The Kozai Mechanism

It has been theorized that the Kozai Mechanism is responsible for the high eccentricities observed in the orbits of exosolar planets. If the parent star has a massive yet unseen substellar companion, orbiting at a great distance, and in an orbit highly inclined to the plane of the planets' orbits, the Kozai Mechanism should induce high eccentricities into the orbits of the planets.

It is also theorized that the Kozai Mechanism may be responsible for the high eccentricities observed in the orbits of many Kuiper Belt Objects such as 2003 UB₃₁₃. The pull of the Milky Way Galaxy causes an object with a high eccentricity to periodically trade this eccentricity for a gain in inclination.

The Jovian system is influenced by the Kozai Mechanism as well. Although Jupiter has many irregular, distant moons orbiting in a myriad of directions, the Kozai Mechanism has set an upper limit of about 60 degrees on their inclinations. Any moon whose inclination exceeded this value would periodically gain an eccentricity so large that its perijove would take it into the region of the Galilean Moons. Its fate would be a collision with one of the Galilean Moons, a collision with Jupiter, or an ejection from the Jovian system.

Jupiter is surrounded by a large system of irregular moons beyond the orbits of the Galilean Moons (purple). Although these moons possess a wide range of inclinations and eccentricities, polar orbits are non-existant.

If the Earth's Moon were in a circular polar orbit, with the same semi-major axis (distance) that it has now, it would quickly be ejected into interplanetary space, or collide with the Earth.

The simulation kozai.gsim places the Moon in a circular polar orbit around Earth. It orbits at a distance of 384,000 kilometers from the Earth's center. The Moon completes many orbits of the Earth with only minimal effect to its orbit. But these effects grow, and soon the Moon's orbit is very elliptical. After 8 years, the Moon slams into the Earth.

The maximum eccentricity attainable through the Kozai mechanism may be approximated from the formula from Takeda and Rasio's 2005 paper.

Here is a javascript calculator that allows you to try different numbers in their formula:

Kozai Eccentricity

The period from minimum eccentricity to maximum and back to minimum may be approximated by the formula from a May 2000 paper by Eric B. Ford, Boris Kozinsky, and Frederic A. Rasio entitled Secular Evolution of Hierarchical Triple Star Systems.

Here is a javascript calculator that allows you to try different numbers in their formula:

Kozai Period

Discuss the Kozai mechanism on Gravity Simulator's web forum