International Journal of Astronomy and Astrophysics, 2017, 7, 45-61
http://www.scirp.org/journal/ijaa
ISSN Online: 2161-4725
ISSN Print: 2161-4717
Existence of Equilibrium Points in the R3BP
with Variable Mass When the Smaller Primary
is an Oblate Spheroid
M. R. Hassan1, Sweta Kumari2, Md. Aminul Hassan3
Department of Mathematics, S. M. College, T. M. Bhagalpur University, Bhagalpur, India
Research Scholar, T. M. Bhagalpur University, Bhagalpur, India
3
GTE, Bangalore, India
1
2
How to cite this paper: Hassan, M.R.,
Kumari, S. and Hassan, Md.A. (2017) Existence of Equilibrium Points in the R3BP with
Variable Mass When the Smaller Primary is
an Oblate Spheroid. International Journal of
Astronomy and Astrophysics, 7, 45-61.
https://doi.org/10.4236/ijaa.2017.72005
Received: February 25, 2017
Accepted: April 9, 2017
Published: April 12, 2017
Copyright © 2017 by authors and
Scientific Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
Abstract
The paper deals with the existence of equilibrium points in the restricted
three-body problem when the smaller primary is an oblate spheroid and the
infinitesimal body is of variable mass. Following the method of small parameters; the co-ordinates of collinear equilibrium points have been calculated,
whereas the co-ordinates of triangular equilibrium points are established by
classical method. On studying the surface of zero-velocity curves, it is found
that the mass reduction factor has very minor effect on the location of the
equilibrium points; whereas the oblateness parameter of the smaller primary
has a significant role on the existence of equilibrium points.
Keywords
Restricted Three-Body Problem, Jean’s Law, Space-Time Transformation,
Oblateness, Equilibrium Points, Surface of Zero-Velocity
1. Introduction
Restricted problem of three bodies with variable mass is of great importance in
celestial mechanics. The two-body problem with variable mass was first studied
by Jeans [1] regarding the evaluation of binary system. Meshcherskii [2] assumed that the mass was ejected isotropically from the two-body system at very
high velocities and was lost to the system. He examined the change in orbits, the
variation in angular momentum and the energy of the system. Omarov [3] has
discussed the restricted problem of perturbed motion of two bodies with variable
mass. Following Jeans [1], Verhulst [4] discussed the two body problem with
slowly decreasing mass, by a non-linear, non-autonomous system of differential
DOI: 10.4236/ijaa.2017.72005 April 12, 2017
M. R. Hassan et al.
equations. Shrivastava and Ishwar [5] derived the equations of motion in the
circular restricted problem of three bodies with variable mass with the assumption that the mass of the infinitesimal body varies with respect to time.
Singh and Ishwar [6] showed the effect of perturbation on the location and
stability of the triangular equilibrium points in the restricted three-body problem. Das et al. [5] developed the equations of motion in elliptic restricted problem of three bodies with variable mass. Lukyanov [7] discussed the stability of
equilibrium points in the restricted problem of three bodies with variable mass.
He found that for any set of parameters, all the equilibriums points in the problem (Collinear, Triangular and Coplanar) are stable with respect to the conditions considered in the Meshcherskii space-time transformation. El Shaboury [8]
discussed the equation of motion of Elliptic Restricted Three-body Problem
(ER3BP) with variable mass and two triaxial rigid bodies. He applied the Jeans
law, Nechvili’s transformation and space-time transformation given by Meshcherskii in a special case.
Plastino et al. [9] presented techniques for the problems of Celestial Mechanics, involving bodies with varying masses. They have emphasized that Newton’s
second law is valid only for the body of fixed masses and the motion of a body
losing mass is isotropically unaffected by this law. Bekov [10] [11] has discussed
the equilibrium points and Hill’s surface in the restricted problem of three bodies with variable mass. He has also discussed the existence and stability of equilibrium points in the same problem. Singh et al. [12] has discussed the non-linear
stability of equilibrium points in the restricted problem of three bodies with variable mass. They have also found that in non-linear sense, collinear points are unstable for all mass ratios and the triangular points are stable in the range of linear stability except for three mass ratios which depend upon β , the constant due to the
variation in mass governed by Jean’s law.
At present, we have proposed to extend the work of Singh [12] by considering
smaller primary as an oblate spheroid in the restricted problem of three bodies
as shown in Figure 1 and to find the co-ordinates of equilibrium points
Li ( i = 1, 2,3, 4,5 ) by the method of small parameters.
2. Equations of Motion
Let m be the mass of the infinitesimal body varying with time. The primaries
of masses µ and 1 − µ are moving on the circular orbits about their centre of
mass as shown in Figure 1. We consider a bary-centric rotating co-ordinate system
( O, xyz ) , rotating relative to inertial frame with angular velocity
ω . The
line joining the centers of µ and 1 − µ is considered as the x -axis and a line
lying on the plane of motion and perpendicular to the x -axis and through the
centre of mass as the y -axis and a line through the centre of mass and perpen-
dicular to the plane of motion as the z -axis. Let ( µ , 0, 0 ) and ( µ − 1, 0, 0 )
respectively be the co-ordinates of the primaries P1 and P2 and ( x, y, z ) be
the co-ordinates of the infinitesimal mass P . The equation of motion of the in-
finitesimal body of variable mass m can be written as
46
M. R. Hassan et al.
Figure 1. Rotating frame of reference in the R3BP in 3-Dimension about Z-axis.
1 − µ
d  dr 
µ
9 Aµ 
−Gm  3 ρ1 + 3 ρ2 +
ρ2  ,
m  =
dt  dt 
ρ2
2 ρ 25 
 ρ1
where
(1)
d ∂
=
+ ω × is the defined operator under consideration,
dt ∂t
dr ∂r
=
+ ω× r .
dt ∂t
(2)
The oblateness parameter of the smaller primary is given by
A=
a2 − c2
,
5R 2
where a and c are the equatorial and polar radii of the oblate primary, R
is the dimensional distance between the primaries,
ρ12 = ( x − µ ) + y 2 + z 2 ,
2
2
2
2
and ρ 2 = ( x − µ + 1) + y + z .
2
Now from Equation (1),
1 − µ
9 Aµ 
µ
∂
 ∂r

