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Chapter III: The UG equations of Astronomical Objects

Section III-1: The UG Morphology Model- The Non-Relativistic Approach

The goal of this chapter is to develop the equations and tools required for the application of the UG theory to the quantitative analysis of large astronomical objects, such as galaxies and nebulae. For convenience, the reader should keep in mind that throughout this chapter, the term ďgalaxyĒ will be broadened to refer to galaxies as well as all types of nebulae, and that the tools and equations developed here are general, and are not limited to galaxies and nebulae. The same tools and equations will be applied in later chapters for the analysis of cosmic voids, planetary rings, as well as to additional astronomical phenomena.

The UG theory is rooted in the assumption that the dominant superheavy particles in a galaxy are produced in areas of extremely high pressure and temperature, and are therefore most likely to be situated at the center, or in orbits at close proximity to the center of the galactic bulge. Each of these superheavy particles can be viewed as part of a point-like group, where a ďgroupĒ is defined as either a single SHP, or a tight group of SHPs of the same mass that share the same orbit, location and velocity.1 Note that the superheavy particles within a single group, or within different SHP groups, are prevented from collapsing into each other by the UG rejection zones generated between them.2

Although there is no reason to assume that a galaxy center should contain only a single type of SHP, it will be demonstrated via the UG theory that even simple configurations can explain a large portion of the observed galactic morphologies. Figure 3-1 provides a schematic of the two simplest interactions possible between an object composed of ordinary particles located within the galaxy plane at time and superheavy particles arranged in either a single or binary grouping. Figure 3-1a demonstrates the interaction between the object and a single SHP group orbiting in circular motion of radius and constant angular velocity around the center of the galaxy. Figure 3-1b presents the interaction between the object and a binary grouping of identical superheavy particles in circular orbit of radius around the center of the galaxy.3 It will be demonstrated that the morphology of a galaxy is determined mainly by the velocities and orbital radii of the dominant SHP groups, while the size of the galaxy is determined mainly by the mass of the dominant SHP type, and to a lesser degree by their velocity.4

Figure 3-1: Provides a schematic of the two simplest interactions possible between an object composed of ordinary particles located within the galaxy plane at time (represented by the red point) and SHPs arranged in either a single or binary grouping. Figure 3-1a demonstrates the interaction between the object and a single group orbiting in circular motion of radius and constant speed around the center of the galaxy. Figure 3-1b presents the interaction between the object and binary SHP groups in circular motion of radius A around the galactic center. The small open circles (cyan) denote the locations of the SHP groups at time , while the full circles (blue) provide the given SHP locations at time , where provides the time required for the gravitational signal to travel from the SHP group to the remote particle at distance . This delay will be demonstrated to play an important role in creating the spiral structure.

For simplicity, the provided analysis of galactic morphology will be guided by the following assumptions:

  1. The unified gravitational formula of the UG potential energy between two interacting fundamental particles moving at non-relativistic speeds relative to the center of the galaxy is given by equation 2-1-1,

  1. A free-falling frame of reference that covers the entire spacetime of a galaxy during the local time of the entire period of observation5 can be regarded as an inertial frame of reference. This assumption is in line with the conclusion of the currently accepted cosmological model, which states that the effects of general relativity are negligible on the scale of galaxies and galactic clusters.6 7 A convenient choice for an inertial frame is to set the origin of the frame at the center of the galaxy. For the case of a flat galactic disk, the axis of the frame can be chosen to be perpendicular to the galactic plane. This specific inertial frame of reference will be referred to as frame .

  2. On their journey from the galaxy to the observer on Earth, photons pass through the gravitational fields of other galaxies. Therefore, the trajectories of the photons may be bent by gravitation, causing a distortion in the perceived morphology of the galaxy via gravitational lensing. The present discussion will be limited to cases where gravitational lensing is too small to significantly affect the perceived morphology of the galaxy.

Radiation redshift, due either to the speed of the observed galaxy relative to the observer, or due to gravitation, affects the wavelength and frequency of galactic radiation, but not the perceived morphology of the galaxy. In addition, the second and third assumptions allow the analysis of galactic morphology to ignore possible distortions due to external gravitational effects on the galaxy, on the observer, or on the spacetime anywhere along the path of the radiation emitted by the galaxy and intercepted by the observer, and to instead rely exclusively on the UG theory and on special relativity within the limits of the given galaxy.8

To further simplify the analysis, the discussion will be restricted to galaxies with the following properties:

  1. The galaxy center contains either a single SHP group or a binary grouping in circular orbit of radius around the galaxy center, where the group(s) consist of a single type of superheavy particles of identical mass.

