Journal of Institute of Science and Technology, 2014, 19(2): 76-78, © Institute of Science and Technology, T.U.
The Stability of Solutions of Sitnikov Restricted Problem of three Bodies When the Primaries are Triaxial Rigid Bodies R.R. Thapa Dept. of Mathematics, P.G.Campus, Biratnagar, Morang, Nepal. Email:
[email protected] ABSTRACT The paper deals with the stability of the solutions of Sitnikov's restricted problem of three bodies if the primaries are triaxial rigid bodies. The infinitesimal mass is moving in space and is being influenced by motion of two primaries (m1>m2). They move in circular orbits without rotation around their centre of mass. Both primaries are considered as axis symmetric bodies with one of the axes as axis of symmetry whose equatorial plane coincides with motion of the plane. The synodic system of co-ordinates initially coincides with inertial system of co-ordinates. It is also supposed that initially the principal axis of the body m1 is parallel to synodic axis and are of the axes of symmetry is perpendicular to plane of motion. Keywords-Axis symmetric bodies; equatorial plane; infinitesimal body; Libration points; synodic axes. INTRODUCTION The Sitnikov problem is a special case of restricted three body problem .It refers to the motion of the test particle along an axis perpendicular to the plane of motion of two equal primaries that move on elliptic orbits. The axis passes through the center of mass of the system.
But
48 2 a 5 10( a7 8 2 a 3 a5
2 0
2 6 a 5 10( 1 3 a2 a2 1 2 a
The stability of motion in the Sitnikov problem have studied by Soulis (2007) and found that as mass of infinitesimal body increases, the domain of allowed motion grows significantly. This paper tries to establish the solutions of Sitnikov restricted problem of these bodies when both the primaries are triaxial rigid body. Now following Thapa and Hassan (2013)
c5 23cos 1024 7
c 2048
cos 3
297 cos 3 8
2 2 0
2 0 2
24 cos 3
cos 5
3cos 5
cos 7 16
547c 7 cos 32768
2 0
2 0
3
and
...
10(
6 2 a 5 a4 6 2 a 5 a4
(1)
2 0
)
)
1
3 a2
10(
1
1
) 1
3 a2
10(
1
1
1
4
2 ) 10(
) 3q
1
) 2
1
1
1
1
1
...
15 2 a2
30 ( a2
1
1
)
1
1
)
)
2
1
216 2 720 (a 6 ) ( a8 a4
76
1
1) 1
36 4 ( a 4a 2 a8
6 2 (a a4
)
1
1
36 2 720 (a 4 ) ( a6 a6
3
3 2 0
6 2 a 5 a2
6 2 (a 2 ) 10( a4
Solution by Lindstedt-Poincare Method z ccos
1
2 48 a 5 a 5 10( a7 8 a2 3
The restricted problem of three bodies if primaries are oblate spheroids where equatorial plane coincides with plane of motion and their stability has been studied by Vidyakin (1974). Subba Rao and Sharma (1975) have studied the stability of libration points. EI-Shaboury (1991) also studied the stability of libration points. Khanna and Bhatnagar (2001) have studied the stability of solutions of Sitnikov restricted problem of the smaller primary is a triaxial rigid body.
c3 cos 32
1
36 2 (a 2 ) 10( a8
) 36 2 (a a6
)
1
)
4 )
720 ( a6
1
1
)
R.R. Thapa
From (1) 3
3c cos 16a 2
z ccos
5c cos 16
cos 3
24 cos 3
c5 15 23cos a 2 128
24 cos 3
1
)
cos 5 (
cos 3
)
1)
1
1
1
c5 144 (2Cos 23cos a 6 1024
c 7 216 6 a8 2048
2nx
(
1
1
24 cos 3
cos 5
3cos 5
1 cos 7 16
z
Since only hence,
a2 4
2
= 2
1
a2 4
2
2
a2 4
4 z2
1
a2 4
z
a2 4
2
3 2
1
4 z
(2)
3 4z2
a2 4
4 z
45( 7 2
4 z2
5 2
2z
7
3 5 2
a2 4
2
15 z 2
z2 2
3z
3z 2 3 2
1
1
z
)
3
z2
1
y
4 z2
1
a2 4
7 2 z z 2
2
...
4 z
3
1
7 2
a2 4
2z
5
a2 4
2
3 2
a2 4
5 2 z z 2 z2
15(
)
x
3 2
z
2z
3
a2 4 5 2
a2 4
2
1 2
a2 4
3 2 z z 2 z2
zz
1 2
3 2
a2 4
4
297 cos 3 8
1
0
y z z
0
x z
)
When
Where
2
3 4
z
a2 4
x y
2
2
0,
)
1 cos 7 16
3cos 5
2 xy
z
z2
0,
2
z2
Stability of the motion The general equations of motion of the infinitesimal mass under the mutual gravitational field of two axis symmetric bodies (triaxial rigid bodies) are
y
2
2
1 cos 7 16
3cos 5
297 cos 3 8
547 216 6 c 7 cos 32768 a8
x
0,
cos 5
393840 c 7 cos ( 32768 a 4
297 cos 3 8
c3 3 (Cos cos a4 8
1
2
2
14769 c 7 cos 4096 a 6
c 7 720 a 4 2048
cos 3 (
x y
c5 9 23cos a 4 256
c 7 216 a 6 2048
2
Also
3
1
)z2
a2 4
2
15 z 2 5 2
a2 4
105(
7 2
4 z
3
4 z2
5 2
a2 4
2
1
a2 4
3
1
1 2
1
a2 4
105(
7 2
4 z2
1
)z4 9 2
1
a2 4
)z4 9 2
Let us write the equations as
z2 5 2
x 2 xy 0 y 2nx 0 z
z
2
f ( x, y , z ) g ( x, y , z )
z
is independent of x and y, but depending upon z
3 z 3 2
a2 4
2
4 z
2
a2 4
15( 5 2
4 z
1 2
1
a2 4
) z3 7 2
h( x, y, z ). x y
0
For the conditions of stable solution, the square of characteristic roots must be less then or equal to zero.