ρ2  ,
 + ω ×  m + mω × r  =−Gm  3 ρ1 + 3 ρ2 +
2 ρ 25 
ρ2
 ∂t
 ∂t

 ρ1
1 − µ
µ
9 Aµ  
⇒ mr + m ( r + ω × r ) + 2m=
ω × r ω 2 m xiˆ + yjˆ − Gm  3 ρ1 + 3 ρ2 +
ρ2  
ρ2
2 ρ 25 
 ρ1

 2
(1 − µ )( x − µ ) iˆ + yjˆ + zkˆ µ

2 ˆ
ˆ
ˆ
ˆ
ˆ
= m ω xi + ω yj −
− 3 ( x − µ + 1) i + yj + zk
 (3)
3
ρ1
ρ2




Aµ

ˆ
ˆ
ˆ
− 9 5 ( x − µ + 1) i + yj + zk  ,

2 ρ2


(
)
{
{
}
}
47
M. R. Hassan et al.
where units are so chosen that the sum of the masses of the primaries and the
gravitational constant G both are unity.
The equations of motion in the Cartesian form are
x +
m
1 ∂U
−
( x − ω y ) + 2ω x =
m
m ∂x
y +
m
1 ∂U
−
( y + ω x ) − 2ω y =
m
m ∂y
z +
m
1 ∂U
z = −
m
m ∂z

,


,


,

(4)
where
 ω2
1 − μ μ 3 Aμ 
U=
−m  ( x 2 + y 2 ) +
+ + 3 .
ρ1
ρ2 2 ρ2 
 2
(5)
(1 − µ )( x − µ ) µ ( x − µ + 1) 9 Aµ ( x − µ + 1) 
1 ∂U
=
−
−
,
ω2 x −
m ∂x
2 ρ 25
ρ13
ρ 23


(1 − µ ) y µ y 9 Aµ y
1 ∂U

−
= ω2 y −
−
−
,

m ∂y
2 ρ 25
ρ13
ρ 23


and


2
z
1
−
µ
(
) µ 9 Aµ z
1 ∂U

.
−
=
−
−
−

m ∂z
ρ13
ρ 23 2 ρ 25
(6)
i.e.,
−
By Jeans law, the variation of mass of the infinitesimal body is given by
dm
m
= −α m n i.e.,
= −α m n −1 ,
dt
m
(7)
where α is a constant coefficient and the value of exponent n ∈ [ 0.4, 4.4]
for the stars of the main sequence.
Let us introduce space time transformations as
=
x ξγ − q =
, y ηγ − q =
, z ζγ − q ,=
dt γ − k dτ , 

m

−q
=
ρ1 r1γ − q , ρ=
r
γ
,
=
γ
<
1,
2
2

m0

(8)
where m0 is the mass of the satellite at t = 0 .
From Equations ((7) and (8)), we get
dγ
= − βγ n −1 ,
dt
where
=
β α=
m0n −1 constant .
Differentiating x, y and z with respect to t twice, we get
=
x ξ ′γ k − q + β qξγ n − q −1 ,
=
y η ′γ k − q + β qηγ n − q −1 ,
=
z ζ ′γ k − q + β qζγ n − q −1 ,
48
(9)
M. R. Hassan et al.

