惑星放射線帯からのシンクロトロン放射 - 地球惑星科学科 - 北海道大学

Synchrotron Radiation fromPlanetary Radiation Belts Kengo Komatsu 2003/01/31

01MeV 10MeV (1MeV 87%20MeV 99.9%)MHz GHz

i 1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 62.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 . . . . . . . . . . . . . . . . . . . . . . 72.1.3 . . . . . . . . . . . . . . . . 142.1.4 (2.49)(2.50) 1 . . . . . . . . . . . . . . . . . 162.1.5 . . . . . . . . . 162.1.6 . . . . . . . . . . . . . . . . 172.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 . . . . . . . . . . . . . . . . . . . . 222.2.3 . . . . . . . . . . . . . . . . . . . 243 254 27 28 29

1 11 1.1 (magnetosphere) (solar wind) (frozen-in) [Alfven,1950] (magnetopause) 10R E (R E ) [Ferraro,1960]( (magnetotail))1.1: [Haltqvist,1999]1972 1973 10 11 1977 1 2 [ ,1997] [ ,1997]1 [ , ,2000]

1 245R J 100R J (R J ) [Rogers,1995]1.2: [Rogers,1995]1 2 1 22.524R S 2 18.5R S (R S ) 2 [ ,1997]60 0.3R U (R U ) (1.3) 2 18R U (R U ) [ ,1997]47 [ ,1997](1.4)1.2 1958 VanAllen [ ,1970]

1 3 B S (T) R PL (km) R MP (R PL )2.6 × 10 −7 2440 1.53.0 × 10 −5 6378 109.6 × 10 −4 71400 554.9 × 10 −5 60250 25 9.1 × 10 −6 25220 18 1.8 × 10 −5 24700 261: (B S ) (R PL ) (R MP ) [Möbius,1994]1.3: [Ness et al,1991]1.4: [Ness et al,1995]

1 41.5: [Schulz et al,1995]1 3 4 (1958 10 12 1959 3 )3 (1958 5) 50000 km [ ,1970]20 MeV 1 MeV 1 MeV (1.6)1.5R E 4R E10 R E [ , ,2000]1.6: NASA 10 MeV 0.5 MeV [/cm 2 s 1 ] [ , ,2000]21 MeV 30 MeV ( H

1 5OS) 1.9R J 10 7 [/ cm 2 s 1 ] [Rogers,1995] [ , ,2000]

2 62 2.1 2.1.1 Maxwell ∂B(x, t)rotE(x, t) + = 0,∂t(2.1)divB(x, t) = 0, (2.2)rotB(x, t) − 1 ∂E(x, t)= µc 2 0 i e (x, t),∂t(2.3)divE(x, t) = 1 ε 0ρ e (x, t) (2.4)Lorentz A(x, t) φ(x, t) ∂A(x, t)E(x, t) = − − gradφ(x, t), (2.5)∂tB(x, t) = rotA(x, t), (2.6)(∇ 2 − 1 )∂ 2A(x, t) = −µc 2 ∂t 2 0 i e (x, t), (2.7)(∇ 2 − 1 )∂ 2φ(x, t) = − 1 ρc 2 ∂t 2 e (x, t), (2.8)ε 0divA(x, t) + 1 c 2 ∂φ(x, t)∂t= 0 (2.9)(2.7) (2.8) ()∫1∞φ(x, t) = d4πε 0∫Vδ t − t ′ ∓ |x−x′ |3 x ′ dt ′ cρ−∞ |x − x ′ e (x ′ , t ′ ), (2.10)|()A(x, t) = µ ∫ ∫ ∞ δ t − t ′ ∓ |x−x′ |0d 3 x ′ dt ′ ci4π V|x − x ′ e (x ′ , t ′ ) (2.11)|−∞ + x ′ t ′ x t = t ′ + |x−x′ |cx ′ t ′ i e (x ′ , t ′ ) x t t = t ′ − |x−x′ | A(x, t) c(2.9) Lorentz

