International Journal of Astronomy and Astrophysics , 2011, 1, 39-44
doi:10.4236/ijaa.2011.12006 Published Online June 2011 (http://www.SciRP.org/journal/ijaa)
Discrete Scale Relativity and SX Phoenicis Variable Stars
Robert Louis Oldershaw
12 Emily Lane, Amherst, Massachusetts, USA
Received April 28, 2011; revised May 29, 2011; accepted June 3, 2011
Abstract
Discrete Scale Relativity proposes a new symmetry principle called discrete cosmological self-similarity
which relates each class of systems and phenomena on a given scale of nature’s discrete cosmological hier-
archy to the equivalent class of analogue systems and phenomena on any other scale. The new symmetry
principle can be understood in terms of discrete scale invariance involving the spatial, temporal and dynamic
parameters of all systems and phenomena. This new paradigm predicts a rigorous discrete self-similarity
between Stellar Scale variable stars and Atomic Scale excited atoms undergoing energy-level transitions and
sub-threshold oscillations. Previously, methods for demonstrating and testing the proposed symmetry princi-
ple have been applied to RR Lyrae,  Scuti and ZZ Ceti variable stars. In the present paper we apply the
same analytical methods and diagnostic tests to a new class of variable stars: SX Phoenicis variables. Dou-
ble-mode pulsators are shown to provide an especially useful means of testing the uniqueness and rigor of
the conceptual principles and discrete self-similar scaling of Discrete Scale Relativity. These research efforts
will help theoretical physicists to understand the fundamental discrete self-similarity of nature, and to model
both stellar and atomic systems with one unified physics.
Keywords: Discrete Scale Relativity, SX Phe Variables, Rydberg Atoms, Quantum Cosmology
1. Introduction
1.1. Preliminary Discussion of Discrete
Cosmological Self-Similarity
The arguments presented below are based on the Self-
Similar Cosmological Paradigm (SSCP) [1-6] which has
been developed over a period of more than 30 years, and
can be unambiguously tested via its definitive predictions
[1,4] concerning the nature of the galactic dark matter.
Briefly, the discrete self-similar paradigm focuses on na-
ture’s fundamental organizational principles and symme-
tries, emphasizing nature’s intrinsic hierarchical organiza-
tion of systems from the smallest observable subatomic
particles to the largest observable superclusters of galaxies.
The new discrete fractal paradigm also highlights the fact
that nature’s global hierarchy is highly stratified. While
the observable portion of the entire hierarchy encompasses
nearly 80 orders of magnitude in mass, three relatively
narrow mass ranges, each extending for only about 5 or-
ders of magnitude, account for ≥ 99% of all mass observed
in the cosmos. These dominant mass ranges: roughly 10–27
g to 10–22 g, 1028 g to 1033 g and 1038 g to 1043 g, are re-
ferred to as the Atomic, Stellar and Galactic Scales, re-
spectively. The cosmological scales constitute the discrete
self-similar scaffolding of the observable portion of na-
ture’s quasi-continuous hierarchy. At present the number
of scales cannot be known, but for reasons of natural phi-
losophy it is tentatively proposed that there are a denu-
merable infinite number of cosmological scales, ordered in
terms of their intrinsic ranges of space, time and mass
scales. A third general principle of the new paradigm is
that the cosmological scales are rigorously self-similar to
one another, such that for each class of fundamental parti-
cles, composite systems or physical phenomena on a given
scale there is a corresponding class of particles, systems or
phenomena on all other cosmological scales. Specific
self-similar analogues from different scales have rigor-
ously analogous morphologies, kinematics and dynamics.
When the general self-similarity among the discrete scales
is exact, the paradigm is referred to as Discrete Scale
Relativity [5] and nature’s global space-time geometry
manifests a new universal symmetry principle: discrete
self-similarity or discrete scale invariance.
