celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

, 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 Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...

and

inclination Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Eart ...

. 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 satellite In astronomy, an irregular moon, irregular satellite, or irregular natural satellite is a natural satellite following an orbit that is irregular in some of the following ways: Distant; inclined; highly elliptical; retrograde. They have often be ...

s of the planets,

trans-Neptunian object A trans-Neptunian object (TNO), also written transneptunian object, is any minor planet in the Solar System that orbits the Sun at a greater average distance than Neptune, which has an orbital semi-major axis of 30.1 astronomical units (AU). ...

s, extrasolar planets, and

multiple star system A star system or stellar system is a small number of stars that orbit each other, bound by gravitational attraction. It may sometimes be used to refer to a single star. A large group of stars bound by gravitation is generally called a ''st ...

s. It hypothetically promotes

black hole merger Black is a color that results from the absence or complete Absorption (electromagnetic radiation), absorption of visible spectrum, visible light. It is an achromatic color, without Colorfulness#Chroma, chroma, like white and grey. It is ofte ...

s. 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

asteroid An asteroid is a minor planet—an object larger than a meteoroid that is neither a planet nor an identified comet—that orbits within the Solar System#Inner Solar System, inner Solar System or is co-orbital with Jupiter (Trojan asteroids). As ...

s perturbed by

Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...

. The citations of the papers by Kozai and Lidov have risen sharply in the 21st century. , 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

\mathcal

, of

canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...

phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

. The canonical coordinates consist of the

generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.p. 397 ...

x_k

in configuration space and their conjugate momenta

p_k

, for

k = 1, ... N

, for the bodies in the system (

N = 3

for the von Zeipel-Kozai–Lidov effect). The number of

(x_k, p_k)

pairs required to describe a given system is the number of its

degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...

. 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 In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as ''form invariance''. Although Hamilton's equations are preserved, it need not ...

. The

equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...

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 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 Computer arithmetic is the scientific field that deals with representation of numbers on computers and corresponding implementations of the arithmetic operations. It includes: *Fixed-point arithmetic *Floating-point arithmetic *Interval arithmet ...

. The practical consequence is that, the

three-body problem In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses orbiting each other in space and then calculate their subsequent trajectories using Newton' ...

cannot be solved analytically for an indefinite amount of time, except in special cases. Instead,

numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...

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 In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...

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, :

\mathcal = \mathcal_ + \mathcal_ + \mathcal_.

The coupling term is then expanded in the orders of parameter

\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'' (

\propto\alpha^2

), ''octupole'' (

\propto\alpha^3

) and ''hexadecapole'' (

\propto\alpha^4

) order terms, :

\mathcal_ = \mathcal_ + \mathcal_ + \mathcal_ + 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.

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.

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 test particle – an idealized point-like object with negligible mass compared to the other two bodies, the ''primary'' and the distant perturber. These assumptions are valid, for instance, in the case of an artificial satellite in a

low Earth orbit A low Earth orbit (LEO) is an geocentric orbit, orbit around Earth with a orbital period, period of 128 minutes or less (making at least 11.25 orbits per day) and an orbital eccentricity, eccentricity less than 0.25. Most of the artificial object ...

that is perturbed by the

Moon The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...

, or a short-period comet that is perturbed by

. Under these approximations, the orbit-averaged equations of motion for the secondary have a

conserved quantity A conserved quantity is a property or value that remains constant over time in a system even when changes occur in the system. In mathematics, a conserved quantity of a dynamical system is formally defined as a function of the dependent vari ...

: 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's

and

relative to the plane of the outer binary: :

L_\mathrm = \sqrt\, \cos i = \mathrm

Conservation of 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

semimajor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...

constant reduces the distance between the objects at

periapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...

, this mechanism can cause comets (perturbed by

) to become sungrazing. Lidov–Kozai oscillations will be present if is lower than a certain value. At the critical value of , a "fixed-point" orbit appears, with constant inclination given by :

i_\mathrm = \arccos \left( \sqrt\, \right) \approx 39.2^\mathsf

For values of less than this critical value, there is a one-parameter family of orbital solutions having the same but different amounts of variation in or . Remarkably, the degree of possible variation in is independent of the masses involved, which only set the timescale of the oscillations.