2 72.1.2 r = r(t) e ρ e (x, t) = eδ 3 (x − r(t)), (2.12)i e (x, t) = eṙ(t)δ 3 (x − r(t)) (2.13)(2.10), (2.11) ()∫e∞φ(x, t) = d4πε 0∫Vδ t − t ′ − |x−x′ |3 x ′ dt ′ cδ 3 (x ′ − r(t ′ )), (2.14)−∞ |x − x ′ |()A(x, t) = µ ∫ ∫ ∞0eδ t − t ′ − |x−x′ |d 3 x ′ dt ′ cṙ(t ′ )δ 3 (x ′ − r(t ′ ))(2.15)4π|x − x ′ |V−∞(2.14) ()φ(x, t) =e ∫ ∞ δ t − t ′ − |x−r(t′ )|dt ′ c(2.16)4πε 0 |x − r(t ′ )|−∞t ′ r(t ′ ) t ′ (2.16) φ(x, t) =f(t ′ ) ≡ t ′ + |x − r(t′ )|c∫ ∞−∞(2.17)g(t ′ )δ (t − f(t ′ )) dt ′ (2.18)t ′ y = f(t ′ ) (2.19)t ′ = h(y) (2.20) (2.18) ∫ ∞−∞dt ′ = dh(y)dyg(h(y))δ(t − y) dh(y)dydy (2.21)dy = g(h(t)) dh(t)dt(2.22)

2 8(2.19) y = t t ′ t ′ 0 t = f(t ′ 0) = t ′ 0 + |x − r(t′ 0)|c(2.23)t ′ 0 = h(t) (2.24)dtdt ′ 0dt ′ 0dtα −1 (t ′ 0) (2.22) = df(t′ 0)dt ′ 0= 1 − 1 x − r(t ′ 0)c |x − r(t ′ 0)| · dr(t′ 0)dt ′ 0= dh(t)dt≡=[ df(t′0 )dt ′ 0[1 − 1 c] −1= dh(t)dtx − r(t ′ 0)|x − r(t ′ 0)| · dr(t′ 0)dt ′ 0(2.25)(2.26)] −1(2.27)∫ ∞−∞[ ] df(tg(t ′ )δ (t − f(t ′ )) dt ′ = g(t ′ ′ −10 )0)(2.28)= e4πε 01|x − r(t ′ 0)|[ df(t′0 )dt ′ 0] −1dt ′ 0= e4πε 01|x − r(t ′ 0)| − 1 c ṙ(t′ 0) · (x − r(t ′ 0))(2.29)(2.15) A(x, t) = µ 0e4πṙ(t ′ 0)|x − r(t ′ 0)| − 1 c ṙ(t′ 0) · (x − r(t ′ 0))t ′ 0 t = t ′ 0 + |x − r(t′ 0)|c(2.30)(2.31)11 ( ) t, t ′ , t ′ 0 t : t ′ : r(t ′ ) t ′ 0: ( . . . (2.31)) t = t ′ r(t ′ ) t = t ′ 0 r(t′ 0) t = t ′ 0 + |x−r(t′ 0 )|c x

