3 The Restricted Three-Body Problem

3. The Restricted Three-Body Problem 1
3 The Restricted Three-Body Problem
3.1 Introduction
2 ,
,
. 3
.
3
, 2
, 3
, 3 (the circular, restricted, three-body
problem) . ,2
, 3 ,
3 .
3
,
, ,
.
. 3 ,
,
. , Jacobi
. 1
Hill
,
.
3
.
3.2 Equations of Motion
, m 1 , m 2 2
. , 2
,
.
ξ, η, ζ , 2
(3.2). ξ ,
t = 0 , m 1 m 2
, η ξ 2
. ζ ξ − η
, 2
. (ξ 1 , η 1 , ζ 1 ), (ξ 2 , η 2 , ζ 2 ),
, ,
. , µ = G(m 1 + m 2 ) = 1
. , m 1 > m 2 ,
¯µ =
m 2
m 1 + m 2
, (3.1)
, m 1 , m 2
µ 1 , µ 2 ,
.
µ 1 = gm 1 = 1 − ¯µ and µ 2 = gm 2 = ¯µ. (3.2)
2
. (inertial system) , sidereal system

3. The Restricted Three-Body Problem 2
3.1: sidereal (ξ, η, ζ) synodic (x, y, z) . P , O 2
, ζ, z
.
.
(ξ, η, ζ) ,
.
¨ξ = µ 1
ξ 1 − ξ
r 3 1
¨η = µ 1
η 1 − η
r 3 1
¨ζ = µ 1
ζ 1 − ζ
r 3 1
+ µ 2
ξ 2 − ξ
r 3 2
+ µ 2
η 2 − η
r 3 2
+ µ 2
ζ 2 − ζ
r 3 2
(3.3)
(3.4)
(3.5)
, (3.2)
r 2 1 = (ξ 1 − ξ) 2 + (η 1 − η) 2 + (ζ 1 − ζ) 2 , (3.6)
r 2 2 = (ξ 2 − ξ) 2 + (η 2 − η) 2 + (ζ 2 − ζ) 2 , (3.7)
. 2
, n
. ,
2
. ,
, . ,
, t = 0 ξ, η, ζ , ζ
n
. x
, m 1 , m 2
, (x 1 , y 1 , z 1 ) = (−µ 2 , 0, 0), (x 2 , y 2 , z 2 ) = (−µ 1 , 0, 0)
. , (3.2), 3.2
r 2 1 = (x + µ 2 ) 2 + y 2 + z 2 , (3.8)
r 2 2 = (x − µ 1 ) 2 + y 2 + z 2 . (3.9)

3. The Restricted Three-Body Problem 3
(x, y, z)
, synodic system .
.
⎛ ⎞ ⎛
⎞ ⎛ ⎞
ξ cos nt − sin nt 0 x
⎜ ⎟ ⎜
⎟ ⎜ ⎟
⎝ η ⎠ = ⎝ sin nt cos nt 0 ⎠ ⎝ y ⎠ . (3.10)
ζ 0 0 1 z
n = 1 , n
. (3.16)(3.18)
.
(3.10) 2
2 .
⎛ ⎞ ⎛
⎞ ⎛ ⎞
˙ξ cos nt − sin nt 0 ẋ − ny
⎜ ⎟ ⎜
⎟ ⎜ ⎟
⎝ ˙η ⎠ = ⎝ sin nt cos nt 0 ⎠ ⎝ ẏ + nx ⎠ . (3.11)
˙ζ 0 0 1 ż
⎛
⎜
⎝
¨ξ
¨η
¨ζ
⎞
⎟
⎠ =
⎛
⎜
⎝
cos nt − sin nt 0
sin nt cos nt 0
0 0 1
⎞ ⎛
⎟ ⎜
⎠ ⎝
ẍ − 2nẏ − n 2 x
ÿ + 2nẋ − n 2 y
¨z
⎞
⎟
⎠ . (3.12)
, Corioli (the Corioli’s acceleration, nx, ny)
(the centrifugal acceleration, n 2 x, n 2 y) .
, (3.3)
(3.5)
,
(ẍ−2nẏ − n 2 x) cos nt − (ÿ + 2nẋ − n 2 y) sin nt =
[
] [ ]
x 1 − x x 2 − x
µ 1
µ 1 + µ
r1
3 2 cos nt + + µ 2
y sin nt,
r2
3 r1
3 r2
3
(3.13)
(ẍ−2nẏ − n 2 x) sin nt + (ÿ + 2nẋ − n 2 y) cos nt =
[
] [ ]
x 1 − x x 2 − x
µ 1
µ 1 + µ
r1
3 2 sin nt + + µ 2
(3.14)
y cos nt,
r2
3 r1
3 r2
3
[ ]
µ 1
¨z = − + µ 2
z. (3.15)
r1
3 r2
3
(3.13) cos nt (3.14) − sin nt
, (3.13)
− sin nt (3.14) cos nt
,
, synodic

3. The Restricted Three-Body Problem 4
.
[
]
ẍ − 2nẏ − n 2 x = −
ÿ + 2nẋ − n 2 y = −
¨z = −
[
[
x + µ 2 x − µ 1
µ 1 + µ 2
µ 1
r1
3
µ 1
r 3 1
r 3 1
+ µ 2
r 3 2
+ µ 2
r 3 2
]
]
r 3 2
(3.16)
y, (3.17)
z. (3.18)
U
.
U = U(x, y, z)
.
ẍ − 2nẏ = ∂U
∂x , (3.19)
ÿ + 2nẋ = ∂U
∂y , (3.20)
¨z = ∂U
∂z , (3.21)
U = n2
2 (x2 + y 2 ) + µ 1
r 1
+ µ 2
r 2
. (3.22)
(x 2 + y 2 )
, 1/r 1 , 1/r 2
.
U ∗ = −U ,
.
ẍ − 2nẏ = − ∂U ∗
∂x , (3.23)
ÿ + 2nẋ = − ∂U ∗
∂y , (3.24)
¨z = − ∂U ∗
∂z . (3.25)
, U
. U
, .
.
3.3 The Jacobi Integral
(3.19) ẋ, (3.20) ẏ, (3.21) ż
,
ẋẍ + ẏÿ + ż¨z = ∂U
∂x ẋ + ∂U
∂y ẏ + ∂U
∂z ż = dU
dt . (3.26)

3. The Restricted Three-Body Problem 5
,
ẋ 2 + ẏ 2 + ż 2 = 2U − C J (3.27)
C J . ẋ 2 + ẏ 2 + ż 2 = v 2 ,
(3.22) ,
C J = n 2 (x 2 + y 2 ) + 2
v 2 = 2U − C J (3.28)
(
µ1
r 1
+ µ 2
r 2
)
− ẋ 2 − ẏ 2 − ż 2 (3.29)
, 2U − v 2 = C J ,
. C J Jacobi (the Jacobi integral) .
.
3
. Jacobi
3
.
C J , 3.1 (ξ, η, ζ)
. (3.10) ,
⎛ ⎞ ⎛
⎞ ⎛ ⎞
x cos nt sin nt 0 ξ
⎜ ⎟ ⎜
⎟ ⎜ ⎟
⎝ y ⎠ = ⎝ − sin nt cos nt 0 ⎠ ⎝ η ⎠ , (3.30)
z
0 0 1 ζ
⎛
⎜
⎝
. ,
⎛
ẋ − nẏ
⎜
⎝ ẏ + nẋ
ż
ẋ − nẏ
ẏ + nẋ
ż
⎞
⎟
⎠ =
⎞
⎟
⎠ =
⎛
⎜
⎝
ẋ
ẏ
ż
⎛
⎜
⎝
⎞
cos nt sin nt 0
− sin nt cos nt 0
0 0 1
⎛
⎟ ⎜
⎠ + n ⎝
⎞ ⎛
⎟ ⎜
⎠ ⎝
sin nt − cos nt 0
cos nt sin nt 0
0 0 1
˙ξ
˙η
˙ζ
⎞
⎟
⎠ , (3.31)
⎞ ⎛
⎟ ⎜
⎠ ⎝
ξ
η
ζ
⎞
⎟
⎠ , (3.32)
, ,
⎛ ⎞ ⎛
ẋ cos nt sin nt 0
⎜ ⎟ ⎜
⎝ ẏ ⎠ = ⎝ − sin nt cos nt 0
ż
0 0 1
⎞ ⎛
⎟ ⎜
⎠ ⎝
˙ξ
˙η
˙ζ
⎞
⎛
⎟ ⎜
⎠ − n ⎝
sin nt − cos nt 0
cos nt sin nt 0
0 0 1
⎞ ⎛
⎟ ⎜
⎠ ⎝
ξ
η
ζ
⎞
⎟
⎠ (3.33)
⎛
A =
⎜
⎝
cos nt sin nt 0
− sin nt cos nt 0
0 0 1
⎞
⎟
⎠ and B =
⎛
⎜
⎝
sin nt − cos nt 0
cos nt sin nt 0
0 0 1
⎞
⎟
⎠ (3.34)

3. The Restricted Three-Body Problem 6
(3.33) ,
⎛
ẋ 2 + ẏ 2 + ż 2 ⎜
= (ẋ ẏ ż) ⎝
⎛
= ( ˙ξ ˙η ˙ζ)A T ⎜
A ⎝
ẋ
ẏ
ż
˙ξ
˙η
˙ζ
⎞
⎟
⎠
⎞
− n(ξ η ζ)B T ⎜
A ⎝
⎟
⎠ − n( ˙ξ ˙η ˙ζ)A T B
⎛
˙ξ
˙η
˙ζ
⎞
⎛
⎜
⎝
ξ
η
ζ
⎟
⎠ + n 2 (ξ η ζ)B T B
= ˙ξ 2 + ˙η 2 + ˙ζ 2 + n 2 ( ˙ξ 2 + ˙η 2 ) + 2n( ˙ξη − ˙ηξ).
⎞
⎟
⎠
⎛
⎜
⎝
ξ
η
ζ
⎞
⎟
⎠
(3.35)
A T , B T A, B
. A, B
, ,
,
. ,
,
x 2 + y 2 + z 2 = ξ 2 + η 2 + ζ 2 (3.29) ,
(
µ1
C J = 2 + µ )
2
+ 2n(ξ ˙η − η
r 1 r ˙ξ) − ˙ξ 2 − ˙η 2 − ˙ζ 2 (3.36)
2
.
.
(
1
2 ( ˙ξ 2 + ˙η 2 + ˙ζ 2 µ1
) − + µ )
2
= h · n − 1 r 1 r 2 2 C J (3.37)
n = (0, 0, n) ,
. 2.9 ,
h
, 3
.
3.2: 2 Jacobi
. k µ 2 = 0.2 . C J
(a)C J = 3.9 , (b)C J = 3.7.

