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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 LidovāKozai oscillations.
1838:
Lidov, Mikhail L. (1961). "ŠŠ²Š¾Š»ŃŃŠøŃ Š¾ŃŠ±ŠøŃ ŠøŃŠŗŃŃŃŃŠ²ŠµŠ½Š½ŃŃ
ŃŠæŃŃŠ½ŠøŠŗŠ¾Š² ŠæŠ¾Š“ Š²Š¾Š·Š“ŠµŠ¹ŃŃŠ²ŠøŠµŠ¼ Š³ŃŠ°Š²ŠøŃŠ°ŃŠøŠ¾Š½Š½ŃŃ
Š²Š¾Š·Š¼ŃŃŠµŠ½ŠøŠ¹ Š²Š½ŠµŃŠ½ŠøŃ
ŃŠµŠ»" [The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies].
117:. 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. It was pointed out in 2019 by Takashi Ito and Katsuhito Ohtsuka that the Swedish astronomer
536:
1147:
835:
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
1007:
677:
1437:{\displaystyle 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}}
1605:
as the first example of an artificial satellite undergoing LidovāKozai oscillations. Launched in 1959 into a highly inclined, eccentric, geocentric orbit, it was the first mission to photograph the
394:
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
900:: 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
364:
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
1549:
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.
2451:
1021:
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
439:
1064:
2947:
1689:
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.
669:
635:
601:
429:
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
166:
2276:
Katz, Boaz; Dong, Subo; Malhotra, Renu (2011). "Long-Term
Cycling of Kozai-Lidov Cycles: Extreme Eccentricities and Inclinations Excited by a Distant Eccentric Perturber".
1559:. Recently, the Hill-stability radius has been found as a function of satellite inclination, also explains the non-uniform distribution of irregular satellite inclinations.
355:
840:, 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.
279:
559:
1700:
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
1613:. (2016) a different mechanism must have driven the decay of the probe's orbit since the LidovāKozai oscillations would have been thwarted by effects of the Earth's
238:
204:
1708:. Since Lidov was the first to publish, many authors use the term LidovāKozai mechanism. Others, however, name it as the KozaiāLidov or just the Kozai mechanism.
309:
1555:
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
54:, or some combination of Kozai, LidovāKozai, KozaiāLidov or von Zeipel-Kozai-Lidov effect, oscillations, cycles, or resonance. This effect causes the orbit's
70:. The process occurs on timescales much longer than the orbital periods. It can drive an initially near-circular orbit to arbitrarily high eccentricity, and
1851:
Lidov, Mikhail L. (1962). "The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies".
920:
417:
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
2877:
Naoz, Smadar; Farr, Will M.; Lithwick, Yoram; Rasio, Frederic A.; Teyssandier, Jean (2011). "Hot
Jupiters from secular planetāplanet interactions".
569:
rapidly, the qualitative behaviour of a hierarchical three-body system is determined by the initial terms in the expansion, referred to as the
3135:
2362:
2000:
1748:
812:{\displaystyle {\mathcal {H}}_{\rm {pert}}={\mathcal {H}}_{\rm {quad}}+{\mathcal {H}}_{\rm {oct}}+{\mathcal {H}}_{\rm {hex}}+O(\alpha ^{5}).}
1696:
held in Moscow on 20ā25 November 1961. His paper was first published in a
Russian-language journal in 1961. The Japanese astronomer
1463:
is mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner orbit;
376:, which relate time derivatives of the coordinates to partial derivatives of the Hamiltonian with respect to the conjugate momenta.
3010:
Blaes, Omer; Lee, Man Hoi; Socrates, Aristotle (2002). "The Kozai
Mechanism and the Evolution of Binary Supermassive Black Holes".
3202:
2085:
Li, Gongjie; Naoz, Smadar; Holman, Matt; Loeb, Abraham (2014). "Chaos in the Test
Particle Eccentric Kozai-Lidov Mechanism".
1499:
75:
207:
1882:
Lidov, Mikhail L. (20ā25 November 1961). "On approximate analysis of the evolution of orbits of artificial satellites".
2688:
Fabrycky, Daniel; Tremaine, Scott (2007). "Shrinking Binary and
Planetary Orbits by Kozai Cycles with Tidal Friction".
2352:
1853:
853:
The simplest treatment of the von Zeipel-LidovāKozai mechanism assumes that one of the inner binary's components, the
1886:. Problems of Motion of Artificial Celestial Bodies. Moscow, USSR: Academy of Sciences of the USSR (published 1963).