2 9(2.29) (2.30) Liénard-Wiechert (2.5)(2.6) E(x, t) B(x, t) x t ′ 0 x (2.14) (2.15) t ′ x R = |x − x ′ | grad x = (grad x R) ∂∂R= x − x′R∂∂R = n ∂∂R(2.32)n = x − x′|x − x ′ | = R R , R = x − x′ (2.33) x’ (2.14) ∫ (e∞grad x φ(x, t) = d4πε 0∫V3 x ′ dt ′ δ(x ′ − r(t ′ ))n ∂( ))δ t − t ′ − R c−∞∂R R⎛ ()∫e ∞=dt ′ n(t ′ ∂) ⎝ δ ⎞t − t ′ − R(t′ )c⎠4πε 0 −∞ ∂R(t ′ ) R(t ′ )∫ [e ∞=dt ′ n(t ′ ) − 1 ()4πε 0 −∞R 2 (t ′ ) δ t − t ′ − R(t′ )c+ 1 () ] ∂R(t ′ ) ∂R(t ′ ) δ t − t ′ − R(t′ )(2.34)cn(t ′ ) = x − r(t′ )|x − r(t ′ )| , R(t′ ) = x − r(t ′ ),R(t ′ ) = |R(t ′ )| r(t ′ ) t ′ n(t ′ ) x (2.15)∂A(x, t)= µ ∫ ∫ ∞0ed 3 x ′ dt ′ ṙ(t′ )δ(x ′ − r(t ′ )) ∂(t∂t 4π V −∞ |x − x ′ | ∂t δ − t ′ − R )c= µ ∫ ∞()0edt ′ ṙ(t′ ) ∂4π R(t ′ ) ∂t δ t − t ′ − R(t′ )(2.35)c−∞

2 10t − t ′ − R(t′ )= X ∂∂t = ∂X ∂∂t ∂X =c∂∂X = ∂R(t′ ) ∂∂X ∂R(t ′ ) = −c ∂∂R(t ′ )(2.35) = − µ 0ec4π∫ ∞−∞()dt ′ ṙ(t′ ) ∂R(t ′ ) ∂R(t ′ ) δ t − t ′ − R(t′ )c(2.36)E(x, t) ∂A(x, t)E(x, t) = − − gradφ(x, t)∂t∫ [e ∞()=dt ′ n(t ′ )4πε 0 R 2 (t ′ ) δ t − t ′ − R(t′ )c−∞n(t ′ ) − 1+ c ṙ(t′ ) () ] ∂R(t ′ ) ∂R(t ′ ) δ t − t ′ − R(t′ )(2.37)c(2.28) t ′ t ′ 0 α−1 (t ′ 0) (2.37) 1 e n(t ′ 0)4πε 0 R 2 (t ′ 0)α(t ′ 0)(2.38) 2 (2.16) y = f(t ′ ) = t ′ + |x−r(t′ )|c===−e ∫ ∞4πε 0−∞∫e ∞4πε 0 c −∞∫e ∞dy ∂ 4πε 0 c −∞ ∂ydt ′ n(t′ ) − 1 c ṙ(t′ )R(t ′ )()∂∂R(t ′ ) δ t − t ′ − R(t′ )cdy dh(y) n(t ′ ) − 1 c ṙ(t′ ) ∂δ (t − y)dy R(t ′ ) ∂y(∂∵∂R(t ′ ) = ∂y)∂∂R(t ′ ) ∂y = −1 ∂c ∂y⎛⎜⎝ dh(y) n(t ′ ) − 1 ⎞c ṙ(t′ )⎟dy R(t ′ ⎠ δ (t − y))⎛e ∂ ⎜⎝ dh(t) n(t ′ 0) − 1 ⎞c ṙ(t′ 0)⎟4πε 0 c ∂t dt R(t ′ ⎠ (2.39)0)

