# Nonlinear Stability of the Triangular Libration Points for Radiating and Oblate Primaries in CR3BP in Nonresonance Condition

Authors: Nutan Singh; A. Narayan
Journal: Advances in Astronomy
 (2015)

---

## Abstract

This paper investigates the existence of resonance and nonlinear stability of the triangular equilibrium points when both oblate primaries are luminous. The study is carried out near the resonance frequency, satisfying the conditionsω1=ω2,ω1=2ω2, and ω1=3ω2 in circular cases by the application of Kolmogorov-Arnold-Moser (KAM) theory. The study is carried out for the various values of radiation pressure and oblateness parameters in general. It is noticed that the system experiences resonance at ω1=2ω2,ω1=3ω2 for different values of radiation pressures and oblateness parameter. The case ω1=ω2 corresponds to the boundary region of the stability for the system. It is found that, except for some values of the radiation pressure, and oblateness parameters and for μ≤μc=0.0385209, the triangular equilibrium points are stable.


---

## Body

## 1. Introduction

The three-body problem, which describes three masses interacting through Newtonian gravity, has attracted many scientists for more than 300 years. The three-body problem is one of the most challenging problems in the history of science. In celestial mechanics the general three-body problem deals with gravitationally interacting astronomical bodies and intends to predict their motions. The restricted three-body problem (R3BP) is a particular case of the general three-body problem. The problem is restricted in the sense that the infinitesimal mass does not influence the motion of two gravitating primaries but is rather influenced by them. In R3BP, there are two possibilities; namely, the two bodies with dominant masses move around their common centre of mass along either circular or elliptic orbits, which lead to the respective circular or elliptic restricted three-body problems (ER3BP). The ER3BP describes the three-dimensional motion of small particle called the infinitesimal mass under the gravitational force of two finite bodies called primaries around their common centre of mass. The ER3BP generalizes the original circular restricted three-body problems, while some useful properties of circular model still can be satisfied in the elliptical case. The ER3BP describes the dynamical system more accurately as the primaries move along the elliptical orbit.Modern applications of the three-body problem and restricted three-body problem have been extended to include the Earth, the Moon, and the artificial satellites as well as recently discovered exoplanets. The study of the stability of an elliptical or circular restricted three- body problem of the Hamiltonian system is generally performed by using Kolmogorov-Arnold-Moser (KAM) theorem. The KAM theory is very useful for studying the global stability in three-body problem. The KAM theorem is suitable for the motion undergoing small perturbations which preserves the features of the unperturbed motion.The nonlinear stability for resonance as well as for the nonresonance cases of the triangular libration points, taking one of the bodies as radiating, was studied by Manju and Choudhry [1]. Kumar and Choudhry [2] investigated the stability of the triangular libration points for nonresonance as well as resonance case, taking both the bodies as radiating in circular restricted three-body problem in the presence of the third and fourth order resonance. Bhatnagar et al. [3] discussed the nonlinear stability of the triangular equilibrium points in circular restricted three bodies, considering bigger primary as a source of radiation. The nonlinear stability of the triangular Lagrangian points, considering the bigger primary as oblate spheroid in circular case, was examined by Markellos et al. [4]. Recently Narayan and Singh [5] studied the nonlinear stability of higher order for both radiating primaries and found that binary systems are stable.The detailed description and the behavior of equilibrium points in ER3BP are touched upon by Danby [6], Bennett [7], Szebebely [8], Markeev [9], Selaru and Cucu-Dumitrescu [10], and Hallan and Rana [11]. The influence of the eccentricity of the orbits of the primaries with or without radiation pressure on the existence and stability of the equilibrium points was studied by Györgyey [12], Grebenikov [13], Kumar and Choudhry [2, 14], Markellos [4, 15], Sahoo and Ishwar [16], Roberts [17], Zimovshchikov and Tkhai [18], Ammar [19], Érdi et al. [20], Kumar and Ishwar [21], Singh and Umar [22, 23], Usha et al. [24], and Narayan and Singh [5, 25, 26].It may be noted that the case when the frequencies are equal to zero or are equal to each other usually corresponds to the boundary of stability of the linear system, since, in the absence of oblateness parameters and radiation pressures the critical value of mass ratio denoted byμc is 0.0385209. Also, if it is considered that ω1>ω2>0, then it is found that the inequality n1ω1+n2ω2≠0, provided n1+n2≤4 (nj are integers), is invalid for ω1=2ω2, ω1=3ω2 which gives rise to the resonance cases.The present paper investigates the existence of the resonance and the stability of the infinitesimal mass about the triangular equilibrium points by taking both primaries as radiating and oblate in nonresonance condition. The existence of the resonance and the stability of infinitesimal near the resonance frequency satisfying the conditionsω1=ω2, ω1=2ω2, ω1=3ω2 are studied in the circular cases. The study is carried out at various values of radiation pressures and oblateness parameters.This paper has been organized in various sections.Section1 gives introduction, Section 2 describes the equations of motion of the problem, and Section 3 deals with the characteristics roots and first order stability of the triangular equilibrium points. The existence of resonance is discussed in Section 4, while Section 5 deals with normalization and higher order stability of the libration points. Finally Section 6 summarizes the discussion and conclusion of the paper.


