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1 /
2
3 ( ) ( )
4 :
5 : f(e, J), f(e) Phase mixing Landau Damping, violent relaxation
6
7 : 2 2
8 :
9 ? m i d 2 x i dt 2 = j i f ij (1) x i m i i f ij j i f ij G f ij = Gm i m j x j x i x j x i 3, (2)
10 ( ) 2 ( ) 3 ( / )
11 YES NO
12 : +9 ( ) 0 ( )
13 ( ) : 3 0.1%
14 19
15 20 20
16 2
17 ( )
18 3 3 (2 )
19 3 3
20 3 2,3 Sun L4 L5 Jupiter
21 2006 IAU Genearal Assembly ( )
22 II 10 Figure-8 Solution Edinburgh Douglas Heggie
23 Figure-8 solution 3 (0.005% )
24
25 1987 2
26 e 2 8.5
27 45 2
28 100
29 : : 1000 ( )
30 : : 10 7 : : 10 5 ( )
31
32 ( )
33 ( )
34
35
36 ( )
37 ( ) SDSS
38 10 15 ( )
39 Central Cluster Genzel et al 2003 K-band shift-and-add image SgrA
40 Surface density Genzel et al M ( ) 0.5 (S1, S2, S )
41 Eisenhauer et al 2005 (10M )
42 20 : : ( ) ( 10 5 ) : : : Arches, Quintuplet ( ) : M82
43 Arches, Quintuplet 30pc 10 4 M ( ) 1-2Myrs
44 d 2 x i dt 2 = j i Gm j x j x i x j x i 3, (3) (1 ) f(x, v, t) :( )
45 ( ) f f + v f Φ = 0, (4) t v f:6 Φ :, 2 Φ = 4πGρ. (5) G ρ ρ = m dvf, (6) m f
46 BBGKY : f t + v f Φ f v 6 Df/Dt = 0:
47 1 w = (x, v, t) Φ = Φ(x, t) ẇ = (ẋ, v) = (v, Φ) (7) f t + 6 (fẇ i ) = 0, (8) i=1 w i 2 w 6 i=1 ẇ i w i = 3 i=1 v i + v i x i v i = 3 i=1 v i Φ = 0 (9) x i
48 2 w f t + v f Φ f v f:6 = 0, (10) Φ :, 2 Φ = 4πGρ. (11) G ρ ρ = m dvf, (12)
49 = f = Φ f 0
50 f (f log f )
51 :
52 =
53 ( ) : N 1/N ( ) N
54 N 1/ N 1 1/ N N
55 N/ log N
56 Φ x, v I d I(x, v) = 0, (13) dt v I Φ I v = 0 (14)
57 : : 1/v 2 + Φ r L = r v
58 I f I 1, I 2,..., I m f = f(i 1, I 2,..., I m )
59 : f OK f I k 0 5
60 3 4 :
61
62 f(e, J) f E J f J f(e, J) 1 d r 2 dr r 2dΦ = 4πG dr f 1 2 v2 + Φ, r v dv, (15) f(e, J)
63 f(e) J < v 2 e >= 1 ρ v 2 e f(v2 + Φ)dv (16) f v v e
64 f(e) ( ) Ψ = Φ + Φ 0, E = E + Φ 0 = Ψ v 2 /2 (17) Φ 0 E > 0 f > 0, E f = 0 v 1 d r 2 dr r 2dΨ dr = 16π 2 G 2Ψ 0 f(ψ 1 2 v2 )v 2 dv = 16π 2 G Ψ 0 f(e) 2(Ψ E)dE. (18) f Ψ
65 Hernquist