3. The Restricted Three-Body Problem 7
Jacobi
. 2 ,
. , 3 , Jacobi
.
Jacobi
, 0
. ,
2U = C J (3.38)
(
n 2 (x 2 + y 2 µ1
) + 2 + µ )
2
= C J (3.39)
r 1 r 2
,
. x − y
.
. 3.3 µ 2 = 0.2, n = 1 . (3.27) Jacobi
2U ≥ C J . (3.39)
,
. , 3 . (3.3)
, .
(3.3a) , , C J
µ 1
, µ 2 ,
. (3.3b) , µ 1
, µ 2
. ,
. Hill (Hill’s stability) .
3.4 Tisserand Relation
a, e, I
. ,
a ′ , e ′ , I ′ . 2
, Jacobi
. Jacobi , C J = 2U −v 2 ,
.3
,
r = (ξ, η, ζ), r = ( ˙ξ, ˙η, ˙ζ) .
(3.37) Jacobi
.
(
1
2 ( ˙ξ 2 + ˙η 2 + ˙ζ 2 µ1
) − + µ )
2
= h · n − 1 r 1 r 2 2 C J (3.40)
r 1 , r 2 ,
. ,
,
. ,
.
g(m Sun + m comet ) ≈ g(m Sun + m Jupitar ) = 1 (3.41)
m Sun , m comet , m Jupitar , , ,
. - 2
(eq:2.34) 1 ,
1 (2.34): v 2 = µ(2/r − 1/a)
˙ξ 2 + ˙η 2 + ˙ζ 2 = 2 r − 1 a , (3.42)

3. The Restricted Three-Body Problem 8
µ = 1 , ,
r 1 ≈ r .
.
h = r × ṙ. (3.43)
ζ
ξ ˙η − η ˙ξ = h cos I. (3.44)
, h 2 = a(1 − e 2 ) . (3.40)
2
r − 1 a − 2√ a(1 − e 2 ) cos I = 2 r − 2µ 2
( 1
r − 1 )
− C J (3.45)
r 2
. ,
, 1/r 2 , µ 2
,
1
2a + √ a(1 − e 2 ) cos I ≈ constant. (3.46)
,
,
.
1
2a + √ a(1 − e 2 ) cos I = 1
2a + √ a ′ (1 − e ′2 ) cos I ′ (3.47)
′
Tisserand (Tesserand relation, Tesserand 1896)
,
,
.
3.3:
. .

3. The Restricted Three-Body Problem 9
3.4: e Jup = 0, e Jup = 0.048 Tisserand . 35 .
3.4 . ,
,
8AU
.
a = 4.81AU,
e = 0.763, I = 7 .47,
a = 10.8AU, e = 0.731, I = 21 .4 .
Tisserand
Jacobi
, 1
. , + √ a(1 − e
2a 2 ) cos I , 0
. 3.4 .
0( ),
0.048 Tisserand
.
, 1 ,
12
.
3.4
, , ,
. 3
,
.
,
.
,
. ,
.
,
, , ,
.

3. The Restricted Three-Body Problem 10
3.5 Lagrangian Equilibrium Points
2 m 1 , m 2
n
,
n
, 2
. ,
, P
(equilibrium point) .
a, b, c , m 1 , 2 , m 2
.
P (3.5). , F 1 , F 2 P m 1 , m 2
. P
, O
b . P −b
,
.
F = F 1 + F 2 . (3.48)
P
Coriolis
.
O
.
b = m 1a + m 2 c
m 1 + m 2
(3.49)
,
m 1 (a − b) = m 2 (b − c). (3.50)
F 1 + F 2
,
m 2 (F 1 × c) + m 1 (F 2 × a) = 0. (3.51)
F 1 , c F 2 , a
, ,
m 2 F 1 c = m 1 F 2 a. (3.52)
3.5: P , m 1 , m 2 . O m 1 , m 2

3. The Restricted Three-Body Problem 11
, F 1 = gm 1 /a 2 , F 2 = gm 2 /c 2
(3.52) a = c
. , 2 3 2 3
.
P
,
n 2 b = F 1 cos β + F 2 cos γ. (3.53)
β F 1 b , γ F 2 b
(3.6).
,
n 2 =
, O, P, 2
3 ,
g
a 2 b 2 (m 1b cos β + m 2 b cos γ). (3.54)
b cos β = a − g cos α,
b cos γ = a − (d − g) cos α,
(3.55)
.d m 1 , m 2 , g m 1 O
. ,
.
,
cos α = d
2a , (3.56)
g = m 2
m 1 + m 2
d, d − g =
m 2
m 1 + m 2
d. (3.57)
(3.54)
n 2 = g(m (
1 + m 2 )
a 2 − m )
1m 2
a 3 b 2 (m 1 + m 2 ) 2 d2 . (3.58)
3.6: P
. m 1 , m 2
2
.
.

3. The Restricted Three-Body Problem 12
. ,
(3.57) ,
b 2 = a 2 + g 2 − 2ag cos α = a 2 + g 2 − gd. (3.59)
, (3.58)
.
b 2 = a 2 − m 1m 2
(m 1 + m 2 ) 2 d2 . (3.60)
n 2 = g(m 1 + m 2 )/a 3 . (3.61)
,
, Kepler 3 ,
n 2 = g(m 1 + m 2 )/d 3 , (3.62)
a = d .
, m 1 , m 2
, m 1 , m 2 ,
. ,
.
Lagrangian equilibrium point, L 4 , L 5 . , 3 L 1 , L 2 , L 3 , m 1 , m 2
.
,
,
.
, ,
.
3.6 Location of Equilibrium Points
3
,
. ,
.
. ,
x − y
, , 2
, n = 1 .
, Brower & Clemence (1961) . U
. (3.8) 2 , (3.9) 3 r 1 , r 2 , µ 1 + µ 2 = 1 ,
(3.22) 4 ,
µ 1 r 2 1 + µ 2 r 2 2 = x 2 + y 2 + µ 1 µ 2 , (3.63)
U = µ 1
( 1
r 1
+ r2 1
2
2 (3.8): r 2 1 = (x + µ 2 ) 2 + y 2 + z 2
3 (3.9): r 2 2 = (x − µ 1 ) 2 + y 2 + z 2
4 (3.22): U = n2
2 (x2 + y 2 ) + µ 1
r 1
+ µ 2
r 2
)
( ) 1
+ µ 2 + r2 2
− 1 r 2 2 2 µ 1µ 2 . (3.64)

3. The Restricted Three-Body Problem 13
x, y
,
. , x, y , r 1 , r 2
.
(3.19) 5 , (3.20) 6 , ẍ = ÿ = ẋ = ẏ = 0
.
.
∂U
∂x = ∂U ∂r 1
∂r 1 ∂x + ∂U ∂r 2
= 0,
∂r 2 ∂x
(3.65)
∂U
= ∂U ∂r 1
∂y ∂r 1 ∂y + ∂U ∂r 2
= 0.
∂r 2 ∂y
(3.66)
U (3.64)
.
µ 1
(
− 1 r 2 1
+ r 1
) x + µ2
r 1
+ µ 2
(− 1 r 2 2
µ 1
(
− 1 r 2 1
+ r 1
) y
r 1
+ µ 2
(
− 1 r 2 2
+ r 2
) x + µ1
r 2
= 0, (3.67)
+ r 2
) y
r 2
= 0. (3.68)
(3.65), (3.65) ,
∂U
∂r 1
= µ 1
(− 1 r 2 1
+ r 1
)
= 0,
∂U
∂r 2
= µ 2
(− 1 r 2 2
+ r 2
)
= 0, (3.69)
, r 1 = r 2 = 1
. ,
r 2 1 = (x + µ 2 ) 2 + y 2 = 1, r 2 2 = (x − µ 1 ) 2 + y 2 = 1 (3.70)
, 2
.
x = 1 √
3
2 − µ 2, y = ±
2 . (3.71)
r 1 = r 2 = 1 , ,µ 1 , µ 2
,
.
(Lagrangian equilibrium point) ,
,L 4 , L 5 .
(3.68) , y = 0 (3.68)
. x
, (3.65)
. ,
(collinear Lagrangian equilibrium point) L 1 , L 2 , L 3 . L 1 µ 1 , µ 2 ,
L 2 µ 2 , L 3 x
. ,
,
.
L 1 ,
r 1 + r 2 = 1, r 1 = x + µ 2 , r 2 = −x + µ 1 ,
5 (3.19): ẍ − 2nẏ = ∂U
∂x
6 (3.20): ÿ + 2nẋ = ∂U
∂y
∂r 1
∂x = −∂r 2
∂x
= 1. (3.72)