Based upon decades of studying the scaling relation-
ships among analogue systems from the Atomic, Stellar

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

R. L. OLDERSHAW41
of variables has the appearance of a heterogeneous col-
lection of discrete periods. The fact that these white dwarf
stars usually have radii well below the Stellar Scale Bohr
radius of 2.76 × 109 cm strongly suggests that they cor-
respond to ions that have lost one or more electrons. The
facts that so many ZZ Cetis are multi-mode pulsators and
that the pulsations have quite low amplitudes suggest that
they are equivalent to highly perturbed ions undergoing
sub-threshold oscillations at allowed transition frequen-
cies for transitions with 2  n  8, 1  n  6, and 1  l 
7. Unique tests of Discrete Scale Relativity using ZZ Ceti
stars are made very difficult by the heterogeneity of this
class of variables. However, the observed and predicted
oscillation periods of a subsample of low-mass He+ ana-
logues showed good correspondence, whereas the ob-
served oscillation periods for a high-mass ZZ Ceti sub-
sample did not match up with the predicted period spec-
trum based on He+ data, as would be expected. More de-
finitive tests of discrete cosmological self-similarity in-
volving ZZ Ceti stars are a future research goal.
1.3. SX Phoenicis Variable Stars
Fortunately, the SX Phoenicis class of variable stars are
high-amplitude, low-l, oscillators that are more like the
RR Lyrae stars and far less heterogeneous than the ZZ
Ceti stars. They are Population II field stars with masses
in a range of roughly 1.4 M to 2.0 M which is similar
to the mass range of
 Scuti stars, but with shorter peri-
ods and higher amplitudes than is typically the case with
the
 Scuti stars of Population I. Given a reasonably ac-
curate mass estimate for a specific SX Phoenicis star,
and a knowledge of its basic oscillation properties, we
can readily identify its Atomic Scale analogue and make
testable predictions about specific energy-level transi-
tions. In the present paper, three SX Phoenicis variables:
DY Pegasi, BL Camelopardalis and QU Sagittae, will be
analyzed and tested using the concepts and quantitative
scaling of Discrete Scale Relativity.
2. DY Pegasi
The SX Phoenicis variable star DY Pegasi has a mass of
approximately 1.5 M and an effective temperature (Teff)
of approximately 7660 Kelvin [12]. The star exhibits a
regular radial-mode pulsation at a frequency of 13.713
cycles per day, with a relatively large amplitude of 0.2455
magnitudes [13]. In Table 1 the basic physical charac-
teristics of DY Peg are summarized.
Given the basic physical properties of DY Peg, we can
use Discrete Scale Relativity (DSR) to determine the spe-
cific Atomic Scale analogue for this specific Stellar Scale
system, and we can identify a very limited set of two spe-
cific atomic E-level transitions that could possibly be
Table 1. Physical properties of DY Peg.
Mass (M) ~1.5 M

Effective Temperature (Teff) ~7660 K
Frequency (  ) 13.713 d–1
Period (P) 0.072926 d = 6300.82 sec
Amplitude (A) 0.2455 mag
Oscillation Mode radial
consistent with a rigorous analogy to the star’s pulsation
characteristics. The DSR analysis can then be tested
quantitatively by the prediction that one, and only one, of
the two-member set of possible atomic E-level transitions
will have a period (p) that is related to the period (P) of
DY Peg in the discrete self-similar manner required by
Equation (2), i.e., P =
 p.
Given the star’s mass of  1.5 M and the DSR rela-
tion: 1 SMU = 0.145 M, one can determine that DY
Pegasi is a 10 SMU system and that the most likely
Atomic Scale analogue with 10 amu is an excited Boron
atom. Given the star’s period of approximately 6300.82
sec, one can determine the relevant value of n using the
Stellar Scale analogue of the n versus p relation for
Rydberg atoms: pn = n3p0, where p0 for the Stellar Scale
is  p0 or 78 sec. The value of n derived by this ap-
proximate method is 4.3, and so one can specify that
the relevant E-level transition is predominantly associ-
ated with n = 4. Given that the pulsation of DY Peg is
primarily a fundamental radial-mode oscillation, one can
safely assume that the relevant E-level transition must
involve l values of 0 or 1.