3207:
1731:
Shevchenko, Ivan I. (2017). "The Lidov-Kozai effect ā applications in exoplanet research and dynamical astronomy".
865:
and the distant perturber. These assumptions are valid, for instance, in the case of an artificial satellite in a
1472:
531:{\displaystyle {\mathcal {H}}={\mathcal {H}}_{\rm {in}}+{\mathcal {H}}_{\rm {out}}+{\mathcal {H}}_{\rm {pert}}.}
3212:
1902:
1142:{\displaystyle i_{\mathrm {crit} }=\arccos \left({\sqrt {{\frac {3}{5}}\,}}\,\right)\approx 39.2^{\mathsf {o}}}
365:
2447:"Extreme trans-Neptunian objects and the Kozai mechanism: Signalling the presence of trans-Plutonian planets"
1570:
A number of moons have been found to be in the LidovāKozai resonance with their planet, including Jupiter's
1498:) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the
398:, including even small uncertainties in determining the initial conditions, and rounding-errors in computer
177:
1678:
1516:
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of the inner and the outer binary and hence small in a hierarchical system. Since the perturbative series
118:
86:
55:
1609:. It burned in the Earth's atmosphere after completing eleven revolutions. However, according to Gkolias
644:
610:
576:
1642:
1495:
169:
135:
1900:
Kozai, Yoshihide (1962). "Secular perturbations of asteroids with high inclination and eccentricity".
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1963:
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94:
63:
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less than this critical value, there is a one-parameter family of orbital solutions having the same
147:
2388:"Generalized Hill-Stability Criteria for Hierarchical Three-Body Systems at Arbitrary Inclinations"
1542:
The LidovāKozai mechanism places restrictions on the orbits possible within a system. For example:
1531:
occurs when the body is farthest from the equatorial plane. This effect is part of the reason that
1476:
426:
389:
369:
314:
82:
31:
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cannot be solved analytically for an indefinite amount of time, except in special cases. Instead,
3053:
3019:
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1953:
1815:
1781:
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430:
403:
385:
211:
90:
3162:
Grebnikov, E. A. (1962). "Conference on General and Applied Problems of Theoretical Astronomy".
3179:
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1996:
1807:
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satellites period of half a (sidereal) day, the Kozai timescale is a little over 4 years; for
566:
407:
358:
243:
98:
3073:"Sur l'application des sĆ©ries de M. Lindstedt Ć l'Ć©tude du mouvement des comĆØtes pĆ©riodiques"
544:
27:
Dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body
3123:
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3037:
2977:
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2184:
2153:
Naoz, Smadar; Farr, Will M.; Lithwick, Yoram; Rasio, Frederic A.; Teyssandier, Jean (2013).
2112:
2043:
1971:
1919:
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1799:
1736:
1034:
896:
Under these approximations, the orbit-averaged equations of motion for the secondary have a
889:
861:ā an idealized point-like object with negligible mass compared to the other two bodies, the
2816:
Lithwick, Yoram; Naoz, Smadar (2011). "The eccentric Kozai mechanism for a test particle".
216:
182:
1697:
1587:
866:
562:
106:
3118:
Nakamura, Tsuko; Orchiston, Wayne, eds. (2017). "The emergence of astrophysics in Asia".
1182:
is independent of the masses involved, which only set the timescale of the oscillations.
288:
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2839:
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1967:
1915:
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1626:
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1022:
399:
2657:
2622:
2606:
2581:
1884:
Proceedings of the Conference on General and Practical Topics of Theoretical Astronomy
3196:
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2847:
2785:
2750:
2348:
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2132:
2116:
1874:
1819:
1686:
1637:
exoplanets orbiting their stars on tight orbits. The high eccentricity of the planet
1583:
1579:
858:
105:
while analyzing the orbits of artificial and natural satellites of planets. In 1962,
102:
39:
17:
3057:
2989:
2802:
2735:
2674:
2331:
2071:
42:
perturbed by a distant third body under certain conditions. It is also known as the
2932:
2307:
1658:
1571:
395:
2981:
1976:
1941:
1654:
1630:
1595:
1564:
1556:
1002:{\displaystyle L_{\mathrm {z} }={\sqrt {1-e^{2}\;}}\,\cos i=\mathrm {constant} }
908:
173:
67:
2524:
2499:
2357:. Princeton Series in Astrophysics. Princeton, NJ: Princeton University Press.
