Abstract
Previous work on the plane circular restricted problem of three bodies (Message 1958, 1959, 1970, and Fragakis 1973) has shown the existence, in association with each of the commensurabilities 2:1 and 3:1 of the orbital periods, of a pair of families of asymmetric periodic solutions, branching from the stable series of symmetric periodic solutions of Poincarés second sort associated with that commensurability. (Each solution of either family is the mirror image, in the line of the two finite bodies, of a member of the other family of solutions associated with the commensurability.) The stability is transferred at the bifurcation to the two series of asymmetric orbits, each of which is therefore stable. Recent numerical integrations carried out by one of us (P.J.M.) have found such asymmetric periodic orbits associated also with the 4:1 commensurability, and quantities describing orbits of one of the two series are given in Table 1, showing the run of such orbits up to a second bifurcation with the same series of symmetric periodic orbits from which it sprang. Quantities describing some members of this series of symmetric orbits are given in Table 2. It is seen that stability is transferred back to the symmetric series at the second bifurcation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Fragakis, C.N., 1973, Astrophysics and Space Science, vol. 22, pp. 421–440, and vol. 23, pp. 17-42.
Message, P.J., 1958, Astronomical Journal, vol. 63, pp. 443–448.
Message, P.J., 1959, Astronomical Journal, vol. 66, pp. 226–236.
Message, P.J., 1970, in “Periodic Orbits, Stability, and Resonances”, (ed. G.E.O. Giacaglia), pp. 19-32.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1978 D. Reidel Publishing Company, Dordrecht, Holland
About this paper
Cite this paper
Message, P.J., Taylor, D.B. (1978). On Asymmetric Periodic Solution of the Plane Restricted Problem of Three Bodies, and Bifurcations of Families. In: Szebehely, V. (eds) Dynamics of Planets and Satellites and Theories of Their Motion. Astrophysics and Space Science Library, vol 72. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9809-4_37
Download citation
DOI: https://doi.org/10.1007/978-94-009-9809-4_37
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-9811-7
Online ISBN: 978-94-009-9809-4
eBook Packages: Springer Book Archive