^{1}

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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.

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 [

Singh and Ishwar [

Plastino et al. [

At present, we have proposed to extend the work of Singh [

Let

where

The oblateness parameter of the smaller primary is given by

where

and

Now from Equation (1),

where units are so chosen that the sum of the masses of the primaries and the gravitational constant

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

Let us introduce space time transformations as

where

From Equations ((7) and (8)), we get

where

Differentiating

Also,

Now,

Putting the values of

where

In order to make the Equation (10) free from the non-variational factor, it is sufficient to put

Thus the System (10) reduces to

where

From System (13),

The Jacobi’s Integral is

For the existence of equilibrium points

For solving the above equations, let us change these equations in Cartesian form as

For the Collinear equilibrium points,

From Equation (17), we get

Let

Thus from Equation (18),

as

For the first equilibrium point

Here, Equation (20) is seven degree polynomial equation in

Thus

Thus the Equation (20) reduces to

Let

Putting the value of

where

Therefore, the co-ordinate of the first equilibrium point

Let

Thus from Equation (18),

Since

where

In terms of

i.e.,

The Equation (26) is a seven degree polynomial equation in

If we put

Here

where

Let

where

Thus the co-ordinate of the second equilibrium point is given by

Let

Thus from Equation (18), we have

When

Let

where

Thus Equation (29) reduced to

By putting values of

where

Thus the co-ordinates of the third equilibrium point is given by

For triangular equilibrium point

Now

Again

Since

For better approximation

For triangular equilibrium points

Now,

Again,

Putting the value of

Thus,

Therefore,

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

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

co-ordinates of

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

From the above facts we concluded that in the perturbed case, first equilibrium point

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