140:
In Hamiltonian mechanics, a physical system is specified by a function, called
3127:
2621:
Gkolias, Ioannis; Daquin, JƩrƓme; Gachet, Fabien; Rosengren, Aaron J. (2016).
2215:
Naoz, Smadar (2016). "The Eccentric Kozai-Lidov Effect and Its Applications".
1772:
Tremaine, Scott; Yavetz, Tomer D. (2014). "Why do Earth satellites stay up?".
1740:
1638:
1614:
1591:
121:
had also studied this mechanism in 1909, and his name is sometimes now added.
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59:
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2189:
2154:
2063:
2386:
Grishin, Evgeni; Perets, Hagai B.; Zenati, Yossef; Michaely, Erez (2017).
1567:, a hypothetical planet orbiting the Sun far beyond the orbit of Neptune.
81:
The effect has been found to be an important factor shaping the orbits of
3024:
1701:
1528:
110:
2908:
421:. The perturber and the centre of mass of the inner binary comprise the
1705:
1536:
1030:
878:
114:
1803:
1692:
Lidov first presented his work on artificial satellite orbits at the
1602:
3072:
1058:, a "fixed-point" orbit appears, with constant inclination given by
396:
enormously sensitive to any slight changes in the initial conditions
3041:
2719:
2639:
2404:
2229:
2030:
1958:
1923:
1694:
Conference on General and Applied Problems of Theoretical Astronomy
2964:
2891:
2830:
2767:
2702:
2580:
BrozoviÄ, Marina; Jacobson, Robert A.; Sheppard, Scott S. (2011).
2465:
2290:
2171:
2099:
1786:
1532:
884:
883:
1677:
The effect was first described in 1909 by the Swedish astronomer
2751:"High-inclination planets and asteroids in multistellar systems"
1468:
870:
410:
are used for forecast-times limited by the available precision.
2445:
de la Fuente Marcos, Carlos; de la Fuente Marcos, Raul (2014).
1780:(8). American Association of Physics Teachers (AAPT): 769ā777.
1502:
orbit that separates librating orbits from oscillating orbits.
357:
pairs required to describe a given system is the number of its
2016:
Musielak, Z.E.; Quarles, B. (2014). "The three-body problem".
541:
The coupling term is then expanded in the orders of parameter
765:
740:
712:
684:
503:
478:
456:
445:
153:
109:
published this same result in application to the orbits of
2948:"Spin-orbit misalignment in the HD 80606 planetary system"
2452:
Monthly Notices of the Royal Astronomical Society: Letters
1735:. Vol. 441. Cham: Springer International Publishing.
1625:
The von Zeipel-LidovāKozai mechanism, in combination with
1190:
The basic timescale associated with Kozai oscillations is
1653:
The mechanism is thought to affect the growth of central
425:. Such systems are often studied by using the methods of
372:
for the system are obtained from the Hamiltonian through
62:, which in turn leads to a periodic exchange between its
1049:
is lower than a certain value. At the critical value of
1995:. Cambridge, UK; New York: Cambridge University Press.
1833:
1831:
1829:
311:
for the von Zeipel-KozaiāLidov effect). The number of
2155:"Secular dynamics in hierarchical three-body systems"
1661:. It also drives the evolution of certain classes of
1199:
1067:
1025:
constant reduces the distance between the objects at
923:
680:
647:
613:
579:
547:
442:
317:
291:
246:
219:
185:
150:
1535:
is dynamically protected from close encounters with
1527:about either 90Ā° or 270Ā°, which is to say that its
1471:'s period of 27.3 days, eccentricity 0.055 and the
402:arithmetic. The practical consequence is that, the
38:is a dynamical phenomenon affecting the orbit of a
1935:
1933:
1482:The period of oscillation of all three variables (
1436:
1178:. Remarkably, the degree of possible variation in
1141:
1001:
811:
663:
629:
595:
553:
530:
349:
303:
273:
232:
198:
160:
3157:
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2755:Monthly Notices of the Royal Astronomical Society
2392:Monthly Notices of the Royal Astronomical Society
2159:Monthly Notices of the Royal Astronomical Society
1942:"The Lidov-Kozai Oscillation and Hugo von Zeipel"
1726:
1724:
1722:
1720:
74:an initially moderately inclined orbit between a
2148:
2146:
2144:
2142:
1681:in his work on the motion of periodic comets in
1515:The von Zeipel-LidovāKozai mechanism causes the
1029:, this mechanism can cause comets (perturbed by
2761:(4). Oxford University Press (OUP): 1721ā1726.
