B 1 B.1.......................... 1 B.1.1................. 1 B.1.2................. 2 B.2........................... 5 B.2.1.......................... 5 B.2.2.................. 6 B.2.3.................. 9 B.3.............................. 11 B.3.1................... 11 B.3.2...................... 12 B.4....................... 17 B.4.1................... 17 B.4.2................... 22 B.5.......................... 28 B.5.1....................... 28 B.5.2...................... 30 B.5.3..................... 33 B.6...................... 36 B.6.1........................ 36 B.6.2................. 37 1

B B.1 g ϕ g = ϕ ϕ B.1.1 M (x, y, z) (r, θ, φ) r r = x 2 + y 2 + z 2 g g = GM r r 3 = GM ( x r 3, y r 3, z r 3 ). (B.1) divergence div g = g x x + g y y + g z z x- g x x = GM ( x ) ( ) 1 x r 3 = GM r 3 3x2 r 5 (B.2) (B.3) div g = g x x + g y y + g z z = 0 (B.4) 1

2 g ϕ ( g = ϕ = ϕ ) x, ϕ y, ϕ z (B.5) div g = 2 ϕ = ϕ = 2 ϕ x 2 + 2 ϕ y 2 + 2 ϕ z 2 = 0 (B.6) B.6 Laplace equation 1 B.1.2 ρ(r) B.1 P dv' O ε P B.1: P ε dv P ϕ ϕ in 1 nabla nabla Laplacian R. Murphy

ϕ out ϕ(r) = ϕ in (r) + ϕ out (r) 3 (B.7) P 2 ϕ out (r) = 0 (B.8) P Gρ(r )dv ϕ in (r) = r r (B.9) in ρ(r ) P ρ(r) r r (r, θ, φ) (r, θ, φ ) dv = r 2 sin θ dr dθ dφ r r = r2 + r 2 2rr cos θ ϕ in (r) = Gρ(r) = 2πGρ(r) ε π 2π 0 0 ε 0 0 r 2 dr π r 2 sin θ dr dθ dφ r2 + r 2 2rr cos θ 0 sin θ dθ r2 + r 2 2rr cos θ (B.10) θ π 0 sin θ dθ = 1 r2 r2 + r 2 2rr cos θ rr + r 2 2rr cos θ π 0 { 2r r > r = 2 r r < r (B.11) r ε 0 r 2 dr π 0 sin θ dθ r2 + r 2 2rr cos θ = 2 r r 0 r 2 dr + ε r 2r dr

4 = 2 3 r2 + ε 2 r 2 = 1 3 r2 + ε 2 (B.12) ε 2 ε 2 2 ϕ in (r) = 2πGρ(r) 2 ( 1 3 r2 ) (B.13) ρ 2 r 2 = 2 (x 2 + y 2 + z 2 ) = 6 2 ϕ(r) = ϕ(r) = 4πGρ(r) (B.14) B.14 Poisson equation (x, y, z) (r, φ, z) (r, θ, φ) 1 r 2 ( r 2 ϕ r r 1 r r ) + 2 ϕ x 2 + 2 ϕ y 2 + 2 ϕ = 4πGρ(x, y, z), z2 (B.15) ( r ϕ ) + 1 2 ϕ r r 2 φ 2 + 2 ϕ z 1 r 2 sin θ ( sin θ ϕ θ θ ) + 2 = 4πGρ(r, φ, z), (B.16) 1 r 2 sin 2 θ 2 ϕ = 4πGρ(r, θ, φ). φ2 (B.17) ϕ(r) ρ(r) ρ(r) ϕ(r)

5 B.2 virial theorem 2 B.2.1 m i r i F i m i d 2 r i dt 2 = m i r i = F i (B.18) d/dt r i = (x i, y i, z i ) F i = (X i, Y i, Z i ) m i ẍ i = X i, m i ÿ i = Y i, m i z i = Z i. (B.19) I N N N I m i r i r i = m i ri 2 = m i (x 2 i + yi 2 + zi 2 ). i=1 i=1 i=1 (B.20) 2 di N dt = 2 m i ṙ i r i, i=1 (B.21) 2 virial vis vires R. J. E. Clausius 1822 1888 the virial of force

6 d 2 I dt 2 N = 2 (m i ṙ i ṙ i + m i r i r i ) (B.22) i=1 2 1 d 2 I 2 dt 2 N = (m i ṙ i ṙ i + m i r i r i ) (B.23) i=1 B.23 ṙ v 2 r a m i r = m i a i i F i 1 d 2 I 2 dt 2 = 2 N i=1 1 2 m iv 2 i + N F i r i i=1 (B.24) B.24 1 T N i=1 1 2 m iv 2 i (B.25) 2 2 N N V r i F i = (x i X i + y i Y i + z i Z i ) (B.26) i=1 i=1 B.2.2 B.24 2 B.26

