ ν· (grrE=
×
H at
= −E
· J, $(20) !
t
the
own
coordinate
atial
structure
electromag"
# ∞)
The
covariant
equation
of event
theaselectromag√
−Fµν I gives
energy
equation
netic
field
∇ν Tµν = energy–momentum
8π
4πus the
1
1
√
√
2 . as+ ∇ ·i where
ν
ν
nates.
k × =F−E
=
1
+
1
−
a
dius netic
ofwhere
horizon
is
r
denotes
=
−F
I
gives
us
the
energy
equation
fieldevent
∇
(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$the
angular
momentum
equation as
√
!
k ∇the
ij =
k"−E
(Espace-like
· D+B
·
H)
+
·
E
×
H
· J,Levi-Civita
(20) so
∂ξt spectively,
is
in
ergosphere,
where
ctor
that
E =1974)では、
D and H
γ
)ϵ
is
the
pseudo-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
· where
EC×· H
=of
−E
t 0. (E
+
β
The
of
boundary
the
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· m
denotes
the angular
equation
as
1 respectively (see Appen
4π # 1 + 1 1 − a 2 cos2 θ.
the4πstationary
is
agnetic
field
in Kerrlimit)
space–time,
$
#
i res =
(
D
×
B)
·
m
+
∇
·
∂
t
where
momentum equation
as −(E · m) D − (H · m)B
1 C · F denotes C Fi ,$and
1 the angular
4π
4π
cs of the
de( Dasymptotes
×which
B)
·MacDonald
mwas
+∇
∂t 2#version
−(E
m)
D−
(H ·shapes
m)B
space–time
to· 4π
the(1982)
flat
The
related
to the
1
$
# ·one.
Thorne
&
and
Thorne
et al. (1986)
developed
thecurrent
3+1 as
4π
$
+ 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( Dergosphere
version
without
introducing
E=or−(ρ
H,E and
showed
$
(E
·
D
+
B
·
H)m
+
J
× B) · m,
(
+
Maxwell
therein). The
4π covariant
4π
1
J
=
α
j
−
ρβ.
2
D
+ have
Bequations,
· H)m
=
−(ρcan
Epaper,
+find
J ×singularB) energy
· m,
(21)
+of (E
the·tothese
expressions
this
such
as equations
(22) and (29).
ild4πI
(KS)
coordinates
noin
=
are
reduced
where
mµ some
=∂
one
the
den$coordinate
ϕ . From
2
1
where 角運動量ﬂux
m = ∂ . From these equations, one can find the energy d
(13)
χν =
erver
s not
sis to
roniti
(see
ity
utiliz(14)
l equaD field
nates
×
E=
y, one
event
s−atathe
2.
lution.
where
= 0.
ϕutilizofequathe
mentum
2 θ.
Dos
field
pulsar
×E=
hapes
y, one
at the
pulsar 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 B
of )=−√α/
the
γ
γ
•  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
~ =where
D
0 we have defined ( ≡ Aϕ . It is easi
of the absolute space. The condition
zero electric
magnetic
whichofmeans
that ( isand
constant
along ea
susceptibilities
for force
general
ionized
plasmas2859
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 = 4π
α B − β × D, 4π
(18)
tains
~p
定常軸対称として
S
Electromotive
force
in
the
BZ
process
2859
The energy equation (20) is reduced
ij k to
CjﬂF"
where
C
×
F
denotes
e
Poyn^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 equaIn
state,
angular
momentum
F
p
4π
4π
H
< 0each B
'FIDOs,
field,
magnetic
density
measured
constant by
along
is, %as
tion
(21) is
reduced tofield and chargeThat
F is
! can be"deduced
! equation
" A
which
by
(25)
together
with
equations
~ pThe
also
described
by
Ei =
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
related
to
the
current
as
measured
by
FIDOs,
j
,
as
poloidal4π
currents which have4πthe component perpendicular to the
~p propertie
J
between
the
field
lines.
These
The
energy
is reduced
to component of the Poyntpoloidal
B equation
field, and(20)
then
the poloidal
Coroniti
θ) (see
ocity
(14)
dinates
e event
1 − a2.
where
y of the
cos2 θ.
shapes
.
ngulars are no
should
romag-
where we have defined ( ≡ Aϕ . It is easily
of the absolute space. The condition
of zero
and magnetic
which means
thatelectric
( is constant
along each
susceptibilities for general fullyThe
ionized
plasmas
following
condition
D · leads
B = 0toand
equation
constitutive 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】　の定常状態は維持できない。　⌦F = 0, H' = tains
0
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 along
field, magnetic field1and
charge
density
as
measured
by each
FIDOs,
is
constant
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 m
2
i
F ∂i (, which
↵
↵
line is by
equipotential,
and %F corresponds to
related 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
energy
equation 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 of
gosphere (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
√
⇒　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
the
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

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