磁場と光子場の中で発達する
電磁カスケード
岡山商科大
中塚隆郎
Contents
¯宇宙カスケードの特色
¯詳しい断面積と宇宙カスケードの数値積分解
¯簡略化断面積と光子ガス中のカスケードの解析解
¯簡略化断面積と磁場中のカスケードの解析解
¯磁場中のカスケードが見えるか？
¯シャワーの解析解による開発プログラムの点検
¯結論と議論
Sep., 2010 – p.2/36
Comparison of the cross-sections
radiation
pair
in matter
in photon gas
in magnetic field
where
and
光子ガスでの断面積の中の同次関数
Homogenious functions,
and
, of the
cross-section for Inverse Compton (left) and photon-photon pair
production (right) are indicated below:
0.1
1
10
100
0
0.2
0.4
0.6
0.8
1
φ(
κ
,v
)
v
=
ε
γ /
ε
e
κ=0.1
κ=0.2
κ=0.5
κ=1.0
κ=2.0
κ=5.0
κ=10
κ=20
κ=50
κ=100
Figure 1:
.1, .2, .5,
, 100, from left to
right.
0.1
1
10
100
0
0.2
0.4
0.6
0.8
1
ψ
(λ
,u
)
u
=
ε
e
/
ε
γ
λ=1.5
λ=3
λ=5
λ=10
λ=30
λ=100
Figure 2:
.1,
.2, .5,
, 100, from bottom to
top.
Sep., 2010 – p.4/36
磁場での断面積の中の同次関数
Homogenious functions,
and
, of the
cross-section for photon radiation (left) and pair production
(right) under the magnetic fields are indicated below:
0
2
4
6
8
10
0
0.2
0.4
0.6
0.8
1
ϕ(
v)
v
phi(x)
Figure
3:
.
0
2
4
6
8
10
0
0.2
0.4
0.6
0.8
1
ϕ(
v)
v
psi(x)
Figure
4:
.
数値積分法で得たカスケードの遷移曲線
0.1
1
10
100
1000
10000
100000
1e+006
1e+007
0
10
20
30
40
50
60
70
80
Number of electrons
cascade length (
t
)
cascades in matter
Our
E
0
/E
= 10^2
Our
E
0
/E
= 10^4
Our
E
0
/E
= 10^6
Our
E
0
/E
= 10^8
CasA
E
0
/E
= 10^2
CasA
E
0
/E
= 10^4
CasA
E
0
/E
= 10^6
CasA
E
0
/E
= 10^8
0.0001
0.001
0.01
0.1
1
10
100
0
1
2
3
4
5
6
7
8
9
Number of electrons
cascade length (
t
)
cascades in photon field
Our
E
0
/E
= 10^1
Our
E
0
/E
= 10^2
Our
E
0
/E
= 10^3
Our
E
0
/E
= 10^4
AP’s
E
0
/E
= 10^1
AP’s
E
0
/E
= 10^3
AP’s
E
0
/E
= 10^4
0.1
1
10
100
1000
10000
100000
0
1
2
3
4
5
6
7
8
Number of electrons
cascade length (
t
)
cascades in magnetic field
Our
E
0
/E
= 10^2
Our
E
0
/E
= 10^4
Our
E
0
/E
= 10^6
Our
E
0
/E
= 10^8
AP’s
E
0
/E
= 10^2
AP’s
E
0
/E
= 10^4
AP’s
E
0
/E
= 10^6
AP’s
E
0
/E
= 10^8
Figure 5:
Transition curves of shower electron developing in matter (left),
photon fields (middle), and magnetic fields (right). Our results in matter (lines)
are compared with the analytical results in Nishimura, and our results in
pho-ton fields with
and in magnetic fields with cascade length defined
by
¾ ¿ ¼ ½¿
(lines) are compared with those indicated in
Aharonian and Plyasheshnikov (dots).
