zbMATH — the first resource for mathematics

The eccentric Kozai-Lidov effect as a resonance phenomenon. (English) Zbl 1382.70008
Summary: Exploring weakly perturbed Keplerian motion within the restricted three-body problem, M. L. Lidov [“The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies”, Planet. Space Sci. 9, No. 10, 719–759 (1962)] and, independently, Y. Kozai [“Secular perturbations of asteroids with high inclination and eccentricity”, Astron. J. 67, 591–598 (1962; doi:10.1086/108790)] discovered coupled oscillations of eccentricity and inclination (the KL cycles). Their classical studies were based on an integrable model of the secular evolution, obtained by double averaging of the disturbing function approximated with its first non-trivial term. This was the quadrupole term in the series expansion with respect to the ratio of the semimajor axis of the disturbed body to that of the disturbing body. If the next (octupole) term is kept in the expression for the disturbing function, long-term modulation of the KL cycles can be established [E. B. Ford, B. Kozinsky and F. A. Rasio, “Secular evolution of hierarchical triple star systems”, Astrophys. J. 535, No. 1, 385–401 (2000); S. Naoz et al., “Hot jupiters from secular planet-planet interactions”, Nature 473, 187–189 (2011; doi:10.1038/nature10076); B. Katz, S. Dong and R. Malhotra, “Long-term cycling of Kozai-Lidov cycles: extreme eccentricities and inclinations excited by a distant eccentric perturber”, Phys. Rev. Lett. 107, No. 18, Article ID 181101, 5 p. (2011; doi:10.1103/PhysRevLett.107.181101)]. Specifically, flips between the prograde and retrograde orbits become possible. Since such flips are observed only when the perturber has a nonzero eccentricity, the term “eccentric Kozai-Lidov effect” (or EKL effect) was proposed by Y. Lithwick and S. Naoz [“The eccentric Kozai mechanism for a test particle”, Astrophys. J. 742, No. 2, Article No. 94, 8 p. (2011; doi:10.1088/0004-637X/742/2/94)] to specify such behavior. We demonstrate that the EKL effect can be interpreted as a resonance phenomenon. To this end, we write down the equations of motion in terms of “action-angle” variables emerging in the integrable Kozai-Lidov model. It turns out that for some initial values the resonance is degenerate and the usual “pendulum” approximation is insufficient to describe the evolution of the resonance phase. Analysis of the related bifurcations allows us to estimate the typical time between the successive flips for different parts of the phase space.
Reviewer: Reviewer (Berlin)

70F07 Three-body problems
70F15 Celestial mechanics
Full Text: DOI
[1] Aksenov, EP, The doubly averaged, elliptical, restricted, three-body problem, Sov. Astron., 23, 236-240, (1979) · Zbl 0424.70011
[2] Antognini, JMO, Timescales of kozai-lidov oscillations at quadrupole and octupole order in the test particle limit, MNRAS, 452, 3610-3619, (2015)
[3] Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. Springer, New York (2006) · Zbl 1105.70002
[4] Bailey, ME; Chambers, JE; Hahn, G, Origin of sungrazers: a frequent cometary end-state, Astron. Astrophys., 257, 315-322, (1992)
[5] Blaes, O; Lee, ME; Socrates, A, The kozai mechanism and the evolution of binary supermassive black holes, Astrophys. J., 578, 775-786, (2002)
[6] Byrd, P.L., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Physicists. Springer, Berlin (1954) · Zbl 0055.11905
[7] Eggleton, PP; Kiseleva-Eggleton, L, Orbital evolution in binary and triple stars, with an application to SS lacertae, Astrophys. J., 562, 1012-1030, (2001)
[8] Farago, F; Laskar, J, High-inclination orbits in the secular quadrupolar three-body problem, MNRAS, 401, 1189-1198, (2010)
[9] Ford, EB; Kozinsky, B; Rasio, FA, Secular evolution of hierarchical triple star systems, Astrophys. J., 535, 385-401, (2000)
[10] Froeschlé, Ch; Scholl, H; Morbidelli, A, Complex dynamical behaviour of the asteroid 2335 James associated with the secular resonances nu5 and nu16-numerical studies and theoretical interpretation, Astron. Astrophys., 249, 553-562, (1991)
[11] Gordeeva, YF, Time-dependence of orbital elements in long-period oscillations in the three-body boundary-value problem, Cosm. Res., 6, 450-454, (1968)
[12] Gronchi, GF; Milani, A, The stable kozai state for asteroids and comets, with arbitrary semimajor axis and inclination, Astron. Astrophys., 341, 928-935, (1999)
[13] Harrington, RS, Dynamical evolution of triple stars, Astron. J., 73, 190-194, (1968)