Timescale

The basic timescale associated with Kozai oscillations is :

T_\mathrm = 2 \pi\,\frac\,\frac\left( 1 - e_2^2 \right)^ = \frac \frac\,\left( 1 - e_2^2\right )^

where indicates the semimajor axis, is orbital period, is eccentricity and is mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner orbit; is the mass of the primary. For example, with

's period of 27.3 days, eccentricity 0.055 and the

Global Positioning System The Global Positioning System (GPS) is a satellite-based hyperbolic navigation system owned by the United States Space Force and operated by Mission Delta 31. It is one of the global navigation satellite systems (GNSS) that provide ge ...

satellites period of half a (sidereal) day, the Kozai timescale is a little over 4 years; for

geostationary orbit A geostationary orbit, also referred to as a geosynchronous equatorial orbit''Geostationary orbit'' and ''Geosynchronous (equatorial) orbit'' are used somewhat interchangeably in sources. (GEO), is a circular orbit, circular geosynchronous or ...

s it is twice shorter. The period of oscillation of all three variables (, , – the last being the

argument of periapsis The argument of periapsis (also called argument of perifocus or argument of pericenter), symbolized as ''ω (omega)'', is one of the orbital elements of an orbiting body. Parametrically, ''ω'' is the angle from the body's ascending node to it ...

) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the separatrix orbit that separates librating orbits from oscillating orbits.

Astrophysical implications

Solar System

The von Zeipel-Lidov–Kozai mechanism causes the argument of pericenter () to librate about either 90° or 270°, which is to say that its periapse occurs when the body is farthest from the equatorial plane. This effect is part of the reason that

Pluto Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of Trans-Neptunian object, bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the Su ...

is dynamically protected from close encounters with

Neptune Neptune is the eighth and farthest known planet from the Sun. It is the List of Solar System objects by size, fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 t ...

. 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 The Hill sphere is a common model for the calculation of a Sphere of influence (astrodynamics), gravitational sphere of influence. It is the most commonly used model to calculate the spatial extent of gravitational influence of an astronomical ...

. Recently, the Hill-stability radius has been found as a function of satellite inclination, also explains the non-uniform distribution of irregular satellite inclinations. The mechanism has been invoked in searches for

Planet Nine Planet Nine is a List of hypothetical Solar System objects, hypothetical ninth planet in the outer region of the Solar System. Its gravitational effects could explain the peculiar clustering of orbits for a group of extreme trans-Neptunian obj ...

, a hypothetical planet orbiting the Sun far beyond the orbit of Neptune. A number of moons have been found to be in the Lidov–Kozai resonance with their planet, including Jupiter's Carpo and

Euporie In Greek mythology, the Horae (), Horai () or Hours (, ) were the goddesses of the seasons and the natural portions of time. Etymology The term ''hora'' comes from the Proto-Indo-European ("year"). Function The Horae were originally the p ...

, Saturn's Kiviuq and Ijiraq, Uranus's

Margaret Margaret is a feminine given name, which means "pearl". It is of Latin origin, via Ancient Greek and ultimately from Iranian languages, Old Iranian. It has been an English language, English name since the 11th century, and remained popular thro ...

, and Neptune's Sao and Neso. Some sources identify the Soviet space probe

Luna 3 Luna 3, or E-2A No.1 (), was a Soviet spacecraft launched in 1959 as part of the Luna programme. It was the first mission to photograph the far side of the Moon and the third Soviet space probe to be sent to the neighborhood of the Moon. The hi ...

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 the

far side of the Moon The far side of the Moon is the hemisphere of the Moon that is facing away from Earth, the opposite hemisphere is the near side. It always has the same surface oriented away from Earth because of synchronous rotation in the Moon's orbit. C ...

. It burned in the Earth's atmosphere after completing eleven revolutions. However, according to Gkolias ''et 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's

oblateness Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f ...