  2. The observer views the galaxy face-on.

  3. Due to its rotation, the galaxy is confined to a flat disk with negligible depth, designated as the plane of the inertial frame .

  4. Tidal forces and other external influences of nearby galaxies are relatively small and can be ignored.

  5. The diameter of the galaxy is very small relative to its distance from the observer.

  6. The orbiting objects are composed of ordinary matter, and their size is negligible relative to their radius of orbit around the galaxy center.

  7. The orbital radii of the objects are significantly greater than the distance between the SHP group(s) and the center of the galaxy.

The above limitations serve to reduce mathematical complexity, and to focus the discussion on the most important factors that influence galaxy morphology. There is nothing preventing the use of the same tools developed here for the case of more complex galaxies that may involve multiple SHP types contained in a number of groups, or for galaxies that are viewed at different orientations (other than face-on observation), or for cases where the galaxies are influenced by external galaxies, or where the orbiting matter is composed of SHPs as well as ordinary particles. The same types of tools can also be used for the analysis of stellar and planetary systems, and will be applied in Chapter V for the more complex analysis of Saturnís rings.

In principle, the effective distance photons travel on their way from the emitting atom within the galaxy toward the observer depends on the location of the galaxy and on its relative velocity compared to the observer as well as on the velocity and the location of the emitting atom within the galaxy.9 Given assumptions 5, 6 and 8 we are assured that the velocity of the particle that emitted an observed photon is perpendicular to the photonís path. Therefore, regardless of the position or the velocity of the emitting atoms within the galaxy disk at the time of the photon emission, the effective distance between the emitting atoms and the observer and the time it take the photons to reach the observer are virtually identical and depend only on the distance and velocity of the galaxy relative to the observer.10 Furthermore, assumptions 5 and 8 and the observation that the object within the galactic disk move at non-relativistic velocities relative to the galactic center, assure us that for all practical purposes, any two photons that are detected simultaneously by the observer have traveled the same distance and the same amount of time regardless of the location from which they were emitted within the given galaxy.11

Galaxy morphology is typically determined by the spatial distribution of the radiation emitted by ordinary matter within the galaxy that is detected by the observer. The observerís perception of galactic shape is strongly affected by the contrast between areas of high radiation (and therefore, high brightness) and areas of low radiation. The amount of radiation emitted by any given region of a galaxy is related to the density of ordinary matter within this region. The density of ordinary matter, and thus the radiation level, is expected to be higher at locations where the overall energy of orbiting ordinary matter has a local minimum. Thus, identifying the local minima, predominantly those that produce sharp brightness contrast to their background, will provide the theoretical morphology of the galaxy.

The task at hand is to use the UG equation to analyze the energy patterns formed by the combined effect of superheavy particles (either stationary or rotating) at the central core of a galaxy and the surrounding ordinary matter; In particular, to identify the minimum points, contours and arcs, and how they change over time. Establishing this task will facilitate in confirming the initial hypothesis, that in all or most cases, the observed shape and properties of a galaxy can be explained by the configuration of its dominant superheavy particles.

The following equations, developed to identify the local minima, will initially use the symbols for the speed of light and for the speed of gravitation. The assumption that the propagation speed of gravity is equal to the speed of light will only be made at a later stage. As gravity propagates at a finite speed , the gravitational signal requires time to reach the orbiting object. Consider an object with an orbital radius of . The gravitational signal (or graviton) detected by the orbiting object at at time in frame S was actually emitted by a superheavy particle at the S position at an earlier time , where the emittance time and location are related by . The period presents a delay that increases with the distance .

The velocity of matter located in the galaxy halo is typically non-relativistic at about a few hundred . For non-relativistic SHP groups with velocities , the entire calculation can take place in the rest frame of the center of the galaxy. In this frame, the potential energy of an object composed of ordinary particles located in the galaxy halo is given approximately by

Equation 3-1-1

where denotes the location of the object at time , and and represent the locations of the two groups of superheavy particles at the time they emitted the gravitational signals intercepted by the object at time . For simplicty, the total mass of ordinary matter in the galaxy, given by , is assumed to be homogeneously distributed around the galaxy center within a radius . The value is a positive number defined as the ratio between the number of SHPs in group 2 and group 1. Setting for the case of a single SHP group rotating around the center of the galaxy, and for the case of identical binary groups, will allow the same set of equations to cover both scenarios. In the case of binary groups where , both groups are assumed to follow a circular orbit of radius around the galaxy center with the same constant speed . It is further assumed that the two SHP groups and the galaxy center are co-linear,12 The last term of 3-1-1, which represents the interaction between the ordinary matter of the galaxy and the ordinary matter of the orbiting object includes the variable , defined as when , and as when .

