Galactic Scale Self-Similarity

I. Introduction

It has been almost two decades since the last SSCP paper specifically devoted to Galactic Scale phenomena (Paper #23 in the “Publications List”), so it is appropriate to review and update our knowledge of Galactic Scale and Metagalactic Scale phenomena within the context of the Self-Similar Cosmological Paradigm.  We will begin with a brief summary of this discrete fractal paradigm.  If you have already read this material on the “Main Concepts” page, please skip ahead to section II. 

The SSCP draws attention to the self-evident, but often under-appreciated, fact that from the smallest observable subatomic particles to the largest cosmological structures, nature is organized in a hierarchical manner.  Although the whole hierarchy, in its full detail, is quasi-continuous, the SSCP emphasizes that the cosmological hierarchy is also highly stratified.  While the observable portion of the entire hierarchy encompasses nearly 80 orders of magnitude in mass, three narrow mass ranges, each extending for only about 5 orders of magnitude, account for ³ 99% of all mass observed in nature.  These dominant mass ranges are referred to as the Atomic, Stellar and Galactic Scales.  They constitute the discrete self-similar scaffolding of the observable portion of the quasi-continuous hierarchy.  The SSCP proposes that nature’s hierarchy extends far beyond our current observational limits on both large and small scales, and probably is completely unbounded in terms of scale, such that there are no largest or smallest objects (or Scales) in nature. 

The SSCP further proposes that the Atomic, Stellar and Galactic Scales, and all other fundamental Scales of nature’s transfinite hierarchy, are rigorously self-similar such that for each class of particle, composite system or phenomena on any given Scale, there is a discrete self-similar analogue on all other Scales.  Mass (M), length (R) and time (T) parameters associated with analogues on neighboring Scales Ψ and Ψ-1 are related by the following set of discrete self-similar transformation equations. 

                                    RΨ = ΛRΨ-1 .                                                      (1)

                                     TΨ = ΛTΨ-1 .                                                     (2)

                                     MΨ = ΛD MΨ-1    .                                               (3)

 The dimensionless constants Λ and D have values of 5.2 x 1017 and 3.174, respectively.  The value of ΛD = 1.70 x 1056.  The symbol Ψ is used as an index for keeping track of Scales, and

                                     Ψ = {…, -2, -1, 0, +1, +2,  …}.                     (4)

Usually the Atomic, Stellar and Galactic Scales are assigned Ψ = -1, Ψ = 0 and Ψ = +1, respectively.    Any dimensional “constant” or parameter appearing in an equation associated with a given Scale must be transformed according to the scaling equations (1) – (3) in order to find the correct value for the counterpart “constant” or parameter on a neighboring Scale.

The self-consistency and universality of the discrete self-similar scaling equations of the SSCP have been subjected to many tests.  A partial list of natural phenomena that have been shown to be consistent with retrodictions and predictions based on the SSCP can be found on the “Successful Predictions/Retrodictions” page of this website.


II. Preview

In this review of Galactic and Metagalactic Scale phenomena we will discover that the entire observable universe constitutes a nearly infinitesimal region of one Metagalactic Scale object.  The galaxies within this region are crammed together at very high densities.  Galactic Scale objects are also chaotically moving at high velocities and this indicates an extremely high ambient temperature.  The combination of very high temperature and density produces frequent galactic interactions and mergers.  The mass spectrum of galaxies is relatively flat, unlike the abundance trends on the Atomic and Stellar Scales, and therefore significant numbers of moderately massive and very massive systems are present.  The evidence for an extremely high-energy environment, the presence of substantial numbers of massive galactic systems, and the strong evidence for global expansion, all suggest a reasonably unique analogy to the interior of a supernova shortly after detonation.  In this analogy, galactic systems play the role of fully ionized Atomic Scale particles and nuclei under very-high-energy plasma conditions.  In what follows we will see that the SSCP analysis leads inevitably to this radical reinterpretation of the standard Big Bang model for the “origin” of the “Universe”. 


III. Properties of Galactic Scale Systems

(a) Numbers of Galaxies: Twenty years ago the canonical value for the total number of galaxies in the observable universe was on the order of 100 billion.  Today the rough estimate is slightly higher, possibly by up to a factor of four, due to the increased detection of small dim galaxies.  In our calculations we will tentatively use a compromise estimate of 2 x 1011 galaxies as the best estimate for the total number of galaxies in the observable universe.

