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| In [[celestial mechanics]], the '''Kozai mechanism''', or the '''Lidov–Kozai mechanism''', refers to the orbit of a satellite that is perturbed by another body orbiting farther out. Due to the perturbation, the orbit of the satellite experiences [[libration]] (oscillation about a constant value) of its [[argument of pericenter]]. As the orbit librates, there is a
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| periodic exchange between its [[inclination]] and its [[Orbital eccentricity|eccentricity]].
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| The effect was described in 1961 by the Soviet specialist in space dynamics [[Michael Lidov]] (Russian: [[:ru:Михаил Львович Лидов]]) while analyzing the orbits of artificial and natural satellites of planets,<ref> {{cite journal |last= Lidov|first= M. L. |year=1962 |title= The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies|journal= Planetary and Space Science|volume=9 |issue= |pages=719–759|doi= 10.1016/0032-0633(62)90129-0|url= |accessdate=4 April 2013|bibcode = 1962P&SS....9..719L }}</ref>
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| and in 1962 by the Japanese astronomer [[Yoshihide Kozai]] while analyzing the orbits of the [[asteroid]]s.<ref>{{cite journal |last=Kozai |first=Y.|year=1962 |title=Secular perturbations of asteroids with high inclination and eccentricity |journal=The Astronomical Journal |volume=67 |issue= |page=591 |publisher= |doi= 10.1086/108790|url=http://adsabs.harvard.edu/cgi-bin/bib_query?1962AJ.....67..591K |accessdate=4 April 2013|bibcode = 1962AJ.....67..591K }}
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| </ref> Since then this effect has been found to be an important factor shaping the orbits of [[irregular satellite]]s of the planets, [[trans-Neptunian object]]s, and a few [[extrasolar planets]] and [[multiple star system]]s.
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| == Kozai mechanism ==
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| 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.
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| Under these approximations, the orbit-averaged equations of motion for the satellite have a [[integral of motion|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 [[Orbital eccentricity|eccentricity]] ''e'' and [[inclination]] ''i'' relative to the plane of the outer binary:
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| :<math> L_z = \sqrt{(1-e^2)} \cos i .</math>
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| Conservation of ''L''<sub>z</sub> 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 comet|sungrazing]].
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| Kozai oscillations will be present if ''L''<sub>z</sub> is lower than a certain value. At the critical value of ''L''<sub>z</sub>, a "fixed-point" orbit appears, with constant inclination given by
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| :<math>\arccos\left(\sqrt\frac{3}{5}\right) \approx 39.2^{o}</math>
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| For values of ''L''<sub>z</sub> less than this critical value, there is a one-parameter family of orbital solutions having the same ''L''<sub>z</sub> 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.<ref name=DEGN/>
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| ==Consequences==
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| The Kozai mechanism causes the [[argument of pericenter]] 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]].
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| The Kozai mechanism places restrictions on the orbits possible within a system, for example
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| *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
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| *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]]
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| ==Timescale==
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| The basic timescale associated with Kozai oscillations is<ref name="DEGN">{{cite book|last=Merritt|first=David|authorlink=David Merritt|title=Dynamics and Evolution of Galactic Nuclei|year=2013|publisher=Princeton University Press|location=Princeton, NJ|page=575|url=http://openlibrary.org/works/OL16802359W/Dynamics_and_Evolution_of_Galactic_Nuclei}}
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| </ref> | |
| :<math>
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| T_\mathrm{Kozai} = 2\pi\frac{\sqrt{GM}}{Gm_2}\frac{a_2^3}{a^{3/2}}\left(1-e_2^2\right)^{3/2} = \frac{M}{m_2}\frac{P_2^2}{P}\left(1-e_2^2\right)^{3/2}
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| </math>
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| 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.
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| The period of oscillation of all three variables (''e'', ''i'', ω) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the [[separatrix (dynamical systems)|separatrix]] orbit that separates librating (Kozai) orbits from oscillating orbits.
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| ==See also==
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| * [[Jacobi integral]]
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| * [[Tisserand's relation]]
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| ==References==
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| {{reflist}}
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| == External links ==
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| *[http://www.orbitsimulator.com/gravity/articles/kozai.html Kozai mechanism visualization]
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| [[Category:Orbital perturbations]]
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