Generalised analytical results on $n$-ejection-collision orbits in the RTBP. Analysis of bifurcations

Abstract

In the planar circular restricted three-body problem and for any value of the mass parameter $μ\in (0,1)$ and $n\ge 1$, we prove the existence of four families of $n$-ejection-collision ($n$-EC) orbits, that is, orbits where the particle ejects from a primary, reaches $n$ maxima in the distance with respect to it and finally collides with the primary. Such EC orbits have a value of the Jacobi constant of the form $C=3μ+Ln^{2/3}(1-μ)^{2/3}$, where $L>0$ is big enough but independent of $μ$ and $n$. In order to prove this optimal result, we consider Levi-Civita's transformation to regularize the collision with one primary and a perturbative approach using an ad hoc small parameter once a suitable scale in the configuration plane and time has previously been applied. This result improves a previous work where the existence of the $n$-EC orbits was stated when the mass parameter $μ>0$ was small enough. In this paper, any possible value of $μ\in (0,1)$ and $n\ge 1$ is considered. Moreover, for decreasing values of $C$, there appear some bifurcations which are first numerically investigated and afterwards explicit expressions for the approximation of the bifurcation values of $C$ are discussed. Finally, a detailed analysis of the existence of $n$-EC orbits when $μ\to 1$ is also described. In a natural way Hill's problem shows up. For this problem, we prove an analytical result on the existence of four families of $n$-EC orbits and numerically we describe them as well as the appearing bifurcations.