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2/1 resonant periodic orbits in three dimensional planetary systems

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Abstract

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50◦–60◦, which may be related with the existence of real planetary systems.

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References

  • Antoniadou K.I., Voyatzis G., Kotoulas T.: On the bifurcation and continuation of periodic orbits in the three body problem. Int. J. Bifurcation Chaos 21, 2211 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Barnes R., Greenberg R., Quinn T.R., McArthur B.E., Benedict G.F: Origin and dynamics of the mutually inclined orbits of υ andromedae c and d. Astrophys. J. 726, 71 (2011)

    Article  ADS  Google Scholar 

  • Beaugé C., Michtchenko T.A., Ferraz-Mello S.: Planetary migration and extrasolar planets in the 2/1 mean-motion resonance. Mon. Notices R. Astron. Soc. 365, 1160–1170 (2006)

    Article  ADS  Google Scholar 

  • Chatterjee S., Ford E.B., Matsumura S., Rasio F.A.: Dynamical outcomes of planet-planet scattering. Astrophys. J. 686, 580–602 (2008)

    Article  ADS  Google Scholar 

  • Correia A.C.M., Laskar J., Farago F., Boué G.: Tidal evolution of hierarchical and inclined systems. Celest. Mech. Dyn. Astron. 111, 105–130 (2011)

    Article  ADS  Google Scholar 

  • Ferraz-Mello S., Beaugé C., Michtchenko T.A.: Evolution of migrating planet pairs in resonance. Celest. Mech. Dyn. Astron. 87, 99–112 (2003)

    Article  ADS  Google Scholar 

  • Froeschlé C., Lega E., Gonczi R.: Fast lyapunov indicators. Application to asteroidal motion. Celest. Mech. Dyn. Astron. 67, 41–62 (1997)

    Article  ADS  MATH  Google Scholar 

  • Goździewski K., Konacki M., Wolszczan A.: Long-term stability and dynamical environment of the psr 1257+12 planetary system. Astrophys. J. 619, 1084–1097 (2005)

    Article  ADS  Google Scholar 

  • Hadjidemetriou, J.D.: Periodic Orbits in Gravitational Systems. In: Steves, B.A., et al. (eds.) Chaotic Worlds: from Order to Disorder in Gravitational N-Body Dynamical Systems, pp. 43–79. Springer, Dordrecht (2006a)

  • Hadjidemetriou J.D.: Symmetric and asymmetric librations in extrasolar planetary systems: a global view. Celest. Mech. Dyn. Astron. 95, 225–244 (2006b)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Hadjidemetriou J., Voyatzis G.: The 2/1 and 3/2 resonant asteroid motion: a symplectic mapping approach. Celest. Mech. Dyn. Astron. 78, 137–150 (2000)

    Article  ADS  MATH  Google Scholar 

  • Hadjidemetriou J.D., Voyatzis G.: On the dynamics of extrasolar planetary systems under dissipation: migration of planets. Celest. Mech. Dyn. Astron. 107, 3–19 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Haghighipour N., Couetdic J., Varadi F., Moore W.B.: Stable 1:2 resonant periodic orbits in elliptic three-body systems. Astrophys. J. 596, 1332–1340 (2003)

    Article  ADS  Google Scholar 

  • Hénon M.: Vertical stability of periodic orbits in the restricted problem. i. equal masses. Astron. Astrophys. 28, 415 (1973)

    ADS  MATH  Google Scholar 

  • Ichtiaroglou S., Michalodimitrakis M.: Three-body problem—the existence of families of three-dimensional periodic orbits which bifurcate from planar periodic orbits. Astron. Astrophys. 81, 30–32 (1980)

    ADS  MATH  Google Scholar 

  • Katopodis K., Ichtiaroglou S., Michalodimitrakis M.: The family i1v of the three-dimensional general three-body problem. Astron. Astrophys. 90, 102–105 (1980)

    MathSciNet  ADS  MATH  Google Scholar 

  • Kotoulas T.A.: The dynamics of the 1:2 resonant motion with neptune in the 3d elliptic restricted three-body problem. Astron. Astrophys. 429, 1107–1115 (2005)

    Article  ADS  Google Scholar 

  • Kotoulas T.A., Voyatzis G.: Three dimensional periodic orbits in exterior mean motion resonances with neptune. Astron. Astrophys. 441, 807–814 (2005)

    Article  ADS  Google Scholar 

  • Laughlin G., Chambers J., Fischer D.: A dynamical analysis of the 47 ursae majoris planetary system. Astrophys. J. 579, 455–467 (2002)

    Article  ADS  Google Scholar 

  • Lee M.H., Peale S.J.: Dynamics and origin of the 2:1 orbital resonances of the GJ 876 planets. Astrophys. J. 567, 596–609 (2002)

    Article  ADS  Google Scholar 

  • Lee M.H., Thommes E.W.: Planetary migration and eccentricity and inclination resonances in extrasolar planetary systems. Astrophys. J. 702, 1662–1672 (2009)

    Article  ADS  Google Scholar 

  • Libert A.-S., Tsiganis K.: Kozai resonance in extrasolar systems. Astron. Astrophys. 493, 677–686 (2009a)