2 11∂∂t = ∂t′ 0∂t∂∂t ′ 0= 1α(t ′ 0)∂∂t ′ 0(2.39) = e 14πε 0 c α(t ′ 0)∂∂t ′ 0⎛n(t⎜′ 0) − 1 ⎞⎝c ṙ(t′ 0)⎟α(t ′ 0)R(t ′ ⎠ (2.40)0)E(x, t) ⎡⎛E(x, t) =e ⎢ n(t ′ 0)⎣4πε 0 R 2 (t ′ 0)α(t ′ 0) + 1 1 ∂ n(t⎜′ 0) − 1 ⎞⎤c α(t ′ ⎝c ṙ(t′ 0)⎟⎥0) α(t ′ 0)R(t ′ ⎠⎦ (2.41)0)∂t ′ 0B(x, t) = rotA(x, t)= µ ∫ ∫ (∞0ed 3 x ′ dt ′ δ(x ′ − r(t ′ ))(n × ṙ(t ′ )) ∂( ))δ t − t ′ − R c4π V −∞∂R R⎛ ()= µ ∫ ∞0edt ′ (n(t ′ ) × ṙ(t ′ ∂)) ⎝ δ ⎞t − t ′ − R(t′ )c⎠4π −∞∂R(t ′ ) R(t ′ )= µ ∫ (()∞δ t − t0e′ − R(t′ )dt ′ (n(t ′ ) × ṙ(t ′ c)) −4π −∞R 2 (t ′ )+ 1 () )∂R(t ′ ) ∂R(t ′ ) δ t − t ′ − R(t′ )cB(x, t) = µ [ṙ(t0e′ ) × n(t ′ 0)4π R 2 (t ′ 0)α(t ′ 0) + 1 (ṙ(t )]′∂ 0 ) × n(t ′ 0)(2.42)cα(t ′ 0) α(t ′ 0)R(t ′ 0)∂t ′ 0 (2.41)(2.42)

2 12 n(t ′ ) dn(t ′ 0)dt ′ 0= d x − r(t ′ 0)dt ′ 0 R(t ′ 0)= −ṙ(t ′ 10)( )R(t ′ 0) + (x − d 1 dR(t′0 )r(t′ 0))dR(t ′ 0) R(t ′ 0) dt ′ 0(= − ṙ(t′ 0)R(t ′ 0) + (x − r(t′ 0)) − 1R 2 (t ′ 0)R(t()′ 0)1= − ṙ(t ′R(t0) + n(t ′ 0)(n(t ′ 0) · ṙ(t ′ 0))′0)= n(t′ 0) × (n(t ′ 0) × ṙ(t ′ 0))R(t ′ 0)) (− (x − r(t′ 0)) · ṙ(t ′ 0))(2.43)(2.41) [E(x, t) =e n(t ′ 0)4πε 0 α(t ′ 0)R 2 (t ′ 0) + n(t′ 0) × (n(t ′ 0) × ṙ(t ′ 0))cα 2 (t ′ 0)R 2 (t ′ 0)( )+ n(t′ 0) ∂ 1cα(t ′ 0) ∂t ′ 0 α(t ′ 0)R(t ′ 0)− 1 ( ) ] ∂ ṙ(t′0 )(2.44)c 2 α(t ′ 0) α(t ′ 0)R(t ′ 0)∂t ′ 01 2 nαR 2 α 2 )E(x, t) =+==n × (n × ṙ)cα 2 R 2n n(n · ṙ) n × (n × ṙ)− +α 2 R2 ( cα 2 R 2 ) cα 2 R 2.. . 1 α = 1 −c n · ṙnα 2 R − ṙ2 cα 2 R 2[e n(t ′ 0)4πε 0 α 2 (t ′ 0)R 2 (t ′ 0) + n(t′ 0)cα(t ′ 0)∂∂t ′ 0(1)α(t ′ 0)R(t ′ 0)ṙ(t ′−0)cα 2 (t ′ 0)R 2 (t ′ 0) − 1c 2 α(t ′ 0)∂∂t ′ 0( ṙ(t′0 )α(t ′ 0)R(t ′ 0)) ] (2.45)(2.42) [B(x, t) = µ 0e ṙ(t ′ 0)4π α 2 (t ′ 0)R 2 (t ′ 0) + 1 ( ) ] ∂ ṙ(t′0 )× n(t ′cα(t ′ 0) α(t0) (2.46)′0)R(t ′ 0)∂t ′ 0