 (b) Radii of Galaxies: The estimated range of radii for galaxies has expanded in the last two decades due to observational improvements that allow astrophysicists to detect smaller galactic objects and trace the full extent of larger ones.  The overwhelming majority of spiral, irregular, dwarf elliptical and giant elliptical galaxies have radii that range from about 5 kpc to 50 kpc, where a kiloparsec equals roughly 3,200 light years or about 3 x 1021 cm.  The SSCP proposes that these most common galaxies are Galactic Scale analogues of fully ionized Atomic Scale nuclei.  If we multiply the radii of the proton and the lead nucleus by Λ2, in accordance with Eq. (1), we should get a range of radii that is comparable to the range for the most common galaxies.  The proton’s charge radius is about 0.8 x 10-13 cm, and multiplying that by Λ2 gives 2.2 x 1022 cm, or about 7 kpc.  Scaling the 8.3 x 10-13 cm radius for the lead nucleus by Λ2 gives 2.2 x 1023 cm, or about 75 kpc.  Given that we are scaling over a size range of just over 35 powers of ten, the agreement between the rough observational estimate of 5-50 kpc and the SSCP prediction of 7-75 kpc based on Atomic Scale data must be judged as being quite impressive.

The full range of radii for Galactic Scale objects is quite a bit broader than the radius range for the dominant galactic populations.  Dwarf spheroidal galaxies have been discovered with radii on the order of 0.1 kpc and some of the gigantic cD galaxies at the centers of large galaxy clusters can have radii on the order of 500 kpc.  There are also systems called ultracompact dwarf galaxies with radii estimated to be less than 0.1 kpc.  Most importantly, there is a huge population of globular clusters, which are less massive than galaxies by a factor of 103 or more.  Their core radii tend to be in the range of 0.002 to 0.007 kpc and their full tidal radii are still only on the order of about 0.1 kpc.  Given the SSCP interpretation of Metagalactic Scale phenomena in terms of the deep interior of a supernova, it is to be expected that the dominant population of Galactic Scale nucleus analogues would be accompanied by the more complete bestiary of subatomic particles.  Such a very-high-energy environment would include nucleons, nuclei, electrons, muons, tau particles, pi mesons, kaons, etas, charmed mesons, hyperons and a host of other particles and resonances.  Therefore we cannot expect the range of radii based on the proton and the lead nucleus radii to account for the full range of radii for all Galactic Scale systems.  In the future, many important tests of the SSCP will revolve around the degree to which Galactic and Metagalactic Scale phenomena manifest discrete self-similarity with well-documented phenomena in the high-energy physics regime.  For the purposes of this review of Galactic and Metagalactic Scale phenomena, we will restrict the discussion to a first approximation analogy to a supernova and concern ourselves mainly with the more common and relatively stable systems that are most readily observable.

 (c) Galaxy Rotation Periods: One might suspect that it would be easy to find data on the rotation periods of various types of galaxies, but that is not the case.  This is largely due to the fact that galactic rotation periods are at least partially a function of radius (differential rotation) and the fact that the outer boundaries of galaxies are hard to define.  Most importantly, the detailed spatial distribution of the primary dark matter component of galaxies is still uncertain.  The fact that the periods are on the order of 108 years rules out the easiest direct methods of determining rotation periods.  We do know that our Galaxy’s rotational period at the Sun’s location is roughly 2 x 108 years, and the best estimate for the overall rotation period of the Milky Way Galaxy is on the order of 4 x 108 years.  Since our Galaxy is considered a typical galaxy, we will adopt the value of 4 x 108 years as the correct order of magnitude for the average rotational period of a galaxy.  As more rotational period estimates become available, they can be included in the discussion.

