A NEW ANALYTIC APPROACH TO THE SITNIKOV PROBLEM

A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (Z(max) = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (-0.4 < e < 0.4). First solutions are searched for the limiting...

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Bibliographic Details
Published in:Celestial mechanics and dynamical astronomy 1992-09, Vol.53 (3), p.267-292
Main Author: Hagel, J.
Format: Article
Language:English
Subjects:
Astronomy
Astronomy & Astrophysics
Celestial mechanics (including n-body problems)
Earth, ocean, space
Exact sciences and technology
Fundamental astronomy
Fundamental astronomy and astrophysics. Instrumentation, techniques, and astronomical observations
Mathematics
Mathematics, Interdisciplinary Applications
Physical Sciences
Science & Technology
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Summary:A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (Z(max) = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (-0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z(3)) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement.
ISSN:0923-2958
1572-9478
DOI:10.1007/BF00052614