66 E F E f(e) = n 3/2 (E > 0) (19) 0 otherwize Ψ ρ = c n Ψ n (Ψ > 0) (20) c n n > 1/2 ρ 1 d r 2 dr r 2dΨ + 4πGc n Ψ n = 0 (21) dr
67 Lane-Emden 1 d r 2 dr r 2dΨ + 4πGc n Ψ n = 0 (22) dr 1 d s 2 ds s 2dψ ds Lane-Emden + ψ n = 0 (23) Lane-Emden
68 n Lane-Emden n = 5 1 φ = (24) s2 c 5 φ 5 r = 0 r 1/r 3 self-consistent
69 Lane-Emden P ρ 1+1/n (25) : ( ) :
70 Hernquist Model 1990 (Hernquist, L., 1990, ApJ 356, 359) Φ = 1 r + a ρ = C (26) a r(r + a) 3 (27)
71 Hernquist Model r 1/4 Hernquist Model r 1/4 ( ) r 1/4 1/4
72 Hernquist Hernquist : ( ) Jaffe Dehnen Tremaine η 3 ( ) Navarro-Frenk-White (NFW) Moore
73 Dehnen Model ρ = C γ = 1: Hernquist model γ = 2: Jaffe model a r γ (r + a) 3 γ (28)
74 NFW Moore NFW (NFW1996) ρ = C a r(r + a) 2 (29) Moore (Moore et al 1999) a ρ = C r 1.5 (r a 1.5 ) (30)
75 f(e) = ρ 1 (2πσ 2 ) 3/2eE/σ2 = ρ 1 exp (2πσ 2 ) 3/2 Ψ v 2 /2 σ 2 (31)
76 ( ) 2 e x2 dx = 1 (32) π 0 ρ = ρ 1 e Ψ/σ2 (33) 1 d r 2dΨ r 2 = 4πGρ (34) dr dr 1 r 2 d dr r 2d log ρ = 4πGσ 2 ρ (35) dr
77 ( 2) ρ = (36) 2πGr 2 (singular isothermal sphere) self consistent : M r r flat rotation curve : ρ 1/r 2 ( ) r singular isothermal σ2
78 dp dr = ρgm r (37) r 2 P = k BT m ρ (38) P M r C d dr r 2d log ρ = 4πGρr 2 (39) dr
79 stellar system stellar system
80 King Model 0 :
81 f(e) = ρ 1 (2πσ 2 ) 3/2eE/σ2 = ρ 1 exp (2πσ 2 ) 3/2 Ψ v 2 /2 σ 2 (40) : (E ) 0
82 Lowered Maxwellian f(e) = ρ 1 (2πσ 2 ) 3/2 (e E/σ2 1) (E > 0) 0 (E 0) (41) E = 0 f = 0 1
83 ρ = 4πρ 1 (2πσ 2 ) 3/2 = ρ 1 2Ψ 0 e Ψ/σ2 erf Ψ σ exp Ψ v 2 /2 1 v 2 dv 4Ψ πσ 2 σ Ψ 3σ 2 (42) erf 1 erf(z) = 2 π z 0 e t2 dt. (43)
84 dψ/dr = 0 Ψ 0
85 Ψ 0 1, 3, 6, 9, 12 r 0 r 0 = 9σ 2 (44) 4πGρ 0
86 ( ) r t ρ 0 ( Ψ 0 ) : King model tidal radius King Model c = log(r t /r 0 ) concentration parameter Ψ 0
87 :
88 X Einstein IPC Hydra A (David et al. ApJ 1990, 356, 32)
89 Chandra
90
91 ρ g ρ s d dr ρ s r 2d log ρ g = 4πGm dr k B T ρ sr 2 (45) ρ g ρ g = Cρ s (46)
