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{{Short description|Dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body}}
{{use dmy dates |date=December 2020}}
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''', or some combination of Kozai, Lidov, and/or von Zeipel. It also termed an effect, oscillations, cycles, or resonance. This effect causes the orbit's [[argument of pericenter]] to [[libration|oscillate about a constant value]], which in turn leads to a periodic exchange between its [[Orbital eccentricity|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 [[Retrograde and prograde motion|prograde and a retrograde motion]].

The effect has been found to be an important factor shaping the orbits of [[irregular satellite]]s of the planets, [[trans-Neptunian object]]s, [[extrasolar planets]], and [[multiple star system]]s.<ref name=Shevchenko-2017 />{{rp|page=v}} It hypothetically promotes [[black hole merger]]s.<ref>{{cite journal |last1=Mazzolari |first1=Giovanni |last2=Bonetti |first2=Matteo |last3=Sesana |first3=Alberto |last4=Colombo |first4=Riccardo M |last5=Dotti |first5=Massimo |last6=Lodato |first6=Giuseppe |last7=Izquierdo-Villalba |first7=David |title=Extreme mass ratio inspirals triggered by massive black hole binaries: from relativistic dynamics to cosmological rates |journal=Monthly Notices of the Royal Astronomical Society |date=7 September 2022 |volume=516 |issue=2 |pages=1959–1976 |doi=10.1093/mnras/stac2255|doi-access=free |arxiv=2204.05343 }}</ref>It was described in 1961 by [[Mikhail Lidov]] while analyzing the orbits of artificial and natural satellites of planets.<ref name=Lidov-1961-1962-1963 /> In 1962, [[Yoshihide Kozai]] published this same result in application to the orbits of [[asteroid]]s perturbed by [[Jupiter]].<ref name=Kozai-1962 /> 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.<ref name=Shevchenko-2017 />{{rp|page=vi}} 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.<ref name=Ito-Ohtsuka-2019 />

==Background==
===Hamiltonian mechanics===
{{main|Hamiltonian mechanics}}
In Hamiltonian mechanics, a physical system is specified by a function, called ''Hamiltonian'' and denoted <math>\mathcal{H}</math>, of [[canonical coordinates]] in [[phase space]]. The canonical coordinates consist of the [[generalized coordinates]] <math>x_k</math> in [[Configuration space (physics)|configuration space]] and their [[conjugate momentum|conjugate momenta]] <math>p_k</math>, for <math>k = 1, ... N</math>, for the {{mvar|N}} bodies in the system (<math>N = 3</math> for the von Zeipel-Kozai–Lidov effect). The number of <math>(x_k, p_k)</math> pairs required to describe a given system is the number of its [[degree of freedom|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.

===The three-body problem===
{{main|Three-body problem|Perturbation theory}}
<!-- Insert illustration of a hierarchical setup showing the notation -->
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 [[Chaos theory|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]] arithmetic. The practical consequence is that, the [[three-body problem]] cannot be solved analytically for an indefinite amount of time, except in special cases.<ref name=Valtonen-2005 />{{rp|page=221}} Instead, [[numerical methods]] are used for forecast-times limited by the available precision.<ref name=Musielak-Quarles-2014 />{{rp|pages=2,10}}

The Lidov–Kozai mechanism is a feature of ''hierarchical'' triple systems,<ref name=Li-etal-2014 />{{rp|page=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 the '''inner binary'''. The perturber and the centre of mass of the inner binary comprise the '''outer binary'''.<ref name=Naoz-etal-2013 />{{rp|at=§I}} 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 (physics)|coupling]] the two orbits,<ref name=Naoz-etal-2013 />

:<math> \mathcal{H} = \mathcal{H}_{\rm in} + \mathcal{H}_{\rm out} + \mathcal{H}_{\rm pert}.</math>

