Necklace Flower Constellations

Transcription

1 David Arnas Martínez Necklace Flower Constellations Departamento Instituto Universitario de Investigación en Matemáticas y sus Aplicaciones Director/es Eva Tresaco Vidaller Antonio Elipe Sánchez

2 Reconocimiento NoComercial SinObraDerivada (by-nc-nd): No se permite un uso comercial de la obra original ni la generación de obras derivadas. Universidad de Zaragoza Servicio de Publicaciones ISSN

3 Tesis Doctoral NECKLACE FLOWER CONSTELLATIONS Autor David Arnas Martínez Director/es Eva Tresaco Vidaller Antonio Elipe Sánchez UNIVERSIDAD DE ZARAGOZA Instituto Universitario de Investigación en Matemáticas y sus Aplicaciones 2018 Repositorio de la Universidad de Zaragoza Zaguan

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11 G M G M L Ω L M N M L Ω L M

12 J2 n J 2

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27 m 1 ẍ 1 = G m 1m 2 x 1 x 2 2 x 1 x 2 x 1 x 2, m 2 ẍ 2 = G m 1m 2 x 2 x 1 2 x 2 x 1 x 2 x 1,

28 G m 1 m 2 x 1 x 2 ẍ 1 ẍ 2 r = µ r r 3, µ = G(M +m 2 ) GM M r h h = r ṙ, r ṙ h ḣ = ṙ ṙ + r ṙ = r ṙ µ (r r) = 0, r3 r = r h r = 0 h v = 0 h r ṙ u x, u y u z r = ( 3µ yz ) ( r 5 3µyz r 5 u x + 3µ xz ) ( r 5 3µxz r 5 u y + 3µ xy ) r 5 3µxy r 5 u z = 0, V V = r r r r dr = µ r 3 dr = µ r.

29 ξ ξ = v2 2 µ r. d ( r ) r (ṙ r) = dt r r 3 = r h = d (ṙ ) h, µ dt µ e = eu e = ṙ h µ r r, h e h = 0,

30 u r u θ u h u r u h u h = h/ h u θ = u h u r r, θ r = ru r, ṙ = ṙu r + r θu θ, ( r = r θ ) 2 r u r + ( θr + 2 θṙ ) u θ, h = hu h = r 2 θuh θ = h r 2, θ e = ( ) θrh µ 1 u r ṙh µ u θ, e ξ ξ = (e2 1)µ 2 2h 2, Ω i z

31 x cos(i) = h u z h, cos(ω) = (u z h) u x u z h, i [0, π) Ω [0, 2π) ω u e u n = u z u h cos(ω) = (u z h) e u z h e, ω [0, 2π) Ω, i ω ν cos(ν) = r e re, r = p 1 + e cos(ν), p = h 2 /µ r 2 = x 2 p + y 2 p x p = r cos(ν) x p y p (u e, u p = (u h u e ), u h ) (1 e 2 )x 2 p + y 2 p + 2epx p p 2 = 0, e a r per r apo a = r per + r apo 2 = p 2 ( e + 1 ) = p 1 e 1 e 2.

32 ṙ = p e sin(ν) ν, (1 + e cos(ν)) 2 sin(ν) = 0 ṙ 2 = h2 r 2, ξ = 1 h 2 2 p 2 (1 + e)2 µ p (1 + e) = 1 2 µ(1 e2 ) = µ p 2a, a e i Ω ω ν E r 2 = a 2 (1 e 2 ) sin 2 (E) + (ae a cos(e)) 2, r = a(1 e cos(e)). ṙ = ae sin(e)ė. ṙ 2 + ( ) h 2 = 2µ r r µ a, ṙ ṙ 2 = µ ( 2 a r 1 (1 ) e2 ) r 2.

33 E µ = (1 e cos(e))ė, a3 M = E e sin(e) = µ a 3 t. M M n M n = µ a 3, M = 0 M = 2π a 3 T = 2π µ, T µ J 2

34 J 2 J 2, J 3, J 4

35 J 2 S m S/m = 0.001m 2 /kg S/m = 0.01m 2 /kg r = µ r r 3 + γ, γ H t H = µ R(a, e, i, ω, Ω, M, t), 2a R

36 da dt de dt di dt dω dt dω dt dm dt = 2 R na M, = 1 e2 R 1 e na 2 e M 2 R na 2 e ω, cos i R = na 2 1 e 2 sin i M 1 R na 2 1 e 2 sin i Ω, 1 e 2 R = na 2 e e cos i R na 2 1 e 2 sin i i, 1 R = na 2 1 e 2 sin i i, = n 2 na R a 1 e2 na 2 e R e. J 2 J 2 R = 3µJ 2 2r ( R r ) 2 ( sin 2 (i) sin 2 (ω + ν) 1 ), 3 R J 2 = R avg = µj 2 a ( R a ) 2 ( 3 4 sin2 (i) 1 ) ( 2 1 (1 e 2 ) 3/2 ), ȧ sec = 0, ė sec = 0, i sec = 0, ( R ω sec = 3 ) 2 4 J 2 a(1 e 2 n ( 4 5sin 2 (i) ), ) Ω sec = 3 ) 2 2( 2 J R a(1 e 2 n cos(i), ) [ µ Ṁ sec = a ( ) 2 4 J R ( 2 2 3sin 2 a(1 e 2 (i) ) 1 e 2], ) ȧ sec ė sec i sec ω sec Ωsec Ṁsec J 2 sin 2 (i) = 4/5

37 ω sec = 0 i = o u r, u θ, u h γ r γ θ γ h ȧ = 2a2 h ( e sin(ν)γ r + p ) r γ θ, ė = 1 h (p sin(ν)γ r + ((p + r) cos(ν) + re) γ θ ), i = r h cos(ω + ν)γ h, Ω = r sin(ω + ν) γ h, h sin(i) ω = 1 eh ( p cos(ν)γ r + (p + r) sin(ν)γ θ ) + r sin(ω + ν) γ h, h tan(i) 1 e 2 Ṁ = n + ((p cos(ν) 2re)γ r (p + r) sin(ν)γ θ ). eh

38 N o N so N c N c [0, N o 1] Ω, M Ω M Ω, M Ω M 2π N o 0 N c N so Ω ij M ij = 2π i 1 j 1, i = 1,, N o j = 1,, N so N c [0, N o 1] Ω ij M ij (i, j) j i a e i ω Ω 11 M N o = 5 3 N so = 3 a = 14419, 944 km i = o e = 0 N c = 3 Ω 11 = 0 M 11 = 0 (Ω, M) (Ω, M)

39 (Ω, M) Ω = 0 Ω M (Ω, M)

40 N o N w N so N c1, N c2, N c3 N c1 [0, N o 1] N c2 [0, N w 1] N c3 [0, N o 1] N o 0 0 Ω ijk i 1 N c3 N w 0 ω ijk = 2π k 1 ; N c1 N c2 N so M ijk j 1 Ω ijk ω ijk M ijk Ω 111 ω 111

41 M 111 i = 1,, N o j = 1,, N so k = 1,, N w i k j Ω ijk ω ijk M ijk [0, 2π] N o = 5 N w = 3 N so = 3 N c1 = 3 N c2 = 2 N c3 = Ω ijk ω ijk M ijk = 2π i 1 k 1 j 1 ; N o N w N so = 45 N o N ω N so = 45 N ω N so = 9 N c1, N c2, N c3 N o N so (Ω, ω, M) N c1 N c2 N c3

42 (Ω, ω, M) a = 11522, 451 km e = 0.25 i = o y = 0

43 n 1 n + 1 G Z n = {1,..., n} G = {1, 2, 4} Z 4 G 1 G 2 = G 1 = G2 s : G 1 = G 2 + s mod (n), s Z n s G = {1, 2, 4} = {1, 2, 3} = {2, 3, 4} = {1, 3, 4} ;

44 K(n) = K(n) = { Z n }/ =, G G Z n Sym(G) r Z n G + r = G Z n Sym(G) = min {1 r n : G + r = G Z n }. r G 1 = G2 Sym(G 1 ) = Sym(G 2 ) Sym : K(n) N G Sym(G). n = 6 G = {1, 3, 5} Z 6

45 Sym(G) = 2 {1, 3, 5} {3, 5, 7} mod (6) {2, 4, 6} G n k N k (n) = 1 φ(d)k n/d, n d n d n φ(d) d d d k = 2 Ω ij M ij M Ω Ω ij M ij G k {1,..., Sym(G) 1} Ω M

46 M = 2π N so k 2π N so N c N o, k N o M Sym(G) kn o N c, Sym(G) kn o N c Sym(G) N o N c G = {1, 2} N so = 4 Sym(G) = 4 N o = 6 N c = 2 k Ω = 2π 4 6k 2, k = {1, 3}

47 G + X + : G X X (g, x) g + x, g 1 + (g 2 + x) = (g 1 + g 2 ) + x 1 G + x = x g 1, g 2 G, x X (x) x + (x) = {g + x g G} X; g (g) X g (g) = {x X g + x = x}. X y = g + x y x + 1 (g), G g G Y Y

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50 L Ω 0 L MΩ L M Ω ij M ij = 2π i 1 j 1, L Ω L M L MΩ i {1,..., L Ω } j {1,..., L M } Ω ij = 2π L Ω (i 1), M ij = 2π L M (j 1) 2π L M L MΩ L Ω (i 1), G M Z LM N M G M G M {1,..., L M }, G M = N M G M G M = {G M (1),..., G M (j ),..., G M (N M )}, 1 G M (1) < < G M (j ) < < G M (N M ) L M, j G M N M j + N M j G M : Z NM Z LM j G M (j ).

