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

Ollé, Mercè; Rodriguez, Oscar Maurício Hernandez; Soler, Jaume

doi:10.1016/j.cnsns.2020.105294

Cited by 12 publications

(20 citation statements)

References 18 publications

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“…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.…”

Section: Introductionmentioning

confidence: 87%

“…With this new change of time only the singularity r 2 = 0 persists; however the z axis (cos ϕ = 0) -on the configuration space-is invariant. For this reason, both the system with this new change of time or system (14) are still not convenient systems to work with orbits that pass through or near the z axis and other strategies are preferable as we will see in the next section.…”

Section: Mcgehee Regularization In the 3d Casementioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Rodriguez

Soler

2021

SSRN Journal

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Section: Introductionmentioning

confidence: 87%

Section: Mcgehee Regularization In the 3d Casementioning

confidence: 99%

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

Rodriguez

Soler

2021

SSRN Journal

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“…For i ∈ {1, 2} consider P i as defined in Equation (43). For x = (x 0 , x 1 , x 2 , x 3 , x 4 ) ∈ R 14 , where x 0 , x 4 ∈ R, x 1 , x 2 , x 3 ∈ R 4 , and j ∈ {4, 5} we define F u i,j , F s i,j : R 16 → R 16 , 22 In each frame m 1 is depicted as a black dot and L 4 as a red dot.…”

Section: Connections Between Collisions and Libration Points L 4 Lmentioning

confidence: 99%

“…These works depend on results from geometry/topology, the calculus of variations, and the KAM theory. For parameter and energy regimes where analytical results are unavailable, numerical studies illuminate the dynamics of the collision set [41,42,43,44,45,46,47,48]. Our work goes towards providing a framework for computer assisted proofs for collision orbits, for the parameter regimes where the perturbative methods cannot be applied.…”

Section: Introductionmentioning

confidence: 99%

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

Capiński¹,

Kepley²,

James³

2022

Preprint

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This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a-posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's assymptotic periodic orbits (transverse homoclinics for L 4,5 ), families of periodic orbits passing through collision, and orbits connecting L 4 to ejection or collision.

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

et al. 2022

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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.

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

Cited by 12 publications

References 18 publications

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

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

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

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

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