The orbiting stars and interstellar gas in a galaxy are expected to gravitate strongly toward regions of lower potential energy, creating areas of increased density in their vicinity. As , the interaction between the galaxyís ordinary matter and the object provides a relatively smooth and slow-changing potential energy curve compared with the rapid oscillations of the potential energy of the object due to its interaction with the galaxyís SHPs. Therefore, as demonstrated in figure 3-2, it is likely that even if the contribution of a galaxyís ordinary matter to the potential energy of an object is significantly larger than the overall contribution of the galaxyís superheavy particles, the locations of the potential energy minima of the interaction are determined almost entirely by the SHP masses, velocities and locations, while the influence of the galaxyís ordinary matter on the minima locations is almost negligible.13

Figure 3-2: Presents the potential energy of a single ordinary particle of mass as a function of its distance from the galaxy center. The red curve provides the potential energy due to the influence of two stationary SHP groups, each containing 54 SHPs of mass , located at a distance of from the galaxy center. The two groups are assumed to be positioned in a linear alignment with the galactic center, on either side of the center. The purple curve provides the potential energy of the particle due to of ordinary matter distributed homogeneously within a sphere of centered around the galaxy center. The combined effect of both SHP groups and of the galaxyís ordinary matter is demonstrated by the blue curve. The black vertical lines indicate the deepest minima contours between , demonstrating that the minima occur almost precisely at the same distances, whether or not the effect of the ordinary matter is included. Ordinary matter is expected to be concentrated in the vicinity of the local minima, with higher densities at the deeper minima. Although the potential energy in this example is dominated by the objectís interaction with ordinary matter (at least at ), the galactic ordinary matter is demonstrated to have little influence on the positions of substantial minima. Note, however, that shallow minima contributed by the two SHP groups may become washed out by the contribution of the galactic ordinary matter. For example, the two shallow minima between and , and the two minima between and 14 in the red curve do not remain minima after the inclusion of the ordinary matter contribution, as shown by the blue curve.

Therefore, locations of high density matter would not be notably affected by the Newtonian term in equation 3-1-1, and can consequently be found in the minima of the following equation:

Equation 3 - 1 - 2

First, the locations at which the gravitational signal was emitted and must be calculated. Due to the finite speed of gravitation, the object located at at time in the inertial frame simultaneously receives the UG gravitational signals that were emitted by the two respective groups at earlier times and by the two respective groups, which were located at and at the time of the signal emissions. Therefore,

Equation 3 -1Ė3a


Equation 3Ė1-3b

Therefore and are explicit functions of , and their dependency on ,, and is only through . Similarly, and are explicit functions of , and thus are indirectly dependent on , , and .

The minus signs preceding the terms and in the equation for the second group are due to the requirement that at any given time, the two groups and their common center of mass are drawn along a straight line, and are thus half a cycle apart. Since the distances between the orbiting object and the two groups in the inertial frame are usually different, the amount of time required for the gravitational signal to propagate from each group to the object will vary, and frequently . However, as the maximum difference between the two groups is , .

The values and are essential for conducting successful calculations of galactic shapes and properties. Unfortunately, finding a direct analytical solution for equations 3-1-3a and 3-1-3b is not that simple. Instead, it is more practical to use an iterative approach, where the first order of titration is given by

Equation 3-1-4

Followed by additional titrations,

and for the second group,

At the final stage, the emission times in frame are assigned the values and . For the following examples, it is assumed that in the case of , a single iteration can provide sufficiently accurate results. Therefore,

Equation 3 -1Ė5


Equation 3-1-6

Given the small value of the constant (), at galactic distance ranges the exponent terms and can be replaced by . As we are looking for the minima of equation 3-1-1, the highest density of matter is expected to be concentrated in the vicinity of the deepest minima, which occur at the lowest points of the following equation:

Equation 3-1-7

Note that at the limit , is always greater than or equal to zero. With and, the lowest minima, and therefore the highest density distribution, will occur in the vicinity of the locations that comply with , where . This will happen at locations where both cosine terms are simultaneously equal to . Note that at higher levels of potential energy, the discussion can be extended to areas of lower density by simply allowing a range of higher values for . This method of dividing the range of possible potential energies into slices of values will be used in the next few chapters to provide the two dimensional contour maps (or isophotes) of the potential energy profile of a galaxy, and to demonstrate the resulting features, such as rings or spiral arms.