51 G M (j ) j {1,..., N M } (a, b) = a mod (b) G M (j ) = G M ( (j + N M, N M )), j = j + L M mod (L M ), S MΩ Z j : (Z LΩ Z NM ) (Z LΩ Z LM ) (i, j ) (i, j), j = G M (j ) + S MΩ (i 1). j 1 = G M (j ) 1 + S MΩ (i 1). j G M = G M + Sym(G M ) Z LM, j 1 = (G M (j ) 1 + S MΩ (i 1), Sym(G M )). Ω ij = 2π L Ω (i 1), M ij = 2π L M ( (G M (j ) 1 + S MΩ (i 1), Sym(G M ))) 2π L M L MΩ L Ω (i 1), G M i j

52 S MΩ S MΩ {0,..., Sym(G M ) 1} Sym(G M ) Ω ij = Ω i(j M +N M ), M ij = M i(j +N M ), Ω ij = Ω (i+lω )j Ω, M ij = M (i+lω )j, M 2π L Ω (i 1) = 2π L Ω (i 1), (G M (j ) 1 + S MΩ (i 1), Sym(G M )) = = (G M ( (j + N M, N M )) 1 + S MΩ (i 1), Sym(G M )). G M (j ) = G M ( (j + N M, N M )) 2π L Ω (i 1) = 2π L Ω (L Ω + i 1) mod (L Ω ), L M 2π M (i+l Ω )j = L M 2π M ij, (G M (j ) 1 + S MΩ (L Ω + i 1), Sym(G M )) L MΩ L Ω (L Ω + i 1) = = (G M (j ) 1 + S MΩ (i 1), Sym(G M )) L MΩ L Ω (i 1). A Z G M (j ) 1 + S MΩ (i 1) + ASym(G M ) = G M (j ) 1 + S MΩ (L Ω + i 1) L MΩ, ASym(G M ) = S MΩ L Ω L MΩ,

53 S MΩ G M L Ω L MΩ Sym(G M ) S MΩ L Ω L MΩ, Sym(G M ) (S MΩ L Ω L MΩ ) G M 14 e = 0 a = km i = o L Ω = 7 N M = 2 N c = {0,..., 6} L M = 20 L Ω L M = 140 N M = 2 70 G M = {1, 2} Z 20 L MΩ = 6 Sym(G M ) = 20 {1, 2} = {1, 2} + 20 mod (20) Sym(G M ) S MΩ L Ω L MΩ 20 7S MΩ 6, S MΩ = 18 (Ω, M) Ω 11 = M 11 = 0 Ω = 0 Ω = 2π L Ω L M (Ω, M) N M = 2

54 (Ω, M) (Ω, M) 90 o

55 G M G M S MΩ G M gcd(sym(g M ), L Ω ) L MΩ gcd(sym(g M ), L Ω ). ASym(G M ) + L Ω S MΩ = L MΩ,

56 A A S MΩ gcd(sym(g M ), L Ω ) L MΩ. (S MΩ ) λ = (S MΩ ) 0 + λ l, l = L Ω (A) λ = (A) 0 λ gcd(sym(g M ), L Ω ), Sym(G M ) gcd(sym(g M ), L Ω ), (S MΩ ) 0 (A) 0 λ Sym(G M ) {1,..., L M }, S MΩ {0,..., Sym(G M ) 1}, L MΩ {0,..., L Ω 1}, S MΩ S MΩ = (Sym(G M ) 1) λ l S MΩ SMΩ (Sym(GM ) 1) gcd(sym(g M ), L Ω ) = = l x x = Sym(G M ) gcd(sym(g M ), L Ω ) gcd(sym(g M), L Ω ) Sym(G M ), gcd(sym(g M ), L Ω ) gcd(sym(gm ), L Ω ) gcd(sym(g M ), L Ω ), Sym(G M ) x x gcd(sym(g M ), L Ω ) [1, Sym(G M )] gcd(sym(g M ), L Ω ) Sym(G M ) (0, 1], gcd(sym(g M ), L Ω ) 1.

57 S MΩ (S MΩ ) λ gcd(sym(g M ), L Ω ), L Ω L MΩ Sym(G M ) G M L Ω L M G M L Ω L M Sym(G M ) {S MΩ, L MΩ } G M L Ω L M L Ω L Ω S MΩ 1L MΩ = ASym(G M ), S MΩ L MΩ gcd(l Ω, 1) ASym(G M ), gcd(l Ω, 1) = 1 ASym(G M ) Sym(G M ) L Ω (S MΩ ) λ = (S MΩ ) 0 + λ, (L MΩ ) λ = (L MΩ ) 0 λl Ω, (S MΩ ) 0 (L MΩ ) 0 λ ASym(G M ) L MΩ {0,..., L Ω 1} A

58 ASym(G M ) min (ASym(G M )) = (L Ω 1), max (ASym(G M )) = (Sym(G M ) 1)L Ω. ASym(G M ) (ASym(G M )) = max (ASym(G M )) min (ASym(G M )) = L Ω Sym(G M ) 1. Sym(G M ) (ASym(G M )) = ASym(G M ), A A = L Ω 1 Sym(G M ). 1 A A A A = 1 L Ω Sym(G M ) = L Ω 1 Sym(G M ). Sym(G M ) 1 (0, 1] L Ω 1 L Ω A A L Ω Sym(G M ) L Ω L M L Ω L M L Ω L M d L M φ(d)2 LM /d, d L M φ(d) d {L MΩ, S MΩ } {L MΩ, S MΩ }

59 N M L Ω L M N M = G M N M L Ω L M L Ω L M L M g=1 g L M L Mg N M (g), (g) g (g) = L M g g N M g L M L M g g 1 g =1 g g N M g (g ). L M + Z LM G M Z LM G = Z LM L M ϕ ϕ : G X X (g, x) x + g mod (L M ). (g) g G (X) g = Sym(G M ) g L M L M g N M

60 g = Sym(G M ) (g) g L M /g L M N M g/l M g P C(g) P C(g) = g N M g L M. L M /g L M N M L M g L M g P C(g) = L M g g N M. g L M L M = 4 N M = 2 g = 4 g = 2 {1, 3} {2, 4} g N M g (g) g g P C(g ) = g L M (g ). g g < g P C(g) = g N M g L M L M g g 1 g =1 g g N M g L M (g ), g g g L M g N M g

61 (g) (g) = L M g g N M g L M L M g g 1 g =1 g g N M g (g ), L M g G = Z LM g N M L M 1 g g N M g L M (g ) L M g g 1 g =1 g g N M g (g ). L M L Ω g L Ω g g N M g L M L M g g 1 g =1 g g N M g (g ). L M L M g=1 g L M L Mg N M L Ω g g N M g L M L M g g 1 g =1 g g N M g (g ), L M

62 L Ω L M L M g=1 g L M L Mg N M (g), (g) G M G Ω N Ω L Ω G Ω {1,..., L Ω }, G Ω = N Ω i G Ω G Ω (i ) = G Ω ( (i + N Ω, N Ω )), i = (i + L Ω, L Ω ). i i G Ω i = G Ω (i ), Ω i j = 2π L Ω (G Ω (i ) 1), M i j = 2π ( (G M (j ) 1 + S MΩ (G Ω (i ) 1), Sym(G M ))) L M 2π L MΩ (G Ω (i ) 1), L M L Ω

63 G Ω N Ω N M G Ω G M Sym(G M ) S MΩ L Ω L MΩ L Ω L M S MΩ L MΩ a e i ω Ω i j M i j a e i ω N Ω, N M L Ω, L M S MΩ L MΩ Ω i j M i j