2 13B(x, t) = 1 c n(t′ 0) × E(x, t) (2.47)(2.45) ddt ′ 0(αR) = dα R + α dRdt ′ 0dt ′ 0ddt ′ 0ṙ = ¨r{ ( d= 1 − n · ṙ )}R − α(n · ṙ)dt ′ 0 c= − 1 ( )n × (n × ṙ)· ṙ + n · ¨r R −(1 − 1 )c Rc n · ṙ (n · ṙ)= − 1 ( )n(n · ṙ)· ṙ + n · ¨r R −(1 − 1 )c Rc n · ṙ (n · ṙ)= 1 c ṙ2 − (n · ṙ) − R (n · ¨r) (2.48)c(2.45) [e (n(t ′E(x, t) =0) − β(t ′ 0))(1 − β 2 (t ′ 0))4πε 0 α 3 (t ′ 0)R 2 (t ′ 0) (2.47) B(x, t) =e4πε 0 c[(β(t ′ 0) × n(t ′ 0))(1 − β 2 (t ′ 0))α 3 (t ′ 0)R 2 (t ′ 0)+ n(t′ 0) × {(n(t ′ 0) − β(t ′ 0)) × ˙β(t ′ 0)}cα 3 (t ′ 0)R(t ′ 0)+ (β(t′ 0) × n(t ′ 0))(n(t ′ 0) · ˙β(t ′ 0)) + ˙β(t ′ 0) × n(t ′ 0)(1 − n(t ′ 0) · ˙β(t ′ 0))cα 3 (t ′ 0)R(t ′ 0)](2.49)](2.50)n(t ′ 0) = R(t′ 0)R(t ′ 0) = x − r(t′ 0)|x − r(t ′ 0)| , (2.51)β(t ′ 0) = ṙ(t′ 0),c(2.52)α(t ′ 0) = 1 − n(t ′ 0) · β(t ′ 0) (2.53)

2 141 ṙ(t ′ 0) ¨r(t ′ 0) R −2 (t ′ 0) 2 R −1 (t ′ 0) ( (2.49),(2.50) 1 )2.1.3 (2.49)(2.50) 2 E(x, t) =e n(t ′ 0) × {(n(t ′ 0) − β(t ′ 0)) × ˙β(t ′ 0)}4πε 0 cα 3 (t ′ 0)R(t ′ 0)(2.54)B(x, t) = 1 c n(t′ 0) × E(x, t) (2.55)Poynting t ′ 0 t x S(x, t) = 1 µ 0E(x, t) × B(x, t) (2.56)= 1µ 0 c E(x, t) × (n(t′ 0) × E(x, t))= 1 {n(t′µ 0 c 0 )(E(x, t)) 2 − E(x, t)(E(x, t) · n(t ′ 0)) }= 1µ 0 c (E(x, t))2 n(t ′ 0) (2.57)= 1 ( ) 2 e n(t ′ (0)n(t ′µ 0 c 4πε 0 c 2 α 6 (t ′ 0)R 2 (t0) × {(n(t ′ 0) − β(t ′ 0)) × ′0)˙β(t) 2′0)}(2.58)t ′ 0 = T 1 t ′ 0 = T 2 (T 2 −T 1 ) x (2.31) t = T 1 + |x−r(T 1)| t = Tc2 + |x−r(T 2)|[(c ) ()]x T 2 + |x−r(T 2)|− T 1 + |x−r(T 1)|(T 2 − T 1 ) cc

2 15t t ′ 0 (T 2 − T 1 ) x E E =E =∫ t=T2 + |x−r(T 2 )|ct=T 1 + |x−r(T 1 )|c∫ t ′0 =T 2t ′ 0 =T 1S(x, t) · n(t ′ 0) dt (2.59)S(x, t) · n(t ′ 0) dt dt ′dt ′ 0 (2.60)0dEdt ′ 0= (S(x, t) · n(t ′ 0)) dtdt ′ 0= (S(x, t) · n(t ′ 0))α(t ′ 0) (2.61)I R(t ′ 0)∫I = S(x, t) · n(t ′ 0)α(t ′ 0)R 2 (t ′ 0) dΩ (2.62)dΩ x (2.58) I = 1µ 0 c 3 ( e4πε 0) 2 ∫dΩ [n(t′ 0) × {(n(t ′ 0) − β(t ′ 0)) × ˙β(t ′ 0)}] 2(1 − n(t ′ 0) · β(t ′ 0)) 5 (2.63)dI(2.62) dΩdIdΩ = S(x, t) · n(t′ 0)α(t ′ 0)R 2 (t ′ 0)= 1µ 0 c (E(x, t))2 α(t ′ 0)R 2 (t ′ 0)= c µ 0(B(x, t)) 2 α(t ′ 0)R 2 (t ′ 0) (2.64)

