Effect of Perturbations in Coriolis and Centrifugal Forces on the Nonlinear Stability of Equilibrium Point in Robe ’ s Restricted Circular Three-Body Problem

The effect of perturbations in Coriolis and cetrifugal forces on the nonlinear stability of the equilibrium point of the Robe’s (1977) restricted circular three-body problem has been studied when the density parameter K is zero. By applying KolmogorovArnold-Moser (KAM) theory, it has been found that the equilibrium point is stable for all mass ratios μ in the range of linear stability 8/9 + (2/3)((43/25) 1 − (10/3) ) < μ < 1, where and 1 are, respectively, the perturbations in Coriolis and centrifugal forces, except for five mass ratios μ1 = 0.93711086 − 1.12983217 + 1.50202694 1, μ2 = 0.9672922 − 0.5542091 + 1.2443968 1, μ3 = 0.9459503− 0.70458206 + 1.28436549 1, μ4 = 0.9660792− 0.30152273 + 1.11684064 1, μ5 = 0.893981− 2.37971679 + 1.22385421 1, where the theory is not applicable.


Introduction
Robe [1] has considered a new kind of restricted three-body problem in which one of the primaries is a rigid spherical shell m 1 filled with a homogeneous incompressible fluid of density ρ 1 .The second primary is a mass point m 2 outside the shell and the third body m 3 is a small solid sphere of density ρ 3 , inside the shell, with the assumption that the mass and radius of m 3 are infinitesimal.He has shown the existence of an equilibrium point with m 3 at the center of the shell, while m 2 describes a Keplerian orbit around it.Further, he has discussed the linear stability of the equilibrium point.Hallan and Rana [2] considered the effect of perturbations , 1 in Coriolis and centrifugal forces, respectively, on the location and linear stability of the equilibrium points in Robe's circular three-body problem when the density parameter K is zero.They have found that (−μ + (μ 1 /(1 + 2μ)), 0, 0) is the only equilibrium point and in the linear sense it is stable for μ c < μ < 1 and unstable for 0 < μ ≤ μ c , where μ c = 8/9+(2/3)((43/25) 1 − (10/3) ).Shrivastava and Garain [3], A. R. Plastino and A. Plastino [4], Giordano et al. [5] have also discussed Robe's problem.But all of them have discussed the linear stability of the equilibrium points.Hallan and Mangang [6] discussed the nonlinear stability of equilibrium point of Robe's restricted three-body problem when K = 0 in the linear stability range 8/9 < μ < 1 and they found that the equilibrium point is stable in nonlinear sense for all mass ratios except for the five mass ratios μ 1 = 0.93711086 . . ., μ 2 = 0.9672922 . . ., μ 3 = 0.9459503 . . ., μ 4 = 0.9660792 . . ., μ 5 = 0.893981 . . ., where the KAM theory is not applicable.Many authors discussed nonlinear stability of equilibrium points.Recently, Elipe and L ópez-Moratalla [7] discussed on the Lyapunov stability of stationary points around a central body.Elipe et al. [8] studied stability of equilibria in two degrees of freedom Hamiltonian system.Elipe et al. [9] discussed nonlinear stability in resonant cases.In the present study, we wish to discuss the effects of perturbations in Coriolis and centrifugal forces on the nonlinear stability of equilibrium point (−μ + (μ 1 /(1 + 2μ)), 0, 0) found by Hallan and Rana [2] in Robe's restricted circular threebody problem by taking the density parameter K as zero by applying Moser's version of the Arnold theorem (KAM Advances in Astronomy theory) and following the procedure as that adopted by Hallan and Mangang [6].
Moser's version [10] of Arnold theorem [11] states the following. If is the normalized Hamiltonian with I 1 , I 2 , I 3 as the action momenta coordinates and ω 1 , ω 2 , ω 3 are the basic frequencies for the linear dynamical system, then on each energy manifold H = in the neighborhood of an equilibrium point, there exist invariant tori of quasiperiodic motions which divide the manifold and consequently the equilibrium point is stable provided that (i) k 1 ω 1 + k 2 ω 2 + k 3 ω 3 / = 0, for all triplets (k 1 , k 2 , k 3 ) of rational integers such that (ii) determinant D / = 0, Applying Arnold's theorem, Leontovich [12] proved that the triangular equilibrium points in the restricted three-body problem are stable for all permissible mass ratios except for a set of measure zero.Deprit and Deprit-Bartholome [13] discussed nonlinear stability of the triangular equilibrium points of the classical restricted three-body problem by applying Moser's theorem.Bhatnagar and Hallan [14] also discussed the nonlinear stability of the triangular equilibrium points in the same problem after considering perturbations in Coriolis and centrifugal forces.In another paper, Bhatnagar and Hallan [15] discussed the nonlinear stability of a cluster of stars sharing galactic rotation.
By applying the Lyapunov theorem [16] to the linear stability result obtained by Hallan and Rana [2] in Robe's restricted three-body problem, we can say that the equilibrium point, (−μ + (μ 1 /(1 + 2μ)), 0, 0), is unstable in the nonlinear sense also for 0 < μ ≤ μ c .Therefore, we will study the nonlinear stability of the equilibrium point for μ c < μ < 1.

