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 $$\mu \in (0,1)$$ 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.

Study of the Ejection/Collision Orbits in the Spatial RTBP Using the McGehee Regularization

“…We also recall that the condition C ≥ C L1 guarantees that there is no possible transit from moving in a neighborhood of one primary to a neighborhood around the other one. Concerning the results from our simulations, we will show how their statement about the existence of two circles of EC orbis is not true, but only six EC orbits can be guaranteed for µ > 0 and C ≥ C L1 fixed: four of them being planar and already known (see [11,14,17]) and two new spatial EC orbits.…”

“…These kind of orbits have been studied in depth only for the planar case (see for example [11,13,14,15,16,17]).…”

“…They are also shown in Figure 5 right in black. (The interested reader is referred to [11,14] for the analytical proof of existence of n-EC orbits, for n ≥ 1). However this approach is not straightforward in the spatial RTBP.…”

“…Actually, numerically, we check that the ejection orbits really pass close to the primary at the first crossing with Σ m . The interested reader is referred to [11,13,14,16,17] for massive numerical simulations on n-EC orbits, for n ≥ 1, in the planar RTBP.…”

“…In particular, this paper is focussed on the EC orbits in the spatial RTBP. At present there are some results about such orbits in the planar RTPB (see [11,13,14,16] and references therein). The EC orbits in the spatial RTBP have been, however, hardly ever analysed.…”