Solving procedure for the motion of infinitesimal mass in BiER4BP

Abstract

In this paper, we present a new ansatz for solving equations of motion for the trapped orbits of the infinitesimal mass m, which is moving near the primary M3 in case of bi-elliptic restricted problem of four bodies (BiER4BP), where three primaries M1, M2, M3 are rotating around their common center of mass on elliptic orbits with hierarchical configuration M3 ≪ M2 ≪ M1. A new type of the solving procedure is implemented here to obtain the coordinates $$ \vec{r} = \;\{ x,y,z\} $$ of the infinitesimal mass m with its orbit located near the primary M3. Meanwhile, the system of equations of motion has been successfully explored with respect to the existence of analytical or semi-analytical (approximated) way for presentation of the solution. We obtain as follows: (1) the solution for coordinate x is described by the key nonlinear ordinary differential equation of fourth order at simplifying assumptions, (2) solution for coordinate y is given by the proper analytical expression, depending on coordinate x and true anomaly f, (3) the expression for coordinate z is given by the equation of Riccati-type—it means that coordinate z should be quasi-periodically oscillating close to the fixed plane $$ \{ x,y,\,0\} $$ .