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Appendix A: The UG Scaling Theorems

Nebulae and galaxies are observed to come in many shapes and sizes. This raises the question of whether the morphology observed for an astronomical object of a given size can recur in astronomical objects on an entirely different distance scale. As satellites, planets, stars, nebulae, galaxies and clusters of galaxies are assumed to be governed by the same gravitational law, it is logical to suppose that the same morphology can be observed across distance scales. According to the UG theory, for any given system consisting of a test object that rotates around a source object with an orbital radius significantly larger than the constant (approximately larger than the size of an atom), the following UG scaling theorem can be applied:


A-1: The First UG Scaling Theorem


Consider a system containing a test object with to groups of particles of mass located at moving at velocities relative to the center of the source object. The source object contains to groups of particles of mass located at moving at velocities of relative to its center.1 The UG potential energy of the test object is therefore given by



The UG potential energy between the two objects of the system scaled via , , and , where and are rational numbers2 and all velocities and remain unaltered, yields



where the angular velocity of any particle in the scaled system is reduced by a factor of ( and ).

Proof:

Denote and , where within the plane defined by the vectors and , is the component that is perpendicular to the vector , is the component that is parallel to the vector within the plane, and is the component that is perpendicular to the plane. Since only the parallel component of is contracted by the relativistic effect, the distance between the two particles, as viewed in the inertial frame of either particle, is given by

with . Thus,



or


On scales significantly larger than nuclear scale, the distance between the two objects is substantially greater than ; hence, becomes indistinguishable from . Therefore,

Equation A-1-1


Scaling , , and and using the identity , while retaining the same sets of velocities and , provides

Equation A-1-3


Thus,

as claimed in the first part of the first theorem.

Note that for the first theorem to hold true, the set of velocities and must remain unchanged.3 In the contrary case, the terms and would cause significant distortion in the scaled system. Therefore, to preserve the morphology, all the particles of the scaled system must travel at the same velocity as their counterparts in the original system. In the case of solid rotation, where a group of particles (within either object of the original system) rotate as a solid sub-object around the center of the source object with an angular momentum , the group of particles travels at a velocity of (or ). As and the angular velocity of the scaled sub-object must be scaled to in order for the velocity of the particles of the scaled system to equal the velocity of the particles of the original system. While different sections of the original system may demonstrate different angular velocities, all the angular velocities of all of the particles in the scaled system must be reduced by the same factor of , proving the second part of the first theorem.


A-2: The Second UG Scaling Theorem

The potential energy is linear with the number of particles of each type.

Therefore, scaling up the number of particles of all the groups within the test or the source objects by factors of and respectively yields



Proof:


Equation A-1-4

Thus,

Note that in contrast to the first theorem, the second theorem is valid at all distances, including the nuclear range where .





1 Note that this description is sufficiently general to describe any real astronomical system, as it is always possible to select the SHP groups in such a way that each one includes only a single particle. Thus, and for all and , and the number of particles in the system is equal to .

2 Although the values of and can mathematically be any real number, and only have a physical meaning as integers or rational numbers, since the number of particles must be given by an integer.

3 Note that in the case where all of the particles of the original system as well as the scaled system are non-relativistic, the theorem can be extended to allow for the scaling of all velocities by a fixed factor , where , , , and . In such a case, the morphology of the system will remain unaltered, while all of its angular velocities will increase by a factor of .



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