=
x ξ ′′γ 2 k − q + βξ ′ ( 2q − k ) γ n + k − q −1 − β 2 qξ ( n − q − 1) γ 2 n − q − 2 ,
y η ′′γ 2 k − q + βη ′ ( 2q − k ) γ n + k − q −1 − β 2 qη ( n − q − 1) γ 2 n − q − 2 ,
=
z ζ ′′γ 2 k − q + βζ ′ ( 2q − k ) γ n + k − q −1 − β 2 qζ ( n − q − 1) γ 2 n − q − 2 .
=
Also,
m
n −1
=
−α m n −1 =
−α ( m0γ ) =
−α m0n −1γ n −1 ,
m
m
i.e.,
= − βγ n −1 .
m
Now,
1 ∂U
1 ∂U ∂ξ
γ q ∂U
γ q −1  ∂U 
=
−
=
−
=
−

,
m ∂x
m0γ ∂ξ ∂x
m0γ ∂ξ
m0  ∂ξ 
1 ∂U
γ q −1 ∂U
−
=
−
,
m ∂y
m0 ∂η
1 ∂U
γ q −1 ∂U
−
=
−
.
m ∂z
m0 ∂ς
−
Putting the values of x , y , z, 
x, 
y, 
z,
m
∂U ∂U ∂U
and
in Equation
,
,
m
∂x ∂y ∂z
(4), we get

ξ ′′ + ( 2q − k − 1) βξ ′γ n − k −1 − 2ωη ′γ − k − ( n − q ) β 2 qξγ 2( n − k −1) 


γ 2 q − 2 k −1 ∂U 
n − q −1
,
− ( 2q − 1) βωηγ
 = − m
∂ξ 
0
η ′′ + ( 2q − k − 1) βη ′γ n − k −1 + 2ωξ ′γ − k − ( n − q ) β 2 qηγ 2( n − k −1) 


γ 2 q − 2 k −1 ∂U 
+ ( 2q − 1) βωξγ n − q −1  = −
,
m0 ∂η 

2 q − 2 k −1
∂U 
ζ ′′ + ( 2q − k − 1) βζ ′γ n − k −1 − ( n − q ) β 2 qζγ 2( n − k −1)  =− γ
,


m0 ∂ζ 
(10)
where
1
1 − µ q +1 µ q +1 3 Aµ 3q +1 
U=
γ + γ + 3 γ
−m0  ξ 2 + η 2 ω 2γ 1− 2 q +
.
r1
r2
2r2
2

(
)
(11)
In order to make the Equation (10) free from the non-variational factor, it is
sufficient to put
n − k − 1= 0, 2q − k − 1= 0, n= 1 ∈ [ 0.4, 4.4] , 

1

i.e., k = 0, q = , n = 1 and β = α

2
(12)
Thus the System (10) reduces to
1 ∂U α 2 
+
ξ ,
m0 ∂ξ
4 
1 ∂U α 2 
+ η, 
η ′′ − 2ωξ ′ = −
m0 ∂η
4 

1 ∂U α 2
+
ζ ′′ = −
ζ, 
m0 ∂ζ
4

i.e. ξ ′′ − 2ωη ′ = −
(13)
49
M. R. Hassan et al.
where
1
(1 − µ ) 32 µ 32 3 Aµ 52 
U=
γ + γ + 3 γ .
− m0  ξ 2 + η 2 ω 2 +
r1
r2
2r2
2

(
)
(14)
From System (13),
−
(
)
(1 − µ ) ξ − µ γ 32
1 ∂U
=
ω 2ξ −
γ
m0 ∂ξ
r13




µ ξ − µ γ + γ 3 9 Aµ ξ − µ γ + γ 5 
−
γ2 −
γ 2 ,

r23
2r25

3
3
5

(1 − µ )η 2 µη 2 9 Aµη 2
2
=
ω η−
γ − 3 γ −
γ ,

3
5
r1
r2
2r2


3
3
5
1
−
µ
ζ
(
) 2 µζ 2 9 Aµζ 2

=
−
γ
−
γ
−
γ
.

r13
r23
2r25
(
−
1 ∂U
m0 ∂η
−
1 ∂U
m0 ∂ζ
)
(
)
(15)
The Jacobi’s Integral is
ω 2
ξ ′2 + η ′2 +=
ζ ′2 2 
 2
+
α2
4
(ξ 2 + η 2 ) +
3
3
5
1 − µ 2 µ 2 3 Aµ 2 
γ + γ + 3 γ 
r1
r2
2r2

(ξ
2
2
+η + ζ
2
) + C.
(16)
3. Existence of Equilibrium Points
′′ η=
′′ ζ=
′′ 0, then
For the existence of equilibrium points ξ=′ η=′ ζ=′ ξ=
from Systems (13) and (15)
(
)
(1 − µ ) ξ − µ γ 23
α2

