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Chaotic motion and the Graziani-Black criterion

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Abstract

Chaotic motion is presented as a fundamental cause for the development of the Laplacian instability described by the Graziani-Black(GB) criterion. While the GB criterion is valid for different sets of finite masses in a general three-body system, a circular planar restricted three-body system is studied here. Poincaré surfaces of section and Liapunov characteristic numbers are used to test for the chaotic motion in this system. A simple description of the chaos detection techniques and the GB criterion is provided.

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References

  1. Szebehely, V.G. and McKenzie, R.: 1981, ‘Stability of Outer Planetary Systems’, Celestial Mechanics 23, 3–7.

    Google Scholar 

  2. Szebehely, V.G. and McKenzie, R.: 1977, ‘Stability of Planetary Systems with Bifurcation Theory’, Astronomical Journal 82 (1), 79–83.

    Google Scholar 

  3. Graziani, F. and Black, D.: 1981, ‘Orbital Stability Constraints on the Nature of Planetary Systems’, Astrophysical Journal 251, 337–341.

    Google Scholar 

  4. Black, D.: 1982, ‘A Simple Criterion for Determining the Dynamical Stability of Three-body Systems’, Astronomical Journal 87 (9), 1333–1337.

    Google Scholar 

  5. Pendleton, Y. and Black, D.: 1982, ‘Further Studies on Criteria for the Onset of Dynamical Instability in General Three-body Systems’, Astronomical Journal 88 (9), 1415–1419.

    Google Scholar 

  6. Szebehely, V.G.: 1967, Theory of Orbits, Academic Press.

  7. Benettin, G. et al.: 1979, Phys. Rev. A19, 2454.

    Google Scholar 

  8. Wolf, A. et al.: 1985, ‘Determining Lyapunov Exponents from a Time Series’, Physica 16D, 285–317.

    Google Scholar 

  9. Smith, R.H.: 1991. ‘The Onset of Chaotic Motion in the Restricted Problem of Three Bodies’, Proceedings of the AAS/AIAA Astrodynamics Conference, Durango Colo., Aug. 1991, AAS 91–387.

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Authors and Affiliations

  1. Department of Aerospace Engineering, University of Texas, 78712, Austin, TX, USA

    R. H. Smith & V. Szebehely

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  1. R. H. Smith
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  2. V. Szebehely
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Smith, R.H., Szebehely, V. Chaotic motion and the Graziani-Black criterion. Celestial Mech Dyn Astr 54, 357–362 (1992). https://doi.org/10.1007/BF00049147

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  • DOI: https://doi.org/10.1007/BF00049147

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