III-2: The UG Morphology Model- The Relativistic Approach

The dynamic calculations required for the case of a group of superheavy particles moving at relativistic velocity relative to the inertial frame of reference are somewhat more complex. Force and potential energy are not invariant under Lorentz transformations, and may change form when viewed in different inertial frames moving at relativistic velocities relative to each other. Therefore, a force law must be defined in a specific inertial frame. Prior to Einsteinís special theory of relativity, Coulombís law was known to accurately provide the electromagnetic force applied on a test charge moving at any constant velocity only when the source charge is at rest. When the source charge is not stationary relative to the observer, it generates a magnetic field that applies an additional force on the test particle, resulting in an overall force which may or may not be a central force. Einstein showed that on the basis of Coulombís force and special relativity alone, one can generate a quantitative description of electric and magnetic interactions between charges moving with arbitrary constant velocities, and that what appears as a purely magnetic field, or as a combination of an electric and a magnetic field, when viewed in one coordinate system may be simply an electric (Coulomb) field when viewed in another coordinate system. The key to developing all of the electromagnetic kinematic and dynamic quantities is to use Coulombís law only when the calculations are performed in the inertial rest frame of the source charge. Thus, when the source charge moves in respect to a given frame, the procedure entails the following three steps:

Step 1: The kinematic and dynamic parameters of the test particle must first be transformed to frame , in which the source charge is at rest.

Step 2: Apply the Coulomb force (or potential energy) equation to the test particle.

Step 3: Transform the particlesí parameters back from frame to frame .

All transformations are done via the Lorentz transformations (French, 1968). The realization that a force is not invariant under Lorentz transformations, and that the same force that appears as a central force from the point of view of one inertial frame may appear as a non-central force and may be described by a different equation form when viewed in a different inertial frame, led to the language used in the second UG postulate given in Chapter II and repeated below:

UG Postulate II:

The Unified Gravitational force is a force between a pair of particles. When viewed at an inertial rest frame of one of the interacting particles (the source particle), the unified gravitational force applied on the second particle (the test particle) is predominantly a central and conserving force that depends exclusively on the absolute distance between the particles and on the product of their masses.

This basic postulation further led, in conjunction with three additional postulates (as well as the application of the principle of Occamís Razor), to a family of possible gravitational equations, where equation 2-1-1 was selected as the simplest potential energy equation that complies with the given postulates. An additional assumption was made at the beginning of this chapter, stating that a free-falling frame of reference that covers the entire spacetime of a given galaxy during the local time of the observation can be regarded as an inertial frame throughout the galaxy, with the exception of the immediate vicinity of collapsing stars.Under the provided postulates and assumptions, the galaxy can be correctly analyzed via equation 2-1-5 and special relativity. As this equation is only valid in the rest frame of a source particle, calculating the UG force applied to a moving test particle requires a procedure similar to the three-step procedure described above for the case of the Coulomb force. However, there is an important difference: the Coulomb force depends on the particle charges, which are the same in any frame of reference. The UG force equation, however, depends on the particle masses, which are not invariant under the Lorentz transformations.

Therefore, the mass of the test particle in equation 2-1-5 provides the mass as viewed by the source particle, which is equal to , where is the rest mass of the test particle and . The same procedure can be used to calculate the potential energy of the test particle via equation 2-1-1.

Therefore, in order to assess the UG effect applied by either SHP group on the object, the calculation must take place in the inertial frame where the group is momentarily at rest, denoted by for the first group and for the second group. The relativistic velocity of the object relative to the SHP groups can theoretically result from either the relativistic velocities of either one of the SHP groups and , or from the relativistic velocity of the object relative to the center of the galaxy, or from both. However, matter in the galaxy disk and halo has typically been observed to travel at non-relativistic velocities between and relative to the galactic center. Therefore, to the extent that relativistic effects occur in galaxies, they must be attributed to the relativistic velocities of their SHP groups. The speed of the object thus becomes negligible compared with the relativistic speed of the groups. In such cases, the relativistic velocity between a given SHP group and a given objectcan be regarded as equal to the velocity of the group. 14

As mentioned above, the UG force or potential energy equations 2-1-5 and 2-1-1 applied on an orbiting object are not invariant under Lorentz transformation, and are assumed to be valid exclusively in the inertial frame of the source at rest. For the relativistic case of a galaxy consisting of two SHP groups, the UG equation of each group must therefore be calculated in different inertial frames; specifically, in inertial frame , where the first group is momentarily at rest at the time , and in inertial frame , where the second group is momentarily at rest at time . As a reminder, and are the time at which the respective groups emitted the gravitational signals, which were simultaneously intercepted by the orbiting object at time . Time and are given by equation 3-1-5 and 3-1-6 respectively.

Calculating the potential energy at any arbitrary point in the inertial frame requires the Lorentz transformation of coordinates from frame to frames and , where the UG equations 2-1-1 or 2-1-5 can be applied. The results are then transformed back to the frame and combined to provide the overall potential energy or force.