 (d) The Galactic Mass Function: One might also assume that the abundances of various classes of galaxies would be well known, but again definitive data are hard to find.  It is commonly reported that smaller, low-mass galaxies outnumber medium-sized galaxies, and that very large galaxies like giant ellipticals and cD galaxies are even more in the minority.  In the future one hopes that detailed data on the shape of the distribution of galaxy masses, as a function of cosmological epoch, will become more readily available.  For now we can say that the for the “recent” cosmological epoch, say galaxies within 50 to 100 Mpc, there are substantial numbers of moderately massive and very massive galaxies.  This is quite unlike the overall situation on the Atomic Scale where low-mass systems (such as protons, electrons, alpha particles and low-mass atoms) are overwhelmingly dominant.

 (e) Galactic “Peculiar Velocities”:  The well known relationship between the observed velocities of galaxies, as measured by their redshifts, and their estimated distances appears to be most naturally explained in terms of a global expansion of the observable portion of the metagalaxy.  Galaxies additionally have sizeable random motions, often referred to as “peculiar velocities,” superimposed upon the global expansionary motion.  These peculiar velocities are helpful in characterizing the ambient temperature of the local Metagalactic Scale environment, since temperatures and velocity distributions are correlated.  In 1986 the best estimate for the average peculiar velocity (<vp>) was about 400 km/sec.  An updated value of <vp> on scales up to about 50 Mpc was reported1 as approximately 700 km/sec, and we will use this estimate in our order-of-magnitude calculations below.  The full distribution of radial peculiar velocities for galaxies within about 50 Mpc ranges from – 2,000 km/sec to + 2000 km/sec.


IV. The Metagalactic Scale Environment

Given the information assembled in Section III and a few additional observations, we can begin to investigate the Metagalactic Scale environment.  According to the SSCP, it is essentially arbitrary whether we treat galaxies as Galactic Scale systems comprising part of a Metagalactic Scale object, as analogues of Stellar Scale systems comprising part of a Galactic Scale object, or as analogues of Atomic Scale systems comprising part of a Stellar Scale object.  We will find that the last analogy: viewing galaxies as if they were Atomic Scale nuclei comprising a small portion of a Stellar Scale system, is quite useful in developing an intuitive feel for what is happening on the largest observable size scales of the cosmos.  At any point we can revert to the standard perspective on galaxies without penalty. 

(a) Size of the Observable Universe:  In 1986 the canonical estimate for the radius of the observable universe (Ru) was roughly 4,000 Mpc, or 4 Gpc.  Today estimates are somewhat higher and there are more candidates for exactly how to determine Ru.  One source noted that the “cosmic light horizon is 4.2 Gpc distant” if we accept the value of 13.7 years as the time since the global expansion began.  A study by the WMAP collaboration published in 2004 estimated that Ru ≈ 24 Gpc, while a value of 14.4 Gpc was reported in the March 2005 issue of Scientific American.  In 2006, a review of cosmology’s progress over the last 100 years noted that our local region of the cosmos had an “observed horizon about 30 Gpc across,” which presumably represents a diameter rather than a radius.  For the purposes of this review we will adopt an approximate value of Ru ≈ 15 Gpc, which is ≈ 4.5 x 1028 cm.  The volume of the observable universe (Vu) would then be (4 /3)π(Ru)3 ≈ 3.9 x 1086 cm3.

(b) The Observable Portion of the Metagalaxy:  An interesting question with a very surprising answer is: “How much of the Metagalactic Scale object that we inhabit can we observe?”  We have just shown above that the radius of the observable universe is ≈ 4.5 x 1028 cm.  Now consider the fact that a median radius for a galaxy (<Rgal>) is about 30 kpc, or on the order of 1023 cm.  The truly remarkable fact about our observational limits is that Ru is only ≈ 105<Rgal>.  Putting our Atomic Scale analogy to good use, we can view galaxies as analogues of atomic nuclei.  Within the context of that analogy, the entire observable universe is not much bigger than a single atom, since atoms are roughly 105 times the size of a typical nucleus.

So if we view our metagalaxy as an analogue of a Stellar Scale system, then we can only observe a volume that is slightly larger than the volume of a single atom of that stellar system.  The inescapable conclusion is that the vast observable universe constitutes a nearly infinitesimal volume of the Metagalactic Scale object that we inhabit.  One needs to think about this, and perhaps do the necessary radius and volume calculations, until the reality of this fact sinks in.

