Axial oscillation of a planetoid in Restricted Three Body Problem : The circular Sitnikov problem

Abstract

Two equally massive primaries are assumed to be moving in circular orbits in Cartesian x-y plane. A planetoid is assumed to be on the z-axis. This is a particular case of the restricted three body problem with mass ratio J,l.=1I2, known as the circular Sitnikov problem. Motion of the planetoid is calculated using LindStedtPoincare' perturbation and Green's function method. It is found that the planetoid oscillates nonlinearly along the z-axis. We present analytic solutions up to 2nd order of approximation and compare the solutions with earlier results of other authors. A solution of the exact problem is also discussed.