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The Theory Advances

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Part of the Astrophysics and Space Science Library book series (ASSL,volume 441)

Abstract

Since the time when the Lidov-Kozai effect was discovered, a substantial and steady progress in the analytical theory of this secular effect has been observed. This Chapter is an attempt to describe this progress. The stellar three-body problem in octupole approximation, the timescales of the Lidov-Kozai effect, and the place of LK-resonance in the general typology of resonances are considered.

Keywords

  • Lidov Kozai
  • Mean Motion Resonance
  • Quadrupole Approximation
  • Outer Binaries
  • Octupole Terms

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…the integrability of the non-restricted

problem under consideration is, in a way,

a happy coincidence.

Lidov and Ziglin (1976)

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Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 4.4
Fig. 4.5
Fig. 4.6
Fig. 4.7

Notes

  1. 1.

    The choice of the “ωX” plane is especially appropriate for studies of a highly-eccentric motion; see Sects. 6.2 and 7.3

  2. 2.

    By the elimination of nodes one implies the elimination of Delaunay variables H i and h i (i = 1, 2) in the Hamiltonian, see Jefferys and Moser (1966); H i and h i are defined by Equations (2.15).

  3. 3.

    See critics of Kozai’s (1962) argumentation and works of other authors in this respect in Naoz et al. (2013a).

  4. 4.

    Indeed, for a bounded motion, the eccentricity is limited by the value of 1 from above, and any increase decelerates on approaching a maximum.

  5. 5.

    The LKE in its dynamics was discussed in Chap. 1

  6. 6.

    Note that there is a misprint in Gordeeva’s (1968) formula (2): \(\beta = 1 -\frac{5} {3}c_{2}\) should be corrected to \(\beta = 1 -\frac{5} {2}c_{2}\).

  7. 7.

    The properties of nonlinear resonance are described in detail in Chirikov’s general review (Chirikov 1982), where the fundamental concepts of nonlinear dynamics are explained in most accessible and, at the same time, rigourous way.

  8. 8.

    Note that the same symbol \(\Omega\) is used throughout this book to traditionally designate the longitude of ascending node.

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Shevchenko, I.I. (2017). The Theory Advances. In: The Lidov-Kozai Effect - Applications in Exoplanet Research and Dynamical Astronomy. Astrophysics and Space Science Library, vol 441. Springer, Cham. https://doi.org/10.1007/978-3-319-43522-0_4

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