Kozai mechanism

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.

Contents

• Consequences
• Timescale
• References

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, and in 1962 by the Japanese astronomer Yoshihide Kozai (Japanese: 古在由秀) 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.

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 L_z 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 L_z is lower than a certain value. At the critical value of L_z, a "fixed-point" orbit appears, with constant inclination given by

For values of L_z less than this critical value, there is a one-parameter family of orbital solutions having the same L_z 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.