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The Lidov-Kozai Effect - Applications in Exoplanet Research and Dynamical Astronomy

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

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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.

…the integrability of the non-restricted

problem under consideration is, in a way,

a happy coincidence.

Lidov and Ziglin (1976)

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Notes

  1. 1.

    The choice of the “ω–X” 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.

References

  • Aksenov, E. P. (1979a) “The doubly averaged, elliptical, restricted, three-body problem.” Sov. Astron., 23, 236–239

    ADS  MATH  Google Scholar 

  • Aksenov, E. P. (1979b) “Trajectories in the doubly-averaged elliptical restricted three-body problem.” Sov. Astron., 23, 351–354

    ADS  MATH  Google Scholar 

  • Antognini, J. M. O. (2015) “Timescales of Kozai–Lidov oscillations at quadrupole and octupole order in the test particle limit.” Mon. Not. R. Astron. Soc., 452, 3610–3619

    Article  ADS  Google Scholar 

  • Arnold, V. I., Kozlov, V. V., & Neishtadt, A. I. (2002) Mathematical Aspects of Classical and Celestial Mechanics (Editorial URSS, Moscow) (in Russian)

    MATH  Google Scholar 

  • Bailey, M. E., Chambers, J. E., & Hahn, G. (1992) “Origin of sungrazers: a frequent cometary end-state.” Astron. Astrophys., 257, 315–322

    ADS  Google Scholar 

  • Breiter, S. (2003) “Extended fundamental model of resonance.” Celest. Mech. Dyn. Astron., 85, 209–218

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Brouwer, D., & Clemence, G. M. (1961) Methods of Celestial Mechanics (Academic, New York)

    MATH  Google Scholar 

  • Chenciner, A., & Montgomery, R. (2000) “A remarkable periodic solution of the three-body problem in the case of equal masses.” Annals of Mathematics – Second Series, 152, 881–901

    Article  MathSciNet  MATH  Google Scholar 

  • Chirikov, B. V. (1959) “Resonance processes in magnetic traps.” Atomnaya Energiya, 6, 630–638 (in Russian) [Chirikov, B. V. (1960) “Resonance processes in magnetic traps.” J. Nucl. Energy, Part C: Plasma Physics, 1, 253–260]

    Google Scholar 

  • Chirikov, B. V. (1979) “A universal instability of many-dimensional oscillator systems.” Phys. Rep., 52, 263–379

    Article  ADS  MathSciNet  Google Scholar 

  • Chirikov, B.V. (1982) “Nonlinear resonances and dynamical stochasticity.” Priroda, No. 7, 15–25 (in Russian)

    Google Scholar 

  • Duquennoy, A., & Mayor, M. (1991) “Multiplicity among solar-type stars in the solar neighbourhood. II — Distribution of the orbital elements in an unbiased sample.” Astron. Astrophys., 248, 485–524

    ADS  Google Scholar 

  • Ferraz-Mello, S. (2007) Canonical Perturbation Theories. Degenerate Systems and Resonance (Springer, New York)

    Book  MATH  Google Scholar 

  • Ferrer, S., & Osácar, C. (1994) “Harrington’s Hamiltonian in the stellar problem of three bodies: reductions, relative equilibria and bifurcations.” Celest. Mech. Dyn. Astron., 58, 245–275

    Article  ADS  MATH  Google Scholar 

  • Ford, E. B., Kozinsky, B., & Rasio, F. A. (2000) “Secular evolution of hierarchical triple star systems.” Astrophys. J., 535, 385–401

    Article  ADS  Google Scholar 

  • Giacaglia, G. E. O. (1968) “Secular motion of resonant asteroids.” Smithsonian Astrophysical Observatory, Special Report No. 278. 70 pp.

    Google Scholar 

  • Giacaglia, G. E. O. (1969) “Resonance in the restricted problem of three bodies.” Astron. J, 74, 1254–1261

    Article  ADS  Google Scholar 

  • Gordeeva, Yu. F. (1968) “The time dependence of elements in the long-period oscillations in the restricted three-body problem.” Cosmic Research, 6, 450–453

    ADS  Google Scholar 

  • Gronchi, G. F., & Milani, A. (1998) “Averaging on Earth-crossing orbits.” Celest. Mech., 71, 109–136

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Gronchi, G. F., & Milani, A. (1999) “The stable Kozai state for asteroids and comets, with arbitrary semimajor axis and inclination.” Astron. Astrophys., 341, 928–935

    ADS  Google Scholar 

  • Harrington, R. S. (1968) “Dynamical evolution of triple stars.” Astron. J., 73, 190–194

    Article  ADS  Google Scholar 

  • Harrington, R. S. (1969) “The stellar three-body problem.” Celest. Mech., 1, 200–209

