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.
In this paper, a material system consisting of two spherically symmetric bodies of comparable masses located inside a gas-dust ball with a spherically symmetric distribution of the density of the medium in it is considered. After choosing the corresponding energy-momentum tensor from the Einstein field equations using the Einstein-Infeld approximation procedure, the metric of the corresponding space-time, the gravitational field created by the «two-body – medium» system are found, and then the equations of motion of the bodies and their center of mass are obtained in Newton’s and post-Newtonian approximations of the general theory of relativity. It is proved that in the case of the indicated density of the medium, the following effect should exist already in the Newtonian approximation. The center of mass of two bodies shifts at a variable speed, although it was at rest in the void. This situation is a consequence of the fact that the two-body-medium system is not closed. For the first time, formulas for calculating the displacement value, which is proportional to the density of the medium in the center of the gas-dust ball and the 5th degree of the distance between the bodies, are derived. Therefore, at large distances between bodies, their center of mass has large displacements (it can reach several million kilometers per revolution of bodies around their center of mass). If the masses of the bodies are equal, their center of mass is at rest if it is at rest in the void.
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.
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.
The motion equations for a system of two bodies moving in a medium are derived in the Cartesian coordinate system in the Newtonian theory. The coordinate system is barycentric, that is, the center of mass of the two-body system is immobile. Using the Einstein – Infeld approximation procedure, the gravitational field created by the “two bodies – medium” system was found from the Einstein field equations, and then the equations of motion of the bodies in this field were obtained.It is shown that in the post-Newtonian approximation of the general theory of relativity, the center of mass of two bodies moving in a gas – dust rarefied medium of constant density, determined by analogy with the Newtonian center of mass, is displaced along the cycloid, although in the Newtonian approximation it is stationary, i.e. the movement along the cycloid occurs with respect to the barycentric Newtonian fixed reference frame. Numerical estimates are given for the magnitude of this displacement. Given a popular value of the medium density ρ = 10–21 g·cm–3 its order can reach 106 km per one rotation of two bodies around their center of mass. In the case of the equality of masses of the bodies, their relativistic center of mass, like their Newtonian center of mass, is immobile.It has been hypothesized that for any elliptical orbits of two bodies and an inhomogeneous distribution of the gas – dust medium the qualitative picture of motion of the relativistic center of mass of the two bodies will not change.
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