Branches and bifurcations of ejection-collision orbits in the planar circular restricted three body problem

TL;DR

The study addresses the existence and global structure of ejection-collision heteroclinic orbits in the planar CRTBP by formulating a constructive, computer-assisted fixed-point problem that leverages Levi-Civita regularization to handle collisions. It proves three main results: (i) energy-parameterized branches for fixed \\mu\\ extending from one primary to the other, (ii) a real-analytic \\mu = 1/2\\ branch with a saddle-node bifurcation at \\C^*\\, and (iii) a similar Earth–Moon \\mu_{em} branch with a saddle-node, all obtained via high-order \\mathcal{X}-Taylor expansions and rigorous a-posteriori validation. The approach is non-perturbative and provides a framework for global continuation of ejection-collision branches, with potential applicability to broader conservative systems and to invariant-manifold/homoclinic–heteroclinic structures in celestial mechanics. The methods combine fixed-point theory, bifurcation analysis, and validated numerics to produce rigorous existence results and usable bifurcation diagrams for these delicate orbital phenomena.

Abstract

The goal of this paper it to prove existence theorems for one parameter families (branches) of ejection-collision orbits in the planar circular restricted three body problem (CRTBP), and to study some of their bifurcations. The orbits considered are ejected from one primary body and collide with the other (as opposed to more local ejections-collision orbits which involve only a single body). We consider branches which are (i) parameterized by the Jacobi integral (energy like quantity conserved by the CRTBP) and (ii) parameterized by the two body mass ratio when energy is fixed. The method of proof is constructive and computer assisted, hence can be applied in non perturbative settings and (potentially) to other conservative systems of differential equations. The main requirement is that the system should admit a change of coordinates which regularizes the singularities (collisions). In the planar CRTBP the necessary regularization is provided by the classical Levi-Civita transformation.

Figures (7)

• Figure 1: Configuration of the Circular Restricted Three Body Problem: the left frame illustrates the motion of the massive primary bodies $m_1$ and $m_2$ in inertial coordinates. The are restricted a-priori to Keplerean circles. The line determined by the position of $m_1$, $m_2$ and the center of mass rotates at a constant speed, and it is possible to change to a co-rotating coordinate system with constant angular velocity, thus removing the motion of primaries. Inserting a massless test particle into the resulting gravitational vortex, one obtains the CRTBP as illustrated in the right frame.
• Figure 2: Branches parameterized by energy: The 5 approximate heteroclinic ejection-collision orbits form the statement of Theorem \ref{['th:other']}. The left frame illustrates trajectories 1 (black) and 2 (red) in the Earth-Moon system, while the right frame illustrates trajectories 3 (black), 4 (red), and 5 (blue) in the equal mass (or Copenhagen) system.
• Figure 3: A global branch of heteroclinic ejection-collision orbits parameterized by mass ratio: The figure illustrates some approximate orbits from Theorem \ref{['th:fixedC']}. We note that, in the CRTBP, the mass parameter $\mu$ is normalized so that it lies in the interval $[0, 1/2]$. This branch starts a $\mu = 1/2$, decreases to $\mu = \mu_* < 1/2$, undergoes a fold bifurcation, and returns to $\mu = 1/2$: hence we follow it over its full lifespan. The blue curve illustrates the solution at the bifurcation parameter (mass ratio) $\mu=\mu^*$, while the red and black the solutions at $\mu=1/2$. The figure illustrates the result of Theorem \ref{['th:fixedC']}, which shows that there exists a one parameter family of ejection-collision orbits, parameterized by $\mu$, which connects the red to the black solution through the saddle node bifurcation at the blue solution. Note that for the orbits in this family, the radial velocity with respect to the large primary vanishes twice, hence the family is dynamically different from the one established in MR0682839.
• Figure 4: Levi-Civita Regularized Coordinates: center top figure depicts the configuration space of the CRTBP, with singularities at $(x,y) = (-1 + \mu, 0)$ and at $(x,y) = (\mu, 0)$ due to collision of the massless particle with the smaller primary $m_2$ or the larger primary $m_1$ respectively. The bottom left and right figures depict the configuration spaces of the Levi-Civita systems associated with regularization of the smaller and larger masses respectively. Note that in the bottom left and right frames the massive bodies $m_2$ and $m_1$ respectively have been moved to the origin. These bodies are depicted as "ghosts" using transparencies to indicate that the resulting fields are perfectly well defined there. However, the double covering introduced by the complex squaring function creates mirror images of the remaining primary, so that the Levi-Civita systems still have a pair of singular point. These mirrored singular points are located at $(x,y) = (\pm 1, 0)$ for the case the regularization at $m_2$ and at $(x,y) = (0, \pm 1)$ for the regularization at $m_1$. The mirrored singularities play no role in our set up, as we employ the Levi-Civita coordinates only in a small neighborhood of the origin.
• Figure 5: Approximate orbits in the configuration space in rotating coordinates of the solution at $C^*$ obtained by Theorem \ref{['th:muonehalf']}. For this family the radial velocity with respect to the first primary does not vanish, hence it is related to the family of MR0682839. However, Theorem \ref{['th:muonehalf']} holds at the highly non-perturbative energy of $\mu = 1/2$.

Theorems & Definitions (12)

• Definition 1: Ejection-collisions
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• Proposition 1
• Lemma 2
• Lemma 3
• Lemma 4