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Herein, the restricted circular three-body problem in homogeneous and inhomogeneous media is considered. Particular attention is paid to libration points. The conditions of their existence or non-existence in the Newtonian and post-Newtonian approximations of the general theory of relativity are derived. Several regularities, new Newtonian and relativistic effects arising due to the impact of the additional relativistic forces on bodies of gravitational fields of mediums in the differential equations of the motion of bodies are indicated. Using the previously derived equations of the motion of two bodies A1, A2 in the medium, the authors substantiated the following statements. In a homogeneous medium (density of the medium ρ = const) in the Newtonian approximation of the general theory of relativity there are ρ-libration points , 1,...,5, moving along the same circles as the Euler and Lagrangian libration points Li but with an angular velocity 0 , greater than the angular velocity ω0 of libration points Li in a vacuum. Bodies A1, A2 also move along their circles with an angular velocity 0 > w When passing from the Newtonian approximation of the general theory of relativity to the post-Newtonian approximation of the general theory of relativity, the centre of mass of two bodies, resting in a homogeneous medium in the Newtonian approximation of the general theory of relativity, must move along a cycloid. The trajectories of the bodies can not be circles, the libration points Li disappear. In the case of an inhomogeneous medium distributed, for example, spherically symmetrically, the centre of mass of two bodies, already in the Newtonian approximation of the general theory of relativity, must move along the cycloid, despite it was at rest in the void. Therefore, bodies A1, A2 must describe loops that form, figuratively speaking, a «lace», as in the case of a homogeneous medium in the post-Newtonian approximation of the general theory of relativity. The figure illustrating the situation is provided. Due to the existence of the «lace» effect, the libration point Li movements are destroyed. In the special case, when the masses of bodies A1, A2 are equal (m1 = m2), the cycloids disappear and all the ρ-libration points exist in homogeneous and inhomogeneous media in the Newtonian and post-Newtonian approximations of the general theory of relativity. Numerical estimates of the predicted patterns and effects in the Solar and other planetary systems, interstellar and intergalactic mediums are carried out. For example, displacements associated with these effects, such as the displacement of the centre of mass, can reach many billions of kilometres per revolution of the two-body system. The possible role of these regularities and effects in the theories of the evolution of planetary systems, galaxies, and their ensembles is discussed. A brief review of the studies carried out by the Belarusian scientific school on the problem of the motion of bodies in media in the general theory of relativity is given.
Herein, the restricted circular three-body problem in homogeneous and inhomogeneous media is considered. Particular attention is paid to libration points. The conditions of their existence or non-existence in the Newtonian and post-Newtonian approximations of the general theory of relativity are derived. Several regularities, new Newtonian and relativistic effects arising due to the impact of the additional relativistic forces on bodies of gravitational fields of mediums in the differential equations of the motion of bodies are indicated. Using the previously derived equations of the motion of two bodies A1, A2 in the medium, the authors substantiated the following statements. In a homogeneous medium (density of the medium ρ = const) in the Newtonian approximation of the general theory of relativity there are ρ-libration points , 1,...,5, moving along the same circles as the Euler and Lagrangian libration points Li but with an angular velocity 0 , greater than the angular velocity ω0 of libration points Li in a vacuum. Bodies A1, A2 also move along their circles with an angular velocity 0 > w When passing from the Newtonian approximation of the general theory of relativity to the post-Newtonian approximation of the general theory of relativity, the centre of mass of two bodies, resting in a homogeneous medium in the Newtonian approximation of the general theory of relativity, must move along a cycloid. The trajectories of the bodies can not be circles, the libration points Li disappear. In the case of an inhomogeneous medium distributed, for example, spherically symmetrically, the centre of mass of two bodies, already in the Newtonian approximation of the general theory of relativity, must move along the cycloid, despite it was at rest in the void. Therefore, bodies A1, A2 must describe loops that form, figuratively speaking, a «lace», as in the case of a homogeneous medium in the post-Newtonian approximation of the general theory of relativity. The figure illustrating the situation is provided. Due to the existence of the «lace» effect, the libration point Li movements are destroyed. In the special case, when the masses of bodies A1, A2 are equal (m1 = m2), the cycloids disappear and all the ρ-libration points exist in homogeneous and inhomogeneous media in the Newtonian and post-Newtonian approximations of the general theory of relativity. Numerical estimates of the predicted patterns and effects in the Solar and other planetary systems, interstellar and intergalactic mediums are carried out. For example, displacements associated with these effects, such as the displacement of the centre of mass, can reach many billions of kilometres per revolution of the two-body system. The possible role of these regularities and effects in the theories of the evolution of planetary systems, galaxies, and their ensembles is discussed. A brief review of the studies carried out by the Belarusian scientific school on the problem of the motion of bodies in media in the general theory of relativity is given.
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.
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