The coupling term is then expanded in the orders of parameter <math>\alpha</math>, defined as the ratio of the [[semi-major axis|semi-major axes]] of the inner and the outer binary and hence small in a hierarchical system.<ref name=Naoz-etal-2013 /> Since the perturbative series [[Convergent 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''
(<math>\propto\alpha^2</math>), ''octupole'' (<math>\propto\alpha^3</math>) and ''hexadecapole'' (<math>\propto\alpha^4</math>) order terms,<ref name=Naoz-2016 />{{rp|pages=4–5}}

:<math> \mathcal{H}_{\rm pert} = \mathcal{H}_{\rm quad} + \mathcal{H}_{\rm oct} + \mathcal{H}_{\rm hex} + O(\alpha^5).</math>

<!-- Plot with octupole turned off -->
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.<ref name=Katz-etal-2011 /><!-- summarize the hex term -->

====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.<ref name=Naoz-2016 />{{rp|page=4}}

==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]] that is perturbed by the [[Moon]], or a [[short-period comet]] that is perturbed by [[Jupiter]].

[[Image:Orbit1.svg|thumb|200px|right|The Keplerian [[orbital elements]].]]
Under these approximations, the orbit-averaged equations of motion for the secondary have a [[integral of motion|conserved 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's [[Orbital eccentricity|eccentricity]] {{mvar|e}} and [[inclination]] {{mvar|i}} relative to the plane of the outer binary:

:<math> L_\mathrm{z} = \sqrt{ 1-e^2 \;}\, \cos i = \mathrm{constant}</math>

Conservation of {{mvar|L}}<sub>{{math|z}}</sub> 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]] constant reduces the distance between the objects at [[periapsis]], this mechanism can cause comets (perturbed by [[Jupiter]]) to become [[Sungrazing comet|sungrazing]].

Lidov–Kozai oscillations will be present if {{mvar|L}}<sub>{{math|z}}</sub> is lower than a certain value. At the critical value of {{mvar|L}}<sub>{{math|z}}</sub>, a "fixed-point" orbit appears, with constant inclination given by

:<math>i_\mathrm{crit} = \arccos \left( \sqrt{\frac{3}{5}\,}\, \right) \approx 39.2^\mathsf{o}</math>

For values of {{mvar|L}}<sub>{{math|z}}</sub> less than this critical value, there is a one-parameter family of orbital solutions having the same {{mvar|L}}<sub>{{math|z}}</sub> but different amounts of variation in {{mvar|e}} or {{mvar|i}}. Remarkably, the degree of possible variation in {{mvar|i}} is independent of the masses involved, which only set the timescale of the oscillations.<ref name=Merritt-2013 />

====Timescale====
The basic timescale associated with Kozai oscillations is<ref name=Merritt-2013 />{{rp|page=575}}
:<math>
T_\mathrm{ Kozai} = 2 \pi\,\frac{\,\sqrt{G\,M\;}\,}{G\,m_2}\,\frac{\,a_2^3\,}{a^{3/2}}\left( 1 - e_2^2 \right)^{3/2} = \frac{\,P_2^2\,}{P}\,\left( 1 - e_2^2\right )^{3/2}
</math>
where {{mvar|a}} indicates the semimajor axis, {{mvar|P}} is orbital period, {{mvar|e}} is eccentricity and {{mvar|m}} is mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner orbit; {{mvar|M}} is the mass of the primary. For example, with [[Moon]]'s period of 27.3 days, eccentricity 0.055 and the [[Global Positioning System]] satellites period of half a (sidereal) day, the Kozai timescale is a little over 4 years; for [[geostationary orbit]]s it is twice shorter.

The period of oscillation of all three variables ({{mvar|e}}, {{mvar|i}}, {{mvar|ω}} – the last being the [[argument of periapsis]]) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the [[separatrix (dynamical systems)|separatrix]] orbit that separates librating orbits from oscillating orbits.

==Astrophysical implications==

===Solar System===
The von Zeipel-Lidov–Kozai mechanism causes the [[argument of pericenter]] ({{mvar|ω}}) to [[Libration|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]] is dynamically protected from close encounters with [[Neptune]].