2623:"From Order to Chaos in Earth Satellite Orbits"
2343:
2341:
2165:(3). Oxford University Press (OUP): 2155ā2171.
1563:The mechanism has been invoked in searches for
2500:"The Orbits of Jupiter's irregular satellites"
2498:BrozoviÄ, Marina; Jacobson, Robert A. (2017).
1467:is the mass of the primary. For example, with
2398:(1). Oxford University Press (OUP): 276ā285.
1645:system is likely due to the Kozai mechanism.
1601:Some sources identify the Soviet space probe
8:
2541:"The orbits of the outer Uranian satellites"
2210:
2208:
1946:Monographs on Environment, Earth and Planets
1895:
1893:
1040:LidovāKozai oscillations will be present if
3122:. Cham: Springer International Publishing.
2217:Annual Review of Astronomy and Astrophysics
914:relative to the plane of the outer binary:
176:. The canonical coordinates consist of the
2582:"The orbits of Neptune's outer satellites"
1247:
957:
413:The LidovāKozai mechanism is a feature of
3023:
2963:
2890:
2829:
2784:
2766:
2701:
2656:
2638:
2633:(5). American Astronomical Society: 119.
2605:
2564:
2523:
2482:
2464:
2421:
2403:
2354:Dynamics and Evolution of Galactic Nuclei
2289:
2284:(18). American Physical Society: 181101.
2228:
2188:
2170:
2098:
2029:
1975:
1957:
1940:Ito, Takashi; Ohtsuka, Katsuhito (2019).
1785:
1424:
1420:
1409:
1404:
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1380:
1374:
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1103:
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290:
245:
224:
218:
190:
184:
152:
151:
149:
1716:
1685:. In 1961, the Soviet space scientist
2539:BrozoviÄ, M.; Jacobson, R. A. (2009).
1733:Astrophysics and Space Science Library
1170:but different amounts of variation in
1133:
7:
1879:(translation of Lidov's 1961 paper)
3120:Historical & Cultural Astronomy
2749:Verrier, P.E.; Evans, N.W. (2009).
2247:10.1146/annurev-astro-081915-023315
664:{\displaystyle \propto \alpha ^{4}}
630:{\displaystyle \propto \alpha ^{3}}
596:{\displaystyle \propto \alpha ^{2}}
2885:(7346). Springer Nature: 187ā189.
1218:
1215:
1212:
1209:
1206:
1083:
1080:
1077:
1074:
995:
992:
989:
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463:
25:
2786:10.1111/j.1365-2966.2009.14446.x
1665:and may play a role in enabling
76:prograde and a retrograde motion
60:oscillate about a constant value
3071:von Zeipel, H. (1 March 1910).
831:The LidovāKozai mechanism is a
2308:10.1103/PhysRevLett.107.181101
2223:(1). Annual Reviews: 441ā489.
2018:Reports on Progress in Physics
1451:indicates the semimajor axis,
803:
790:
561:, defined as the ratio of the
374:Hamilton's canonical equations
344:
318:
161:{\displaystyle {\mathcal {H}}}
101:. It was described in 1961 by
1:
2048:10.1088/0034-4885/77/6/065901
2024:(6). IOP Publishing: 065901.
350:{\displaystyle (x_{k},p_{k})}
97:. It hypothetically promotes
2952:Astronomy & Astrophysics
2566:10.1088/0004-6256/137/4/3834
1977:10.6084/m9.figshare.19620609
1875:10.1016/0032-0633(62)90129-0
1841:Iskusstvennye Sputniki Zemli
2982:10.1051/0004-6361/200912463
2658:10.3847/0004-6256/152/5/119
2607:10.1088/0004-6256/141/4/135
1854:Planetary and Space Science
1774:American Journal of Physics
3229:
2946:PONT; et al. (2009).
2848:10.1088/0004-637x/742/2/94
2117:10.1088/0004-637x/791/2/86
1506:Astrophysical implications
383:
133:
3128:10.1007/978-3-319-62082-4
3077:Astronomische Nachrichten
3012:The Astrophysical Journal
2824:(2). IOP Publishing: 94.
2818:The Astrophysical Journal
2690:The Astrophysical Journal
2093:(2). IOP Publishing: 86.