Starting with group 1, the first task is to calculate the magnitude and direction of its velocity at time . As assumed above, the coordinates of the first group within the frame at the emission time are given by

Equation 3-1-8

and and

Consequently, the velocity of the SHP group within the frame is given by the derivative of equation 3-1-8,

Equation 3-1-9


Equation 3-1-10

Note that the angular velocity may be positive for counterclockwise rotation and negative for clockwise rotation.

As discussed, applying the UG equation 2-1-1 requires the use of the inertial frame where the source mass, in this case group 1, is momentarily at rest. Note that as group 1 travels in circular motion around the center of the galaxy, it is accelerating. Therefore, the group will remain at rest in frame for only an infinitesimal period of time.

The distance in frame between the frame location of the orbiting object () at the interception time , and the frame location of group 1 at the time of emission is given by

Equation 3-1-11

where , and are given by equation 3-1-3a. Since group 1 is at rest in frame at time , travels at a velocity given by equation 3-1-9 and 3-1-10 relative to the inertial frame . Therefore, the distance between the orbiting object and group 1 in the inertial frame is almost always contracted. In calculating the distance , it is more convenient to separately calculate the parallel and vertical components of the velocity of group 1 within the rotation plane .

The component of the distance in the frame that is parallel to the velocity of group 1 at time is given by

where the value of , as given by equation 3-1-10. Additionally, in circular motion the velocity of the SHP group is perpendicular to the vector connecting the rotating group to the center of the circle. Therefore, . Consequently,

Equation 3-1-12

Since is defined to be perpendicular to , and both are contained in the galaxy plane ,

Equation 3-1-13

Since the distance is perpendicular to the velocity of group 1, it is not altered by the Lorentz transformation from the frame to the frame of reference. However, the parallel component is contracted via division by , where . In addition, the velocity is the relative velocity between group 1 and the object, and . Therefore,

Equation 3-1-14


Using equations 3-1-13 and 3-1-14

Equation 3-1-15

Substituting equations 3-1-11 and 3-1-12 into equation 3-1-15,

Equation 3-1-16

where the term is the same as in the non-relativistic case, and the term provides the relativistic distance contraction. Applying equation 3-1-8, and the fact that ,

Equation 3-1-17a

In the case of a binary grouping, a phase shift of must be added to the operand of each cosine and sine term of the second group and the time should be replaced by to provide

Equation 3-1-17b

When only the contribution of group 1 is taken into account, the objectís energy as viewed at the inertial frame (at a point denoted by the coordinates ) is given by

Equation 3-1-18a

where the mass of the objectís ordinary matter (in the inertial frame) can be substituted by the product of its rest mass and .

At time the second group travels at a velocity of relative to the inertial frame . Therefore, the objectís velocity relative to group 2 is (and ). When only the contribution of group 2 is taken into consideration, a similar analysis of the energy of an object in the inertial frame of group 2 is provided by

Equation 3-1-18b

To find the overall energy of the object in the frame, the energy due to group 1 in frame and the energy due to group 2 in frame (including the objectís rest energy) must first be identified and transformed via Lorentz transformations, i.e. = , where or . The total energy can then be derived by adding and , and since the energy derived from the objectís rest mass was counted twice (once in either inertial frame), it must also be subtracted once. In addition, as stated above, the velocity of the object in frame (typically less than ) is non-relativistic, and therefore in the case of relativistic superheavy particles, . Consequently, and .15 This approximation serves to simplify the math, as it removes the need to know the exact direction of the velocity at the time ; however, it also eliminates the non-relativistic kinetic energy of the object, which must therefore be added back into the equation. Taking the above, as well as the contribution of the influence of non-relativistic ordinary matter into consideration, the overall energy of the object in the inertial frame is given by

Equation 3-1-19

Using the identity leads to

Equation 3-1-20

Recall that the morphology of a galaxy is determined by the distribution of the radiation detected by an observer. The observerís perception of morphology is strongly affected by the areas of high radiation (or brightness), and by their contrast with the background level of brightness. The amount of radiation emitted by any given area of the galaxy is largely related to its size and the density of its ordinary matter. The density, however, is expected to be higher at locations where the total energy of the object has a local minimum, particularly at the relatively deep minima, which are significantly lower than their neighboring minima (as shown in figure 3-2). Since the term in equation 3-1-20 is independent of location (and time), it does not have any effect on the location of the minima and can be removed. The term will cause the total energy minima to depart slightly from the potential energy minima, and will shift the orbits of objects in the outward direction, away from the potential energy minimum contours (thus creating a force to balance the centrifugal force). Adhering to the same logic used in equation 3-1-7, the Newtonian term bears little influence on the location of the minima. Thus, the high density concentrations should occur where

Equation 3-1-21a

where , and where and can be replaced by 1, resulting in a non-negative energy value for . 16

Equation 3-1-21b

Equation 3-1-21b holds true in the immediate vicinity of the coordinates at which both and at the inertial frame , where and are given by equations 3-1-17a and 3-1-17b using and respectively. As expected, when applied to non-relativistic SHP velocity , the relativistic equation 3-1-21 provides identical results to the non-relativistic equation 3-1-7, since at non-relativistic velocities

, and.

