Lissajous and Halo Orbits in the Restricted Three-Body Problem

Abstract

We study the dynamics near the collinear Lagrangian points of the spatial, circular, restricted three-body problem. Following a standard procedure, we reduce the system to the center manifold and we analyze the Lissajous orbits as well as the halo orbits, the latter ones arising from bifurcations of the planar Lyapunov family of periodic orbits. To obtain the Lissajous orbits, we perform a classical perturbation theory and we provide a formal approximate solution under suitable non-degeneracy and non-resonance conditions. As for the halo orbits, we construct a normal form adapted to the synchronous resonance: introducing a detuning, measuring the displacement from the resonance, and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place. Except for a particular case, the analytical values obtained after a second order resonant perturbation theory are in very good agreement (in some cases up to the fourth decimal digit) with the numerical values found in the literature.

Appendix: Proof of Proposition 2

The proof of Proposition 2 can be found in several papers (see, e.g., Jorba and Masdemont 1999). For self-consistency, we give in this Appendix some details of the proof.

Let $P=(P_1,P_2,P_3)$, $Q=(Q_1,Q_2,Q_3)$ and let us introduce a generating function $G=G(P,Q)$, that we expand as a sum of homogeneous polynomials $G=\sum _{k\ge 3} G_k$ with

$$\begin{aligned} G_k(P,Q)=\sum _{k_p,k_q\in {{\mathbb {Z}}}^3,\ |k_p|+|k_q|=k}g_{k_p,k_q}P^{k_p}Q^{k_q}, \end{aligned}$$

where $|k_p|=\sum _{j=1}^3 |k_{pj}|$ (similarly for $k_q$), while $P^{k_p}$ stands for $P_1^{k_{p1}} P_2^{k_{p2}} P_3^{k_{p3}}$ (similarly for $Q^{k_q}$). At each order $k$, the terms $G_k$ are defined in order to separate the center and hyperbolic directions, so to obtain a first integral which admits the center manifold as level surface. This can be achieved by eliminating all monomials such that the first component of $k_p$ is different from the first component of $k_q$, say $k_{p1}\not =k_{q1}$. Precisely, denote by $H_{2q}$ the quadratic part in (4.1). The generating function $G$ induces a transformation of coordinates, such that the new Hamiltonian $\hat{H}$ is given by

$$\begin{aligned} \hat{H}=H_2^{(c)}+\left\{ H_2^{(c)},G\right\} +{1\over {2!}}\left\{ \{H_2^{(c)},G\},G\right\} +\cdots , \end{aligned}$$

where $\{\cdot ,\cdot \}$ denotes the Poisson brackets. Let us start to determine the third-order term $G_3$ of $G$. Let $\hat{H}=\sum _{k\ge 2}\hat{H}_k$, where $\hat{H}_k$ are homogeneous polynomials of degree $k$. Equating terms of the same degree in $P$, $Q$, we obtain that

$$\begin{aligned} \hat{H}_2&= H_{2q}\nonumber \\ \hat{H}_3&= H_3+\left\{ H_{2q},G_3\right\} \nonumber \\ \hat{H}_4&= H_4+\left\{ H_3,G_3\right\} +{1\over {2!}}\left\{ \{H_{2q},G_3\},G_3\right\} ,\ldots \end{aligned}$$

(6.12)

We determine $G_3$ is such a way to eliminate all monomials of the form $P^{k_p}Q^{k_q}$ with $k_{p1}\not =k_{q1}$. Expanding $H_3$ as

$$\begin{aligned} H_3(P,Q)=\sum _{k_p,k_q\in {{\mathbb {Z}}}^3,\ |k_p|+|k_q|=3}h_{k_p,k_q}^{(3)}\ P^{k_p}Q^{k_q}, \end{aligned}$$

then from the second of (6.12) we obtain that $G_3$ is given by

$$\begin{aligned} G_3(P,Q)=-\sum _{k_p,k_q\in {{\mathbb {Z}}}^3,\ |k_p|+|k_q|=3,\ k_{p1}\not =k_{q1}}{{h_{k_p,k_q}^{(3)}}\over {\langle k_p-k_q,\omega \rangle }}\ P^{k_p}Q^{k_q}, \end{aligned}$$

where $\langle \cdot ,\cdot \rangle$ denotes the scalar product and $\omega \equiv (\lambda _x,i\omega _y,i\omega _z)$. Thus, we have obtained that the new Hamiltonian has the desired form (4.2) up to the third order. Iterating the procedure up to the order $N$ and determining $G_4,\ldots ,G_N$ as we did for $G_3$, we obtain the Hamiltonian (4.2), where the polynomials $\tilde{H}_n$ will depend on $Q_1, P_1$, only through the product $Q_1P_1$.