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66 L Ω L ω L M L MΩ, L Mω, L ωω L MΩ [0, L Ω 1] L Mω [0, L w 1] L ωω [0, L Ω 1] L Ω 0 0 L ωω L ω 0 L MΩ L Mω L M Ω ijk ω ijk M ijk = 2π i 1 k 1 j 1 Ω ijk ω ijk M ijk {Ω 000, ω 000, M 000 } (i, j, k) i [1, L Ω ] k [1, L ω ] j [1, L M ] Ω ijk ω ijk M ijk [0, 2π] ; Ω ijk = 2π L Ω (i 1), ω ijk = 2π L ω (k 1) 2π L ω L ωω L Ω (i 1), M ijk = 2π (j 1) 2π L Mω (k 1) 2π L M L M L ω L M ( LMΩ L Ω L Mω L ω ) L ωω (i 1), L Ω ω ijk i = 1 i = L Ω +1 i [1, L Ω + 1] k ω ijk ω (i+lω )jk

67 M ijk M (i+lω )jk M ijk M ij(k+lω) G M N M = G M G M Z LM N M L M G M N M G M = (G M (1),, G M (j ),, G M (N M )), 1 G M (1) G M (j ) G M (N M ) L M, j N M j + N M j : Z NM Z LM j G M (j ). G M (j ) j {1,, N M } (a, b) = a mod (b) G M (j ) = G M ( (j + N M, N M )), j = j + L M mod (L M ), G ω N ω = G ω L ω G M G ω = (G ω (1),, G ω (k ),, G ω (N ω )), 1 G ω (1) G ω (k ) G M (N ω ) L ω,

68 k N ω : Z Nω Z Lω k G ω (k ), G ω (k ) k {1,, N ω } G ω (k ) = G ω ( (k + N ω, N ω )), k = k + L ω mod (L ω ), (i, j, k ) (i, j, k) : Z LΩ Z NM Z Nω Z LΩ Z LM Z Lω (i, j, k ) (i, j, k), S ωω S MΩ S Mω k j k = G ω (k ) + S ωω (i 1), j = G M (j ) + S Mω (k 1) + S MΩ (i 1). k 1 = G ω (k ) 1 + S ωω (i 1), j 1 = G M (j ) 1 + S Mω (k 1) + S MΩ (i 1). k 1 = G ω (k ) 1 + S ωω (i 1) mod Sym(G ω ), j 1 = G M (j ) 1 + S Mω (k 1) + S MΩ (i 1) mod Sym(G M ).

69 j k k k j 1 = G M (j ) 1 + S Mω (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) + + S MΩ (i 1) mod Sym(G M ), j Ω ij k = 2π (i 1), L Ω ω ij k = 2π [ (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) L ωω L ω M ij k = 2π L M [ ( G M (j ) 1 + S Mω ( G ω (k ) S ωω (i 1), Sym(G ω ) ) ) + S MΩ (i 1), Sym(G M ) ] (i 1) L Ω, L Mω L ( ω LMΩ (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) L ) ] Mω L ωω (i 1), L ω L Ω L Ω G M G ω M ij k S ωω, S Mω, S MΩ S ωω [0, Sym(G ω ) 1], S Mω [0, Sym(G M ) 1], S MΩ [0, Sym(G M ) 1],

70 Ω ij k = Ω i(j +N M )k, ω ij k = ω i(j +N M )k, M ij k = M i(j +N M )k, Ω ij k ω ij k M ij k Ω ij k = Ω ij(k +N ω), ω ij k = ω ij (k +N ω), M ij k = M ij (k +N ω), k ω ij k k + N ω ω ij k + 2π = ω ij (k +N ω), [ 2π (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) L ] ωω (i 1) + 2π = L ω L Ω = 2π [ (G ω (k + N ω ) 1 + S ωω (i 1), Sym(G ω )) L ] ωω (i 1), L ω L Ω

71 (G ω (k + N ω ) 1 + S ωω (i 1), Sym(G ω )) (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) = L ω, M ij k [ 2π ( G M (j ) 1 + S Mω ( G ω (k ) 1 + L M + S ωω (i 1), Sym(G ω ) ) ) + S MΩ (i 1), Sym(G M ) L Mω L ( ω LMΩ (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) L ) ] Mω L ωω (i 1) = L ω L Ω L Ω = 2π [ ( G M (j ) 1 + S Mω ( G ω (k + N ω ) 1 + L M + S ωω (i 1), Sym(G ω ) ) ) + S MΩ (i 1), Sym(G M ) L Mω L ( ω LMΩ (G ω (k + N ω ) 1 + S ωω (i 1), Sym(G ω )) L ) ] Mω L ωω (i 1) ; L ω L Ω L Ω ( G M (j ) 1 + S Mω (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) + ) + S MΩ (i 1), Sym(G M ) L Mω (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) = L ( ω = G M (j ) 1 + S Mω (G ω (k + N ω ) 1 + S ωω (i 1), Sym(G ω )) + ) + S MΩ (i 1), Sym(G M ) L Mω (G ω (k + N ω ) 1 + S ωω (i 1), Sym(G ω )). L ω Sym(G M ) ASym(G M ) = S Mω L ω L Mω ; A Sym(G M ) S Mω L ω L Mω, Sym(G M ) (S Mω L ω L Mω )

72 S Mω Ω ij k = Ω (i+l Ω )j k, ω ij k = ω (i+l Ω )j k, M ij k = M (i+l Ω )j k. Ω ij k = 2π L Ω (i 1) = 2π L Ω (i 1) + 2π mod (2π), L ω 2π ω ij k = L ω 2π ω (i+l Ω )j k, (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) L ωω L Ω (i 1) = = (G ω (k ) 1 + S ωω (i 1) + S ωω L Ω, Sym(G ω )) L ωω L Ω (i 1) L ωω, (G ω (k ) 1 + S ωω (i 1) + S ωω L Ω, Sym(G ω )) (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) = L ωω, Sym(G ω ) BSym(G ω ) = S ωω L Ω L ωω ; B Sym(G ω ) S ωω L Ω L ωω.

73 S ωω Sym(G ω ) L Ω L ωω L M 2π M ij k = L M 2π M (i+l Ω )j k, ( ) G M (j ) 1 + S Mω (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) + S MΩ (i 1), Sym(G M ) ( ) LMΩ (i 1) = L Mω (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) L Mω L ωω L ω L Ω L ω L Ω ( = G M (j ) 1 + S Mω (G ω (k ) 1 + S ωω (i 1) + S ωω L Ω, Sym(G ω )) + ) + S MΩ (i 1) + S MΩ L Ω, Sym(G M ) (G ω (k ) 1 + S ωω (i 1) + S ωω L Ω, Sym(G ω )) ( LMΩ L Ω L Mω L ω L ωω L Ω ) (i 1) L Mω L ω ( L MΩ L MωL ωω L ω ), ( G M (j ) 1 + S Mω (G ω (k ) 1 + S ωω (i 1)+ ) + S ωω L Ω, Sym(G ω )) + S MΩ (i 1) + S MΩ L Ω, Sym(G M ) ( G M (j ) 1 + S Mω (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) + ) + S MΩ (i 1), Sym(G M ) = L MΩ. Sym(G ω ) CSym(G M ) = S MΩ L Ω (L MΩ S Mω L ωω ), C Sym(G M ) S MΩ L Ω (L MΩ S Mω L ωω ). S MΩ S Mω

74 G Ω ij k = 2π (i 1), L Ω ω ij k = 2π [ (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) L ωω L ω M ij k = 2π L M [ ( G M (j ) 1 + S Mω ( G ω (k ) S ωω (i 1), Sym(G ω ) ) ) + S MΩ (i 1), Sym(G M ) ] (i 1) L Ω, L Mω L ( ω LMΩ (G ω (k ) 1 + S ωω (i 1), Sym(G ω )) L ) ] Mω L ωω (i 1), L ω L Ω L Ω S ωω S Mω S Mω Sym(G ω ) S ωω L Ω L ωω, Sym(G M ) S Mω L ω L Mω, Sym(G M ) S MΩ L Ω (L MΩ S Mω L ωω ). Sym(G) S MΩ L Ω N c, G N c L MΩ

75 L Ω = 7 N ω = 2 N M = 3 L ω = 6 L M = 9 G M = 10 G ω = 3 G M G ω L 2 Ω L ω = 8820 L 2 Ω N ω = 98 L MΩ = 4 L Mω = 3 L ωω = 6 G M = {1, 4, 7} G ω = {1, 4} Sym(G M ) = 3 Sym(G ω ) = 3 Sym(G ω ) S ωω L Ω L ωω 3 7S ωω 6, S ωω = 0 Sym(G M ) S Mω L ω L Mω 3 6S Mω 3, S Mω = 0, 1, 2 Sym(G M ) S MΩ L Ω (L MΩ S Mω L ωω ) 3 7S MΩ (4 6S Mω ), S MΩ = 1 S Mω = 0, 1, 2 S Mω S MΩ S Mω S ωω = 0 S Mω = 2 S MΩ = 1 (Ω, ω, M)