Extrasolar planets

The von Zeipel-Lidov–Kozai mechanism, in combination with tidal friction, is able to produce

Hot Jupiter Hot Jupiters (sometimes called hot Saturns) are a class of gas giant exoplanets that are inferred to be physically similar to Jupiter (i.e. Jupiter analogue, Jupiter analogues) but that have very short orbital periods (). The close proximity to t ...

s, which are

gas giant A gas giant is a giant planet composed mainly of hydrogen and helium. Jupiter and Saturn are the gas giants of the Solar System. The term "gas giant" was originally synonymous with "giant planet". However, in the 1990s, it became known that Uranu ...

exoplanets orbiting their stars on tight orbits. The high eccentricity of the planet

HD 80606 b HD 80606 b (also Struve 1341 Bb or HIP 45982 b) is an eccentric Jupiter, eccentric hot Jupiter 217 light-years from the Sun in the constellation of Ursa Major. HD 80606 b was discovered orbiting the star HD 80606 in April 2001 by a team led by Mi ...

in the HD 80606/80607 system is likely due to the Kozai mechanism. 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.

Black holes

The mechanism is thought to affect the growth of central

black holes A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

in dense

star cluster A star cluster is a group of stars held together by self-gravitation. Two main types of star clusters can be distinguished: globular clusters, tight groups of ten thousand to millions of old stars which are gravitationally bound; and open cluster ...

s. It also drives the evolution of certain classes of

binary black hole A binary black hole (BBH), or black hole binary, is an astronomical object consisting of two black holes in close orbit around each other. Like black holes themselves, binary black holes are often divided into binary stellar black holes, formed e ...

s and may play a role in enabling

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''. 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. 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. His paper was first published in a Russian-language journal in 1961. The Japanese astronomer Yoshihide Kozai was among the 1961 conference participants. Kozai published the same result in a widely read English-language journal in 1962, using the result to analyze orbits of

s perturbed by

. 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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Planetary and Space Science ''Planetary and Space Science'' (P&SS), published 15 times per year, is a peer-reviewed scientific journal established in 1959. It publishes original research articles along with short communications (letters). The main topic is Solar System proc ...