The two energy-level transitions for Boron that would
fit our n and l requirements are the [1s22s25s → 1s22s24s]
transition and the [1s22s24p → 1s22s24s] transition [13].
The latter transition has a  E of 2776.826 cm–1, which
corresponds to an oscillation frequency (  ) of 8.303 ×
1013 sec–1 and an oscillation period (p) of 1.204 × 10–14
sec. The question then is whether P =  p for this specific
E-level transition. In fact,  p = 6263.02 sec, which
agrees with the star’s P of 6300.82 sec at the 99.4% level.
The [5s → 4s] transition yields a predicted P of 3386.74
sec which does not agree with 6300.82 sec. Therefore,
one can conclude that the [1s22s24p → 1s22s24s] transi-
tion for Boron is the unique E-level transition identified
by Discrete Scale Relativity as the Atomic Scale ana-
logue for the high-amplitude fundamental oscillation
exhibited by DY Pegasi. The factor of 0.006 different
between P and
 p has three possible origins. Firstly,
ambient Stellar Scale electromagnetic fields can shift the
E-levels of systems in Rydberg states. Secondly, the
mass of DY Peg indicates that it is an analogue to 10B
while the atomic data used in our calculations is based on
the more common isotope 11B. Finally, and perhaps most
likely, there is a small inevitable uncertainty in the value
of  due to the fact that it is determined empirically and

R. L. OLDERSHAW 42
Table 2. Physical properties of the atomic scale analogue of
DY Pegasi
Mass 10 amu (Boron)
n ~4
l ~0
E 2776.826 cm–1
 8.303 × 1013 sec–1
p 1.204 × 10–14 sec
Mode ~radial
Transition [1s22s24p → 1s22s24s]
is therefore still a first approximation [3]. The physical
properties of the proposed Atomic Scale analogue system
for the DY Peg system are given in Table 2.
For Atomic Scale systems it is well-known that their
energies and frequencies are related by Planck’s law: E =
h
 . Discrete Scale Relativity predicts that an analogous
relationship should be equally valid for the Stellar Scale
analogues of Atomic Scale systems. Therefore DSR pre-
dicts that high-amplitude variable stars that constitute
systems undergoing bona fide E-level transitions should
obey a general law of the form: E = H
 , where H =

4.174h  5.86 × 1047 erg·sec. This highly important pre-
diction could be tested if an independent method of de-
termining E for the variable stars could be identified. In
that case E/
 should always be an integral multiple of H.
Future research efforts should explore the possibilities of
using stellar masses, effective temperatures, luminosities,
spectroscopic properties, etc. to uniquely identify and
quantify individual discrete E-levels for stars.
3. The SX Phoenicis Variable of QU Sagittae
It is statistically unlikely that the discrete self-similarity
between the physical characteristics and pulsation phe-
nomena of DY Peg and Boron [4 p → 4 s] was a fortui-
tous coincidence, but it is desirable to rule out that possi-
bility more definitively. It is proposed here that the fol-
lowing successful applications of the same DSR methods
to two additional SX Phoenicis variables, each a dou-
ble-mode pulsator, will remove any reasonable doubt
about the fact that that these SX Phe stars and their
Atomic Scale Boron analogues share a discrete self-simi-
larity which is unique and quantitative. The second SX
Phe star to be discussed in this paper is located in a binary
star system known as QU Sagittae [14]. This SX Phe va-
riable is a double-mode oscillator with a relatively
low-amplitude of about 0.024 magnitudes and a relatively
high (> 0.8) period ratio that suggests a nonradial pulsa-
tion mode for at least one of the oscillations. The primary
period Pα is 2407.825 sec and the secondary period Pβ is
2167.206 sec. Given the Pn = n3P0 relation used above,
one may calculate an approximate value of n  3.1 for this
multi-mode oscillator. The high period ratio suggests that
one should conservatively anticipate values of l in the 0 -
2 range [14].