γ
+ ω2 ξ −

r13
 4

−
(
µ ξ −µ γ + γ
r23
)γ
3
2
−
(
9 Aµ ξ − µ γ + γ
2r25
)γ
5
2
0,
=
α2
(1 − µ )η 23 µη 23 9 Aµ 52
2
ω
η
γ − 3 γ − 5 ηγ =
0,
+
−


r13
r2
2r2
 4

α2 
(1 − µ ) ζ 32 µζ 32 9 Aµ 52
ζ
γ − 3 γ − 5 ζγ = 0.
and 
−

r13
r2
2r2
 4 
For solving the above equations, let us change these equations in Cartesian
form as
α2

(1 − µ )( x − µ ) µ ( x − µ + 1) 9 Aµ ( x − µ + 1) 
+ ω2  x −
−
−
=
0, 

ρ13
ρ 23
2 ρ 25
 4



µ
1
y
−
α2

(
)
µ
µ
y
9
A
y

0,
+ ω2  y −
−
−
=

3
3
5
4
ρ
ρ
ρ
2



1
2
2

and


α2 
(1 − µ ) z µ z 9 Aµ z

0.
−
−
=

z−
ρ13
ρ 23 2 ρ 25

 4 
50
(17)
M. R. Hassan et al.
4. Existence of Collinear Equilibrium Points
For the Collinear equilibrium points, y= z= 0, then
ρ1 = x − µ , ρ 2 = x − µ + 1 .
From Equation (17), we get
α2

(1 − µ )( x − µ ) µ ( x − µ + 1) 9 Aµ ( x − µ + 1)
+ ω2  x −
−
−
=
0 . (18)

3
3
5
x−µ
x − µ +1
2 x − µ +1
 4

Let L1 (ξ1 , 0, 0 ) be the first collinear equilibrium point lying to the left of the
second primary M 2 ( µ − 1, 0, 0 ) i.e., ξ1 < µ − 1 as shown in Figure 2 then
ξ1 < µ ,
⇒ ξ1 − µ + 1 < 0 and ξ1 − µ < 0,
⇒ ξ1 − µ + 1 = − (ξ1 − µ + 1) and ξ1 − µ = − (ξ1 − µ ) .
Thus from Equation (18),
α2

(1 − µ )
9 Aµ
µ
+ ω 2  ξ1 +
+
+
=
0.
(19)

2
2
4
(ξ1 − µ ) (ξ1 − µ + 1) 2 (ξ1 − µ + 1)
 4

as ξ1 < µ − 1 , so let ξ1 = µ − 1 − ρ where ρ is a small quantity.
For the first equilibrium point L1 (ξ1 , 0, 0 ) , we have
α2

(1 − µ ) µ 9 Aµ
⇒
+ ω2  ( µ −1− ρ ) +
+ 2+
=0,
2
4
( −1 − ρ ) ρ 2 ρ
 4

α2

(1 − µ ) µ 9 Aµ
⇒
+ ω2  ( µ −1− ρ ) +
+ 2+
=0,
2
4
(1 + ρ ) ρ 2 ρ
 4

α2

2
2
+ ω 2  ( µ − 1 − ρ )(1 + ρ ) ρ 4 + 2 (1 − µ ) ρ 4
 4

(20)
+2 µρ (1 + ρ ) + 9 Aµ (1 + ρ ) =
0.
2
2
2
Here, Equation (20) is seven degree polynomial equation in ρ , so there are
seven values of ρ . If we put µ = 0 then from Equation (20), we get
α2

3
−2 
+ ω 2  (1 + ρ ) ρ 4 + 2 ρ 4 =
0.
4


(21)
Figure 2. Locations of collinear and triangular equilibrium points.
51
M. R. Hassan et al.
Thus ρ = 0, 0, 0, 0 , are four roots of Equation (21) when µ = 0 , so
1
 14 
=
=
ρ 4 o ( µ ) i.e.,
ρ o  µ=
υ µ4 ⇒ =
µ υ4 .
 o (υ ) i.e., =
 
Thus the Equation (20) reduces to
α2

2
+ ω 2  υ 4 − 1 − ρ (1 + ρ ) ρ 4 + 2 1 − υ 4 ρ 4
2
4


2
2
0.
+2υ 4 ρ 2 (1 + ρ ) + 9 Aυ 4 (1 + ρ ) =
(
)
(
)
(22)
Let ρ =a1υ + a2υ 2 + a3υ 3 + a4υ 4 + a5υ 5 + a6υ 6 + a7υ 7 +  where
a1 , a2 , a3 , a4 , a5 , a6 , a7  are small parameters, then
ρ 2 = a12υ 2 + 2a1a2υ 3 + ( a22 + 2a1a3 )υ 4 + 2 ( a1a4 + a2 a3 )υ 5


