(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x

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1 Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k 3 5 β v c, γ v c η αβ k α k β η αβ k α k β η k k + η k k + η 33 k k + η 33 k 3 k 3 ω c + k Const. ω c k 6

2 θ x ω/c k k ω c cos θ, k ω c cos θ Eq. Eq.3 k x ωt k x + k x + k 3 x 3 ωt γ k + βk x + k x + k 3 x 3 γc k + βk t k x ω t k x + k x + k 3 x 3 ω t γ k k x + k x + k 3 x 3 γc k βk t ω γc k + βk ω γc c + βω cos θ γω + β cos θ 7 c ω γc k βk ω γc c βω c cos θ γω cos θ 8. O ν O Eq.8 ν ν ν cos θ θ π v ν ν ν + β + β ν Red Shit θ v ν ν ν + β ν Blue Shit 9. θ π/ v ν ν γ ν

3 3 Beaming E ω p k E ω p α c, p c, k k α Eq. η αβ p α p β E c p or E c p O O u u u u + v + vu /c, u γ u + vu /c, u3 γ u 3 + vu /c v u u u u x u u + v + vu, u /c u γ + vu /c tan θ u u γ u sin θ u cos θ + v 3 sin θ u u, cos θ u u, u u u + u 4 γ, v/c O u c tan θ u u γ sin θ cos θ + v/c c γ v γ θ π/ γ 5 sin θ γ 6 O θ γ Beaming 4 ε mc,,,, pα γi c, ε c n i p α ei

4 E ε p α e c, p, p α γ c, ε c n n i n p α e p α γi + pα ei pα γ p α γi + pα ei pα γ + pα e 7 m c η αβ p α e pβ e pe pei + p γi p γ 8 n i,, n cos θ, sin θ, ε p α ei mc,,,, pα γi c, ε c,,, p α e E c, p, p α e ε c, ε c cos θ, ε c sin θ, p e p ei + p γi + p γ + p γi p ei p ei p γ p γi p γ ε c mc+mc ε c ε ε c c + ε ε c c cos θ mε+mε + εε c cos θ mε+mε + ε } mc cos θ ε ε + ε mc cos θ 9 ε hν, ε hν, mc hν c λ c/ν Eq.9 hc λ + hc λ hc λ λ c hc cos θ λ + λ c λ λ cos θ λ λ λ c cos θ λ c λ c h mc.43 Å Eq.9 Eq. 5 inverse Compton scattering O inverse Compton scattering O

5 5. O x v O n i cos θ i, sin θ i cos φ i, sin θ i sin φ i n cos θ, sin θ cos φ, sin θ sin φ p α ei p α γi p α e p γ α Eq.8 ε p eiα mc,,,, p γiα E p α e c, p p e α c, ε c cos θ i, ε c sin θ i cos φ i, ε c sin θ i sin φ i ε c, ε c cos θ, ε c sin θ cos φ, ε c sin θ sin φ, ε c mc + mc ε p ei p ei + p γi + p γ + p γi p ei p ei p γ p γi p γ c mε + mε ε ε c ε c mε + mε + ε ε c cos Θ ε c + ε ε c c cos θ i cos θ i + ε ε c c sin θ i sin θ cos φ i cos φ + ε ε c + cos θ i cos θ + sin θ i sin θ cos Θ cos Θ cos θ i cos θ + sin θ i sin θ ε ε + ε cos Θ mc cos φ i cos φ + sin φ i sin φ cos φ i cos φ + sin φ i sin φ } cos θ i cos θ + sin θ i sin θ cos φ i φ cos Θ ε ε ε /mc Eq.3 [ ] ε ε ε cos Θ mc c sin θ i sin θ sin φ i sin φ ε mc ε ε O γ ε ε Eq.7,8 ε εγ cos θ i εγ cos θ i + cos θ i γ 5

6 γ β v c γ γ cos θ i ε εγ ε ε ε ε γ + β cos θ ε γ + cos θ cos θ 6 γ + cos θ ε ε γ ε ε ε ε γ ε γ εγ 7 γ ε ε εγ 5 ev ev γ ev 5.3 p α γi εi c, ε i c n i, p α γ ε c, ε c n p α ei Ei E c, p i, p α e c, p 7 Eq.8 ε i c c + ε i c p i n i ε c c + ε c p i n ε i ε c c + ε i ε c c n i n ε i + ε i cp i n i + ε ε i ε Ei cp c i n + ni n c ε i + ε i cp i n i + ε c Ei cp i n + ε i ni n } cp i n i ε ε i 8 cp i n + ε i ni n p i n i p i cos θ i, p i n p i cos θ, β v i c n i n cos Θ cos θ i cos θ + sin θ i sin θ cos φ i φ

7 ε ε i cos θ i cos θ + ε i cos Θ mc p i v i c 9 γ θ γ, cos Θ Eq.9 cos θ + ε i cos Θ cos θ + ε i cos Θ γ θ + o θ + ε i + γ + γ + ε i + o γ γ + ε i ε i mc γ cos θ i 3 ε cos θ i cos θ + ε i cos Θ ε i cos θ i γ + ε i ε i γ γ ε i 3 Eq.3 γε i mc γε i 6 6. Klein-Nishina x j incident θ, φ dω j scattered θ, φ dσ dσ Ω j scattered r dω j incident 3 r r dω j scattered r dω dσ σ dσ Ω jscattered r dω 33 j incident j scattered r dω σ j incident 34 j incident σ

