Existence of Equilibrium Points in the R 3 BP with Variable Mass When the Smaller Primary is an Oblate Spheroid

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.


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 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 ( )

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 ( )

P 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 Figure 1.Rotating frame of reference in the R3BP in 3-Dimension about Z-axis.
where d is the defined operator under consideration, dt t The oblateness parameter of the smaller primary is given by where a and c are the equatorial and polar radii of the oblate primary, R is the dimensional distance between the primaries, ( ) 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 ( ) where ( ) i.e., By Jeans law, the variation of mass of the infinitesimal body is given by where α is a constant coefficient and the value of exponent [ ] for the stars of the main sequence.
Let us introduce space time transformations as where 0 m is the mass of the satellite at 0 t = .
From Equations (( 7) and ( 8)), we get where Differentiating , x y and z with respect to t twice, we get k q n k q n q k q n k q n q k q n k q n q x q k q n q y q k q n q z q k q n q ( ) where ( ) In order to make the Equation ( 10) free from the non-variational factor, it is Thus the System (10) reduces to where From System (13), ( The Jacobi's Integral is ( ) ( )

Existence of Equilibrium Points
For the existence of equilibrium points 0, For solving the above equations, let us change these equations in Cartesian form as

Existence of Collinear Equilibrium Points
For the Collinear equilibrium points, 0, From Equation (17), we get 1 ,0,0 L ξ be the first collinear equilibrium point lying to the left of the second primary ( ) 1 0 and 0, 1 1 and .
For the first equilibrium point ( ) Thus the Equation (20) reduces to , , , , , , a a a a a a a  are small parameters, then ( ) Putting the value of , , , , , , , ρ ρ ρ ρ ρ ρ ρ  in Equation ( 22) and equating the co-efficient of different powers of υ to zero, we get the values of the parameters as where ( ) Therefore, the co-ordinate of the first equilibrium point ( ) where ρ is a small quantity.
In terms of ρ , the Equation ( 25) can be written as The Equation ( 26) is a seven degree polynomial equation in ρ , so there are seven values of ρ in Equation ( 26).
If we put 0 µ = in Equation (26), we get ( ) Here 4 0 i.e., 0, 0, 0, 0 ρ ρ = = are the four roots of Equation ( 27) when 0 µ = , so ρ we can choose as some order of µ i.e., ( ) ( ) where ( ) Thus the co-ordinate of the second equilibrium point is given by 3 ,0,0 L ξ be the third equilibrium point right to the first primary, then Thus from Equation (18), we have When 0 µ = , then Equation (29) reduced to ( ) , , , , , , ρ ρ ρ ρ ρ ρ ρ in Equation ( 29) and equating the co-efficient of different powers of υ , we get , and so on where ( ) Thus the co-ordinates of the third equilibrium point is given by

Existence of Triangular Equilibrium Points
For triangular equilibrium point 0, 0 x y ≠ ≠ and 0 z = then from the System (17), we have Since 0 1 A <  , hence for the first approximation, if we put 0 A = , then from Equations ((34) and ( 35)), we get For better approximation 0 A ≠ , then the above solutions can be written as For triangular equilibrium points

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 given in Equations ( 8) and ( 9).The Jacobi's integral has been derived in Equation (16).
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 ( ) ,0,0 µ and ( ) 1,0,0 µ − respectively be the co-ordinates of the primaries 1 P and 2

Figure 2 .
Figure 2. Locations of collinear and triangular equilibrium points.
b b b b  are small parameters.Putting the values of 26) and equating the coefficients of different powers of υ , we get

a
in Equation (35), we get

,
where α is a constant and the interval [ ] 0.4, 4.4 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