One other amazing thing to consider is that if we could observe a Galactic Scale analogue of a hydrogen atom in a moderately low energy state (say n = 6; l = 0), then that single galactic system would entirely fill the observable universe.  However, the very-high-energy environment of the observable portion of our metagalactic object precludes the possibility of observing any Galactic Scale systems in low energy states.

(c) The Local Metagalactic Scale Density:  Continuing with the analogy in which we view galaxies as Atomic Scale nuclei and our local Metagalactic Scale region as a nearly infinitessimal portion of a Stellar Scale object, Ru can be converted to a RΨ=0 analogue by dividing Ru by Λ2 to equal 1.66 x 10-7 cm.  We have seen that this is about the radius of a moderately excited hydrogen atom with n = 6.  The relevant volume based on this radius can then be calculated.

                        VΨ=0 = (4/3)π(RΨ=0)3 = 1.92 x 10-20 cm3.                                 (5)

In this volume we find on the order of 2 x 1011 nucleus analogues with masses that vary from that of the proton at 1 atomic mass unit (amu), or about 1.67 x 10-24 g, to that of the lead nucleus with a mass of over 200 amu.  To calculate an approximate density we need to choose an average mass (<MΨ=-1>) for our nucleus analogues.  Since our knowledge of the distribution of Galactic Scale masses is marginal, we will calculate 3 density values based on <MΨ=-1> estimates of 4 amu, 16 amu and 64 amu.

If <MΨ=-1> corresponds to the mass of a He++ nucleus, then we calculate a density (ΦΨ=0) appropriate to the proposed galaxies-as-nuclei analogy as follows.

                        ΦΨ=0 = (2 x 1011)(<MΨ=-1>) ¸ (4/3)π(RΨ=0)3                           (6)

                                  = (2 x 1011)(6.68 x 10-24 g) ¸ 1.92 x 10-20 cm3

                                  = 6.97 x 107 g/cm3.

If <MΨ=-1> equals 16 amu instead of 4 amu, then ΦΨ=0 would be increased to 2.79 x 108 g/cm3.  If <MΨ=-1> equals 64 amu, then ΦΨ=0 is increased to 1.11 x 109 g/cm3. 

Therefore the density of galactic systems in the observable portion of our metagalactic system corresponds to a density that is in the range of roughly 108 g/cm3 to 109 g/cm3 within the context of our analogy that treats galaxies as nuclei.  This density range is not as high as the densities in the interior of a nucleus (~1014 g/cm3) or a neutron star, but a density on the order of 108 g/cm3 is still extremely high.  For comparison, very dense white dwarf stars, which are the precursors of type-I supernovae, have densities on the order of 106 g/cm3, which is equal to over 2,000 lbs per teaspoon.

(d) The Local Metagalactic Scale Temperature:  Our next goal is to derive an estimate for the ambient temperature (TΨ=+2) that applies to the observable portion of our metagalactic system.  It should be fully acknowledged that our estimate can only be a very crude approximation because of the fairly high levels of uncertainty associated with various Galactic and Metagalactic Scale parameters.  However, a crude approximation is better than none at all, and even in its rough form the estimate may tell us something quite important about our local Metagalactic Scale region.

In the case of a gas of Atomic Scale particles at approximate thermal equilibrium, there is a relationship between the average kinetic energy (<Ekin>) of the particles and the ambient temperature (T):

                                    <Ekin> = (3/2)kT  ,                                                       (7)

where k is Boltzmann’s constant (1.38 x 10-16 g cm2/ sec2 oK).  The very-high-energy state and global expansion that characterize the observable portion of our metagalactic system surely deviate from thermal equilibrium conditions, but we are only looking for a crude approximation of TΨ=+2.  We can express Eq. (7) as

                                    (1/2)m<v>2  = (3/2)kT   ,                                             (8)

where m is the mass of an individual particle and <v> is the average velocity of the particles.  Eq. (8) can then be rearranged to give

                                    T = (1/3)(m/k)<v>2  .                                                   (9)

Continuing with our analogy between galaxies and atomic nuclei, and remembering that the average peculiar velocities for galaxies is about 700 km/sec, we can solve Eq. (9) for three hypothetical cases wherein all of the nuclei have masses of 4 amu, 16 amu or 64 amu.  If m = 4 amu, then

            T = (1/3)[(6.68 x 10-24 g) ¸ (1.38 x 10-16 g cm2/ sec2 oK)][7 x 107 cm]2            (10)

               = 7.9 x 107 oK  .

