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

Abstract

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 is zero. By applying Kolmogorov-Arnold-Moser (KAM) theory, it has been found that the equilibrium point is stable for all mass ratios in the range of linear stability , where and are, respectively, the perturbations in Coriolis and centrifugal forces, except for five mass ratios , , , + 1.11684064, + 1.22385421, where the theory is not applicable.

1. Introduction

Robe [1] has considered a new kind of restricted three-body problem in which one of the primaries is a rigid spherical shell filled with a homogeneous incompressible fluid of density . The second primary is a mass point outside the shell and the third body is a small solid sphere of density , inside the shell, with the assumption that the mass and radius of are infinitesimal. He has shown the existence of an equilibrium point with at the center of the shell, while 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 , 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 is zero. They have found that is the only equilibrium point and in the linear sense it is stable for and unstable for , where . 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 in the linear stability range and they found that the equilibrium point is stable in nonlinear sense for all mass ratios except for the five mass ratios , , , , , where the KAM theory is not applicable. Many authors discussed nonlinear stability of equilibrium points. Recently, Elipe and López-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 found by Hallan and Rana [2] in Robe's restricted circular three-body problem by taking the density parameter as zero by applying Moser's version of the Arnold theorem (KAM 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 , , as the action momenta coordinates and , , are the basic frequencies for the linear dynamical system, then on each energy manifold 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) for all triplets of rational integers such that(ii) determinant ,

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, , is unstable in the nonlinear sense also for . Therefore, we will study the nonlinear stability of the equilibrium point for .

2. First-order Normalization

Using nondimensional variables and a synodic system of coordinates and considering perturbations , , respectively, in Coriolis and centrifugal forces, the equations of motion of Robe's restricted problem, when density parameter and eccentricity , are [2]where , , , , (), = mass of the second primary, = mass of the first primary along with the mass of the fluid inside it.

Lagrangian of the problem isThere is only one equilibrium point , where [2]. Shifting the origin to and expanding in Taylor series expansion and neglecting second and higher degree terms in , , the Lagrangian can be written aswhereTo the first order, Lagrange's equations of motion areThe characteristic equation of the first two equations iswhere .

The characteristic equation of the third equation isEquation (9) has pure imaginary roots if[2] and it is obvious that (10) has pure imaginary roots. The four characteristic roots of (9) are , and the two characteristic roots of (10) are , where , , represent the perturbed basic frequencies of the linear dynamical system. We can writewhere , , represent the unperturbed basic frequencies of the linear dynamical system such thatFrom (13), we see that , therefore, we have .

Following the method given by Whittaker [17], we use the canonical transformation from the phase space into the phase space product of the angle coordinates , , and action momenta , , given bywhereThe transformation changes the second-order part of the Hamiltonian into the normal formThe general solution of the corresponding equations of motion are

3. Second-order Normalization

We wish to perform Birkhoff's normalization for which the coordinates are to be expanded in double D'Alembert's series:where the homogeneous components , , of degree are of the formThe double summation over the indices , , and is such that (a) runs over those integers in the interval that have the same parity as , (b) runs over those integers in the intervals that have the same parity as , (c) runs over those integers in the interval that have the same parity as . , , and are to be regarded as constants of integration and , , and are to be determined as linear functions of time such thatwhere , , are of the form

As shown by Hallan and Mangang [6], the first-order components , , and are the values of and given by (15). , and are the solutions of the partial differential equationswhereand , , are obtained from , , , respectively, by substituting the first-order components for , , .

Equation (23) can be solved for , , by using the formulaewhereThe second-order components , , and are as follows:whereand , , , , , are given in the appendix. We have checked that , , and transform , the third-order part of the Hamiltonian, to zero.

4. Second-order Coefficient in The Frequencies

Proceeding as in the work of Hallan and Mangang [6], the third-order components , , and in the coordinates , , and the second-order polynomials , , and in the frequencies , , and satisfy the partial differential equationswhereand are the homogeneous components of order 3 obtained, respectively, from , , by substitutingThe components , and are not required to be found out. We find the coefficients of , () on the right-hand side of (29). They are the critical terms as .

We eliminate these terms by choosing properly the coefficients in the polynomialsWe find thatwhere

If the normalized Hamiltonian is written asthen, from Hamilton's equations of motionand (21), we find that

5. Stability

Now we apply Moser's modified form of Arnold's theorem [11] to discuss the nonlinear stability. We haveThe condition (i) of the theorem is satisfied provided the basic frequencies do not satisfy the equations (I),(II),(III),(IV),(V),(VI),(VII),(VIII),(IX),(X). Out of these ten equations (I)–(X) in , , , (IX) and (X) along with (12) and (13) do not give the values of in the interval . The remaining eight from (I) to (VIII) are the resonance cases. Taking any of the equations from (I) to (VIII) and eliminating , , from that equation as well as (12) and (13), the eliminant is an equation in . Solving those equations, we get only five roots in the range . They areFor these values of , the condition (i) of the theorem does not hold.

The determinant occurring in the condition (ii) of the theorem iswhere , , , , , are the cofactors of , , , , , , respectively, in the determinant if the value of , in the range , does not satisfy the equation obtained by eliminating , , from the equationand (12) and (13).

Using Mathematica 5.1, the eliminant is , where

So condition (ii) of the theorem is not satisfied for those values of which satisfy the equationand also for the value , where , and consequently is not defined. The roots of the equation when are seventeen in number, [6], out of which nine are real and they areWhen , let the roots be (). Putting these roots in (44) and solving for , after neglecting higher-order terms in , , we haveNone of these roots lie in the range . Hence, the equilibrium point is stable in the nonlinear sense in the range of linear stability for all values of except , , , , , where the KAM theory is not applicable and consequently no conclusion about stability can be drawn for the five mass ratios. The result is in agreement with that result found out by Hallan and Mangang [6] when there is no perturbations in Coriolis and centrifugal forces ().

Appendix

Acknowledgment

The authors are thankful to the Council of Scientific and Industrial Research (C.S.I.R.), Government of India, for providing financial support for this research work.