## 2. Equations of Motion

The differential equations of the motion of the infinitesimal mass in elliptical restricted three-body problem under radiating primaries in pulsating system as given by Narayan and Shrivastava [27] are(1)x′′-2y′=ϕΩx;y′′-2x′=ϕΩy,where the force function is defined as(2)
Ω
=
x
2
+
y
2
2
+
1
1
+
3
A
1
+
A
2
/
2
1
-
μ
Q
1
r
1
+
μ
Q
2
r
2
+
1
-
μ
Q
1
A
1
2
r
1
3
+
μ
A
2
Q
2
2
r
2
3
,
r
1
2
=
x
+
μ
2
+
y
2
;
r
2
2
=
x
-
1
+
μ
2
+
y
2
;
ϕ
=
1
1
+
e
cos
⁡
f
,where Qi=1-βi  (i=1,2) is the radiation pressure, f is a true anomaly of the primaries, A1, A2 are the oblateness parameter, e is the eccentricity of the orbits, and μ is the mass ratio defined as(3)μ=m2m1+m2;1-μ=m1m1+m2;0≤μ≤12.The coordinates of the triangular equilibrium points L4 and L5 as given by Narayan and Shrivastava [27] are(4)
x
0
=
1
2
-
μ
+
A
1
2
-
A
2
2
+
β
2
3
-
β
1
3
-
1
2
A
1
β
1
+
1
2
·
A
2
β
2
;
y
o
=
±
3
2
1
-
A
1
3
-
A
2
3
-
2
β
1
9
-
2
β
2
9
-
1
3
A
1
β
1
-
1
3
A
2
β
2
.


