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Perturbation Theory in Celestial Mechanics

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Perturbation theory aims to find an approximate solution of nearly-integrable systems, namely systems which are composed by an integrable partand by a small perturbation. The key point of perturbation theory is the construction of a suitable canonical transformation which removes theperturbation to higher orders. A typical example of a nearly-integrable system is provided by a two-body model perturbed by thegravitational influence of a third body whose mass is much smaller than the mass of the central body. Indeed, the solution of the three-bodyproblem greatly stimulated the development of perturbation theories. The solar system dynamics has always been a testing ground for such theories,whose applications range from the computation of the ephemerides of natural bodies to the development of the trajectories of artificial satellites.

Introduction

The two-body problem can be solved by means of Kepler's laws, according to which for negative energies the...

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Abbreviations

KAM theory:

Provides the persistence of quasi-periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.?e. a non-degeneracy of the integrable system and a diophantine condition of the frequency of motion. It yields a constructive algorithm to evaluate the strength of the perturbation ensuring the existence of invariant tori.

Perturbation theory:

Provides an approximate solution of the equations of motion of a nearly-integrable system.

Spin-orbit problem:

A model composed of a rigid satellite rotating about an internal axis and orbiting around a central point-mass planet; a spin-orbit resonance means that the ratio between the revolutional and rotational periods is rational.

Three-body problem:

A system composed by three celestial bodies (e.?g. Sun-planet-satellite) assumed to be point-masses subject to the mutual gravitational attraction. The restricted three-body problem assumes that the mass of one of the bodies is so small that it can be neglected.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy

    Alessandra Celletti

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  1. Alessandra Celletti
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Celletti, A. (2009). Perturbation Theory in Celestial Mechanics . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_397

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