Celestial Mechanics and Dynamical Astronomy

, Volume 85, Issue 2, pp 145–174 | Cite as

Three-Dimensional p–q Resonant Orbits Close to Second Species Solutions



The purpose of this paper is to study, for small values of μ, the three-dimensional pq resonant orbits that are close to periodic second species solutions (SSS) of the restricted three-body problem. The work is based on an analytic study of the in- and out-maps. These maps are associated to follow, under the flow of the problem, initial conditions on a sphere of radius μα around the small primary, and consider the images of those initial points on the same sphere. The out-map is associated to follow the flow forward in time and the in-map backwards. For both mappings we give analytical expressions in powers of the mass parameter. Once these expressions are obtained, we proceed to the study of the matching equations between both, obtaining initial conditions of orbits that will be 'periodic' with an error of the order μ1−α, for some α∈(1/3,1/2). Since, as μ → 0, the inner solution and the outer solution will collide with the small primary, these orbits will be close to SSS.

restricted three-dimensional three-body problem close encounters periodic orbits resonance 


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  1. Barrabés, E.: 2001, 'Òrbites de segona espècie del problema espacial de 3 cossos', PhD Thesis, Universitat Autònoma de Barcelona, Barcelona, Spain.Google Scholar
  2. Barrabés, E. and Gómez, G.: in press, 'spatial p-q resonant orbits of the RTBP', Celest. Mech. & Dyn. Astr. Google Scholar
  3. Font, Q., Nunes, A. and Simó, C.: 2002, 'Consecutive quasi-collisions in the planar circular RTBP', Nonlinearity 15, 115-142.Google Scholar
  4. Hénon, M.: 1997, Generating Families in the Restricted Three-Body Problem, Springer-Verlag, Berlin.Google Scholar
  5. Stiefel, E. and Scheifele, G.: 1971, Linear and Regular Celestial Mechanics, Die Grundlehren der Mathematischen Wissenschaften, Band 174, Springer-Verlag, Berlin, Heidelberg, New York, IX.Google Scholar
  6. Szebehely, V.: 1967, Theory of Orbits. The Restricted Problem of Three Bodies, Academic Press, New York.Google Scholar
  7. Yen, C.L.: 1985, 'Ballistic Mercury orbiter mission via Venus and Mercury gravity assist', AAS/AIAA Astrodynamics Specialist Conference, Paper AAS 85-346.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  1. 1.Departament d'Informàtica i Matemàtica AplicadaUniversitat de GironaGironaSpain
  2. 2.IEEC & Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain, e-mail

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