92 T g T s T g /T s = 1/β d dr r 2d log ρ g = β[ (45) ] (47) dr ρ g = Cρ β s (48) β
93 ρ = ρ 0 [1 + (r/r c ) 2 ] 3/2 (49) β Sarazin X-ray emissions from clusters of galaxies 5.5 β (48) ρ s ρ s r < 2.5r c
94 β 1 ρ = ρ 0 [1 + (r/r c ) 2 ] 3/2 (50) ρ = ρ 0 [1 + (r/r c ) 2 ] 3β/2 (51) X X = 2 I(r) = 2 0 ρ( r 2 + z 2 ) 2 dz (52) I(r) = C[1 + (r/r c ) 2 ] 3β+1/2 (53)
95 β : (Jones and Forman 1984) Hydra A
96 β 2 Ikebe et al. 1997, ApJ 481, 660
97 β excess Cooling flow β β β 2
98 Cooling flow cluster β β isothermal
99
100
101
102 ρ f f(e) f(e) ρ(r) = 4π Ψ 0 f(e) 2(Ψ E)dE. (54) Ψ r ρ Ψ Ψ 1 dρ 8π dψ = Ψ 0 f(e)de Ψ E (55)
103 ρ f 1 dρ 8π dψ = Ψ 0 f(e)de Ψ E (56) Abel f(e) = 1 8π 2 f(e) = 1 8π 2 E 0 d 2 ρ dψ 2 d de E 0 dρ dψ dψ + 1 E Ψ E dψ E Ψ (57) dρ dψ Ψ=0 (58)
104 :Jaffe model Jaffe model ρ = r 4 J r 2 (r + r J ) 2, Ψ = ln r r + r J r Ψ ρ (59) ρ = (e Ψ/2 e Ψ/2 ) 4 (60) 58 x = E Ψ E = Er j /GM f(e) = M 2π 3 (GMr J )3/2 [F ( 2E) 2F ( E) 2F + ( E) + F + ( 2E)] (61)
105 F ± = e x2 x 0 e±x 2 dx (62)
106 Jeans Equations f t + v f Φ f v = 0, (63) 3 0
107 ( ) 2 ν = fd 3 v; v i = 1 ν fvi d 3 v (64) ν t + (ν v i) x i = 0 (65)
108 1 Collisionless Boltzman v j fvj d 3 v + f v i v j d 3 v Φ f vj d 3 v = 0, (66) t x i x i v i i v j f 1 (ν v j ) vj f v i d 3 v = t + (νv iv j ) x i v j v i fd 3 v = δ ij ν (67) + ν Φ x j = 0 (68)
109 v i v j v i v j ν/ t ν v j t v (ν v i ) j + (νv iv j ) x i x i + ν Φ x j = 0 (69) σ 2 ij = (v i v i )(v j v j ) (70)
110 Jeans ν v j t + ν v v j i = ν Φ (νσ2 ij ) (71) x i x j x i Lagrange stress tensor σij 2 stress tensor σ 2 I I
111 M/L Jeans equation d(νv 2 r ) dr + ν r [ 2v 2 r ( v 2 θ + v2 φ)] = ν dφ dr (72) 1 d(νvr 2) ν dr M(r) = rv2 r G = dφ dr d ln ν d ln r + d ln v2 r d ln r (73) (74)
112 M/L M(r) = rv2 r G d ln ν d ln r + d ln v2 r d ln r (75) ν luminosity
113 M/L M/L M/L
114 : M STScI
115 ( ) 3 2 M X... X ( )
116
117 (1)
118 (2) 1. M/L χ 2!
119 : Sosin & King 1997, : Guhathakurta et al
120 ( ) M/L :
