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The linear stability of the triangular points was studied for the Robes restricted three-body problem when the bigger primary (rigid shell) is oblate spheroid and the second primary is radiating. The critical mass obtained depends on the oblateness of the rigid shell and radiation of the second primary as well as the density parameter k. The stability of the triangular points depends largely on the values of k. The destabilizing tendencies of the oblateness and radiation factors were enhanced when k > 0 and weakened for k < 0.

Robe [

Robes model may be useful for studying the small oscillations of the earth’s inner core by taking into consideration the moon’s attraction. The model is also applicable to the study of the motion of the artificial satellite under the influence of the earth’s attraction.

Robes problem has been modified to define a new problem (Shrivastava and Garain [

In our model we consider a rigid shell which is oblate spheroid and the second primary which radiates to study the effect of oblateness of the first primary and radiation of the second primary on the stability of the triangular equilibrium points of the Robes restricted three-body problem.

The paper consists of four sections. Section one establishes the relevant equations of motion that incorporates the effect of buoyancy force using some basic assumptions. In the second section we obtained the equilibrium points. Section three deals with the variational equations of motion of the problem and solutions of the resulting characteristic equation obtained. In section four, we obtained the critical mass of the mass parameter. This is followed by the conclusion on the findings.

Let the mass of the rigid shell be and the point mass be. Let the density of the incompressible fluid inside the shell be and that of the infinitesimal mass be and it’s mass. Let denote the oblateness coefficient of the first primary such that and the radiation force of the second primary which given by such that.

Let, and be the centers of, and respectively such that and . Let G be the gravitational constant and the coordinates of the infinitesimal mass. Let the line joining and be the. Then the total potential acting on is

where

Let the coordinates of and be and respectively. In the dimensionless rotational coordinate system we choose the unit of mass to be the sum of the masses of the primaries (and). We take the unit of length equal to the distance between the primaries and is chosen such that.

The equations of motion of the infinitesimal body are (AbdulRaheem and Singh, [

where

Equilibrium points exist when

For we have

The triangular points are given by the equations

, ,.

That is

and

Equations (9) and (10) give

Knowing and from Equations (11) and (12) the exact coordinates of the triangular points are obtained by using Equation (5) for and.

Thus

When the bigger primary is not oblate, the smaller primary is not radiating and

We assume the solutions of equations (11) and (12) are

where and are very small perturbations. Using Equations (11) and (12) and restricting ourselves to linear terms in, and, we obtain

The coordinates obtained in Equation (16) are the triangular points and are denoted by and.

Putting, in Equation (3), in order to study the motion near the triangular points and, we obtain the variational equations of motion as

The characteristic equation is

where the superscript 0 indicates that the partial derivatives are evaluated at the triangular points, and are given by

where

Each, is very small.

The characteristic equation becomes

Its roots are

We observe that the roots are functions of, , , and they depends upon the nature of the discriminant and is given by

Three cases can be discussed for:

1) When, we have that the roots are negative showing that the triangular points are linearly stable.

2) When, we have that the real parts of two of the four roots are positive and equal, showing that the triangular points are unstable.

3) When, we have that the double roots give secular terms, showing that the triangular points are unstable.

The solution of the equation gives the critical mass value of the mass parameter. That is

where and

. Restricting ourselves to linear terms in

, and, and neglecting the product, we find that

where

Equation (22) gives the critical mass value of the mass parameter. It reflects the effect of the oblateness of the first primary (rigid mass) and the radiation of the second primary on the critical mass of the Robes restricted three-body problem, indicating a destabilizing effect on the triangular equilibrium points.

The destabilizing tendencies of both the oblateness and radiating factors are further enhanced when and weakened when .

When we confirm the result of Abdul Raheem and Singh (2006) for, and.

When, and we obtain the critical mass value of the classical restricted three-body problem.

The effect of oblateness of the first primary (rigid shell) and radiation of the second primary on the stability of the triangular equilibrium points of the Robes restricted three-body problem was studied. The value of the critical mass value obtained depends on the oblateness coefficient of the rigid shell, radiation factor of the second primary and the density of the fluid and that of the infinitesimal mass in the shell.

It was observed that the oblateness and radiation factors have destabilizing tendencies on the triangular equilibrium points. These destabilizing tendencies are further enhanced or weakened, depending on whether the density of the fluid in the shell is less than that of the infinitesimal mass or the density of the infinitesimal mass is less than that of the fluid.