Received 16 July 2015; accepted 21 December 2015; published 24 December 2015

ABSTRACT

Effect of perturbations in Coriolis and centrifugal forces on the non-linear stability of the libration point 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 is stable for all mass ratios in the range of linear stability except for the three mass ratios, and, which depend upon the perturbations and in the Coriolis and centrifugal forces respectively and the parameters and which depend upon the semi-axes of the triaxial rigid bodies and p, the radiation parameter.

Keywords:

Restricted Three Body Problem, Axis Symmetric Bodies; Non-Linear Stability, Libration Point, Double D’Alembert’s Series Method

1. Introduction

We propose to study the effect of perturbations in Coriolis () and centrifugal forces () on the non-linear stability of libration point () when both the primaries are axis symmetric bodies and the bigger primary is a source of radiation. We use Moser’s conditions by employing the iterative scheme of Henrard (Deprit and Deprit-Bartholome [1] ), for transforming the involved Hamiltonian to the Birkhoff’s normal form with the help of double D’Alembert’s series. In the year 1983, Bhatnagar and Hallan [2] investigated the perturbation effects in Coriolis and centrifugal forces in the non-linear aspect of stability of. Rajiv Aggarwal et al. [3] studied the non-linear stability of in the restricted three body problem for radiated axes symmetric primaries with resonances. Mamta Jain and Rajiv Aggarwal [4] investigated the existence of non-collinear libration points and their stability (in linear sense) in the circular restricted three body problem, in which they had considered the smaller primary as an oblate spheroid and the bigger one as a point mass including the effect of dissipative force especially Stokes drag. Bhavneet Kaur and Rajiv Aggarwal [5] studied the Robe’s restricted problem of 2 + 2 bodies when the bigger primary was a Roche ellipsoid. Jagadish Singh [6] investigated the combined effects of perturbations, radiation and oblateness on the non-linear stability of triangular points. We have extended this study by taking the primaries as axis symmetric bodies. In the present paper, our aim is to examine the effect of perturbations in Coriolis and centrifugul forces in the non-linear stability of the libration point of the restricted three body problem when both the primaries are axis symmetric bodies and the bigger primary is a source of radiation with its equatorial plane coincident with the plane of motion.

2. Equations of Motions and Linear Stability

We shall use dimensionless variables and adopt the notation and terminology of Szebehely [7] . The equations of motion of the infinitesimal mass in a synodic co-ordinate system () are

where

is the distance between the primaries, and () being the masses of the primaries. and are the semi-axes of the axis symmetric body of mass and and are the semi-axes of the axis symmetric body of mass. The configuration is given in Figure 1. Since, we will reject second and higher order terms in and.

We adopt the method used by Bhatnagar and Hallan [2] and give perturbation in Coriolis and centrifugal forces with the help of the parameters and respectively. The unperturbed value of each is unity. Consequently we take the equations of motion as

where

Equations of motion of mass can be put in the form

(1)

where

3. Location of Libration Point of L₄

At, we have

On solving above equations, we get

(2)

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

Shift the origin to and expanding in power series of x and y, we get

(3)

where

Hamiltonian function H corresponding to above Lagrangian is given by:

(4)

where

, , ,

and

To investigate the linear stability of the motion, as in Whittaker [8] , we consider the following set of linear equations in the variables x and y

(5)

where

The Equation (5) has a nonzero solution if and only if, which implies that

(6)

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

If then, it is bounded, hence stable when where

(7)

Let the roots of characteristic Equation (6) be. These are long term and short term perturbed frequencies, which are given by

(8)

Here represent the perturbed basic frequencies. The unperturbed basic frequencies, are given by

We may write

(9)

by taking perturbations in the Coriolis and centrifugal forces. Here are to be determined so that Equations (8) are satisfied. Simple calculations give

where

4. Determination of the Normal Co-Ordinates

To express in normal form, we consider the set of linear Equation (5), the solution of which can be obtained as

We use the canonical transformations from the phase space () into the phase space of the angles () and the action moment as () i.e.

(10)

where

Following the procedure of Bhatnagar and Hallan [2] , we get the normal form of the Hamiltonian

Taking

,

Equations of motion

become

The general solution of the equations of the motion is

,

,

.