    Article  ADS  MATH  Google Scholar 

  • Henrard, J., & Lemaître, A. (1983) “A second fundamental model for resonance.” Celest. Mech., 30, 197–218

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Holman, M., Touma, J., & Tremaine, S. (1997) “Chaotic variations in the eccentricity of the planet orbiting 16 Cygni B.” Nature, 386, 254–256

    Article  ADS  Google Scholar 

  • Jefferys, W. H., & Moser, J. (1966) “Quasi-periodic solutions for the three-body problem.” Astron. J., 71, 568–578

    Article  ADS  MathSciNet  Google Scholar 

  • Katz, B., Dong, S., & Malhotra, R. (2011) “Long-term cycling of Kozai–Lidov cycles: Extreme eccentricities and inclinations excited by a distant eccentric perturber.” Phys. Rev. Lett., 107, 181101 (5pp)

    Article  ADS  Google Scholar 

  • Kholshevnikov, K. V. (1997) “D’Alembertian functions in celestial mechanics.” Astron. Rep., 41, 135–142

    ADS  Google Scholar 

  • Kholshevnikov, K. V. (2001) “The Hamiltonian in the planetary or satellite problem as a D’Alembertian function.” Astron. Rep., 41, 135–142, 45, 577–579

    Google Scholar 

  • Kinoshita, H., & Nakai, H. (2007) “General solution of the Kozai mechanism.” Celest. Mech. Dyn. Astron., 98, 67–74

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kirkwood, D. (1867) Meteoric Astronomy (Lippincott, Philadelphia)

    Google Scholar 

  • Kolomensky, A. A., & Lebedev, A. N. (1966) Theory of Cyclic Accelerators (Wiley, New York)

    Google Scholar 

  • Kozai, Y. (1962) “Secular perturbations of asteroids with high inclination and eccentricity.” Astron. J., 67, 591–598

    Article  ADS  MathSciNet  Google Scholar 

  • Kozai, Y. (1979) “Secular perturbations of asteroids and comets.” In: Dynamics of the Solar System, ed. by Duncombe, R. L. (Reidel, Dordrecht) pp. 231–237

    Google Scholar 

  • Kozai, Y. (1985) “Secular perturbations of resonant asteroids.” Celest. Mech., 36, 47–69

    Article  ADS  MATH  Google Scholar 

  • Krasinsky, G. A. (1972) “Critical inclinations in planetary problems.” Celest. Mech., 6, 60–83

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Krasinsky, G. A. (1974) “Stationary solutions of the averaged three-body problem and some problems of planet motion stability.” In: The Stability of the Solar System and of Small Stellar Systems (IAU Symp. No. 62), ed. by Kozai, Y. (Springer, Berlin) pp. 95–116

    Google Scholar 

  • Krymolowski, Y., & Mazeh, T. (1999) “Studies of multiple stellar systems – II. Second-order averaged Hamiltonian to follow long-term orbital modulations of hierarchical triple systems.” Mon. Not. R. Astron. Soc., 304, 720–732

    Article  ADS  Google Scholar 

  • Lemaître, A. (1984) “High-order resonances in the restricted three-body problem.” Celest. Mech. Dyn. Astron., 32, 109–126

    Article  MathSciNet  MATH  Google Scholar 

  • Li, G., Naoz, S., Kocsis, B., & Loeb, A. (2014a) “Eccentricity growth and orbit flip in near-coplanar hierarchical three-body systems.” Astrophys. J., 785, 116 (8pp)

    Article  ADS  Google Scholar 

  • Li, G., Naoz, S., Holman, M., & Loeb, A. (2014b) “Chaos in the test particle eccentric Kozai–Lidov mechanism.” Astrophys. J., 791, 86 (10pp) [Erratum: (2015) Astrophys. J., 802, 71 (1p)]

    Google Scholar 

  • Li, G., Naoz, S., Kocsis, B., & Loeb, A. (2015) “Implications of the eccentric Kozai–Lidov mechanism for stars surrounding supermassive black hole binaries.” Mon. Not. R. Astron. Soc., 451, 1341–1349

    Article  ADS  Google Scholar 

  • Lichtenberg, A. J., & Lieberman, M. A. (1992) Regular and Chaotic Dynamics. 2nd edition (Springer-Verlag, New York)

    Book  MATH  Google Scholar 

  • Lidov, M. L. (1961) “Evolution of artificial planetary satellites under the action of gravitational perturbations due to external bodies.” Iskusstviennye Sputniki Zemli (Artificial Satellites of the Earth), 8, 5–45 (in Russian)

    Google Scholar 

  • Lidov, M. L. (1962) “The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies.” Planet. Space Sci., 9, 719–759 (An English translation of Lidov’s (1961) article.)