Cascades in photon gas
Diffusion equation
¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼for the differential energy spectra,
and
, with
where
,
, and
denote the energies of the shower electron,
shower photon, and background photon in units of
, and
Approximating the homogenious functions
0.1
1
10
100
0
0.2
0.4
0.6
0.8
1
φ(
κ
,v
)
v
=
ε
γ /
ε
e
κ=0.1
κ=0.2
κ=0.5
κ=1.0
κ=2.0
κ=5.0
κ=10
κ=20
κ=50
κ=100
1
Figure 6:
.
0.1
1
10
100
0
0.2
0.4
0.6
0.8
1
ψ
(λ
,u
)
u
=
ε
e
/
ε
γ
λ=1.5
λ=3
λ=5
λ=10
λ=30
λ=100
1
Figure 7:
.
Sep., 2010 – p.8/36
Applying Mellin transforms
½we have the differential-difference equations,
with
and the differential spectra of shower particles become
½ ½
We derive the approximated solution by dividing
with
equal stepsizes,
,
with
and
. Then we have
where
Sep., 2010 – p.10/36
Applying the inverse Mellin transforms, we have the approximated
differential electron spectrum
!
as
! !" ½ ½ #where
denotes the 1,2 element of
.
have poles at
-1, 0, k-2, so we have the differential energy
spectrum of electron,
! $ $ $using residues
$at
.
residuess
$Exact solutions for the differential
spectra and the transition curves
At the limit of
, we have the exact solution for the differential electron
spectrum
!.
The differential spectra and the integral spectra of electron components and
photon components for gamma-initiated shower so obtained are
! ¼ % Æ ¼
where
&
denotes the exponential integral function,
& ½ #
Sep., 2010 – p.12/36
Differential spectra of electrons and
photons for cascades in photon gas
1e-005
0.0001
0.001
0.01
0.1
1
1e-005
0.0001
0.001
0.01
0.1
1
κ
×
π
(
κ
,t)
κ
/
κ
0
t=.
01
t=.
02
t=.
05
t=
0.1
t=
0.2
t=
0.5
t=
1.0
t=
2.0
t=
5.0
Figure 8:
-weighted
dif-ferential energy spectrum of
electrons for gamma-initiated
showers of energy
.
1e-005
0.0001
0.001
0.01
0.1
1
1e-005
0.0001
0.001
0.01
0.1
1
λ
×
γ
(
λ
,t)
λ
/
κ
0
t=.
01
t=.
02
t=.
05
t=
0.1
t=
0.2
t=
0.5
t=
1.0
t=
2.0
t=
5.0
Figure 9:
-weighted
differential energy spectrum
of photon for gamma-initiated
showers of energy
.
Transition curves of electrons and
pho-tons for cascades in photon gas
0
1
2
3
4
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Π
(κ
0
/κ
,t)
radiation length (
t
)
Anal. κ
0
/
κ=10
1
Anal. κ
0
/
κ=10
2
Anal. κ
0
/
κ=10
3
Anal. κ
0
/
κ=10
4
Num. κ
0
/
κ=10
1
Num. κ
0
/
κ=10
2
Num. κ
0
/
κ=10
3
Num. κ
0
/
κ=10
4
Figure 10:
Transition
curves
of
electrons
with
10, 10
, 10
, 10
for
gamma-initiated
showers
of
energy
. Analytical results
(linrs)
are
compared
with
numerical results(dots).
0
1
2
3
4
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Γ(
κ
0
/λ
,t)
radiation length (
t
)
Anal. κ
0
/
κ=10
1
Anal. κ
0
/
κ=10
2
Anal. κ
0
/
κ=10
3
Anal. κ
0
/
κ=10
4
Num. κ
0
/
κ=10
1
Num. κ
0
/
κ=10
2
Num. κ
0
/
κ=10
3
Num. κ
0
/
κ=10
4
Figure 11:
Transition
curves of photons with
10, 10
, 10
, 10
for
gamma-initiated showers of energy
.