8 hν mc σ σ T 8 3 πr, r e mc 35 r Klein-Nishina [ ] 3 + x x + x ln + x σ σ T ln + x x x x 3 + 3x } + x x hν/mc / x Eq.36 x σ σ T x x 33 x x4 + o x 4, x x Eq.36 x σ 3 [ 8 σ T + ln x + x 9 x x 54 + ln x 7 ] 3 x + o, 4 x 4 x ε ε ε + dε dω d j incident ε x, ε, Ω d j incident c ε dεdω, σ σ T 39 x, p d3 p ε x, ε, Ω dεdω ε dω Eq.34 dω σ T d j incident cσ T ε dεdω 4

9 6.5 ε ε P cσ T ε ε dεdω 4 Power P P de/ dp de P dω dω dω O de cσ T ε ε dε 43 ε mc, ε ε de de O Eq.43 de de cσ T ε dε cσ T ε ε εdε /ε ε dε ε ε dε ε ε ε dε ε de cσ T ε ε dε ε Eq.5 de cσ T γ cos θ ε ε dε O de P compt dω cσ T γ cos θ ε ε dεdω 44

10 6.7 ε x, Ω ε x, Ω ε x 45 Eq.43 π π cos θ dω π cos θ sin θ dθ π sin θ β sin θ cos θ + β cos θ sin θ dθ π π sin θ β sin θ β cos 3 θ } ] π β β dθ π [ cos θ + cos θ 3 3 cos3 θ 4π + β 3 P compt cσ T γ cσ T γ + β U ph, 3 cos θ ε ε dεdω cσ T γ + β 3 U ph 4π 4π ε ε dε ε ε dε 46 U ph cσ T U ph 47 Eq.46 Eq.47 dw cσ T γ + β compt 3 U ph cσ T U ph cσ T γ U ph + β 3 γ 4 3 cσ T γ β U ph 48 dw R 4 cσ T U ph compt 3 γ β 49 γ, β R β γ, β R γ 7 Rybicki & Lightman979 j ν σ ν J ν j ν α ν J ν, J ν 4π α ν N α ν Nσ ν I ν dω

11 I V Iε ε hν ε I V /hν n ν ε cn ν K K ε F δ ε ε 5 x γmc K ε, µ ε/ɛ ε, µ ε ε, µ ε ε, µ ε ε ε ε ε F δ ε ε ε F δ ε ε 5 ε µ cos θ K x Eq.6 Eq.5 ε, µ ε F ε δ ε ε ε F ε δ γε + βµ ε ε F ε δ γβε µ ε γε } γβε ε F µ ε γβε δ ε γε γβε Thomsom dσ /dω j ν 5 ε g ε g ε + N σ T ε, µ dµ N σ T ε F ε γβ 53 ε ε N Eq.6 ε ε γ + β < ε < ε γ Eq.53 g ε N σ T ε F g ε ε γβ, i ε γ + β < ε < ε γ, otherwise 54 hν g ν g ν /ν g ν g ε, µ ε ε g ε N σ T ε F ε γβ Nσ T ε F ε γ β, N γn 55 ε ε γε µ

12 ε γ + β µ < ε ε < γ µ 56 g ε, µ Nσ T ε F g ε, µ ε γ β, i ε γ + β µ < ε ε < γ µ, otherwise 57 Eq.56 ε < ε > ε γ µ ε + β µ > ε µ β ε γ + β µ ε µ < µ β } + β ε } ε ε ε µ β ε ε + β } < µ < β ε } ε 58 < µ < Eq.58 ε } < or < β ε β ε /ε ε } + β ε ε < ε + β or ε > + β ε g ε, µ g ε, µ < µ < ε }, β ε or + β < ε <. ε 59 ε } + β < µ <, β ε or < ε < + β ε. 6 Eq.55 µ Eq.59 gε } β ε ε β Nσ T F 4ε γ β gε + dµ g ε, µ Nσ T ε F ε γ β dµ Nσ T ε F 4ε γ β ] [ + β ε ε, or [ ε } ] + β ε + β < ε ε < 6 Eq.6 gε β ε Nσ T F 4ε γ β } ε +β Nσ T ε F ε γ β dµ Nσ T ε F 4ε γ β ] [ + β ε ε [ ε }] + β β ε, or < ε < + β ε 6

13 γ Eq.6 gε Nσ [ T F + β ε ] 4ε γ β ε 3Nσ T F ε 4ε γ 3 4γ ε 3Nσ T F hx 4ε γ Nσ T F 4ε γ + β ε Nσ T F ε β ε β 4ε γ ε γ 3Nσ T F 4ε γ 3 x, x ε 4γ ε 63 < x < γ 64 hx x ln x + x + x, < x < 65 Eq.6 Eq.6 gε [ + β ε ε 4γ gε β Nσ T F 4γ β 5 ] ε ] [ + β ε ε, i + β < ε ε <, i < ε < + β ε 4 Β. g Ε Ε N ΣT F 3 Β.5 Β Ε Ε β ε < ε

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

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