76 (Ω, ω, M) e = 0.3 i = o a = 12, 770 km

77

78 n

79 J 2 J 2

80 a M ω Ω {V a, V Ω, V ω, V M } V a V Ω V ω V M {V a, V Ω, V ω, V M } {V a, V Ω, V ω, V M } [0, 1] {r, i, k, j} L a L Ωa L Ω 0 0 L ωa L ωω L ω 0 L Ma L MΩ L Mω L M V a ijkr V Ω ijkr V ω ijkr V M ijkr = r i k j ; L a L Ω L ω L M r = {1,..., L a } i = {1,..., L Ω } k = {1,..., L ω } j = {1,..., L M } L Ωa {0,..., L a 1} L ωa {0,..., L a 1} L ωω {0,..., L Ω 1} L Ma {0,..., L a 1} L MΩ {0,..., L Ω 1} L Mω {0,..., L ω 1}

81 V a ijkr = 1 L a r, V Ω ijkr = 1 L Ω i L Ωa L a L Ω r, Vijkr ω = 1 k L ωω i 1 L ω L ω L Ω L ω Vijkr M = 1 j L Mω k 1 L M L M L ω 1 L M ( LMa L a L MΩ L Ω L M ( Lωa L Ωa L a L a ( LMΩ L ωω L Ωa L Ω L Ω L Mω L ω L a L Mω L ω ( Lωa L a ) r, L ωω L Ω ) i L Ωa L a L ωω L Ω )) r, [ ] 1 Vijkr a = r, 1, L a [ ( 1 Vijkr Ω = i L ) ] Ωa r, 1, L Ω L a [ ( 1 Vijkr ω = k L ( ωω Lωa i L ) ) ] ωω L Ωa r, 1, L ω L ω L a L Ω L a [ ( 1 Vijkr M = j L ( Mω LMΩ k L ) Mω L ωω i L M L ω L Ω L ω L Ω ( LMa L MΩ L Ωa L ( Mω Lωa L )) ) ] Ωa L ωω r, 1, L a L Ω L a L ω L a L a L Ω [a, b] a b L a L a L Ω L a L Ω L ω L a L Ω L ωm L a Vijkr a = [r, L a ], L Ω L a Vijkr Ω = [L a i L Ωa r, L Ω L a ], L ω L Ω L a Vijkr ω = [L Ω L a k L ωω L a i (L ωa L Ω L ωω L Ωa ) r, L ω L Ω L a ], L M L ω L Ω L a Vijkr M = [L ω L Ω L a j L Mω L Ω L a k (L MΩ L ω L a L Mω L ωω L a ) i (L Ma L ω L Ω L MΩ L Ωa L ω L Mω (L ωa L Ω L Ωa L ωω )) r, L M L ω L Ω L a ]. {N a, N Ω, N ω, N M } N a ijkr = L a V a ijkr, Nijkr Ω = L Ω L a Vijkr Ω Nijkr ω = L ω L Ω L a Vijkr ω N M ijkr = L M L ω L Ω L a V M ijkr,

82 Nijkr a = [r, L a ], Nijkr Ω = [L a i L Ωa r, L Ω L a ], Nijkr ω = [L Ω L a k L ωω L a i (L ωa L Ω L ωω L Ωa ) r, L ω L Ω L a ], Nijkr M = [L ω L Ω L a j L Mω L Ω L a k (L MΩ L ω L a L Mω L ωω L a ) i (L Ma L ω L Ω L MΩ L Ωa L ω L Mω (L ωa L Ω L Ωa L ωω )) r, L M L ω L Ω L a ], {N a, N Ω, N ω, N M } N a {1,..., L a }, N Ω {1,..., L Ω L a }, N ω {1,..., L ω L Ω L a }, N M {1,..., L M L ω L Ω L a }, L a L Ω = 4 N Ω = {1, 2, 3, 4} Ω 1 = 0 o Ω 2 = 30 o Ω 3 = 90 o Ω 4 = 180 o N p N d N p N d T c = N p T = N d T d, T T d

83 T c N d N p N p J2 J 2 J 2 J 2 [ µ ȧ sec = 0; n = a ( ) 2 4 J R ( 2 2 3sin 2 a(1 e 2 (i) ) 1 e 2] ; ) ( R ) 2 n ( 5cos 2 (i) 1 ) ; ė sec = 0; ω sec = 3 4 J 2 a(1 e 2 ) i sec = 0; Ωsec = 3 ) 2 2( 2 J R a(1 e 2 n cos(i); ) µ R n Ω sec = Ω sec0 ω sec = ω sec0 T c Ω sec0 ω sec0 T c = N p 2π n = N dt d.

84 [ 2π N p µ = T d N d a ( ) 2 4 J R ( 2 2 3sin 2 a(1 e 2 (i) ) 1 e 2], ) ω sec0 = 3 ( ) 2 4 J R 2π N p ( 2 5cos 2 a(1 e 2 (i) 1 ), ) T d N d Ω sec0 = 3 ) 2 2( 2 J R 2π N p a(1 e 2 cos(i), ) T d N d N a ω sec0 Ω sec0 cos(i) = ω sec0 ± ω sec Ω 2 sec0 5 Ω, sec0 ω sec0 Ω sec0 Ω sec0 1 e 2 = R a 3 2 J 2 2π N p cos(i), Ω sec0 T d N d a 3 T d N d 2 = µ 1 Ω sec0 T d N d 2 3 sin 2 (i) R 3 J 2 2π N p cos(i), 2π N p 4π N p cos(i) a 2 Ω sec0 T d N d N a L a = 2

85 L Ω = 3 L ω = 3 L M = 3 N d = 3 i = o e = 0.1 N p = {8, 6} km km Ω ijkr = 2π N ijkr Ω ; ω ijkr = 2π N ijkr ω ; L a L Ω L a L Ω L ω N M ijkr M ijkr = 2π. L a L Ω L ω L M L ij L 21 = 0 L 32 = 2 L 31 = 1 L 43 = 0 L 42 = 2 L 41 = 1

86 N a G Ω G ω G M

87 G G Ω {1,..., L Ω }, G ω {1,..., L ω }, G M {1,..., L M }, G Ω = N Ω G Ω G ω = N ω G ω G M = N M G M G Ω = {G Ω (1),..., G Ω (i ),..., G Ω (N Ω )}, G ω = {G ω (1),..., G ω (k ),..., G ω (N ω )}, G M = {G M (1),..., G M (j ),..., G M (N M )}, 1 G Ω (1) < < G Ω (i ) < < G Ω (N Ω ) L Ω, 1 G ω (1) < < G ω (k ) < < G ω (N ω ) L ω, 1 G M (1) < < G M (j ) < < G M (N M ) L M, i k j G Ω G ω G M G : Z La Z NΩ Z Nω Z NM Z La Z LΩ Z Lω Z LM (r, i, k, j ) (r, G Ω (i ), G ω (k ), G M (j )). G i {1,..., N Ω } k {1,..., N ω } j {1,..., N M } S Ωr S ωr S ωω S Mr S MΩ S Mω i = G Ω (i ) + S Ωr r, k = G ω (k ) + S ωr r + S ωω i, j = G M (j ) + S Mr r + S MΩ i + S Mω k. Sym(G) Sym(G) = min {1 r n : G + r G}. i = [G Ω (i ) + S Ωr r, Sym(G Ω )], k = [G ω (k ) + S ωr r + S ωω i, Sym(G ω )], j = [G M (j ) + S Mr r + S MΩ i + S Mω k, Sym(G M )].

88 S Ωr {0,..., Sym(G Ω ) 1}, S ωr, S ωω {0,..., Sym(G ω ) 1}, S Mr, S MΩ, S Mω {0,..., Sym(G M ) 1}. i k j k i = [G Ω (i ) + S Ωr r, Sym(G Ω )], k = [G ω (k ) + S ωr r + S ωω [G Ω (i ) + S Ωr r, Sym(G Ω )], Sym(G ω )], j = [G M (j ) + S Mr r + S MΩ [G Ω (i ) + S Ωr r, Sym(G Ω )] + + S Mω [G ω (k ) + S ωr r+ + S ωω [G Ω (i ) + S Ωr r, Sym(G Ω )], Sym(G ω )], Sym(G M )], Ni a j k r = [r, L a ], Ni Ω j k r = [L a [G Ω (i ) + S Ωr r, Sym(G Ω )] L Ωa r, L Ω L a ], Ni ω j k r = [L Ω L a [G ω (k ) + S ωr r + S ωω [G Ω (i )+ + S Ωr r, Sym(G Ω )], Sym(G ω )] L ωω L a [G Ω (i )+ + S Ωr r, Sym(G Ω )] (L ωa L Ω L ωω L Ωa ) r, L ω L Ω L a ], Ni M j k r = [L ω L Ω L a [G M (j ) + S Mr r + S MΩ [G Ω (i )+ + S Ωr r, Sym(G Ω )] + S Mω [G ω (k ) + S ωr r+ + S ωω [G Ω (i ) + S Ωr r, Sym(G Ω )], Sym(G ω )], Sym(G M )] L Mω L Ω L a [G ω (k ) + S ωr r + S ωω [G Ω (i )+ + S Ωr r, Sym(G Ω )], Sym(G ω )] (L MΩ L ω L a L Mω L ωω L a ) [G Ω (i ) + S Ωr r, Sym(G Ω )] (L Ma L ω L Ω L MΩ L Ωa L ω L Mω (L ωa L Ω L Ωa L ωω )) r, L M L ω L Ω L a ].