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]]. Recently, the Hill-stability radius has been found as a function of satellite inclination, also explains the non-uniform distribution of irregular satellite inclinations.<ref name=Grishin-etal-2017 />

The mechanism has been invoked in searches for [[Planet Nine]], a hypothetical planet orbiting the Sun far beyond the orbit of Neptune.<ref name=Marcos-Marcos-2014 />

A number of moons have been found to be in the Lidov–Kozai resonance with their planet, including Jupiter's [[Carpo (moon)|Carpo]] and [[Euporie (moon)|Euporie]],<ref name="Brozović-Jacobson-2017"/> Saturn's [[Kiviuq (moon)|Kiviuq]] and [[Ijiraq (moon)|Ijiraq]],<ref name=Shevchenko-2017 />{{rp|page=100}} Uranus's [[Margaret (moon)|Margaret]],<ref name="Brozović-Jacobson-2009"/> and Neptune's [[Sao (moon)|Sao]] and [[Neso (moon)|Neso]].<ref name="Brozović-etal-2011"/>

Some sources identify the Soviet space probe [[Luna 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 the [[far side of the Moon]]. It burned in the Earth's atmosphere after completing eleven revolutions.<ref name=Shevchenko-2017 />{{rp|pages=9–10}} 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]].<ref name=Gkolias-etal-2016 />

===Extrasolar planets===
The von Zeipel-Lidov–Kozai mechanism, in combination with [[tidal friction]], is able to produce [[Hot Jupiter]]s, which are [[gas giant]] exoplanets orbiting their stars on tight orbits.<ref name=Fabrycky-Tremaine-2007 /><ref name=Verrier-Evans-2009 /><ref name=Lithwick-Naoz-2011 /><ref name=Naoz-etal-2011 /> The high eccentricity of the planet [[HD 80606 b]] in the [[HD 80606 and HD 80607|HD 80606/80607]] system is likely due to the Kozai mechanism.<ref>{{Cite journal |last=Pont |first=Frédéric J. |display-authors=etal<!-- 26 other authors --> |title=Spin-orbit misalignment in the HD 80606 planetary system |journal=Astronomy & Astrophysics |volume=502 |issue=2 |pages=695–703 |url=http://www.aanda.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/aa/abs/2009/29/aa12463-09/aa12463-09.html |accessdate=7 February 2013 |doi=10.1051/0004-6361/200912463 |year=2009 |arxiv=0906.5605 |bibcode=2009A&A...502..695P |s2cid=55219971 }}</ref>

===Black holes===
The mechanism is thought to affect the growth of central [[black holes]] in dense [[star cluster]]s. It also drives the evolution of certain classes of [[binary black hole]]s<ref name=Naoz-etal-2013 /> and may play a role in enabling [[black hole merger]]s.<ref name=Blaes-etal-2002 />

==History and development==
<!-- Insert image -->
The effect was first described in 1909 by the Swedish astronomer [[Edvard Hugo von Zeipel|Hugo von Zeipel]] in his work on the motion of periodic comets in ''Astronomische Nachrichten''.<ref>{{Cite journal |last=von Zeipel |first=Hugo |date=1910-03-01 |title=Sur l'application des séries de M. Lindstedt à l'étude du mouvement des comètes périodiques |url=https://zenodo.org/record/2517416 |journal=Astronomische Nachrichten |volume=183 |issue=22 |pages=345–418 |doi=10.1002/asna.19091832202 |bibcode=1910AN....183..345V |issn=0004-6337 }}</ref><ref name=Ito-Ohtsuka-2019 />  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.<ref name=Lidov-1961-1962-1963 /><ref name=Nakamura-Orchiston-2017 />{{rp|page=88}}