## 3. Characteristics Roots and First Order Stability of the Triangular Equilibrium Points

The stability of the elliptical restricted three-body problem is restricted to planar case only. Since the nature of stability aboutL4 and L5 is similar it is sufficient to study the stability only about L4. The Hamiltonian as described by Narayan and Shrivastava [27] is given by(5)
H
=
p
1
2
+
p
2
2
2
+
p
1
q
2
-
p
2
q
1
+
q
1
2
+
q
2
2
e
cos
⁡
f
2
1
+
e
cos
⁡
f
-
1
1
+
e
cos
⁡
f
1
+
3
A
1
+
A
2
/
2
1
-
μ
Q
1
r
1
+
μ
Q
2
r
2
+
1
-
μ
Q
1
A
1
2
r
1
3
+
μ
A
2
Q
2
2
r
2
3
,where p1, p2, q1, q2 are the variations in the coordinates L4.By substitutingQi=1-βi  (i=1,2) we have(6)
H
=
p
1
2
+
p
2
2
2
+
p
1
q
2
-
p
2
q
1
+
q
1
2
+
q
2
2
e
cos
⁡
f
2
1
+
e
cos
⁡
f
-
1
1
+
e
cos
⁡
f
1
+
3
A
1
+
A
2
/
2
1
-
μ
1
-
β
1
r
1
+
μ
1
-
β
2
r
2
+
1
-
μ
1
-
β
1
A
1
2
r
1
3
+
μ
A
2
1
-
β
2
2
r
2
3
.Now expand the Hamiltonian function as(7)H=∑K=0∞HK=H0+H1+H2+H3+⋯+H4+⋯,where HK is the sum of the terms of the Kth degree which is homogeneous in the variables p1, p2, q1, q2. H0 is constant and H1=0. Consider (8)H2=p12+p222+p1q2-p2q1+ecos⁡f21+ecos⁡f1+3A1+A2/2q12+q22-121+ecos⁡fH20q12-H11q1q2-H02q22,where (9)
H
20
=
-
-
1
8
+
9
8
A
1
-
9
16
A
2
-
73
24
A
1
β
1
+
11
48
A
2
β
2
-
27
16
μ
A
1
+
27
16
μ
A
2
+
17
8
μ
β
1
-
7
9
μ
β
2
+
β
2
2
-
13
β
1
8
,
(10)
H
11
=
3
3
4
+
17
4
A
1
+
7
8
A
2
+
139
24
A
1
β
1
+
15
8
A
2
β
2
-
41
8
μ
A
1
-
41
8
μ
A
2
-
1
6
μ
β
1
+
μ
β
2
-
3
μ
2
-
β
2
3
-
7
β
1
12
,
(11)
H
02
=
5
8
+
27
8
A
1
+
21
16
A
2
+
107
16
A
1
β
1
+
11
16
A
2
β
2
-
33
8
μ
A
1
+
33
16
μ
A
2
-
7
8
μ
β
1
+
11
8
μ
β
2
-
β
2
2
+
7
β
1
8
,
(12)
H
3
=
-
1
6
1
+
e
cos
⁡
f
1
+
3
A
1
+
A
2
/
2
q
1
3
21
8
+
15
A
1
4
-
295
16
A
2
-
93
β
1
4
-
65
β
2
6
-
365
β
1
A
1
24
-
155
β
1
A
2
16
+
μ
-
57
8
-
15
A
1
4
-
305
A
2
16
-
29
β
1
4
+
51
β
2
8
-
365
β
1
A
1
24
+
85
β
1
A
2
16
-
17
β
2
A
1
8
-
5
β
2
A
2
3
+
3
q
1
2
q
2
-
3
8
+
21
A
1
8
-
61
8
A
2
-
113
β
1
24
+
7
β
2
4
-
53
β
1
A
1
24
+
61
β
1
A
2
8
+
μ
3
17
A
1
16
+
9
A
2
16
+
47
β
1
6
-
13
β
2
4
-
53
β
1
A
1
24
-
113
β
1
A
2
24
-
91
β
2
A
1
16
-
β
2
A
2
48
+
3
q
1
q
2
2
-
33
8
-
315
A
1
8
+
5
12
A
2
+
149
β
1
24
+
5
β
2
8
-
265
β
1
A
1
8
-
5
β
1
A
2
12
+
μ
33
4
-
245
A
1
8
+
263
A
2
8
+
77
β
1
48
-
311
β
2
16
-
55
β
1
A
1
4
+
3805
β
1
A
2
96
+
55
β
2
A
1
8
-
1441
β
2
A
2
16
+
q
2
3
-
9
8
-
51
A
1
4
-
69
16
A
2
-
13
β
1
8
-
11
β
2
4
+
151
β
1
A
1
16
+
65
β
1
A
2
16
+
μ
195
A
1
16
-
33
A
2
2
+
81
β
1
16
+
31
β
2
8
-
151
β
1
A
1
16
+
117
β
1
A
2
32
-
317
β
2
A
1
32
+
5
β
2
A
2
4
,
(13)
H
4
=
-
1
24
1
+
e
cos
⁡
f
1
+
3
A
1
+
A
2
2
q
1
4
-
111
16
-
1155
A
1
16
+
9815
16
A
2
+
397
β
1
2
-
25
β
2
2
-
1235
β
1
A
1
16
-
1185
β
1
A
2
2
+
μ
-
1575
16
-
395
A
1
48
-
29545
A
2
48
+
9269
β
1
16
+
439
β
2
144
-
1234
β
1
A
1
16
-
6325
β
1
A
2
12
+
6755
β
2
A
1
48
+
605
β
2
A
2
48
+
4
3
q
1
3
q
2
-
75
16
-
281
A
1
16
-
21
2
A
2
+
5
β
1
16
+
21
β
2
6
+
351
β
1
A
1
16
+
21
β
1
A
2
2
+
μ
75
8
+
5669
A
1
208
-
117
A
2
8
-
235
β
1
48
-
6121
β
2
560
-
7
β
1
A
1
2
+
319
β
1
A
2
32
+
2357
β
2
A
1
156
-
83
β
2
A
2
4
+
6
q
1
2
q
2
2
123
16
+
4865
A
1
48
-
105
8
A
2
+
235
β
1
12
+
75
β
2
12
-
4295
β
1
A
1
48
+
105
β
1
A
2
8
+
μ
-
123
8
-
5735
A
1
48
+
1601
A
2
16
-
421
β
1
48
-
31
β
2
8
+
4865
β
1
A
1
48
+
245
β
1
A
2
32
+
45
β
2
A
1
8
-
539
β
2
A
2
4
+
4
3
q
1
q
2
3
135
16
+
1531
A
1
16
+
15
4
·
A
2
-
69
β
1
16
-
21
β
2
5
+
23
β
1
A
1
2
-
15
β
1
A
2
4
+
μ
-
135
8
-
337
A
1
2
-
8997
A
2
112
+
259
β
1
16
+
293
β
2
40
+
1391
β
1
A
1
16
-
3005
β
1
A
2
32
+
1315
β
2
A
1
116
-
175
β
2
A
2
48
+
q
2
4
9
16
+
1125
A
1
16
-
1575
32
A
2
+
1781
β
1
16
-
45
β
2
2
-
1045
β
1
A
1
8
+
1575
β
1
A
2
32
+
μ
-
495
A
1
8
-
3167
A
2
32
-
1917
β
1
16
+
26
β
2
+
1045
β
1
A
1
8
-
14765
β
1
A
2
96
-
1325
β
2
A
1
32
-
6775
β
2
A
2
48
.Restricting to H2 alone, the characteristics equation can be given in the form(14)λ2-1-2H20-2λ-H112λ-H11λ2-1+2H02=0,where H20, H11, and H02 are given by ((9), (10), and (11)). After further calculations the characteristics equation reduces to the form(15)λ4+λ2+27μ1-μ4·1-1256A1+403A2+56β1-3518β2=0.If ω1 and ω2 are the frequencies then putting λ2=-ω2 in (15), roots can be written as(16)
ω
1
2
=
-
λ
1,2
2
=
1
2
1
+
1
-
27
μ
1
-
μ
·
1
-
125
6
A
1
+
40
3
A
2
+
5
6
β
1
-
35
18
β
2
1
/
2
,
ω
2
2
=
-
λ
3,4
2
=
1
2
1
-
1
-
27
μ
1
-
μ
·
1
-
125
6
A
1
+
40
3
A
2
+
5
6
β
1
-
35
18
β
2
1
/
2
.The correlation between μ and ω1 and ω2 is shown in Figures 1–5. It is found that for fixed values of radiation pressure ω1 increases with an increase in μ whereas ω2 decreases.Figure 1
Correlation betweenμ and ω1 and ω2 with β1=0.01, β2=0.02, A1=0.01, and A2=0.02.Figure 2
Correlation betweenμ and ω1 and ω2 with β1=0.01, β2=0.02, A1=0.03, and A2=0.04.Figure 3
Correlation betweenμ and ω1 and ω2 with β1=0.01, β2=0.02, A1=0.04, and A2=0.05.Figure 4
Correlation betweenμ and ω1 and ω2 with β1=0.01, β2=0.02, A1=0.05, and A2=0.06.Figure 5
Correlation betweenμ and ω1 and ω2 with β1=0.01, β2=0.02, A1=0.001, and A2=0.002.


