Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem

Abstract

We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in [11], which is used to model small oscillations of the earth's inner core taking into account the moon attraction. For the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem, earlier results of such linear stability problem depend on a lot of numerical computations, while we give an analytic approach to it. The linearized Hamiltonian system near the elliptic relative equilibrium point in our problem coincides with the linearized system near the Euler elliptic relative equilibria in the classical three-body problem except for the rang of the mass parameter. We first establish some relations from the linear stability problem to symplectic paths and its corresponding linear operators. Then using the Maslov-type ω-index theory of symplectic paths and the theory of linear operators, we compute ω-indices and obtain certain properties of the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem. Figure 1: The Robe's restricted three-body problem considered: m 1 is a spherical shell filled with a fluid of density ρ 1 ; m 2 a mass point outside the shell and m 3 a small solid sphere of density ρ 3 inside the shell. Later on, A. R. Plastino and A. Plastino ([10]) studied the linear stability of the equilibrium points and the connection between the effect of the buoyancy forces and a perturbation of a Coriolis force. In 2001, P. P. Hallen and N. Rana ([1]) found other new equilibrium points of the restricted problem and discussed their linear stabilities. K. T. Singh, B. S. Kushvah and B. Ishwar ([ 13]) examined the stability of triangular equilibrium points in Robe's generalized restricted three body problem where the problem is generalized in the sense that a more massive primary has been taken as an oblate spheroid.However, in [11], for the elliptic equilibrium points, the bifurcation diagram of linear stability was obtained by numerical methods. In [10,1,12,13], the authors studied the linear stability of some kinds of equilibrium points, but their studies did not contain the elliptic case.On the other hand, in [4, 5] of 2009-2010, X. Hu and S. Sun found a new way to relate the stability problem to the iterated Morse indices. Recently, by observing new phenomenons and discovering new properties of elliptic Lagrangian solutions, in the joint paper [2] of X. Hu, Y. Long and S. Sun, the linear stability of elliptic Lagrangian solutions is completely solved analytically by index theory (cf. [6]) and the new results are related directly to (β, e) in the full parameter rectangle. Inspired by the analytic method, Q. Zhou and Y. Long in [14] studied the linear stability of elliptic triangle solutions of the charged three-body problem.Recently, in [15,16], Q. Zhou and Y. Long studied the linear stability of elliptic Euler-Moulton solutions of n-body problem for n = 3 and for general n ≥ 4, respectively. Also, the linear stability of Euler collision so...

Trace estimation of a family of periodic Sturm-Liouville operators with application to Robe’s restricted three-body problem

“…For the elliptic case, the studies of the linear stability of equilibrium point are much more complicated than that of the circular case, thus in [12], the bifurcation diagram of linear stability was obtained just by numerical methods. Recently, in [14], the author and Y. Zhang have studied the linear stability of the elliptic equilibrium point along the xy-plane analytically by using the Maslov-type index theory. For completeness, in this section, we will study the linear stability of such equilibrium point along the z-axis analytically.…”