Kozai mechanism

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In celestial mechanics, the Kozai mechanism, or the Lidov–Kozai mechanism, is a perturbation of the orbit of a satellite by the gravity of another body orbiting farther out, causing libration (oscillation about a constant value) of the orbit's argument of pericenter. As the orbit librates, there is a periodic exchange between its inclination and its eccentricity.

The effect was described in 1961 by the Soviet specialist in space dynamics Michael Lidov (Russian: Михаил Львович Лидов) while analyzing the orbits of artificial and natural satellites of planets,[1][2] This result was reported by Lidov at the Conference on General and Practical Topics of Theoretical Astronomy, held in Moscow on 20 - 25 November 1961. Among the participants of that conference was a Japanese astronomer Yoshihide Kozai (Japanese: 古在由秀) who soon published this same result, in application to the orbits of the asteroids.[3] Since then, this effect has been found to be an important factor shaping the orbits of irregular satellites of the planets (the moons in the Kozai resonance being Jupiter's Carpo and Euporie,[4] Saturn's Kiviuq and Ijiraq,[5] Uranus's Margaret,[6] and Neptune's Sao and Neso;[7] the effect also explains the non-uniform distribution of irregular satellite inclinations), trans-Neptunian objects, and a few extrasolar planets and multiple star systems.

In the hierarchical, restricted three-body problem, it is assumed that the satellite has negligible mass compared with the other two bodies (the "primary" and the "perturber"), and that the distance between the primary and perturber is much greater than the distance from the primary to the satellite. These assumptions would be valid, for instance, in the case of an artificial satellite in a low-Earth orbit that is perturbed by the moon, or a short-period comet that is perturbed by Jupiter.

Under these approximations, the orbit-averaged equations of motion for the satellite have a conserved quantity: the component of the satellite's orbital angular momentum parallel to the angular momentum of the primary/perturber angular momentum. This conserved quantity can be expressed in terms of the satellite's eccentricity e and inclination i relative to the plane of the outer binary:

Conservation of Lz means that orbital eccentricity can be "traded for" inclination. Thus, near-circular, highly inclined orbits can become very eccentric. Since increasing eccentricity while keeping the semimajor axis constant reduces the distance between the objects at periapsis, this mechanism can cause comets (perturbed by Jupiter) to become sungrazing.

Lidov-Kozai oscillations will be present if Lz is lower than a certain value. At the critical value of Lz, a "fixed-point" orbit appears, with constant inclination given by

For values of Lz less than this critical value, there is a one-parameter family of orbital solutions having the same Lz but different amounts of variation in e or i. Remarkably, the degree of possible variation in i is independent of the masses involved, which only set the timescale of the oscillations.[8]


The Lidov–Kozai mechanism causes the argument of pericenter (ω) to librate about either 90° or 270°, which is to say that its periapse occurs when the body is farthest from the equatorial plane. This effect is part of the reason that Pluto is dynamically protected from close encounters with Neptune.

The Lidov–Kozai mechanism places restrictions on the orbits possible within a system, for example

  • for a regular moon: if the orbit of a planet's moon is highly inclined to the planet's orbit, the eccentricity of the moon's orbit will increase until, at closest approach, the moon is destroyed by tidal forces
  • for irregular satellites: the growing eccentricity will result in a collision with a regular moon, the planet, or alternatively, the growing apocenter may push the satellite outside the Hill sphere

The mechanism has been invoked in searches for Planet X, hypothetical planets orbiting the Sun beyond the orbit of Neptune.[9]


The basic timescale associated with Kozai oscillations is[8]

where a indicates semimajor axis, P is orbital period, e is eccentricity and m is mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner (satellite) orbit; M is the mass of the primary. The period of oscillation of all three variables (e, i, ω – the last being the argument of periapsis) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the separatrix orbit that separates librating (Kozai) orbits from oscillating orbits.

See also[edit]


  1. ^ Lidov, Michael L. "On approximate analysis of the evolution of orbits of artificial satellites". Problems of Motion of Artificial Celestial Bodies. Proceedings of the Conference on General and Practical Topics of Theoretical Astronomy, held in Moscow on 20 - 25 November 1961. Publication of the Academy of Sciences of the USSR, Moscow 1963. 
  2. ^ Lidov, Michael L. (October 1962). "The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies (English translation of the paper published in: Iskusstvennye sputniki Zemli, 1961. № 8. pp. 5-45)". Planetary and Space Science. 9 (10): 719–759. Bibcode:1962P&SS....9..719L. doi:10.1016/0032-0633(62)90129-0. 
  3. ^ Kozai, Yoshihide (November 1962). "Secular perturbations of asteroids with high inclination and eccentricity". The Astronomical Journal. 67: 591. Bibcode:1962AJ.....67..591K. doi:10.1086/108790. 
  4. ^ Brozović, Marina; Jacobson, Robert A. (9 March 2017). "The Orbits of Jupiter's Irregular Satellites". The Astronomical Journal. 153:147. Bibcode:2017AJ....153..147B. doi:10.3847/1538-3881/aa5e4d. 
  5. ^ Shevchenko, Ivan (2016). The Lidov-Kozai Effect - Applications in Exoplanet Research and Dynamical Astronomy. Springer-Verlag. p. 101. doi:10.1007/978-3-319-43522-0. ISBN 978-3-319-43520-6. 
  6. ^ Brozovic, M.; Jacobson, R. A. (4 March 2009). "The Orbits of the Outer Uranian Satellites". The Astronomical Journal. 137 (4): 3834–42. Bibcode:2009AJ....137.3834B. doi:10.1088/0004-6256/137/4/3834. Retrieved 19 July 2017. 
  7. ^ Brozović, Marina; Jacobson, Robert A.; Sheppard, Scott S. (10 March 2011). "The Orbits of Neptune's Outer Sallites". The Astronomical Journal. 141 (4). Bibcode:2011AJ....141..135B. doi:10.1088/0004-6256/141/4/135. Retrieved 19 July 2017. 
  8. ^ a b Merritt, David (2013). Dynamics and Evolution of Galactic Nuclei. Princeton Series in Astrophysics. Princeton, NJ: Princeton University Press. p. 575. ISBN 978-0-691-12101-7. OCLC 863632625. 
  9. ^ de la Fuente Marcos, Carlos; de la Fuente Marcos, Raul (September 1, 2014) [3 June 2014]. "Extreme trans-Neptunian objects and the Kozai mechanism: signalling the presence of trans-Plutonian planets". Monthly Notices of the Royal Astronomical Society: Letters. 443 (1): L59–L63. arXiv:1406.0715Freely accessible. Bibcode:2014MNRAS.443L..59D. doi:10.1093/mnrasl/slu084. 

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