3. The Restricted Three-Body Problem 14
, (3.67) ,
) (
1
µ 1
(−
(1 − r 2 ) + 1 − r 2 2 − µ 2 − 1 r2
2
+ r 2
)
= 0 (3.73)
,
µ 2
= 3r 3 (1 − r 2 + r2/3)
2
2
µ 1 (1 + r 2 + r2)(1 2 − r 2 ) , (3.74)
3
.
,
( ) 1/3 µ2
α = , (3.75)
3µ 1
r 2
, r 2 = α
. α Hill (µ 2
)
. (3.74)
,
α = r 2 + 1 3 r2 2 + 1 3 r3 2 + 53
81 r4 2 + O(r 5 2) (3.76)
. 2.5 Lagrange’s inversion method
, r 2 α
. (3.76) (2.89) 7
,
. φ
.
,
r 2 = α + (−1/3)φ(r 2 ), (3.77)
φ(r 2 ) = r 2 2 + r 3 2 + 53
27 r4 2 + O(r 5 2). (3.78)
[φ(α)] 2 = α 4 + 2α 5 + O(α 6 ), (3.79)
d
dα [φ(α)]2 = 4α 3 + 10α 4 + O(α 5 ), (3.80)
d 2
[φ(α)] 3 = α 6 + O(α 7 ), (3.81)
dα 2 [φ(α)]3 = 30α 4 + O(α 5 ). (3.82)
(2.90) 8
, r 2
.
r 2 = α +
∞∑ (−1/3) j d j−1
[φ(α)]j
j! dαj−1 j=1
= α − 1 3 α2 − 1 9 α3 − 23
81 α4 + O(α 5 ). (3.83)
7 (2.89): ζ = z + eφ(ζ)(e < 1)
8 (2.90): ζ = z + ∑ ∞
j=1 ej d j−1
j! dz
[φ(z)] j .
j−1

3. The Restricted Three-Body Problem 15
L 2 ,
r 1 − r 2 = 1, r 1 = x + µ 2 , r 2 = x − µ 1 , ∂r 1
∂x = ∂r 2
= 1.
∂x
(3.84)
, (3.67) ,
) (
1
µ 1
(−
(1 + r 2 ) + 1 + r 2 2 + µ 2 − 1 r2
2 )
+ r 2 = 0 (3.85)
. α
,
.L 3 ,
µ 2
= 3r 3 (1 + r 2 + r2/3)
2
2
µ 1 (1 + r 2 ) 2 (1 − r2) , (3.86)
3
α = r 2 − 1 3 r2 2 + 1 3 r3 2 + 1
81 r4 2 + O(r2) 5 (3.87)
r 2 = α + 1 3 α 2 − 1 9 α3 − 31
81 α4 + O(α 5 ) (3.88)
r 2 − r 1 = 1, r 1 = −x − µ 2 , r 2 = −x + µ 1 , ∂r 1
∂x = ∂r 2
∂x
= −1. (3.89)
, (3.67) r 2 ,
(
µ 1 − 1 )
)
1
+ r
r1
2 1 + µ 2
(−
(1 + r 1 ) + 1 + r 2 1 = 0 (3.90)
,r 1 = 1 + β ,
µ 2
= − 12
µ 1 7 β + 144
49 β2 − 1567
β = − 7 12
µ 2
µ 1
= (1 − r3 1)(1 + r 1 ) 2
r 3 1(r 2 1 + 3r 1 + 3) . (3.91)
(
µ2
µ 1
)
+ 7 12
343 β3 + O(β 4 ), (3.92)
( ) 2 µ2
− 13223 ( ) 3 ( ) 4 µ2 µ2
+ O
(3.93)
µ 1 20736 µ 1 µ 1
3.7 , µ 2 = 0.2, 3 Jacobi
, Lagrangian
points . 3.8 ,
C J
3 .
3.7, 3.8 , L 1 Jacobi
(C J = 3.805, µ 2 = 0.2),
. 3.2
, L 1 , 2
. L 2 (C J = 3.552, µ 2 = 0.2)
. C J < C L2
,
,

3. The Restricted Three-Body Problem 16
.
L 3 (C J = 3.197, µ 2 = 0.2) .L 4 , L 5 (C J = 2.84, µ 2 = 0.2)
C J . , Jacobi C J < C L4,5 ,
.
. Jacobi
.
Lagrangian equilibrium point
.
L 5 . ,
Jacobi ,
. x, y ,
r 1 , r 2 (3.39) 9
, Jacobi
. O(µ 2 ) .
C L1 ≈ 3 + 3 4/3 µ 2/3
2 − 10µ 2 /3, (3.94)
C L2 ≈ 3 + 3 4/3 µ 2/3
2 − 14µ 2 /3, (3.95)
C L3 ≈ 3 + µ 2 , (3.96)
C L4 ≈ 3 − µ 2 , (3.97)
C L5 ≈ 3 − µ 2 . (3.98)
(3.8) 10 , (3.9) 11 , (3.39) , y −y
.
x
. , L 4 , L 5 Jacobi
.
µ 2 = m 2 /(m 1 + m 2 ) , Pluto-Charon ,
∼ 10 −1 . - ∼ 10 −2 . ,
- , -
µ 2 ,
. Lagrangian equilibrium
point
, µ 2
.
(3.94), (3.95) , µ 2 → 0 C L1 → C L2 . (3.83), (3.88) ,
µ 2 → 0 , O(α 2 )
. L 1 , L 2 µ 2
. L 3 1 + β
. β < 0 (3.93)
. µ 2 → 0 , L 3
.
µ 1
, r = (1 − µ 2 + µ 2 2) 1/2
. µ 2 , µ 1
, µ 2 → 0 , µ 1
.
3.6 µ 2 = 0.01( -
)
Lagrangian equilibrium
point . L 1 , L 2 µ 2 ,
, L 3
. 3.9
µ 1
. 1%
,
.
9 (3.39): n 2 (x 2 + y 2 ) + 2(µ 1 /r 1 + µ 2 /r 2 ) = C J
10 (3.8): r 2 1 = (x + µ 2 ) 2 + y 2 + z 2
11 (3.9): r 2 2 = (x − µ 1 ) 2 + y 2 + z 2

3. The Restricted Three-Body Problem 17
3.7: Lagrangian equilibrium point ( ) µ 2 = 0.2
.
, L 1 , L 2 , L 3
3 Jacobi (3.805,
3.552, 3.197)
. O 2
.
3.8: C J = 2U
3 , Lagrangian equilibrium point
. L 1 , L 2 , L 3 ,
.
C J
.

3. The Restricted Three-Body Problem 18
3.9: Lagrangian equilibrium point ( ) µ 2 = 0.01
.
.
3.10: 5 Lagrangian equilibrium point
. µ 2 log
.

3. The Restricted Three-Body Problem 19
µ 2 → 0
, µ 2 = 10 −1
µ 2 = 10 −10
(3.10). , L 3 , L 4 , L 5 , L 1 ,
L 2
.
.

3. The Restricted Three-Body Problem 20
3.7 Stability of Equilibrium Points
,
.
,
.
,
.
, ,
.
.
, . , 2
, ,
,
, , .
. , 2
. , ,
.
. L 4 , L 5 , C J =
2U − v 2
. U
U ∗ (U ∗ = −U) , L 4 , L 5 C J
. ,
,
. 3.11 µ 2 = 0.1 2
. (3.16) 12 , (3.17) 13
(ẍ, ÿ)
.
, ,
.
3.11
, L 4 C J
3.11: 2
(C J = 2.95, C J = 3.09, µ 2 = 0.1)
.
12 (3.16): ẍ − 2nẏ − n 2 x = −[µ 1 (x + µ 2 )/r 3 1 + µ 2 (x − µ 1 )/r 3 2]
13 (3.16): ÿ + 2nẋ − n 2 y = −[µ 1 /r 3 1 + µ 2 /r 3 2]y

3. The Restricted Three-Body Problem 21
, L 4
. ,
. ,
,
. ,
(linear stability analysis)
.
(x 0 , y 0 ) ,
. (X, Y ) .
(x, y) = (x 0 + X, y 0 + Y )
. (3.19) 14 , (3.20) 15 ,
.
( ) ( ( )) ( ( ))
∂U ∂ ∂U
∂ ∂U
Ẍ − 2nẎ ≈ + X
+ Y
.
∂x
0
∂x ∂x
0
∂y ∂x
0
( ( )
∂ 2 U ∂ 2 U
= X + Y
(3.99)
∂x∂y
∂x 2 )0
( ) ( ( ))
∂U ∂ ∂U
Ÿ + 2nẊ ≈ + X
∂y
0
∂x ∂y
( ) ( ∂ 2 U ∂ 2 U
= X + Y
∂x∂y
0
0
0
+ Y
( ∂
∂y
( )) ∂U
,
∂y
0
. (3.100)
∂y
)0
2
0
. , (∂U/∂x) 0 = (∂U/∂y) 0 = 0
(∂U/∂x = ∂U/∂y = 0
).
, (X, Y )
.
(linearise)
.
.
n = 1, ,
( ∂ 2 U
U xx =
Ẍ − 2Ẏ = XU xx + Y U xy , Ÿ + 2Ẋ = XU xy + Y U yy . (3.101)
, U
∂x
)0
2 xy =
( ) ( ∂ 2 U
∂ 2 U
, U yy = . (3.102)
∂x∂y
0
∂y
)0
2
.
,
.
.
,
.
⎛ ⎞ ⎛
⎞ ⎛ ⎞
Ẋ 0 0 1 0 X
Ẏ
⎜ ⎟
⎝ Ẍ ⎠ = 0 0 0 1
Y
⎜
⎟ ⎜ ⎟
(3.103)
⎝ U xx U xy 0 2 ⎠ ⎝ Ẋ ⎠
Ÿ U xy U yy −2 0 Ẏ
14 (3.19): ẍ − 2nẏ = ∂U
∂x
15 (3.20): ÿ + 2nẋ = ∂U
∂y

