Generalized Analytical Results on n-Ejection–Collision Orbits in the RTBP. Analysis of Bifurcations

Abstract

AbstractIn the planar circular restricted three-body problem and for any value of the mass parameter $$\mu \in (0,1)$$ μ ∈ ( 0 , 1 ) and $$n\ge 1$$ n ≥ 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 $$C=3\mu +Ln^{2/3}(1-\mu )^{2/3}$$ C = 3 μ + L n 2 / 3 ( 1 - μ ) 2 / 3 , where $$L>0$$ L > 0 is big enough but independent of $$\mu $$ 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 $$\mu >0$$ μ > 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 $$\mu \rightarrow 1$$ μ → 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.