2087:The Astrophysical Journal
1741:10.1007/978-3-319-43522-0
1683:Astronomische Nachrichten
1473:Global Positioning System
869:that is perturbed by the
844:Overview of the mechanism
3097:10.1002/asna.19091832202
2627:The Astronomical Journal
2586:The Astronomical Journal
2545:The Astronomical Journal
2525:10.3847/1538-3881/aa5e4d
2504:The Astronomical Journal
1991:Valtonen, M. J. (2005).
1903:The Astronomical Journal
1552:For irregular satellites
366:canonical transformation
274:{\displaystyle k=1,...N}
2974:2009A&A...502..695P
2278:Physical Review Letters
2239:2016ARA&A..54..441N
1867:1962P&SS....9..719L
1673:History and development
1546:For a regular satellite
554:{\displaystyle \alpha }
178:generalized coordinates
87:trans-Neptunian objects
1993:The Three-Body Problem
1952:(1). Terrapub: 1-113.
1517:argument of pericenter
1438:
1143:
1003:
893:
813:
665:
631:
597:
555:
532:
380:The three-body problem
351:
305:
285:bodies in the system (
275:
234:
200:
162:
119:Edvard Hugo von Zeipel
56:argument of pericenter
44:von Zeipel-Kozai-Lidov
3203:Orbital perturbations
2484:10.1093/mnrasl/slu084
2423:10.1093/mnras/stw3096
1629:, is able to produce
1496:argument of periapsis
1494:ā the last being the
1479:it is twice shorter.
1439:
1144:
1004:
887:
877:that is perturbed by
827:Secular approximation
814:
666:
632:
598:
556:
533:
352:
306:
276:
235:
233:{\displaystyle p_{k}}
201:
199:{\displaystyle x_{k}}
170:canonical coordinates
163:
136:Hamiltonian mechanics
130:Hamiltonian mechanics
95:multiple star systems
52:KozaiāLidov mechanism
48:LidovāKozai mechanism
18:Kozai-Lidov mechanism
2190:10.1093/mnras/stt302
1607:far side of the Moon
1477:geostationary orbits
1459:is eccentricity and
1197:
1065:
921:
678:
645:
611:
577:
545:
440:
315:
289:
244:
217:
183:
148:
83:irregular satellites
3176:1962SvA.....6..440G
3089:1910AN....183..345V
3034:2002ApJ...578..775B
2909:10.1038/nature10076
2901:2011Natur.473..187N
2840:2011ApJ...742...94L
2777:2009MNRAS.394.1721V
2712:2007ApJ...669.1298F
2649:2016AJ....152..119G
2598:2011AJ....141..135B
2557:2009AJ....137.3834B
2516:2017AJ....153..147B
2475:2014MNRAS.443L..59D
2414:2017MNRAS.466..276G
2300:2011PhRvL.107r1101K
2181:2013MNRAS.431.2155N
2109:2014ApJ...791...86L
2040:2014RPPh...77f5901M
1968:2019MEEP....7....1I
1916:1962AJ.....67..591K
1796:2014AmJPh..82..769T
1455:is orbital period,
1414:
1379:
1338:
1289:
849:Test particle limit
427:perturbation theory
390:Perturbation theory
370:equations of motion
304:{\displaystyle N=3}
208:configuration space
32:celestial mechanics
1667:black hole mergers
1663:binary black holes
1621:Extrasolar planets
1434:
1400:
1365:
1324:
1275:
1139:
999:
898:conserved quantity
894:
875:short-period comet
809:
661:
627:
593:
551:
528:
404:three-body problem
386:Three-body problem
359:degrees of freedom
347:
301:
271:
230:
196:
158:
99:black hole mergers
91:extrasolar planets
3208:Orbital resonance
3137:978-3-319-62080-0
2364:978-0-691-12101-7
2002:978-0-521-85224-1
1804:10.1119/1.4874853
1750:978-3-319-43520-6
1385:
1310:
1268:
1248:
1116:
1113:
958:
408:numerical methods
212:conjugate momenta
16:(Redirected from
3220:
3188:
3187:
3164:Soviet Astronomy
3159:
3150:
3149:
3115:
3109:
3108:
3068:
3062:
3061:
3027:
3025:astro-ph/0203370
3007:
3001:
3000:
2998:
2996:
2967:
2943:
2937:
2936:
2894:
2874:
2868:
2867:
2833:
2813:
2807:
2806:
2788:
2770:
2746:
2740:
2739:
2705:
2696:(2): 1298ā1315.
2685:
2679:
2678:
2660:
2642:
2618:
2612:
2611:
2609:
2577:
2571:
2570:
2568:
2551:(4): 3834ā3842.