3. The Restricted Three-Body Problem 22
, 2 2
, 4 1
.
.
⎛
X = ⎜
⎝
X
Y
Ẋ
Ẏ
Ẋ = AX, (3.104)
⎞
⎛
⎞
0 0 1 0
⎟
⎠ and A = 0 0 0 1
⎜
⎟
⎝ U xx U xy 0 2 ⎠ . (3.105)
U xy U yy −2 0
, (3.104) , A n × n , X n-
,
.
x
,
Ax = λx, (3.106)
λ
. x A
(eigenvector), λ
. A
, (3.106)
x A
, x
,
.
(3.104)
, A
. (3.106)
,
(A − λI)x = 0, (3.107)
. I n × n
. n
, A − λI 0
,
.
det(A − λI) = 0. (3.108)
. λ n
, n
.
n
, λ (3.106) , x
.
, (3.104)
. X X i
. ,
, Ẋ , X, Y , Ẏ
. ,
,
, . X , Ẏ i Y i
, n Y i
.
. X Y .
Y = BX, (3.109)
B
.
X = B −1 Y , Ẋ = B −1 Ẏ , (3.110)
, (3.104)
.
B −1 Ẏ = AB −1 Y . (3.111)

3. The Restricted Three-Body Problem 23
, B ,
Ẏ = BB −1 Ẏ = BAB −1 Y . (3.112)
(3.112) , Y
. B Λ
.
BAB −1 = Λ. (3.113)
Λ
. B −1
A
, Λ A
.
⎛
⎞
λ 1 0 · · · 0
0 λ 2 · · · 0
Λ = ⎜
⎝
.
. . ..
⎟
. ⎠
0 0 · · · λ n
(3.114)
,
Ẏ = ΛY . (3.115)
,
Ẏ i = λ i Y i . (3.116)
(3.116)
Y i = c i e λ it . (3.117)
c i n
. Y i
, X i
. (3.110) ⎛
X = B −1 Y = B −1 ⎜
⎝
⎞
c 1 e λ 1t
c 2 e λ 2t
⎟
. ⎠
c n e λnt
(3.118)
n
c i , (3.118) n
.
,
. n = 4 ,
.
−λ 0 1 0
0 −λ 0 1
det(A − λI) =
= 0 (3.119)
U xx U xy −λ 2
∣ U xy U yy −2 −λ ∣

3. The Restricted Three-Body Problem 24
.
λ 4 + (4 − U xx − U yy )λ 2 + U xx U yy − U 2 xy = 0. (3.120)
, λ 4 , λ 2 2
. , ,
[
1
λ 1,2 = ±
2 (U xx + U yy − 4)
− 1 2
[
(4 − Uxx − U yy ) 2 − 4(U xx U yy − U 2 xy) ] 1/2
[
1
λ 3,4 = ±
2 (U xx + U yy − 4)
+ 1 2
[
(4 − Uxx − U yy ) 2 − 4(U xx U yy − U 2 xy) ] 1/2
] 1/2
(3.121)
] 1/2
(3.122)
(3.118) , 4 A
, . (3.118) X, Ẋ
.
X =
4∑
ᾱ j e λjt , Ẋ =
j=1
4∑
ᾱ j λ j e λjt . (3.123)
j=1
ᾱ j
. X Y , Ẏ
, ¯β j ,
Y =
4∑
¯β j e λjt , Ẏ =
j=1
4∑
¯β j λ j e λjt , (3.124)
j=1
. 4
, ¯β j ᾱ j . ¯βj
ᾱ j (3.101) . (3.101) X, Y , Ẏ
,
4∑
(ᾱ j λ 2 j − 2 ¯β j λ j − U xx ᾱ j − U xy ¯βj )e λjt = 0. (3.125)
j=1
¯β j
,
¯β j = λ2 j − U xx
2λ j + U xy
ᾱ j , (3.126)
. t = 0 ,
X = X 0 , Y = Y 0 , Ẋ = Ẋ0, Ẏ = Ẏ0 , 4 ᾱ j 4
.
4∑
ᾱ j = X 0 ,
j=1
4∑
λ j ᾱ j = Ẋ0,
j=1
4∑
¯β j = Y 0 ,
j=1
4∑
λ j ¯βj = Ẏ0, (3.127)
j=1

3. The Restricted Three-Body Problem 25
(3.123), (3.123) , ᾱ j , ¯β j
,
. ,
.
Ā = µ 1
+ µ 2
(3.128)
(r1) 3 0 (r
[ 2) 3 0
µ1
¯B = 3 + µ ]
2
y
(r1) 5 0 (r2) 5 0 2 (3.129)
[
0
]
(x 0 + µ 2 ) (x 0 − µ 1 )
¯C = 3 µ 1 + µ
(r1) 5 2 y
0 (r2) 5 0 , (3.130)
0
]
(x 0 + µ 2 )
¯D = 3
[µ 2 (x 0 − µ 1 ) 2
1 + µ
(r1) 5 2 . (3.131)
0 (r2) 5 0
U xx = 1 − Ā + ¯D, (3.132)
U yy = 1 − Ā + ¯B, (3.133)
U xy = ¯C. (3.134)
. , λ j , ᾱ j , ¯β j
, , X, Y , Ẋ, Ẏ
.
(3.121),(3.122) ,
λ 1,2 = ±(j 1 + ik 1 ), λ 3,4 = ±(j 2 + ik 2 ). (3.135)
. j 1 , k 1 , j 2 , k 2
, i = √ −1 .
,
e λjt
, 2 e +(j+ik)t e −(j+ik)t
. j = 0 ,
. ,
e +ikt , e −ikt
,
. j
,
.
.
.

3. The Restricted Three-Body Problem 26
3.7.1 The Collinear Points
Lagrangian Point
L 1 , L 2 , L 3
.
, y 0 = 0,
(r1) 2 0 = (x 0 + µ 2 ) 2 , (r2) 2 0 = (x 0 − µ 1 ) 2 ,
U xx = 1 + 2Ā, U yy = 1 − Ā, U xy = 0. (3.136)
λ 4 + (2 − Ā)λ2 + (1 + Ā − 2Ā2 ) = 0, (3.137)
.
, µ 2 = 0.01 L 1 , 3.6
. 3.6 , x 0 = 0.848, y 0 = 0. X 0 = Y 0 = 10 −5 , Ẋ 0 = Ẏ0 = 0, ,
±2.90, ±2.31i .
L 1
. ᾱ j ,
¯β j
, X(t), Y (t)
.
X(t) =6.99 × 10 −6 e −2.90t + 4.96 × 10 −6 e +2.90t
+ 1.96 × 10 −6 cos 2.32t + 2.54 × 10 −6 sin 2.32t,
Y (t) =3.25 × 10 −6 e −2.90t − 2.31 × 10 −6 e +2.90t
+ 9.06 × 10 −6 cos 2.32t + 6.96 × 10 −6 sin 2.32t,
(3.138)
2 X, Y
. µ 2
1/2.90 .
3.12:
( ) ( ) . logµ 2

3. The Restricted Three-Body Problem 27
L 1
3.12 . µ 2 0.1 0.001
.
, ±j, ±ik . j, k
. L 2 , L 3
.
, ,
.
.
(λ 1 λ 2 )(λ 3 λ 4 ) = 1 + Ā − 2Ā2 (3.139)
(3.121), (3.122) , λ 1 = −λ 2 , λ 3 = −λ 4 .
(
) , λ 2 1 = λ 2 2 < 0, λ 2 3 = λ 2 4 < 0
. ,
1 + Ā − 2Ā2 = (1 − Ā)(1 + 2Ā) > 0 .
−1/2 < Ā < 1
. , (3.128)
r 1 , r 2
, µ 2 < 1/2 , Ā > 1 .
µ 2
. , (3.138)
,
.
, x
. 2 ,
,
. , 2
2
.
, 2
1
. x
,
.
,
,
(Szebehely 1967). , - L 1
SOHO(Domingo et al., 1995)
,
.
3.7.2 The Triangular Points
Lagrangian Point
L 4 , L 5
. , r 1 = r 2 = 1,
x = 1/2 − µ 2 , y = ± √ 3/2 ,
U xx = 3/4, U yy = 9/4, U xy = ±3 √ 3(1 − 2µ 2 )/4. (3.140)
.
λ 4 + λ 2 + 27 4 µ 2(1 − µ 2 ) = 0. (3.141)
.
. L 4 , µ 2 = 0.01 , x 0 = 0.49, y 0 = √ 3/2
.
, ±0.963i, ±0.268i ,

