In celestial mechanics, the Kozai mechanism is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions. The mechanism is also named von Zeipel–Kozai–Lidov, Lidov–Kozai, Kozai–Lidov, etc., and may be termed an effect, oscillation, cycle, or resonance. This effect causes the orbit's argument of pericenter to oscillate about a constant value, which in turn leads to a periodic exchange between its eccentricity and inclination. The process occurs on timescales much longer than the orbital periods. It can drive an initially near-circular orbit to arbitrarily high eccentricity, and flip an initially moderately inclined orbit between a prograde and a retrograde motion. The effect has been found to be an important factor shaping the orbits of irregular satellites of the planets, trans-Neptunian objects, extrasolar planets, and multiple star systems. It hypothetically promotes black hole mergers. It was described in 1961 by Mikhail Lidov while analyzing the orbits of artificial and natural satellites of planets. In 1962, Yoshihide Kozai published this same result in application to the orbits of asteroids perturbed by Jupiter. The citations of the papers by Kozai and Lidov have risen sharply in the 21st century. As of 2017, the mechanism is among the most studied astrophysical phenomena. It was pointed out in 2019 by Takashi Ito and Katsuhito Ohtsuka that the Swedish astronomer Edvard Hugo von Zeipel had also studied this mechanism in 1909, and his name is sometimes now added.
Background
Hamiltonian mechanics
In Hamiltonian mechanics, a physical system is specified by a function, called Hamiltonian and denoted H {\displaystyle {\mathcal {H}}} , of canonical coordinates in phase space. The canonical coordinates consist of the generalized coordinates x k {\displaystyle x_{k}} in configuration space and their conjugate momenta p k {\displaystyle p_{k}} , for k = 1 , . . . N {\displaystyle k=1,...N} , for the N bodies in the system ( N = 3 {\displaystyle N=3} for the von Zeipel-Kozai–Lidov effect). The number of ( x k , p k ) {\displaystyle (x_{k},p_{k})} pairs required to describe a given system is the number of its degrees of freedom. 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.
Three-body problem
The dynamics of a system composed of three bodies system acting under their mutual gravitational attraction is chaotic: its behavior over long periods of time is enormously sensitive to any slight changes in the initial conditions. This exposes computations to rapid deterioration from uncertainties in those conditions, in determining them, and then preserving them from rounding away in computer arithmetic. The practical consequence is that, the three-body problem cannot be solved analytically for an indefinite amount of time, except in special cases. Instead, numerical methods are used for forecast-times limited by the available precision. The Lidov–Kozai mechanism is a feature of hierarchical triple systems, 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. 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,
H = H i n + H o u t + H p e r t . {\displaystyle {\mathcal {H}}={\mathcal {H}}_{\rm {in}}+{\mathcal {H}}_{\rm {out}}+{\mathcal {H}}_{\rm {pert}}.}
The coupling term is then expanded in the orders of parameter α {\displaystyle \alpha } , defined as the ratio of the semi-major axes of the inner and the outer binary and hence small in a hierarchical system. Since the perturbative series converges rapidly, the qualitative behaviour of a hierarchical three-body system is determined by the initial terms in the expansion, referred to as the quadrupole ( ∝ α 2 {\displaystyle \propto \alpha ^{2}} ), octupole ( ∝ α 3 {\displaystyle \propto \alpha ^{3}} ) and hexadecapole ( ∝ α 4 {\displaystyle \propto \alpha ^{4}} ) order terms,
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