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

• Published in: Journal of nonlinear science 2023-02, Vol.33 (1), Article 17
• Main Authors: M-Seara, T., Ollé, M., Rodríguez, Ó., Soler, J.
• Format: Article
• Language: English
• Subjects: Analysis; Bifurcations; Classical Mechanics; Economic Theory/Quantitative Economics/Mathematical Methods; Ejection; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Orbits; Parameters; Theoretical; Three body problem
• ISSN: 0938-8974; 1432-1467
• DOI: 10.1007/s00332-022-09873-y

In the planar circular restricted three-body problem and for any value of the mass parameter μ ∈ ( 0 , 1 ) and 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 μ + L n 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. 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 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.