This “New Development” is the last segment of a 3-part study of discrete self-similar relationships between well-defined classes of variable stars and their specific Atomic Scale analogues. The first part, entitled “RR Lyrae Stars, Helium Atoms and Energy State Transitions,” demonstrated that RR Lyrae variables are the Stellar Scale equivalents of 4He atoms undergoing single-level transitions between excited states with 7 ≤ n ≤ 10 and l ≤ 1. A very large sample of RR Lyrae oscillation periods, and a smaller high resolution sample, were used to document strong correlations between observed Stellar Scale periods and Atomic Scale transition periods, when the latter measurements are scaled up in accordance with the discrete self-similar Scale transformation equations of the Self-Similar Cosmological Paradigm (SSCP).
The second part of the series, entitled “Evidence For Discrete Cosmological Self-Similarity In Delta Scuti Variable Stars,” demonstrated that the δ Scuti class is analogous to more massive atoms such as carbon, oxygen and nitrogen in excited states characterized by 3 ≤ n ≤ 6 and 0 ≤ l ≤ n-1. The high degree of heterogeneity in this class of variable stars prevented a straightforward period comparison test for large samples of δ Scuti stars and C/N/O atoms, as was possible in the RR Lyrae case. However, a specific δ Scuti star undergoing pure radial double-mode oscillations was shown to be quantitatively self-similar to its Atomic Scale counterpart: a 12C atom undergoing a radial mode transition between the n = 5 and n = 4 energy levels.
In this final part of the 3-part study devoted to the discrete cosmological self-similarity of variable stars, we turn our attention to ZZ Ceti variable stars. The complex multi-mode oscillation behavior of ZZ Cetis, their substantial mass range and their broad period range pose a considerable challenge to analysis in terms of the SSCP.
The SSCP emphasizes nature’s intrinsic and well-stratified hierarchical organization, proposing that the hierarchy is divided into discrete cosmological Scales, of which we can currently observe the Atomic, Stellar and Galactic Scales. The SSCP also proposes that the discrete Scales are rigorously self-similar to one another, such that for each class of fundamental particle, composite system or physical phenomenon on any given Scale there are self-similar analogues on all other Scales. At present the number of Scales cannot be known, but for reasons of natural philosophy it is tentatively proposed that there are a denumerably infinite number of Scales, ordered in terms of their intrinsic ranges of space, time and mass scales. The spatial (R), temporal (T) and mass (M) parameters of discrete self-similar analogues on neighboring Scales Ψ and Ψ-1 are related by the following set of discrete self-similar Scale transformation equations.
RΨ = ΛRΨ-1 , (1)
TΨ = ΛTΨ-1 and (2)
MΨ = ΛD MΨ-1 , (3)
where Λ and D are empirically determined dimensionless scale factors equal to 5.2 x 1017 and 3.174, respectively. The value of ΛD is 1.70 x 1056. The symbol Ψ is used as an index to distinguish different Scales, such that
Ψ =
and the Stellar Scale is usually assigned Ψ = 0. Thus, Atomic Scale systems and phenomena are designated by Ψ = -1 and Galactic Scale systems and phenomena are assigned Ψ = +1. The fundamental self-similarity of the SSCP and the recursive character of the discrete scaling equations suggest that nature is an infinite discrete fractal, in terms of its morphology, kinematics and dynamics. A more detailed review of the SSCP can be found in papers #1 and #2 of the “Selected Papers” section of this website.
There are two main goals to be achieved in the present study. Firstly,
using information about the general properties of ZZ Ceti stars,
we hope to identify the Atomic Scale analogues of ZZ Cetis and
to understand the physical states and processes associated with ZZ Ceti phenomena. Secondly, we attempt
to define and pursue empirical tests that will help in convincing us that
we have correctly analyzed a unique manifestation of discrete cosmological
self-similarity between Atomic and Stellar Scale systems.
ZZ Ceti stars are variable white dwarf stars, also known as DAV stars.1 Their all-important mass range2 is roughly from 0.53 M¤ to at least 1.10 M¤. Typical radii for white dwarf stars are on the order of 109 cm, or 104 km. Most of their mass is thought to be in a core object composed of He, C, O, and heavier atoms.3 Surrounding the core is a thin envelope of pure H whose mass has a canonical value4 of ≈ 10-4 Mstar, or roughly 6 x 10-5 M¤ to 8.7 x 10-5 M¤ depending on modeling assumptions.1,2,5 It is thought that the ZZ Ceti pulsations “are driven in the partial hydrogen ionization zone.”1 ZZ Ceti variables are multi-periodic, non-radial pulsators with an average of 3 independent low-amplitude oscillation modes per star, and ≥ 20 modes in one star is not uncommon. ZZ Ceti periods can be amazingly stable6 or they can vary significantly within a fraction of a day. The period range for ZZ Cetis is about 100 – 1200 sec and a histogram of observed oscillation periods for a sample3 of 82 ZZ Cetis is shown in Figure 1.