First-Order Normalization
Using nondimensional variables and a synodic system of coordinates (x, y, z) and considering perturbations , 1 , respectively, in Coriolis and centrifugal forces, the equations of motion of Robe's restricted problem, when density parameter K = 0 and eccentricity e = 0, are [2] ẍ where mass of the first primary along with the mass of the fluid inside it.
Lagrangian L of the problem is ( There is only one equilibrium point (−μ + p, 0, 0), where p = μ 1 /(1 + 2μ) [2].Shifting the origin to (−μ + p, 0, 0) and expanding in Taylor series expansion and neglecting second and higher degree terms in , 1 , the Lagrangian can be written as where To the first order, Lagrange's equations of motion are The characteristic equation of the first two equations is where p = μ/(1 + 2μ).

Advances in Astronomy 3
The characteristic equation of the third equation is Equation ( 9) has pure imaginary roots if [2] and it is obvious that (10) has pure imaginary roots.
The four characteristic roots of ( 9) are ±iω 1 , ±iω 2 and the two characteristic roots of (10) are ±iω 3 , where ω 1 , ω 2 , ω 3 represent the perturbed basic frequencies of the linear dynamical system.We can write , where ω 1 , ω 2 , ω 3 represent the unperturbed basic frequencies of the linear dynamical system such that From ( 13), we see that Following the method given by Whittaker [17], we use the canonical transformation from the phase space (x, y, z, p x , p y , p z ) into the phase space product (ϕ 1 , ϕ 2 , ϕ 3 , I 1 , I 2 , I 3 ) of the angle coordinates ϕ 1 , ϕ 2 , ϕ 3 and action momenta I 1 , I 2 , I 3 given by where , Advances in Astronomy , The transformation changes the second-order part of the Hamiltonian into the normal form The general solution of the corresponding equations of motion are

Second-Order Normalization
We wish to perform Birkhoff 's normalization for which the coordinates (x, y, z) are to be expanded in double D'Alembert's series: where the homogeneous components B 1,0,0 n , B 0,1,0 n , B 0,0,1 n of degree n are of the form 0< ,m<n The double summation over the indices i, j, and k is such that (a) i runs over those integers in the interval 0 ≤ i ≤ n − − m that have the same parity as n − − m, (b) j runs over those integers in the intervals − ≤ j ≤ that have the same parity as , (c) k runs over those integers in the interval −m ≤ k ≤ m that have the same parity as m.I 1 , I 2 , and I 3 are to be regarded as constants of integration and ϕ 1 , ϕ 2 , and ϕ 3 are to be determined as linear functions of time such that where f 2n , g 2n , h 2n are of the form As shown by Hallan and Mangang [6], the first-order components B 1,0,0 1 , B 0,1,0 1 , and B 0,0,1 1 are the values of x, y, and z given by (15).B 1,0,0 2 , B 0,1,0 2 , and B 0,0,1 2 are the solutions of the partial differential equations where and X 2 , Y 2 , Z 2 are obtained from ∂L 3 /∂x, ∂L 3 /∂y, ∂L 3 /∂z, respectively, by substituting the first-order components for x, y, z.

Stability
Now we apply Moser's modified form of Arnold's theorem [11] to discuss the nonlinear stability.We have The condition (i) of the theorem is satisfied provided the basic frequencies do not satisfy the equations Out of these ten equations (I)-(X) in ω 1 , ω 2 , ω 3 , (IX) and (X) along with ( 12) and ( 13) do not give the values of μ in the interval μ c < μ < 1.The remaining eight from (I) to (VIII) are the resonance cases.Taking any of the equations from (I) to (VIII) and eliminating ω 1 , ω 2 , ω 3 from that equation as well as ( 12) and ( 13), the eliminant is an equation in μ.Solving those equations, we get only five roots in the range μ c < μ < 1.They are For these values of μ, the condition (i) of the theorem does not hold.The determinant D occurring in the condition (ii) of the theorem is where D / = 0 if the value of μ, in the range μ c < μ < 1, does not satisfy the equation obtained by eliminating ω 1 , ω 2 , ω 3 from the equation and ( 12) and ( 13).