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In celestial mechanics, the Kozai mechanism is a secular perturbative effect resulting in the periodic sychronised changes of the eccentricity and the inclination of the orbit of the perturbed body.

The effect was described in 1962 by Yoshihide Kozai analysing the orbits of the asteroids. Since then the Kozai resonance have been found to be an important factor shaping the orbits in the solar system (irregular satellites of the planets, trans-Neptunian objects, a few extrasolar planets and multiple star systems).

Contents 1 Kozai resonance 2 Consequences 3 External links 4 References

[edit] Kozai resonance

For a body with semimajor axis <math>a\,\!</math>, eccentricity <math>e\,\!</math>, and inclination <math>i\,\!</math>, relative to the orbit of the perturbing larger body the following value is conserved

<math> \sqrt{(1-e^2)} \cos i</math>

Consequently, perturbations may lead to the resonance between the orbit inclination and eccentricity. Near circular, highly inclined orbits can thus become very eccentric (in exchange for lower inclination). As example, such mechanism can produce Sun-grazing comets.¹

Typically, for the objects on small inclination orbits, the perturbations result in the precession of the pericenter. Starting with some value of the angle, the precession is replaced by libration around 90° or 270°, i.e. the pericenter is forced to oscillate around one these values. This minimal required inclination angle, called the Kozai angle , is

<math>arccos(\sqrt\frac{3}{5}) \approx 39.2^{o}</math>

For retrograde satellites the angle is 140.8°.

Physically, the effect is related to the angular momentum transfer: the expression "conserved" is actually the normal component of the angular momentum (see also Jacobi integral and Tisserand's relation).

[edit] Consequences

The Kozai resonance 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 satelites: as above, 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 is also believed to produce Sun-grazing comets.

¹Large eccentricity with constant semimajor axis means small perihelion.

[edit] External links

• Orbit simulator [1]

[edit] References

• Murray, Dermot Solar System Dynamics, Cambridge University Press, ISBN 0-521-57597-4
• Innanen et al. The Kozai Mechanism and the stability of planetary orbits in binary star systems, The Astronomical Journal,113 (1997).

fi:Kozain resonanssi