Celestial mechanics

, Volume 42, Issue 1–4, pp 53–79 | Cite as

Area-preserving mappings and deterministic chaos for nearly parabolic motions

  • T. Y. Petrosky
  • R. Broucke


The present work investigates a mechanism of capturing processes in the restricted three-body problem. The work has been done in a set of variables which is close to Delaunay's elements but which allows for the transition from elliptic to hyperbolic orbits. The small denominator difficulty in the perturbation theory is overcome by embedding the small denominator in an analytic function through a suitable analytic continuation. The results indicate that motions in nearly parabolic orbits can become chaotic even though the model is deterministic. The theoretical results are compared with numerical results, showing an agreement of about one percent. Some possible applications to cometary orbits are also given.


Analytic Function Perturbation Theory Theoretical Result Analytic Continuation Deterministic Chaos 
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  1. Arnold, V.I., Usp. Mat. Nank.18, 9 (1963) [Russ. Math. Surv.18, (6), 85, (1963)].Google Scholar
  2. Balescu, R.,Statistical Mechanics of Charged Particles, (Interscience, New York, 1963).Google Scholar
  3. Bilo, e.H. and Van de Hulst, H.C.,Bull. Astron. Inst. Netherlands 15, 119 (1960).Google Scholar
  4. Birkoff, G.D.,Dynamical Systems, (A.M.S. Publications, Providence, 1927).Google Scholar
  5. Cary, J.R.,Phys. Rep. 79, 129 (1981).Google Scholar
  6. Chirikov, B.V. and Vecheslavov, V.V., “Chaotic Dynamics of Comet Haley”, (Preprint 86-184, Institute of Nuclear Physics, Novosibirsk, 1986).Google Scholar
  7. Deprit, A.,Celestial Mechanics 1, 12 (1969).Google Scholar
  8. Dewar, R.L.,J. Phys. A9, 2043 (1976).Google Scholar
  9. Escobal, P.,Methods of Orbit Determination (Krieger, New York, 1976).Google Scholar
  10. Everhart, E.,Astron. J. 73, 1039 (1968).Google Scholar
  11. Everhart, E.,Astron. J. 74, 735 (1969).Google Scholar
  12. Everhart, E.,Astron. J. 75, 258 (1970).Google Scholar
  13. Everhart, E.,Astron. J. 78, 329 (1973)Google Scholar
  14. Gldberger, M. and Watson, r.,Collision Theory, (Wiley, New York, 1964).Google Scholar
  15. Greence, J.M.,J. Math. Phys. 20, 1183 (1980).Google Scholar
  16. Hori, G., Publications of the Astronomical Society of Japan18, 287 (1966).Google Scholar
  17. Kolmogorov, A.N.,Dok. Akad. Nank. 98, 527 (1954). English translation: Los Alamos Scientific Laboratory Translation No. LA-TR-71-67.Google Scholar
  18. Lighthill, J.,Proc. R. Soc. Lond. A407, 35 (1986).Google Scholar
  19. Lyttleton, R.A. and Hammersley, J.M.,M. N. Roy. Astr. Soc. 127, 257 (1964).Google Scholar
  20. Marsden, B.G.,Catalog of Cometary Orbits, International Astronomical Union, 1986.Google Scholar
  21. Moser, J.,Nach. Akad. Wiss. Goettingen Math. Phys. K1. 2 1, 1 (1962).Google Scholar
  22. Oort, J.H.,Bull. Astron. Inst. Neth. 11, 91 (1950).Google Scholar
  23. Petrosky, T.Y.,Phys. Rev. A. 29, 2078 (1984).Google Scholar
  24. Petrosky, T.Y.,Phys. Rev. A 32, 3716 (1985).Google Scholar
  25. Petrosky, T.Y.,Phys. Lett. A117, 328 (1986).Google Scholar
  26. Petrosky, T.Y. and I. Prigogine, PNAS (to appear, 1987).Google Scholar
  27. Poincaré, H.,Les Methodes Nouvelles de la Mechanique Celeste, (Dover, New York, 1967; English translation NASA Technical Translton F-451.)Google Scholar
  28. Prigogine, I.,Non-Equilibrium Statistical Mechanics, (Interscience, New York, 1962).Google Scholar
  29. Prigogine, I., Grecos, A. and George, Cl.,Celestial Mechanics 16, 489 (1977).Google Scholar
  30. Résibois, R. and I. Prigogine,Bull. Classe Sci., Acad. Roy. Belg.46, 53 (1960).Google Scholar
  31. Strömgre, E.,Publ. KObs. No. 19 (1914).Google Scholar
  32. Szebehely, V.,Theory of Orbits, (Academic Press, New York, 1967).Google Scholar
  33. Whittaker, E.T.,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th Edition. (Cambridge University Press, Cambridge, 1959).Google Scholar
  34. Yabushita, S.,Astron. and Astrophys. 16, 471 (1972a).Google Scholar
  35. Yabushita, S.,Astron. and Astrophys. 20, 205 (1972b).Google Scholar
  36. Yabushita, S.,Mon. Not. R. Astr. Soc. 187, 445 (1979).Google Scholar

Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • T. Y. Petrosky
    • 1
  • R. Broucke
    • 2
  1. 1.Center for Studies in Statistical MechanicsThe University of Texas at AustinAustin
  2. 2.Department of Aerospace EngineeringThe University of Texas at AustinAustin

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