## 4. Existence of Resonance in Circular Cases

In order to discuss the existence of resonance, we consider the following three cases.Case 1.
The first case is whenω1=ω2; that is, (17)λ1,22=λ3,42.Solving the above equation we obtain (18)1-27μ1-μ1-1256A1+403A2+56β1-3518β2=0;that is,(19)271-1256A1+403A2+56β1-3518β2μ2-271-1256A1+403A2+56β1-3518β2μ+1=0.Solving for μ we obtain (20)μ=9±691-125/69A1+80/69A2+5/69β1-35/207β218.Since μ≤1/2, the positive sign is inadmissible. Hence the region of stability in the first approximation can be written as(21)0<μ<9-691-125/69A1+80/69A2+5/69β1-35/207β218.Thus, the value of μ responsible for stable equilibrium points is given by (22)μc=0.0385208965+0.83601A1+0.535048A2-0.0334405β1+0.0780279β2.It is clear from (22) that in the absence of radiation pressures and oblateness parameters the critical value of mass ratio is 0.0385208965 when ω1=ω2, which usually corresponds to a boundary of the region of stability of the system.Case 2.
It is whenω1=2ω2; that is,(23)λ1,22=4λ3,42;that is,(24)
1
+
1
-
27
μ
1
-
μ
·
1
-
125
6
A
1
+
40
3
A
2
+
5
6
β
1
-
35
18
β
2
1
/
2
=
4
1
-
1
-
27
μ
1
-
μ
·
1
-
125
6
A
1
+
40
3
A
2
+
5
6
β
1
-
35
18
β
2
1
/
2
.Solving the above equation for μ the resonance value is obtained as (25)μ02=12±14-166751-125A1/6+40A2/3+5β1/6-35β2/18.Case 3.
It is whenω1=3ω2; that is,(26)λ1,22=9λ3,42;that is,(27)
1
+
1
-
27
μ
1
-
μ
·
1
-
125
6
A
1
+
40
3
A
2
+
5
6
β
1
-
35
18
β
2
1
/
2
=
9
1
-
1
-
27
μ
1
-
μ
·
1
-
125
6
A
1
+
40
3
A
2
+
5
6
β
1
-
35
18
β
2
1
/
2
.Solving the above equation for μ the resonance value is obtained as (28)μ03=12±14-96751-125A1/6+40A2/3+5β1/6-35β2/18.Table 1 shows the values of μ corresponding to ω1=ω2, ω1=2ω2, and ω1=3ω2. Figures 6–10 show correlation between μ and A1 by varying β1, β2, and A2. Correlation between μ and β1 is depicted in Figure 11 by varying β2 and taking A1=0.01 and A2=0.02. Figure 12 shows combined figures between μ and β1 by taking β2=0.01, A1=0.01, and A2=0.01, 0.02, 0.03.Table 1
Values ofμ01, μ02, and μ03 for different values of radiation pressure and oblateness parameter.
β
1
β
2
A
1
A
2
μ
01
μ
02
μ
03
0.01
0.01
0.01
0.02
0.0367
0.032
0.0129
0.01
0.01
0.02
0.03
0.0397
0.0250
0.0139
0.01
0.01
0.04
0.05
0.0473
0.0297
0.0165
0.01
0.02
0.01
0.02
0.0374
0.0236
0.0131
0.01
0.02
0.02
0.03
0.0405
0.0255
0.0142
0.01
0.02
0.04
0.05
0.0485
0.0305
0.0169
0.01
0.03
0.01
0.02
0.0367
0.0232
0.0129
0.01
0.03
0.02
0.03
0.0441
0.0278
0.0154
0.01
0.03
0.04
0.05
0.0498
0.0312
0.173
0.01
0.04
0.01
0.02
0.0390
0.0246
0.0137
0.01
0.04
0.02
0.03
0.0423
0.0266
0.0148
0.01
0.04
0.04
0.05
0.0511
0.0321
0.0178
0.01
0.05
0.01
0.02
0.0398
0.0251
0.0139
0.01
0.05
0.02
0.03
0.0433
0.0272
0.0151
0.01
0.05
0.04
0.05
0.0525
0.0329
0.0182
0.01
0.06
0.01
0.02
0.0406
0.0256
0.0142
0.01
0.06
0.02
0.03
0.0443
0.0279
0.0155
0.01
0.06
0.04
0.05
0.0540
0.0338
0.0187
0.01
0.07
0.01
0.02
0.0415
0.0262
0.0145
0.01
0.07
0.02
0.03
0.0453
0.0285
0.0158
0.01
0.07
0.04
0.05
0.0556
0.0348
0.0193
0.01
0.08
0.01
0.02
0.0425
0.0267
0.0149
0.01
0.08
0.02
0.03
0.0465
0.0292
0.0162
0.01
0.08
0.04
0.05
0.0573
0.0358
0.0198
0.01
0.09
0.01
0.02
0.0434
0.0273
0.0152
0.01
0.09
0.02
0.03
0.0476
0.0299
0.0166
0.01
0.09
0.04
0.05
0.0590
0.0369
0.0204
0.87
0.79
0.01
0.02
0.1835
0.1074
0.0572
0.87
0.79
0.02
0.03
0.3131
0.1648
0.0846
0.87
0.79
0.04
0.05
0.5
–
1.1902
i
0.5
–
0.9037
i
0.5
–
0.5916
i
0.72
0.29
0.01
0.02
0.0351
0.0221
0.0123
0.72
0.29
0.02
0.03
0.0378
0.0238
0.0133
0.72
0.29
0.04
0.05
0.0409
0.0258
0.0143
0.62
0.39
0.01
0.02
0.0476
0.0299
0.0166
0.62
0.39
0.02
0.03
0.0527
0.0331
0.0183
0.62
0.39
0.04
0.05
0.0671
0.0418
0.231
0.52
0.49
0.01
0.02
0.0742
0.0461
0.0254
0.52
0.49
0.02
0.03
0.0875
0.0540
0.0296
0.52
0.49
0.04
0.05
0.1367
0.0823
0.0445
0.82
0.29
0.01
0.02
0.0325
0.0205
0.0115
0.82
0.29
0.02
0.03
0.0348
0.0220
0.0122
0.82
0.29
0.04
0.05
0.0405
0.0255
0.0142
0.92
0.29
0.01
0.02
0.0303
0.0192
0.0107
0.92
0.29
0.02
0.03
0.0323
0.0204
0.0114
0.92
0.29
0.04
0.05
0.0371
0.0234
0.0130Figure 6
Correlation betweenμ and A1  and  β1=0.01, β2=0.02, and A2=0.01.Figure 7
Correlation betweenμ and A1  and  β1=0.01, β2=0.02, and A2=0.02.Figure 8
Correlation betweenμ and A1  and  β1=0.01, β2=0.02, and A2=0.03.Figure 9
Correlation betweenμ and A1  and  β1=0.01, β2=0.02, and A2=0.04.Figure 10
Correlation betweenμ and A1  and  β1=0.01, β2=0.02, and A2=0.05.Figure 11
Correlation betweenβ1 and μ for A1=0.01, A2=0.02, and β2=0.02, 0.04, 0.06, 0.08.Figure 12
Combined figure betweenμ and β1 (β2=0.01, A1=0.01, and A2=0.01, 0.02, 0.03).