7 B.2: B.2 i m i j m j r ij j i F ij (X ij, Y ij, Z ij ) X ij = Gm im j r 2 ij (x i x j ), Y ij = Gm im j (y i y j ) r ij rij 2, Z ij = Gm im j r ij rij 2 (z i z j ) r ij (B.27) x i X ij + y i Y ij + z i Z ij i j ( X ij, Y ij, Z ij ) x j X ij y j Y ij z j Z ij v = (x i x j )X ij + (y i y j )Y ij + (z i z j )Z ij = Gm im j (x i x j ) 2 rij 2 Gm im j (y i y j ) 2 r ij rij 2 Gm im j r ij rij 2 (z i z j ) 2 r ij = Gm im j r ij (B.28) i > j V = N i>j Gm i m j r ij = Ω (B.29)

8 3 B.24 T Ω 1 d 2 I = 2T + Ω (B.30) 2 dt2 d 2 I/dt 2 = 0 2T + Ω = 0 Virial Theorem (B.31) T + Ω = E (B.32) E 3 i F i F i = N j=1 (j i) Gm i m j (r i r j ) r i r j 3 B.24 2 N N N Gm F i r i = i m j (r i r j ) ri r i r j 3 i=1 i=1 j=1 (j i) N N = 1 Gm i m j (r i r j ) 2 2 r i r j 3 i=1 j=1 N Gm i m j = r i r j i>j Ω.

B.32 B.31 T = E, (B.33) Ω = 2E (B.34) M v T = N i=1 1 2 m iv 2 i = 1 2 M v2 (B.35) R Ω = GM 2 R 9 (B.36) v 2 = GM R E = 1 2 2 Ω = GM 2R (B.37) (B.38) B.2.3 B.2 m H N M = m H N T gas k 1 (3/2)kT gas N T = 3 2 kt gasn (B.39)

10 B.3: c V γ c V T gas = R g T/(γ 1) R g = k/m H U = 1 kt gas M (B.40) γ 1 m H B.39 B.40 M = m H N T = 3 (γ 1)U 2 (B.41) B.31 3(γ 1)U + Ω = 0 (B.42) U + Ω = E (B.43) B.42 E = U + Ω = (3γ 4)U = 3γ 4 3(γ 1) Ω (B.44)

11 B.3 B.3.1 λ ν λ 0 ν 0 shift 3C 273 quasar 11.6 Hα 656.3 nm 3C 273 Hα 760.0 nm 3C 273 Hα 656.3 nm 760.0 nm λ λ 0 1 λ λ 0 z = λ λ 0 1 = λ λ 0 λ 0 = ν 0 ν 1 (B.45) z λ λ 0 z λ λ 0 z redshift z blue shift

12 B.3.2 3 1 1905 Doppler effect B.4 B.4 θ v v cos θ radial velocity θ τ n c cτ λ 0 = cτ n (B.46)

13 B.4: t v ct + vt cos θ λ = ct + vt cos θ n B.47 B.46 (B.47) 1 + z = λ = t (1 + β cos θ) (B.48) λ 0 τ β = v/c B.48 t/τ t = γτ = τ 1 β 2 B.5 (B.49) 1 + z = λ λ 0 = ν 0 ν = 1 + β cos θ 1 β 2. (B.50)

14 5 4 v/c 0.9 3 0.7 1+z 2 1 0.5 0.3 0.1 0 0 30 60 90 120 150 180 θ 赤方偏移青方偏移 B.5: v/c θ z θ = 0 v c = 2z + z2 2 + 2z + z 2 (B.51) v c B.50 z 2 z = v cos θ. (B.52) c 10 M r

15 座標時間 t λ 固有時間 τ λ0 r B.6: τ t τ = t 1 r g r, r S = 2GM c 2 (B.53) B.6 r S = 2GM/c 2 Schwarzschild radius τ = 1 100 100 Hz 100 t = 2 3 gravitational redshift 1 + z = λ λ 0 = ν 0 ν = 1 1 r S r. (B.54) 4.1 kev 6.7 kev FeXXV z

16 3 a cosmological redshift λ 0 a(τ) λ a(t) 1 + z = λ λ 0 = ν 0 ν = a(t) a(τ). (B.55) a(t) λ 現在 t (t=137 億年 ) a(τ) λ0 τ 晴れ上がり (t=38 万年 ) ビッグバン (t=0) B.7: t τ a(τ) t τ B.55 1 + z = 1 + da(t)/dt (t τ) (B.56) a(t) (da/dt)/a = H (t τ) = r/c z = H c r. (B.57)