    Article  ADS  MATH  Google Scholar 

  • Libert A.-S., Tsiganis K.: Trapping in high-order orbital resonances and inclination excitation in extrasolar systems. Mon. Notices R. Astron. Soc. 400, 1373–1382 (2009b)

    Article  ADS  Google Scholar 

  • Marchal C.: The Three-Body Problem. Elsevier, Amsterdam (1990)

    MATH  Google Scholar 

  • Marzari F., Weidenschilling S.J.: Eccentric extrasolar planets: the jumping jupiter model. Icarus 156, 570–579 (2002)

    Article  ADS  Google Scholar 

  • Michalodimitrakis M.: General three-body problem—families of three-dimensional periodic orbits. ii. Astrophys. Space Sci. 65, 459–475 (1979a)

    Article  ADS  MATH  Google Scholar 

  • Michalodimitrakis, M.: On the continuation of periodic orbits from the planar to the three-dimensional general three-body problem. Celest. Mech. 19, 263–277 (1979b); 1979/4/1

  • Michalodimitrakis M.: General three-body problem—families of three-dimensional periodic orbits. i. Astron. Astrophys. 81, 113–120 (1980)

    ADS  MATH  Google Scholar 

  • Michalodimitrakis M.: The families c 1 v and m 2 v of the three-dimensional general three-body problem. Astron. Astrophys. 93, 212–218 (1981)

    MathSciNet  ADS  Google Scholar 

  • Michtchenko, T.A., Ferraz-Mello, S.: The periodic and chaotic regimes of motion in the exoplanet 2/1 mean-motion resonance. In: Cincotta, P.M., et al. (eds.) Proceedings of Third La Plata International School on Astronomy and Geophysics: Chaos, diffusion and Non-integrability in Hamiltonian Systems Applications to Astonomy Assoc. Argent. Astron., La Plata (2011)

  • Michtchenko T.A., Beaugé C., Ferraz-Mello S.: Stationary orbits in resonant extrasolar planetary systems. Celest. Mech. Dynam. Astron. 94, 411–432 (2006a)

    Article  ADS  MATH  Google Scholar 

  • Michtchenko T.A., Ferraz-Mello S., Beaugé C.: Modeling the 3-d secular planetary three-body problem. Discussion on the outer υ and romedae planetary system. Icarus 181, 555–571 (2006b)

    Article  ADS  Google Scholar 

  • Michtchenko T.A., Beaugé C., Ferraz-Mello S.: Dynamic portrait of the planetary 2/1 mean-motion resonance—i. Systems with a more massive outer planet. Mon. Notices R. Astron. Soc. 387, 747–758 (2008a)

    Article  ADS  Google Scholar 

  • Michtchenko T.A., Beaugé C., Ferraz-Mello S.: Dynamic portrait of the planetary 2/1 mean-motion resonance—ii. Systems with a more massive inner planet. Mon. Notices R. Astron. Soc. 391, 215–227 (2008b)

    Article  ADS  Google Scholar 

  • Psychoyos D., Hadjidemetriou J.D.: Dynamics of 2/1 resonant extrasolar systems application to HD82943 and gliese876. Celest. Mech. Dyn. Astron. 92, 135–156 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Schwarz R., Bazso Á., érdi B., Funk B.: Stability of the lagrangian point L 4 in the spatial restricted three-body problem—application to exoplanetary systems. Mon. Notices R. Astrono. Soc. 427, 397–402 (2012)

    Article  ADS  Google Scholar 

  • Thommes E.W., Lissauer J.J.: Resonant inclination excitation of migrating giant planets. Astrophys. J. 597, 566–580 (2003)

    Article  ADS  Google Scholar 

  • Varadi F.: Periodic orbits in the 3:2 orbital resonance and their stability. Astron. J. 118, 2526–2531 (1999)

    Article  ADS  Google Scholar 

  • Veras D., Armitage P.J.: The dynamics of two massive planets on inclined orbits. Icarus 172, 349–371 (2004)

    Article  ADS  Google Scholar 

  • Voyatzis G.: Chaos, order, and periodic orbits in 3:1 resonant planetary dynamics. Astrophys. J. 675, 802–816 (2008)

    Article  ADS  Google Scholar 

  • Voyatzis G., Hadjidemetriou J.D.: Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance. Celest. Mech. Dyn. Astron. 93, 263–294 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Voyatzis G., Kotoulas T., Hadjidemetriou J.D.: On the 2/1 resonant planetary dynamics—periodic orbits and dynamical stability. Mon. Notices R. Astron. Soc. 395, 2147–2156 (2009)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    K. I. Antoniadou & G. Voyatzis

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  1. K. I. Antoniadou
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  2. G. Voyatzis
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Correspondence to K. I. Antoniadou.

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Antoniadou, K.I., Voyatzis, G. 2/1 resonant periodic orbits in three dimensional planetary systems. Celest Mech Dyn Astr 115, 161–184 (2013). https://doi.org/10.1007/s10569-012-9457-4

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  • DOI: https://doi.org/10.1007/s10569-012-9457-4

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