A High Order Perturbation Analysis of the Sitnikov Problem

Author(s)

Christoph Lhotka, Johannes Hagel

Abstract

The Sitnikov problem is one of the most simple cases of the elliptic restricted

three body system. A massless body oscillates along a line (z) perpendicular to a plane

(x, y) in which two equally massive bodies, called primary masses, perform Keplerian

orbits around their common barycentre with a given eccentricity e. The crossing point

of the line of motion of the third mass with the plane is equal to the centre of gravity

of the entire system. In spite of its simple geometrical structure, the system is nonlinear

and explicitly time dependent. It is globally non integrable and therefore represents

an interesting application for advanced perturbative methods. In the present work a high

order perturbation approach to the problem was performed, by using symbolic algorithms

written in Mathematica. Floquet theory was used to derive solutions of the linearized

equation up to 17th order in e. In this way precise analytical expressions for the stability

of the system were obtained. Then, applying the Courant and Snyder transformation

to the nonlinear equation, algebraic solutions of seventh order in z and e were derived

using the method of Poincar¿e¿Lindstedt. The enormous amount of necessary computations

were performed by extensive use of symbolic programming. We developed automated

and highly modularized algorithms in order to master the problem of ordering an increasing

number of algebraic terms originating from high order perturbation theory.

Key words: Courant and Snyder transformation, Floquet theory, Mathematica, perturbation

Organisation(s)

Department of Astrophysics

Journal

Celestial Mechanics and Dynamical Astronomy: an international journal of space dynamics

Volume

93

Pages

201-228

No. of pages

27

ISSN

0923-2958

Publication date

2005

Peer reviewed

Yes

Austrian Fields of Science 2012

1010 Mathematics, 1020 Computer Sciences, 103003 Astronomy

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