Kozai mechanism
In
The effect has been found to be an important factor shaping the orbits of
Background
Hamiltonian mechanics
In Hamiltonian mechanics, a physical system is specified by a function, called Hamiltonian and denoted , of canonical coordinates in phase space. The canonical coordinates consist of the generalized coordinates in
The coordinate pairs are usually chosen in such a way as to simplify the calculations involved in solving a particular problem. One set of canonical coordinates can be changed to another by a canonical transformation. The equations of motion for the system are obtained from the Hamiltonian through Hamilton's canonical equations, which relate time derivatives of the coordinates to partial derivatives of the Hamiltonian with respect to the conjugate momenta.
The three-body problem
The dynamics of a system composed of three bodies system acting under their mutual gravitational attraction is complex. In general, the behaviour of a three-body system over long periods of time is
The Lidov–Kozai mechanism is a feature of hierarchical triple systems,[7]: 86 that is systems in which one of the bodies, called the "perturber", is located far from the other two, which are said to comprise the inner binary. The perturber and the centre of mass of the inner binary comprise the outer binary.[8]: §I Such systems are often studied by using the methods of perturbation theory to write the Hamiltonian of a hierarchical three-body system as a sum of two terms responsible for the isolated evolution of the inner and the outer binary, and a third term coupling the two orbits,[8]
The coupling term is then expanded in the orders of parameter , defined as the ratio of the
For many systems, a satisfactory description is found already at the lowest, quadrupole order in the perturbative expansion. The octupole term becomes dominant in certain regimes and is responsible for a long-term variation in the amplitude of the Lidov–Kozai oscillations.[10]
Secular approximation
The Lidov–Kozai mechanism is a secular effect, that is, it occurs on timescales much longer compared to the orbital periods of the inner and the outer binary. In order to simplify the problem and make it more tractable computationally, the hierarchical three-body Hamiltonian can be secularised, that is, averaged over the rapidly varying mean anomalies of the two orbits. Through this process, the problem is reduced to that of two interacting massive wire loops.[9]: 4
Overview of the mechanism
Test particle limit
The simplest treatment of the von Zeipel-Lidov–Kozai mechanism assumes that one of the inner binary's components, the secondary, is a
Under these approximations, the orbit-averaged equations of motion for the secondary have a
Conservation of Lz 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
Lidov–Kozai oscillations will be present if Lz is lower than a certain value. At the critical value of Lz, a "fixed-point" orbit appears, with constant inclination given by
For values of Lz less than this critical value, there is a one-parameter family of orbital solutions having the same Lz 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.[11]
Timescale
The basic timescale associated with Kozai oscillations is[11]: 575
where a indicates the semimajor axis, P is orbital period, e is eccentricity and m is mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner orbit; M is the mass of the primary. For example, with Moon's period of 27.3 days, eccentricity 0.055 and the Global Positioning System satellites period of half a (sidereal) day, the Kozai timescale is a little over 4 years; for geostationary orbits it is twice shorter.
The period of oscillation of all three variables (e, i, ω – the last being the
Astrophysical implications
Solar System
The von Zeipel-Lidov–Kozai mechanism causes the
The Lidov–Kozai mechanism places restrictions on the orbits possible within a system. For example:
- For a regular satellite
- 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. Recently, the Hill-stability radius has been found as a function of satellite inclination, also explains the non-uniform distribution of irregular satellite inclinations.[12]
The mechanism has been invoked in searches for Planet Nine, a hypothetical planet orbiting the Sun far beyond the orbit of Neptune.[13]
A number of moons have been found to be in the Lidov–Kozai resonance with their planet, including Jupiter's Carpo and Euporie,[14] Saturn's Kiviuq and Ijiraq,[1]: 100 Uranus's Margaret,[15] and Neptune's Sao and Neso.[16]
Some sources identify the Soviet space probe
Extrasolar planets
The von Zeipel-Lidov–Kozai mechanism, in combination with
Black holes
The mechanism is thought to affect the growth of central
History and development
The effect was first described in 1909 by the Swedish astronomer Hugo von Zeipel in his work on the motion of periodic comets in Astronomische Nachrichten.[24][25] In 1961, the Soviet space scientist Mikhail Lidov discovered the effect while analyzing the orbits of artificial and natural satellites of planets. Originally published in Russian, the result was translated into English in 1962.[3][26]: 88
Lidov first presented his work on artificial satellite orbits at the Conference on General and Applied Problems of Theoretical Astronomy held in Moscow on 20–25 November 1961.[27] His paper was first published in a Russian-language journal in 1961.[3] The Japanese astronomer Yoshihide Kozai was among the 1961 conference participants.[27] Kozai published the same result in a widely read English-language journal in 1962, using the result to analyze orbits of asteroids perturbed by Jupiter.[4] Since Lidov was the first to publish, many authors use the term Lidov–Kozai mechanism. Others, however, name it as the Kozai–Lidov or just the Kozai mechanism.
References
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Shevchenko, Ivan I. (2017). "The Lidov-Kozai effect – applications in exoplanet research and dynamical astronomy". Astrophysics and Space Science Library. Vol. 441. Cham: Springer International Publishing. ISSN 0067-0057.
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Tremaine, Scott; Yavetz, Tomer D. (2014). "Why do Earth satellites stay up?". American Journal of Physics. 82 (8). American Association of Physics Teachers (AAPT): 769–777. S2CID 119298013.
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Lidov, Mikhail L. (1961). "Эволюция орбит искусственных спутников под воздействием гравитационных возмущений внешних тел" [The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies]. Iskusstvennye Sputniki Zemli (in Russian). 8: 5–45.
Lidov, Mikhail L. (1962). "The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies". . (translation of Lidov's 1961 paper)
Lidov, Mikhail L. (20–25 November 1961). "On approximate analysis of the evolution of orbits of artificial satellites". Proceedings of the Conference on General and Practical Topics of Theoretical Astronomy. Problems of Motion of Artificial Celestial Bodies. Moscow, USSR: Academy of Sciences of the USSR (published 1963). - ^ a b
Kozai, Yoshihide (1962). "Secular perturbations of asteroids with high inclination and eccentricity". doi:10.1086/108790.
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