Effect of Perturbations in Coriolis and Centrifugal Forces on the Non-Linear Stability of L 4 in the Photogravitational Restricted Three Body Problem

Effect of perturbations in Coriolis and centrifugal forces on the non-linear stability of the libration point L4 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 L4 is stable for all mass ratios in the range of linear stability except for the three mass ratios c1 μ , c2 μ and c3 μ , which depend upon the perturbations 1 ε and 2 ε in the Coriolis and centrifugal forces respectively and the parameters A A A 1 2 3 , , and A4 which depend upon the semi-axes a b c a b c 1 1 1 2 2 2 , , ; , , of the triaxial rigid bodies and p, the radiation parameter.


Introduction
We propose to study the effect of perturbations in Coriolis ( 1 ε ) and centrifugal forces ( 2 ε ) on the non-linear stability of libration point ( 4 L ) 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 4 L .Rajiv Aggarwal et al. [3] studied the non-linear stability of 4 L 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 4 L 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.

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 3 m in a synodic co-ordinate system ( , x y ) are 2 , 2 , x y x ny y nx

, , , , 1 p A A A A <
 , we will reject second and higher order terms in

, , , p A A A and 4
A .We adopt the method used by Bhatnagar and Hallan [2] and give perturbation in Coriolis and centrifugal forces with the help of the parameters α and β respectively.The unperturbed value of each is unity.Consequently we take the equations of motion as Equations of motion of mass 3 m can be put in the form 2 , 2 , x y x n y y n x α α where Figure 1.Configuration of the photogravitational restricted problem with both the primaries axis symmetric bodies and the bigger primary is source of radiation.

Location of Libration Point of L4
At 4 L , we have 0, 0 and 0.
x y y Ω = Ω = ≠ On solving above equations, we get ( ) ( ) The Lagrangian (L) of the system of equations ( 1) is Shift the origin to 4 L and expanding in power series of x and y, we get

L x y A A xy xy x
Hamiltonian function H corresponding to above Lagrangian is given by: ,

H H H H H H
where 0 0 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 The Equation ( 5) has a nonzero solution if and only if det 0 A = , which implies that Let the discriminant of the characteristic Equation ( 6) be denoted by D.
λ < , it is bounded, hence stable when 0 Let the roots of characteristic Equation ( 6) be 1 2 and ± .These are long term and short term perturbed frequencies, which are given by Here 1 2 , ω ω ′ ′ represent the perturbed basic frequencies.The unperturbed basic frequencies 1 2 , ω ω , are given by We may write ( ) ( ) by taking perturbations in the Coriolis and centrifugal forces.Here 1 2 1 2 , , , p p q q are to be determined so that Equations (8)

Determination of the Normal Co-Ordinates
To express 2 H 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 ( , , , where Following the procedure of Bhatnagar and Hallan [2], we get the normal form of the Hamiltonian The general solution of the equations of the motion is

Second Order Normalization
Now, to perform Birkhoff's normalization, the coordinates ( , x y ) are to be expanded in double D'Alembert series: , , , , , , , , , I I 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 0 i n m ≤ ≤ − that have the same parity as n m − 2) j runs over those integers in the interval m j m − ≤ ≤ that have the same parity as m.
1 I and 2 I are to be regarded as constants of integration and 1 2 , φ φ are to be determined as linear functions of time (t) such that ( ) ( ) , , , .
where 2n f and 2n g 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 , 0.
Where the left hand members stand for the Poisson's brackets with respect to the phase variables ( )

I
are the values of x and y given by Equation (10) The values of 13 14 21 22 23 24 , , , , , J J J J J J can be obtained from Appendix.Proceeding as in Deprit and Deprit-Bartholome [1], it is observed that the second order components 0,1 2 B and 1,0 2 B are solutions of the partial differential equations , .( ) 1, 2, , 6 7,8, ,10 1, 2, , 6 7,8, ,10 The values of all , , , i j i j r r s s for 1, 2, , 6; 7,8, ,10 i j = =   can be obtained from the authors on request as the expressions are very long and contained in large number of pages.

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 1 2 1 2 1 2 and I I is zero.That is 3 0 H = .

Second Order Coefficient in the Frequencies
Following the iterative procedure of Henrard, the third order homogeneous components 1,0 3 B and 0,1 3 B in Equation (11) can be obtained by partial differential equations The values of , , ij ij ij p p q ′ are given in Appendix.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:    ,  .

Stability
While evaluating 1,0 0,1 1,0 , , B B B and 0,1 3 B the condition (i) of Moser's theorem as in Moser [9] is assumed.Now we verify that this condition is satisfied.The condition is 1 1 2 2 0 k k We note that the inequalities (13) are violated when 2 and 3 Putting these values in second of Equations ( 8), we get and solving for µ , denoting this value by 1 c µ , we get ( ) Hence for the value µ of mass ratios, condition (i) of Moser's theorem is not satisfied.The normalized Hamiltonian up to fourth order is ( ) where , , A B C are given by Equation (12).Now after simplification, the determinant D occurring in condition (ii) of Moser's theorem is given by: , and m m can be obtained from Bhatnagar and Hallan [2].It is seen that the condition (ii) of Moser's theorem is satisfied i.e. 0 D ≠ if in the interval 0 c µ µ < < , mass ratio does not take the value ( )

Conclusions
The abscissa of 4 L is independent of the perturbation in Coriolis ( 1 ε ) and centrifugal forces ( 2 ε ) and ordinate of L 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 ( , , A B C ) in the polynomials ( 2 2 and f g ) occurring in the frequencies 1 φ  and 2 φ  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 ( , , A B C ) increase (Equation (12)).µ µ µ (values of µ 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 1 2 , c c µ µ decrease if parameters of axis symmetric bodies ( 1 2 3 4 , , , A A A A ) and radiation pressure (p) increase (Equations ( 15) and ( 16)).
Moser's second condition is violated for unperturbed problem (i.e. for 1 By taking both the primaries as axis symmetric bodies and the bigger mass as a source of radiation, the triangular point 4 L 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 (      q q q q p p p p p q q γ γ γ γ µ µ  , , , , p p p p p ′ ′ ′ ′ ′ and 26 p′ can be obtained from the author on request as the expressions are very long and contained in large number of pages.

1 , a b and 1 c
the masses of the primaries.1 are the semi-axes of the axis symmetric body of mass 1 m and 2 2 , a b and 2 c are the semi-axes of the axis symmetric body of mass 2 m .The configuration is given in Figure 1.
p ) into the phase space of the angles ( 1 2 , φ φ ) and the action moment as ( 1 2 , I I ) i.e.X JT = (10) xy m y m x m x y m xy m y y y

ξ ς η ξ η σ σ
from the author on request as the expressions are again very long and contained in large number of pages.Coefficients of

3 c µ increases if 2 A
(17)).It may also be observed that value of of the bigger primary and p increase.If 1 are satisfied.Simple calculations give )