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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.
This paper investigates the degree of influence of the gravitational field of dark matter on the laws of motion of bodies in a medium in a restricted two-body problem, when a test body (planet, asteroid, artificial satellite of a star, in particular, the Sun, etc.) has its own rotation, i. e. own angular momentum impulse. The study was carried out within the framework of the post-Newtonian approximation of the general theory of relativity. In accordance with the latest experimental data, hypotheses about the average densities of dark matter ρD.M. and visible matter ρvis. in planetary systems are accepted. In particular, in the Solar system the following is accepted: ρD.M ≈ 2,8 · 10–19 g · cm–3, ρvis ≈ 3 · 10–20 g · cm–3 and ρΣ = ρvis + ρD.M ≈ 3,1 · 10–19 g · cm–3. In the post-Newtonian approximation of the general theory of relativity, the equation for the trajectory of a rotating test body with respect to ρΣ is derived, and working formulas are obtained that give the laws of secular changes in the direction of the vector of the proper angular momentum impulse of the test body and the modulus of this vector. It is shown that accounting ρD.M changes the magnitude of the periastron shift. For example, in the Solar System when taking into account ρvis, all the planets except Pluto have a directly shifted perihelion in the post-Newtonian approximation of the general theory of relativity. When taking into account ρΣ the planets from Mercury to Saturn included, they have a direct shift of perihelion, and Uranus, Neptune, Pluto have the reverse (against the planets in orbit). There is also a secular change in the eccentricity of the orbit. The formula is derived that can be used to calculate the secular deviation of the translational motion of a rotating body from motion in a plane. Accounting ρΣ enhances deviation. It is emphasized that all the noted effects for planetary systems in the vicinity of neutron stars, radio pulsars and other dense objects can be many orders of magnitude greater than in the solar system.
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