    Google Scholar 

  • Lidov, M. L. (1963a) “Evolution of the orbits of artificial satellites of planets as affected by gravitational perturbation from external bodies.” AIAA Journal, 1, 1985–2002 (An English translation of Lidov’s (1961) article.)

    Google Scholar 

  • Lidov, M. L., & Ziglin, S. L. (1974) “The analysis of the restricted circular twice-averaged three body problem in the case of close orbit.” Celest. Mech., 9, 151–173

    Article  ADS  MATH  Google Scholar 

  • Lidov, M. L., & Ziglin, S. L. (1976) “Non-restricted double-averaged three body problem in Hill’s case.” Celest. Mech., 13, 471–481

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Lithwick, Y., & Naoz, S. (2011) “The eccentric Kozai mechanism for a test particle.” Astrophys. J., 742, 94 (8pp)

    Article  ADS  Google Scholar 

  • Malhotra, R. (1998) “Orbital resonances and chaos in the Solar system.” In: Solar System Formation and Evolution (ASP Conf. Ser. 149), ed. by Lazzaro, D., Vieira Martins, R., Ferraz-Mello, S., Fernández, J., & Beaugé, C. (Astron. Soc. of the Pacific, San Francisco) pp. 37–63

    Google Scholar 

  • Malhotra, R. (2012) “Orbital resonances in planetary systems.” In: Encyclopedia of Life Support Systems by UNESCO. Volume 6.119.55 Celestial Mechanics. 31 pp.

    Google Scholar 

  • Marchal, C. (1990) The Three-Body Problem (Elsevier, Amsterdam)

    MATH  Google Scholar 

  • Mardling, R. A., & Aarseth, S. J. (2001) “Tidal interactions in star cluster simulations.” Mon. Not. R. Astron. Soc., 321, 398–420

    Article  ADS  Google Scholar 

  • Markeev, A. P. (1990) Theoretical Mechanics (Nauka Publishers, Moscow) (in Russian)

    MATH  Google Scholar 

  • Mazeh, T., & Shaham, J. (1979) “The orbital evolution of close triple systems — the binary eccentricity.” Astron. Astrophys., 77, 145–151

    ADS  Google Scholar 

  • McMillan, S., Hut, P., & Makino, J. (1991) “Star cluster evolution with primordial binaries. II – Detailed analysis.” Astrophys. J., 372, 111–124

    Article  ADS  Google Scholar 

  • Michel, P., & Thomas, F. (1996) “The Kozai resonance for near-Earth asteroids with semimajor axes smaller than 2 AU.” Astron. Astrophys., 307, 310–318

    ADS  Google Scholar 

  • Moiseev, N. D. (1945a) “On some basic simplified schemes of celestial mechanics obtained by means of averaging the restricted circular three-body problem. Part 1.” Publications of the Sternberg State Astronomical Institute (Moscow State Univ.), 15, issue 1, pp. 75–99 (in Russian, the issue was typeset in 1941)

    Google Scholar 

  • Moiseev, N. D. (1945b) “On some basic simplified schemes of celestial mechanics obtained by means of averaging the restricted circular three-body problem. Part 2.” Publications of the Sternberg State Astronomical Institute (Moscow State Univ.), 15, issue 1, pp. 100–117 (in Russian, the issue was typeset in 1941)

    Google Scholar 

  • Morbidelli, A. (2002) Modern Celestial Mechanics (Taylor and Francis, London)

    Google Scholar 

  • Murray, C. D., & Dermott, S. F. (1999) Solar System Dynamics (Cambridge Univ. Press, Cambridge)

    MATH  Google Scholar 

  • Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., & Teyssandier, J. (2011) “Hot Jupiters from secular planet-planet interactions.” Nature, 473, 187–189

    Article  ADS  Google Scholar 

  • Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., & Teyssandier, J. (2013a) “Secular dynamics in hierarchical three-body systems.” Mon. Not. R. Astron. Soc., 431, 2155–2171

    Article  ADS  Google Scholar 

  • Naoz, S., Kocsis, B., Loeb, A., & Yunes, N. (2013b) “Resonant post-newtonian eccentricity excitation in hierarchical three-body systems.” Astrophys. J., 773, 187 (16pp)

    Article  ADS  Google Scholar 

  • Poincaré, H. (1899) Les Méthodes Nouvelles de la Mécanique Céleste III (Gauthier–Villars, Paris)

    MATH  Google Scholar 

  • Prokhorenko, V. I. (2002a) “Investigation of periods of evolution for elliptical orbits in the double-averaged Hill problem.” Cosmic Research, 40, 48–54