These stellar characteristics allow a very straightfor-
ward and quantitative test of the DSR analysis of this
double-mode SX Phe star. If the analysis is unique and
correct, then Boron will have two energy-level transi-
tions associated with n  3 and 0  l  2 whose oscilla-
tion periods are related to Pα and Pβ by the DSR scaling
for temporal periods (Equation 2). In fact, we find that
the [1s22s24s → 1s22s2p2] transition has a Eα = 7152.55
cm–1 and a pα = 4.676 × 10–15 sec. The value of
 pα
equals 2431.48 sec, which agrees with Pα at the 99%
level. Likewise, we find that the [1s22s2p2 → 1s22s23s]
transition has a Eβ of 7817.35 cm–1 and a pβ = 4.278 ×
10–15 sec. The value of
 pβ equals 2224.71 sec, which
agrees with Pβ at the 97% level.
In the case of the SX Phoenicis star in the QU Sge bi-
nary, we have used the quantitative Stellar Scale oscilla-
tion properties of the star and DSR to successfully pre-
dict two analogous Atomic Scale oscillation periods for
Boron associated with transitions involving n  3 and 0 
l  2. Two additional interesting properties should be
noted: for this particular set of analogues the amplitudes
of the stellar oscillations are quite low and the two iden-
tified E-level transitions share the [1s22s2p2] level. A
question for future research is whether this variable star
is in a superposition of two linked but sub-threshold
transition periods, or whether this star is simultaneously
undergoing two separate but correlated transitions. The
long-term evolution of the double-mode pulsation should
provide clues to answering this question.
4. BL Camelopardalis
The final SX Phoenicis variable star to be discussed in
this paper, BL Cam, is an interesting and uniquely diag-
nostic multi-mode SX Phe star [15]. The fundamental
radial-mode oscillation has a relatively large amplitude
of about 0.146 magnitudes and a period (Pα) of 3378.24
sec, or approximately 0.0391 days. Given the Pn  n3P0
relation as a rough guide to the relevant value of n for
this star, we calculate that n  4. The large amplitude and
unambiguous radial-mode of this oscillation clearly point
to l = 0. Therefore we can predict that Boron will have a
low-l energy-level transition strongly associated with the
[1s22s24s] level, and with a scaled oscillation period of
 pα  3378 sec. In fact, we find that the [1s22s25s →
1s22s24s] transition has a pα of 6.513 × 10–15 sec and a
 pα = 3386.74 sec. The agreement between Pα for this
SX Phoenicis star and  pα for the relevant E-level transi-
tion in Boron is at the 99.8% level.
The unique and quantitative specificity of the proposed
discrete self-similarity between BL Cam and a Boron

R. L. OLDERSHAW43
atom undergoing a [5s → 4s] transition between Rydberg
states can be further verified by taking advantage of the
multi-mode pulsation of this star. Several of the additional
pulsation periods appear to be close multiples (e.g., 2Pα)
and combinations (e.g., Pα + Pβ) which are not particularly
diagnostic. However, one pulsation period (Pβ) at 2727.65
sec (or about 0.03157 days) clearly appears to be an inde-
pendent oscillation period that does provide a confirma-
tory test of the DSR analysis. Using the Pn  n3P0 relation
we find that this oscillation is associated with n  3. Since
it is a relatively low amplitude oscillation (0.007 mag),
and since the ratio of the fundamental period and this sec-
ondary period indicates that it is a radial first overtone
pulsation [15]. We can expect l  1 for this oscillation.
Therefore we can predict that Boron will have an E-level
transition associated with n  3 and l  1 that has an oscil-
lation period pβ which scales to approximately  pβ  2728
sec in accordance with Equation (2). We may further ex-
pect, on the basis of finding that the two modes of the
double-mode pulsator of QU Sge appeared to be linked
transitions (i.e., they shared a common E-level), that the
two oscillation periods of BL Cam are also linked. In fact,
our highly unique 4-part prediction is vindicated in a con-
vincing manner. The [1s22s24s → 1s22s23p] transition for
Boron is: 1) associated with n  3, 2) involves a change of
l from l = 0 to l = 1, 3) is linked to the fundamental oscil-
lation period through the common 4s E-level, and 4) has a
 pβ of 2719.11 sec, which agrees with the predicted pe-
riod of 2727.65 sec at the 99.7% level.