89 ( ) Ni a j k (r+l, N Ω a) i j k (r+l, N ω a) i j k (r+l, N M a) i j k (r+l a) = = ( Ni a j k r, N i Ω j k r, N i ω j k r, N i M j k r), ( ) N(i a +N Ω )j k r, N (i Ω +N Ω )j k r, N (i ω +N Ω )j k r, N (i M +N Ω )j k r = = ( Ni a j k r, N i Ω j k r, N i ω j k r, N i M j k r), ( ) Ni a j (k +N, N Ω ω)r i j (k +N, N ω ω)r i j (k +N, N M ω)r i j (k +N ω)r = = ( Ni a j k r, N i Ω j k r, N i ω j k r, N i M j k r), ( ) Ni a (j +N M )k r, N i Ω (j +N M )k r, N i ω (j +N M )k r, N i M (j +N M )k r = = ( N a i j k r, N Ω i j k r, N ω i j k r, N M i j k r), Sym(G Ω ) S Ωr L r L Ωr, Sym(G ω ) S ωω L Ω L ωω, Sym(G ω ) S ωr L r (L ωr S ωω L Ωr ), Sym(G M ) S Mω L ω L Mω, Sym(G M ) S MΩ L Ω (L MΩ S Mω L ωω ), Sym(G M ) S Mr L r (L Mr S MΩ L Ωr S Mω L ωr ), a > Ω > ω > M

90 G M = {1} N M = 1 G ω = {1, 2, 3} G ω = {1, 2, 3} Sym(G M ) = 3 1 2S Ωr 0, S Ωr = 0 1 3S ωω 2, S ωω = 0 S ωr = 0 1 2S ωr (1 0), 3 3S Mω 0, S Mω = 0, 1, 2 3 3S MΩ (2 2S Mω ), S Mω = 1 S MΩ = 0, 1, 2 3 2S Mr (1 0 1), S Mr = 0 S MΩ S MΩ S MΩ = 0, 1, 2 S MΩ = 2

91 M ω M ω M Ω M r

92 x z L a = 1 L ω = 1

93 J 2 J 2

94

95 n n n (n 1) n

96 n n n n Z n R n : Z n R n k V, k V k i i Z n V i [0, 1] i = 1,..., n {V i V V i [0, 1]} i {k i k k i Z} L ii V i {k j j i} L ii i k i L ii k i {V (k)} {V (k )} {V (k)} {V (k )} {V (k )} = {V (k)}. V (k ) = V (k), V i (k ) = V i (k), i {1,..., n} V (k) V i ({1,..., k m 1, k m, k m+1,..., k n }) = = V i ({1,..., k m 1, k m + L mm, k m+1,..., k n }) i, m {1, 2,..., n}, k V i (k m ) = V i (k m + L mm ).

97 L ij n n n V i = 1 i 1 k i (L ij V j ), 1, L ii (a, b) b a L ij Z n R n j=1 : Z n R n {α 1,..., α i,..., α n } {V 1,..., V i,..., V n }, n P 1,1 P 1,2 P 1,n V 1 α 1 P 2,1 P 2,2 P 2,n V 2 α = 2, P n,1 P n,2 P n,n P i,j V i P i,j α i L 1, V 1 k 1 L 2,1 L 2,2 0 0 V 2 k 2 =, L n 1,1 L n 1,2 L n 1,n 1 0 V n 1 k n 1 L n,1 L n,2 L n,n L ij Z k i Z P ij α i i L ij V j = k i, V i V i = 1 i 1 k i L ij V j, L ii j=1 j=1 V n V n α n k n

98 n V i (k m ) = V i (k m + L mm ), m > i V i k j j i m = i 1 L ii V i (k i ) = V i (k i + L ii ) i {1, 2,..., n}, i 1 k i L ij V j = 1 i 1 k i + L ii L ij V j, L ii j=1 j=1 L ii = 0 V i = 1 i 1 k i L ij V j, L ii, L ii j=1 k i = k i + L ii mod (L ii ). k i Z Lii k i = k i + L ii mod (L ii ). m < i 1 i 1 k i L ij V j (k m ) = 1 L ii j=1 L ii i 1 k i L ij V j (k m + L mm ), j=1 V j (k m ) = V j (k m + L mm ) j, m {1, 2,..., n},

99 L ii V i = 1 i 1 k i L ij V j, 1. L ii k i = L ii i Z n j=1 L ii L ii V i {k j j i} {L ij i j} k i L ij k i k i {1, 2,..., L ii } k m k m Z k m km Z k m m 1 L mm k m m 1 j=1 L mj V j, 1 = 1 L mm k m m 1 j=1 L mj V j, 1. 1 k m L mm m 1 j=1 L mj V j + A = 1 L mm k m m 1 j=1 L mj V j, A k m + AL mm = km, k m Z Lmm = {1, 2,..., L mm } km Z k m {1, 2,..., L mm }

100 n L ij L ij {0, 1,..., L jj 1}. L L i 1 L ij = L ij + A p L pj, p=j A p n n L ij [0, L jj 1] i L ij L L L L n L ij = L ij + A p L pj, p=1 A p n i 1 L ij = L ij + A p L pj, p=j L ij = L ij + i 1 p=j+1 A p L pj + A j L jj,

101 A j L ij = L ij + i 1 p=j+1 A p L pj mod (L jj ), L ij {0, 1,..., L jj 1}, L ij {0, 1,..., L jj 1} i (L ii ) V i = 1 i 1 k i (L ij V j ), 1, L ii j=1 i (L ii ) n n i i=1 j=1 L ii n L ii L ij i j L ij i i 1 L jj j=1 n i 1 i=1 j=1 L jj

102 n n 1 n 2 n (n 1) L jj L jj L jj = L (n 1) 11 L (n 2) 22 L (1) (n 1)(n 1), j=1 j=1 n i=1 j=1 L (n i) ii = n n i L ii. i=1 j=1 n G Z m, m G 1 G 2 G 1 G 2 {G 1 } = {G 2 }. G 1 G 2 = s Z m G 1 = G2 s : G 1 G 2 + s mod (m), s m

103 G Sym(G) Sym(G) = min {1 r n : G + r G}. G {s : Sym(G) s} G G (Sym(G) 1) S S {0, 1,..., (Sym(G) 1)}. G i i G i N i N i i {k j j i} G i Z Lii G i {1,..., L ii }, G i = N i G i G i = {G i (1),..., G M (k i ),..., G i (N i )}, 1 G i (1) G i (k i ) G i (N i ) L ii, k i {1,..., N i} N i k i + N i k i G i G i : Z Ni Z Lii k i G i (k i ). G i (ki ) G i (ki ) = G i ( (ki + N i, N i )),

104 n i k i = k i + L ii mod (L ii ), i Sym(G i ) G i G i G i + ASym(G i ), A G i = G i + Sym(G i ) mod (Sym(G i )). L ii Sym (G) Sym (G) Sym (G) L ii S n n S ij Z G i j G i G j j < i V i k i n i L ij V j = k i ; j=1 i 1 k i = G i (ki ) + S ij k j, Sym (G i ), j=1 G i i k i S ij G i j k i G i (ki ) : (Z N1 Z N2 Z Nn ) (Z L11 Z L22 Z Lnn ) (k 1, k 2,..., k n) (k 1, k 2,..., k n ),

105 k i i 1 k i = G i (ki ) + S ij k j, G i (ki ) i 1 j=1 S ijk j G i i 1 k i = G i (ki ) + S ij k j, Sym(G i ), j=1 j=1 i L ij V j = k i ; j=1 i 1 k i = G i (ki ) + S ij k j, Sym (G i ). j=1 L ij S ij j > i i L ij V j = k i ; j=1 i 1 k i = G i (ki ) + S ij k j, Sym (G i ), Sym(G i ) S ijl jj L ij j=1 i 1 q=j+1 S iq L qj. i V i (k j ) = V i (k j + L jj ) i, j {1, 2,..., n},

