A New Analytic Approach to the Sitnikov Problem
Abstract
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.
 Publication:

Celestial Mechanics and Dynamical Astronomy
 Pub Date:
 September 1992
 DOI:
 10.1007/BF00052614
 Bibcode:
 1992CeMDA..53..267H
 Keywords:

 Elliptical Orbits;
 Orbital Mechanics;
 Perturbation Theory;
 Three Body Problem;
 Canonical Forms;
 Equations Of Motion;
 Harmonic Oscillators;
 Hill Method;
 Kepler Laws;
 Astrophysics;
 Sitnikov problem;
 perturbation theory