2 17(2.63) ( ) 2dIdΩ = 1 e [n × (n × ˙β)] 2µ 0 c 3 4πε 0 (1 − n · β) 5= e2 ˙v 2 (t ′ 0) sin 2 θ(2.69)16π 2 ε 0 c 3 (1 − β cos θ) 5θ n ˙β β = v/c → 0 (2.7)2.1.6 β ˙β x−z t ′ 0 z x P n n = (sin θ cos φ, sin θ sin φ, cos φ) β = (0, 0, β) β ˙β α/c ˙β = (α/c, 0, 0)s = n − β (2.63) [n × {(n − β) × ˙β}] 2 = [n × (s × ˙β)] 2 = [s(n · ˙β) − ˙β(n · s)] 2s 2 = n 2 − 2n · β + β 2 = 1 − 2β cos θ + β 2 ,= s 2 (n · ˙β) 2 − 2(n · ˙β)(s · ˙β)(n · s) + ˙β 2 (n · s) 2(2.70)(n · β) = β cos θ, (n · ˙β) = (α/c) sin θ · cos φ, (n · s) = n · (n − β) = 1 − β cos θ,(s · ˙β) = (n · ˙β) = (α/c) sin θ · cos φ[n × {(n − β) × ˙β}] 2 = α2c 2 [(1 − 2β cos θ + β 2 ) sin 2 θ · cos φ−2 sin 2 θ · cos 2 φ(1 − β cos θ) + (1 − β cos θ) 2]= α2c 2 [(1 − β cos θ) 2 − (1 − β 2 ) sin 2 θ · cos 2 φ ]( ) 2 [dIdΩ =α2 eµ 0 c ·1·1 − (1 − ]β2 ) sin 2 θ cos 2 φ3 4πε 0 (1 − β cos θ) 3 (1 − β cos θ) 2(2.71)

2 18 z (2.8)() (2.71) I = e2 α 26πε 0 c 3 1(1 − β 2 ) 2 (2.72)

2 191.5beta=010.5Intensity(y) [J/s sr]0-0.5-1-1.5-1.5 -1 -0.5 0 0.5 1 1.5Intensity(x) [J/s sr]beta=0.532Intensity(y) [J/s sr]10-1-2-3-1 0 1 2 3 4 5 6Intensity(x) [J/s sr]1e+07beta=0.995e+06Intensity(y) [J/s sr]0-5e+06-1e+070 5e+06 1e+07 1.5e+07Intensity(x) [J/s sr]2.7: β ˙β ((2.69) ) β ≃ 0 β = 0.5 β = 0.99

2 202beta=01.51Intensity(y) [J/s sr]0.50-0.5-1-1.5-2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Intensity(x) [J/s sr]beta=0.542Intensity(y) [J/s sr]0-2-40 2 4 6 8Intensity(x) [J/s sr]beta=0.99400000200000Intensity(y) [J/s sr]0-200000-4000000 200000 400000 600000 800000 1e+06Intensity(x) [J/s sr]2.8: ((2.71) ) β ≃ 0 β = 0.5 β = 0.99

