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, ...

, the Kozai mechanism is a dynamical phenomenon affecting the orbit of a

binary system A binary system is a system of two astronomical bodies which are close enough that their gravitational attraction causes them to orbit each other around a barycenter ''(also see animated examples)''. More restrictive definitions require that th ...

perturbed by a distant third body under certain conditions. It is also known as the von Zeipel-Kozai-Lidov, Lidov–Kozai mechanism, Kozai–Lidov mechanism, or some combination of Kozai, Lidov–Kozai, Kozai–Lidov or von Zeipel-Kozai-Lidov effect, oscillations, cycles, 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 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 Ea ...

. 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 a distant, inclined, and often eccentric and retrograde orbit. They have been captured by their parent planet, unlike regular s ...

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 a semi-major axis of 30.1 astronomical units (au). Typically ...

extrasolar planets An exoplanet or extrasolar planet is a planet outside the Solar System. The first possible evidence of an exoplanet was noted in 1917 but was not recognized as such. The first confirmation of detection occurred in 1992. A different planet, in ...

, and

multiple star system A star system or stellar system is a small number of stars that orbit each other, bound by gravitational attraction. A large group of stars bound by gravitation is generally called a ''star cluster'' or ''galaxy'', although, broadly speaking ...

s. It hypothetically promotes 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 Yoshihide Kozai (1 April 1928 – 5 February 2018) was a Japanese astronomer specialising in celestial mechanics. He is best known for discovering, simultaneously with Michael Lidov, the Kozai mechanism, for which he received the Imperial Pri ...

published this same result in application to the orbits of

asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...

s perturbed by

Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousand ...

. The citations of the initial papers by Kozai and Lidov have risen sharply in the 21st century. , the mechanism is among the most studied astrophysical phenomena.

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 cl ...

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...

. 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. 39 ...

x_k

in configuration space and their

conjugate momenta 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 cl ...

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 Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

. 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. It need not preserve the form of the Hamiltonian itself. Canon ...

. 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.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...

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 enormously sensitive to any slight changes in the initial conditions, including even small uncertainties in determining the initial conditions, and rounding-errors in computer

floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...

arithmetic. The practical consequence is that, the

three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...

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 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 A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end mov ...

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 orbit around Earth with a period of 128 minutes or less (making at least 11.25 orbits per day) and an eccentricity less than 0.25. Most of the artificial objects in outer space are in LEO, with an altitude never m ...

that is perturbed by the

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

, or a

short-period comet A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena are ...

that is perturbed by

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

conserved quantity In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conserved quantities, and conserved quantities are ...

: the component of the secondary's orbital angular momentum parallel to the angular momentum of the primary / perturber angular momentum. This conserved quantity can be expressed in terms of the secondary's eccentricity 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 lo ...

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. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...

, 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\,\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), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite ...

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 geosynchronous orbit in altitu ...

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'', is one of the orbital elements of an orbiting body. Parametrically, ''ω'' is the angle from the body's ascending node to its periap ...

) 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 S ...

is dynamically protected from close encounters with

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

. 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 of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sph ...

. 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 hypothetical 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 objects (ETNOs), bodies beyond Neptune that orbit ...

, 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, Euporie or Euporia is the goddess of abundance. She is one of the third generation of Horae. Sources * Gaius Julius Hyginus, ''Fabulae Gaius Julius Hyginus (; 64 BC – AD 17) was a Latin author, a pupil of the scholar A ...

, Saturn's

Kiviuq Kiviuq (also spelled "Qiviuq", "Kiviok" and other variants) is a legendary hero of the epic stories of the Inuit of the Arctic regions of northern Canada, Alaska and Greenland. Kiviuq is an eternal Inuit wanderer. Spirits, giants, cannibals, bea ...

and Ijiraq, Uranus's

Margaret Margaret is a female first name, derived via French () and Latin () from grc, μαργαρίτης () meaning "pearl". The Greek is borrowed from Persian. Margaret has been an English name since the 11th century, and remained popular through ...

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

Luna 3 Luna 3, or E-2A No.1 ( rus, Луна 3}) 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 th ...

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 lunar hemisphere that always faces away from Earth, opposite to the near side, because of synchronous rotation in the Moon's orbit. Compared to the near side, the far side's terrain is rugged, with a multitu ...

. 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 ...

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 but that have very short orbital periods (). The close proximity to their stars and high surface-atmosphere t ...

s, which are

gas giant A gas giant is a giant planet composed mainly of hydrogen and helium. Gas giants are also called failed stars because they contain the same basic elements as a star. Jupiter and Saturn are the gas giants of the Solar System. The term "gas giant" ...

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

HD 80606 b HD 80606 b (also Struve 1341 B b or HIP 45982 b) is an eccentric and hot Jupiter 190 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 Michel Mayor a ...

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

Black holes

The mechanism is thought to affect the growth of central

black holes A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...

in dense

star cluster Star clusters are large groups of stars. Two main types of star clusters can be distinguished: globular clusters are tight groups of ten thousand to millions of old stars which are gravitationally bound, while open clusters are more loosely cl ...

s. It also drives the evolution of certain classes of

binary black hole A binary black hole (BBH) is a system consisting of two black holes in close orbit around each other. Like black holes themselves, binary black holes are often divided into stellar binary black holes, formed either as remnants of high-mass binar ...

s and may play a role in enabling black hole mergers.

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

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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{{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 , 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=Z.E. , last2=Quarles , first2=B. , 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 , 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 journal , last1=Tremaine , first1=Scott , last2=Yavetz , first2=Tomer D. , year=2014 , title=Why do Earth satellites stay up? , journal=American Journal of Physics , publisher=American Association of Physics Teachers (AAPT) , volume=82 , issue=8 , pages=769–777 , issn=0002-9505 , doi=10.1119/1.4874853 , arxiv=1309.5244 , bibcode=2014AmJPh..82..769T , s2cid=119298013 {{cite book , last=Valtonen , first=M. J. , 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=P.E. , last2=Evans , first2=N.W. , 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 , arxiv=0812.4528 , bibcode=2009MNRAS.394.1721V , s2cid=18302413 Orbital perturbations Orbital resonance