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

We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in [ 13 ], which is used to model small oscillations of the earth's inner core taking into account the moon's attraction. Earlier results for the linear stability of the elliptic equilibrium point in Robe's restricted problem depend on a lot of numerical computations, while we give an analytic approach to it. The linearized Hamiltonian system near the elliptic 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 range of the mass parameter. We first establish some relations of the linear stability problem to the properties of some symplectic paths and some corresponding linear operators. Then using the Maslov-type \begin{document} $ω$ \end{document} -index theory of symplectic paths and the theory of linear operators, we compute \begin{document} $ω$ \end{document} -indices and obtain certain properties of the linear stability of the elliptic equilibrium point of Robe's restricted three-body problem.


Introduction and main results.
A new kind of restricted three-body problem that incorporates the effect of buoyancy forces was introduced by Robe in 1977. In [13], he regarded one of the primaries as a rigid spherical shell m 1 filled with a homogenous 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 the equilibrium point with m 3 at the center of the shell, where m 2 describes a Keplerian orbit around it, see Figure 1.
Furthermore, he discussed two cases of the linear stability of the equilibrium point of such a restricted three-body problem. In the first case, the orbit of m 2 around m 1 is circular and in the second case, the orbit is elliptic, but the shell is empty (that is no fluid inside it) or the densities of m 1 and m 3 are equal. In the second case, we use "elliptic equilibrium point" to call the equilibrium point. In each case, the domain of stability has been investigated for the whole range of parameters occurring in the problem.
Later on, A. R. Plastino and A. Plastino ([12]) studied the linear stability of the equilibrium point 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 ( [3]) found other new equilibrium points of the restricted problem and discussed their linear stabilities. K. T. Singh, B. S. Kushvah and B. Ishwar ( [15]) examined the stability of triangular equilibrium point 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, 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 [13], the bifurcation diagram of linear stability was obtained just by numerical methods. In [12,3,14,15], the authors studied the stability of equilibrium points in a modified problem, but their studies did not contain the elliptic case.
On the other hand, in [6,7] 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 phenomena and discovering new properties of elliptic Lagrangian solutions, in the joint paper [4] of X. Hu, Y. Long and S. Sun, the linear stability of elliptic Lagrangian solutions is completely solved analytically by index theory (cf. [8]) and the new results are related directly to (β, e) in the full parameter rectangle. Here β is the mass parameter which was given by (1.4) of [4], and e is the common eccentricity of the elliptic orbit of each body. Inspired by the analytic method, Q. Zhou and Y. Long in [16] studied the linear stability of elliptic triangle solutions of a charged three-body problem.
Recently, in [17,18], 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 solutions of 3-body problem was studied by X. Hu and Y. Ou in [5].
In the current paper, we study an analytical approach to the linear stability of equilibrium point of the Robe's restricted three-body problem. We related their linear stabilities to the Maslov-type and Morse indices of them. For such elliptic equilibrium point, we use index theory to compute the Maslov-type indices of the corresponding symplectic paths and determine their stability properties. Following Robe's notation in [13], various forces acting on m 3 are: (1) The attraction of m 2 , (2) The gravitational force of attraction of fluid density ρ 1 .
Let the orbital plane of m 2 around m * 1 (that is the shell with its fluid) be taken as the xy-plane and let the origin of the coordinate system be at the center of the mass, O, of the two primaries. The equation of motion of m 3 is where Assume that m 2 describes around m * 1 an elliptical orbit of eccentricity e and long semi-axis a, the distance l between the two bodies is given by where θ is the true anomaly. The equations of motion in the xy-plane obtained by Robe are: where V is given by x 1 and x 2 being the x coordinates of M 1 and M 2 : Here Robe used a non-uniformly rotating and pulsating coordinate system. In [13], H. Robe studied the equilibrium point at the center of m 1 in the circular case and in the elliptic case. He also studied the linear stability of the above two cases of such equilibrium point. But for the elliptic case, only numerical results was obtained. Later on, in [3], P. P. Hallen and N. Rana studied the existence of all the equilibrium points in the Robe's restricted three-body problem. They found that, in the case of equilibrium points with circular, there are serval different situations depending on K, and the linear stability of such equilibrium points were carefully studied. More details can be seen in [3].