    Article  ADS  Google Scholar 

  • Prokhorenko, V. I. (2002b) “Investigation of the time of ballistic existence for elliptic orbits evolving under the influence of gravitational perturbations of external bodies” Cosmic Research, 40, 264–273

    Article  ADS  Google Scholar 

  • Prokhorenko, V. I. (2015) “On the application of qualitative methods of perturbation theory in solving practical problems of selection and correction of the orbits of the satellites of the planets given the secular and long-period components of the evolution under the influence of external gravitational perturbations.” In: Space Ballistics — from its Origin to the Future. (Institute of Space Research, Russian Academy of Sciences, Moscow) pp. 130–161 (in Russian)

    Google Scholar 

  • Quinn, T., Tremaine, S., & Duncan, M. (1990) “Planetary perturbations and the origin of short-period comets.” Astrophys. J., 355, 667–679

    Article  ADS  Google Scholar 

  • Riddle, R. L., Tokovinin, A., Mason, B. D., et al. (2015) “A survey of the high order multiplicity of nearby Solar-type binary stars with Robo–AO.” Astrophys. J., 799, 4 (21pp)

    Article  ADS  Google Scholar 

  • Shevchenko, I. I. (2000) “Geometry of a chaotic layer.” J. Exp. Theor. Phys., 91, 615–625

    Article  ADS  Google Scholar 

  • Shevchenko, I. I. (2007) “On the Lyapunov exponents of the asteroidal motion subject to resonances and encounters.” In: Near Earth Objects, our Celestial Neighbors: Opportunity and Risk (Proc. IAU Symp. 236), ed. by Milani, A., Valsecchi, G. B., & Vokrouhlický, D. (Cambridge Univ. Press, Cambridge) pp. 15–29

    Google Scholar 

  • Shevchenko, I. I. (2015) “Chaotic zones around gravitating binaries.” Astrophys. J., 799, 8 (7pp)

    Article  ADS  Google Scholar 

  • Shinkin, V. N. (1995) “The integrable cases of the general spatial three-body problem at third-order resonance.” Celest. Mech. Dyn. Astron., 62, 323–334

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Söderhjelm, S. (1984) “Third-order and tidal effects in the stellar three-body problem.” Astron. Astrophys., 141, 232–240

    ADS  MATH  Google Scholar 

  • Šuvakov, M., & Dmitrašinović, V. (2013) “Three classes of Newtonian three-body planar periodic orbits.” Phys. Rev. Lett., 110, 114301 (4pp)

    Article  Google Scholar 

  • Thomas, F., & Morbidelli, A. (1996) “The Kozai resonance in the outer solar system and the dynamics of long-period comets.” Celest. Mech. Dyn. Astron., 64, 209–229

    Article  ADS  MATH  Google Scholar 

  • Tokovinin, A. A. (1997) “On the multiplicity of spectroscopic binary stars.” Astron. Lett., 23, 727–730

    ADS  Google Scholar 

  • Tokovinin, A. (2014) “From binaries to multiples. II. Hierarchical multiplicity of F and G dwarfs.” Astron. J., 147, 87 (14pp)

    Google Scholar 

  • Tokovinin, A., Thomas, S., Sterzik, M., & Udry, S. (2006) “Tertiary companions to close spectroscopic binaries.” Astron. Astrophys., 450, 681–693

    Article  ADS  Google Scholar 

  • Vashkovyak, M. A. (1981a) “Evolution of orbits in the restricted circular twice-averaged three-body problem. I — Qualitative investigations.” Cosmic Research, 19, 1–10

    ADS  Google Scholar 

  • Vashkovyak, M. A. (1981b) “Evolution of orbits of asteroids not belonging to the main belt.” Cosmic Research, 19, 357–365

    Google Scholar 

  • Vashkovyak, M. A. (1982) “Evolution of orbits in the two-dimensional restricted elliptic twice-averaged three-body problem.” Cosmic Research, 20, 236–244

    Google Scholar 

  • Vashkovyak, M. A. (1984) “Integrable cases of the restricted twice-averaged three-body problem.” Cosmic Research, 22, 260–267

    Google Scholar 

  • Williams, J. G., & Benson, G. S. (1971) “Resonances in the Neptune–Pluto system.” Astron. J., 76, 167–177

    Article  ADS  Google Scholar 

  • Ziglin, S. L. (1975) “Secular evolution of the orbit of a planet in a binary-star system.” Sov. Astron. Lett., 1, 194–195

    ADS  Google Scholar 

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  1. Pulkovo Observatory of the Russian Academy of Sciences, St. Petersburg, Russia

    Ivan I. Shevchenko

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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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