This empirical support for the theoretical hypothesis of
discrete cosmological self-similarity between BL Cam
and an excited Boron atom undergoing E-level transitions
in the 3  n  5 range, with 0  l  1, certainly appears to
be quite definitive. Coincidence is not a viable explana-
tion for these successful results, and the 4-part specificity
of the empirical testing would appear to minimize the
possibility of subjectivity or arbitrariness in the analysis.
5. Conclusions
Given the masses and period data for three SX Phoenicis
variables, we have been able to make predictions about
specific energy-level transitions of a specific type of ex-
cited atom (Boron). Technically these predictions are di-
agnostic retrodictions since the Atomic Scale data was
published before the tests were conducted. For each of the
five indentified pulsation periods, the Stellar Scale oscilla-
tion period was approximately  times larger than the
uniquely specified Atomic Scale oscillation period, with
an average agreement at the 99% level. Two of the SX
Phe stars analyzed here were double-mode pulsators that
allowed additional degrees of specificity in testing the
discrete cosmological self-similarity between SX Phe stars
and excited Boron atoms oscillating at published E-level
transition frequencies. The probability is exceedingly
small that five multi-part tests, involving mass, n, l, and
frequency values, could succeed in this manner by chance
or even by the use of subjective methodology. Therefore, a
high degree of discrete self-similarity between these stel-
lar/atomic analogues from neighboring Scales of nature’s
discrete cosmological hierarchy appears to have been veri-
fied quantitatively.
Some additional comments and questions for future
research are as follows.
1) Clearly stellar mass is, or should be, the definitive
parameter in classifying variable stars, as is the case with
atoms. Although periods, pulsation modes, temperatures
and spectroscopic properties are useful diagnostic char-
acteristics, variable stars should be ordered and classified
primarily by their masses in rigorous analogy to the
Atomic Scale classification of atomic elements.
2) It will be of interest to explore the apparent cou-
pling of double-mode pulsations, wherein a shared
E-level is identified. This phenomenon has been docu-
mented for QU Sge, BL Cam and the δ Scuti star GSC
00144-03031 [10]. Is this a common characteristic of all
double-mode Stellar Scale variables? What physical
mechanism could explain this coupling of simultaneous
oscillation phenomena? Is an analogous Atomic Scale
phenomena known?
3) Are the low-amplitude oscillations of multi-mode
variable stars sub-threshold oscillations that would be
virtually unobservable in Atomic Scale systems, since
they would be below the energy threshold required for
emission of quanta?
4) Planck discovered the famous relation E = h  for
atoms, and Discrete Scale Relativity predicts an equivalent
relation E = H  for the high-amplitude variable stars. We
can measure stellar  values quite accurately and we can
predict that H = ( 
D + 1h) or about 5.86 × 1047 erg·sec in
accordance with the discrete scale invariance of DSR [5].
If there were a way to define the Stellar Scale E-levels, the
proposed E = H  relation could be empirically tested.
Taken together the evidence for discrete self-similarity
involving RR Lyrae stars, δ Scuti stars, ZZ Ceti stars, SX
Phoenicis stars, and their respective Atomic Scale ana-
logues are very strong and cannot be due to coincidence
or subjective methods. Discrete scale invariance allows
one to investigate equivalent systems that differ in
lengths and temporal periods by a scale factor of 5.2 ×
1017, and that differ in masses and energies by a scale
factor of 1.70 × 1056. If the reality of Discrete Scale
Relativity were established and accepted, the resulting
paradigmatic advance would create a very rich potential
for new breakthroughs in our understanding of stellar

R. L. OLDERSHAW 44
and atomic systems.
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