We focus on the case when there is no fluid inside the shell or when the densities of m 1 and m 3 are equal (ρ 1 = ρ 3 ), i.e., K = 0. Here K reflect the joint effect of the gravitational force F A and the buoyancy force F B . K = 0 implies that the total effect of the fluid is disappeared. By (17)-(19) in [13],the linearized equations of motion around the equilibrium point are: which is a set of linear homogeneous equations with periodic coefficients of periodic 2π. Now we study equations (9)-(10) by another form. Let (W 1 , W 2 , w 1 , w 2 ) T = (ẋ − y,ẏ + x, x, y) T and t = θ, then we have Let J = 0 −I n I n 0 be the standard symplectic matrix for any n ∈ N, and then (11) can be written asẇ = JB(t)w, where w = (W 1 , W 2 , w 1 , w 2 ) T . When µ = β + 1, B(t) of (12) coincides with B(t) of (2.35) in [17]. Thus a lot of results which were developed in [17] can be applied to this paper.
For any γ ∈ P 2π (2n) we define ν ω (γ) = ν ω (γ(2π)) and , the usual homotopy intersection number, and the orientation of the joint path γ * ξ n is its positive time direction under homotopy with fixed end points. When γ(2π) ∈ Sp(2n) 0 ω , we define i ω (γ) be the index of the left rotation perturbation path γ − with > 0 small enough (cf. Def. 5.4.2 on p.129 of [11]). The pair (i ω (γ), ν ω (γ)) ∈ Z × {0, 1, . . . , 2n} is called the index function of γ at ω. When ν ω (γ) = 0 (ν ω (γ) > 0), the path γ is called ω-non-degenerate (ω-degenerate). For more details we refer to the Appendix or [11]. Based on the above notation, we will give some statements on the stability and instability of the periodic solutions of the Hamiltonian systems via indices of the orbits. Recall that, for M ∈ Sp(2n), it is linearly stable if M j ≤ C for some constant C and all j ∈ N. Note that this implies M is diagonalizable and the eigenvalues of M are all on the unit circle U of the complex plane. We call M to be spectrally stable if all its eigenvalues are on the unit circle. Definition 1.1. Given a T -periodic solution z(t) to a first order Hamiltonian system with fundamental solution γ(t), we say z is spectrally stable (linearly stable) if γ(T ) is spectrally stable (linearly stable, respectively).
In the literature there are many papers concerning the stability of the periodic solutions of the Hamiltonian system using the Maslov-type index [2,9,11]. The complete iteration formula developed by Y. Long and his collaborators is a very effective tool for this purpose.
The following two theorems describe the main results proved in this paper.
Remark 2. When (µ, e) is in Region II or IV, i.e., the shaded regions in Figure 2, then γ µ,e (2π), and hence the elliptic equilibrium point is linear stable. Comparing Figure 3 of [13] with our Figure 2, I is the point µ = 8/9, F is the point µ = µ * in the µ-axis and the shaded regions represent the same parameter regions. In Robe's figure, the boundary point G of the right shaded region is inside Θ. But in our theorem, we can strictly obtain that G is just the point (µ, e) = (1, 1).
The paper is organized as follows. In Section 2, we associate γ µ,e (t), the fundamental solution of the system (13), with a corresponding second order self-adjoint operator A(µ, e). Some connections between γ µ,e (t) and A(µ, e) are given there. In Section 3, we compute the ω-indices along the three boundary segments of (µ, e) rectangle [0, 1] × [0, 1). In Section 4, the non-decreasing property of ω-index is proved in Lemma 4.1 and Corollary 1. Also Theorem 1. 2. Associate γ µ,e (t) with a second order self-adjoint operator A(µ, e). In the Appendix, we give a brief review on the Maslov-type ω-index theory for ω in the unit circle of the complex plane following [11]. In the following, we use notation introduced there. Let and set where a · b denotes the inner product in R 2 . By Legendrian transformation, the corresponding Hamiltonian function to system (13) is Now let γ = γ µ,e (t) be the fundamental solution of the (13) satisfies: In order to transform the Lagrangian system (26) to a simpler linear operator corresponding to a second order Hamiltonian system with the same linear stability as γ µ,e (2π), using R(t) and R 4 (t) = diag(R(t), R(t)) as in Section 2.4 of [4], we let One can show by direct computation that Note that R 4 (0) = R 4 (2π) = I 4 , so γ µ,e (2π) = ξ µ,e (2π) holds and the linear stabilities of the systems (27-28) and (30) are precisely the same. By (29) the symplectic paths γ µ,e and ξ µ,e are homotopic to each other via the homotopy h( (4) which is homotopic to the constant loop γ µ,e (2π), we have γ µ,e ∼ 1 ξ µ,e by the homotopy h. Then by Lemma 5.2.2 on p.117 of [11], the homotopy between γ µ,e and ξ µ,e can be realized by a homotopy which fixes the end point γ µ,e (2π) all the time. Therefore by the homotopy invariance of the Maslov-type index (cf. (i) of Theorem 6.2.7 on p.147 of [11]) we obtain Note that the first order linear Hamiltonian system (30) corresponds to the following second order linear Hamiltonian system For (µ, e) ∈ [0, 1] × [0, 1), the second order differential operator corresponding to (32) is given by where S(t) = cos 2t sin 2t sin 2t − cos 2t , defined on the domain D(ω, 2π) in (132). Then it is self-adjoint and depends on the parameters µ and e. By Lemma 5.6, we have for any µ and e, the Morse index φ ω (A(µ, e)) and nullity ν ω (A(µ, e)) of the operator A(µ, e) on the domain D(ω, 2π) satisfy In the rest of this paper, we shall use both of the paths γ µ,e and ξ µ,e to study the linear stability of γ µ,e (2π) = ξ µ,e (2π). Because of (31), in many cases and proofs below, we shall not distinguish these two paths.
