Abstract
Hill’s lunar problem is a model for the motion of the moon around the earth under the additional influence of the sun. However, it also models the relative motion of a pair of co-orbital satellites near their close encounters. In spite of the simplicity of the differential equations their solutions show a remarkable degree of complexity. In this paper we will discuss the asymptotic behavior of the solutions and outline adequate methods for their numerical integration. Then, based on the notion of the Poincaré map, some particular periodic solutions will be considered. Finally, for a family of homoclinic solutions the intersection angle α in the range │α│ ∈ (10−8, 10−2) between invariant manifolds is numerically calculated.
This is a preview of subscription content, log in via an institution to check access.
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
Hénon, M., 1969: Numerical exploration of the restricted problem V. Hill’s case: periodic orbits and their stability. Astron. and Astrophys. 1, 223–238.
Hénon, M. and Petit, J.M., 1986: Series expansion for encounter-type solutions of Hill’s problem. Celest. Mech. 38, 67–100.
Hill, G.W., 1886: On the part of the motion of the lunar perigee which is a fraction of the mean motions of the sun and moon. Acta Math. 8, 1; also: Collected Mathematical works of G.W. Hill. Vol. 1. 1, p. 243. Cargenie Inst. of Wash., Washington, D.C., 1905.
Lasagni, F., 1988: Canonical Runge-Kutta methods. ZAMP 39, 952–953.
Levi-Civita, T., 1920: Sur la régularisation du problème des trois corps. Acta Math. 42, 99–144.
McLachlan, R.I. and Atela, P., 1992: The accuracy of symplectic integrators. Nonlinearity 5, 541–562.
McLachlan, R.I., 1994: On the numerical integration of ordinary differential equations by symmetric composition methods. To appear in SIAM J. Sci. Comp.
Ruth, R.D., 1983: A canonical integration technique. IEEE Trans. Nucl. Sci. NS-30, 2669–2671.
Sanz-Serna, J.M., 1988: Runge-Kutta schemes for Hamiltonian systems. BIT 28, 877–883.
Siegel, C.L. and Moser, J.K., 1971: Lectures on Celestial Mechanics. Springer-Verlag, Berlin.
Spirig, F. and Waldvogel, J., 1985: The three-body problem with two small masses: A singular-perturbation approach to the problem of Saturn’s co-orbiting satellites. In: V. Szebehely (ed.), Stability of the solar system and its minor natural and artificial bodies. Reidel, 53–63.
Spirig, F. and Waldvogel, J., 1991: Chaos in co-orbital motion. In: A.E. Roy (ed. ), Predictability, stability, and chaos in N-body dynamical systems.
Trotter, H.F., 1959: On the product of semi-group of operators. Proc. Am. Math. Soc. 10, 545–551.
Waldvogel, J. and Spirig, F., 1988: Co-orbital satellites and Hill’s lunar problem. In: A.E. Roy (ed.), Long-term dynamical behavior of natural and artificial N-body systems. Kluwer, 223–243.
Yoshida, H., 1993: Recent progress in the theory and application of symplectic integrators. Celest. Mech. 56, 27–43.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media New York
About this chapter
Cite this chapter
Waldvogel, J., Spirig, F. (1995). Chaotic Motion in Hill’s Lunar Problem. In: Roy, A.E., Steves, B.A. (eds) From Newton to Chaos. NATO ASI Series, vol 336. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1085-1_20
Download citation
DOI: https://doi.org/10.1007/978-1-4899-1085-1_20
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-1087-5
Online ISBN: 978-1-4899-1085-1
eBook Packages: Springer Book Archive