^{1}

^{*}

^{2}

^{3}

Effect of perturbations in Coriolis and centrifugal forces on the non-linear stability of the libration point
L
_{4} in the restricted three body problem is studied when both the primaries are axis symmetric bodies (triaxial rigid bodies) and the bigger primary is a source of radiation. Moser’s conditions are utilized in this study by employing the iterative scheme of Henrard for transforming the Hamiltonian to the Birkhoff’s normal form with the help of double D’Alembert’s series. It is found that
L
_{4} is stable for all mass ratios in the range of linear stability except for the three mass ratios
*μ*
_{c1},
*μ*
_{c2} and
*μ*
_{c3}, which depend upon the perturbations
*ε*
_{1} and
* ε*
_{1} in the Coriolis and centrifugal forces respectively and the parameters
*A*
_{1},
*A*
_{2},
*A*
_{3} and
*A*
_{4} which depend upon the semi-axes a
_{1},b
_{1},c
_{1};a
_{2},b
_{2},c
_{2} of the triaxial rigid bodies and p, the radiation parameter.

We propose to study the effect of perturbations in Coriolis (

We shall use dimensionless variables and adopt the notation and terminology of Szebehely [

where

We adopt the method used by Bhatnagar and Hallan [

where

Equations of motion of mass

where

At

On solving above equations, we get

The Lagrangian (L) of the system of equations (1) is

Shift the origin to

where

Hamiltonian function H corresponding to above Lagrangian is given by:

where

and _{}

To investigate the linear stability of the motion, as in Whittaker [

where

The Equation (5) has a nonzero solution if and only if

Let the discriminant of the characteristic Equation (6) be denoted by D.

If

Let the roots of characteristic Equation (6) be

Here

We may write

by taking perturbations in the Coriolis and centrifugal forces. Here

where

To express

We use the canonical transformations from the phase space (

where

Following the procedure of Bhatnagar and Hallan [

Taking

Equations of motion

become

The general solution of the equations of the motion is

Now, to perform Birkhoff’s normalization, the coordinates (

where

The double summation over the indices i and j is such that:

1) i runs over those integers in the interval

2) j runs over those integers in the interval

_{ }are to be regarded as constants of integration and

where

According to Deprit and Deprit Bartholome [

Where the left hand members stand for the Poisson’s brackets with respect to the phase variables

The values of

Proceeding as in Deprit and Deprit-Bartholome [

where

Now

where

The values of all

Following the procedure of Bhatnagar and Hallan [^{rd} order term in

Following the iterative procedure of Henrard, the third order homogeneous components

where

The values of

The partial derivatives in the last two equations have been obtained by substituting

We find that

After simplification the values of A, B and C are given by:

The values of

While evaluating

We note that the inequalities (13) are violated when

Case (i)

We get

Putting these values in second of Equations (8), we get

Putting

Case (ii)

Proceeding as in case (i), we get

Hence for the value

where

Now after simplification, the determinant D occurring in condition (ii) of Moser’s theorem is given by:

That is

Substituting the values of

where

The abscissa of

With the increase of perturbation in Coriolis force, the range of linear stability increases whereas if we increase perturbation in centrifugal force, the range of stability decreases (Equation (7)).

Values of second order coefficients (

Values of

It may be observed that values of

Moser’s second condition is violated for unperturbed problem (i.e. for

It may also be observed that value of

By taking both the primaries as axis symmetric bodies and the bigger mass as a source of radiation, the triangular point

The results of Jagadish Singh [

All the results of Bhatnagar and Hallan [

KavitaChauhan,S. N.Rai,RajivAggarwal, (2015) Effect of Perturbations in Coriolis and Centrifugal Forces on the Non-Linear Stability of L_{4} in the Photogravitational Restricted Three Body Problem. International Journal of Astronomy and Astrophysics,05,275-290. doi: 10.4236/ijaa.2015.54031

Values of

Values of