Figure 1. Distribution of periods for a combined sample of 82 ZZ Ceti variables from a paper by Mukadam et al.3 Data include all independent periods measured for these stars and some stars are observed multiple times by one or more research groups.
Because ZZ Ceti pulsation behavior is typically multi-periodic, non-radial, non-sinusoidal and low-amplitude, their unprocessed frequency spectra can include independent pulsations, various harmonics, combination frequencies, difference frequencies and aliases. In spite of many factors that might serve to mask discretization in ZZ Ceti frequency spectra, Figure 1 is certainly suggestive of a heterogeneous collection of discrete periods. Bergeron et al7 comment that “these pulsators show typically only a few excited modes out of the very rich g-mode spectra available to them.” Table 1 summarizes the properties of ZZ Ceti variables.
Mass Range | 0.53 M¤ to 1.10 M¤ |
Radii |
~ 109 cm |
Pulsation Properties |
Multi-mode, non-radial, low amplitude, non-sinusoidal |
Number of Periods |
Average = 3, Maximum ≥ 20 |
100 to 1200 sec |
Several hundred ZZ Ceti stars have been identified
to date and since they are relatively faint stars, they are all within 400
parsecs of the Sun5. Therefore observed ZZ Cetis are
Population I members of the Galaxy’s disk population, in contrast
to most classical variable stars, which tend to be Population II objects
and are more typically found throughout the Galaxy, including its halo and
globular clusters.
Of the three classes of variable stars that have been studied in this series (RR Lyraes, δ Scutis and ZZ Cetis), the ZZ Ceti class poses the biggest challenges to analysis within the context of the SSCP. Even within the context of conventional astrophysics the ZZ Cetis remain relatively enigmatic. Here we must be satisfied with achieving a less comprehensive and quantitative understanding than was possible in the case of RR Lyrae stars. However, in spite of the difficult nature of the subject, we will succeed in constructing and testing a first approximation SSCP assessment of ZZ Ceti phenomena that will serve as the foundation for future refinements in studies of the discrete cosmological self-similarity of ZZ Ceti stars.
The key parameter in SSCP investigations of variable stars is the mass range of the relevant systems. The approximate ZZ Ceti mass range is 0.53 M¤ ≤ M ≤ 1.10 M¤, and Eq. (3) can be used to calculate that the masses of Atomic Scale analogues range from 4 to 8 atomic mass units (amu). This mass range includes 4He, 6He, 6Li, 7Li, 8Li, 7Be, 8Be and 8B. Given reasonable assumptions based on observed abundances for various atoms and their isotopes, we can expect 4He to be the most common Atomic Scale analogue of ZZ Ceti stars, with a smaller contribution from 7Li, and probably only minor or trace contributions from 6He, 6,7Li, 7,8Be, 8B and heavier atoms.
Unlike RR Lyraes and δ Scutis, which are analogues of atoms in neutral states, ZZ Cetis appear to be analogues of positive ions. Two physical properties lead to this conclusion. Firstly, the extremely small size of white dwarf stars and Eq. (1) require small tightly bound wavefunctions that are only found in certain ionic states, such as He+ and more massive ions with few remaining electrons.8 Secondly, the unique structure of ZZ Ceti stars, especially the thin pure H outer envelope with a mass on the order of 10-4 M¤, also argues for one or two tightly bound electrons, since the mass of the Stellar Scale electron has been determined9 previously to be ≈ 7 x 10-5 M¤. Therefore, the SSCP proposes that the majority of ZZ Ceti stars are Stellar Scale analogues of 4He+, 7Li+ and 7Li++.