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&nbsp;November 1961.<ref name=Grebenikov-1962 /> His paper was first published in a Russian-language journal in 1961.<ref name=Lidov-1961-1962-1963 /> The Japanese astronomer [[Yoshihide Kozai]] was among the 1961 conference participants.<ref name=Grebenikov-1962 /> Kozai published the same result in a widely read English-language journal in 1962, using the result to analyze orbits of [[asteroid]]s perturbed by [[Jupiter]].<ref name=Kozai-1962 /> 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==
{{reflist|1=25em|refs=
<ref name="Blaes-etal-2002">{{cite journal | last1=Blaes | first1=Omer | last2=Lee | first2=Man Hoi | last3=Socrates | first3=Aristotle | year=2002 | title=The Kozai Mechanism and the Evolution of Binary Supermassive Black Holes | journal=The Astrophysical Journal | volume=578 | issue=2 | pages=775–786 | issn=0004-637X | doi=10.1086/342655 | arxiv=astro-ph/0203370 | bibcode=2002ApJ...578..775B | s2cid=14120610}}</ref>

<ref name="Brozović-Jacobson-2017">{{cite journal |last1=Brozović |first1=Marina |last2=Jacobson |first2=Robert A. |year=2017 |title=The Orbits of Jupiter's irregular satellites |journal=The Astronomical Journal |volume=153 |issue=4 |page=147 |doi=10.3847/1538-3881/aa5e4d |bibcode = 2017AJ....153..147B |doi-access=free}}</ref>

<ref name="Brozović-Jacobson-2009">{{cite journal |last1=Brozović |first1=Marina |last2=Jacobson |first2=Robert A. |year=2009 |title=The orbits of the outer Uranian satellites |journal=The Astronomical Journal |volume=137 |issue=4 | pages=3834–3842 |doi=10.1088/0004-6256/137/4/3834 |doi-access=free |bibcode = 2009AJ....137.3834B}}</ref>

<ref name="Brozović-etal-2011">{{cite journal |last1=Brozović |first1=Marina |last2=Jacobson |first2=Robert A. |last3=Sheppard |first3=Scott S. | year=2011 |title=The orbits of Neptune's outer satellites |journal=The Astronomical Journal |volume=141 |issue=4 |page=135 |doi=10.1088/0004-6256/141/4/135 |doi-access=free |bibcode=2011AJ....141..135B}}</ref>

<ref name="Fabrycky-Tremaine-2007">{{cite journal | last1=Fabrycky | first1=Daniel | last2=Tremaine | first2=Scott | year=2007 | title=Shrinking Binary and Planetary Orbits by Kozai Cycles with Tidal Friction | journal=The Astrophysical Journal | volume=669 | issue=2 | pages=1298–1315 | issn=0004-637X | doi=10.1086/521702 | arxiv=0705.4285 | bibcode=2007ApJ...669.1298F | s2cid=12159532}}</ref>

<ref name="Grebenikov-1962">{{cite journal | last=Grebenikov | first= Evgenii Aleksandrovich | year=1962 | title=Conference on General and Applied Problems of Theoretical Astronomy | journal=Soviet Astronomy | volume=6 | page=440 | issn=0038-5301 | bibcode=1962SvA.....6..440G}}</ref>

<ref name="Gkolias-etal-2016">{{cite journal | last1=Gkolias | first1=Ioannis | last2=Daquin | first2=Jérôme | last3=Gachet | first3=Fabien | last4=Rosengren | first4=Aaron J. | year=2016 | title=From Order to Chaos in Earth Satellite Orbits | journal=The Astronomical Journal | publisher=American Astronomical Society | volume=152 | issue=5 | page=119 | issn=1538-3881 | doi=10.3847/0004-6256/152/5/119 | arxiv=1606.04180 | bibcode=2016AJ....152..119G | s2cid=55672308 | doi-access=free}}</ref>

