Simplification of Galactic Dynamic Equations

Abstract

Galactic dynamics is the foundation for simulating galactic structure and for solving other problems. However, the traditional dynamic equations include some unreasonable assumptions and are therefore scientifically invalid. In this paper, by introducing the following three working assumptions, we established the galactic dynamics of high precision and convenient formalism. 1. In the research of large-scale structure, the retarded potential of the gravitational field should be taken into account, and the weak field and low velocity approximation of Einstein’s field equation should be adopted. 2. The stars in a fully developed galaxy should be zero-pressure and inviscid fluid, and the equation of motion is different from that of ordinary continuum mechanics. Stars move along geodesics. 3. The structure of the galaxy is only related to the total mass density distribution. The equation of state of dark halo is different from that of ordinary luminous interstellar matter, so their trajectories are also very different. In a galaxy, the dark halo and the ordinary matter are automatically separated. The total mass density distribution can be presupposed according to the observation data, and then it can be determined by comparing the solution of the equations with the observed data. These assumptions and treatments are supported by theory and observation. The variables of the equations of simplified galactic dynamics are separated from each other, and the equations are well-posed and can be solved according to a definite procedure. The solution explains the Tully–Fisher relation. Therefore, this simplified dynamic equation system provides a more reasonable and practical framework for the further study of galactic structure, and can solve many practical problems. In addition, it is closely related to the study of dark matter halo in galaxy.

Explanation of Two Important Empirical Relations for Galaxies

“…From the above discussion, we learn that general relativity is a successful theory with profound philosophical thought and mathematical foundation, and has been verified by a large number of experiments and observations, so it is a basic theory that cannot be easily details of the mass and velocity in a spiral pattern, this will greatly increase their credibility. However, according to the author's experience, the modified schemes can hardly achieve this goal for the following reasons: Galactic spiral arms are places in which stars are dense, where the stars have the lowest velocity, the least kinetic energy, and thus the highest potential energy [25]. But this phenomenon is contradictory to the assumption that there is no dark matter, because if the dark halo is absent, then the place where matter is dense must have low potential energy and high speed.…”

“…Due to the complexity of the nonlinear partial differential equations, general relativity is rarely used in the study of galactic structures. In the paper [25], the author simplifies the galactic dynamical equations by introducing the following three working hypotheses. These three assumptions are as follows: 1) Einstein field equation is correct, and the galactic dynamics should be its weak field and low-speed approximation.…”

“…In which g µν µν =   is the scalar curvature. In the classical approximation for the total energy-momentum, tensor takes the following form [25] [39] [40] [41]:…”