Astrophysics and Space Science

, Volume 349, Issue 2, pp 657–676 | Cite as

Resonant periodic orbits in the exoplanetary systems

  • K. I. AntoniadouEmail author
  • G. Voyatzis
Original Article


The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60 of mutual planetary inclination, but in most families, the stability does not exceed 20–30, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.


Extrasolar planets General three body problem Mean motion resonances Periodic orbits Planetary systems 



This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Thales. Investing in knowledge society through the European Social Fund.


  1. Anglada-Escudé, G., Arriagada, P., Vogt, S.S., Rivera, E.J., Butler, R.P., Crane, J.D., Shectman, S.A., Thompson, I.B., Minniti, D., Haghighipour, N., Carter, B.D., Tinney, C.G., Wittenmyer, R.A., Bailey, J.A., O’Toole, S.J., Jones, H.R.A., Jenkins, J.S.: Astrophys. J. Lett. 751, 16 (2012) ADSCrossRefGoogle Scholar
  2. Antoniadou, K.I., Voyatzis, G.: Celest. Mech. Dyn. Astron. 115, 161 (2013) ADSCrossRefMathSciNetGoogle Scholar
  3. Beaugé, C., Ferraz-Mello, S., Michtchenko, T.A.: Astrophys. J. 593, 1124 (2003) ADSCrossRefGoogle Scholar
  4. Callegari, N., Rodríguez, Á.: Celest. Mech. Dyn. Astron. (2013). doi: 10.1007/s10569-013-9496-5 zbMATHGoogle Scholar
  5. Campanella, G.: Mon. Not. R. Astron. Soc. 418, 1028 (2011) ADSCrossRefGoogle Scholar
  6. Campanella, G., Nelson, R.P., Agnor, C.B.: Mon. Not. R. Astron. Soc. 433, 3190 (2013) ADSCrossRefGoogle Scholar
  7. Ferraz-Mello, S., Beaugé, C., Michtchenko, T.A.: Celest. Mech. Dyn. Astron. 87, 99 (2003) ADSCrossRefGoogle Scholar
  8. Ferraz-Mello, S., Michtchenko, T.A., Beaugé, C.: In: Steves, B.A., Maciejewski, A.J., Hendry, M. (eds.) Regular Motions in Extra-Solar Planetary Systems, p. 255. Springer, Berlin (2006) Google Scholar
  9. Froeschlé, C., Lega, E., Gonczi, R.: Celest. Mech. Dyn. Astron. 67, 41 (1997) ADSCrossRefzbMATHGoogle Scholar
  10. Goździewski, K., Konacki, M., Maciejewski, A.J.: Astrophys. J. 645(1), 688 (2006) ADSCrossRefGoogle Scholar
  11. Hadjidemetriou, J.D.: Celest. Mech. Dyn. Astron. 95, 225 (2006) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. Hadjidemetriou, J.D., Voyatzis, G.: Celest. Mech. Dyn. Astron. 107, 3 (2010) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. Hénon, M.: Astron. Astrophys. 28, 415 (1973) ADSzbMATHGoogle Scholar
  14. Ichtiaroglou, S., Michalodimitrakis, M.: Astron. Astrophys. 81, 30 (1980) ADSzbMATHGoogle Scholar
  15. Ichtiaroglou, S., Katopodis, K., Michalodimitrakis, M.: Astron. Astrophys. 70, 531 (1978) ADSzbMATHMathSciNetGoogle Scholar
  16. Laskar, J., Correia, A.C.M.: Astron. Astrophys. 496, 5 (2009) ADSCrossRefGoogle Scholar
  17. Lee, M.H., Peale, S.J.: Astrophys. J. 567, 596 (2002) ADSCrossRefGoogle Scholar
  18. Libert, A.-S., Henrard, J.: Astron. Astrophys. 461, 759 (2007) ADSCrossRefGoogle Scholar
  19. Lovis, C., Ségransan, D., Mayor, M., Udry, S., Benz, W., Bertaux, J.-L., Bouchy, F., Correia, A.C.M., Laskar, J., Lo Curto, G., Mordasini, C., Pepe, F., Queloz, D., Santos, N.C.: Astron. Astrophys. 528, 112 (2011) ADSCrossRefGoogle Scholar
  20. Marchal, C.: The Three-Body Problem. Elsevier, Amsterdam (1990) zbMATHGoogle Scholar
  21. Michalodimitrakis, M.: Celest. Mech. 19, 263 (1979) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  22. Michtchenko, T.A., Beaugé, C., Ferraz-Mello, S.: Celest. Mech. Dyn. Astron. 94, 411 (2006) ADSCrossRefzbMATHGoogle Scholar
  23. Niedzielski, A., Goździewski, K., Wolszczan, A., Konacki, M., Nowak, G., Zieliński, P.: Astrophys. J. 693(1), 276 (2009) ADSCrossRefGoogle Scholar
  24. Psychoyos, D., Hadjidemetriou, J.D.: In: Knežević, Z., Milani, A. (eds.) IAU Colloq. Dynamics of Populations of Planetary Systems, vol. 197, p. 55 (2005) Google Scholar
  25. Rein, H., Papaloizou, J.C.B., Kley, W.: Astron. Astrophys. 510, 4 (2010) ADSCrossRefGoogle Scholar
  26. Skokos, C.: Physica D 159, 155 (2001) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. Steffen, J.H., Fabrycky, D.C., Ford, E.B., Carter, J.A., Désert, J.-M., Fressin, F., Holman, M.J., Lissauer, J.J., Moorhead, A.V., Rowe, J.F., Ragozzine, D., Welsh, W.F., Batalha, N.M., Borucki, W.J., Buchhave, L.A., Bryson, S., Caldwell, D.A., Charbonneau, D., Ciardi, D.R., Cochran, W.D., Endl, M., Everett, M.E., Gautier, T.N., Gilliland, R.L., Girouard, F.R., Jenkins, J.M., Horch, E., Howell, S.B., Isaacson, H., Klaus, T.C., Koch, D.G., Latham, D.W., Li, J., Lucas, P., MacQueen, P.J., Marcy, G.W., McCauliff, S., Middour, C.K., Morris, R.L., Mullally, F.R., Quinn, S.N., Quintana, E.V., Shporer, A., Still, M., Tenenbaum, P., Thompson, S.E., Twicken, J.D., Van Cleve, J.: Mon. Not. R. Astron. Soc. 421, 2342 (2012) ADSCrossRefGoogle Scholar
  28. Steffen, J.H., Fabrycky, D.C., Agol, E., Ford, E.B., Morehead, R.C., Cochran, W.D., Lissauer, J.J., Adams, E.R., Borucki, W.J., Bryson, S., Caldwell, D.A., Dupree, A., Jenkins, J.M., Robertson, P., Rowe, J.F., Seader, S., Thompson, S., Twicken, J.D.: Mon. Not. R. Astron. Soc. 428, 1077 (2013) ADSCrossRefGoogle Scholar
  29. Voyatzis, G.: Astrophys. J. 675, 802 (2008) ADSCrossRefGoogle Scholar
  30. Voyatzis, G., Hadjidemetriou, J.D.: Celest. Mech. Dyn. Astron. 93, 263 (2005) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  31. Voyatzis, G., Kotoulas, T., Hadjidemetriou, J.D.: Mon. Not. R. Astron. Soc. 395, 2147 (2009) ADSCrossRefGoogle Scholar
  32. Wittenmyer, R.A., Horner, J., Tinney, C.G.: Astrophys. J. 761, 165 (2012) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of PhysicsAristotle University of ThessalonikiThessalonikiGreece

Personalised recommendations