Abstract
The purpose of this paper is to show, on the basis of Newtonian mechanics (in Euclidean space), that the core disks of spiral galaxies (the central disks in galactic cores that are perpendicular to the axes of rotation) rotate in the same fashion as a phonograph turntable, if the mass densities in the cores of such galaxies remain uniform. On the basis of the hypothesis of uniform mass density in the core, it is then shown that the density of mass in the shell (the entire domain outside of the core) must remain inversely proportional to the square of radial distance from the axis of rotation and that the angular velocity in the shell annulus (annulus in the shell that contains the spiral forms) is inversely proportional to radial distance, or that the circumferential velocity on the shell disk is independent of radial distance from the core axis. The equation of motion for the shell disk is then obtained and it is concluded that the spiral shaped lanes are not trajectories. But it is shown that any bar-shaped feature crossing the shell annulus and core disk, collinear with the core centre, will become distorted, due to the above angular velocity distribution in the shell disk, assuming the form of two, symmetrically disposed, Archimedean spirals, while the portion of the bar inside the core remains undistorted and merely rotates.
Highlights
Aims and Scope The purpose of this paper is to show that, if the mass density is assumed uniform within the core of a spiral galaxy, by Newtonian Mechanics and Euclidean Geometry, the mass density in the shell must vary as the inverse square of radial distance, while the angular velocity in this domain remains inversely proportional to radial distance, if the core-shell in
From Equations (3.16) and (3.18), (3.21) and (3.22), we arrive at the following theorem which is concerned with the conservation of angular momentum: Theorem 5.1 In a hypothetical body of mass-points acted upon by gravitation and inertia alone, the mass density of which remains uniform within a central core, the areal velocity, as observed from the rotating core disk which is perpendicular to the axis of rotation, equals the areal velocity, as measured in an external fixed frame of reference, from which is subtracted the quantity ρ 2Ω, while the operator d/dt, applied to vectors, is changed to ∂ ∂t
From Equations (4.23) and (5.4), regarding the conservation of energy, we have: Theorem 5.2 In a hypothetical body of mass-points acted upon by gravitation and inertia alone, the mass density of which remains uniform within a central core, the kinetic energy per unit mass, as observed from the rotating core disk, perpendicular to the axis of rotation equals the kinetic energy per unit mass from which must be subtracted the quantity 1 ρ 2Ω2 when the operator d/dt applied to vectors is replaced by ∂ ∂t
Summary
These distributions of matter, in turn, will allow the equations of movement for the core and shell disks (perpendicular to the axis of rotation) to be derived and these equations do not lead to spiral trajectories any radial line of material points crossing the shell and passing through the core centre will be distorted, it will be shown, into a symmetric pair of spirals of Archimedes. The results of McGaugh, Rubin and Blok [1] for the edge-on galaxy F568-3 show that the circumferential velocity is proportional to radial distance from the axis of rotation out to a radius of about 14 seconds of arc This marks the boundary to the core where an abrupt change occurs to a constant velocity of circulation of about 110 k∙ms−1. There is a small region near the axis of rotation marked by substantial departures from linearity [4]-[6]
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