Generalized 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 mu in (0,1) and nge 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 (Euclidean) 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=3mu +Ln^{2/3}(1-mu )^{2/3}, where L>0 is big enough but independent of mu 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 mu >0 was small enough. Moreover, for decreasing values of C, there appear some bifurcations which are first numerically investigated and afterward 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 mu rightarrow 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.