Robert L. Oldershaw
Amherst College
Amherst, MA 01002

Suggested running head: Fractal Paradigm Test


Abstract.  The incidence of planetary systems orbiting red dwarf stars with masses ≤ 0.25 M provides a crucial observational test for the Self-Similar Cosmological Paradigm.  The discrete self-similarity of the paradigm requires that there are anomalously few planets associated with these lowest mass red dwarf stars.  Ongoing observational programs are currently testing this prediction and decisive results should be available in the near future.


Key words: Cosmology, Fractals, Red Dwarf Stars, Planets, Self-Similarity


1. Introduction

The past decade has been an exciting time for those who study planetary systems.  Astronomers have gone from a sample of one system to a cornucopia of more than 50 planetary systems (see:, and the discovery rate continues to increase.  Within the next 5 years the sample of observed planetary systems should become large enough to permit reliable statistical conclusions about their general properties.  Some of the initial results have defied conventional expectations.  For example, one of the earliest observations of an extrasolar planetary system (Wolszczan and Frail, 1992) involved two planets orbiting a pulsar!  Prior to that observation most astronomers would have regarded the possibility of pulsar/planet systems with considerable skepticism.  Another unexpected result was the large number of systems with massive planets orbiting relatively near to their parent stars (Marcy and Butler, 1998).  Given these initial results, one may safely anticipate further surprises as more properties of planetary systems are discovered.  In this note we discuss a prediction of an anomalous deficit of planetary systems orbiting the lowest-mass red dwarf stars.  The prediction comes from a fractal paradigm called the Self-Similar Cosmological Paradigm.


Fractal cosmological models would appear to be worthy of serious consideration given the ubiquity of fractal phenomena in nature.  Galaxy distributions, trees, solar intensity fluctuations, clouds, star clustering, base-pairing in DNA, river systems, stock exchange fluctuations, atom/molecule distributions in the interstellar medium, nervous systems, Brownian motion and fluid turbulence are just a few of the examples wherein fractal or self-similar phenomena are found (Mandelbrot, 1983).  In fact it is difficult to identify parts of nature that do not involve some form of self-similar structures and/or processes.  Therefore, it would seem that fractal cosmological paradigms for the whole cosmos should be explored.  Two examples of such efforts are the Scale Relativity theory of Nottale (1996, 2001) and the present author’s (1987) Self-Similar Cosmological Paradigm (SSCP) which leads to the definitive prediction discussed below.  A two-part review of the SSCP is available (Oldershaw, 1989a,b) and the author’s website (Oldershaw, 2000) contains further discussion of the SSCP and various tests of its predictions.  A very brief summary of its main ideas is given below.


It is virtually self-evident that nature is organized into a highly stratified hierarchical arrangement.  Nearly all of the observable mass in the cosmos is bound up in galaxies.  These galaxies, in turn, are primarily comprised of stellar-mass objects, in extended, compact and ultracompact states.  Stellar objects, in turn, are composed of ultramicroscopic building blocks: atoms and subatomic particles.  The SSCP postulates that this stratified hierarchical arrangement of galactic, stellar and atomic scales continues well beyond our anthropocentric observational limits to include many additional cosmological scales, perhaps a denumerably infinite number of them.  A second key idea is the principle of cosmological self-similarity: for each class of well-defined systems on a given cosmological scale, there is an analogous class of systems with self-similar morphologies and kinematics on every other cosmological scale.  Properties of self-similar analogues on neighboring cosmological scales, as well as dimensional constants, obey the following scale transformation equations:

Rn = ΛRn-1


Tn = ΛTn-1 and


Mn = ΛDMn-1



where R,T and M are lengths, temporal periods and masses, respectively, for analogue systems on neighboring cosmological scales n and n-1, and where Λ (≈ 5.2 x 1017) and D (≈ 3.174) are empirically determined dimensionless constants.


It must be emphasized that the term “self-similar” does not usually mean “exactly the same except in scale.”  The self-similarity of fractal structures can range from the exact self-similarity of certain mathematical examples to the statistical self-similarity of many natural fractals (Mandelbrot, 1983).  The degree of self-similarity among the three observable cosmological scales cannot yet be fully determined empirically, but it appears to exceed statistical self-similarity.  Identified analogues on different cosmological scales appear to be quite similar in both morphological and kinematical properties, usually obeying Eqs. (1) - (3) to a factor of 2, or better (Oldershaw, 1989a).


