Kozai mechanism

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In celestial mechanics, the Kozai mechanism is a secular perturbative effect on certain orbits. It causes changes in the eccentricity and the inclination of the orbit of the perturbed body to be periodic and synchronized, and also causes libration (i.e., oscillation about a constant value) of the argument of pericenter.

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

1 Kozai resonance
2 Consequences
3 References and footnotes
4 External links

[edit] Kozai resonance

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

$\sqrt{(1-e^2)} \cos i$

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 argument of 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

$\arccos(\sqrt\frac{3}{5}) \approx 39.2^{o}$

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 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.

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 satellites: 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.

[edit] References and footnotes

^ Large eccentricity with constant semimajor axis means small perihelion.

Y. Kozai, Secular perturbations of asteroids with high inclination and eccentricity, Astronomical Journal 67, 591 ADS
C. Murray and S. Dermott 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).