Skip to main content
SpringerLink
Log in

Periodic orbits of the planetary type and their stability

  • Long Term Evolution of Planetary Systems
  • Session on Periodic Orbits
  • Published:
Celestial mechanics Aims and scope Submit manuscript

Abstract

A review is presented of periodic orbits of the planetary type in the general three-body problem and fourbody problem and the restricted circular and elliptic tnreebody problem. These correspond to planetary systems with one Sun and two or three planets (or a planet and its satellites), the motion of asteoids and also planetary systems with two Suns. The factors which affect the stability of the above configurations are studied in connection with resonance or additional perturbations. Finally, the correspondence of the periodic orbits in the restricted three-body problem with the fixed points obtained by the method of averaging or the method of surface of section is indicated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Arnold, V.I.: 1978, Mathematical Methods of Classical Mechanics, Springer-Verlag.

  • Arnold, VI. and Avez, A.: 1967, Problémes Ergodiques de la Mechanique Classique, Gauthier-Villars.

  • Bien, R.: 1980, “Stationary Solutions in simplified Resonance Cases of the Restricted Three-body Problem”, Celes. Mech.21, 157–161.

    Google Scholar 

  • Bozis, G. and Hadjidemetriou, J. D.: 1976, “On the Continuation of Periodic Orbits from the Restricted to the General Three-Body Problem”, Celes. Mech.13, 127–136.

    Google Scholar 

  • Broucke, R.: 1969, “Stability of Periodic Orbit in the Elliptic Restricted Three-Body Problem”, A. I. A. A.7, 1003–1009.

    Google Scholar 

  • Colombo, G., Franklin, F. A., Munford, C. M.: 1968, “On a Family of Periodic Orbits of the Restricted Three-Body Problem and the Question of the Gaps in the Asteroid Belt and in Saturn's Rings”, Astron. J.73, 111–123.

    Google Scholar 

  • Delibaltas, P.: 1976, “Families of Periodic Orbits in the General Three-Body Problem for the Sun-Jupiter-Saturn Mass Ratio and their Stability”, Astrophys. Space Sci.45, 207–233.

    Google Scholar 

  • Ferraz-Mello, S.: 1988 The High Eccentricity Libration of the Hildas”, A. J. (to appear).

  • Frangakis, C. N.: 1973, “Phase Plane of the Commensurable Restricted Three-Body Problem”, Astrophys. Space Science22, 421–440.

    Google Scholar 

  • Giffen, R.: 1973, “A study of Commensurable Motion in the Asteroidal Belt”, Astron. Astrophys.23, 387–403.

    Google Scholar 

  • Grigorelis, F.: “A Contribution to the General Four-body Problem”, Doctor's thesis, Univ. of Thessaloniki.

  • Guillaume, P.: 1969, “Families of Symmetric Periodic Orbits of the Restricted Three Body Problem, when the Perturbing Mass is Small”, Astron. Astrophys.3, 57–76.

    Google Scholar 

  • Hadjidemetriou, J. D.: 1976a, “Families of Periodic Planetary-type Orbits in the Three-Body Problem and their Stability”, Astrophys. Space Sci.40, 201–224.

    Google Scholar 

  • Hadjidemetriou, J. D.: 1976b, “Families of Periodic Orbits in the General N-Body Problem”, in V. Szebehely (ed.) Long Time Predictions in Dynamics, D. Reidel, 223–240.

  • Hadjidemetriou, J. D.: 1979, “Instabilities in Periodic Planetary-type Orbits”, in V. Szebehely (ed.) Instabilities in Dynamical Systems, D. Reidel, 135–163.

  • Hadjidemetriou, J. D.: 1980, “The Restricted Planetary 4-Body Problem”, Celes. Mech.21, 63–71.

    Google Scholar 

  • Hadjidemetriou, J. D.: 1981, “The Present Status of Periodic Orbits”, Celes. Mech.23, 277–288.

    Google Scholar 

  • Hadjidemetriou, J. D.: 1982, “On the Relation Between Resonance instability in Planetary Systems”, Celes. Mech.27, 305–322.

    Google Scholar 

  • Hadjidemetriou, J. D.: 1984, “Periodic Orbits”, Celes. Mech.34, 379–393.

    Google Scholar 

  • Hadjidemetriou, J. D.: 1985a, “The Stability of Resonant Orbits in Planetary Systems”, in S. Ferraz-Mello and W. Sessin (eds) Resonances in the Motion of Planets, Satellites and Asteroids, Univ. de São Paulo, 1–18.

  • Hadjidemetriou, J. D.: 1985b “Hamiltonian Perturbations in Dynamical Systems and Applications in Planetary Systems”, in V. Szebehely (ed) Stability of the Solar System, 213–225.

  • Hadjidemetriou, J. D.: 1986, “A Hyperbolic Twist Mapping Model for the study of Asteroid Orbits near the 3:1 Resonance ZAMP37, 776–796.

    Google Scholar 

  • Hadjidemetriou, J. D.: 1988a,, “Resonances in the Solar System and in Planetary Systems”, Proceeding of the 10nth Regional IAU European Meeting, Prague.

  • Hadjidemetriou, J. D.: 1988b, “Algebraic Mappings near a Resonance with an Application to Asteroid Motion”, in A. E. Roy (ed) Long Term Dynamical Behaviour of Natural and Artificial N-body Systems, D. Reidel.

  • Hadjidemetriou, J. D. and Michalodimitrakis, M.: 1981, “Periodic Planetary-type Orbits of the General 4-Body Problem with an Application to the Satellites of Jupiter”, Astron. Astrophy.93, 204–211.

