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Chaotic Scattering in the Restricted Three‐Body Problem II. Small mass parameters

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Abstract

We study the scattering motion of the planar restricted three‐body problem for small mass parameters μ. We consider the symmetric periodic orbits of this system with μ = 0 that collide with the singularity together with the circular and parabolic solutions of the Kepler problem. These divide the parameter space in a natural way and characterize the main features of the scattering problem for small non‐vanishing μ. Indeed, continuation of these orbits yields the primitive periodic orbits of the system for small μ. For different regions of the parameter space, we present scattering functions and discuss the structure of the chaotic saddle. We show that for μ < μc and any Jacobi integral there exist departures from hyperbolicity due to regions of stable motion in phase space. Numerical bounds for μc are given.

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References

  • Benet, L.: 1996, PhD Thesis, University of Basel.

  • Benet, L., Trautmann, D. and Seligman, T. H.: 1997, Celest. Mech. Dynam. Astron. 66, 203.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Benet, L., Jung, C., Papenbrock, T. and Seligman, T. H.: 1998, Physica D, (in press).

  • Bollt, E. M. and Meiss, J. D.: 1995, Phys. Lett. A 204, 373.

    Article  ADS  Google Scholar 

  • Bruno, A. D.: 1994, The Restricted Three-Body Problem, Walter de Gruyter, Berlin.

    Google Scholar 

  • Chaos 3, No.4: 1993.

  • Coe, C. J.: 1932, Trans. Am. Math. Soc. 34, 811.

    Article  MATH  MathSciNet  Google Scholar 

  • Gaspard, P.: 1991, Quantum Chaos, Proc. of the International Summer School of Physics ‘Enrico Fermi’ (Course CXIX). in: G. Casati, I. Guarneri and U. Smilankski (eds), Varena.

  • Guillaume, P.: 1969, Astron. Astrophys. 3, 57.

    MATH  MathSciNet  ADS  Google Scholar 

  • Greenberg, R. and Brahic, A. (eds): 1984, Planetary Rings, University of Arizona Press, Tucson.

    Google Scholar 

  • Guckenheimer, J. and Holmes, P.: 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.

    Google Scholar 

  • Heggie, D. C.: 1988, The Few Body Problem, in: M. J. Valtonen (ed.), Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Hénon, M.: 1965a, Ann. Astrophys. 28, 499.

    MATH  ADS  Google Scholar 

  • Hénon, M.: 1965b, Ann. Astrophys. 28, 992.

    MATH  ADS  Google Scholar 

  • Hénon, M.: 1968, Bull. Astron. (serie 3) 3, 377.

    MATH  Google Scholar 

  • Hénon, M.: 1981, Nature 293, 33.

    Article  ADS  Google Scholar 

  • Hénon, M.: 1997, Generating Families in the Restricted Three-Body Problem, Lecture Notes in Physics m52, Springer-Verlag, New York.

    Google Scholar 

  • Hénon, M. and Guyot, M.: 1970, Periodic Orbits, Stability and Resonances, in: G. E. O. Giacaglia(ed.), Reidel, Dordrecht.

    Google Scholar 

  • Hitzl, D. L. and Hénon, M.: 1977, Celest. Mech. 15, 421.

    Article  MATH  ADS  Google Scholar 

  • Jung, C. and Lipp, C.: 1997, J. Phys A (submitted).

  • Lipp, C., Jung, C. and Seligman, T. H.: 1996, Proc. of the Fourth Int. Wigner Symp., in: N. M. Atakashiev, T. H. Seligman and K. B. Wolf (eds), World Scientific, Singapore.

    Google Scholar 

  • Koopman, B. O.: 1927, Trans. Am. Math. Soc. 29, 287.

    Article  MATH  MathSciNet  Google Scholar 

  • Murison, M. A.: 1989, Astron. J. 98, 2346.

    Article  ADS  Google Scholar 

  • Petit, J. M. and Hénon, M.: 1986, Icarus 66, 536.

    Article  ADS  Google Scholar 

  • Petit, J. M. and Hénon, M.: 1989, Dynamics of Stochastic Processes, in: R. Lima, L. Steit and R. Vilela Mendes (eds), Springer-Verlag, New York.

    Google Scholar 

  • Poincaré, H.: 1899, Les Méthodes Nouvelles de la Mécanique Céleste, vol. 1–3, Gauthier-Villars, Paris.

    Google Scholar 

  • Roy, A. E. (ed.): 1988, Long-Term Dynamical Behavior of Natural and Artificial N-Body Systems, Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Rückerl, B. and Jung, C.: 1994a, J. Phys. A: Math. Gen. 27, 55.

    Article  MATH  ADS  Google Scholar 

  • Rückerl, B. and Jung, C.: 1994b, J. Phys. A: Math. Gen. 27, 6741.

    Article  MATH  ADS  Google Scholar 

  • Smilansky, U.: 1992, Proc. of the 1989 Les Houches Summer School, in: M. J. Giannoni, A. Voros and J. Zinn-Justin (eds), North-Holland, Amsterdam.

    Google Scholar 

  • Szebehely, V.: 1967, Theory of Orbits, Academic Press, New York.

    Google Scholar 

  • Valtonen, M. J. and Innanen, K. A.: 1982, Astron. J. 255, 307.

    Article  ADS  Google Scholar 

  • Whittaker, E. T.: 1904, ATreatise in the Analytical Dynamics ofParticles&Rigid Bodies, Cambridge Mathematical Library, Cambridge, 1989.

    Google Scholar 

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Authors and Affiliations

  1. Centro Internacional de Ciencias, A. P. 48‐3, MEX‐62251, Cuernavaca, México

    L. Benet & T. H. Seligman

  2. Institut für theoretische Physik, Universität Basel, Klingelbergstr., CH‐4056, Basel, Switzerland

    D. Trautmann

Authors
  1. L. Benet
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  2. T. H. Seligman
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  3. D. Trautmann
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Benet, L., Seligman, T.H. & Trautmann, D. Chaotic Scattering in the Restricted Three‐Body Problem II. Small mass parameters. Celestial Mechanics and Dynamical Astronomy 71, 167–189 (1998). https://doi.org/10.1023/A:1008335232601

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