Abstract
Planet crossing orbits give rise to mathematical singularities that make it not possible to apply the classical averaging principle to study the qualitative evolution of Near Earth Asteroids (NEAs). Recently this principle has been generalized to deal with crossings in a mathematical model with the planets on circular coplanar orbits. More accuracy is needed to compute the averaged evolution of planet crossing orbits for different purposes: computing reliable crossing times for the averaged motion, writing more precise proper elements and frequencies for NEAs, etc. In this paper we present the generalization of the averaging principle using a model where the eccentricity and the inclination of the planets are taken into account.
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
Arnold, V. I., Kozlov, V. V. and Neishtadt, A. I.: 1997, Mathematical Aspects of Classical and Celestial Mechanics,Springer.
Bonanno, C.: 1999, ‘An analytical approximation for the MOID and its consequences’, Astron. Astrophys. 360, 411–416.
Carpino, M., Milani, A. and Nobili, A. M: 1987, ‘Long-term numerical integrations and synthetic theories for the motion of the outer planets’, Astron. Astrophys. 181, 182–194.
Demidovic, B. P. and Maron I. A.: 1966, Foundations of Numerical Mathematics,SNTL Praha. Fleming, W. H.: 1964, Functions of Several Variables,Addison-Wesley.
Gronchi, G. F.: 2002, ‘Theoretical and Computational Aspects of Collision Singularities in the N-body Problem’, Ph.D. Thesis, University of Pisa.
Gronchi, G. F. and Michel, P.: 2001, ‘Secular orbital evolution, proper elements and proper frequencies for Near-Earth Asteroids: a comparison between semianalytic theory and numerical integrations’, Icarus 152, 48–57.
Gronchi, G. F. and Milani, A.: 1998, ‘Averaging on Earth-crossing orbits’, Celest. Mech. & Dyn. Astr. 71 /2, 109–136.
Gronchi, G. F. and Milani, A.: 1999, ‘The stable Kozai state for asteroids and comets with arbitrary semimajor axis and inclination’, Astron. Astrophys. 341, 928–935.
Gronchi, G. F. and Milani, A.: 2001, ‘Proper elements for Earth crossing asteroids’, Icarus 152, 58–69.
Kozai, Y: 1962. ‘Secular perturbations of asteroids with high inclination and eccentricity’, Astron. J. 67, 591–598.
Lidov, M. L. and Ziglin, S. L.: 1974, ‘The analysis of restricted circular twice-averaged three body problem in the case of closed orbits’, Celest. Mech. 9, 151–173.
Valsecchi, G. B., Jopek, T. J. and Froeschlè, Cl.: 1999, ‘Meteoroid streams identification: a new approach’, MNRAS 304, 743–750
Valsecchi, G. B., Milani, A., Gronchi, G. F. and Chesley, S. R.: 2001, ‘Resonant returns to close approaches: analytical theory’, Astron. Astrophys. (submitted).
Wetherill, G. W.: 1967, ‘Collisions in the asteroid belt’, J. Geophys. Res. 72, 2429–2444.
Williams, J. G.: 1969, ‘Secular Perturbations in the Solar System’, Ph.D. Thesis, University of California, Los Angeles.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Gronchi, G.F. (2002). Generalized Averaging Principle and the Secular Evolution of Planet Crossing Orbits. In: Celletti, A., Ferraz-Mello, S., Henrard, J. (eds) Modern Celestial Mechanics: From Theory to Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2304-6_7
Download citation
DOI: https://doi.org/10.1007/978-94-017-2304-6_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6078-5
Online ISBN: 978-94-017-2304-6
eBook Packages: Springer Book Archive