The SSCP has passed a series of 20 retrodictive tests (Oldershaw, 1989a), and has gained recent support for some of its unique predictions.


  1. The SSCP predicted (Oldershaw, 1987) that the microlensing groups MACHO, EROS and OGLE would find galactic dark matter objects with typical masses of ≈ 0.15M¤ and ≈ 0.58 M¤. Existing data are consistent with this prediction, given the current levels of uncertainty in this estimate (Zhao and de Zeeuw, 1998; Sadoulet, 1999).  For a recent review of mass estimates for galactic MACHOs see
    <>, or Oldershaw (2002).

  2. The SSCP predicted (Oldershaw, 1989b) an unexpected turnover in the stellar mass function just below ≈ 0.15M¤, and such a turnover has been found for many stellar populations (Travis, 1994; Paresce and De Marchi, 2000; Hillenbrand and Carpenter, 2000).

  3. The SSCP predicted (Oldershaw, 1989b, 1996) that planets would be found orbiting compact and ultracompact stellar objects.

    This unexpected phenomenon has been verified for several pulsars (Wolszczan and Frail, 1992; Thorsett, 1994).

  4. The SSCP predicted (Oldershaw, 1989b, 2002) that the mass functions for galactic bulge and halo microlenses would be very similar to typical stellar mass functions.  This unexpected phenomenon has been observed and called to attention (Gould, 1999; Alcock et al., 2000).

2. The prediction

Given the mass and radius of the ground state hydrogen atom, Eqs. (1) and (3) suggest that a red dwarf star with a mass of ≈ 0.15M should be morphologically and kinematically self-similar to a neutral hydrogen atom with its electron in a low energy state (Oldershaw, 1989a).  If such a star had a stable planetary system, then this would correspond to a hydrogen atom with two bound electrons, one in a low energy state and one in a very highly excited Rydberg state.  Such a system, an H- or hydride ion, does exist in nature but it is unstable and relatively rare.  Therefore, the SSCP makes the following definitive (i.e., prior, unique, quantitative, and testable) prediction.  Stably bound planetary systems should be anomalously rare (<1%) for red dwarf stars with masses below 0.25M. Existing theories of stellar evolution and planet formation, on the other hand, do not predict any cutoff for planetary systems orbiting stars with masses between 0.25M and the hydrogen-burning limit at 0.08M (Armitage and Bonnell, 2002), and any postdiction of such a finding might appear rather ad hoc.  Since significant fractions of G and K stars appear to have planetary systems, the predicted cutoff at ≈ 0.2M for M stars would appear to be a unique and unexpected phenomenon. 


Some caveats must be kept in mind for satisfactory tests of this prediction.

  1. We must be reasonably certain that the test stars have masses in the appropriate mass range.  Stars with masses ≥ 0.29M are predicted to have a normal incidence of planets.

  2. Multiple star systems and ultracompact objects (low mass neutron stars or black holes) do not qualify as valid test systems.  The prediction only applies to single red dwarf stars.

  3. To avoid confusion with temporary “gravitational capture” systems, we concentrate on reasonably stable planetary systems with R ≤ 20 AU.

  4. In order to adequately test the prediction, a large and heterogeneous sample of stars with masses known to 15%, or better, is required.  NASA’s “Nearby Stars” project (stars within 25 pc) is one potential source for the required sample (see

    A related, but less definitive, prediction is that the number (Np) of planets per planetary system should be roughly proportional to parent star masses (Mâ) above 0.29M, or

    Np µ Mâ.


3. Conclusions

If the definitive prediction of a deficit of planetary systems for the lowest mass red dwarf stars is verified, then this would represent significant support for the Self-Similar Cosmological Paradigm.  If the predicted deficit is not observed after adequate testing, then this finding would indicate a low degree of cosmological self-similarity and would cast doubt upon the validity of the SSCP.


There is some tentative evidence in favor of the predicted deficit.  Of the extrasolar planetary systems discovered so far, F dwarfs account for ≈ 9%, G dwarfs account for ≈ 64% and K dwarfs account for ≈ 24%, but M dwarfs only account for ≈ 3%.  In fact, none of the 102 planets that have been identified at this writing, have been found orbiting stars with masses lower than ≈ 0.3 M¤.  Selection effects probably exert an influence on the available results, however, so we must wait for larger and convincingly unbiased samples.  Given the increasing rate at which extrasolar planetary systems are being discovered, an unambiguous verdict on the prediction discussed above should be available in the near future.



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