Modeling the Evolution of a Cluster of Gravitating Bodies Taking Into Account Their Absolutely Inelastic Collisions

Abstract

Numerical simulation of evolution of a cluster of a finite number of gravitating bodies interacting only by their intrinsic gravity has been carried out. The goal of the study was to reveal the main characteristic phases of the spatial distribution of material bodies constituting the cluster. In solving the problem, the possibility of interbody collisions was taken into account, the collisions being assumed to be absolutely inelastic. Forces external to the body cluster under consideration were ignored. Among all the internal force factors acting within the cluster, only the gravitational interaction was taken into account. The total mass of all the gravitating bodies of the cluster was assumed to remain constant during the entire evolution. The Cauchy problem with natural initial conditions was considered. To check the process of solution, the so-called rotation curve was used which represents the current radial distribution of orbital velocities of the cluster bodies. The numerical analysis showed time variations of the model cluster rotation curve and, particularly, the fact that the rotation curve horizontal section is only a short moment in evolution of the gravitating bodies cluster. The results obtained within the scope of classical mechanics show that it is possible to represent all the rotation curve variations for the observed galaxies without appealing to the hypothesis of non-observable gravitating "dark matter".

On the effect of the central body small deformations on its satellite trajectory in the problem of the two‐body gravitational interaction

“…Diligently following this principle, we succeeded, as we hope, in solving several problems with history, namely, in revealing the sources of Chandler's wobble of the Earth's rotation pole [15], in answering the question of why nobody succeeded in refining the gravitation constant [16], in proving the real existence of the gravitational dipole and formulating the definition of the gravitating mass [17], in explaining physical nature of the "plateau" in the galaxy rotation curve without using such a metaphysical concept as "dark matter" [18].…”

On the effect of the central body small deformations on its satellite trajectory in the problem of the two-body gravitational interaction

“…Here J a , J b , J c are the ellipsoid's principal moments of inertia, G is the scaledimension factor (gravitational constant), m A is the ellipsoid's gravitating mass, r, ϕ, λ are the spherical coordinates of material point B, i.e., distance, latitude and longitude, respectively. Let us make some assumptions that will help simplify expression (11) for the potential of central body A which is a three-axis ellipsoid 1) ϕptq " 0 (trajectory B lies in plane Oxy) 2) a ą b " c (ellipsoid of revolution) Among a great variety of solutions at λ 0 " 0, the following value of the deformation coefficient k was chosen: k « 1.00044 (18) at which the so-called "anomalous" shift of the Mercury perihelion ∆ψ " 0.1 2 takes place. This means that the Sun's oblateness in the ecliptic plane is about 1 {1516, namely, semi-major axis a exceeds semi-axis b by « 459 km.…”