If m = 16 amu, then T = 3.2 x 108 oK, and if m = 64 amu, then T = 1.3 x 109 oK.

Assuming that we are justified in using our atomic/galactic analogy, and that the deviation from thermal equilibrium conditions is not introducing fatal errors, our crude approximation of TΨ=+2 is roughly 108 oK to 109 oK.  This enormous temperature is quite consistent with expectations based on the hypothesis that the observable portion of the metagalaxy is self-similar to the interior of a supernova shortly after detonation.  Whereas our numerical results are subject to various sources of errors, the general conclusion that TΨ=+2 is very high is supported by the very large peculiar velocities of galaxies, the very high density of galactic “particles”, the fully ionized states of all observable Galactic Scale systems, the frequent and apparently violent interactions among galaxies, and the global expansion of the observable portion of the metagalaxy.

(e) Expansion Timescale for Our Local Metagalactic Region:  Cosmologists are fairly confident that the global expansion of the observable universe began approximately 13.7 billion years ago.  This sounds like a very long period of time, but relative to the Galactic Scale it only amounts to about 30 rotation periods of our Galaxy.  Within the context of our galactic/atomic analogy, wherein galaxies are treated as Atomic Scale particles and the local region of the metagalactic object is regarded as the interior of a supernova, the time (tΨ=0) since the detonation occurred can be calculated as follows.

                         tΨ=0 = tΨ=+2 ¸ Λ2                                                                                            (11)                                

                                = (13.7 x 109 yr)(3.15 x 107 sec/yr) ¸ 2.7 x 1035

                                 = 1.6 x 10-18 sec  .

Therefore, when we view the expansion of the observable universe in terms of the supernova analogy, then on the order of 10-18 seconds have passed since the detonation.  This is consistent with our findings that the observable Metagalactic Scale region still has extremely high temperature and density values. 

(f) Plasma-Like Distributions of Galaxies and Galactic Clusters:  One of the most remarkable metagalactic phenomena, and one that has been well documented throughout the observable universe, is the unique distribution of galaxies.  Astronomers initially guessed that the distribution would be fairly random on moderately large scales and approximately homogeneous as one approached the largest observable scales.  However, observations gradually showed that the actual distribution of galaxies is quite different from those early assumptions.  Astrophysicists now know that the distribution of galaxies and clusters of galaxies is highly filamentous, with Galactic Scale systems preferentially occurring in vast filaments and thin sheets.  The high-density filaments and sheets tend to surround “voids” wherein the mass density is quite low.  Thus the overall structure in the observable portion of our metagalaxy has been described using terms like: “Swiss cheese”, “honey comb”, “filamentary”, “cosmic webs”, “walls”, “bubble-like voids”, “fractal networks”, etc.  This enigmatic distribution has not been convincingly explained by conventional astrophysics, but our supernova analogy offers an appealing interpretation.  The same mix of filaments, thin sheets and voids is regularly seen in the most common state of matter in the observable universe: plasmas.  When high-energy conditions lead to ionization of neutral matter, the resultant mix of electrons, nuclei and charged ions manifests this highly unique filament/sheet/void morphology due to a complex interplay between the attraction among unlike charges and the repulsion among like charges.  Prime examples, which we have all seen before, are the beautiful (and now highly detailed) photographs of the Crab Nebula supernova remnant.  In this case, the supernova remnant has expanded and cooled for hundreds of years, but the imprint of the plasma morphology is still readily observable.  In the case of the observable universe, the ambient Metagalactic Scale temperature, density and pressure are enormously higher, but the filament/sheet/void character of the distribution of galaxies and galaxy clusters is highly self-similar to the distributions of very-high-energy plasma particles.  This interpretation is certainly in keeping with predictions based on our Metagalactic Scale supernova analogy.  Put the other way around, if galactic systems did not have this unique plasma-like distribution, but rather had a very homogeneous distribution, then the Metagalactic Scale supernova analogy would have had a serious flaw.