## 5. Normalization and Higher Order Stability of the Libration Points

In order to investigate the stability the HamiltonianH is normalized by Birkhoff’s method to the following form: (29)H=ω1t1-ω2t2+c20t12+c11t1t2+c02t22+ot1+t25/2,where (30)2ti=pi2+qi2,i=1,2.If H2 is Hamiltonian of 2nd order, defined by (8), (9), (10), and (11) and is of a positive definite form, then the equilibrium position is stable by virtue of Lyapunov theorem [28]. Otherwise the problem of stability can be solved by KAM theorem as given by Arnold [29, 30]. To apply KAM theorem, linear canonical transformation of variations as given by Manju and Choudhry [1] is used which is given as (31)q1,q2,p1,p2=q1′,q2′,p1′,p2′N,where (32)N=a1a1c1-a1c1a11-ω12b1a2a2c2-a2c2a21-ω22b20a1b1a11-b1a1c10-a2b2-a21-b2-a2c2,where(33)a1=122l1ω12-1/21/2;a2=122l2ω22-1/21/2,l1=1+ω12+2H02;l2=1+ω22+2H02b1=2l1;b2=2l2;c1=-H11l1;c2=-H11l2.The transformation (31) reduces the Hamiltonian to the following form:(34)H=12p1′2+ω12q1′2-12p2′2+ω22q2′2+∑α+γ=3∞hα1α2γ1γ2q1′α1q2′α2p1′γ1p2′γ2,whereα=α1+α2;γ=γ1+γ2.The Hamiltonian H3 can be expanded in the following form:(35)H3=H0003p23+H0030p13+H0300q23+H3000q13+H2100q12q2+H2010q12p1+⋯+H1011q1p1p2.Similarly H4 can be expanded as(36)H4=H0004p24+H0040p14+H0400q24+H4000q14+⋯+H1111q1q2p1p2.The coefficient of third and fourth order terms of hα1α2γ1γ2′ and hα1α2γ1γ2 given by Kumar and Choudhry [2] can be listed as (37)
h
0030
=
a
1
3
b
1
3
H
0300
;
h
3000
=
a
1
3
H
3000
+
c
1
H
2100
+
c
1
2
H
2100
+
c
1
3
H
0300
;
h
1020
=
a
1
3
b
1
2
H
1200
+
3
c
1
H
0300
;
h
2010
=
a
1
3
b
1
H
2100
+
2
c
1
H
1200
+
3
c
1
2
H
0300
;
h
0120
=
a
1
2
a
2
b
1
2
H
1200
+
3
c
2
H
0300
;
h
2001
=
-
a
1
2
a
2
b
2
H
2100
+
2
c
1
H
1200
+
3
c
1
2
H
0300
;
h
1011
=
-
2
a
1
2
a
2
b
1
b
2
H
1200
+
3
c
1
H
0300
;
h
1110
=
2
a
1
2
a
2
b
1
H
2100
+
c
1
+
c
2
H
1200
+
3
c
1
c
2
H
0300
;
h
0021
=
-
3
a
1
2
b
1
2
b
2
H
0300
;
h
2100
=
a
1
2
a
2
3
H
3000
+
2
c
1
+
c
2
H
2100
+
c
1
c
1
+
2
c
2
H
1200
+
3
c
1
2
c
2
H
0300
;
h
1002
=
a
1
b
1
2
a
2
2
H
1200
+
3
c
1
H
0300
;
h
0210
=
a
1
a
2
2
b
1
H
2100
+
2
c
2
H
1200
+
3
c
2
2
H
0300
;
h
0012
=
3
a
1
a
2
2
b
2
2
b
1
H
0300
;
h
1200
=
a
2
2
a
1
3
H
3000
+
c
1
+
2
c
2
H
2100
+
c
2
2
c
1
+
c
2
H
1200
+
3
c
1
c
2
2
H
0300
;
h
0111
=
-
2
a
1
2
a
2
2
b
1
b
2
H
1200
+
3
c
2
H
0300
;
h
1101
=
-
2
a
2
2
a
1
b
2
H
2100
+
c
1
+
c
2
H
1200
+
3
c
1
c
2
H
0300
;
h
0102
=
a
2
3
b
21
2
H
1200
+
3
c
2
H
0300
;
h
0201
=
-
a
2
3
b
2
H
2100
+
2
c
2
H
1200
+
3
c
2
2
H
0300
;
h
0102
=
a
2
3
b
2
2
H
1200
+
3
c
2
H
0300
;
h
0003
=
-
a
2
3
b
2
3
H
0300
;
h
0300
=
a
2
3
H
3000
+
c
2
H
2100
+
c
2
2
H
1200
+
c
2
3
H
0300
;
h
0040
=
a
1
4
b
1
4
H
0400
;
h
4000
=
a
1
4
H
4000
+
c
1
H
3100
+
c
1
2
H
2200
+
c
1
3
H
1300
+
c
1
4
H
0400
;
h
2020
=
a
1
4
b
1
2
H
2200
+
3
c
1
H
1300
+
6
c
1
2
H
0400
;
h
0022
=
6
a
1
2
a
2
2
b
1
2
b
2
2
H
0400
;
h
2200
=
a
1
2
a
2
2
6
H
4000
+
3
c
1
+
c
2
H
3100
+
c
1
2
+
4
c
1
c
2
+
c
2
2
H
2200
+
3
c
1
c
2
c
1
+
c
2
H
1300
+
6
c
1
2
c
2
2
H
0400
;
h
2002
=
a
1
2
a
2
2
b
2
2
H
2200
+
3
c
1
H
1300
+
6
c
1
2
H
0400
;
h
0220
=
a
1
2
a
2
2
b
1
2
H
2200
+
3
c
2
H
1300
+
6
c
2
2
H
0400
;
h
0004
=
a
2
4
b
2
4
H
0400
;
h
0400
=
a
2
4
H
4000
+
c
2
H
3100
+
c
2
2
H
2200
+
c
2
3
H
1300
+
c
2
4
H
0400
;
h
0202
=
a
2
4
b
2
2
H
2200
+
3
c
2
H
1300
+
6
c
2
2
H
0400
;
h
0013
=
-
4
a
1
a
2
3
b
1
b
2
3
H
0400
;
h
1300
=
a
1
a
2