121 ( ) = f
122 : M/L
123 M15 ( ) Baumgardt et al., ApJ 2003, 582, L21
124
125 M/L
126 M
127 M/L M/L Dull et al. ApJ 1997
128 M/L
129 Dull et al.
130 M/L Dull et al. M/L 3
131 Dull et al.
132
133 pc arcmin
134 68 ν ρ x k xk (ρ v j ) t d 3 x = x k (ρv i v j ) x i d 3 x ρx k Φ x j d 3 x (76) xk (ρv i v j ) x i d 3 x = δ ki ρv i v j d 3 x = 2K kj (77) K W
135 ( ) σ 2 K jk = T jk Π jk (78) T jk = 1 2 ρ vj v k d 3 x, Π jk = ρσ 2 jk d3 x, (79) j, k k, j 1 d 2 dt ρ(xk v j + x j v k )d 3 x = 2T jk + Π jk + W jk (80)
136 ( 2) I jk = ρx j x k d 3 x (81) di jk dt = ρ(x k v j + x j v k )d 3 x (82) 1 d 2 I jk = 2T 2 dt 2 jk + Π jk + W jk (83)
137 I 0 T, Π K 2 W Φ W = ρ k x k Φ x k d 3 x = ρ(x)ρ(x ) k x k (x k x k ) x x 3 d 3 xd 3 x x x (84) W = 1 2 ρ(x)ρ(x ) (x k x k )2 k x x 3 d3 xd 3 x = 1 2 ρφd 3 x (85) W
138 ( ) 2K + W = 0 (86) E E = K + W E = K = W/2 (87)
139 King model K = E
140 z W xx = W yy ; W ij = 0 (i j) (88) 2T xx + Π xx + W xx = 0; 2T zz + Π zz + W zz = 0 (89) 2 2 2T xx + Π xx 2T zz + Π zz = W xx W zz (90)
141 T zz = 0 2T xx = 1 2 ρv 2 φ d3 x = 1 2 Mv2 0 (91) v 0 Π xx = Mσ 2 0 (92) (r.m.s) z x δ Π zz = (1 δ)πxx (93)
142 2 v 2 0 σ 2 0 = 2(1 δ) W xx W zz 2 (94) W xx /W zz : ρ = ρ(m); m 2 = a i=1 x 2 i a 2 i (95) W ii /W jj a ρ(m)
143 3 Binney and Tremaine( BT) δ ε1 b/a v 0 σ 0
144 4 BT : : ε v 0 σ 0 : δ = 0
145 5 δ = 0
146 6 Okumura et al. (1991, PASJ 43, 781)
147 7 2 4 r p /r h = 2.6 v 0 /σ 0
148 CBE
149 ρ + (ρv) = 0 (96) t v t + (v )v = 1 p Φ (97) ρ 2 Φ = 4πGρ (98)
150 ρ, p, v, Φ ρ = ρ 0 + ρ v s ρ 1 t + (ρ 0v 1 ) + (ρ 1 v 0 ) = 0 (99) v 1 t + (v 0 )v 1 + (v 1 )v 0 = ρ 1 p ρ p 1 Φ 1 0 ρ 0 (100) 2 Φ 1 = 4πGρ 1 (101) dp p 1 = ρ 1 = v 2 s dρ ρ 1 (102)
151 0 2 ρ 1 t + ρ 1 (ρ 0 v 1 ) = 0 (103) v 1 t = 1 p 1 Φ 1 ρ 0 (104) 2 ρ 1 2 ρ 1 t 2 v2 s 2 ρ 1 4πGρ 0 ρ 1 = 0 (105)
152 2 ρ 1 t 2 v2 s 2 ρ 1 4πGρ 0 ρ 1 = 0 (106) 2
153 ρ 1 = Ce i(k x ωt) (107) ω 2 = v 2 s k2 4πGρ 0 (108)
154 ( ) k 2 J = 4πGρ 0 v 2 s (109) k > k J ω k = k J ω = 0 k < k J ω
155 ( 2) 1/k J
156 f f + v f Φ = 0, (110) t v f 6 Φ 2 Φ = 4πGρ. (111) G ρ ρ = m dvf, (112)
157 f 0 + f 1 Φ 0 + Φ 1 0 f 1 t + v f 1 Φ 0 f 1 v Φ 1 f 0 v = 0, (113) 2 Φ 1 = 4πG f 1 dv. (114)
158 Jeans f 0 Φ 0 f 1 t + v f 1 Φ 1 f 0 = 0, (115) v f 1 = f a (v) exp[i(k x ωt)] (116) Φ 1 = Φ a exp[i(k x ωt)] (117)
159 ( ω + v k)f a (v) Φ a k f 0 v = 0 (118) k 2 Φ a = 4πG f a dv (119) f a Φ a 1 + 4πG k 2 k f 0 v dv = 0 (120) k v ω f 0 k ω
160 1 + 4πG k 2 k f 0 v dv = 0 (121) k v ω k v ω = 0 0