V. The Supernova Analogy for Galactic and Metagalactic Phenomena

As we have seen throughout the proceeding sections, our exploration of the large-scale structure within the observable universe leads us in the direction of an analogy to a Stellar Scale supernova.  There is fairly strong evidence for discrete self-similarity between galaxies and Atomic Scale nuclei based on the radii, rotation periods, oscillation periods, shapes (spheroid, ellipsoid and prolate) and high velocities of galaxies.  The inferred density and temperature for the relatively infinitesimal observable portion of our Metagalactic Scale system are extremely high.  We also noted that the distribution of Galactic Scale masses is skewed towards significantly higher masses than is the case for the observable portions of the Atomic or Stellar Scales.  When these lines of evidence are combined with the compelling evidence for global expansion of the observable portion of the metagalaxy, it appears that the supernova analogy is by far the most probable explanation of Galactic and Metagalactic Scale phenomena, if the symmetry principle we refer to as discrete cosmological self-similarity is valid.

Supernovae are divided into two general classes: Type-I supernovae are thought to be the result of the explosion of a carbon-oxygen white dwarf star that has become unstable due to mass accretion, and Type-II supernovae are thought to result from the collapse and explosion of very massive stars that can no longer resist gravitational collapse, leading to detonation and the formation of an ultracompact central object.  Within the two general categories of supernovae there are many sub-categories that classify supernovae according to more specific characteristics.  At present our knowledge of Galactic and Metagalactic Scale phenomena is not sufficiently certain and detailed for us to specify a type and sub-type for our supernova analogy.  In the 1986 paper on Galactic Scale phenomena (#23 in the “Publications List ” section), it was argued that density estimates favored a Type-II supernova.  However, those density estimates were ΦΨ=+2 estimates and the variation of Φ with Scale was not sufficiently taken into account.  The density estimates derived above are the correct ones for use in the supernova analogy.  It is quite likely that as our observational knowledge of Galactic Scale phenomena becomes more quantitative and statistically refined, the specific type and sub-type of supernova that is analogous to the expansion of our Metagalaxy will become increasingly well defined.

The “tiny” observable portion of our local Metagalactic Scale environment clearly shows signs that it is in the very-high-energy domain, and so it needs to be studied from the perspective of high-energy physics.  We can expect the presence of both stable and unstable “particles”.  Recent studies suggest that the average galaxy has undergone between one and three merger events since detonation and this shows that the interaction probabilities are very high.  It would appear that the application of high-energy particle physics to Galactic and Metagalactic Scale phenomena, when suitably scaled according to the principle of discrete cosmological self-similarity, has a rich potential for discovery and unification.  Using the SSCP’s supernova analogy it should be possible to combine cosmology, supernova astrophysics and subatomic physics to achieve a truly advanced understanding of cosmological phenomena.


VI. The Galactic Scale Electron Analogue

One subject that has been conspicuously absent from most of the above discussion on Galactic Scale phenomena is the identity of a key player in the drama: the Galactic Scale electron analogue.  If the supernova analogy is correct and Galactic Scale systems are analogues to subatomic particles in a very-high-energy plasma, then it seems mandatory that large numbers of Galactic Scale electron analogues should be readily observable.  Based on the principle of discrete cosmological self-similarity, we would expect this important class of Galactic Scale systems to have the following properties.

(1) Ubiquity: Galactic Scale electron analogues should be present in large numbers and should be distributed ubiquitously throughout the observable universe.

 (2) Re-,Ψ=+1 » 4 Parsecs: The SSCP has shown that fundamental particles on all Scales have radii on the order of their Schwarschild radii, RΨ = 2GΨMΨ/c2, where GΨ is the properly scaled gravitational constant, MΨ is the particle mass and c is the velocity of light.  The approximate radius estimate for the Atomic Scale electron is 4 x 10-17 cm.  Multiplying this value by Λ2 yields a value of roughly 1.2 x 1019 cm, or about 3.9 parsecs, as the radius estimate for the Galactic Scale electron analogue.

(3) M e-,Ψ=+1 < 10-3 Mgal: Given that the electron has a mass of about 1836 times less than the proton mass, we can expect that the mass of the Galactic Scale electron analogue will definitely be less than 10-3 times the average mass for a galaxy, but not much less than 10-5 times the average mass for a galaxy.  Because of the very high levels of uncertainty in Galactic Scale mass estimates, this criterion can only involve very rough estimates.