3
4
H
4000
+
c
1
+
3
c
2
H
3100
+
4
c
1
c
2
3
H
0400
+
2
c
2
c
1
+
c
2
H
2200
+
c
2
2
3
c
1
+
c
2
H
1300
;
h
1102
=
a
1
a
2
3
b
2
2
2
H
2200
+
3
c
1
+
c
2
H
1300
+
12
c
1
c
2
H
0400
;
h
0211
=
-
2
a
2
3
a
1
b
1
b
2
H
2200
+
3
c
2
H
1300
+
6
c
2
2
H
0400
;
h
0112
=
3
a
1
a
2
3
b
1
b
2
2
H
1300
+
4
c
1
H
0400
;
h
1003
=
-
a
1
a
2
3
b
2
3
H
1300
+
4
c
1
H
0400
;
h
1201
=
-
a
1
a
2
3
b
2
3
H
3100
+
2
c
1
+
2
c
2
H
2200
+
3
c
2
2
c
1
+
c
2
H
1300
+
12
c
1
c
2
2
H
0400
;
h
0310
=
a
1
a
2
3
b
1
H
3100
+
2
c
2
H
2200
+
3
c
2
2
H
1300
+
4
c
2
2
H
0400
.Again the Hamiltonian is reduced to a more convenient form which is suitable for further investigation by using the following canonical transformation:(38)q1′=12q1′′+iω1p1′′;p1′=12iω1q1′′+p1′′;q2′=-12q2′′+1ω2p2′′;p2′=-12iωq2′′+ip2′′.Thus, the Hamiltonian (34) may be written as(39)H=iω1q1′′p1′′+iω2q2′′p2′′+∑α+γ=3∞hα1α2γ1γ2′q1′′α1q2′′α2p1′′γ1p2′′γ2.As explained by Kumar and Choudhry [2], consider the following.Ifhα1α2γ1γ2′=xα1α2γ1γ2+iyα1α2γ1γ2 then (40)x0030=h0030-1ω12h2010;y0030=1ω1h1020-1ω13h3000;x1020=-12h1020-32ω12h3000;y1020=3ω12h0030+12ω1h2010;x0120=-ω22h0021+12ω1h1110+ω22ω1h2001;y0120=-12h0120-ω22ω1h1011+12ω12h2100;x1011=-ω1h0021-1ω1h2001;y0021=h0021+1ω1ω2h1110-1ω12h2001;x1002=-ω12ω2h0111-12h1002+12ω22h1200;y1002=-ω12h0012+ω12ω12h0210+12ω22h1101;x0012=-h0012+1ω22h0210-1ω1ω2h1101;y0012=1ω2h0111-1ω1h1002+1ω1ω22h1200;x0111=ω2ω1h1002+1ω1ω2h1200;y0111=-ω2h0012-1ω2h0210;x0201=-ω24h0102-34ω2h0300;y0201=-3ω224h0003+14h0201;x0003=-1ω2h0102+1ω23h0300;y0003=-h0003+1ω22h0201.The other ten coefficients hα1α2γ1γ2′ of third order terms of (39) are obtained by the formula(41)hα1α2γ1γ2′=yα1α2γ1γ2+ixα1α2γ1γ2-ω12γ1-α1-ω22γ2-α2.If we denote (α1-γ1)+(α2-γ2)=aαγ and consider α1-γ1+α2-γ2=3 and if it is found that aαγ ≠ an integer, then it implies that resonance of the third order is absent (Kumar and Choudhry [14]).Using Birkhoff’s transformation(qj′′,pj′′)→(qj′′′,pj′′′), all the third order terms from the Hamiltonian (39) are nullified provided third order resonance does not occur. This transformation is introduced by means of the generating function which is given as follows:(42)s=q1′′p1′′′+q2′′′p2′′′+s3+s4,where(43)qi′′′=qi′′+∂s3∂pi′′′+∂s4∂pi′′′;pi′′=pi′′′+∂s3∂qi′′+∂s4∂qi′′;i=1,2.From (39) and (43), expanding and equating the terms of the same degree on the two sides, we obtain(44)H2′q1′′,q2′′,p1′′′,p2′′′=H2q1′′,q2′′,p1′′′,p2′′′;H3′q1′′,q2′′,p1′′′,p2′′′=∑i=12-∂s3∂pi′′′∂H2′∂pi′′′+∂s3∂qi′′∂H2∂pi′′′+H3q1′′,q2′′,p1′′′,p2′′′;∑i=12-∂s4∂pi′′′∂H2′∂qi′′′+∂s4∂qi′′∂H2∂pi′′′+K4=0;H4′=∑i=12-∂s3∂pi′′′∂H3′∂qi′′+∂s3∂qi′′∂H3∂pi′′′+H4-K4,where K4 is the term other than the homogeneous ones in q1p1 and q2p2. In (44) the new variables q1′′′ and p2′′′can be replaced by q1′′ and p2′′ on both sides of (44) by implicit function theorem.Since our system is autonomous∂s3/∂t=∂s4/∂t=0.If we put(45)H3=∑α+γ=3∞hα1α2γ1γ2q1′′′α1q2′′′α2p1′′′γ1p2′′′γ2,s3=∑α+γ=3∞gα1α2γ1γ2q1′′′α1q2′′′α2p1′′′γ1p2′′′γ2,using (44), we get(46)gα1α2γ1γ2=ihα1α2γ1γ2′α1-γ1ω1+α2-γ2ω2.With the help of third equation of (44), the new Hamiltonian, inclusive of the fourth order terms, is given