Oscillatory collision approach in the Earth-Moon restricted three body problem

Abstract

We consider the Earth-Moon planar circular restricted three body problem and present a proof of the existence orbits, which approach arbitrarily close to one of the primary masses, and at the same time after each approach they move away from the mass to a prescribed distance. In other words the orbits oscillate between being arbitrarily close to collision and away from it. We achieve our goal with the use of topological tools combined with rigorous interval computations. We use the Levi-Civita regularization and validate that the dynamics in the regularized coordinates leads to a good topological alignment between various sets. We then perform shadowing arguments that this leads to the required dynamics in the original coordinates of the system.

Figures (10)

• Figure 1: Lyapunov orbits in the regularized coordinates, where the collision is at the origin (left) and in the original coordinates, where the collision is $(x_2,0)=(-\mu_1,0)$ at the tip of the "left wedge" (right). The ejection/collision orbit is depicted in red. The orbits pass close to the Moon, which is at $(1,0)$ on the left plot and at $(x_1,0)=(\mu_2,0)$ on the right plot.
• Figure 2: A depiction of a covering relation $N \xRightarrow{f} M$. The exit sets $N_c^-$ and $M_c^-$ are in red and blue, respectively. On the right we have the final result of the homotopy from Definition \ref{['def:covering-relation']}.
• Figure 3: The local coordinates given by $\psi_0$, in which the collision curve is the straight line depicted in pink. Here we also plot a self $S$-symmetric h-set $Q$ from Lemma \ref{['lem:Q-self-S-symmetric']}.
• Figure 4: The collision/ejection orbit (red) and its homoclinic (green) in the regularized coordinates (projection on coordinates $u, v$).
• Figure 5: The collision/ejection orbit (red) and the homoclinic orbit (green) in standard coordinates.

Theorems & Definitions (57)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 4
• Definition 1
• Definition 2