106 n j i V i (k j ) = V i (k j + L jj ) i j, j > i j = i j < i i L ip V p (k j ) = k i (k j ) p=1 i 1 k i (k j ) = G i (ki ) + S ip k p (k j ) + A i Sym (G i ), p=1 j V p (k j ) k p (k j ) A i i L ip V p (k j + L jj ) = k i (k j + L jj ) p=1 i 1 k i (k j + L jj ) = G i (ki ) + S ip k p (k j + L jj ) + B i Sym (G i ), p=1 j V q (k j + L jj ) q j k p (k j + L jj ) k p B i j > i j k i i i i L ip V p (k j + L jj ) L ip V p (k j ) = L ip [V p (k j + L jj ) V p (k j )] = 0, p=1 p=1 p=1 k p (k j + L jj ) = k p (k j ) p < j. i 1 0 = S ip [k p (k j + L jj ) k p (k j )] + (B i A i )Sym (G i ) p=1

107 j = i j L jp [V p (k j + L jj ) V p (k j )] = L jj, p=1 L jj = k j (k j + L jj ) k j (k j ) j 1 L jj [V j (k j + L jj ) V j (k j )] + L jp [V p (k j + L jj ) V p (k j )] = L jj, p=1 V j (k j + L jj ) V j (k j ) = 1 i V i 1 V i (k i + L ii ) = V i (k i ) j 1 L jj = S jp [k p (k j + L jj ) k p (k j )] + (B j A j )Sym (G j ), p=1 L jj = (B j A j )Sym (G j ), Sym (G j ) L jj j < i i i L iq V q (k j + L jj ) L iq V q (k j ) = k i (k j + L jj ) k i (k j ), q=1 q=1 i L iq [V q (k j + L jj ) V q (k j )] = k i (k j + L jj ) k i (k j ), q=1 + j 1 L iq [V q (k j + L jj ) V q (k j )] + L ij [V j (k j + L jj ) V j (k j )] + q=1 i q=j+1 L iq [V q (k j + L jj ) V q (k j )] = k i (k j + L jj ) k i (k j ),

108 n L ij = k i (k j + L jj ) k i (k j ). i 1 k i (k j + L jj ) k i (k j ) = S ip [k p (k j + L jj ) k p (k j )] + (B j A j )Sym (G j ), p=1 i 1 L ij = S ip [k p (k j + L jj ) k p (k j )] + (B j A j )Sym (G j ). p=j L ij = S ij [k j (k j + L jj ) k j (k j )] + + i 1 p=j+1 S ip [k p (k j + L jj ) k p (k j )] + (B j A j )Sym (G j ), k j (k j + L jj ) k j (k j ) = L jj, j k p (k j + L jj ) k p (k j ) = L pj L ij = S ij L jj + i 1 p=j+1 Sym(G i ) S ij L jj L ij S ip L pj + (B j A j )Sym (G j ), i 1 q=j+1 S iq L qj, Sym(G i ) S i 1 ijl jj L ij S iq L qj. q=j+1

109 n G i i {1,..., n} G i L ii i {1,..., n} N i = G i L ii i {1,..., n} L G N G {1,..., L} G = N = N L L g=1 g L L g N 1 g g N L g g 1 g =1 g g L g N g L (g ). g = Sym(G) (g ) g (g) = L g g N L g g 1 g =1 g g L g N (g ), g L + Z L G Z L G = Z L L ϕ ϕ : G X X (g, x) x + g mod (L). (g) g G (X)

110 n g = Sym(G) g L L g N g = Sym(G) (g) g L/g L Ng/L g P C(g) P C(g) = g N L g. L/g L N L g L g P C(g) = L g g N L g. L = 6 N = 3 g = 6 g = 3 {1, 4} {2, 5} {3, 6} g N g (g) g g P C(g ) = g (g ). L g g < g P C(g) = g g 1 N g L g (g ), L g g g L g N g g =1 g g L g N

111 (g) (g) = L g g N L g g 1 g =1 g g L g N g L (g ), g G = Z L 1 L L M g=1 g L M L Mg N M L M g=1 g L M L g N 1 g g N L g (g), g 1 g =1 g g L g N g L (g ). G i i {1,..., n} G i S ij G i i {1,..., n} gcd(sym(g i ), L jj ) L ij i 1 q=j+1 S ij L qj, {i, j : i > j},

112 n n i 1 gcd(sym(g i ), L jj ), i=1 j=1 ASym(G i ) + L jj S ij = L ij i 1 q=j+1 S ij L qj, A A S ij gcd(sym(g i ), L jj ) L i 1 ij S ij L qj. q=j+1 (S ij ) λ = (S ij ) 0 + λ l, l = L jj (A) λ = (A) 0 λ gcd(sym(g i ), L jj ), Sym(G i ) gcd(sym(g i ), L jj ), (S ij ) 0 (A) 0 λ S ij S ij {0,..., Sym(G i ) 1} S ij Sym(Gi ) 1 (Sym(Gi ) 1) gcd(sym(g i ), L jj ) + 1 = + 1 = l Sym(G i ) gcd(sym(gm ), L Ω ) = gcd(sym(g i ), L jj ) + 1 = Sym(G M ) = gcd(sym(g i ), L jj ). S ij gcd(sym(g i ), L jj ) L ij i 1 q=j+1 S ij L qj, {i, j : i > j}, S ij n i 1 gcd(sym(g i ), L jj ). i=1 j=1

113 Sym(G i ) = 1 n i 1 gcd(1, L jj ) = 1. i=1 j=1 G i L ii i {1,..., n} G i L ii Sym(G i ) {S ij, L ij } G i L ii n n i L ii. i=1 j=1 i S ij L ij L jj S ij 1L ij = ASym(G M ) i 1 q=j+1 S iq L qj, S iq L qj q > j S ij L ij gcd(l jj, 1) ASym(G M) i 1 q=j+1 (S ij ) λ = (S ij ) 0 + λ, S iq L qj (L ij ) λ = (L ij ) 0 + λl jj, (S ij ) 0 (L ij ) 0 λ

114 n A L ij {0,..., L jj 1} A L ij S ij A A min = 1 i 1 Sym(G i ) S iq L qj (L jj 1), A max = 1 Sym(G i ) q=j+1 i 1 q=j+1 S iq L qj + L jj (Sym(G i ) 1), Ljj (Sym(G i ) 1) + (L jj 1) A max A min + 1 = + 1 = L jj Sym(G i ) A L jj {L ij, S ij } L ij {0,..., L jj 1} n n i L ii, i=1 j=1 G i L ii N i L ii i {1,..., n} G i N i = G i

115 L ii i {1,..., n} i n N i n i=1 L (n i) ii L ii g=1 g L ii L ii g N i 1 g g N i g L ii g 1 g =1 g g L ii g N i g (g ), L ii (g ) i g (g) = L g g N L g g 1 g =1 g g L g N (g ), i P i = L ii g=1 g L ii L ii g N i 1 g g N i g L ii g 1 g =1 g g L ii g N i g L g (g ), L ii (g) = L ii g g N i g L ii g 1 g =1 g g L ii g N i g (g ). L ii n P i. i=1 n n P i i=1 i=1 L (n i) ii.

116 n n P i L (n i) ii, n i=1 L (n i) ii L ii g=1 g L ii L ii g N i 1 g i=1 g N i g L ii g 1 g =1 g g L ii g N i g (g ), L ii N i L ii N i = L ii i V i i L ij V j = k i, i 1 p=1 L pp i j=1 j=1 i 1 L pp V j = k i L pp, L ij i 1 p=1 i 1 L pp V i + L ii i 1 p=1 j=1 L ij i 1 p=1 i 1 L pp V j = k i L pp, p=1 p=1 i i 1 i 1 j i 1 L pp V i + L pp V j = k i L pp. p=1 L ij L mm j=1 m=j+1 p=1 p=1

117 V i N i V i i N i = L pp V i. p=1 i 1 N i + L ij N j j=1 i 1 m=j+1 i 1 L mm = k i L pp, p=1 k i i 1 i 1 N i = k i L pp L ij N j p=1 j=1 i 1 m=j+1 L mm. V i N i = [ i p=1 L ppv i N i 0, ] i p=1 L pp N i = k i i 1 i 1 L pp L ij N j p=1 j=1 i 1 m=j+1 L mm, i p=1 L pp. N i Z i i 1 N i + L ij N j j=1 i 1 m=j+1 i 1 L mm = k i L pp ; p=1 i 1 k i = G i (ki ) + S ij k j, Sym (G i ), j=1 N i [ N i 1,..., ] i p=1 L pp