3. The Restricted Three-Body Problem 28
. X(t) =3.45 × 10 −5 cos 0.268t − 2.45 × 10 −5 cos 0.963t,
+ 3.07 × 10 −4 sin 0.268t − 8.55 × 10 −5 sin 0.963t,
Y (t) =5.20 × 10 −5 cos 0.268t − 4.20 × 10 −5 cos 0.963t,
− 1.76 × 10 −4 sin 0.268t + 4.90 × 10 −5 sin 0.963t,
(3.142)
.
, 1/0.268, 1/0.963 2
.
3.13
. , logµ 2 ≈ −1.4
, logµ 2
. , ±j ± ik ,
. , µ 2
, ,
,
.
3.13
,
. U xx , U yy ,
U xy (3.121), (3.122)
,
.
√
−1 − √ 1 − 27(1 − µ 2 )µ 2
λ 1,2 = ±
√ (3.143)
2
λ 3,4 = ±
√
−1 + √ 1 − 27(1 − µ 2 )µ 2
√
2
(3.144)
,
.
1 − 27(1 − µ 2 )µ 2 ≥ 0
µ 2 ≤ 27 − √ 621
54
≈ 0.0385 (3.145)
, k 1 , k 2 , λ 1,2 = ±ik 1 , λ 3,4 = ±ik 2
. , a j , b j
, α j = ā j + i¯b j , X(t)
.
X(t) =(ā 1 + i¯b 1 )e +ik 1t + (ā 2 + i¯b 2 )e −ik 1t
+ (ā 3 + i¯b 3 )e +ik 2t + (ā 4 + i¯b 4 )e −ik 2t .
(3.146)
Y (t)
. (3.127) X, Y , Ẋ, Ẏ
, ā 1 = ā 2 =
a 1 , ā 3 = ā 4 = a 2 , ¯b 1 = −¯b 2 = b 1 , ¯b 3 = −¯b 4 = b 2
, e ,
.
X(t) =(a 1 + ib 1 )e +ik 1t + (a 1 − ib 1 )e −ik 1t
+ (a 2 + ib 2 )e +ik 2t + (a 2 − ib 2 )e −ik 2t ,
(3.147)

3. The Restricted Three-Body Problem 29
e iθ = cos θ + i sin θ
,
.
X(t) = 2a 1 cos k 1 t + 2a 2 cos k 2 t − 2b 1 sin k 1 t − 2b 2 sin k 2 t. (3.148)
,
,
. ,
, .
(3.145)
, , ±(j ± ik) (
3.13
). ,
X(t) =(ā 1 + i¯b 1 )e (j+ik)t + (ā 1 − i¯b 1 )e (j−ik)t
+ (ā 2 + i¯b 2 )e (−j+ik)t + (ā 2 + i¯b 2 )e (−j−ik)t ,
(3.149)
X(t) = 2(a 1 e jt + a 2 e −jt ) cos kt − 2(b 1 e jt + b 2 e −jt ) sin kt. (3.150)
.
.Y , Ẋ, Ẏ . .
.
(µ 2 < 0.0385)
. (3.143), (3.144) ,
√
λ 1,2 ≈ ± −1 + 27 √
4 µ 2, λ 3,4 ≈ ± − 27
4 µ 2. (3.151)
, µ 2
,
2
. L 4 , L 5
2 3.13:
( ) ( ) . logµ 2 .
logµ 2 ≈ −1.4 .

3. The Restricted Three-Body Problem 30
3.14: (3.142) . ( ) epicenter ( )
. epicenter
, L 4
.
epicenter
.
,
. (3.145)
.
(resonances)
,
, (3.145)
(Deprit & Deprit-Bartholome 1967 ).
3.8 Motion near L 4 and L 5
, µ 2
, 2π
. 3.7
, 2π/λ 1,2 , 2π/λ 3,4 .
2
.
• 2π/λ 1,2 ≈ 2π ( µ 2
.),
• 2π/λ 3,4 , (libration)
,
.
2
ᾱ j , ¯β j
,
. ,
epicenter , , epicenter
( )
(3.14).
(3.142)
3.15
2 .
30
. X
. (X ′ , Y ′ )
.
( ) (
) ( )
X ′ (t) cos 30 − sin 30 X(t)
=
. (3.152)
Y ′ (t) sin 30 cos 30 Y (t)

3. The Restricted Three-Body Problem 31
3.15: L 4
.
(3.142) , t = 0 25π (µ 2
12.5 )
. 3.8
epicenter
.
X(t), Y (t) (3.142) ,
X(t) ≈3.54 × 10 −4 sin 0.268t − 9.85 × 10 −5 sin 0.963t,
Y (t) ≈6.23 × 10 −5 cos 0.268t − 4.86 × 10 −5 cos 0.963t,
(3.153)
,
. 2 , (2.40)
16
,
. epicenter ,
. 3.10
associated zero-velocity
curve
, , ,
b/a = (3µ 2 ) 1/2
.
epicenter
(epicyclic) ,
,
2:1
. ,
, 2.6 2
the guideing center approximation
. , 2e,
e
. e ≈ 5 × 10 −5 .
,
, µ 2 ” ” (epicenter
) 2:1
” ”
.
, , ,
epicenter
.
16 (2.40) : (¯x/a) 2 + (ȳ/b) 2 = 1

3. The Restricted Three-Body Problem 32
3.9 Tadpole and Horseshoe Orbits
L 4 , L 5
, .
,
. ,
. (3.2
(3.16) 17 , (3.17) 18 )
3.16 L 4 µ 2 = 0.001 2 ,
.
-
. 3.16a , 3.16b
L 4 .
L 4
. 3.16a
86 , 3.16b 115
. 2 L 4
, 2
, 3.16 ,
µ 1
.
Tadpole .
. Tadpole orbit L 5 .
,
L 4 , L 5
.
, L 4 , L 5
. (Horseshoe orbit)
. 3.17 2
. Taylor (1981)
. 3.17b ,
.
3.16, 3.17 epicenter 3.9
.
3.16: µ 2 = 0.001 , L 4
Tadpole (L 4
x 0 = 1/2 − µ 2 , y 0 = √ 3/2).
µ 1 , µ 2 . (a)
x = x 0 + 0.0065, y = y 0 + 0.0065, ẋ = ẏ = 0 , µ 2 15
. (b) x = x 0 + 0.008, y = y 0 + 0.008, ẋ = ẏ = 0 , µ 2 15.5
.
17 (3.16): ẍ − 2nẏ − n 2 x = −[µ 1 (x + µ 2 )/r 3 1 + µ 2 (x − µ 1 )/r 3 2]
18 (3.17): ÿ + 2nẋ − n 2 y = −[µ 1 /r 3 1 + µ 2 /r 3 2]y

3. The Restricted Three-Body Problem 33
3.17: µ 2 = 0.000953875 2
(Taylor 1981). (a) :
x = −0.97668, ẏ = −0.06118, y = ẋ = 0 (b) : x = −1.02745, ẏ = 0.04032, y =
ẋ = 0.
, 3.10
, µ 2
2
,
. 3.16, 3.17
,
. , L 4 , L 5
,
. , Jacobi L 4 , L 5
,
,
.
Tadpole
. 3.18 , µ 2 = 10 −3 , 3 (
1 , Tadpole 2 )
. θ , , µ 1 µ 2
. 3
e ≈ 0 .
, L 4 , L 5
, Tadpole
.
, L 3
,θ = 180 ,
. r
, a
, e ≈ 0 . 3.16, 3.17
.
µ 2 = 10 −3
a, e 3.19 . µ 2
a .
a a > 1 a < 1 ,
(3.19a). a = 1 + ∆a , θ = 180 ∆a
,
. ∆a = 0.020 . , 1
∆a = −0.0143 . , 2
∆a = 0.0198 ,
. , |∆a|
1
. 3.15 ,

3. The Restricted Three-Body Problem 34
3.18: µ 2 = 10 −3 3
a θ
.
,
.
µ 2 100
.
.
(Dermotto et al., 1980,
Dermott & Marray 1981a). 3.19b µ 2 e , 1
.
, phase effects
(L 4 , L 5 , Tisserand
3.19: 3.18
(a) a . (b) e
. , µ 2 = 10 −3 .

3. The Restricted Three-Body Problem 35
3.20: µ 2 = 10 −3 3
a θ
.
,
. Tadpole
. µ 2 100
.
.
,
). ,
impulse .
3.20, 3.21 . µ 2 = 10 −6
. µ 2
,
a = 1.002 (∆a 1
3.21: 3.20
(a) a . (b) e
. , µ 2 = 10 −6 .

3. The Restricted Three-Body Problem 36
). 3.20 θ a
. a , 3.18 , L 4 , L 5 ( θ = 60 , 300 )
. , 2
. , L 3
, µ 2
, θ = 180 L 3
. , 2 Tadpole , 3.18 , 45 135
, a 3.18
. , Tadpole
, µ 2
. Tadpole ,
,
Tadpole .
µ 2 ,
3.21
.
∆a = 0.00200 , 1
∆a = −0.00199, 2
∆a = 0.00200
.
, µ 2
. e
,
. ,
µ 2
.
,
,
.
L 4 , L 5
. ,
L 3
(Tadpole ).
, L 4 , L 3 , L 5
( ). ,
, epicenter ,
(circulate) . Jacobi
µ 2
.

