Regularisation in Ejection-Collision Orbits of the RTBP

Abstract

Numerical explorations confirm well-known analytical results on the existence of ejection-collision orbits in the restricted three-body problem for very restrictive values of the Jacobi constant C. For different values of C some new types of ejection-collision orbits are found. The concept of n-ejection-collision orbit is introduced and numerical explorations are carried out which show a very rich dynamics when Hill regions contain both primaries. Complete details on the numerical methods and the bifurcations of the different families of orbits are given in the references.

Analytical and numerical results on families of n-ejection-collision orbits in the RTBP

“…As far as we know, the most complete papers about the numerical computation of n-EC orbits, for n = 1, ...., 25, the continuation of families and appearance of bifurcation orbits in the RTBP are [18] and [19]. We also mention the analysis of n-EC orbits in other contexts, for example see [17] in the atomic physics or in [1] in the collinear four-body problem.…”

“…The paper is organized as follows: In Section 2, a short summary of the main properties of the RTBP are recalled, and in particular the equations of motion in the usual synodical (rotating) coordinates (with two singularities associated with collision with each primary) and in the rotating Levi-Civita ones (where the collision with the big primary has been regularized) are provided. In Section 3, the comparison between McGehee regularization and Levi-Civita one is shortly discussed (see more details in [19]) and the main geometrical ideas involved in the proof of the existence of n-EC orbits are presented. Section 4 contains the statement of the theorem and the core of the proof.…”

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

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

“…By contrast, Levi-Civita local regularization results more suitable from the numerical point of view, since the pass through collision is a regular point, but the expression of the resulting ODE is more intricate and a double covering of the phase space intrinsically appears. See [15] where a comparison between both choices is carried out and some other regularizations are also mentioned.…”

Transit regions and ejection/collision orbits in the RTBP

“…However, dealing with the n-EC orbits for n > 1, as far as we know, only few papers can be mentioned. On the numerical side, [23] and [24], where the authors compute, analyse and describe the continuation of families of n-EC orbits for n = 1, ..., 25 and do compare the pros and cons of taking into account the Levi-Civita regularization versus the McGehee's one.…”