Breakdown of homoclinic orbits to L3 in the RPC3BP (II): an asymptotic formula
The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/394711 |
| Acceso en línea: | https://hdl.handle.net/2117/394711 https://dx.doi.org/10.1016/j.aim.2023.109218 |
| Access Level: | acceso abierto |
| Palabra clave: | Hamiltonian systems Exponentially small phenomena Splitting of separatrices Celestial mechanics Coorbital motions L3 Lagrange point Classificació AMS::37 Dynamical systems and ergodic theory Classificació AMS::70 Mechanics of particles and systems::70M20 Orbital mechanics Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1, . . . , L5. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. In this paper, we obtain an asymptotic formula for the distance between the stable and unstable manifolds of L3 for small values of the mass ratio 0 < \mu \leqslant 1. In particular we show that L3 cannot have (one round) homoclinic orbits. If the ratio between the masses of the primaries \mu is small, the hyperbolic eigenvalues of L3 are weaker, by a factor of order \sqrt{\mu }, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to \sqrt{\mu } . Thus, classical perturbative methods (i.e. the Melnikov-Poincaré method) can not be applied. |
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