3. The Restricted Three-Body Problem 37
3.10 Orbits and Zero-Velocity Curves
,
. 3.15
, 30
. ,
, 30 ,
. L 4
x = 1/2 − µ 2 , y = √ 3/2
. (3.8) 19 , (3.9) 20 , x → (1/2 − µ 2 ) + x, y √ 3/2 + y
. , 30
, x → √ 3x ′ /2 + y ′ /2,
y → −x ′ /2 + √ 3y ′ /2 ,
(x ′ , y ′ ) .
,
r 2 1 = 1 + 2y ′ + x ′2 + y ′2 (3.154)
r 2 2 = 1 − √ 3x ′ + y ′ + x ′2 + y ′2 (3.155)
. (3.22) 21
, C J = 2U ,
.
C J = 1 − µ 2 − √ ( 1 −
3µ 2 x ′ + (2 − µ 2 )y ′ + x ′2 + y ′2 µ2
+ 2 + µ )
2
, (3.156)
r 1 r 2
, O(µ 2 2)
. (3.156)
,
.
C J ≈ 3 − µ 2 + 9 4 µ 2x ′2 + 3y ′2 (3.157)
3
, , (i)µ 2
, (ii) 3.9
µ 2 , µ 2 y ′
. µ 2 x
.
. ,
C J = 3 + γµ 2 (3.158)
. γ
, L 4 ,L 5 -1 ((3.97) 22 ,
(3.98) 23 ). (3.157) ,
x ′2
(4/9)(1 + γ) + y ′2
(µ 2 /3)(1 + γ)
= 1. (3.159)
. , (2.40) 24
,
, , a ′ = (2/3) √ 1 + γ, b ′ = √ µ 2 /3 √ 1 + γ , L 4
19 (3.8): r 2 1 = (x + µ 2 ) 2 + y 2 + z 2
20 (3.9): r 2 2 = (x − µ 1 ) 2 + y 2 + z 2
21 (3.22): U = n 2 (x 2 + y 2 )/2 + µ 1 /r 1 + µ 2 /r 2
22 (3.97) : C L4 ≈ 3 − µ 2 ,
23 (3.98) : C L5 ≈ 3 − µ 2
24 (2.40) : ( ) ¯x 2 (
a + ȳ
) 2
b = 1.

3. The Restricted Three-Body Problem 38
. b ′ /a ′ = (1/2) √ 3µ 2
, µ 2 → 0
,
. b ′ /a ′ 3.8 guiding center
2 ,
.
µ 2 ≪ µ 1/2
2 ≪ µ 1/3
2 ≪ 1
. ,
, ,
.
, (3.94)-(3.98) 25
, tadpole
, γ
.
−1 ≤ γ ≤ +1. (3.160)
L 4 , L 5
C J , L 3
.
,
C J = 3 + ζµ 2/3
2 + O(µ 2 ) (3.161)
where
0 ≤ ζ ≤ 3 4/3 . (3.162)
L 3 C J . L 1 , L 2
. ,
γ ζ
tadpole
.
U
(3.64) 26 , n = 1 ,
. (3.28) ,
2U = 2 µ 1
r 1
+ µ 1 r 2 1 + 2 µ 2
r 2
+ µ 2 r 2 2 − µ 1 µ 2 , (3.163)
v 2 = ṙ 2 + (r ˙θ) 2 = 2U − C J (3.164)
µ 2 ≪ 1
, r 1 r
.
,
,
, δr ≪ 1 ,
r = 1 + δr. (3.165)
, ˙θ = 0, ṙ ≠ 0 µ 2
|ṙ| ≪ |r ˙θ| . ,
,
25
(3.94) :C L1 ≈ 3 + 3 4/3 µ 2/3
2 − 10µ 2 /3
(3.95) :C L2 ≈ 3 + 3 4/3 µ 2/3
2 − 14µ 2 /3,
(3.96) :C L3 ≈ 3 ( + µ 2 , ) ( ) 26 1
(3.64) : U = µ 1 r 1
+ r2 1 1
2
+ µ 2 r 2
+ r2 2
2
− 1 2 µ 1µ 2 .
v ≈ r ˙θ = − 3 δr, (3.166)
2

3. The Restricted Three-Body Problem 39
.
, (3.163) 2U
. O(µ 2 )
,
2U = 3 + 3δr 2 + µ 2 H, (3.167)
where
H = 2 r 2
+ r 2 2 − 4, (3.168)
, v 2 = 2U − C J (3.164)
.
9
4 δr2 = 3 + 3δr 2 + µ 2 H − C J . (3.169)
v 2 = 0 ,
.
0 = 3 + 3δr 2 zv + µ 2 H zv − C J (3.170)
zv
,
.
C J
, (3.169), (3.170)
δr 2 = (2δr zv ) 2 + 4 3 µ 2(H zv − H). (3.171)
tadpole ,
, H zv −H ∼ δr, µ 2 δr ≪ δr 2 . guiding
center ,
δr = 2δr zv , (3.172)
, guiding center
2
(Dermott & Murray 1981a).
(3.161), (3.162) ,
3 + 3 4 δr2 + µ 2 H = 3 + ζµ 2/3
2 + O(µ 2 ),
3 + 3δr 2 zv + µ 2 H zv = 3 + ζµ 2/3
2 + O(µ 2 ),
(3.173)
.
, µ 2 2 ≫ µ 2
,
µ 2/3
δr = 2δr zv = 2(ζ/3) 1/2 µ 1/3
2 . (3.174)
tadpole
(3.158), (3.160) ,
3 + 3 4 δr2 + µ 2 H = 3 + γµ 2 , 3 + 3δr 2 zv + µ 2 H zv = 3 + γµ 2 . (3.175)
δr ≈ 2δr zv = 2[(γ − H)/3] 1/2 µ 1/2
2 , (3.176)

3. The Restricted Three-Body Problem 40
. , guiding center
2
.
L 4 , L 5
3.8 . ,δr ≈ 2δr zv ,
guiding center
,
2 .
3 , , (3.153) 27
,
.
X ′ (t) = a sin λ 3 (t − t 3 ) − 2e sin λ 1 (t − t 1 )
Y ′ (t) = (3µ 2 ) 1/2 a cos λ 3 (t − t 3 ) − e cos λ 1 (t − t 3 )
(3.177)
λ 1 , λ 3 2
, 4 a, e, t 1 , t 3 , X ′ (0), Y ′ (0),
Ẋ ′ (0), Ẏ ′ (0)
. 2 , e guiding center
e
. 3 ,
, L 4 , L 5
guiding center
,
.
.
L 4 , L 5
, L 3
.
Jacobi
. (3.173), (3.175)
.
3
4 δr2 + µ 2 H(θ) = constant, (3.178)
θ µ 1
µ 2
, µ 1
2 3
r 2 = 2 sin(θ/2)
,
(
H(θ) = sin
2) θ −1
− 2 cos θ − 2, (3.179)
. (3.178) , tadpole γµ 2 ,
2 .
2 (r i , θ i ), (r j , θ j )
.
.
ζµ 2/3
δr 2 i − δr 2 j = − 4 3 µ 2[H(θ i ) − H(θ j )] (3.180)
tadpole
. 3.22 H(θ)
θ
. tadpole
, δr i = δr j = 0
, H(θ i ) = H(θ j ) .
2 θ min ,
θ max , µ 2
, , 2
D = θ max − θ min
. 3.22 H(θ) , 3.18, 3.20 tadpole
.
27 X(t) ≈3.54 × 10−4 sin 0.268t − 9.85 × 10 −5 sin 0.963t,
Y (t) ≈6.23 × 10 −5 cos 0.268t − 4.86 × 10 −5 cos 0.963t.

3. The Restricted Three-Body Problem 41
3.22: (3.179) H(θ) , tadpole
θ
. L 4 θ , L 5 θ
.
θ i = 180 , δr i = 0
. critical tadpole
. H(180
) = 1 , (3.180)
δr 2 j = 4 3 µ 2[1 − H(θ j )], (3.181)
. H(θ) ≥ −1 tadpole
.
( ) 1/2 8
δr ≤ δr crit = µ 1/2
2 . (3.182)
3
, µ 2
, e
, δr(
µ 2 ) 1
tadpole
.
θ i = 180 δr i = δr 180 ≠ 0
. , ζ ≤ 3 3/4
, δr j = 0, θ j = θ min
.
.
δr 2 180 = 4 3 µ 2[H(θ min ) − 1], (3.183)
,
µ 2
. ,
θ min
µ 2
. tadpole
H(θ min ) = 1
δr 180 = 0, θ min , , µ 2 θ min = 23.5
.

3. The Restricted Three-Body Problem 42
3.11 Trojan Asteroids and Satellites
-
(3.145)
. -
. Table 3.1 -
Tadpole
. Trojan asteroids(
)
.1998
Trojan asteroids
450 , L 4
“Greeks”, L 5
“Trojan” .
3.23 1997 12
-
.(a)
, (b) -
,
. -
, , -
.
,
-
.
, (coorbital satellites) .
coorbital
Janus Epimetheus
,
.
-
-
. tadpole
.
3.9
. (3.174),
(3.176)
∼ µ 1/3
2 tadpole ∼ µ 1/2
2
. 2
R ∼ µ 1/6
2
, R µ 2
. µ 2 ≈ 10 −3 R ≈ 0.3, µ 2 ≈ 10 −9 R ≈ 0.03
.
.
, 3.23: (a)1997 12 18 0:00
.
. (b)
, .
.

