Kozai mechanism

In celestial mechanics, the Kozai mechanism or Lidov–Kozai mechanism or Kozai–Lidov mechanism, also known as the Kozai, Lidov–Kozai or Kozai–Lidov effect, oscillations, cycles or resonance, is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions, causing 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. In celestial mechanics, the Kozai mechanism or Lidov–Kozai mechanism or Kozai–Lidov mechanism, also known as the Kozai, Lidov–Kozai or Kozai–Lidov effect, oscillations, cycles or resonance, is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions, causing 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 is hypothesized to enable black hole mergers. It was first 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 initial papers by Kozai and Lidov have risen sharply in the 21st century. As of 2017, the mechanism is among the most studied astrophysical phenomena. In Hamiltonian mechanics, a physical system is specified by a function, called Hamiltonian and denoted H { extstyle {mathcal {H}}} , of canonical coordinates in phase space. The canonical coordinates consist of the generalized coordinates x i { extstyle x_{i}} in configuration space and their conjugate momenta p i { extstyle p_{i}} . The number of ( x i , p i ) { extstyle (x_{i},p_{i})} pairs required to describe a given system is the number of its degrees of freedom. The coordinates 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. An elliptical orbit in three dimensions is uniquely described by a set of six coordinates, called orbital elements. The traditional choice are the Keplerian elements, which consist of the eccentricity, semimajor axis, inclination, longitude of the ascending node, argument of periapsis, and true anomaly. In celestial mechanics calculations, it is common to use a set of orbital elements introduced in nineteenth century by Charles-Eugène Delaunay. The Delaunay elements form a canonical set of action-angle coordinates and consist of the mean anomaly l { extstyle l} , the argument of periapsis g { extstyle g} and the longitude of the ascending node h { extstyle h} , along with their conjugate momenta denoted by L { extstyle L} , G { extstyle G} , and H { extstyle H} , respectively. 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 depends sensitively on the initial conditions. Thus, the three-body problem, the problem of determining the motions of the three bodies, cannot be solved analytically except in special cases. Instead, numerical methods are used. 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, The coupling term is then expanded in the orders of parameter α { extstyle 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 { extstyle propto alpha ^{2}} ), octupole ( ∝ α 3 { extstyle propto alpha ^{3}} ) and hexadecapole ( ∝ α 4 { extstyle propto alpha ^{4}} ) order terms, 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. 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 wires.