    Google Scholar 

  • Hadjidemetriou, J. D. and Ichtiaroglou, S.: 1984, “A qualitative study of the Kirkwood gaps in the asteroids”, Astron. Astrophys.131, 20–32.

    Google Scholar 

  • Hadjidemetriou, J. D. and Ichtiaroglou,, S: 1988 “Perturbed Keplerian Orbits”, in preparation.

  • Hagel, J. and Dvorak, R.: 1988 “An Analytical Study of Stable Planetary Orbits in the Circular Restricted Problem”, Celes. Mech. (to appear).

  • Hadrava, P. and Kadrnoska, J.: 1986, “Comparison of Analytical and Numerical Calculations of the First Class Periodic Orbits”, Celes. Mech.39, 267–283.

    Google Scholar 

  • Hénon, M.: 1965, “Exploration Numerique du Problème Resteint, II”, Ann. Astrophys.28, 992.

    Google Scholar 

  • Hénon, M. and Guyot, M.: 1970, “Stability of Periodic Orbits in the Restricted Problem”, in G.E.O. Giacaglia (ed) Periodic Orbits, Stability and Resonances, D. Reidel, 349–374.

  • Henrard, J. and Lemaitre, A.: 1983, “A Mechanism of Formation for the Kirkwood Gaps”, Icarus55, 482–494.

    Google Scholar 

  • Henrard, J. and Lemaitre, A.: 1986, “A Perturbation Method for Problems with two Critical Arguments”, Celes. Mech.39, 213–239.

    Google Scholar 

  • Henrard, J. and Lemaitre, A.: A Perturbative Treatment of the 2/1 Jovian Resonance”, Icarus,69, 266–279.

  • Ichtiaroglou, S., Katopodis, K. and Michalodimitrakis, M.: 1980, “Restricted Problem. Families of Vertical Critical Periodic Orbits”, Astron. Astrophys.90, 324–326.

    Google Scholar 

  • Ichtiaroglou, S., Katopodis, K. and Michalodimitrakis, M.: 1986, “Stability of resonant orbits in planetary systems with spheroidal star”, Astron. Astrophys.169, 355–359.

    Google Scholar 

  • Ichtiaroglou, S., Katopodis, K. and Michalodimitrakis, M.: 1988, “Periodic Orbits in Three-dimensional Planetary Systems”, Celes. Mech.

  • Kribbel, J. and Dvorak, R.: 1988, this volume.

  • Kwok, J. H. and Nacozy, P. E.: 1982, “Periodic Orbits of the Elliptic Restricted Problem for the Sun-Jupiter-Saturn System”, Celes. Mech.27, 27–39.

    Google Scholar 

  • Lemaitre, A.: 1984, “High-order Resonances in the Restricted Three-Body Problem”, Celes. mech.32, 109–126.

    Google Scholar 

  • Lichtenberg, A. J. and Lieberman, M. A. 1983, Regular and Stochastic Motion, Springer-Verlag.

  • Markellos, V. V., 1974a, “Numerical Investigation of the Planar Restricted Three-body Problem”, Celes. Mech.9, 365–380

    Google Scholar 

  • Markellos, V. V., 1974b, “Numerical Investigation of the Planar Restricted Three-Body Problem”, Celes. Mech.10, 87–134.

    Google Scholar 

  • Message, P. J.: 1970, “On Asymmetric Periodic Solutions of the Plane Restricted problem of Three Bodies”, in G. E. O. Giacaglia (ed) Periodic Orbits, Stability and Resonances, D. Reidel, 19–32.

  • Message, P. J.: 1980, “On the Existence of Periodic Solutions of Poincaré's Second Sort in the General Problem of Three Bodies Moving in Plane”, Celes. Mech.21, 55–61.

    Google Scholar 

  • Message, D. J.: 1982, “On the Existence of Periodic Solutions of Poincaré's Third Sort in the General Problem of Three-Bodies in Three Dimensions”, Celes. Mech.,28, 107–118.

    Google Scholar 

  • Message, P. J. and Taylor, D. B.: 1978 “On Asymmetric Periodic Solutions of the Plane Restricted Problem of Three Bodies and Bifurcations of Families”, in V. Szebehely (ed), Dynamics of Planets and Satellites and Theories of their Motion, D. Reidel, 319–323.

  • Roy, A. E.: 1982, “Orbital Motion” (2nd ed), Adam Hilger

  • Schubart, J.: 1964, “Long-period Effects in Nearly Commensurable cases of the Restricted Three-Body Problem”, Smithsonian Astrophys. Obs. Report No 149.

  • Szebehely, V.: 1967, Theory of Orbits, Academic Press.

  • Taylor, D. M.: 1983, “Families of Asymmetric Periodic Solutions of the Restricted Problem of Three Bodies”, Celes. Mech.29, 51–74.

    Google Scholar 

  • Wisdom, J.: 1985, “A Perturbative Treatment of Motion near the 3/1 Commensurability”, Icarus63, 272–289.

    Google Scholar 

  • Wisdom, J.: 1986, “Canonical Solution of the Two Critical Argument Problem”, Celes. Mech.,38, 175–180.

    Google Scholar 

  • Yakubovich, V. and Starzhinskii, V. M.: 1975, Linear Differential Equations with Periodic Coefficients, Vol 1, Halsted Press.

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Mechanics, University of Thessaloniki, GR-540 06, Thessaloniki, Greece

    John D. Hadjidemetriou

Authors
  1. John D. Hadjidemetriou
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hadjidemetriou, J.D. Periodic orbits of the planetary type and their stability. Celestial Mechanics 43, 371–390 (1987). https://doi.org/10.1007/BF01234581

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01234581

Keywords

Search

Navigation