2 212.2 2.2.1 f T ω 0 = 2π/T n ( ) ∞∑f(t) = f n e −inω 0tn=−∞(2.1)f n f n = 1 T∫ T/2−T/2f(t) e inω 0t dt (2.2)f(t) f −n = f ∗ n (2.3) (2.1) 2 f n f −n = |f n | 2 2 () f 2 =∞∑n=−∞|f n | 2 = 2∞∑|f n | 2 (2.4)n=1f(t) t = ±∞ f(t) =∫ ∞−∞−iωt dωf ω e2π(2.5)f ω f ω =∫ ∞−∞(2.3) f(t)e iωt dt (2.6)f −n = f ∗ n (2.7)

2 222.2.2 ρ e i e ρ e i e ρ e i e (2.10) φ ρ e φ ω e −iωt ρ ω e −iωt′ ()∫φ ω e −iωt 1∞= d4πε o∫Vδ t − t ′ − |x−x′ |3 x ′ dt ′ cρ−∞ |x − x ′ ω e −iωt′|1= d 3 x ′ ρ ω|x−x′ ||x − x ′ | e−iω(t− ) c (2.8)4πε o∫Vk = ω/c φ ω = 14πε o∫Vρ ωd 3 x ′||x − x ′ | eik|x−x′(2.9)(2.11) A i e A ω e −iωt i ω e −iωt′ A ω = µ ∫0d 3 x ′ i ω|4π |x − x ′ | eik|x−x′ (2.10)ρ ω Vρ ω =∫ ∞−∞dt ′ ρ e e iωt′ (2.11)(2.9) φ ω = 14πε o∫V∫ ∞−∞ρ|x − x ′ | ei(ωt′ +k|x−x ′ |) d 3 x ′ dt ′ (2.12)(2.12) φ ω =e ∫ ∞4πε o−∞(2.10) A ω = µ 0e4π∫ ∞−∞1|x − r(t ′ )| ei(ωt′ +k|x−r(t ′ )|) dt ′ (2.13)ṙ(t ′ )|x − r(t ′ )| ei(ωt′ +k|x−r(t ′ )|) dt ′ (2.14)

2 23(T = 2π/ω 0 ) nω 0φ n =e4πε o TA n = µ 0e4πT∫ T0∫ T01|x − r(t ′ )| einω 0(t ′ + |x−r(t′ )|c ) dt ′ (2.15)ṙ(t ′ )|x − r(t ′ )| einω 0(t ′ + |x−r(t′ )|c ) dt ′ (2.16)|x − x ′ | ≃ |x| − x · x′|x|(2.17) (2.10) A ω ≃ µ 0 e ik|x| ∫i ω e −ik·x′ d 3 x ′ (2.18)4π |x| Vi ω i ω =∫ ∞−∞dt ′ i e e iωt′ (2.19)A ω = µ 0 e ik|x| ∫ ∞eṙ(t ′ )e i(ωt′ −k·r(t ′)) dt ′ (2.20)4π |x| −∞= µ 0e e ik|x| ∫e i(ωt′ −k·r(t ′)) dr(t ′ ) (2.21)4π |x|A n = µ 0e e ik|x| ∮e i(nω 0t ′ −k·r(t ′)) dr(t ′ ) (2.22)4πT |x|(2.11) (2.17) A(x, t) = µ ∫04π V≃ µ 04π1|x|d 3 x ′ i e(x ′ , t − |x−x′ |)c|x − x ′ |∫V(d 3 x ′ i e x ′ , t − |x|c + |x · x′ |c|x|)(2.23)

