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The existence and stability of stationary solutions of the restricted three body problem under the effect of the dissipative force, Stokes drag, are investigated. It is observed that there exist two non collinear stationary solutions. Further, it is also found that these stationary solutions are unstable for all values of the parameters.

Two finite masses, called primaries, are moving in circular orbits around their common centre of mass, and an infinitesimal mass is moving in the plane of motion of the primaries. To study the motion of the infinitesimal mass is called the restricted three body problem. [_{1}, ….. L_{5}), which were the stationary solutions of the restricted problem. Out of them, three are collinear and two are non collinear. The collinear libration points are unstable for all values of mass parameter

As we know, dissipative forces are those where there is a loss of energy such as friction and one of the most important mechanisms of dissipation is the Stokes drag which is a force experienced by a particle moving in a gas, due to the collisions of the particle with the molecules of the gas.

[

where

is the keplerian angular velocity at distance

A number of authors have investigated the location and stability of the equilibrium point in the presence of specific dissipative forces. [_{4} and L_{5} are unstable to this type of drag force. In their studies of the motion of dust particles in the vicinity of the Earth, [_{4} and L_{5} are asymptotically stable with this kind of dissipation. It has been shown by [_{4} and L_{5} is provided in [

Furthermore, [

In the present paper, we study the same problem but with the effects of stokes drag instead Poynting Robertson drag on noncollinear libration points L_{4} and L_{5} in the restricted three body problem.

Suppose

In the synodic axes the equation of motion of

where

Its components along the synodic axes (x, y) are

where

The equations of motion of

where

n = Mean motion, G = Gravitational constant,

Using [

where

The Stokes drag effect is of the order of

The solutions (x, y) of Equations (2) and (3) with

and

Here, if we take_{i} (i = 1, 2, 3, 4, 5). The L_{i} (i = 1, 2, 3) are three collinear libration points which lie along the x-axis and L_{i} (i = 4, 5) are the two non collinear libration points which make the equilateral triangles with the primaries. Due to the presence of the Stokes drag force, it is clear from Equations (6) and (7) that collinear equilibrium solution does not exist. Since there is a possibility of non collinear libration points under the effect of drag forces, so we restrict our analysis to these points. Their locations when

Now, we suppose that the solution of Equations (6) and (7) when

Making the above substitutions in Equations (6) and (7), and applying Taylors series expansion around the libration points by using that

and

After substituting the values of the constants

Hence, putting the values of

Here, the shifts in L_{4} and L_{5} are of

We write the variational equations by putting

Let us consider the trial solution of Equations (11) and (12),

where

Now, from Equations (13) and (14), we derive the following simultaneous linear equations

and

The simultaneous linear Equations (15) and (16) can be written as

where

and

Neglecting terms of

This quadratic Equation (24) has the general form

where

Here

By assuming

The four classical solutions for L_{4 }and L_{5 }to

Since we are primarily interested in the stability of L_{4} and L_{5} under the effects of a drag force, we restrict our analysis to these points. The four roots of the classical characteristic equation can be written as

where

is a real quantity for L_{4} and L_{5}. Using the values of

With the introduction of drag we assume a solution of the form

where

Substituting these in Equation (25), and neglecting products of

(i) The stability of

For

On putting the values of

Now, putting these values of

whose roots are

Also on taking

whose roots are

If

According to [

where

But here in our case of Stokes drag _{4} is not asymptotically stable. Further one of the roots of _{4 }is not stable. Thus we conclude that L_{4 }is neither stable nor asymptotically stable and hence linearly unstable.

Similarly, we conclude that _{ }is neither stable nor asymptotically stable and hence linearly unstable.

We have studied the existence of the triangular libration points and their linear stability by using Stokes drag. We have shown that there exist two noncollinear stationary points

In the classical case i.e. when

In the case of Stokes drag, we have derived a set of linear equations in terms of

Further, we have derived the approximate expressions for

Using the [

is a real quantity for L_{4} and L_{5} in the classical case. After substituting the values of

Further to investigate the stability of the shifted points, by using [

The condition _{4} and L_{5} are not asymptotically stable. Further, we have seen that one of the roots of _{4} and L_{5} are not stable. Hence, due to Stokes drag, L_{4} and L_{5} are neither stable nor asymptotically stable but unstable whereas in the classical case L_{4} and L_{5} are stable for the mass ratio