Kozai mechanism
Encyclopedia
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 asteroid
s. Since then this effect has been found to be an important factor shaping the orbits of irregular satellite
s of the planets, trans-Neptunian object
s, and a few extrasolar planets and multiple star systems.
and inclination
relative to the orbit around the larger body, the following value is conserved:
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
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).
is dynamically protected from close encounters with Neptune
.
The Kozai resonance places restrictions on the orbits possible within a system, for example
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
, the Kozai mechanism, or the Lidov-Kozai mechanism, causes a periodic exchange between the inclination
Inclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...
and eccentricity
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...
of an orbit. That is, libration
Libration
In astronomy, libration is an oscillating motion of orbiting bodies relative to each other, notably including the motion of the Moon relative to Earth, or of Trojan asteroids relative to planets.-Lunar 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 asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s. Since then this effect has been found to be an important factor shaping the orbits of irregular satellite
Irregular satellite
In astronomy, an irregular moon is a natural satellite following a distant, inclined, and often eccentric and retrograde orbit. They are believed to have been captured by their parent planet, unlike regular satellites, which form in situ....
s of the planets, trans-Neptunian object
Trans-Neptunian object
A trans-Neptunian object is any minor planet in the Solar System that orbits the Sun at a greater distance on average than Neptune.The first trans-Neptunian object to be discovered was Pluto in 1930...
s, and a few extrasolar planets and multiple star systems.
Kozai resonance
For an orbiting body with eccentricityOrbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...
and inclinationInclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...
relative to the orbit around the larger body, the following value is conserved: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
Sungrazing comet
A sungrazing comet is a comet that passes extremely close to the Sun at perihelion – sometimes within a few thousand kilometres of the Sun's surface. While small sungrazers can be completely evaporated during such a close approach to the Sun, larger sungrazers can survive many perihelion passages...
.
Typically, for the objects in low-inclination orbits, the perturbations result in the precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
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
For retrograde satellites the angle is 140.8°.
Physically, the effect is related to angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
transfer; the quantity conserved is actually the normal component of the angular momentum (see also Jacobi integral
Jacobi integral
In celestial mechanics, Jacobi's integral is the only known conserved quantity for the restricted three-body problem problem [1]; unlike in the two-body problem, the energy and momentum of the system are not conserved separately and a general analytical solution is not possible...
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 PlutoPluto
Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun...
is dynamically protected from close encounters with Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...
.
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 sphereHill sphereAn astronomical body's Hill sphere is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own...