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Orbital Structure of the Two Fixed Centres Problem

Published online by Cambridge University Press:  12 April 2016

A. Cordero ,
J. Martínez Alfaro  and
P. Vindel
A. Cordero
Affiliation:
Dpt. Matematica Aplicada.Universidad Politécnica de Valencia. Valencia.Spaine-mail:acordero@mat.upv.es
J. Martínez Alfaro
Affiliation:
Dpt. Matemàtica Aplicada. Facultat de Matemàtiques.Universitat de València. València.Spaine-mail:Jose.MtnezAlfaro@uv.es
P. Vindel
Affiliation:
Dpt. Matemàtiques.Universitat Jaume I. Castellò.Spaine-mail:vindel@nuvol.uji.es

Extract

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The set of orbits of the Two Fixed Centres problem has been known for a long time (Chartier, 1902, 1907; Pars, 1965), since it is an integrable Hamiltonian system.

We consider a plane that contains the fixed masses. Denote by φ the angle denned by this plane and the one that contains also the third body. The momentum pφ is a first integral of the system and when pφ is different from zero, the manifold generated by the generalized coordinates and momenta are two copies of the three-dimensional sphere S3. If pφ = 0, that is to say when the planet crosses the line joining both suns, the motion is restricted to a planar one. All the equilibrium points appears in this case and therefore the phase spaces are more complex. We restrict our attention to this case which has two degrees of freedom.

It is again a Bott-integrable Hamiltonian system. The set of periodic orbits of this systems can be studied from a subset of them, the Non-Singular Morse-Smale type orbits (see Casasayas, 1992). It is proved in Campos (1997) that a small perturbation of a Bott-integrable Hamiltonian system transforms it into a Non-Singular Morse-Smale system. The NMS periodic orbits belong to both the NMS system and the Hamiltonian one. Moreover, The NMS p.o. can be continued to nearly Hamiltonian systems. For instance, in our case to the Restricted Three Body Problem and in the study of the motion of a material point moving inside the gravitational field generated by two stars. This approximation is also useful when the motion of an artificial satellite around a spheroidal body is considered.

Type
Extended Abstracts
Information
International Astronomical Union Colloquium , Volume 172: Impact of Modern Dynamics in Astronomy , 1999 , pp. 463 - 464
Copyright
Copyright © Kluwer 1999

References

Campos, B., Martínez Alfaro, J. and Vindel, P.: 1996, Orbital Structure in the Two Fixed Centres Problem, Proceedings of the Third International Workshop on Positional Astronomy and Celestial Mechanics, pp. 313–323.Google Scholar
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