161 ( ) x 1 + 4πG k 2 k f 0 v x dv = 0 (122) kv x ω ω = 0 k 2 J = 4πG f 0 v x v x dv (123) f 0 v x = 0 ω = 0 k J
162 f 0 f 0 (v) = ρ 0 exp (2πσ 2 ) 3/2 v 2 /2 (124) σ 2 e v2 2σ 2 dv = 2πσ 2 (125) k 2 J = 4πGρ 0 σ 2 (126)
163 ω = iγ πGρ 0 kσ 3 vx e v2 x /2σ2 kv x iγ dv x = 0 (127) v x (kv x + iγ)e v2 x /2σ2 k 2 v 2 x + γ2 dv x = 0 (128)
164 ( ) 0 x 2 e x2 x 2 + β 2dx = 1 1 π 2 2 πβeβ2 [1 erf(β)] (129) k 2 = k 2 J πγ 1 exp 2kσ γ 2 2k 2 σ 2 1 erf γ 2kσ (130) k ω (γ) k < k J γ k = k J k = 0
165 ( 2)
166 van Kampen mode van Kampen (1955, Physica, 21, 949)
167 van Kampen mode( ) ( ω + v k)f a (v) + 4πG k k f 0 2 v fa dv = 0 (131) y, z ( ω + kv x )g(v x ) = 8π2 Gv x f 0 (v x ) gdv x = 0 (132) k g f a y, z f 0
168 van Kampen mode( 2) g g(v x ) = 8π2 G k v x f 0 (v x ) ω kv x (133) gdv x = 1 8π2 G k vx f 0 (v x ) ω kv x dv x = 1 (134)
169 van Kampen mode( 3) v = ω/k g g(v x ) = 8π2 Gv x f 0 (v x ) k P 1 + λδ(ω kv x ) ω kv x (135) P δ λ ω k > k J k van Kampen mode
170 van Kampen mode( 4) g(v x ) = 8π2 Gv x f 0 (v x ) k P 1 + λδ(ω kv x ) ω kv x (136) δ v x = ω/k
171 Phase Mixing 1 (overdensity)
172 Phase Mixing ( ) v v v x x x singular van Kampen mode phase mixing
173 Landau Damping ω k f 1 (x, v, t) = g(v) exp[ik(x vt)] (137) : f a (v) exp[i(kx ωt)]
174 Landau Damping ( ) : ω v van Kampen mode singular
175 Landau Damping ( 2) f 1 (x, v, t) = g(v) exp[ik(x vt)] (138) g(v) ρ 1 (x, t) = e ikx g(v)e ivt dv (139) g(v) = 1/2v 0 v < v 0 0 otherwize 1/t (140)
176 ( )Landau Damping g(v)
177 f 1 (x, v, t) = g(v) exp[ik(x vt)] (141) v v g(v) g(v)
178 ( ) f
179 S = f ln fdxdv (142) g(x, v; h) g(x, v; h)dxdv = 1 (143) lim h 0 fg(x0 x, v 0 v; h)dxdv = f(x 0, v 0 ) (144) δ
180 ˆf h ˆf h = fg(x x 1, v v 1 ; h)dx 1 dv 1 (145) Ŝ = ˆf ln ˆfdxdv (146)
181 Ŝ
182 Wave-particle interaction /
183 Wave-particle interaction ( ) v w v p v p v w
184 Wave-particle interaction ( 2)
185 Dynamical Friction 0 OK
186 Dynamical Friction ( )
187 Dynamical Friction ( 2) = dynamical friction 3
188 Dynamical Friction ( 3) Dynamical Friction
189 Violent relaxation Landau damping phase mixing Landau damping
190 r 1/4 Hernquist Profile
191 Violent Relaxation Lynden-Bell (1967) violent relaxation
192 Violent Relaxation ( ) Lynden-Bell
193 Violent Relaxation ( ) phase mixing Lynden-Bell Lynden-Bell Maxwell-Boltzman f Lynden-Bell
194 violent relaxation
195 van Albada (1982, MNRAS 201, 939) : 1982 N = 5000 :
196 N(E) N(E) : N(E)dE = dn