Section III-3: The Creation and Motion of SHP groups

Current theories commonly attribute the creation of planets, stars and galaxies to the gravitational collapse of clouds of gas. According to UG Postulate IV, the extreme temperature and pressure conditions that exist at the cores of large astronomical bodies produce superheavy particles. As theorized here, the vast amount of energy that is required for the creation of massive superheavy particles is likely to originate from the energy released by the collapse of ordinary matter towards the center of the astronomical bodies, and by the high level of pressure and temperature at their central core.17 By the time that an astronomical body reaches a steady state condition, its center comprises of a dense core that contains a significant portion of its ordinary matter, which rotates as a rigid body with constant angular velocity around its axis of rotation. The assumed circular orbits of the SHP groups around the center of the astronomical body may or may not be located within the central core. In either case, the interaction between orbiting SHP groups and the core ordinary matter produces rotating zones, with maxima and minima contours that intersect with the volume of the core. When the angular velocity of any of the SHP groups is equal to , these maxima and minima rotate in unison with the ordinary matter of the central core, allowing both the core and the SHP groups to maintain their angular velocity. Conversely, when the angular velocity of any of the SHP groups orbiting the center of the astronomical body varies from , the angular velocity of the resultant zonal pattern differs from the angular velocity of the core ordinary matter. In such a scenario, ordinary matter within the core will periodically either pass or be overtaken by the maxima and minima of the rotating zone structure. The forces that result from such encounters apply strong torques which accelerate (or decelerate) the rotational velocity of the SHP group, forcing it to converge to the angular velocity of the central core. Therefore, a group composed of superheavy particles of mass in a circular orbit of radius must rotate at the same angular velocity as the central core.18 Consequently, the speed of any SHP group of SHP mass with an orbital radius of is given by 19

Equation 3-3-1

This provides two very important rules that apply to superheavy particles in circular orbit around the center of an astronomical body with a massive rotating central core, where the zonal oscillation ranges of these SHPs are longer than the radius of their orbit around the center of the astronomical body:

Rule 1: In a steady state condition, all orbiting superheavy particles fulfilling the above conditions share the same angular momentum , where is the angular velocity of the central core of the astronomical body.

Rule 2: As a consequence of equation 3-3-1, and the requirement that the velocity of a superheavy particle cannot surpass the speed of light, the orbital radii of all superheavy particles that fulfill the above conditions must be shorter than .

There are a few important questions regarding the nature and characteristics of superheavy particles that must be addressed. What is the mechanism that allows for and enables the generation of superheavy particles? What mechanism forces superheavy particles into nearly circular orbits around the center of the galaxy and accelerates them to relativistic velocities? What prevents their immediate annihilation or decay?

The process by which SHPs are created may be similar to the process that generates a particle and an anti-particle of the same mass, such as an electron and positron pair from photons. If that is the case, the momentum and energy of the newly created SHP and anti-SHP are determined by the energy and momentum of the high-energy photons from which they originated. Newly created SHPs that do not have sufficient kinetic energy to escape the (UG) gravitation of the central core will enter an orbit around it. However, as the superheavy particles settle into orbits, they are accelerated over a relatively short period of time by the mechanisms described above, and forced to move at an angular velocity equal to the angular velocity of the rotating central core. Therefore, according to equation 3-3-1, when the orbital radii of the SHPs are sufficiently large, their velocities become relativistic. Note that complete stability of a superheavy particle orbit can be achieved only if the orbit becomes almost exactly circular (note that the orbit may become slightly deformed by relativistic effects). A non-circular orbit will create a wobbling effect of the zonal pattern relative to the rotating ordinary particles within the planetís central core. This wobbling effect will generate strong torques that force the SHP into a circular orbit where its velocity is perpendicular to the distance vector between the SHP location and the center of the core, and where the angular velocity of the zonal maxima and minima that cross the volume of the core is exactly the same as the angular velocity of the core particles. Upon entering into orbit, the SHPís velocity is still only a small fraction of its final speed when its angular momentum becomes equal to . Consequently, when a superheavy particle enters an orbit with relatively low velocity, other superheavy particles along the same orbit with an angular velocity of are moving at much higher speeds, and are therefore able to catch up and bond20 with the new particle within a very short period of time, generating a group of superheavy particles. This process may be repeated many times as the SHP groups grow to include multiple superheavy particles.