+ a + 2a1a5 + 2a2 a4 υ + 2 ( a2 a5 + a1a6 + a3 a4 )υ +  ,

3
3 3
2
4
2
2
5
3
2
6
ρ = a1 υ + 3a1 a2υ + 3 a1a2 + a1 a3 υ + a2 + 3a1 a4 + 6a1a2 a3 υ 

+ 3 a1a32 + a22 a3 + a12 a5 + 2a1a2 a4 υ 7 +  ,


ρ4 =
a14υ 4 + 4a13 a2υ 5 + 2 3a12 a22 + 2a13 a3 υ 6


3
3
2
7
+ 4 a1a2 + a1 a4 + 3a1 a2 a3 υ +  ,


ρ5 =
a15υ 5 + 5a14 a2υ 6 + 5 2a13 a22 + a14 a3 υ 7 +  ,


ρ6 =
a16υ 6 + 6a15 a2υ 7 +  ,

7
7 7
=
ρ
a1 υ + .

(
2
3
(
(
)
6
)
(
(
7
)
)
(
(
)
)
(23)
)
Putting the value of ρ , ρ 2 , ρ 3 , ρ 4 , ρ 5 , ρ 6 , ρ 7 , in Equation (22) and
equating the co-efficient of different powers of υ to zero, we get the values of
the parameters as







 (24)




1


− α 2 + 6 A + 4  a17 − 9a15 a2 + 15a14 a3 + 30a13 a22 + 6a12 a2 a3 − 2a1a23  , 
 
2

a5 , a6 , a7 , ,

1
 18 A  4
=
a1  2
 ,
α − 6 A 
=
a2 Q 18 Aa1 − 3 α 2 + 6 A + 4 a15  ,


3

a=
Q ( 9 A + 2 ) a12 + α 2 + 4ω 2  a16 − 6a12 a22 − 15a14 a2  + 12a12 a22  ,
3
2



a4 = Q  4a13 + 2 ( 9 A + 2 ) a1a2 + 18 Aa2
)
(
(
(
)
)
where Q = −
(
1
)
2 α + 6 A a13
2
.
Therefore, the co-ordinate of the first equilibrium point L1 (ξ1 , 0, 0 ) is given
by
ξ1 = µ − 1 − ρ
= µ − 1 − ( a1υ + a2υ 2 + a3υ 3 + a4υ 4 + a5υ 5 + a6υ 6 + )
1
2
3
5
6


= µ − 1 − a1 µ 4 + a2 µ 4 + a3 µ 4 + a4 µ + a5 µ 4 + a6 µ 4 + 


∞
n
ξ1 = µ − 1 − ∑ an µ 4 .
n =1
52
M. R. Hassan et al.
Let L2 (ξ 2 , 0, 0 ) be the second collinear equilibrium point between the two
primaries P1 and P2 then µ − 1 < ξ 2 < µ
⇒ ξ 2 − µ + 1 > 0 and ξ 2 − µ < 0.
⇒ ξ 2 − µ + 1 = ξ 2 − µ + 1 and ξ 2 − µ = − (ξ 2 − µ ) .
Thus from Equation (18),
α2

1− µ
µ
9 Aµ
0.
+ ω 2  ξ2 +
−
−
=
(25)

2
2
4
4
(ξ 2 − µ ) (ξ2 − µ + 1) 2 (ξ2 − µ + 1)


Since ξ 2 > µ − 1 hence let ξ 2 = µ − 1 + ρ , thus
ξ 2 − µ + 1 = ρ ⇒ ξ 2 − µ = −1 + ρ ,
where ρ is a small quantity.
In terms of ρ , the Equation (25) can be written as
α2

1− µ
µ 9 Aµ
+ ω2  ( µ −1+ ρ ) +
− 2 − 4 =0,

2
( ρ − 1) ρ ρ
 4

i.e.,
α2

2
+ ω 2  ( µ − 1 + ρ ) ρ 4 ( ρ − 1) + 2 (1 − µ ) ρ 4
2
 4

−2 µρ
2
( ρ − 1)
2
(26)
− 9 Aµ ( ρ − 1) =
0.
2
The Equation (26) is a seven degree polynomial equation in ρ , so there are
seven values of ρ in Equation (26).
If we put µ = 0 in Equation (26), we get
α2

3
+ ω 2  ρ 4 ( ρ − 1) + 2 ρ 4 =
2
0.
 4

(27)
Here ρ 4 = 0 i.e., ρ = 0, 0, 0, 0 are the four roots of Equation (27) when
µ = 0 , so ρ we can choose as some order of µ i.e.,
 14 
=
ρ 4 o ( µ ) ⇒=
ρ o  µ=
 o (υ ) ,
 