197 : violent relaxation
198
199
200 N(E)
201 Lynden-Bell violent relaxation
202 Violent relaxation violent relaxation 2
203 N(E) Lynden-Bell IAU Symposium 127 Structure and Synamics of Elliptical Galaxies Scott Tremaine (Conference Summary) W. Jaffe (Poster) Jaffe
204 N(E) (2) 0 0
205 N(E) (2) : 0
206 0 phase volume 0 N(E) 0 van Albada N(E) E = 0 r N(E) = N 0 + N 0 E N 0 > 0
207 N(E) N(E) Φ = M/r (147) E = M 2r (148)
208 dm = 4πr 2 ρdr = N(E)dE (149) 148 de dr = M 2r 2 (150) ρ = MN(E) 8πr 4 (151) 4 4
209 : Jaffe (1987)
210 E (r, r+ dr) P (E, r)dr 4πr 2 ρ = 0 E r P (E, r)n(e)de (152) E r r M/r P (E, r) P (E, r) = P 0 (r/r E )/r E (153) r E = M/E
211 x = re/m ρ = M 0 4πr P 4 1 0(x)xN(Mx/r)dx (154) r radial orbit J N(E, J) J OK
212 N(E) N(E) E 0 ρ r 4 Hernquist model Jaffe model ρ r 4
213 10 Navarro ApJ 1997, 490, 493) CDM CDM ρ 1 r (1 + r ) 2 (155) NFW
214 Navarro CDM Fukushige and Makino (ApJ 1997, 477 L9) Moore et al. (ApJ 1998, 499, 5L ) Navarro 1.5 Navarro 1 Fukushige Moore 100
215 2 half mass radius 0 1.5
4 19
I / 19 8 1 4 19 : : f(e, J), f(e) Phase mixing Landau Damping, violent relaxation : 2 2 : ( ) http://antwrp.gsfc.nasa.gov/apod/ap950917.html ( ) http://www-astro.physics.ox.ac.uk/~wjs/apm_grey.gif
A 99% MS-Free Presentation
A 99% MS-Free Presentation 2 Galactic Dynamics (Binney & Tremaine 1987, 2008) Dynamics of Galaxies (Bertin 2000) Dynamical Evolution of Globular Clusters (Spitzer 1987) The Gravitational Million-Body Problem
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
Contents 1 Jeans (
Contents 1 Jeans 2 1.1....................................... 2 1.2................................. 2 1.3............................... 3 2 3 2.1 ( )................................ 4 2.2 WKB........................
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n
. X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n
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
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
1 ( ) Einstein, Robertson-Walker metric, R µν R 2 g µν + Λg µν = 8πG c 4 T µν, (1) ( ds 2 = c 2 dt 2 + a(t) 2 dr 2 ) + 1 Kr 2 r2 dω 2, (2) (ȧ ) 2 H 2
1 ( ) Einstein, Robertson-Walker metric, R µν R 2 g µν + Λg µν = 8πG c 4 T µν, (1) ( ds 2 = c 2 dt 2 + a(t) 2 dr 2 ) + 1 Kr 2 r2 dω 2, (2) (ȧ ) 2 H 2 = 8πG a 3c 2 ρ Kc2 a 2 + Λc2 3 (3), ä a = 4πG Λc2 (ρ
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
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