Kozai mechanism

In celestial mechanics, the Kozai mechanism, or the Lidov–Kozai mechanism, causes a periodic exchange between the inclination and eccentricity of an orbit. That is, libration (oscillation about a constant value) occurs in the argument of pericenter.

The effect was described in 1961 by the Soviet specialist in space dynamics Michael Lidov (Russian: Михаил Львович Лидов) while analysing the orbits of artificial and natural satellites of planets, and in 1962 by the Japanese astronomer Yoshihide Kozai while analyzing the orbits of the asteroids. Since then this effect 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.

Kozai resonance

For an orbiting body in the restricted three-body problem with eccentricity $e\,\!$ and inclination $i\,\!$ relative to the fundamental plane, the following value is conserved:

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

Which is to say that orbital eccentricity can be traded for inclination, and perturbation may lead to an interchange between the two. Thus, near-circular, highly-inclined orbits can become very eccentric in exchange for lower inclination. Since increasing eccentricity while keeping the semimajor axis constant reduces the distance between the objects at periapsis, this mechanism can cause comets to become sungrazing.

Typically, for the objects in low-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°, and the pericenter (point of closest approach) is forced to oscillate around one of these values. The minimum required inclination angle, called the Kozai angle, is

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

For retrograde satellites the angle is 140.8°.

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

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: 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

References and footnotes

• M. L. Lidov, The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Iskusstv. Sputn. Zemli, № 8, c. 5, 1961 (Russian). English transl.: Planet. Space Sci., vol. 9, p. 719, 1962
• 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).

External links

• Kozai mechanism visualization