Kozai effect

The Kozai mechanism , also Kozai mechanism or Kozai resonance called describes in the celestial mechanics a periodic train disorder that a change in the eccentricity and the orbit inclination causes (inclination) of the faulty object. The effect is named after Yoshihide Kozai ( 古在由秀 , Kozai Yoshihide ), who discovered it in 1962 while analyzing asteroid orbits . He is by Hugo von Zeipel , who treated him in 1910 and Mikhail Lvovich Lidow , who also discovered it parallel to Kozai in the Soviet Union, also of Zeipel-Lidov-Kozai mechanism , Lidow-Kozai mechanism or effect Kozai-Lidow- called .

Impact and Importance

The Kozai resonance leads to restrictions of the possible paths in a system, for example:

for regular moons : if the orbit of a moon is strongly inclined compared to that of its planet , the eccentricity of the lunar orbit increases until the moon is destroyed by tidal forces .
for irregular moons : as above, only that the increasing eccentricity leads to a collision with a regular moon or to the satellite being ejected from the Hill sphere .

The Kozai effect is therefore considered to be an important factor in the formation of the orbits of some bodies in the solar system (irregular satellites of the planets, trans-Neptunian objects ). It is also used to explain the following observations:

the high number of hot Jupiters among the exoplanets
the frequency of triple systems with a close binary star system and orbital times of less than 5 days near the sun

the emergence of blue stragglers .

Since an increase in the eccentricity with the large orbit half-axis remaining the same results in the periapsis of the orbit being reduced, the Kozai mechanism can also cause the orbits of comets to change over time so that they plunge into the sun .

Explanation

A three-body system is considered, which consists of a central body (e.g. sun), a relatively large body (e.g. planet) surrounding it and a small body (e.g. asteroid) which also orbits the central body. The small body, which runs on an elliptical orbit with an inclination relative to the orbit of the planet and with an eccentricity around the central body, has orbital elements that are secularly disturbed by the large revolving body . In the perturbation theory approach, the following value is constant over time: ${\ displaystyle i}$ ${\ displaystyle e}$

{\ displaystyle \ Theta: = (1-e ^ {2}) \ cos ^ {2} i = {\ text {const.}}}

This constant of movement enables an exchange relationship between inclination and eccentricity: if the inclination decreases, the eccentricity increases and vice versa. Almost circular orbits with high inclination can therefore be changed to very eccentric orbits with low inclination.

If the initial inclination is large enough (i.e. at least as large as the Kozai angle, see below), then a Kozai resonance results . H. a resonant exchange or a periodic, opposite fluctuation of inclination and eccentricity between minimum and maximum values. At the same time there is libration of the pericenter , i. that is, the argument of the pericenter oscillates around a constant value.
However, if the inclination and eccentricity of the small body are quite small, the result of such a disturbance is no resonant exchange between eccentricity and inclination, but only a secular progression of the argument of the pericenter, i.e. H. a perihelion .

Kozai angle

The Kozai angle , i.e. the minimum initial inclination angle required for a Kozai resonance with an initially almost circular orbit ( ), depends on the distance between the disturbing planet and the small body. If this distance is very large, one finds: ${\ displaystyle i_ {0}}$ ${\ displaystyle e = 0}$

{\ displaystyle \ Rightarrow i_ {0} = \ arccos \ left ({\ sqrt {\ frac {3} {5}}} \ right) \ approx 39 {,} 2 ^ {\ circ}}

The transition between perihelion and Kozai effect therefore takes place at a value for the constant of movement of maximal

{\ displaystyle \ Theta _ {0} = {\ frac {3} {5}} = 0 {,} 6}

If the disturbing large body is closer to the orbit of the small body, the Kozai angle decreases and the limit value increases accordingly . ${\ displaystyle \ Theta _ {0}}$

For retrograde , i.e. H. Satellites running "backwards" around the central body have inclination values between 90 ° and 270 °. In this case, the Kozai-angle is a maximum value and is for distant bluff body at . ${\ displaystyle 180 ^ {\ circ} -39 {,} 2 ^ {\ circ} = 140 {,} 8 ^ {\ circ}}$

Sources and literature

Y. Kozai, Secular perturbations of asteroids with high inclination and eccentricity, Astronomical Journal 67, 591 (1962) 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).
Benjamin J. Shappe, Todd A. Thompson: The Mass-Loss induced eccentric Kozai Mechanism: A new Channel for the Production of Close Compact Object-Stellar Binaries. In: Astrophysics. Solar and Stellar Astrophysics . 2012, arxiv : 1204.1053v1 .
Takashi Ito, Katsuhito Ohtsuka: The Lidov – Kozai Oscillation and Hugo von Zeipel . arxiv : 1911.03984 .