In regard to the question of how superheavy particles remain stable, avoiding either annihilation or decaying into smaller particles, all known particles aside from protons and electrons (as well as their anti-particles) are unstable when they are free or un-bonded. However, the neutron, which is also unstable when free, is known to become stable when it is bonded to a proton(s). The fact that SHPs are bonded to the central core of the planet or to other SHPs (within a group) may explain how they remain stable and avoid decay. Moreover, the strong rejection zones between SHPs and anti-SHPs may keep them apart and prevent their annihilation while in orbit around the same center of rotation.

Section III-4: The Issue of the Tail Wagging the Dog

At first sight the UG theory seems to pose the inherent problem of the ďtail wagging the dog.Ē As will be shown in the following chapters, UG calculations suggest that SHP groups of a total mass of the order of few hundreds of kilograms dictate the overall structure of Saturnís ring and satellite system, which amounts to an overall mass of approximately . Similarly, SHP groups of a total mass of the order of to will be shown to determine the overall morphology of a galaxy of a mass of about to . The force that a superheavy particle of a mass of is capable of exerting on ordinary matter within its zonal oscillation range at distances of is about times larger than the Newtonian force applied by a point-like sphere of ordinary matter of a total mass of from the same distance. 21

However, as the mass of the SHP groups is negligible in comparison to the mass of ordinary matter within a galactic disk, or within a system of planetary rings and satellites, the SHP orbits should be profoundly affected by the gravitational influence of ordinary matter. Furthermore, the overall SHP effect exerted on the heavier ordinary matter should be minimal, as the overall mass of the ordinary matter of a galactic or planetary system is larger by many orders of magnitude. Yet, as will be seen, the model here assumes, for example, that in the case of planetary systems, the orbits of SHP groups are completely unaffected by the matter in the rings and satellites, while the rings and the satellite orbits are curved by the effect of the SHP groups.

The logic behind this assumption is quite simple. The SHP groups are not free, and are held in circular orbit by the central rotating core of the planet. As the superheavy particles transfer energy and angular momentum to orbiting matter outside of the massive central core, they may lose angular momentum and energy to the orbiting objects, yet are prevented from slowing down or leaving their orbit around the central core, which is few orders of magnitude heavier than the overall mass of the planetís ring and satellite system. Moreover, the same mechanism which led to equation 3-3-1 () forces the SHPs to rotate around the center of the planet at the exact same angular velocity as the central core. In addition, the SHP group is prevented from significantly reducing or increasing its radius of orbit by the nearby maxima that confined its orbit. Therefore, the energy and angular momentum lost (or gained) by the SHP to the ring and satellite system must be replenished immediately by the far more massive rotating central core. Consequently, the SHP group will maintain the same angular velocity as that of the rotating center and the same orbit and speed. An analogy to this concept is the image of a free rigid rod of almost no mass pushing a small ship. As the mass of the rod is negligible compared with the mass of the ship, the force acting between the two objects will essentially influence the momentum and velocity of the rod, bearing virtually no effect on the momentum and velocity of the ship. However, if the rod is attached to a much larger ship that is using the rod to push the small ship, the opposite effect will occur, and there will be minimal change in the momentum and speed of the rigid rod, which is now part of the much larger ship, while the momentum and velocity of the small ship will change significantly.

1 Note that more than one SHP group can share the same space, orbit and velocity. This may occur when superheavy particles of different masses are bonded together to form a single object.

2 While the zonal maxima are the main elements preventing the different individual superheavy particles from collapsing into each other, the conservation of angular momentum also plays a role.

3 Note that there is a distinction between binary SHP groups and any two groups arbitrarily selected. Binary SHP groups share the same orbit, and are always co-linear with the center of the galaxy. Therefore, in the case of binary SHPs, if the spherical coordinates of the first group at any given time are , the coordinates of the second group at time must be . In contrast, in the case of two arbitrary groups, the second group can be located almost anywhere.

4 The dominant SHP types are those which exert the greatest effect on the local morphology or on the properties of the galaxy. Whether a specific type of SHP is dominant in a given area of the galaxy depends on the SHP mass and prevalence relative to other SHPs, on the location of the given area, and on the velocity and distribution of the superheavy particles.