17 B.4 B.4.1 B.8 Lorentz force cyclotron radiation 4 1 m q v B m dv dt = q c v B c (B.58) v v v v = v + v 0 B.58 dv dt dv dt = 0, (B.59) = q mc v B (B.60) 4

18 B v B.8: B.59 B.60 B.9 r v = v ω = v /r m = m e q = e B.61 m e rω 2 = m e v 2 r = e c v B (B.61) r = r L m ecv eb = 5.69 v gauss 10 8 cm s 1 B cm (B.62) Larmor radius/gyro radius ω = ω L eb B = 1.76 107 rad s 1 (B.63) m e c gauss

19 r v F B B v F q=+e q=-e B.9: cyclotron frequency P = 2π/ω L = 2πm e /eb ν = ν L ω L 2π = eb B = 2.80 106 2πm e c gauss Hz (B.64) m m e = 9.1094 10 28 g q e = 4.8032 10 10 esu v c = 3.0 10 10 cm s 1 B 1gauss 17 m 1.76 10 7 rad/s 2.8 MHz z (x, y, z) B.60 dv x dt = ω Lv y, dv y dt = ω Lv x, dv z dt = 0 (B.65) v x = v cos(ω L + α), v y = v sin(ω L + α), v z = v, (B.66)

20 x = x 0 + r L sin(ω L + α), y = y 0 + r L cos(ω L + α), z = z 0 + v t. (B.67) v B.10: 2 B.10 θ dω 3 dp dω = e4 B 2 v 2 8πm 2 c 5 ( 1 + cos 2 θ ) (B.68) power

21 B.68 P = dp dω dω = π 2π 0 0 dp sin θdθdφ dω (B.69) P = 2e4 B 2 v 2 3m 2 ec 5 ( ) 2 B (v = 1.58 10 15 gauss c ) 2 erg s 1 (B.70) 4 n (v/c) 2n B.11 Iν νl 2νL 3νL ν B.11: 1

22 B.4.2 synchrotron radiation 5 B.12 B v 1 B.12: m q v B d dt (γmv) = q c v B (B.71) γ Lorentz factor v = v γ = 1 1 v2 /c 2 (B.72) 5

1 1 γm v v v v = v + v 0 v B.71 23 dv dt dv dt = 0, (B.73) = q γmc v B (B.74) r v = v ω = v /r m = m e q = e γm e rω 2 = γm e v 2 r = e c v B (B.75) B.75 r = r B γm ecv eb = 5.69 10 8 v gauss cm s 1 γ cm (B.76) B γ ω = ω B eb B 1 = 1.76 107 rad s 1 (B.77) γm e c gauss γ

24 1/γ ν = ν B ω B 2π = eb B 1 = 2.80 106 2πγm e c gauss γ Hz 2 (B.78) B.13 θ 1/γ (B.79) v B.13: 3 power

25 γ 2 P = 2e4 B 2 γ 2 v 2 3c 5 m 2 e ( ) 2 B (v = 1.58 10 15 gauss c ) 2 γ 2 erg s 1. (B.80) α v v = v sin α v 2 = 1 v 2 sin 2 αdω = 2 4π 3 v2 (B.81) B.80 ( P = 4e4 9c 3 m 2 B 2 v ) ( ) 2 2 B (v ) 2 γ 2 = 1.1 10 15 γ 2 erg s 1. (B.82) e c gauss c ( e 2 ) 2 = 6.6520 10 25 cm 2 (B.83) σ T 8π 3 m e c 2 U mag B2 8π B.82 (B.84) P = 4 3 σ Tcγ 2 U mag (B.85) B.85 B.5.3 γ E = γm e c 2 (B.86) t syn γm ec 2 P ( B 7.8 10 8 gauss ) 2 1 γ s. (B.87)

26 4 1 B.14 B.78 ν B = ν L /γ 1/γ B.14 Iν Iν νl/γ νl ν 0.29 1 ν/νc B.14: 1 1 ν c = 3 2 ν Lγ 2 = 3 2 ν Bγ 3 = 3eB 4πmc γ2 m e B m gauss γ2 MHz (B.88) 0.29ν c ν c 6.27 10 12 m ( ) 2 e B E MHz (B.89) m gauss ev

27 power law spectrum B.15 放射強度の対数 振動数の対数 B.15: E = γmc 2 E E + de N(E) N(E)dE = KE p de (B.90) P tot B (1+p)/2 ν (1 p)/2 (B.91) p N E p α S ν ν α α = p 1 2 (B.92)

28 B.5 B.5.1 hν mv 2 /2 mc 2 scattering Thomson scattering electron scattering B.16: B.16

E = E 0 cos(k r ωt + α) dσ dω = erg s 1 erg s 1 cm 2 (B.93) ν hν n θ ν hν n B.17 29 ν' ν n n' θ B.17: ν = ν dσ dω = 1 ( ) e 2 2 [ 1 + (n n 2 m e c 2 ) 2] = 1 ( ) e 2 2 ( 1 + cos 2 2 m e c 2 θ ) = 1 ( 2 r2 e 1 + cos 2 θ ) (B.94)

