New Developments 2003

A Good Test of Stellar/Atomic Self-Similarity [December, 2003]

The principle of cosmological self-similarity is the foundation that underlies the discrete fractal paradigm discussed at this website. It asserts that the phenomena occurring at each discrete cosmological scale, i.e., the atomic, stellar and galactic scales, are self-similar, and so “wholes” are analogous to the “parts” that are their discrete building blocks. In this paradigm, stellar objects have morphologies and internal motions that mimic their atomic scale counterparts; properties such as the sizes, rotation periods, and magnetic moments of analogues from different scales obey a set of simple scale transformation equations (see “Basic Concepts”).

An excellent test of the paradigm is as follows. The mass spectrum of atoms has a large peak at 1 atomic mass unit (AMU), a smaller peak at 4 AMUs and a set of even smaller peaks at 2, 3, 6, 7, 8, 9, 10, … AMUs. Note that there is no peak at 5 AMUs! This is a very striking and unexpected break in the otherwise uniform pattern of discrete peaks separated by 1 AMU.

If the Self-Similar Cosmological Paradigm is valid, then the stellar scale should manifest a similar pattern in its mass spectrum. Using the self-similar transformation equations presented in Paper #1 (or in "Basic Concepts"), one can calculate that 1 AMU corresponds to about 0.145 M on the stellar scale. Therefore, the SSCP predicts that the stellar scale mass function will have a main peak at 0.145 M and a series of smaller peaks at multiples of 0.145 M. The exception is that there should be no peak at about 0.73 M (see paper #6), which reflects the lack of an atomic scale peak at 5 AMUs.

Most astrophysicists would be extremely skeptical of this quasi-discrete type of mass spectrum that the SSCP predicts. However there is some empirical evidence that appears to support it, and this is discussed in several papers at this website. One ongoing observational program called the RECONS project, which is measuring accurate fundamental parameters for nearby stars, should be able to test the SSCP prediction. If we look at the first tentative data release: <>, the graph shows a series of peaks with the correct uniform spacing and an apparent gap near the expected position. However, all the peaks and the gap seem to be uniformly shifted to slightly lower masses, by about 0.05 M.

A new and more accurate analysis of the most current mass data by the RECONS team is due to be available in early 2004. If the SSCP is valid, then the quasi-discrete stellar mass spectrum, with peaks at multiples of 0.145 M and a conspicuous gap at about 0.73 M, should be even more pronounced. It can also be predicted that some form of systematic error will be found to have shifted the original mass values downwards by roughly 0.05 M.

A Narrow Mass Peak at » 0.2 M¤ for the Galactic Dark Matter [October, 2003]

Probably the most definitive prediction of the Self-Similar Cosmological Paradigm is that the mass spectrum of the galactic dark matter has dominant peaks at » 0.15 M¤ and » 7x10-5 M¤.

Microlensing experiments have been running for over a decade in an effort to find, or rule out, stellar-mass dark matter (SMDM) as the primary source of galactic mass. Results suggest that much of the galactic dark matter is in the form of stellar-mass objects, with optical depths exceeding those predicted for previously known stellar populations by a factor of 3 to 5. Other than that, little is known about these enigmatic dark objects.

One unusual finding is that the apparent masses for the SMDM seem to vary with the direction in which we search! Typical lens masses for microlensing toward the Bulge, LMC and SMC were found to be about 0.15, 0.4 and 2.5 M¤, respectively. An idea that may remove this odd disparity has been proposed by Prof. K. Saha (see astro-ph/0302325 at

Saha argues on the basis of several lines of evidence that the lenses involved in the Magellanic Cloud observations are members of those galaxies, not located within our galactic Halo, as is generally believed. If one accepts the “self-lensing” hypothesis, then the typical masses towards the LMC and the SMC are both » 0.2 M¤. Since the typical mass for Bulge events is also estimated to be » 0.2 M¤, the possibility exists of a uniform class of SMDM objects with a sharp mass peak at one of the SSCP’s predicted values.

It will be interesting to see if other microlensing experiments have similar results when Sahu’s “self-lensing” idea is included in the analysis?

Stellar/Atomic Analogies [July, 2003]

The idea of strong analogies and direct morphological and kinematic self-similarity between atoms and stellar systems is one of the most troubling concepts for people considering the Self-Similar Cosmological Paradigm.

Why is this? Niels Bohr won the Nobel Prize in 1922 for his model of the hydrogen atom, which was based on a direct analogy to the Solar System. However, subsequent developments in Quantum Mechanics moved ever farther away from stellar/atomic analogies and they became synonymous with naivete, hard-headedness and/or mental impairment.

But the problem was in trying to compare atoms in low energy states with a stellar system (the Solar System) that was in an extremely high-energy state (principal quantum number n > 100).

Look at what happens when the atomic counterpart in the comparison is also in a very high-energy state. The following excerpts are from a recent paper by Kalinski et al in Physical Review A 67, 032503, 2003.

"We predict the existence of a self-sustained one-electron wave packet moving on a circular orbit in the helium atom. The wave packet is localized in space, but does not spread in time. This is a realization within quantum theory of a classical object that has been called a "Rutherford atom," a localized planetary electron on an unquantized circular orbit under the influence of a massive charged core."

"[W]e provide the first demonstration of the existence of what has been called [14] a "Rutherford atom," i.e., the wave function for a single electron moving on an unquantized stable and nonspreading planetary orbit about a massive charged core."

When a proper comparison is made between high-energy state atoms and high-energy state stellar systems, the stellar/atomic analogies and self-similarity are quite strong. This is not a naïve idea and one should not have to feel defensive about the concept. It is those who would deny strong stellar/atomic analogies who are in error.

"The authority of a thousand is not worth the humble reasoning of a single individual." - Galileo