Motion stability of the system of two bodies and their mass center in an inhomogeneous medium

Abstract

Within the framework of Newtonian celestial mechanics, a material system is considered. It consists of two spherically symmetrical bodies of comparable masses moving inside a gas dust ball with a spherically symmetrical density distribution of the medium in it. Problems are formulated and solved. They give an answer to the degree of influence of the gravitational field of an inhomogeneous medium on the motion stability of bodies and their mass center relative to the coordinates of the bodies, the coordinates of their mass center, as well as on the orbital stability according to Lyapunov. Additionally, the problems of the motion stability of bodies in the sense of Lagrange and Poisson are considered. It is proved that the gravitational field of a spherically symmetrically distributed medium transforms the considered motions, which are stable in vacuum, into unstable ones in the sense of Lagrange, Poisson, Lyapunov. Some numerical estimates related to instabilities are presented. They show that for popular pairs of stars and pairs of galaxies in an inhomogeneous medium, their additional displacements of the order of many millions of kilometers arise. When dark matter is taken into account, the displacements should not be an order of magnitude greater than the last estimate. The noted instabilities are a consequence of a secular displacement along the cycloid or deformed cycloid of the mass center of the system of two bodies and the absence of a barycentric coordinate system when taking into account the influence of the gravitational field of a spherically symmetrically distributed medium on the motion of bodies (the considered material system is not closed). It is proved that for this system, circular and elliptical orbits of bodies cannot exist. Instead of these orbits, we have “turns” shown in the figure given in the article. In planetary systems (such as the Solar System) immersed into an inhomogeneous medium, the displacements of the mass centers are negligible and therefore we can assume that circular and elliptical orbits can practically exist.