[14] Howard, JE; Humpherys, J, Nonmonotonic twist maps, Physica D, 80, 256-276, (1995) · Zbl 0888.58055
[15] Innanen, KA; Zheng, JQ; Mikkola, S; Valtonen, MJ, The kozai mechanism and the stability of planetary orbits in binary star systems, Astron. J., 113, 1915-1919, (1997)
[16] Ivanov, PB; Polnarev, AG; Saha, P, The tidal disruption rate in dense galactic cusps containing a supermassive binary black hole, MNRAS, 358, 1361-1378, (2005)
[17] Katz, B; Dong, S; Malhotra, R, Long-term cycling of kozai-lidov cycles: extreme eccentricities and inclinations excited by a distant eccentric perturber, Phys. Rev. Lett., 107, 181101, (2011)
[18] Kinoshita, H; Nakai, H, General solution of the kozai mechanism, Celest. Mech. Dyn. Astron., 98, 67-74, (2007) · Zbl 1116.70025
[19] Kozai, Y, Secular perturbations of asteroids with high inclination and eccentricity, Astron. J., 67, 591-598, (1962)
[20] Kozai, Y; Duncombe, RL (ed.), Secular perturbations of asteroids and comets, 231-237, (1979), Dordrecht
[21] Kozai, Y, Secular perturbations of resonant asteroids, Celest. Mech., 36, 47-69, (1985) · Zbl 0577.70014
[22] Li, G; Naoz, S; Holman, M; Loeb, A, Chaos in the test particle eccentric kozai-lidov mechanism, Astrophys. J., 791, 86, (2014)
[23] Li, G; Naoz, S; Kocsis, B; Loeb, A, Eccentricity growth and orbit flip in near-coplanar hierarchical three-body systems, Astrophys. J., 785, 116, (2014)
[24] Libert, A-S; Henrard, J, Exoplanetary systems: the role of an equilibrium at high mutual inclination in shaping the global behavior of the 3-D secular planetary three-body problem, Icarus, 191, 469-485, (2007)
[25] Lidov, ML, The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies, Planet. Space Sci., 9, 719-759, (1962)
[26] Lidov, ML; Ziglin, SL, The analysis of restricted circular twice-averaged three body problem in the case of close orbits, Celest. Mech., 9, 151-173, (1974) · Zbl 0355.70007
[27] Lithwick, Y; Naoz, S, The eccentric kozai mechanism for a test particle, Astrophys. J., 742, 94, (2011)
[28] Luo, L; Katz, B; Dong, S, Double-averaging can fail to characterize the long-term evolution of lidov-kozai cycles and derivation of an analytical correction, MNRAS, 458, 3060-3074, (2016)
[29] Morozov, AD, Degenerate resonances in Hamiltonian systems with 3/2 degrees of freedom, Chaos, 12, 539-548, (2002) · Zbl 1080.37575
[30] Naoz, S, The eccentric kozai-lidov effect and its applications, Ann. Rev. Astron. Astrophys., 54, 441-489, (2016)
[31] Naoz, S; Farr, WM; Lithwick, Y; Rasio, FA; Teyssandier, J, Hot jupiters from secular planet-planet interactions, Nature, 473, 187-189, (2011)
[32] Naoz, S; Farr, WM; Lithwick, Y; Rasio, FA; Teyssandier, J, Secular dynamics in hierarchical three-body systems, MNRAS, 431, 2155-2171, (2013)
[33] Nesvorny, D; Alvarellos, JLA; Dones, L; Levison, HF, Orbital and collisional evolution of the irregular satellites, Astron. J., 126, 398-429, (2003)
[34] Shevchenko, I.I.: The Lidov-Kozai Effect—Applications in Exoplanet Research and Dynamical Astronomy. Springer, Berlin (2016)
[35] Subr, L; Karas, V, On highly eccentric stellar trajectories interacting with a self-gravitating disc in sgr A, Astron. Astrophys., 433, 405-413, (2005)
[36] Thomas, P; Morbidelli, A, The kozai resonance in the outer solar system and the dynamics of long-period comets, Celest. Mech. Dyn. Astron., 64, 209-229, (1996) · Zbl 1002.70560
[37] Triaud, AHMJ; Collier Cameron, A; Queloz, D; Anderson, DR; Gillon, M; Hebb, L; etal., Spin-orbit angle measurements for six southern transiting planets. new insights into the dynamical origins of hot jupiters, Astron. Astrophys., 524, a25, (2010)
[38] Vashkovyak, MA, Evolution of orbits in the restricted circular twice-averaged-three body problem. I-qualitative investigations, Cosm. Res., 19, 1-10, (1981)
[39] Vashkovyak, MA, An investigation of the evolution of some asteroid orbits, Cosm. Res., 24, 255-267, (1986)
[40] Vashkovyak, MA, Evolution of the orbits of distant satellites of uranus, Astron. Lett., 25, 476-481, (1999)
[41] Vashkovyak, MA, Orbital evolution of new distant Neptunian satellites and omega-librators in the satellite systems of saturn and Jupiter, Astron. Lett., 29, 695-703, (2003)
[42] Wen, L, On the eccentricity distribution of coalescing black holes binaries driven by kozai mechanism in global clusters, Astrophys. J., 598, 419-430, (2003)
[43] Ziglin, SL, Secular evolution of the orbit of a planet in a binary-star system, Sov. Astron. Lett., 1, 194-195, (1975)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.