Note that A(µ, e) is the same as A(β, e) when β = µ − 1 in [17] with different parameter ranges, then by Lemma 4.2 in [17] and modifying its proof to the different range of parameters, we get the following important lemma: Lemma 4.1. (i) For each fixed e ∈ [0, 1), the operatorĀ(µ, e) is non-increasing with respect to µ ∈ (0, 1) for any fixed ω ∈ U. Specially is a non-negative definite operator for µ 0 ∈ (0, 1).
Then under similar steps to those of Lemma 6.2 and Theorem 6.3 in [4], we can prove the theorem. By (62), −1 is a double eigenvalue of the matrix γ µ * ,e (2π), then the two curves bifurcate out from (µ * , 0) when e > 0 is small enough.

Now we can give
The proofs of Theorem 1.3 parts (i)-(iii).
Now we can give The Proofs of Theorem 1.3 parts (iv)-(x). Some arguments below are use the methods in the proof of Theorem 1.2 in [4].
If Case (A) happens, for this µ 0 , by a similar argument of Theorem 1.7 in [4], there exists e 0 > 0 sufficiently close to 1 such that γ µ,e (2π) is hyperbolic for all (µ, e) in the region (0, µ l (ê i )] × [e 0 , 1). Then by the monotonicity of Case (A) we obtain Therefore (µ l (ê i+m ),ê i+m ) will get into this region for sufficiently large m ≥ 1, which contradicts the definition of µ l (ê i+m ). If Case (B) happens, the proof is similar. Thus (v) holds.
If γ ∈ P * τ,ω (2n), define where the right hand side of (114) is the usual homotopy intersection number, and the orientation of γ * ξ n is its positive time direction under homotopy with fixed end points. If γ ∈ P 0 τ,ω (2n), we let F(γ) be the set of all open neighborhoods of γ in P τ (2n), and define i ω (γ) = sup is called the index function of γ at ω.

5.2.
Morse index of Lagrangian system. Our purpose here is to give the relation for the Morse index and the Maslov-type index which covers the applications to our problem.
For T > 0, suppose x is a critical point of the functional where L ∈ C 2 ((R/T Z) × R 2n , R) and satisfies the Legendrian convexity condition L p,p (t, x, p) > 0. It is well known that x satisfies the corresponding Euler-Lagrangian equation: x(0) = x(T ),ẋ(0) =ẋ(T ).
We define the ω-Morse index φ ω (x) of x to be the dimension of the largest negative definite subspace of the index form I which was defined on D(ω, T ) × D(ω, T ). Moreover, F (x) is a self-adjoint operator on L 2 ([0, T ], R n ) with domain D(ω, T ). We also define ν ω (x) = dim ker(F (x)).
In general, for a self-adjoint operator A on the Hilbert space H , we set ν(A) = dim ker(A) and denote by φ(A) its Morse index which is the maximum dimension of the negative definite subspace of the symmetric form A·, · . Note that the Morse index of A is equal to the total multiplicity of the negative eigenvalues of A.
A generalization of the above lemma to arbitrary boundary conditions is given in [6]. For more information on these topics, we refer to [11].