1
where µ 4 = υ i.e., µ = υ 4 .
Let ρ =b1υ + b2υ 2 + b3υ 3 + b4υ 4 + b5υ 5 + b6υ 6 + b7υ 7 +. where
b1 , b2 , b3 , b4 , b5 , b6 , b7  are small parameters. Putting the values of
ρ 1 , ρ 2 , ρ 3 , ρ 4 and µ = υ 4 in Equation (26) and equating the coefficients of
different powers of υ , we get
1

9A

4

b1 =  −
,

2

 2 +α + 6A 

2
4
b2 R 3 α + 4 + 6 A b1 + 18 A ,
=



 (28)
2
6
4
2 2
2 2
2


b3 R α + 4 + 6 A −3b1 + 15b4 b2 − 6b1 b2 + 12b1 b2 − ( 9 A + 2 ) b1 + 18 Ab2 , 
=



 α 2 + 4 + 6 A b17 − 18b15b2 + 15b14 b3 + 30b12 b22 − 4b1b23 − 12b12 b2 b3 



b4 = R
,
3
2
3

 +8 b1b2 + 3b1 b2 b3 + 4b1 − 2 ( 9 A + 2 ) b1b2 + 18 Ab3


(
(
(
(
)
)(
)(
)
)
)
53
M. R. Hassan et al.
where R =
(
1
.
)
4 α + 2 + 6 A b13
2
Thus the co-ordinate of the second equilibrium point is given by
ξ 2 =−
µ 1 + b1υ + b2υ 2 + b3υ 3 + b4υ 4 + b5υ 5 + b6υ 6 + b7υ 7 + 
1
2
3
5
6
= µ − 1 + b1 µ 4 + b2 µ 4 + b3 µ 4 + b4 µ 1 + b5 µ 4 + b6 µ 4 + 
∞
n
ξ 2 = µ − 1 + ∑ bn µ 4 .
n =1
Let L3 (ξ3 , 0, 0 ) be the third equilibrium point right to the first primary, then
ξ3 > ξ 2 = µ − 1 − ρ ,
⇒ ξ3 > µ − 1 + ρ ,
⇒ ξ3 = µ − 1 + 2 ρ
Thus from Equation (18), we have
α2

1− µ
µ
9 Aµ
+ ω 2  ( µ − 1 + 2ρ ) −
−
−
=0,

2
2
4
( 2 ρ − 1) ( 2 ρ ) 2 ( 2 ρ )
 4

α2

1− µ
µ
9 Aµ
+ ω 2  ( µ − 1 + 2ρ ) −
− 2−
=0,

2
4
4
ρ
32
ρ4
( 2 ρ − 1)



α2

2
+ ω 2  ( 2 ρ − 1 + µ )( 2 ρ − 1) ρ 4 − 32 (1 − µ ) ρ 4 
32 

 4

2
2

2
0.
−8µρ ( 2 ρ − 1) − 9 Aµ ( 2 ρ − 1) =

(29)
When µ = 0 , then Equation (29) reduced to
α2

3
+ ω 2  ( 2 ρ − 1) ρ 4 − 32 ρ 4 =
32 
0.
 4

(30)
ρ 4 = 0 i.e., ρ = 0, 0, 0, 0 are the four roots of the Equation (29) when µ = 0 ,
 1
4
4
so ρ=
O(µ ) ⇒ =
ρ O  µ 4=
 O (υ ) say when µ = υ .
 
2
Let ρ =c1υ + c2υ + c3υ 3 + c4υ 4 +
where c1 , c2 , c3 , c4 , are small parameters.
Thus Equation (29) reduced to
(
8 α 2 + 4ω 2
)( 2ρ − 1 + υ ) ( 2ρ − 1)
4
2
ρ 4 − 32 (1 − υ 4 ) ρ 4
0
−8υ 4 ρ 2 ( 2 ρ − 1) − 9 Aυ 4 ( 2 ρ − 1) =
2
2
By putting values of ρ , ρ 2 , ρ 3 , ρ 4 , ρ 5 , ρ 6 , ρ 7 in Equation (29) and equating
the co-efficient of different powers of υ , we get







2
2
2
2
4
2
2
2
=
c3 S 16 α + 4ω c1 15c1 c2 − 6c1 − 3c2 − 192c1 c2 − 3 Ac2  ,
 (31)

