Incelestial mechanics, theKozai mechanism is a dynamical phenomenon affecting the orbit of abinary system perturbed by a distant third body under certain conditions. The mechanism is also namedvon Zeipel-Kozai-Lidov,Lidov–Kozai,Kozai–Lidov, etc., and may be termed aneffect,oscillation,cycle, orresonance. This effect causes the orbit'sargument of pericenter tooscillate about a constant value, which in turn leads to a periodic exchange between itseccentricity andinclination. The process occurs on timescales much longer than the orbital periods. It can drive an initially near-circular orbit to arbitrarily high eccentricity, andflip an initially moderately inclined orbit between aprograde and a retrograde motion.

The effect has been found to be an important factor shaping the orbits ofirregular satellites of the planets,trans-Neptunian objects,extrasolar planets, andmultiple star systems.^[1]: v It hypothetically promotesblack hole mergers.^[2] It was described in 1961 byMikhail Lidov while analyzing the orbits of artificial and natural satellites of planets.^[3] In 1962,Yoshihide Kozai published this same result in application to the orbits ofasteroids perturbed byJupiter.^[4] The citations of the papers by Kozai and Lidov have risen sharply in the 21st century. As of 2017^[update], the mechanism is among the most studied astrophysical phenomena.^[1]: vi It was pointed out in 2019 by Takashi Ito and Katsuhito Ohtsuka that the Swedish astronomerEdvard Hugo von Zeipel had also studied this mechanism in 1909, and his name is sometimes now added.^[5]

In Hamiltonian mechanics, a physical system is specified by a function, calledHamiltonian and denoted${\displaystyle \mathcal{H}}$, ofcanonical coordinates inphase space. The canonical coordinates consist of thegeneralized coordinates${\displaystyle x_k}$ inconfiguration space and theirconjugate momenta${\displaystyle p_k}$, for${\displaystyle k=1,...N}$, for theN bodies in the system (${\displaystyle N=3}$ for the von Zeipel-Kozai–Lidov effect). The number of${\displaystyle (x_k,p_k)}$ pairs required to describe a given system is the number of itsdegrees 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 acanonical transformation. Theequations of motion for the system are obtained from the Hamiltonian throughHamilton's canonical equations, which relate time derivatives of the coordinates to partial derivatives of the Hamiltonian with respect to the conjugate momenta.

The dynamics of a system composed of three bodies system acting under their mutual gravitational attraction ischaotic: 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 incomputer arithmetic. The practical consequence is that, thethree-body problem cannot be solved analytically for an indefinite amount of time, except in special cases.^[6]: 221 Instead,numerical methods are used for forecast-times limited by the available precision.^{[7]: 2, 10}

The Lidov–Kozai mechanism is a feature ofhierarchical triple systems,^[8]: 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 theinner binary. The perturber and the centre of mass of the inner binary comprise theouter binary.^[9]: §I Such systems are often studied by using the methods ofperturbation 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 termcoupling the two orbits,^[9]

${\displaystyle \mathcal{H}={\mathcal{H}}_{\mathrm{i}\mathrm{n}}+{\mathcal{H}}_{\mathrm{o}\mathrm{u}\mathrm{t}}+{\mathcal{H}}_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}}.}$

The coupling term is then expanded in the orders of parameter${\displaystyle \alpha }$, defined as the ratio of thesemi-major axes of the inner and the outer binary and hence small in a hierarchical system.^[9] Since the perturbative seriesconverges rapidly, the qualitative behaviour of a hierarchical three-body system is determined by the initial terms in the expansion, referred to as thequadrupole(${\displaystyle \propto {\alpha }^2}$),octupole (${\displaystyle \propto {\alpha }^3}$) andhexadecapole (${\displaystyle \propto {\alpha }^4}$) order terms,^{[10]: 4–5}

${\displaystyle {\mathcal{H}}_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}}={\mathcal{H}}_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}+{\mathcal{H}}_{\mathrm{o}\mathrm{c}\mathrm{t}}+{\mathcal{H}}_{\mathrm{h}\mathrm{e}\mathrm{x}}+O({\alpha }^5).}$

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.^[11]

The Lidov–Kozai mechanism is asecular 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 besecularised, 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.^[10]: 4

The simplest treatment of the von Zeipel-Lidov–Kozai mechanism assumes that one of the inner binary's components, thesecondary, is atest particle – an idealized point-like object with negligible mass compared to the other two bodies, theprimary and the distant perturber. These assumptions are valid, for instance, in the case of an artificial satellite in alow Earth orbit that is perturbed by theMoon, or ashort-period comet that is perturbed byJupiter.