2 24(2.6) B = rotA≃ − 1 c n × ∂ ∂t A (2.24)1/|x| 2 1/|x| (2.55) 1c n × E = −1 c n × Ȧ)n × (n × E) = n ×(Ȧ × n)E =(Ȧ × n × n (2.25)BEA B ω e −iωt E ω e −iωt A ω e −iωt e −iωt B ω = ik × A ω , E ω = ic2ω k × (A ω × k) (2.26)k = kn 2.2.3 (2.4) (2.64) B 2 ω = nω 0 dI ndΩ dI ndΩ = 2c |B n | 2 α(t ′µ0)R 2 (t ′ 0)0= 2cµ 0|A n × k| 2 α(t ′ 0)R 2 (t ′ 0) (2.27)(2.22) (γ ≡ √1−β 1 ≫ 1) 2( )µ0 e 2I n = 0.52 ω0n 2 1/3 , (1 ≪ n ≪ γ 3 ) (2.28)I n = 12 √ π(µ0 e 24π)ω 2 04π( ) 1/2 ne (−2/3)(n/γ3) , (n ≫ γ 3 ) (2.29)γn n ω 0

3 253 1MeV 2R E B E = 3.8 × 10 −6 [T] T = m ec 2√ (3.1)1 − β2√( )me c 2 2β = 1 −(3.2)Tβ E = 0.86 m e m e = 9.1 × 10 −31 [kg]ω 0 ω 0 = eBm e√1 − β2(3.3) ω 0E = 3.4 × 10 5 [Hz] e e = 1.6 × 10 −19 [C]() r λ r λ = v ω 0(3.4) r λE = 8.7 × 10 2 [m] (2.72) I E = 6.8 × 10 −25 [J/s] 20MeV 2R J B J = 1.2 × 10 −4 [T] (3.2) β J = 0.99968 (3.3) ω 0J = 5.3 × 10 5 [Hz] (3.4) r λJ = 5.7 × 10 2 [m] I J = 3.6 × 10 −19 [J/s]

3 2614Earth5Earth124Intensity [x10^-26 J/s Hz]1086Intensity [x10^-26 J/s Hz]324120-1 0 1 2 3 4 5 6 7 8n00 10 20 30 40 50 60 70 80n3.9: 1MeV B E = 3.8 × 10 −6 [T] (1 ≪ n ≪ γ 3 ∼ 8) (2.29) (n ≫ γ 3 ∼ 8) (2.29)γ 3 ∼ 8 (2.29)(2.29) 50Jupiter200Jupiter454015035Intensity [x10^-25 J/s Hz]3025201510Intensity [x10^-26 J/s Hz]10050500 10000 20000 30000 40000 50000 60000n0100000 200000 300000 400000 500000 600000 700000 800000n3.10: 20MeV B J = 1.2 × 10 −4 [T] (1 ≪ n ≪ γ 3 ∼ 62000) (2.29) (n ≫ γ 3 ∼ 62000) (2.29)

4 274 (2) E e (MeV) B PL (T) β ω 0 (Hz) I(J/s) 1 3.8 × 10 −6 0.86 3.4 × 10 5 6.8 × 10 −25 20 1.2 × 10 −4 0.99968 5.7 × 10 5 3.6 × 10 −192: ( 2 )E e B

29Baumjohann, W., Treumann, R. A., 1999: BASIC SPACE PLASMA PHYSICS.Imperial College Press, 329pp.Bergstralh, J. T., 1991: Uranus. Unibersity of Arizona Press, 1076pp.Chen, F. F., 1977: . , 299pp.Cruikshank, D. P., 1995: Neptune and Triton. University of Arizona Press,1249pp.Dessler,A.J.,1983: Physics of the Jovian Magnetosphere. Cambridge UniversityPress, 564pp.Kluwer Aca-Hultqvist, B., 1999: SPACE SCIENCE REVIEWS (chapter 1).demic Publishers, 482pp.Landau, L., Lifshitz, E., 1978: . , 450pp.Möbius, E., 1994: Sources and Acceleration of Energetic Particles in PlanetaryMagnetospheres. The Astrophysical Journal Supplement Series, 90, 521-530. , 1997: 12 . , 478pp. , 1970: . , 484pp. , , 2000: . , 302pp.Panofsky, E., Phillips, M., 1968: . , 551pp.Rogers, J. H., 1995: The Giant Planet Jupiter.418pp.Cambridge University Press, , 1999: 3 . , 463pp.