32c13 − 3 Ac3 − 80 Ac1c2 − 384 c1c23 + 3c12 c2 c3
 
 ,
c4 = S 
 −16 α 2 + 4ω 2 2c1c23 + 6c12 c2 c3 − 15c14 c3 − 30c12 c22 + 36c15 c2 − 4c17  

and so on


1

4
9A
 ,
c1 =  −
 8 α 2 + 4ω 2 + 4 


2
2
5

=
c2 S  48 α + 4ω c1  ,
(
)
)
) (
(
(
(
54
)(
(
)
)
)
M. R. Hassan et al.
where S =
(
1
)
32 α + 4ω 2 + 4 c13
2
.
Thus the co-ordinates of the third equilibrium point is given by
∞
∞
n
ξ3 = µ − 1 − 2 ρ = µ − 1 + 2∑ cnυ n = µ − 1 + 2∑ cn µ 4
=
n 1=
n 1
5. Existence of Triangular Equilibrium Points
For triangular equilibrium point x ≠ 0, y ≠ 0 and z = 0 then from the System
(17), we have
a0 x −
(1 − µ )( x − µ ) µ ( x − µ + 1)
−
ρ
3
1
a0 −
ρ
1− µ
ρ13
−
3
2
−
9 Aµ ( x − µ + 1)
2 ρ 25
µ 9 Aµ
−
=
0.
ρ 23 2 ρ 25
=
0.
(32)
(33)
Now Equation ( 32 ) − ( 33) × ( x − µ + 1) gives
a0 −
1
ρ
3
2
1
= 0 i.e.,
ρ 23
= a0 .
(34)
Again Equation ( 32 ) − ( 33) × ( x − µ ) gives
1
ρ
3
1
+
9A
1
9A
− a0 = 0 i.e., 3 + 5 = a0 .
5
2 ρ2
ρ2 2 ρ2
(35)
Since 0 < A  1 , hence for the first approximation, if we put A = 0 , then
from Equations ((34) and (35)), we get
1
1
α2
ρ1
ρ2
4
=
3
= a0 = 1 +
3
 α2 
⇒ ρ1 = ρ 2 = 1 +

4 

−
,
1
3
= 1−
1 2
α < 1.
12
For better approximation A ≠ 0 , then the above solutions can be written as
α2
α2
+ α1 and ρ 2 =−
1
+ α 2 where 0 < α1 , α 2  1 .
12
12
For triangular equilibrium points z = 0 , then
ρ1 =−
1
ρ12 = ( x − µ ) + y 2 and ρ 22 = ( x − µ + 1) + y 2 .
2
2
Now,
ρ 22 = ( x − µ ) + y 2 + 2 ( x − µ ) + 1,
2
ρ 22 − ρ12 = 2 ( x − µ ) + 1.
1 
⇒ ( x − µ ) = α 2 − α1 − , 
2 

1
⇒ x = µ − + α 2 − α1 

2
(36)
Again,
55
M. R. Hassan et al.
ρ12 =( x − µ ) + y 2 ,
2
2

α2  
1
y 2 = 1 + α1 −
 −  α 2 − α1 −  ,
12
2

 
2
3
α
y 2 = + α1 + α 2 −
,
4
6
2
3 2
2α 2 
y=
±
1 + (α1 + α 2 ) −
.
2  3
9 
(37)
Putting the value of ρ 2 and a0 in Equation (34), we get α 2 = −
the value of ρ 2 and a0 in Equation (35), we get
A
. Putting
2
15
α2
.
A−
2
3
α1 =
1+
Thus,
α 2 + α1 = 1 + 7 A −
α2
3
and α 2 − α1 = −1 − 8 A +
α2
3
.
(38)
Therefore,
x= µ−
3
α2
,
− 8A +
2
3
 2α 2 
,
−
9 