2536:
2530:
2529:
2527:
2495:
2489:
2488:
2486:
2468:
2442:
2436:
2435:
2425:
2407:
2383:
2377:
2376:
2345:
2336:
2335:
2293:
2273:
2267:
2266:
2232:
2212:
2203:
2202:
2192:
2174:
2150:
2137:
2136:
2102:
2082:
2076:
2075:
2033:
2013:
2007:
2006:
1988:
1982:
1981:
1979:
1961:
1937:
1928:
1927:
1897:
1888:
1887:
1878:
1848:
1835:
1824:
1823:
1789:
1769:
1763:
1762:
1728:
1590:, and Neptune's
1522:
1493:
1489:
1485:
1466:
1462:
1458:
1454:
1450:
1443:
1441:
1440:
1435:
1433:
1432:
1428:
1419:
1415:
1413:
1408:
1386:
1381:
1378:
1373:
1362:
1357:
1356:
1352:
1343:
1339:
1337:
1332:
1311:
1309:
1308:
1304:
1291:
1288:
1283:
1272:
1269:
1267:
1266:
1265:
1251:
1249:
1239:
1235:
1223:
1222:
1221:
1181:
1177:
1173:
1168:
1164:
1159:
1155:
1148:
1146:
1145:
1140:
1138:
1137:
1136:
1123:
1119:
1117:
1114:
1106:
1104:
1088:
1087:
1086:
1056:
1052:
1047:
1043:
1019:
1015:
1012:Conservation of
1008:
1006:
1005:
1000:
998:
959:
956:
955:
940:
935:
934:
933:
913:
906:
890:orbital elements
818:
816:
815:
810:
802:
801:
783:
782:
781:
769:
768:
758:
757:
756:
744:
743:
733:
732:
731:
716:
715:
705:
704:
703:
688:
687:
670:
668:
667:
662:
660:
659:
636:
634:
633:
628:
626:
625:
602:
600:
599:
594:
592:
591:
560:
558:
557:
552:
537:
535:
534:
529:
524:
523:
522:
507:
506:
496:
495:
494:
482:
481:
471:
470:
469:
460:
459:
449:
448:
433:the two orbits,
356:
354:
353:
348:
343:
342:
330:
329:
310:
308:
307:
302:
284:
280:
278:
277:
272:
239:
237:
236:
231:
229:
228:
205:
203:
202:
197:
195:
194:
167:
165:
164:
159:
157:
156:
85:of the planets,
21:
3228:
3227:
3223:
3222:
3221:
3219:
3218:
3217:
3213:Kozai mechanism
3193:
3192:
3191:
3161:
3160:
3153:
3138:
3117:
3116:
3112:
3083:(22): 345ā418.
3070:
3069:
3065:
3009:
3008:
3004:
2994:
2992:
2945:
2944:
2940:
2876:
2875:
2871:
2815:
2814:
2810:
2748:
2747:
2743:
2687:
2686:
2682:
2620:
2619:
2615:
2579:
2578:
2574:
2538:
2537:
2533:
2497:
2496:
2492:
2444:
2443:
2439:
2385:
2384:
2380:
2365:
2347:
2346:
2339:
2275:
2274:
2270:
2214:
2213:
2206:
2152:
2151:
2140:
2084:
2083:
2079:
2015:
2014:
2010:
2003:
1990:
1989:
1985:
1939:
1938:
1931:
1899:
1898:
1891:
1881:
1880:
1861:(10): 719ā759.
1850:
1849:
1837:
1836:
1827:
1771:
1770:
1766:
1751:
1730:
1729:
1718:
1714:
1698:Yoshihide Kozai
1679:Hugo von Zeipel
1675:
1651:
1623:
1520:
1513:
1508:
1491:
1487:
1483:
1464:
1460:
1456:
1452:
1448:
1393:
1389:
1388:
1363:
1317:
1313:
1312:
1292:
1273:
1257:
1252:
1236:
1200:
1195:
1194:
1188:
1179:
1175:
1171:
1169:
1166:
1162:
1160:
1157:
1153:
1127:
1102:
1098:
1068:
1063:
1062:
1057:
1054:
1050:
1048:
1045:
1041:
1020:
1017:
1013:
947:
924:
919:
918:
911:
904:
867:low Earth orbit
851:
846:
829:
793:
762:
737:
709:
681:
676:
675:
671:) order terms,
651:
643:
642:
617:
609:
608:
583:
575:
574:
563:semi-major axes
543:
542:
500:
475:
453:
438:
437:
392:
384:Main articles:
382:
334:
321:
313:
312:
287:
286:
282:
242:
241:
220:
215:
214:
186:
181:
180:
146:
145:
138:
132:
127:
107:Yoshihide Kozai
36:Kozai mechanism
28:
23:
22:
15:
12:
11:
5:
3226:
3224:
3216:
3215:
3210:
3205:
3195:
3194:
3190:
3189:
3151:
3136:
3110:
3063:
3042:10.1086/342655
3018:(2): 775ā786.