Under these approximations, the orbit-averaged equations of motion for the secondary have aconserved quantity: the component of the secondary's orbital angular momentum parallel to the angular momentum of the primary / perturber orbit. This conserved quantity can be expressed in terms of the secondary'seccentricitye andinclinationi relative to the plane of the outer binary:

${\displaystyle L_{\mathrm{z}}=\sqrt{1-e^2}\cos i=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$

Conservation ofL_z means that orbital eccentricity can be "traded for" inclination. Thus, near-circular, highly inclined orbits can become very eccentric. Since increasing eccentricity while keeping thesemimajor axis constant reduces the distance between the objects atperiapsis, this mechanism can cause comets (perturbed byJupiter) to becomesungrazing.

Lidov–Kozai oscillations will be present ifL_z is lower than a certain value. At the critical value ofL_z, a "fixed-point" orbit appears, with constant inclination given by

${\displaystyle i_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=\arccos \left(\sqrt{\frac{3}{5}}\right)\approx {39.2}^{\mathsf{o}}}$

For values ofL_z less than this critical value, there is a one-parameter family of orbital solutions having the sameL_z but different amounts of variation ine ori. Remarkably, the degree of possible variation ini is independent of the masses involved, which only set the timescale of the oscillations.^[12]

The basic timescale associated with Kozai oscillations is^[12]: 575

${\displaystyle T_{\mathrm{K}\mathrm{o}\mathrm{z}\mathrm{a}\mathrm{i}}=2\pi \frac{\sqrt{GM}}{G{m}_2}\frac{a_2^3}{a^{3/2}}{\left(1-e_2^2\right)}^{3/2}=\frac{M}{m_2}\frac{P_2^2}{P}{\left(1-e_2^2\right)}^{3/2}}$

wherea indicates the semimajor axis,P is orbital period,e is eccentricity andm 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, withMoon's period of 27.3 days, eccentricity 0.055 and theGlobal Positioning System satellites period of half a (sidereal) day, the Kozai timescale is a little over 4 years; forgeostationary orbits it is twice shorter.

The period of oscillation of all three variables (e,i,ω – the last being theargument of periapsis) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for theseparatrix orbit that separates librating orbits from oscillating orbits.

The von Zeipel-Lidov–Kozai mechanism causes theargument of pericenter (ω) tolibrate about either 90° or 270°, which is to say that itsperiapse occurs when the body is farthest from the equatorial plane. This effect is part of the reason thatPluto is dynamically protected from close encounters withNeptune.

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 theHill 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.^[13]

The mechanism has been invoked in searches forPlanet Nine, a hypothetical planet orbiting the Sun far beyond the orbit of Neptune.^[14]

A number of moons have been found to be in the Lidov–Kozai resonance with their planet, including Jupiter'sCarpo andEuporie,^[15] Saturn'sKiviuq andIjiraq,^[1]: 100 Uranus'sMargaret,^[16] and Neptune'sSao andNeso.^[17]

Some sources identify the Soviet space probeLuna 3 as the first example of an artificial satellite undergoing Lidov–Kozai oscillations. Launched in 1959 into a highly inclined, eccentric, geocentric orbit, it was the first mission to photograph thefar side of the Moon. It burned in the Earth's atmosphere after completing eleven revolutions.^{[1]: 9–10} However, according to Gkoliaset al.. (2016) a different mechanism must have driven the decay of the probe's orbit since the Lidov–Kozai oscillations would have been thwarted by effects of the Earth'soblateness.^[18]

The von Zeipel–Lidov–Kozai mechanism, in combination withtidal friction, is able to produceHot Jupiters, which aregas giant exoplanets orbiting their stars on tight orbits.^{[19][20][21][22]} The high eccentricity of the planetHD 80606 b in theHD 80606/80607 system is likely due to the Kozai mechanism.^[23]KELT-19 Ab is likely to have the evidence of Kozai mechanism due to primordial misalignments between the planetary orbital axis and stellar spin axis with its orbit flipped.^[24]

The mechanism is thought to affect the growth of centralblack holes in densestar clusters. It also drives the evolution of certain classes ofbinary black holes^[9] and may play a role in enablingblack hole mergers.^[25]

The effect was first described in 1909 by the Swedish astronomerHugo von Zeipel in his work on the motion of periodic comets inAstronomische Nachrichten.^[26][5] In 1961, the Soviet space scientistMikhail 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][27]: 88}

Lidov first presented his work on artificial satellite orbits at theConference on General and Applied Problems of Theoretical Astronomy held in Moscow on 20–25 November 1961.^[28] His paper was first published in a Russian-language journal in 1961.^[3] The Japanese astronomerYoshihide Kozai was among the 1961 conference participants.^[28] Kozai published the same result in a widely read English-language journal in 1962, using the result to analyze orbits ofasteroids perturbed byJupiter.^[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.

• Orbital perturbations
• Orbital resonance
• Kozai mechanism

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