118 n n n n

119

120 a e i ω Ω M x = (r cos f, r sin f, 0), r r = a ( 1 e 2) 1 + e cos f,

121 f R 3 R 1 x ECI = R 3 (Ω) R 1 (i) R 3 (ω) x, x ECEF = R 3 ( ψ G0 ω t) x ECI, ψ G0 t = 0 ω r cos f x ECEF = R 3 ( ψ G0 ω t) R 3 (Ω) R 1 (i) R 3 (ω) r sin f, 0 r cos f x ECEF = R 3 (Ω ψ G0 ω t) R 1 (i) R 3 (ω) r sin f. 0 a e i ω a e i ω Ω 0 t 0 x 0 r cos f x 0 ECEF = R 3 (Ω 0 ψ G0 ω t) R 1 (i) R 3 (ω) r sin f, 0 r f t t 0 t 1 t 0

122 Ω = ω (t 1 t 0 ) x 1 t 1 r cos f x 1 ECEF = R 3 (Ω 0 ψ G0 ω (t 1 t 0 + t)) R 1 (i) R 3 (ω) r sin f, 0 r f t 1 + t x 1 ECI = R 3 (ψ G0 + ω t) x 1 ECEF, x 1 ECI = R 3 (Ω 0 ω (t 1 t 0 )) R 1 (i) R 3 (ω) r cos f r sin f 0. {a, e, i, ω, Ω 0, M 0 } {a, e, i, ω, Ω 1, M 1 } M 0 M 1 x 0 ECEF (t + (t 1 t 0 )) = x 1 ECEF (t) t R, t 1 t Ω 1 = Ω 0 ω (t 1 t 0 ). M = n(t + t 0 τ), τ t 0 n n = µ a 3, µ τ 0 τ 1 M 1 = n(t + t 1 τ),

123 t 1 1 M 1 = M 0 + n(t 1 t 0 ). M 1 Ω 1 ( M 1 = M 0 + n ) Ω 0 n Ω 1 ; ω ω M 1 (Ω 1 ) n/ω (Ω, M) (Ω, M) t q t 1 N st q [1, N st ] q ( M q = M 0 + n ) Ω 0 n Ω q ; ω ω r cos f x q ECI = R 3 (Ω 0 ω (t q t 0 )) R 1 (i) R 3 (ω) r sin f. 0 Ω q q Ω q = Ω 0 ω (t q t 0 ).

124 t q t 0 t t q ω Ω q f q t q t q x q ECI = R 3 (Ω 0 ω (t q t 0 )) R 1 (i) R 3 (ω) a ( 1 e 2) 1 + e cos f q cos f q a ( 1 e 2) 1 + e cos f q sin f q 0. t q Ω q = Ω 0 ω (t q t 0 ); M q = M 0 + n(t q t 0 ); t q Ω 0 t 0 M 0 a = km e = 0.4 i = o Ω 0 = 0 M 0 = 0 t 0 = 0 t q = 300(q 1); q [1, 5] 300 Ω q = ω t q ; M q = nt q ;

125 Ω q deg M q deg N t k [1, N t ] N s N s = N st N t, N st k0 k 00

126 r cos f x kq ECEF = R 3 (σ) R 1 (i) R 3 (ω) r sin f, 0 σ = Ω 0 + Ω k ψ G0 ω (t kq t 0 + t), Ω k t kq r f t kq t kq k q Ω k t kq f kq q k x kq ECI = R 3 (Ω kq ) R 1 (i) R 3 (ω) a ( 1 e 2) 1 + e cos f kq cos f kq a ( 1 e 2) 1 + e cos f kq sin f kq 0, Ω kq = Ω 0 + Ω k ω (t kq t 0 ), (Ω, M) Ω kq = Ω 0 + Ω k ω (t kq t 0 ), M kq = M 0 + n(t kq t 0 ), Ω kq M kq M kq = ( M 0 + n ) Ω 0 + n Ω k n Ω kq ; ω ω ω 11 k = 1, q = 1 Ω 1

127 (Ω, M) 880km a = 7260 km i = o Ω k = 2π k 1 N t, t kq = 2π q 1 N st ; N t = 3 N st = 5 Ω kq = 2π k 1 N t M kq = 2π q 1 N st, 2πω n q 1 N st, k [1, N t ] q [1, N st ] (Ω, M)

128 (Ω, M) Ω kq Ω kq

129 t kq Ω k t kq Ω k t kq t k t q t kq = t k + t q, t q t k k t k Ω k t q Ω kq = Ω 0 + Ω k ω (t k + t q t 0 ), k t k = Ω k ω, t k Ω kq = Ω 0 ω (t q t 0 ). Ω kq k t q q k f kq t q + t k (Ω, M) Ω kq = Ω 0 ω (t q t 0 ), ( ) Ωk M kq = M 0 + n + t q t 0, ω ( M kq = M 0 + n ) Ω 0 + n Ω k n Ω kq. ω ω ω t q t k

130 (Ω, M) a = km e = 0 i = 50 o Ω 0 = 0 M 0 = 0 t 0 = 0 t q Ω k k = 1 k = 2 k = 3 q = 2

131 q = 3 q = 1 t q t 1 = 0 s t 2 = 600 s t 3 = 600 s Ω k Ω 1 = 0 Ω 2 = ω t 2 Ω 3 = ω t 3 (k, q) q k (1, 1) (1, 2) (1, 3) (2, 2) (3, 3) Ω kq deg M kq deg

132 x 0 t = t q t 0, t 0 t q q [1, N st ] x q ECI ṽ q ECI x q ECEF v q ECEF x q ECEF = R 3 ( ψ G0 ω (t q t 0 )) x q ECI, v q ECEF = R 3 ( ψ G0 ω (t q t 0 )) ṽ q ECI ω x q ECEF. x q ECI v q ECI x q ECI = R 3 (ψ G0 ) x q ECEF, v q ECI = R 3 (ψ G0 ) v q ECEF + ω x q ECI.

133 t kq Ω k a 0 e 0 i 0 w 0 Ω k0 = Ω 0 + Ω k ω (t k0 t 0 ), Ω k0 k [1, N t ] t 0 t k0 t kq t k0 N t t = t kq t 0 x kq ECEF = R 3 ( ψ G0 ω (t kq t 0 )) x kq ECI, v kq ECEF = R 3 ( ψ G0 ω (t kq t 0 )) ṽ kq ECI ω x kq ECEF ; x kq ECI = R 3 (ψ G0 ) x kq ECEF, v kq ECI = R 3 (ψ G0 ) v kq ECEF + ω x kq ECI. x kq ECI v kq ECI N t N s t k t q t kq = t k + t q t k Ω k

134 t k ω ω Ω Ω t k = Ω k ω Ω k0, Ω k0 k t k N s N t N st T c T c = N p T Ω = N d T ΩG, N p N d T Ω T ΩG N st q [1, N st ] t q T c t q < T c t q = (q 1) T c N st. N t k [1, N t ]

135 Ω k = Ω 0 + (k 1) 2π N t, Ω k = Ω k Ω 0 = (k 1) 2π N t. t k t k = (k 1) 2π N t ω Ω k0, t kq = t k + t q t kq = (q 1) T c 2π + (k 1) ( N st N t ω Ω ). k0 N d N p N p = 2 N d = 3 N p = 4 N d = 6 N p N t N f N p N t Ω k = Ω 0 + (k 1) 2π, N t N f t kq = (q 1) T c 2π + (k 1) ( N st N t N f ω Ω ), k0

136 N p N d N st N t (ψ r, ϕ r ) f = π x y z v x v y v z

137 t 0 t 0 = 0 N p = 2 N d = 1 i = o e = 0.5 N t = 6 N st = 4 N s = N t N st = 24 N t = 6 N p = 2 N f = 2 ϕ r = o ψ r = o

138

139 o 705 ± 5 km a = km e = 0 i = o x y z v x v y v z 16 16

140 16 16

141

142

143

144 a e i ω Ω M ω µ R J 2 J 2 T c T c = N p T Ω = N d T ΩG, N p N d T Ω n M o ω T Ω = 2π Ṁ + ω = 2π n + M o + ω.

145 T ΩG T ΩG = 2π ω Ω, n + M o + ω = N ( p ω N Ω ), d J 2 n = M o = ω = µ a 3 ; 3J 2 nr 2 4a 2 (1 e 2 ) 2 1 e 2 ( 2 3 sin 2 i ) = 3J 2 nr 2 4a 2 (1 e 2 ) 2 ( 4 5 sin 2 i ) = 3J 2nR 2 4a 2 (1 e 2 ) 2 1 e 2 ( 3 cos 2 i 1 ) ; 3J 2nR 2 4a 2 (1 e 2 ) 2 ( 5 cos 2 i 1 ) ; Ω = 3J 2nR 2 2a 2 (1 e 2 2 cos i. ) [ µ 3J 2 R 2 [ (2 a sin 2 4a 2 (1 e 2 ) 2 i ) ] ] 1 e sin 2 i = µ = N p 3J 2 N d ω a + 3 R2 2a 2 (1 e 2 ) 2 cos i, µ [a 2 + 3J 2R 2 [ (2 3 sin 2 4 (1 e 2 ) 2 i ) 1 e sin 2 i 2 N ]] p cos i = N p ω a 7/2. N d N d k 1 k 2 k 1 = N d µ ; N p ω k 2 = k 1 3J 2 R 2 4 (1 e 2 ) 2 [ (2 3 sin 2 i ) 1 e sin 2 i 2 N ] p cos i, N d a 7/2 k 1 a 2 k 2 = 0.

