Restricted three-body problem with variable mass has an important role in celestial mechanics. The phenomenon of isotropic radiation or absorption in stars was studied by the leading scientists to formulate the restricted three-body problem with variable mass. The two body problem with variable mass was studied by Jeans  regarding the evaluation of binary system. Meshcherskii  assumed that the mass is ejected isotropically from the two body system at very high velocities and is lost to the system. He examined the change in orbits, the variation in angular momentum and the energy of the system. Shrivastava and Ishwar  derived the equations of motion of the circular restricted three-body problem 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 due to oblateness on the existence and stability of the triangular libration points in the restricted three-body problem.
Das et al.  developed the equations of motion of elliptic restricted three-body problem with variable mass. Lukyanov  discussed the stability of libration points in the restricted three-body problem with variable mass. He has found that for any set of parameters, all the libration points in the problem (Collinear, Triangular) are stable with respect to the conditions considered by the Meshcherskii’s space-time transformation. El Shaboury  had established the equations of motion of elliptic restricted three-body problem (ER3BP) with variable mass with two triaxial rigid primaries. He has applied the Jeans law, Nechvili’s transformation and space-time transformation given by Meshcherskii in a special case.
Singh et al.  have discussed the non-linear stability of libration points in the restricted three-body problem with variable mass. They have 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 depending upon the mass variation parameter governed by Jean’s law. Hassan et al.  studied the existence of libration points with variable mass in the R3BP when the smaller primary is an oblate spheroid. They found that Jacobi constant shows no effect in the position of libration points, but for , slight shifting of libration points is found due to oblateness only not due to the mass reduction factor .
In present work, we have established coordinates of five libration points in the R3BP with variable mass when smaller primary is a triaxial rigid body by small parameter method  and the method used by Hassan et al.  .
2. Equations of Motion
Let the two primaries of non-dimensional masses and be moving on the circular orbits about their centre of mass. In Figure 1, we consider a bary-centric coordinate system rotating relative to the inertial frame with angular velocity . The line joining the centers of and of the primaries 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 perpendicular to the plane of motion as the z-axis. Let and respectively be the coordinates of and and be the coordinates of the infinitesimal body of variable mass m at P.
The equation of motion of the infinitesimal body of variable mass m can be written as
Figure 1. Configuration of R3BP when smaller primary is triaxial.
the differential operators are given by the relations
where are the semi-axes of the triaxial rigid body, R is the dimensional distance between the centre of the primaries. Thus using Equation (2) in (1), we get
Choosing units of mass and distance in such way that and , then the equations of motion of the infinitesimal body in cartesian form can be written as:
By Jeans law, the variation of mass of the infinitesimal body is given by
where is a constant and the value of exponent for the stars of the main sequence (from Observational facts).
Let us introduce Meshcherskii’s space time transformations  as:
is the mass of the infinitesimal body when and is the pseudo time.
From Equation (7) and Equation (8), we get
Differentiating with respect to t twice and using fourth equation of (8), we get
where dot represents differentiation with respect to real time t and prime represents the differentiation with respect to pseudotime .
Replacing the values of Equation (10) in Equation (4) to obtain
As the mass of the infinitesimal body is variable, so only the variational factors but not the non-variational factors should be taken into consideration in the equations of motion of the infinitesimal body. Thus to avoid the non-variational factors, we have
Thus the Equations (11) reduced to
The Jacobi integral in Meshcherskii’s space is
whereas the Jacobi integral in the rotating frame is
3. Libration Points
Since in the vicinity of the libration points (Lagrangian points), no translatory motion exists, only vibrational motion exists, hence velocity and acceleration components must vanish at these points i.e., .
Thus from Equations (15), we have
For solving the above equations in the rotating frame , we apply the inverse transformation in the above equations to get
4. Collinear Libration Points
As we know that all the three collinear libration points lie on the x-axis (the line joining the centre of the first and second primary) so and hence from Equation (2)
Thus from Equations (17), we have
Let be the first collinear libration point lying to the left of the second primary then
Thus from Equation (18),
As , so let where is a very small positive quantity.
From Equation (19)
Equation (22) is seven degree polynomial equation in , so there are seven values of . If we put then from Equation (22), we get
Here gives four roots of Equation (23) when but we know that so , so there must be some order relation between and
i.e., can be expressed as the order of i.e., where .
Thus the Equation (20) reduces to
As so can be expressed as
where are small parameters  , then
Using above quantities of Equation (25) and in Equation (24) and equating the coefficients of different powers of to zero, we get the values of small parameters as
Therefore the first coordinate of the first libration point is given by
Here depends upon the mass parameter of the primaries, the small parameters , mass variation parameter , angular velocity and triaxiality parameters and . It is to be noted that each small parameters depends upon the preceeding small parameters and other parameters like etc. i.e., .
Thus from Equation (27), it is clear that in the classical case the coordinate of libration point depends upon the mass parameter only but under perturbation it depends upon the parameters as well as .