3002:
2958:(2): 695ā703.
2938:
2869:
2808:
2741:
2720:10.1086/521702
2680:
2613:
2572:
2531:
2490:
2459:(1): L59āL63.
2437:
2378:
2363:
2349:Merritt, David
2337:
2268:
2204:
2138:
2077:
2008:
2001:
1983:
1929:
1924:10.1086/108790
1889:
1843:(in Russian).
1825:
1764:
1749:
1715:
1713:
1710:
1674:
1671:
1650:
1647:
1643:HD 80606/80607
1627:tidal friction
1622:
1619:
1561:
1560:
1553:
1550:
1547:
1512:
1509:
1507:
1504:
1445:
1444:
1431:
1427:
1423:
1418:
1412:
1407:
1403:
1399:
1396:
1392:
1384:
1377:
1372:
1368:
1360:
1355:
1351:
1347:
1342:
1336:
1331:
1327:
1323:
1320:
1316:
1307:
1303:
1299:
1295:
1287:
1282:
1278:
1264:
1260:
1255:
1246:
1242:
1232:
1229:
1226:
1220:
1217:
1214:
1211:
1208:
1203:
1187:
1184:
1165:
1156:
1152:For values of
1150:
1149:
1135:
1130:
1126:
1122:
1112:
1109:
1101:
1097:
1094:
1091:
1085:
1082:
1079:
1076:
1071:
1053:
1044:
1023:semimajor axis
1016:
1010:
1009:
997:
994:
991:
988:
985:
982:
979:
976:
972:
969:
966:
963:
954:
950:
946:
943:
938:
932:
927:
888:The Keplerian
850:
847:
845:
842:
828:
825:
820:
819:
808:
805:
800:
796:
792:
789:
786:
780:
777:
774:
767:
761:
755:
752:
749:
742:
736:
730:
727:
724:
721:
714:
708:
702:
699:
696:
693:
686:
658:
654:
650:
624:
620:
616:
590:
586:
582:
550:
539:
538:
527:
521:
518:
515:
512:
505:
499:
493:
490:
487:
480:
474:
468:
465:
458:
452:
447:
400:floating point
381:
378:
346:
341:
337:
333:
328:
324:
320:
300:
297:
294:
270:
267:
264:
261:
258:
255:
252:
249:
227:
223:
193:
189:
155:
134:Main article:
131:
128:
126:
123:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3225:
3214:
3211:
3209:
3206:
3204:
3201:
3200:
3198:
3185:
3181:
3177:
3173:
3169:
3165:
3158:
3156:
3152:
3147:
3143:
3139:
3133:
3129:
3125:
3121:
3114:
3111:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3078:
3074:
3067:
3064:
3059:
3055:
3051:
3047:
3043:
3039:
3035:
3031:
3026:
3021:
3017:
3013:
3006:
3003:
2991:
2987:
2983:
2979:
2975:
2971:
2966:
2961:
2957:
2953:
2949:
2942:
2939:
2934:
2930:
2926:
2922:
2918:
2914:
2910:
2906:
2902:
2898:
2893:
2888:
2884:
2880:
2873:
2870:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2832:
2827:
2823:
2819:
2812:
2809:
2804:
2800:
2796:
2792:
2787:
2782:
2778:
2774:
2769:
2764:
2760:
2756:
2752:
2745:
2742:
2737:
2733:
2729:
2725:
2721:
2717:
2713:
2709:
2704:
2699:
2695:
2691:
2684:
2681:
2676:
2672:
2668:
2664:
2659:
2654:
2650:
2646:
2641:
2636:
2632:
2628:
2624:
2617:
2614:
2608:
2603:
2599:
2595:
2591:
2587:
2583:
2576:
2573:
2567:
2562:
2558:
2554:
2550:
2546:
2542:
2535:
2532:
2526:
2521:
2517:
2513:
2509:
2505:
2501:
2494:
2491:
2485:
2480:
2476:
2472:
2467:
2462:
2458:
2454:
2453:
2448:
2441:
2438:
2433:
2429:
2424:
2419:
2415:
2411:
2406:
2401:
2397:
2393:
2389:
2382:
2379:
2374:
2370:
2366:
2360:
2356:
2355:
2350:
2344:
2342:
2338:
2333:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2292:
2287:
2283:
2279:
2272:
2269:
2264:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2231:
2226:
2222:
2218:
2211:
2209:
2205:
2200:
2196:
2191:
2186:
2182:
2178:
2173:
2168:
2164:
2160:
2156:
2149:
2147:
2145:
2143:
2139:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2106:
2101:
2096:
2092:
2088:
2081:
2078:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2032:
2027:
2023:
2019:
2012:
2009:
2004:
1998:
1994:
1987:
1984:
1978:
1973:
1969:
1965:
1960:
1955:
1951:
1947:
1943:
1936:
1934:
1930:
1925:
1921:
1917:
1913:
1909:
1905:
1904:
1896:
1894:
1890:
1885:
1876:
1872:
1868:
1864:
1860:
1856:
1855:
1846:
1842:
1834:
1832:
1830:
1826:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1793:
1788:
1783:
1779:
1775:
1768:
1765:
1760:
1756:
1752:
1746:
1742:
1738:
1734:
1727:
1725:
1723:
1721:
1717:
1711:
1709:
1707:
1704:perturbed by
1703:
1699:
1695:
1690:
1688:
1687:Mikhail Lidov
1684:
1680:
1672:
1670:
1668:
1664:
1660:
1659:star clusters
1656:
1648:
1646:
1644:
1640:
1636:
1632:
1628:
1620:
1618:
1616:
1612:
1608:
1604:
1599:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1568:
1566:
1558:
1554:
1551:
1548:
1545:
1544:
1543:
1540:
1538:
1534:
1530:
1526:
1518:
1510:
1505:
1503:
1501:
1497:
1480:
1478:
1474:
1470:
1429:
1425:
1421:
1416:
1410:
1405:
1401:
1397:
1394:
1390:
1382:
1375:
1370:
1366:
1358:
1353:
1349:
1345:
1340:
1334:
1329:
1325:
1321:
1318:
1314:
1305:
1301:
1297:
1293:
1285:
1280:
1276:
1262:
1258:
1253:
1244:
1240:
1230:
1227:
1224:
1201:
1193:
1192:
1191:
1185:
1183:
1128:
1124:
1120:
1110:
1107:
1099:
1095:
1092:
1089:
1069:
1061:
1060:
1059:
1038:
1036:
1032:
1028:
1024:
970:
967:
964:
961:
952:
948:
944:
941:
936:
925:
917:
916:
915:
910:
903:
899:
891:
886:
882:
880:
876:
872:
868:
864:
860:
859:test particle
856:
848:
843:
841:
839:
834:
826:
824:
806:
798:
794:
787:
784:
759:
734:
706:
674:
673:
672:
656:
652:
648:
640:
622:
618:
614:
606:
588:
584:
580:
572:
568:
564:
548:
525:
497:
472:
450:
436:
435:
434:
432:
428:
424:
420:
416:
411:
409:
405:
401:
397:
391:
387:
379:
377:
375:
371:
367:
362:
360:
339:
335:
331:
326:
322:
298:
295:
292:
268:
265:
262:
259:
256:
253:
250:
247:
225:
221:
213:
209:
191:
187:
179:
175:
171:
143:
137:
129:
124:
122:
120:
116:
113:perturbed by
112:
108:
104:
103:Mikhail Lidov
100:
96:
92:
88:
84:
79:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
40:binary system
37:
33:
19:
3167:
3163:
3119:
3113:
3080:
3076:
3066:
3015:
3011:
3005:
2993:. Retrieved
2955:
2951:
2941:
2882:
2878:
2872:
2821:
2817:
2811:
2758:
2754:
2744:
2693:
2689:
2683:
2630:
2626:
2616:
2589:
2585:
2575:
2548:
2544:
2534:
2507:
2503:
2493:
2456:
2450:
2440:
2395:
2391:
2381:
2353:
2281:
2277:
2271:
2220:
2216:
2162:
2158:
2090:
2086:
2080:
2021:
2017:
2011:
1992:
1986:
1949:
1945:
1907:
1901:
1883:
1858:
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