as(47)H′=iω1q1′′′p1′′′+iω2q2′′′p2′′′-c20q1′′′p1′′′2+c11q1′′′p1′′′q2′′′p2′′′-c02q2′′′p2′′′2+⋯,where(48)K4=H4q1′′′,q2′′′,p1′′′,p2′′′-h2020′q1′′′2p1′′′2-h1111′q1′′′p1′′′q2′′′p2′′′-h0202′q2′′′2p2′′′2;c20=-h2020′-3ω128x00302+y00302-32x10202+y10202+12x10112+y10112-ω122ω22ω1-ω2x01202+y01202+ω12ω282ω1+ω232x00212+y00212,c11=-h1111′-2ω22ω1ω1-2ω2x10022+y10022+ω1ω222ω1+2ω2x00122+y00122-ω2ω1222ω1+ω2x00212+y00212-2ω122ω1-ω2x01202+y01202+2x0111x1020+y0111y1020-4ω2x0201y1011+x1011y0201;c02=h0202′+3ω228x00032+y00032+6ω22x02012+y02012-ω222ω1ω1-2ω2x10022+y10022-12x01112+y01112-ω1ω228ω1+2ω2x00122+y00122;h2020′=-32ω12h0040-32ω12h4000-12h2020;h1111′=ω1ω2h0022+1ω1ω2h2200+ω1ω2h0220+ω2ω1h2002;h0202′=-32ω22h0004-32ω22h0400-12h0202;now, KAM theorem is applied which is best suited for the problem which is being defined and is stated as follows.If the Hamiltonian of the perturbed motion satisfy the following given conditions mentioned below as:(
1
) The characteristics equation of the system with H2 has pure imaginary roots and the frequencies ω1, ω2 satisfy the inequality(49)n1ω1+n2ω2≠0;whenn1+n2≤4njare integers.(
2
)  Consider D=c20ω22+c11ω1ω2+c02ω12≠0, as given by Kumar and Choudhry [2].If all the above mentioned conditions are satisfied, then the equilibrium points are stable.The value ofD is calculated with the help of the following formula:(50)D=c20ω22+c11ω1ω2+c02ω12,where c20, c11, and c02 are defined in (48).Table2 shows the values D for different values of radiation pressures and oblateness parameters.Table 2
Values ofD  at different values of eccentricity for different values of radiation pressure and oblateness parameter.
μ
β
1
β
2
A
1
A
2
Values ofD
0.001
0.87
0.29
0.01
0.02
−3.1202 × 109
0.001
0.87
0.29
0.02
0.03
−4.8371 × 109
0.001
0.87
0.29
0.04
0.05
−1.1754 × 107
0.005
0.82
0.29
0.01
0.02
−2.7755 × 107
0.005
0.82
0.29
0.02
0.03
−4.2670 × 107
0.005
0.82
0.29
0.04
0.05
−6.5897 × 108
0.010
0.72
0.29
0.01
0.02
−3.8443 × 106
0.010
0.72
0.29
0.02
0.03
−5.8280  × 106
0.010
0.72
0.29
0.04
0.05
−1.3814  × 107
0.015
0.62
0.39
0.01
0.02
−1.5405 × 106
0.015
0.62
0.39
0.02
0.03
−2.4736 × 106
0.015
0.62
0.39
0.04
0.05
−6.9668 × 106
0.025
0.52
0.49
0.01
0.02
−5.7810 × 105
0.025
0.52
0.49
0.02
0.03
−1.0457  × 106
0.025
0.52
0.49
0.04
0.05
−4.5781 × 106
0.035
0.92
0.29
0.01
0.02
−7.8501 × 104 +1.1316 × 104i
0.035
0.92
0.29
0.02
0.03
−1.4121 × 105 +1.4800 × 104i
0.035
0.92
0.29
0.04
0.05
−9.9125 × 104
0.038
0.72
0.49
0.01
0.02
6.8892 × 105
0.038
0.72
0.49
0.02
0.03
−4.8483 × 106
0.038
0.72
0.49
0.04
0.05
−2.3721 × 106
0.038
0.62
0.39
0.01
0.02
6.9437 × 104
0.038
0.62
0.39
0.02
0.03
3.2725 × 105
0.038
0.62
0.39
0.04
0.05
−2.5116 × 106
0.001
0.01
0.01
0.01
0.02
−3.2004 × 107
0.001
0.01
0.01
0.02
0.03
−5.4067 × 107
0.001
0.01
0.01
0.04
0.05
−1.5692 × 108
0.005
0.01
0.02
0.01
0.02
−3.1222 × 105
0.005
0.01
0.02
0.02
0.03
−5.2414 × 107
0.005
0.01
0.02
0.04
0.05
−1.5044 × 106
0.010
0.01
0.03
0.01
0.02
−3.0522 × 105
0.010
0.01
0.03
0.02
0.03
−8.5874 × 104
0.010
0.01
0.03
0.04
0.05
−2.3880 × 105
0.015
0.01
0.04
0.01
0.02
−2.2203 × 104
0.015
0.01
0.04
0.02
0.03
−3.5166 × 104
0.015
0.01
0.04
0.04
0.05
−9.2124 × 104
0.025
0.01
0.05
0.01
0.02
−4.5169 × 105
0.025
0.01
0.05
0.02
0.03
−3.3881 × 104
0.025
0.01
0.05
0.04
0.05
−3.2969 × 104
0.030
0.01
0.06
0.01
0.02
3.4841 × 104
0.030
0.01
0.06
0.02
0.03
1.7250 × 104
0.030
0.01
0.06
0.04
0.05
−4.8836 × 104
0.035
0.01
0.07
0.01
0.02
492.1240
0.035
0.01
0.07
0.02
0.03
2.5112 × 104
0.035
0.01
0.07
0.04
0.05
5.7581 × 105
0.038
0.01
0.08
0.01
0.02
38.4651
0.038
0.01
0.08
0.02
0.03
1.0408 × 103
0.038
0.01
0.08
0.04
0.05
3.9251 × 104