5. Second Order Normalization

Now, to perform Birkhoff’s normalization, the coordinates () are to be expanded in double D’Alembert series:

(11)

where are homogenious functions of degree n in and are in the form

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

1) i runs over those integers in the interval that have the same parity as

2) j runs over those integers in the interval that have the same parity as m.

and are to be regarded as constants of integration and are to be determined as linear functions of time (t) such that

where and are of the form

According to Deprit and Deprit Bartholome [1] , the canonical character of the transformation will be ensured formally by requesting that the double D’Alembert series satisfy the identities

Where the left hand members stand for the Poisson’s brackets with respect to the phase variables. The first order components and in and are the values of x and y given by Equation (10)

The values of can be obtained from Appendix.

Proceeding as in Deprit and Deprit-Bartholome [1] , it is observed that the second order components and are solutions of the partial differential equations

where

and are obtained by

Now

where

The values of all for can be obtained from the authors on request as the expressions are very long and contained in large number of pages.

6. Third-Order Terms in H

Following the procedure of Bhatnagar and Hallan [2] , Hamiltonian H given by Equation (4) transforms to the Hamiltonian in which the 3^rd order term in is zero. That is.

7. Second Order Coefficient in the Frequencies

Following the iterative procedure of Henrard, the third order homogeneous components and in Equation (11) can be obtained by partial differential equations

where

The values of are given in Appendix.

The partial derivatives in the last two equations have been obtained by substituting and in and. Now choosing

We find that

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

(12)

The values of, can be obtained from the author on request as the expressions are again very long and contained in large number of pages. Coefficients of can be obtained by Bhatnagar and Hallan [2] .

8. Stability

While evaluating and the condition (i) of Moser’s theorem as in Moser [9] is assumed. Now we verify that this condition is satisfied. The condition is for all pairs () of rational integers such that

(13)

We note that the inequalities (13) are violated when

(14)

Case (i).

We get

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

Putting and solving for, denoting this value by, we get

(15)

Case (ii)

Proceeding as in case (i), we get, where

(16)

Hence for the value and of mass ratios, condition (i) of Moser’s theorem is not satisfied. The normalized Hamiltonian up to fourth order is

where are given by Equation (12).

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

That is

Substituting the values of from Equation (12) and using the Equation (8) and Equation (9), we obtain

, are given in the Appendix. Values of can be obtained from Bhatnagar and Hallan [2] . It is seen that the condition (ii) of Moser’s theorem is satisfied i.e. if in the interval, mass ratio does not take the value

(17)

where

9. Conclusions

The abscissa of is independent of the perturbation in Coriolis () and centrifugal forces () and ordinate of is affected by perturbation in centrifugal force (Equation (2)).

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 () in the polynomials () occurring in the frequencies and are affected by the perturbations in Coriolis and centrifugal forces. It is observed that if perturbation in Coriolis and centrifugal forces increase then values of second order coefficients () increase (Equation (12)).

corresponds to the resonance cases and. Their values are given in Equation (13).

Values of (values of at which Moser’s theorem is not applicable) increase if perturbation in Coriolis force increases and decrease if perturbation in centrifugal force increases (Equations (15)-(17)).

It may be observed that values of decrease if parameters of axis symmetric bodies () and radiation pressure (p) increase (Equations (15) and (16)).

Moser’s second condition is violated for unperturbed problem (i.e. for) when (Equation (17)).

It may also be observed that value of increases if of the bigger primary and p increase. If increase, value of decreases (Equation (17)).

By taking both the primaries as axis symmetric bodies and the bigger mass as a source of radiation, the triangular point is stable in the range of linear stability except for the three mass ratios given in Equations (15)-(17) at which Moser’s theorem does not apply.

The results of Jagadish Singh [6] can be deduced by taking and.

All the results of Bhatnagar and Hallan [2] can be deduced by taking .

Cite this paper

KavitaChauhan,S. N.Rai,RajivAggarwal, (2015) Effect of Perturbations in Coriolis and Centrifugal Forces on the Non-Linear Stability of L₄ in the Photogravitational Restricted Three Body Problem. International Journal of Astronomy and Astrophysics,05,275-290. doi: 10.4236/ijaa.2015.54031

References

Appendix

and can be obtained from R and respectively by replacing, , , , and.

Values of and can be obtained from Hallan and Bhatnagar (983).

Values of, and can be obtained from the author on request as the expressions are very long and contained in large number of pages.