Let be the second collinear libration point between the two primaries and then .
Thus from Equation (18), we have
Since hence let , thus , is a very small quantity so it can be chosen as some order of .
In terms of , the Equation (28) can be written as
The Equation (30) is a seven degree polynomial equation in , so there are seven values of in Equation (30).
If we put in Equation (30), we get
Here also gives four roots of Equation (31) for , so as earlier case let
where are small parameters.
Putting the values of and in Equation (29) and equating the coefficients of different powers of , we get
Similar to Equation (27), the coordinates of the Second libration point is given by
Let be the third libration point right to the First primary, then
Let then and .
Thus from Equation (18), we have
when , then Equation (33) reduced to
As in Equation (33) let , whereas are small parameters. Thus Equation (30) reduced to
By putting values of and in Equation (33) and equating the coefficients of different powers of , we get
and so on ,
Similar to Equation (27), the coordinates of the third libration point is given by
5. Triangular Libration Points
For triangular libration points, and then from the system (17) we have
where from Equation (2)
Now Equation (35) Equation (36) gives
and Equation (35) Equation (36) gives
For the first approximation, suppose , then and from Equation (38) and Equation (39), we get
For better approximation let , then the above solutions can be written as
From Equation (37) ,
Also from the first equation of (37), we have
From Equation (37), we have
So from Equation (38) and Equation (39), we get
Neglecting higher order terms and coupling terms of , we have
are the triangular libration points.
6. Discussions and Conclusion
We have studied the existence of coplanar libration points in the restricted three-body problem with variable mass and smaller primary as a triaxial rigid body as shown in Figure 1. By taking the mass ratio and the mass variation parameter as the fixed quantities, the variation of mass reduction factor of the infinitesimal body is taken into consideration and studied the effect of on the existence of coplanar libration points.
In Figure 2, the classical case has been discussed for in which all the five libration points exist. The triangular libration points and form equilateral triangle with the primaries. In Figure 3, taking perturbing parameters , then only three collinear libration points exist and no triangular points exist. The libration points and are located at the extreme points of the loop of the lamniscate shaped oval and this oval is again enveloped by a bigger loop. This development of loops is due to the non-zero values of triaxiality parameters and .
Figure 2. Locations of libration points for (classical case).
Figure 3. Locations of libration points for (perturbed case).
In Figure 4, two collinear libration points and exist when and , which contradicts theoretical evolution of the existence of the five libration points. In Figure 5, four coplanar points and exist for where and are collinear and and are non-collinear which don’t form the equilateral triangle with the primaries. The existence of and to the right of the origin is a contradiction to the theoretical evolution of the existence of libration points in the classical case of Figure 2 (Theory of Orbits  ).
Figure 4. Locations of libration points for (perturbed case).
Figure 5. Locations of libration points for (perturbed case).
In Figure 6, when , all the five libration points exist with a difference. In Figure 6, the angular displacement of and relative to is more than that in Figure 5. Further when , angular displacement of and relative to is more in Figure 7 than that in Figure 6 and similar case is repeated in Figure 8 for . Thus due to the variational parameters and triaxiality parameters and , the location of triangular libration points and has been shifted from left to right and
Figure 6. Locations of libration points for (perturbed case).
Figure 7. Locations of libration points for (perturbed case).
Figure 8. Locations of libration points for (perturbed case).
the angular distances of and relative to increase with the decrease of . From the above discussions, we conclude that for and for , all the five libration points exist with an increase in angular displacement of and relative to with the decrease of and shifting of and from positive to negative side of the x-axis.
 Shrivastava, A.K. and Ishwar, B. (1983) Equations of Motion of the Restricted Three-Body Problem with Variable Mass. Celestial Mechanics and Dynamical Astronomy, 30, 323-328.
 Singh, J. and Ishwar, B. (1985) Effect of Perturbations on the Stability of Triangular Points in the Restricted Three-Body Problem with Variable Mass. Celestial Mechanics and Dynamical Astronomy, 35, 201-207.
 Das, R.K., Shrivastav, A.K. and Ishwar, B. (1988) Equations of Motion of Elliptic Restricted Three-Body Problem with Variable Mass. Celestial Mechanics and Dynamical Astronomy, 45, 387-393.
 El-Shaboury, S.M. (1990) Equations of Motion of Elliptically-Restricted Problem of a Body with Variable Mass and Two Triaxial Bodies. Astrophysics and Space Science, 174, 291-296.
 Singh, J. (2008) Non-Linear Stability of Libration Points in the Restricted Three-Body Problem with Variable Mass. Astrophysics and Space Science, 314, 281-289.
 Hassan, M.R., Kumari, S. and Hassan, M.A. (2017) Existence of Libration Points in the R3BP with Variable Mass when the Smaller Is an Oblate Spheroid. International Journal of Astronomy and Astrophysics, 7, 45-61.