Sitnikov-type solution for the motion of infinitesimal mass in BiER4BP

Abstract

In this paper, we present a new ansatz for approximated solving equations of motion of the infinitesimal mass m in case of bi-elliptic restricted problem of four bodies (BiER4BP) (where three primaries M1, M2, M3 are rotating around their common centre 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 of the infinitesimal mass m. Meanwhile, the system of equations of motion has been successfully explored with respect to the existence of semi-analytical (approximated) way for presentation of the solution. We obtain as follows: 1) the solution for coordinates {x, y} = {0, 0} is approximately satisfied both the first and second equations of motion if we take into consideration assumption {M3, M2} << M1, 2) the expression for coordinate z(f) is given by the equation of 2-nd order, which describes Sitnikov-type approximated solution. It means that test particle is moving along the z-axis, outward the common barycenter of the system (but perpendicular to the plane of the mutual rotation of all the primaries).