146 a j+1 = a j a7/2 j k 1 a 2 j k 2, 7 2 a5/2 j 2k 1 a j a j+1 a j < tol tol a 0 [ a 3 0 µ N p = N (Nd ) ] 2 1/3 d µ = a 0 =. ω J 2 T Ω N p ω 2 J 2 i = o N p = 14 N d = 1 ϕ r = o N ψ r = o W

147 N p = 14 N d = 1 i = o T c Δψ ψ

148 2N p T c ψ a 0 N p N d ψ 0 ψ 0 a ψ 0 = ω (T c0 T c ) = ω N p (T 0 T ), N p T 0 T a 0 a ψ 0 = ω N p 2π a 3 0 µ 2π a 3. µ dψ dψ 0 = ψ 0 T c. dψ 0 ( ) dψ 0 = ω 1 a3 a 3. 0 a 0 dψ 0 a ( a = a 0 1 dψ ) 2/3 0. ω ( a i = a i 1 1 dψ ) 2/3 i 1, ω

149 a j = a (j 1) dψ (j 1) ( a(j 1) a (j 2) ) ( dψ(j 1) dψ (j 2) ), a j dψ j < ϵ ϵ N d = 1 N p = 14 N p = 14 N d = 1 e = 0.0 i = o

150 J < km km/day km km/day < x y z v x v y v z

151 N p N d 0 o 0 o w = 90 o 0 o N p N d km N p = 15 N d = km/day km km 0.55 km 1 km

152 Deviation in Equator (km/day) e = 0.00 e = 0.01 e = 0.02 e = 0.03 e = 0.04 e = Inclination (deg) N p = 15 N d = Difference in semi major axis (km) e = 0.00 e = e = 0.02 e = e = 0.04 e = Inclination (deg) N p = 15 N d = 1 N p = 3 N d = 1 8 km/day

153 Deviation in Equator (km/day) e = 0.0 e = 0.1 e = 0.2 e = 0.3 e = 0.4 e = Inclination (deg) N p = 3 N d = km Difference in semi major axis (km) e = 0.0 e = 0.1 e = 0.2 e = 0.3 e = 0.4 e = Inclination (deg) N p = 3 N d = 1

154 N p = 1 N d = 1 Deviation in Equator (km/day) e = 0.0 e = 0.1 e = 0.2 e = 0.3 e = 0.4 e = Inclination (deg) N p = 1 N d = 1 Difference in semi major axis (km) e = 0.0 e = 0.1 e = 0.2 e = 0.3 e = 0.4 e = Inclination (deg) N p = 1 N d = 1

155 Ω sec = ω yr, Ω sec yr = Ω sec = 3 2 J R 2 µ 2 a 7/2 (1 e 2 2 cos i. )

156 N p N d ( 1 e 2 ) ( 2 3 N = 2 J 2 yr R 2 7 p Nd 7 ω 4 µ 2 ) 1/3 cos i. r p = a (1 e), N p N d e max = 1 ( N 2 p ω 2 N 2 d µ ) 1/3 r p. ( e min = 3 N 1 2 J 2 yr R 2 7 p N 7 d ω 4 ) 1/3 µ 2. cos i max = 2 ( ) 1 e 2 2 ( min N 7 d 3 J 2 yr R Np 7 cos i min = 2 ( ) 1 e 2 2 ( max N 7 d 3 J 2 yr R Np 7 ω 4 µ 2 ) 1/3 ω 4 µ 2 ) 1/3.

157 J 2 J 3 J 2 J 3 ė = 3 2 [ ẇ = µ a 3 3J 3 (1 e 2 ) 2 ( R ) 3 J 3 (1 e 2 ) a µ R 2 a 3 a 2 ( 2 sin i cos ω )] [ ( sin2 i 1 5 ) 4 sin2 i, 1 + J 3 R J 2 2a sin ω 1 e 2 ( sin 2 i e 2 cos 2 i e sin i )]. ω = π/2, 3π/2 ( 1 e 2 ) e = 1 J 3 R sin 2 i e 2 cos 2 i sin ω, 2 J 2 a sin i a i

158 Ω kq = Ω 00 ω (t q t 0 ), M kq = M 00 + n (t k + t q t 0 ) ; Ω 00 M 00 t 0 t q t k k Ω kq M kq q k 2π t k = (k 1) ( N t N f ω Ω ), k0 t q = (q 1) T c N st, N t N st Ωk0 k0 N f N p N t N f = gcd(n p, N t ) k [1, N t ] q [1, N st ] N p = 59 N d = 4

159 N t = 1 N st = 4 N st /gcd(n st, N d ) x y z v x v y v z J 2

160

161

162

163

164 Ω kq = Ω k ω (t kq t 0 ), M kq = n(t kq t 0 ), (t kq t 0 ) Ω k Ω ij = 2π L Ω (i 1), M ij = 2π L M (j 1) 2π L M L MΩ L Ω (i 1), L Ω L M [( ) (i 1) Ω k = 2π (t kq t 0 ) = 2π n 1 ω n [ (j 1) L M L MΩ L M L Ω L MΩ L M (i 1) L Ω ] (j 1) L M + ω mod (2π), n ], (i 1) = G Ω 1, (j 1) = G M 1 + S MΩ (G Ω 1),

165 [( Ω k = 2π 1 ω n (t kq t 0 ) = 2π n L MΩ L M ) (GΩ 1) L Ω [ (GM 1 + S MΩ (G Ω 1)) L M + ω n ] (G M 1 + S MΩ (G Ω 1)) L MΩ L M (G Ω 1) L Ω mod (2π), ]. L M n ω N p N d n = 2π N p T c, ω = 2π N d T c, T c [( Ω k = 2π 1 N ) d L MΩ (GΩ 1) + N ] d (G M 1 + S MΩ (G Ω 1)) N p L M L Ω N p L M (t kq t 0 ) = T [ ] ( c (GM 1 + S MΩ (G Ω 1)) Tc + A N p N p L M L MΩ L M (G Ω 1) L Ω mod (2π), ), A (t kq t 0 ) = (q 1) T c = T [ c (GM 1 + S MΩ (G Ω 1)) L ] ( ) MΩ (G Ω 1) Tc + A, N st N p L M L M L Ω N p N st = L Ω L M (q 1) (q 1) = N st N p [ (GM 1 + S MΩ (G Ω 1)) L M L ] MΩ (G Ω 1) + A L M L Ω ( Nst N p ), (q 1) = 1 [ ] ( ) LΩ L M (G M 1 + S MΩ (G Ω 1))L Ω (G Ω 1)L MΩ + A. N p N p (q 1) L MΩ

166 Ω q = 2πN d (q 1) N st, M q = 2πN p (q 1) N st,

167 ±7.5 o 185 km 705 km ± 5 km 38.5 o o 32.9 o o

168 ±0.15 o 1982 kg 4 m 2.7 m 705 km ±0.1 o 530 kg 2.3 m 1.4 m 705 km 999 kg 2.3 m 2.3 m 2.8 m 1.5 km ±0.01

169 705 km N p N d N p = 233 N d = km o o

170 51.43 o 360 o / o 7 L Ω = 7 N Ω = 3 0 = 2πN d (q 1) N st mod (2π), N d L M = N d = 16 N M = 2 (Ω, M) L Ω L M L MΩ

171 L MΩ = 2 (Ω, M) G M G M = {1, 9} G M (1) = 1 G M (2) = 9 8 7S MΩ 2, S MΩ = o

172 G Ω = {1, 2, 7} G Ω (1) = 1 G Ω (2) = 2 G Ω (3) = 7 G Ω (1) G Ω (2) G Ω (3) G M (1) G M (2) G M (1) G M (2) G M (1) G M (2) Ω o o o o o o M o o o o o o (Ω, M) q {1,..., L Ω L M }

173 G Ω (1) G Ω (2) G Ω (3) G M (1) G M (2) G M (1) G M (2) G M (1) G M (2) q J 2

174 J 2 ±0.01 o ±1.5 km

175

176

177 v

178 v

179

180 v 0.02 o v = m/s

181 ±0.01 o

182

183

184

185

186 n

187

188 n

189

190 th J 2

191

192 J 2