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  regarding the evaluation of binary system. Meshcherskii  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  has discussed the restricted problem of perturbed motion of two bodies with variable mass. Following Jeans  , Verhulst  discussed the two body problem with slowly decreasing mass, by a non-linear, non-autonomous system of differential equations. Shrivastava and Ishwar  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  showed the effect of perturbation on the location and stability of the triangular equilibrium points in the restricted three-body problem. Das et al.  developed the equations of motion in elliptic restricted problem of three bodies with variable mass. Lukyanov  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  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.  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   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.  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, the constant due to the variation in mass governed by Jean’s law.
At present, we have proposed to extend the work of Singh  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 by the method of small parameters.
2. Equations of Motion
Let be the mass of the infinitesimal body varying with time. The primaries of masses and 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, rotating relative to inertial frame with angular velocity. The line joining the centers of and is considered as the -axis and a line lying on the plane of motion and perpendicular to the -axis and through the centre of mass as the -axis and a line through the centre of mass and perpendicular to the plane of motion as the -axis. Let and respectively be the co-ordinates of the primaries and and be the co-ordinates of the infinitesimal mass. The equation of motion of the infinitesimal body of variable mass can be written as
Figure 1. Rotating frame of reference in the R3BP in 3-Dimension about Z-axis.
The oblateness parameter of the smaller primary is given by
where and are the equatorial and polar radii of the oblate primary, is the dimensional distance between the primaries,
Now from Equation (1),
where units are so chosen that the sum of the masses of the primaries and the gravitational constant both are unity.
The equations of motion in the Cartesian form are
By Jeans law, the variation of mass of the infinitesimal body is given by
where is a constant coefficient and the value of exponent for the stars of the main sequence.
Let us introduce space time transformations as
where is the mass of the satellite at.
From Equations ((7) and (8)), we get
Differentiating and with respect to twice, we get
Putting the values of and in Equation (4), we get
In order to make the Equation (10) free from the non-variational factor, it is sufficient to put
Thus the System (10) reduces to
From System (13),
The Jacobi’s Integral is
3. Existence of Equilibrium Points
For the existence of equilibrium points then from Systems (13) and (15)
For solving the above equations, let us change these equations in Cartesian form as
4. Existence of Collinear Equilibrium Points
For the Collinear equilibrium points, then .
From Equation (17), we get
Let be the first collinear equilibrium point lying to the left of the second primary as shown in Figure 2 then,
Thus from Equation (18),
as, so let where is a small quantity.
For the first equilibrium point, we have
Here, Equation (20) is seven degree polynomial equation in, so there are seven values of. If we put then from Equation (20), we get
Figure 2. Locations of collinear and triangular equilibrium points.
Thus, are four roots of Equation (21) when, so i.e.,.
Thus the Equation (20) reduces to
Let where are small parameters, then
Putting the value of in Equation (22) and equating the co-efficient of different powers of to zero, we get the values of the parameters as
Therefore, the co-ordinate of the first equilibrium point is given by
Let be the second collinear equilibrium point between the two primaries and then
Thus from Equation (18),
Since hence let, thus
where is a small quantity.
In terms of, the Equation (25) can be written as
The Equation (26) is a seven degree polynomial equation in, so there are seven values of in Equation (26).
If we put in Equation (26), we get
Here are the four roots of Equation (27) when, so we can choose as some order of i.e.,
Let where are small parameters. Putting the values of and in Equation (26) and equating the coefficients of different powers of, we get
Thus the co-ordinate of the second equilibrium point is given by
Let be the third equilibrium point right to the first primary, then
Thus from Equation (18), we have
When, then Equation (29) reduced to
are the four roots of the Equation (29) when, so say when
where are small parameters.
Thus Equation (29) reduced to
By putting values of in Equation (29) and equating the co-efficient of different powers of, we get
Thus the co-ordinates of the third equilibrium point is given by
5. Existence of Triangular Equilibrium Points
For triangular equilibrium point and then from the System (17), we have
Since, hence for the first approximation, if we put, then from Equations ((34) and (35)), we get
For better approximation, then the above solutions can be written as and where.
For triangular equilibrium points, then
Putting the value of and in Equation (34), we get. Putting the value of and in Equation (35), we get
6. Surface of Zero?Velocity
Figure 3. Zero velocity curve (ZVC) for (classical case).
Figure 4. Zero velocity curve for (classical case).
Figure 5. Zero velocity curve for (classical case).
Figure 6. Zero velocity curve for (perturbed case).
Figure 7. Zero velocity curve for (perturbed case).
Figure 8. Zero velocity curve (ZVC) for (perturbed case).
Figure 9. 3 Dimensional view of ZVC of Figure 6.
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
, where is a constant and the interval
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).
In section 3, the equations for solving equilibrium points, have been derived in Equation (17) by putting in Equation (13). Again the equations for equilibrium points, have been transformed to original frame which are given in Equation (18). In section 4, for collinear equilibrium points we put, then from Equation (17), we get only one Equation (19). Applying small parameter method, we established the
co-ordinates of as in terms of order of. 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 and in classical case and perturbed case.
From the above facts we concluded that in the perturbed case, first equilibrium point shifted away from the second primary whereas shifted towards the first primary but 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 and, we can say that for and, the triangular equilibrium configuration is maintained but for in both classical and perturbed cases, the equilateral triangular configuration is not maintained. Whatever be the analytical changes in the co-ordinates of and that is due oblateness not due to the mass reduction factor of the infinitesimal body.