Dynamics of a small planetoid in Newtonian gravity field of Lagrangian configuration of three primaries

Abstract

Novel method for semi-analytical solving of equations of a trapped dynamics for a planetoid m 4 close to the plane of mutual motion of main bodies around each other (in case of a special type of Bi-Elliptic Restricted 4-Bodies Problem) is presented. We consider here three primaries m 1, m 2, m 3 orbiting around their center of mass on elliptic orbits which are permanently forming Lagrangian configuration of an equilateral triangle. Our aim is to obtain approximate coordinates of quasi-planar trajectory of the infinitesimal planetoid m 4, when the primaries have masses equal to 1/3 (not stable configuration of the Lagrange solution). Results are as follows: (1) equations for coordinates {x¯,y¯} are described by system of coupled second-order ODEs with respect to true anomaly f and (2) expression for z¯ stems from solving second-order Riccati ordinary differential equation that determines the quasi-periodical oscillations of planetoid m 4 not far from invariant plane {x¯,y¯,0} .