3 2
4α 2 
±
y=
1 + (1 + 7 A ) −
.
2  3
9 
±
y=
α2
3  2
1 + 1 + 7 A −
2  3 
3

3
α2
3 2
4α 2  
L4,5
, ±
=  µ − − 8 A +
1 + (1 + 7 A ) −

2
3
2  3
9  

6. Surface of Zero–Velocity
Figure 3. Zero velocity curve (ZVC) for C = 1 (classical case).
56
(39)
(40)
M. R. Hassan et al.
Figure 4. Zero velocity curve for C = 2 (classical case).
Figure 5. Zero velocity curve for C = 3 (classical case).
Figure 6. Zero velocity curve for C = 1 (perturbed case).
57
M. R. Hassan et al.
Figure 7. Zero velocity curve for C = 2 (perturbed case).
Figure 8. Zero velocity curve (ZVC) for C = 3 (perturbed case).
Figure 9. 3 Dimensional view of ZVC of Figure 6.
58
M. R. Hassan et al.
Figure 10. 3 Dimensional view of ZVC of Figure 7.
Figure 11. 3 Dimensional view of ZVC of Figure 8.
7. Discussions and Conclusions
In section 2, the equations of motion of the infinitesimal body with variable
mass have been derived under the gravitational field of one oblate primary and
other spherical. By Jean’s law, the time rate mass variation is defined as
dm
= −α m n , n ∈ [ 0.4, 4.4] , where α is a constant and the interval [ 0.4, 4.4]
dt
in which exponent of the mass of the stars of the main sequence lies. The System
(4) is transformed to space-time co-ordinates by the space-time transformations
given in Equations (8) and (9). The Jacobi’s integral has been derived in Equation (16).
59
M. R. Hassan et al.
In section 3, the equations for solving equilibrium points, have been derived
′′ 0, η=′ η=
′′ 0, ζ=′ ζ=
′′ 0 in Equation
in Equation (17) by putting ξ=′ ξ=
(13). Again the equations for equilibrium points, have been transformed to
original frame
( 0, xyz )
which are given in Equation (18). In section 4, for col-
linear equilibrium points we put y= z= 0 , then from Equation (17), we get
only one Equation (19). Applying small parameter method, we established the
1
co-ordinates of L1 , L2 , L3 as ξ1 , ξ 2 , ξ3 in terms of order of µ 4 . In section 5,
the co-ordinates of equilateral triangular equilibrium points have been calculated by the classical method. In section 6, zero-velocity curves in Figures 3-8
and its 3-dimensional surface in Figures 9-11 have been drawn for=
C 1,=
C 2
and C = 3 in classical case and perturbed case.
From the above facts we concluded that in the perturbed case, first equili-
brium point L1 (ξ1 , 0, 0 ) shifted away from the second primary P2 ( µ − 1, 0, 0 )
L2 (ξ 2 , 0, 0 ) shifted towards the first primary P1 ( µ , 0, 0 ) but
L3 (ξ3 , 0, 0 ) is not influenced by the perturbation which can be seen in Figures
6-8. So far, the matter is concerned with the influence of perturbation on the
co-ordinates of L4 and L5 , we can say that for C = 1 and C = 3 , the triangular equilibrium configuration is maintained but for C = 2 in both classical
and perturbed cases, the equilateral triangular configuration is not maintained.
Whatever be the analytical changes in the co-ordinates of L1 , L2 , L3 , L4 and L5
that is due oblateness not due to the mass reduction factor α of the infinitesimal body.
whereas
References
60
[1]
Jeans, J.H. (1928) Astronomy and Cosmogony. Cambridge University Press, Cambridge.
[2]
Meshcherskii, L.V. (1949) Studies on the Mechanics of Bodies of Variable Mass,
Gostekhizdat, Moscow.
[3]
Omarov, T.B. (1963) The Restricted Problem of Perturbed Motion of Two Bodies
with Variable Mass. Soviet Astronomy, 8, 127.
[4]
Verhulst, F. (1972) Two-Body Problem with Slowly Decreasing Mass. Celestial Mechanics and Dynamical Astronomy, 5, 27-36. https://doi.org/10.1007/bf01227820
[5]
Das, R.K., Shrivastav, A.K. and Ishwar, B. (1988) Equations of Motion of Elliptic
Restricted Problem of Three Bodies with Variable Mass. Celestial Mechanics and
Dynamical Astronomy, 45, 387-393. https://doi.org/10.1007/BF01245759
[6]
Shrivastava, A.K. and Ishwar, B. (1983) Equations of Motion of the Restricted
Problem of Three Bodies with Variable Mass. Celestial Mechanics and Dynamical
Astronomy, 30, 323-328. ttps://doi.org/10.1007/BF01232197
[7]
Lukyanov, L.G. (1990) The Stability of the Libration Points in the Restricted
Three-Body Problem with Variable Mass. Astronomical Journal, 67, 167-172.
[8]
El-Shaboury, S.M. (1990) Equations of Motion of Elliptically-Restricted Problem of
a Body with Variable Mass and Two Tri-Axial Bodies. Astrophysics and Space
Science, 174, 291-296. https://doi.org/10.1007/BF00642513
[9]
Plastino, A.R. and Muzzio, J.C. (1992) On the Use of and Abuse of Newton’s Second
Law for Variable Mass Problems. Celestial Mechanics and Dynamical Astronomy,
M. R. Hassan et al.
53, 227-232. https://doi.org/10.1007/BF00052611
[10] Bekov, A.A. (1993) The Libration Points and Hill Surfaces in the Restricted Problem
of Three Variable Mass Bodies. Proceedings of the International Astronomical Union Colloquium 132, New Delhi, 10-13 October 1993, 277-288.
https://doi.org/10.1017/S0252921100066173
[11] Bekov, A.A. (1993) Problem of Physics of Stars and Extragalactic Astronomy. 91-114.
[12] Singh, J. and Ishwar, B. (1985) Effect of Perturbations on the Stability of Triangular
Points in the Restricted Problem of Three Bodies with Variable Mass. Celestial Mechanics and Dynamical Astronomy, 35, 201-207.
https://doi.org/10.1007/bf01227652
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