R. L. OLDERSHAW

40

and Galactic Scales [1-6], a close approximation to na-

ture’s discrete self-similar scale transformation equations

for the length (L), time (T) and mass (M) parameters of

analogue systems on neighboring cosmological scales

and

–1, as well as for all dimensional constants, are as

follows.

L

=

L

–1 (1)

T

=

T

–1 (2)

M

=

D M

–1 (3)

The self-similar scaling constants

and D have been

determined empirically and are equal to 5.2 × 1017 and

3.174, respectively [1-6]. The value of

D is 1.70 × 1056.

Different cosmological scales are designated by the dis-

crete index

( , –2, –1, 0, 1, 2, ) and the Atomic,

Stellar and Galactic Scales are usually assigned

= –1,

= 0 and

= +1, respectively.

The fundamental self-similarity of the SSCP and the

recursive character of the discrete scaling equations sug-

gest that nature is an infinite discrete fractal, in terms of

its morphology, kinematics and dynamics. The underly-

ing principle of the paradigm is discrete scale invariance

and the physical embodiment of that principle is the dis-

crete self-similarity of nature’s physical systems. Perhaps

the single most thorough and accessible resource for ex-

ploring the SSCP and Discrete Scale Relativity is the

author’s website [6].

1.2. Discrete Self-Similarity of Variable Stars

and Excited Atoms

Discrete Scale Relativity hypothesizes that each well-

defined class of systems on a given cosmological scale

has a discrete self-similar class of analogue systems on

any other cosmological scales

x. Given their mass

ranges, radius ranges, frequency ranges, morphologies

and spherically harmonic oscillation phenomena, Dis-

crete Scale Relativity uniquely, unambiguously and

quantitatively identifies variable stars as discrete scale

invariant analogues of excited atoms undergoing energy-

level transitions, or oscillating with sub-threshold ampli-

tudes at the allowed frequencies of a limited set of en-

ergy-level transitions. The latter subclass of multiple-

period/low-amplitude oscillators can be interpreted as

systems in sub-threshold superposition states.

In four previous papers the discrete self-similarity

among three different classes of variable stars and their

classes of analogue systems on the Atomic Scale was

quantitatively demonstrated and empirically tested. Paper

(I) [7] identified RR Lyrae variables as analogues of he-

lium atoms on the basis of their narrow mass range.

Given that one atomic mass unit (amu) equals approxi-

mately 1.67 × 10–24 g, one can use Equation (3) to de-

termine that one stellar mass unit (SMU) equals ap-

proximately 0.145 M. The average mass of a RR Lyrae

star (0.6 M) can then be divided by 0.145 M/SMU to

yield a value of 4 SMU. Using the period (P) distribu-

tion for RR Lyrae variables, the pn n3p0 relation for

Rydberg atoms, and the temporal scaling of Equation (2),

one can identify the relevant principal quantum numbers

(n) for RR Lyrae stars as predominantly n = 7 to n = 10.

Because RR Lyrae variables are primarily fundamental

radial-mode oscillators, we can assume that the most

probable angular quantum numbers (l) are l = 0 or l = 1.

It was further assumed that the analogue transitions are

most likely to be single-level transitions. To test the

foregoing conceptual and quantitative analysis of the

discrete self-similarity between RR Lyrae stars and neu-

tral helium atoms in Rydberg states undergoing sin-

gle-level transitions between n = 7 and n = 10, with l = 0

or 1, a high resolution [8] period spectrum for 84 RR

Lyrae stars was compared with a predicted period spec-

trum derived by scaling the empirical helium transition

data in accordance with Equation (2). Ten separate peak

and gap structures identified in the stellar period spec-

trum were found to correspond quantitatively to coun-

terpart peaks and gaps in the scaled helium period spec-

trum.

In Paper (II) [9] the same analysis was extended to a

very large sample of over 7,600 RR Lyrae stars, and

again the quantitative match between the observed and

predicted period spectra supported the proposed discrete

self-similarity hypothesis, albeit with some loss of reso-

lution due to the sample size.

In Paper (III) [10] the same analytical approach was

applied to high-amplitude

Scuti variables. This class of

variable stars was found to correspond to a heterogene-

ous class of systems with masses ranging from 10 SMU

to 17 SMU. The period distribution indicated transitions

in the n = 3 to n = 6 range and l values were again in the

0 to 1 range. A specific

Scuti variable, GSC 00144 -

03031, was identified as a neutral carbon atom undergo-

ing a 12C [1s22s22p5p 4p, (J = 0), 1S] transition. Once

again, there was an apparently unique agreement be-

tween the observed oscillation period for the variable star

and the predicted period derived from the atomic data for

the identified atomic analogue.

Finally, in Paper (IV) [11] the enigmatic class of ZZ

Ceti variable stars was analyzed within the context of

Discrete Scale Relativity. Two properties of this class of

variable stars argue that it is much more heterogeneous

than the RR Lyrae class or the

Scuti stars. Firstly, the

mass range of roughly 0.6 M to 1.10 M corresponds to

atoms with masses in the 4 to 8 amu range, i.e., 4,6He,

6,7,8Li, 7,8Be and 8B, with the overwhelming majority ex-

pected to be analogues of 4He and 7Li. Secondly, the

broad and relatively erratic period spectrum for this class

Copyright © 2011 SciRes. IJAA