30 B.18 ( ) e 2 2 r e = m e c 2 = 2.8179 10 13 cm (B.95) classical electron radius B.18: B.94 dσ dσ σ = dω dω = sin θdθdφ dω = 8π ( ) e 2 2 3 m e c 2 = 8π 3 r2 e σ T = 6.6520 10 25 cm 2 (B.96) σ T m e = 9.1094 10 28 g e = 4.8032 10 10 esu c = 3.0 10 10 cm s 1 B.5.2 hν

31 m e c 2 Compton scattering hν' hν v' θ φ B.19: ν hν θ ν hν ϕ v B.19 γ = 1/ 1 (v /c) 2 hν + m e c 2 = hν + γ m e c 2, (B.97) hν c = hν c cos θ + γ m e v cos ϕ, (B.98) 0 = hν c sin θ γ m e v sin ϕ (B.99) B.98 B.99 2 ( ) 2 ( ) hν c hν hν 2 c cos θ + c sin θ = γ 2 m 2 ev 2 = γ 2 m 2 ec 2 m 2 ec 2 (B.100)

32 B.97 ( ) 2 ( ) hν c hν hν 2 ( ) 2 hν c cos θ + c sin θ = c + m ec hν m 2 c ec 2 (B.101) m e hν m e hν = hν c hν (1 cos θ) c (B.102) m e c 2 hν m ec 2 = 1 cos θ, (B.103) hν ν ν = 1 + hν (B.104) 2 (1 cos θ) m e c B.103 (B.104 B.104 λ = c/ν λ λ = λ C (1 cos θ) (B.105) λ C h = 0.002426 nm m e c (B.106) B.103 B.105 B.102 ν ν = ν ν ν hν m e c 2 (B.107)

33 6 B.5.3 m e v 2 /2 m e c 2 inverse Compton scattering ν hν v ν hν v B.20 0 ν > ν 0 ν > ν ν < ν ν T 6 dσ dω = 3 ( ν ) 2 ( ν 16π σ T ν ν + ν ) ν 1 + cos2 θ ) σ T (1 2hν σ ( m e c) 2 [ ( ) ] 3 8 σ mec 2 T ln 2hν hν m e c 2 + 2 1 hν m e c 2 hν m e c 2 σ T (B.108) (B.109)

34 v hν v' hν' B.20: 1 ν ν = hν m e c 2 + 4kT m e c 2 (B.110) 1 2 hν > 4kT hν < 4kT 1 ( e 2 ) 2 = 6.6520 10 25 cm 2 (B.111) σ T 8π 3 m e c 2 U rad 2h ν 3 n(ν)dν (B.112)

n 35 P = 4 3 σ Tcγ 2 U rad (B.113) B.113 B.4.2 2 1 4kT/m e c 2 power law spectrum B.21 放射強度の対数 振動数の対数 B.21: p N E p α S ν ν α α = p 1 2 (B.114)

36 B.6 B.6.1 1 1 B.22 B.22 re -e a0 a0 +e B.22: 10 11 s

Bohr radius a 0 h 2 4π 2 m e e 2 = 5.2918 10 9 cm = 0.5 Å r e e2 m e c 2 = 2.8179 10 13 cm 37 (B.115) (B.116) 1 1 m e = 9.1094 10 28 g, (B.117) e = 4.8032 10 10 esu (B.118) h = 6.6261 10 27 erg s (B.119) B.6.2 e e B.23 r r n v v n d 2 r m e dt 2 = e2 r 2 + m v 2 e r = 0 (B.120) B.120 1 2 j n j = m e rv = n h 2π (B.121)

38 v +e r -e B.23: E = 1 2 m ev 2 e2 (B.122) r B.122 1 2 B.121 B.24 λ 2πr = nλ (B.123) λ = h m e v (B.124) B.121 B.120 B.122 B.120 B.121 r = r n = h 2 4π 2 m e e 2 n2 = a 0 n 2 = 0.053n 2 nm (B.125)

39 r B.24: v = v n = 2πe2 h 1 n = 2.2 108 1 n cm s 1 (B.126) a 0 = 0.053 nm B.125 B.126 B.122 E = E n = 2π2 m e e 4 h 2 1 n 2 = e2 2a 0 1 n 2 = 13.59 1 n 2 ev (B.127) E n E n hν = E n E n (B.128) 1 λ = ν c = 2π2 m e e 4 ( 1 ch 3 n 2 1 ) n 2 (B.129)