## 6. Discussion and Conclusion

The stability of the triangular equilibrium points in CR3BP is investigated when both primaries are radiating and oblate, under the nonresonance case using KAM theory. Recently Narayan and Singh [5] studied nonlinear stability considering both primaries as radiating in ER3BP and found that the binary systems are stable. It has been observed that in general the stability character remains the same even if oblateness factor is considered apart from radiation factor in circular cases. The following observation has been made assuming both primaries as radiating and oblateness:(i)
It is found thatω1=ω2 corresponds to the boundary region of the stability for the system, whereas the other two cases ω1=2ω2 and ω1=3ω2 correspond to the resonant cases. It has been observed that resonance of the third and the fourth order exists for all values of μ, β1, β2, A1, and A2 taken.(ii)
It is noticed that for fixed values of radiation pressures and by varying oblateness parametersω1 increases by increasing μ whereas ω2 decreases and becomes equal to the critical value (i.e., μ=μc).(iii)
From Table2 it is clear that for μ=0.038, β1=0.72, β2=0.49, A1=0.04, A2=0.05 and β1=0.62, β2=0.39, A1=0.01, A2=0.02, there is change in the sign of value of D.This change in sign suggests that for fixed μ=0.038 and by slightly changing radiation pressure and oblateness parameters D vanishes, where KAM theory is not applicable. It is also concluded that except at this point the equilibrium points are stable. Consider the following:(i)
It is clear from Table2 that for μ=0.025, β1=0.01, β2=0.05, A1=0.04, A2=0.05 and μ=0.030, β1=0.01, β2=0.06, A1=0.01, A2=0.02 again there is change in sign of D. This suggests that for slight variation in radiation pressure and oblateness parameters and for 0.025≤μ≤0.030, the value of D must be zero.(ii)
It is observed that forμ=0.030, β1=0.01, β2=0.06, A1=0.02, A2=0.03 and μ=0.030, β1=0.01, β2=0.06, A1=0.04, A2=0.05D changes sign. This implies that when μ=0.030, β1=0.01, and β2=0.06 and for slight variation in oblateness parameters, D vanishes.(iii)
Also, it is found that whenμ=0.038, β1=0.62, β2=0.39, A1=0.02, A2=0.03 and μ=0.038, β1=0.01, β2=0.06, A1=0.04, A2=0.05, there is change in the sign of D.From the above discussion the following conclusion can be drawn.In all the above four cases considered, it is noticed that KAM theory fails. From the above discussion it is also clear that by varying one of the parameters and taking another parameter as fixed, there may be many cases where KAM theory fails and their motion can be found numerically which needs further investigation. It is observed that, except for some values of radiation pressure and oblateness parameters and forμ<0.0385209 the triangular equilibrium points are stable.



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*Source: 850252-2015-08-24.xml*