5 The observation time is the period of time observed from the point of view of the galaxy, rather than the time measured by the observer. As the observed galaxy is located far from Earth, the detected radiation must have been emitted long ago (for example, several billions years ago). In addition, the radiation detected on Earth is redshifted, since at the time of emission the galaxy was moving away from Earth at high speed relative to our galaxy. Therefore, from the point of view of the observer on Earth, the local clock on Earth runs faster than the same clock in the observed galaxy. The time period studied from the inertial frame of the galaxy should thus be shorter than the length of time perceived by earthbound observation.

6See sections VII-1 to VII-3.

7 This assumption does not hold true in the immediate vicinity of a collapsed star, such as a neutron star or a black hole.

8 Strong external gravitational fields applied on the galaxy create tidal forces. Therefore, in the presence of strong external gravitational fields, no single inertial frame can cover the entire galaxy. The influence of substantial external gravitational fields on the spacetime along the path of the radiation will result in a gravitational lensing effect, which may distort the observed morphology of the galaxy. In addition, strong external gravitation applied on the observer will result in strong accelerations, and may cause additional morphology distortions.

9 The dependency on the relative velocity of the galaxy and the emitting atom is due to relativistic distance contraction.

10This does not include the case of radiation emitted from an atom in the vicinity of a collapsed star.

11 The discussion is limited to galactic morphologies driven by self-generated radiation, rather than by absorption or reflection of externally generated radiation.

12In cases involving two or more SHP groups that are not co-linear with the center of the galaxy, or in cases of groups that rotate at different speeds or at different orbital radii, the potential energy of the orbiting object can be calculated by obtaining the sum of the contributions of all SHP groups, where equation 3-1-1 is applied for all individual (non-binary) groups with a value of .

13 It will be demonstrated that in the case of relativistic SHP groups, the locations of the objectís potential energy minima may depend on the velocity of the SHP as well.

14 The mechanism by which the superheavy particles are accelerated to relativistic velocities will be discussed at the end of this chapter.

15 According to Lorentz transformations, and . Typically, , while is assumed to be relativistic. Therefore, , , and consequently . The same holds true for and .

16 Theoretically, the lowest minima of equation 3-1-21a are negative, however their amplitudes are completely negligible since by many orders of magnitude (typically by a factor of ).

17 In the case of galaxies, stars, and possibly large planets, some of the energy can be attributed to nuclear reactions deep within their cores.

18 Note that the large forces that equalize the angular velocity of the SHP group with that of the central rotating core can only attain sufficient strength if the center of the planet is within the zonal oscillation range of the groupís superheavy particles. This will occur only if or . To avoid violating special relativity, the velocity of the SHP group must be lower than the speed of light, leading to . Consequently, a group of superheavy particles of mass cannot maintain a circular orbit of radius , and will either be pulled toward the core or ejected outward to an orbit beyond the zonal oscillation range of the given SHP-ordinary matter interaction. Beyond this range the UG gravitational force reduces to the Newtonian force, and the SHP group may orbit at an angular velocity significantly lower than .

19 This raises the question of whether the UG force is sufficiently strong to keep a relativistic SHP group in a circular orbit. Assume, for example, that the central core rotates at rotations per second and that the orbital radius of the SHP group is . The resultant speed of the group is thus equal to . The force required to balance the centrifugal force at non-relativistic velocities is equal to (note that the relativistic correction may change this number. However, for the purpose of estimating the order of magnitude of the acceleration, the non-relativistic force is sufficiently accurate). Assuming a very small central core (of radius , the acceleration of the superheavy particles is given by , where provides the overall mass of the ordinary matter in the central core, where the value of is of the order of , where , and where . Therefore, the terms and (which may be of the order of provide the SHP group with the enormous acceleration levels needed to maintain a circular orbit at relativistic velocities. However, this logic cannot be applied to objects composed entirely of ordinary matter, or for SHPs with an oscillation range that is substantially shorter than their radii of orbit around the center of the galaxy. In the latter case, the forces enacted by the rotating center on the SHP group at distance are outside of the zonal range of the interacting SHP-ordinary particle pairs. Hence, , and . The UG force equation therefore converges to the Newtonian force equation, and is not sufficiently strong to keep relativistic SHP groups in a circular orbit. In the specific case of an object composed exclusively of ordinary matter, the force exerted on the object by the ordinary matter of the central core is also Newtonian, and the UG force generated by the SHPs is proportional to , and therefore smaller by a factor of (of the order of less than ). Hence, the object does not experience sufficiently large forces or torques to force the SHP group to move at the angular velocity of the central core . Consequently, ordinary matter within this range of distances will not be able to keep pace with the rotation rate of the central core, and will orbit at a much lower angular velocity.

20 via the UG force acting between two superheavy particles.

21 These calculated numbers are even larger when the relativistic corrections are taken into account.

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