<ref name="Grishin-etal-2017">{{cite journal | last1=Grishin | first1=Evgeni | last2=Perets | first2=Hagai B. | last3=Zenati | first3=Yossef | last4=Michaely | first4=Erez | year=2017 | title=Generalized Hill-Stability Criteria for Hierarchical Three-Body Systems at Arbitrary Inclinations | journal=Monthly Notices of the Royal Astronomical Society | publisher=Oxford University Press (OUP) | volume=466 | issue=1 | pages=276–285 | issn=1365-2966 | doi=10.1093/mnras/stw3096 | doi-access=free | arxiv=1609.05912 | bibcode=2017MNRAS.466..276G}}</ref>

<ref name="Ito-Ohtsuka-2019">{{cite journal |last1=Ito |first1=Takashi |last2=Ohtsuka |first2=Katsuhito |year=2019 |title=The Lidov-Kozai Oscillation and Hugo von Zeipel |journal=Monographs on Environment, Earth and Planets |volume=7 | issue=1 | page=1-113 |publisher=Terrapub |doi=10.6084/m9.figshare.19620609 | arxiv=1911.03984 | doi-access=free |bibcode=2019MEEP....7....1I}}</ref>

<ref name="Katz-etal-2011">{{cite journal | last1=Katz | first1=Boaz | last2=Dong | first2=Subo | last3=Malhotra | first3=Renu | year=2011 | title=Long-Term Cycling of Kozai-Lidov Cycles: Extreme Eccentricities and Inclinations Excited by a Distant Eccentric Perturber | journal=Physical Review Letters | volume=107 | issue=18 | page=181101 | publisher=American Physical Society | issn=0031-9007 | doi=10.1103/PhysRevLett.107.181101 | pmid=22107620 | arxiv=1106.3340 | bibcode=2011PhRvL.107r1101K | s2cid=18317896}}</ref>

<ref name="Kozai-1962">{{cite journal | last = Kozai | first = Yoshihide | year = 1962 | title = Secular perturbations of asteroids with high inclination and eccentricity | journal = [[The Astronomical Journal]] | volume = 67 | page=591 | doi = 10.1086/108790 | bibcode = 1962AJ.....67..591K}}</ref>

<ref name="Li-etal-2014">{{cite journal | last1=Li | first1=Gongjie | last2=Naoz | first2=Smadar | last3=Holman | first3=Matt | last4=Loeb | first4=Abraham | year=2014 | title=Chaos in the Test Particle Eccentric Kozai-Lidov Mechanism | journal=The Astrophysical Journal | volume=791 | issue=2 | page=86 | publisher=IOP Publishing | issn=1538-4357 | doi=10.1088/0004-637x/791/2/86 | arxiv=1405.0494 | bibcode=2014ApJ...791...86L | s2cid=118866046}}</ref>

<ref name="Lidov-1961-1962-1963">{{cite journal | last = Lidov | first = Mikhail L. | year = 1961 | title = Эволюция орбит искусственных спутников под воздействием гравитационных возмущений внешних тел | trans-title = The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies | journal = Iskusstvennye Sputniki Zemli | volume = 8 | pages = 5–45 | language = ru}} <br/> {{cite journal |last=Lidov |first=Mikhail L. |year=1962 |title=The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies |journal=[[Planetary and Space Science]] |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) <br/> {{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}}</ref>

<ref name="Lithwick-Naoz-2011">{{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}}</ref>

<ref name="Marcos-Marcos-2014">{{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}}</ref>

<ref name="Merritt-2013">{{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}}</ref>

<ref name="Musielak-Quarles-2014">{{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}}</ref>

<ref name="Nakamura-Orchiston-2017">{{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}}</ref>

<ref name="Naoz-etal-2011">{{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}}</ref>

<ref name="Naoz-etal-2013">{{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}}</ref>

<ref name="Naoz-2016">{{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}}</ref>

<ref name="Shevchenko-2017">{{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}}</ref>

<ref name="Valtonen-2005">{{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}}</ref>

<ref name="Verrier-Evans-2009">{{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}}</ref>
}}

[[Category:Orbital perturbations]]
[[Category:Orbital resonance]]
[[Category:Kozai mechanism]]