A new analytic approach to the Sitnikov problem

Details A new analytic approach to the Sitnikov problem A new analytic approach to the Sitnikov problem

Sep 1992

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1992cemda..53..267h&link_type=abstract

Celestial Mechanics and Dynamical Astronomy (ISSN 0923-2958), vol. 53, no. 3, 1992, p. 267-292.

Astronomy and Astrophysics

Astronomy

19

Elliptical Orbits, Orbital Mechanics, Perturbation Theory, Three Body Problem, Canonical Forms, Equations Of Motion, Harmonic Oscillators, Hill Method, Kepler Laws

Scientific paper

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 (e between values of -0.4 and 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-cubed) is dealt with as perturbation to the linear solution. First, a transformation is introduced 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 resulting analytical solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement.

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