3. The Restricted Three-Body Problem 43
,
tadpole
. Dermott & Murray(1981a) 2 µ 2
|∆a|
, , Γ = T/µ 3/5
2
.
3.12 Janus and Epimetheus
tadpole
. , 3.9
, L 4 , L 5 , L 3
.
,
,
.
1980 Voyager I
, Janus Epimetheus 2
.
Janus, Epimetheus , a J = 151472km, a E =
151422km ,
175km, 105km . 2
, ,
1
. Janus-Epimetheus µ 2 = 5 × 10 −9 ,
δr = 3 × 10 −4 , (3.182) δr > δr crit ≈ 17km, (3.174) ζ = 0.02 < 3 4/3
, Epimetheus
.
,
Epimetheus , Janus
(
0.25
.), , 2
. ,
, 2 ,
. 2 ,
2
, Janus Epimetheus . W J , W E Janus , Epimetheus
,
,
, .
m J W J = m E W E (3.184)
3.5
. m 2 < m 1 2 m 1 , m 2 . 2 ,
,
m c
(3.24). C 1 m 1 m c ,
C
, C 1 , C, m 2
. m 2 m c , m 1
. C R 2
T 2 .
2.9 ((2.149) 28 )
. (2.145)
v 29 e 2 = 0
,
ȧ 2 = 2T 2 /n 2 , (3.185)
28 dh
(2.149) :
dt = r ¯T
29 da
(2.145) :
dt = 2 √
a 3/2
[ ¯Re sin f + ¯T (1 + e cos f)].
µ(1−e 2 )

3. The Restricted Three-Body Problem 44
3.24:
m 1 , m 2
. m c . C 1 m 1 m c . C .
a 2 , n 2 m 2
. guiding center epicyclic ,
guiding center
.
,
.
m 2
m c , m 1
. 3.24 ,
T 2 = ( −Gm 1 /r 2 2)
sin γ, +
(
−Gmc /r 2) sin β, (3.186)
C 1 m 2
,
and
sin γ =
sin β =
m c r
cos θ m c + m 1 r 0 2
(3.187)
m 1 r
sin θ (3.188)
m c + m 1 r 0
r
= m [
c + m 1
a ∗ 4 m ( )
1 mc + m 1
sin 2 θ ] −1/2
≈ m c + m 1
, (3.189)
r 0 m c m c m c 2 m c
.
θ , m 1 ≪ m c
,
. T 2 = − ( gm 1 /r 2) ¯H(θ), (3.190)
¯H(θ) =
cos(θ/2)
4 sin 2 − sin θ (3.191)
(θ/2)

3. The Restricted Three-Body Problem 45
¯H(θ) = −(1/2)dH(θ)/dθ . m 1
,
T 1 = + ( gm 2 /r 2) ¯H(θ), (3.192)
.
, 2
.
, (3.185)
s = a 2 − a 1 , (3.193)
ṡ = ˙θds/dθ = −2(T 1 − T 2 )/n, (3.194)
n
. 3 ˙θ/n = −(3/2)s/a. a
. ds/dθ (s i , θ i ), (s j , θ j ) ,
( si
) 2 ( sj
) 2 8
− =
a a 3 g m ∫
1 + m 2
4
θ
n 2 a 3 i θ j ¯H(θ)dθ = −
3 g m 1 + m 2
[H(θ
n 2 a 3 i ) − H(θ j )] , (3.195)
. (3.191) , ¯H(θ) ±60
. n
,
1 r . n n 2 r 3 = g(m c + m 1 + m 2 ). (3.196)
r ≈ a ,
, m c + m 1 + m 2 ≈ m c
, (3.195) ( si
) 2 ( sj
) 2 4 m 1 + m 2
− = − [H(θ
a a 3 n 2 a 3 i ) − H(θ j )] (3.197)
. (3.180) m 2 ≪ m 1
. 2
. (3.182), (3.183) ( ( ) 1/2 ( ) ( s 8 m1 + m 2
=
1/2) (3.198)
a)
crit 3 m c
and
( s180
a
) ( )
2 4 m1 + m 2
= [H(θ min ) − 1]. (3.199)
3 m c
Janus Epimetheus
.
, s j = ∆a 0 = 50km , θ i = 180 . (3.197) m 2 /m 1 = 0.25
,
, s θ
. a J = a + ∆a J , a E = a + ∆a E , (3.184) ,
, ∆a J ∆a E
(
3.25).
Janus
.
,
,
. 3
,
±60
(3.18, 3.20).
, ±60
.

3. The Restricted Three-Body Problem 46
3.25: Janus-Epimetehus
∆a J /∆ 0 , ∆a E /∆ 0 .
∆ 0 = 50km 180
2
.
. Janus
Epimetheus
±60
.
3.26: Janus-Epimetheus 2
.
. ,
∼ 0.25 .

3. The Restricted Three-Body Problem 47
3.27: 2
. Janus-Epimetheus
.
3.26 2
.
. Janus Epimetheus 10 km , 40 km . ,
150432 km .
,
. , 4
, .
, Dermott & Murray (1981b) Janus Epimetheus
,
. (3.199)
. 3.27 2 guiding center
. s 180 /a = 3.32 × 10 −4 .
,
0.001
. Nicholson et
al.(1992)
,
. Janus Epimetheus 5.64
, ,
0.65 ± 0.08g cm −3 , 0.63 ± 0.11g cm −3
.Voyger
,
,
.

3. The Restricted Three-Body Problem 48
3.13 Hill’s Equation
3 ,
(3.9 ).
2
.
,
,
.
, ,
. µ 1 ≈ 1 ,x − y
(3.16)
30 , (3.17) 31 ,
ẍ − 2ẏ − x = − x r 3 1
ÿ + 2ẋ − y = − y r 3 1
x − 1
− µ 2 , (3.200)
r2
3
y
− µ 2 , (3.201)
r2
3
. , x x → x + 1 , ∆ = r 2 .
(
L 1 , L 2 )
, x, y, ∆ O(µ 1/3
2 )
. µ 2
, r 1 ≈ (1 + 2x) 1/2 .
(3.200), (3.201)
ẍ − 2ẏ =
(
3 − µ 2
∆ 3 )
x = ∂U H
∂x , (3.202)
ÿ − 2ẋ = − µ 2
∆ 3 y = ∂U H
∂y , (3.203)
where
U H = 3 2 x2 + µ 2
∆ , and ∆2 = x 2 + y 2 , (3.204)
. Jacobi
,
C H = 3x 2 + 2 µ 2
∆ − ẋ2 − ẏ 2 , (3.205)
. Jacobi (3.29) z
, n = 1
.
(
C J = x 2 + y 2 µ1
+ 2 + µ )
2
− ẋ 2 − ẏ 2 , (3.206)
r 1 r 2
(3.202), (3.203) Hill (Hill’s euation) , Hill
. (3.202) , 3∆ 3 = µ 2 ,
. 30 (3.16): ẍ − 2nẏ − n 2 x = −[µ 1 (x + µ 2 )/r 3 1 + µ 2 (x − µ 1 )/r 3 2]
31 (3.17): ÿ + 2nẋ − n 2 y = −[µ 1 /r 3 1 + µ 2 /r 3 2]y

3. The Restricted Three-Body Problem 49
3.28: µ 2 = 0.1 L 1 , L 2
. Hill ,
µ 2
.
. , Hill (Hill’s sphere)
.
∆ H =
( µ2
) 1/3
. (3.207)
3
(3.202), (3.203) ẋ = ẏ = ẍ = ÿ = 0 (x ≠ 0) , L 1 , L 2
. (3.202) ∆ L1,2 = (µ 2 /3) 1/3 , (3.205) C H = 3 4/3 µ 2/3
2
. L 1 , L 2 (3.207) Hill
. ,
C H = ζµ 2/3
2 (3.208)
, ζ < 3 4/3
. L 1 , L 2
3.28 .
,
.
ẋ = ẍ = ÿ = 0 , x, ẏ
.
ẏ 2 = 3x 2 − ζµ 2/3
2 . (3.209)
, (3.205), (3.208) , , ∆
. , x zv
x
, ẏ = 0
,
x 2 zv = 1 3 ζµ2/3 2 , (3.210)

3. The Restricted Three-Body Problem 50
3.29: Hill
( ) ( )
.
. n 2 a 3 = 1 , ẏ = −(3/2)x
,
x 2 = 4 3 ζµ2/3 2 , (3.211)
, x = 2x zv
2 . 3.10
. 3.29
.
, guiding center
(Dermott & Murray 1981a, b).
, Tisserand
. a 1 = 1 + ∆a 1 ,
e = ∆e 1 ,
a 2 = 1 + ∆a 2 , e = ∆e 2 . ∆a 1 , ∆a 2 ,
∆e 1 , ∆e 2
. 3.4 Tisserand
32 1
1 + ∆a + 2(1 + ∆a)1/2 (1 − ∆e 2 ) 1/2 ≈ constant, (3.212)
.
3
4 ∆a2 − ∆e 2 ≈ constant, (3.213)
or
3
4 ∆a2 1 − ∆e 2 1 ≈ 3 4 ∆a2 2 − ∆e 2 2. (3.214)
32 T isserand :
1
2a + √ a(1 − e 2 ) cos I = 1
2a ′ + √ a ′ (1 − e ′2 ) cos I ′