Analytical results (linrs) are
compared with numerical
re-sults(dots).
Cascades in magnetic field
Akhiezer equation
¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼We approximate the homogenious functions as
0
2
4
6
8
10
0
0.2
0.4
0.6
0.8
1
φ(
v)
v
phi(x)
phic(x)
Figure 12:
.
0
2
4
6
8
10
0
0.2
0.4
0.6
0.8
1
ψ
(u
)
u
psi(x)
psic(x)
Figure 13:
.
Sep., 2010 – p.16/36
The differential-difference equations for cascades in the
magnetic fields are
with
Then we have the Mellin transform functions,
½where
Applying the inverse Mellin transforms, we have the differential electron
spectrum
!for gamma-initiated shower as
! ½ !" ½ ½ #
where
denotes the 1,2 element of
.
residues
in magnetic fields
have poles at
and
.
residues
$ ' ' ' ' ' ' ' ' 'Exact solutions for the differential
spectra and the transition curves
The differential spectra and the integral spectra of electron components and
photon components for gamma-initiated shower so obtained are
! ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ( ) )
where we define
( 'and
) 'Æ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ( ) )
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ( () ) ( ) () ( ) ( ( ( ( )
Sep., 2010 – p.22/36
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ( () ) ( ) () ( ) ( ( ( ( )
where
( ½Differential spectra of electrons and
photons for cascades in magnetic field
1e-006
1e-005
0.0001
0.001
0.01
0.1
1
10
100
1e-0081e-0071e-0061e-0050.00010.001 0.01
0.1
1
E
×
π
(E,t
)
E/E
0
t=.
01
t=.
02
t=.
05
t=
0.1
t=
0.2
t=
0.5
t=
1.0
t=
2.0
t=
5.0
Figure 14:
-weighted
differential energy spectrum of
electrons for gamma-initiated
showers of energy
.
1e-006
1e-005
0.0001
0.001
0.01
0.1
1
10
100
1e-0081e-0071e-0061e-0050.00010.001 0.01
0.1
1
W
×
γ
(W,t
)
W/E
0
t=.
01
t=.
02
t=.
05
t=
0.1
t=
0.2
t=
0.5
t=
1.0
t=
2.0
t=
5.0
Figure 15:
W-weighted
differential energy spectrum
of photon for gamma-initiated
showers of energy
.
Transition curves of electrons and
pho-tons for cascades in magnetic field
0.1
1
10
100
1000
10000
100000
0
1
2
3
4
5
6
7
8
Π
(E
0
/E,t
)
radiation length (
t
)
Anal.
E
0
/E=
10
2
Anal.
E
0
/E=
10
4
Anal.
E
0
/E=
10
6
Anal.
E
0
/E=
10
8
Figure 16:
Transition
curves
of
electrons
with
10, 10
, 10
, 10
for
gamma-initiated
showers
of
energy
.
0.1
1
10
100
1000
10000
100000
0
1
2
3
4
5
6
7
8
Γ(
E
0
/W,t
)
radiation length (
t
)
Anal.
E
0
/W=
10
2
Anal.
E
0
/W=
10
4
Anal.
E
0
/W=
10
6
Anal.
E
0
/W=
10
8
Figure 17:
Transition
curves
of
photons
with
10, 10
, 10
, 10
for gamma-initiated showers of
energy
.
IACTs
データに磁場カスケード型スペ
クトルを探す
簡略化断面積で計算した磁場シャワーでは発達の深さによ
らず、低エネルギーで微分スペクトルが
乗を示す。
これは
の極の寄与である。
この様子は詳しい断面積の数値積分解でも変わらない。
こんなガンマ線天体を探してみる。
ICRC2011
の以下のスライドの中にスペクトルのべきの近い
ものが見られる、
P. Colin
の
Id:1092
のスライド
