SITNIKOV FIVE BODY PROBLEM FORMING SQUARE CONFIGURATION UNDER PERTURBATION OF PHOTOGRAVITATION: AN ELLIPTIC CASE

Abstract

This paper deals with averaging the equation of motion of the elliptic Sitnikov restricted five-body problem when all the primaries as source of same radiation pressure. We assumed that the primaries are at the vertices of a square so the distances of the primaries from centre of mass are time depending. Next we have developed averaged equation of motion by applying the Van der Pol transformation and averaging technique of Guckenheimer and Holmes (Nonlinear Oscillations, Dynamical System Bifurcations of Vector Fields, Springer, Berlin (1983). In addition to the resonance criterion at the 3/2 commensurability we have chosen, is the angular velocity of the coordinate system. Furthermore we have linearized the equation of motion to obtain the Hill’s type equation, then by
using the Floquet’s theory (Jose and Saletan 1998) to find the approximate solution. Finally the solutions are obtained by the Poincare surface of sections. It is shown that chaotic region emerging from the destroyed invariant tori, can easily be seen for certain eccentricities.