(4) Stability: To the degree that it is self-similar to the Atomic Scale electron, the Galactic Scale electron analogue should be a relatively “stable” particle.

(5) Attraction/Repulsion: Just as electrons are attracted to positively charged particles and repelled from negatively charged particles, the Galactic Scale electron analogues should be attracted to Galactic Scale nuclei (e.g., galaxies), but repel each other.

(6) Central Singularity: The Galactic Scale electron analogue should have a central singularity that contains the overwhelming majority of its Galactic Scale mass.  From the point of view of a Metagalactic Scale system, it should behave as a “point-like” particle.

These six criteria put strong constraints on any candidate for the Galactic Scale electron analogue, and in fact there is only one possible candidate: globular clusters.  Below we will briefly discuss the general agreement between the properties of globular clusters and the six criteria for the Galactic Scale electron analogue.

(1) Globular Clusters Are Ubiquitous: There is a conspicuous presence of large numbers of globular clusters in all well-observed parts of the observable universe.

(2) Half-Light Radii Of About 4 Parsecs: Given the structure of globular clusters: a very dense central core and a spherical halo of stars, there are several ways to measure their radii (Rgc).  The most physically meaningful estimates are the core radius (Rc) and the half-light radius (R1/2), while the tidal radius (Rt) is a less diagnostic measure.  Studies2 of the globular clusters that are associated with our Milky Way Galaxy have found a mean R1/2 value of 4.4 parsecs and a median R1/2 of about 3.0 parsecs.  The core radii for globular clusters in our Local Group of galaxies are typically in the 2 pc to 7 pc range.  Therefore, there is a good agreement between Rgc and the predicted Re-,Ψ=+1 value of ≈ 4 parsecs.

(3) Mgc < 10-3 Mgal: The mean mass (Mgc) for globular clusters in our Galaxy has been estimated2 at about 2 x 105 M8.  Galaxy mass estimates tend to range from about 107 M8 to 1013 M8.  If we assume that <Mgal> ~ 1010 M8, then <Mgc> ~ 2 x10-5<Mgal>, which is within the expected range.  According to the SSCP, Galactic Scale mass estimates are severely hampered by three problems: (a) they are based on the use of GΨ=0, instead of the correct Galactic Scale gravitational constant (GΨ=+1 = GΨ=0 ¸ ΛD-1), (b) the estimates do not adequately take into account the singular nature of Galactic Scale mass distributions, and (c) the estimates completely ignore electromagnetic contributions to Galactic Scale mass/energy calculations.  Given these problems, arguments based on conventional Galactic Scale mass estimates must be limited to crude approximations of relative mass relationships, such as order of magnitude mass ratios for different types of galactic systems.

(4) Globular Clusters Are Stable: Globular clusters are regarded as among the oldest and most “pristine” of all Galactic Scale systems.  Therefore, relative to Galactic Scale considerations, globular clusters have a high degree of stability.

(5) Globular Clusters Are Attracted To Galaxies:  Most globular clusters are associated with galaxies, which the SSCP identifies as analogues of positively charged nuclei.  It also appears that tightly bound systems of two or more globular clusters are not observed, and may not exist at all.  Therefore it appears that globular clusters are attracted to galaxies, but tend to avoid close contact with other globular clusters.  These basic tendencies agree with the attraction/repulsion behavior expected for electron analogues.

(6) Central Singularities In Globular Clusters:  The number of globular clusters with ultracompact objects detected in their cores has steadily increased over the years, and some astrophysicists hypothesize that a central singularity may be a universal characteristic of globular clusters.

Given the general agreement between the six criteria for Galactic Scale electron analogues and the basic properties of globular clusters, it seems very likely that globular clusters are the self-similar Galactic Scale analogues of electrons.  An interesting test of this idea is as follows.  The number of globular clusters associated with a galaxy can vary from zero to hundreds, and even to thousands in the case of giant galaxies.  If our analogy between Atomic Scale electrons and Galactic Scale globular clusters is valid, then the distribution of electrons in very-high-energy Atomic Scale plasmas will be found to be compatible with the large numbers of globular clusters that are associated with high-mass galaxies.



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