, volume=9 , issue=10 , pages=719–759 , doi=10.1016/0032-0633(62)90129-0 , bibcode = 1962P&SS....9..719L , lang=en (translation of Lidov's 1961 paper)
{{cite conference , last=Lidov , first= Mikhail L. , date=20–25 November 1961 , title=On approximate analysis of the evolution of orbits of artificial satellites , conference=Problems of Motion of Artificial Celestial Bodies , place=Moscow, USSR , book-title=Proceedings of the Conference on General and Practical Topics of Theoretical Astronomy , publication-date=1963 , publisher=Academy of Sciences of the USSR {{cite journal , last1=Lithwick , first1=Yoram , last2=Naoz , first2=Smadar , year=2011 , title=The eccentric Kozai mechanism for a test particle , journal=The Astrophysical Journal , volume=742 , issue=2 , page=94 , publisher=IOP Publishing , issn=0004-637X , doi=10.1088/0004-637x/742/2/94 , arxiv=1106.3329 , bibcode=2011ApJ...742...94L , s2cid=118625109 {{cite journal , last1 = de la Fuente Marcos , first1 = Carlos , last2 = de la Fuente Marcos , first2 = Raul , year = 2014 , title = Extreme trans-Neptunian objects and the Kozai mechanism: Signalling the presence of trans-Plutonian planets , journal = Monthly Notices of the Royal Astronomical Society: Letters , volume = 443 , issue = 1 , pages = L59–L63 , arxiv = 1406.0715 , doi = 10.1093/mnrasl/slu084 , doi-access = free , bibcode = 2014MNRAS.443L..59D {{cite book , last = Merritt , first = David , author-link = David Merritt , title = Dynamics and Evolution of Galactic Nuclei , year = 2013 , series = Princeton Series in Astrophysics , publisher = Princeton University Press , place = Princeton, NJ , isbn = 978-0-691-12101-7 , oclc = 863632625 , url = https://openlibrary.org/works/OL16802359W/Dynamics_and_Evolution_of_Galactic_Nuclei {{cite journal , last1=Musielak , first1=Zdzislaw E. , last2=Quarles , first2=Billy , year=2014 , title=The three-body problem , journal=Reports on Progress in Physics , volume=77 , issue=6 , page=065901 , publisher=IOP Publishing , issn=0034-4885 , pmid=24913140 , arxiv=1508.02312 , doi=10.1088/0034-4885/77/6/065901 , bibcode=2014RPPh...77f5901M , s2cid=38140668 {{cite book , chapter=The emergence of astrophysics in Asia , year=2017 , editor1-last=Nakamura , editor1-first=Tsuko , editor2-last=Orchiston , editor2-first=Wayne , title=Historical & Cultural Astronomy , publisher=Springer International Publishing , location=Cham , isbn=978-3-319-62080-0 , issn=2509-310X , doi=10.1007/978-3-319-62082-4 , ref={{sfnref , Springer International Publishing {{full citation, date=December 2020, reason=Missing author of chapter {{cite journal , last1=Naoz , first1=Smadar , last2=Farr , first2=Will M. , last3=Lithwick , first3=Yoram , last4=Rasio , first4=Frederic A. , last5=Teyssandier , first5=Jean , year=2011 , title=Hot Jupiters from secular planet–planet interactions , journal=Nature , volume=473 , issue=7346 , pages=187–189 , publisher=Springer Nature , issn=0028-0836 , doi=10.1038/nature10076 , pmid=21562558 , arxiv=1011.2501 , bibcode=2011Natur.473..187N , s2cid=4424942 {{cite journal , last1=Naoz , first1=Smadar , last2=Farr , first2=Will M. , last3=Lithwick , first3=Yoram , last4=Rasio , first4=Frederic A. , last5=Teyssandier , first5=Jean , year=2013 , title=Secular dynamics in hierarchical three-body systems , journal=Monthly Notices of the Royal Astronomical Society , volume=431 , issue=3 , pages=2155–2171 , publisher=Oxford University Press (OUP) , issn=1365-2966 , doi=10.1093/mnras/stt302 , doi-access=free , arxiv=1107.2414 , bibcode=2013MNRAS.431.2155N {{cite journal , last=Naoz , first=Smadar , year=2016 , title=The Eccentric Kozai-Lidov Effect and Its Applications , journal=Annual Review of Astronomy and Astrophysics , volume=54 , issue=1 , pages=441–489 , publisher=Annual Reviews , issn=0066-4146 , doi=10.1146/annurev-astro-081915-023315 , arxiv=1601.07175 , bibcode=2016ARA&A..54..441N , s2cid=119214240 {{cite book , last=Shevchenko , first=Ivan I. , year=2017 , chapter=The Lidov-Kozai effect – applications in exoplanet research and dynamical astronomy , title=Astrophysics and Space Science Library , volume=441 , publisher=Springer International Publishing , place=Cham , isbn=978-3-319-43520-6 , issn=0067-0057 , doi=10.1007/978-3-319-43522-0 {{cite book , last1=Valtonen , first1=Mauri J. , last2=Karttunen , first2=Hannu , year=2005 , title=The Three-Body Problem , publisher=Cambridge University Press , place=Cambridge, UK; New York , isbn=978-0-521-85224-1 {{cite journal , last1=Verrier , first1=Patricia Eleanor , last2=Evans , first2=Neil Wyn , year=2009 , title=High-inclination planets and asteroids in multistellar systems , journal=Monthly Notices of the Royal Astronomical Society , publisher=Oxford University Press (OUP) , volume=394 , issue=4 , pages=1721–1726 , issn=0035-8711 , doi=10.1111/j.1365-2966.2009.14446.x , doi-access=free , arxiv=0812.4528 , bibcode=2009MNRAS.394.1721V , s2cid=18302413 Orbital perturbations Orbital resonance Kozai mechanism