3. The Restricted Three-Body Problem 51
.
( ∆a 1 ≈ −∆a 2 ) , ∆e 1 ≈
∆e 2 .
Hill
. (3.202), (3.203) ∆
,
, (3.205) (3.208) ,
ẍ − 2ẏ = 3x, (3.215)
ÿ − 2ẋ = 0, (3.216)
ẋ 2 + ẏ 2 = 3x 2 − ζµ 2/3
2 , (3.217)
. guiding center
n = 1,
, x = ∆a + e sin t, ẋ = e cos t, ẍ = −e sin t
. (3.215), (3.216) ÿ = −2e cos t,
ẏ = − 3 ∆a − 2e sin t, (3.218)
2
.
guiding center
,
guiding center
. (3.217) x, ẋ, ẏ ,
e 2 cos 2 t +
(
− 3 2 ∆a − 2e sin t ) 2
= 3(∆a + e sin t) 2 − ζµ 2/3
2 , (3.219)
.
3
4 ∆a2 − ∆e 2 = ζµ 2/3
2 , (3.220)
. .
Tisserand
.
y (
, guiding center ) ,
∆ min
. 3.29 ,
y
. 1 + ∆a 0
. Hill
(x 0 , y 0 ) , (3.218)
(3.205)
ẏ 0 = − 3 2 x 0 = − 3 2 ∆a 0 (3.221)
|∆a 0 | = 2(ζ/3) 1/2 µ 1/3
2 (3.222)
y
, x = 0, ẏ =
0, ẋ = ẋ min , y = ∆ min . ẋ min
ẋ . (3.205)
ẋ min = 2 µ 2
∆ min
− ζµ 2/3
2 . (3.223)

3. The Restricted Three-Body Problem 52
3.30:
Hill
. , L 1 , L 2
(x ′ , y ′ ) = (±1, 0)
.
y ′ = ±200 ẋ ′ = 0
.
.
ẋ 2 min ≪ ẏ0(Dermott 2 & Murray 1981a) ,
y min ≈ (2/ζ) 1/2 µ 1/3
2 , (3.224)
, ∆a
y min ≈ 8 3 ∆2 0µ 2 (3.225)
.
Hill
µ 1/3
2 .
(3.202), (3.203) m 1/3
2 x → x ′ (µ 2 /3) 1/3 y → y ′ (µ 2 /3) 1/3 ∆ → ∆ ′ (µ 2 /3) 1/3
,
.
ẍ ′ − 2ẏ ′ = 3x ′ (
1 − 1
∆ ′3 )
, (3.226)
ÿ ′ + 2ẋ ′ = −3 y′
, (3.227)
∆
′3
.
.
, L 1 , L 2 µ 2
. 3.30
.

3. The Restricted Three-Body Problem 53
,
y ′ . y ′
(| x′ |≪ 1.7)
” ” ,
. , |x ′ |
,
. |x ′ |
,
.

3. The Restricted Three-Body Problem 54
3.14 The Effects of Drag
,
,
.
Poyntingrobertson(PR)
(
) ,
. PR ,
.
, ,
(∼ 10 −6 m).
PR ,
.
F = − βGm c
a 2 r 2 (
ẋ − y + x r 2 (xẋ + yẏ), ẏ + x + y r 2 (xẋ + yẏ) )
(3.228)
β
, m c , a
, . β
. (3.228) r·ṙ
, 2 PR (Schuerman 1980).
Dermott & Gold (1977)
, µ 2 < 10 −10
, .
Dermott et al.(1980)
. PR
(Burns et al. 1979). , 3.31:
. ,
. e ≈ 0 ,
r = 1, θ = 0, 360
.

3. The Restricted Three-Body Problem 55
,
, 1
. (3.31, 3.9 )
L 4 , L 5
, ,
,
,
.
.
,
.
. ,
.
, 2
. (i)
, (ii)
.
,
µ 1 , µ 2 , F = (F x , F y )
. F = (F x , F y )
, k |F| = O(k) .
3.14.1 Analysis of the Jacobi Constant
3.2 ,
.
ẍ − 2ẏ = ∂U
∂x + F x, (3.229)
ÿ + 2ẋ = ∂U
∂y + F y. (3.230)
(3.229) ẋ, (3.230) ẏ
,
(
ẋẍ + ẏÿ − ẋ ∂U
∂x + ẏ ∂U )
= ẋF x + ẏF y . (3.231)
∂y
. , Jacobi C J = 2U − ẋ 2 − ẏ 2
,
dC J
dt
= −2(ẋF x + ẏF y ) (3.232)
. ẋF x + ẏF y
F
.
k < 0
, C˙
J ẋF x + ẏF y
. 3.3
, C J
, L 4 , L 5
.
, F = kv = k(ẋ, ẏ) ,
ẋF x + ẏF y = k(ẋ 2 + ẏ 2 ) < 0. (3.233)
(nebular drag
) ,
, L 4 , L 5
(Jeffreys 1929).

3. The Restricted Three-Body Problem 56
PR
, F = k(ẋ − y, ẏ + x)/r 2 , (3.234)
,
ẋF x + ẏF y = k(ẋ 2 + ẏ 2 ) − k(ẋy − ẏx). (3.235)
. ,
.
,
.
Jacobi
.
3.14.2 Linear Stability of the L 4 and L 5 Points
, . L 4 , L 5 ,
,
. Murray (1994b)
5
k , L 4 , L 5
.
( ) 27
λ 4 + a 3 λ 3 + (1 + a 2 )λ 2 +
4 µ 2 + a 0 = 0. (3.236)
a i (i = 0, 1, 2, 3) O(k) ,
.
a 0 = 9 4 k x,x + 3 4 k y,y ∓ 3√ 3
4 (k x,y + k y,x ) (3.237)
a 1 = 9 4 k x,ẋ + 3 4 k y,ẏ + 2(k x,y − k y,x ) ∓ 3√ 3
4 (k x,ẏ + k y,ẋ ) (3.238)
a 2 = −k x,x − k y,y + (k x,ẏ − k y,ẋ ) (3.239)
a 3 = −(k x,ẋ − k y,ẏ ), (3.240)
(3.237), (3.238)
L 4 , L 5 . ,
k x,x , k y,x · · ·
.
[ ] [ ]
[ ] [ ]
∂Fx
∂Fy
∂Fx
∂Fy
k x,x = , k y,x = , k x,ẋ = , k y,ẋ = ,
∂x
0
∂x
0
∂ẋ
0
∂ẋ
0
[ ] [ ]
[ ] [ ] (3.241)
∂Fx
∂Fy
∂Fx
∂Fy
k x,y = , k y,y = , k x,ẏ = , k y,ẏ =
∂y
0
∂y
0
∂ẏ
0
∂ẏ
0
0 ,
. ,
O(k) ,
, x 0 , y 0
.

3. The Restricted Three-Body Problem 57
3.32: L 4 , L 5
. ,
. (a) (
). (b)guiding center
,
(
).(c)guiding center
,
(
).(d)
( )
(3.236), (3.237) , a i O(µ 2 )
4 ,
((3.141) 33 ).
, (3.145) 34 ,
.
, 3.32
4
.
1.
, .
2. guiding center
, , .
3. guiding center
, , .
4.
, .
Murray (1994b) , µ 2 → 0
λ
,
0 < a 1 < a 3 , (3.242)
. a 1 , a 3 , (3.238), (3.240)
. ,
, a 0 , a 2
, L 4 , L 5
k x,x , k y,y
. L 4 , L 5
2
.
F = k(ẋ, ẏ)
, (3.241)
,
33 (3.141) : λ 4 + λ 2 + 27µ 2 /4(1 − µ 2 ) = 0
34 (3.145) : µ 2 ≤ 27−√ 621
54
≈ 0.0385
k x,ẋ = k, k y,ẏ = k, (3.243)

3. The Restricted Three-Body Problem 58
,
a 1 = 3k, a 3 = −2k, (3.244)
. k < 0 a 1 < 0 , (3.242)
.
, Jacobi
,
.
PR , F = k(ẋ − y, ẏ + x)/r 2 , (3.241)
k x,ẋ = k y,x = k y,ẏ = k, k x,y = −k (3.245)
,
a 1 = −5k, a 3 = −2k, (3.246)
. PR
.
F = k(ẋ − y, ẏ + x)
, (3.241)
,
k x,ẋ = k y,x = k y,ẏ = k, k x,y = −k, (3.247)
,
a 1 = −k, a 3 = −2k. (3.248)
(3.242) ,
. ,
, Tadpole
.
3.14.3 Inertial Drag Forces
Murray (1994b)
.
F i = kVg(x, y, ẋ, ẏ), (3.249)
, k < 0 , V = (ẋ−y, ẏ+x)
. g(x, y, ẋ, ẏ)
. Murray (1994b) inertial
drag . k
r 2 r 3 1r 3 2 + µ 2 r 3 1(µ 1 x − r 2 ) − µ 1 r 3 2(µ 2 x + r 2 ) = 0, (3.250)
. 5
. 3.33a , µ 2 = 0.2
.
µ 2 → 0
, Murray L 3 , L 4 , L 5
, L 1 ,
L 2
. g ∗ = g(x, y, 0, 0) = g ∗ (r)

3. The Restricted Three-Body Problem 59
(
) , µ 2 L 4 , L 5
.
(
)
¯k
1
= sin θ
µ 2 (2 − 2 cos θ) − 1 . (3.251)
3/2
, ¯k = kg ∗ (1) .
3.33b
. θ = 108.4,
θ = 251.6 , ¯k/µ 2 = −0.7265, ¯k/µ 2 = +0.7265
. L 3 ,
L 4 , 180 , 60
¯k/µ 2 = −0.7265 θ = 108.4
. g r
.
, g ∗ r
, L 4 L 5
.
inertial drag
L 4 , L 5
3.14.2
. , Murray (1994b) inertial drag
.
F i = kVV i r i , (3.252)
i, j . Murray , k < 0
.
0 < 1 − i + 2j < 2 + i (3.253)
, i = 0, j = n
,
−1/2 < n < 1/2
L 4 ,
L 5
.
. Murray (1994b) , , L 4 , L 5
.
.
3.33: (a) (3.249) µ 2 = 0.2
. (b)¯k/µ 2 (¯k
,
) L 3 , L 4 , L 5
θ(30¡θ¡330) .
. k < 0
, k > 0
.