Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-06-13T12:22:10.833Z Has data issue: false hasContentIssue false
  • English
  • Français

Families of Periodic Planetary-Type Orbits in the N-Body Problem and Their Application to the Solar System

Published online by Cambridge University Press:  12 April 2016

J. D. Hadjidemetriou  and
M. Michalodimitrakis
J. D. Hadjidemetriou
Affiliation:
University of Thessaloniki, Thessaloniki, GREECE
M. Michalodimitrakis
Affiliation:
University of Thessaloniki, Thessaloniki, GREECE

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new approach to the study of the Solar System and planetary systems in general is proposed, through the use of periodic planetary-type orbits of the general N-body problem. In such an orbit, one body (called Sun) has a large mass and the rest N-l bodies (called planets) have small but not negligible masses and it can be proved that monoparametric families of periodic orbits of the N-body problem exist in a rotating frame of reference, all being of the planetary type.

Two cases are studied in detail, N=3 and N=4. In N=3, apart from a general discussion, we present a detailed analysis of the Sun-Jupiter-Saturn system and a study is made on which configurations with the masses of these two planets, or a multiple of them, are stable or unstable. Also, part of a family is shown to represent the Jupiter family of comets. It was found that commensurabilities are not in general associated with instabilities. For N=4 we present three families of periodic orbits. The motion corresponding to a branch of one of the above families has many similarities with the actual motion of the three inner satellites of Jupiter.

It is shown that there exist many commensurable cases in the obtained periodic orbits and that the resonant orbits increase as the number of bodies increases. Based on these results, an attempt is made to explain the existence of commensurabilities in the Solar System.

Finally, it is mentioned that a periodic motion of the planetary type can be used as a reference orbit for accurate computations for the actual motions of the planets or satellites of the Solar System. In this way the small divisor difficulties existing in the classical approach will not appear.

Type
Part V. Gravitational Problem of three or more bodies
Information
International Astronomical Union Colloquium , Volume 41: Dynamics of Planets and Satellites and Theories of their Motion , 1978 , pp. 263 - 281
Copyright
Copyright © Reidel 1978

References

1. Brower, D. and Clemence, G.M.: 1961, Orbits and Masses of Planets and Satellites, in Kuiper, G.P. and Middlehurst, B.M (eds.), The Solar System, The University of Chicago Press.Google Scholar
2. Delibaltas, P.: 1976, Astrophys. Space Sience 45, 207.CrossRefGoogle Scholar
3. Hadjidemetriou, J.D.: 1975a, Celes. Mech. 12, 155.CrossRefGoogle Scholar
4. Hadjidemetriou, J.D.: 1975b, Celes. Mech. 12, 255.CrossRefGoogle Scholar
5. Hadjidemetriou, J.D.: 1976a, Astrophysics Space, Science 40, 201.CrossRefGoogle Scholar
6. Hadjidemetriou, J.D.: 1976b, in Szebehely, V. and Tapley, B.O. (eds). Long-Time Predictions in Dynamics, D. Reidel Publ. Co., 223.CrossRefGoogle Scholar
7. Hadjidemetriou, J.D.: 1976c, The Existence of Families of Periodic Orbits in the N-Body Problem, Celes. Mech. (to appear).CrossRefGoogle Scholar
8. Hadjidemetriou, J. D. and Christides, Th.: 1975, Celes. Mech. 12, 175.CrossRefGoogle Scholar
9. Hadjidemetriou, J. D. and Michalodimitrakis, M.: Families of Periodic Planetary-Type Orbits in the N-Body Problem and their Stability (in preparation).Google Scholar
10. Hagihara, Y.: 1961, The Stability of the Solar System, in Kuiper, G.P. and Middlehurst, B.M. (eds.). The Solar System, The University of Chicago Press.Google Scholar
11. Moser, J.: 1973, Stable and Random Motions in Dynamical Systems, Princeton Univ. Press, ch. I.Google Scholar
12. Peek, B.M.: 1958, The Planet Jupiter, Faber and Faber, London, p. 256.Google Scholar