0 z
z
z2
3 z 2
a 4
3 2
4 z2
15( 2
a 4
5 2
1
4 z2
1)z 2
a 4
2 i
i.e.
3
But
7 2
2 1
2 2
When 77
for i 1, 2,3.
0
0
11
4n 2 2 3
21 0 zz
.
12
2ni,
0. 22
2ni.
The Stability of Solutions of Sitnikov Restricted Problem of three Bodies When the Primaries are Triaxial Rigid Bodies
Now, 3 4 z02
1
0 zz
a2 4
z02
3 2
1 a 2
3
4z 1 a 1
8 1 a3
1
2
3
1
z04 1
3
1
9 2
When a 2 18 z02 3 2
1 7 2
2 0 2
4z 1 a
1
4
8 a3
1
96 zo a3 a
1 2
4 z02 a2
3
z0 a
1
24 a5
1 14
a
z0 a
1 10
2
2
...
a2
1
2
2
1
5
2
z0 a
2
...
3
8 2 a 18 z02 a5
3 2
1
2
2
1
a2 3 2
1
2
2
1
2
18
18
18
3 18
to
a2
3 18
, so that the condition of
EI-Shaboury, S.M. 1991. Equilibrium solutions of the restricted problem of 2+2 axisymmetric rigid bodies. Celest. Mech. Dyn. Astron. 50: 199-208. 1
2
2
z0 a
2
Khanna, M. and Bhatnagar, K.B. 2001. Existence of Libration points in the restricted three body problem when the primaries are triaxial rigid bodies. Indian J. pure appl. Math. 32: 125-141. Soulis, P.S. 2007. Stability of motion in the Sitnikov problem. Celest. Mech. Dyn. Astron. 99: 129-148. Subba Rao, P.V. and Sharma, R.K. 1975. A note on stability of the triangular points of equilibrium in the restricted three body problem. Astronomy and Astrophysics 43: 381-383. Thapa, R.R. and Hassan, M.R. 2013. Proceedings: Modern trends in science and Technology. Nepal Physical Society and Research Council of Science and Technology, Biratnagar, Nepal, 129-148p. Vidyakin, V.V. 1994. Criterion of instability of translational – rotational motion of perfectly rigid bodies, Soviet Astronomy 36: 214.
...
24
8 2 a 18 z02 a5
z0
REFERENCES
24 a5
1 8a 2 144 z02 a5
2
The author is grateful to respected guide Dr. M.R. Hassan, Dept. of Mathematics, S.M. College, Bhagalpur, T.M.B. University, Bhagalpur, 812007, India, for his inspiration to complete this work.
...
...
1
2
ACKNOWLEDGEMENTS
2
2
480 5 a5
0
stability are satisfied by the solution of the Sitnikov motion.
...
10 a2
2
2
If the primaries are triaxial rigid bodies then the motion of the third body depends on parameter , distance ‘a’ between the primaries, constant c and . In this case, the infinitesimal 2 1 2 2 1 2 mass m3 will perform oscillatory motion in between
2
z0 a
1
CONCLUSION
9 2
2
2
144 z0 a3 a
2
2
2
2
18
of mass 0.
Neglecting higher order infinitesimal above the second order 0 zz
1
1
Thus the infinitesimal mass will perform oscillatory a2 3 motion from to a 2 3 about the centre
5 2
7 2
4 z02 a2
1
z 1 18 0 a
z 24 4 0 a3 a 3
1
1
a
3
1
z0 a
1
2
2
a2 3 2
z02
4 z02 a2
2
z 24 4 0 a3 a
1
48 z0 a3 a
32 4 zo2 a5
1
3
105 128 a5
48 z0 a3 a
3 4
3
29 a9
15 32 z0 a5 a
8 a3
a2 4
18
128 a7
480 z0 a5 a
z02
z04
a2 3 2
3 2
2
8 a3
a 2
1
9 2
4 z02 a2
48 z0 a3 a
8 a3
4z 1 a
3 7
1
z )
4 z02 a2
1
15 2 z0 4 105 4
a 2
a2 4
15 2 z0 4
5 2
2 0 2
105 4
1 7 2
4 0
1
9
5
3
1
z02
4 z02
3 4
3 2
2 0 2
105 ( 4 a 2
a2 4
4 z02
3
15 2 z0 4
5 2
0 Hence if a 2 18 z02 3 0 then 0 and 32 0 , zz both are imaginary. Thus all the characteristic 31 & 32 roots of the coefficient matrix A are either zero or imaginary. Hence the matrix of Sitnikov is stable.
2
78