Having defined the primary Atomic Scale analogue systems and their ionic states, we would now like to explore the relevant range of n and Δn values that correspond to the oscillations observed for ZZ Ceti stars. In order to make progress in this direction we need to adopt a crucial hypothesis about the nature of ZZ Ceti oscillations. We will hypothesize that the dominant independent periods of ZZ Cetis (not including harmonics, aliases, or combination/difference periods) correspond to specific allowed transition periods of their Atomic Scale analogues. It is nearly inconceivable that multi-mode low-amplitude ZZ Cetis are actually undergoing discrete energy level transitions, since that would appear to require from 3 to 20 (or more) simultaneous transitions in systems most typically possessing one electron! It seems more reasonable and natural to interpret ZZ Ceti stars in terms of Atomic Scale analogues that are highly perturbed systems in a high-energy Galactic disk environment characterized by relatively strong electric, magnetic, pressure, temperature and radiation fields. Within the context of the SSCP, ZZ Cetis in the Galactic disk are analogous to He and Li ions within the equatorial disk of a Stellar Scale ultracompact object such as a neutron star or black hole. Although the multi-mode oscillations may not represent full-fledged energy level transitions, our working hypothesis here is that the perturbed ZZ Cetis are “ringing” at discrete transition frequencies that are rigorously self-similar to the corresponding discrete transition periods of their Atomic Scale analogues, but with amplitudes that are below the threshold for actual energy level transitions. This critical hypothesis will be empirically tested below.
The effort to determine relevant n and Δn values for the Atomic Scale analogues of RR Lyrae and δ Scuti stars was greatly facilitated by approximate relationships among the radii, periods and principle quantum numbers of Rydberg atoms. However, these helpful relationships are not applicable in the case of ZZ Ceti stars because they do not appear to be analogues of Rydberg atoms, i.e., neutral atoms with the single outermost electron excited to relatively high values of n. To make progress toward n and Δn estimates it is necessary to take an entirely empirical approach. We proceed by calculating the oscillation periods for all energy level transitions for 4He+ ions between 1 ≤ n ≤ 8, including Δn values of 1 ≤ Δn ≤ 6. Our effort is greatly simplified by the fact that the 4He+ ion has a remarkably degenerate energy level structure. This means that to a very good approximation (≤ 1% errors) we can treat the energy levels as discrete single-valued functions of n, i.e., with virtually no level splitting due to other quantum numbers. Table 2 presents the calculations based on data found in a standard source10 for atomic data, and lists predicted Stellar Scale oscillation periods derived from 4He+ transition periods and Eq. (2).
Transition |
Observed Atomic Scale Period (sec) |
Predicted
Stellar Scale Period (sec) |
||
6 ↔ 1 |
5 |
426717.129 |
7.812 x 10-17 |
40.6 |
4 ↔ 1 |
3 |
411477.158 |
8.101 x 10-17 |
42.1 |
2 ↔ 1 |
1 |
329179.744 |
1.013 x 10-16 |
52.7 |
8 ↔ 2 |
6 |
102871.310 |
3.240 x 10-16 |
168.5 |
7 ↔ 2 |
5 |
100771.958 |
3.308 x 10-16 |
172.0 |
6 ↔ 2 |
4 |
97537.385 |
3.417 x 10-16 |
177.7 |
5 ↔ 2 |
3 |
92172.941 |
3.616 x 10-16 |
188.1 |
4 ↔ 2 |
2 |
82297.414 |
4.050 x 10-16 |
210.6 |
3 ↔ 2 |
1 |
60961.198 |
5.468 x 10-16 |
284.3 |
8 ↔ 3 |
5 |
41910.112 |
7.954 x 10-16 |
413.6 |
7 ↔ 3 |
4 |
39810.760 |
8.373 x 10-16 |
435.4 |
6 ↔ 3 |
3 |
36576.187 |
9.113 x 10-16 |
473.9 |
5 ↔ 3 |
2 |
31211.743 |
1.068 x 10-15 |
555.3 |
4 ↔ 3 |
1 |
21336.216 |
1.562 x 10-15 |
812.4 |
8 ↔ 4 |
4 |
20573.896 |
1.620 x 10-15 |
|
7 ↔ 4 |
3 |
18474.544 |
1.804 x 10-15 |
938.2 |
6 ↔ 4 |
2 |
15239.971 |
2.187 x 10-15 |
1137.4 |
5 ↔ 4 |
1 |
9875.529 |
3.375 x 10-15 |
1755.2 |
8 ↔ 5 |
3 |
10698.369 |
3.116 x 10-15 |
1620.2 |
7 ↔ 5 |
2 |
8599.017 |
3.876 x 10-15 |
2015.7 |
6 ↔ 5 |
1 |
5364.444 |
6.214 x 10-15 |
3231.2 |
8 ↔ 6 |
2 |
5333.925 |
6.249 x 10-15 |
|
7 ↔ 6 |
1 |
3234.573 |
1.031 x 10-14 |
5358.8 |
Figure 2. The histogram of ZZ Ceti periods from Fig. 1 is overlain with red lines designating the positions of scaled transition periods of 4He+ in the 100 to 1200 sec range (see Table 1).
Our largely heuristic SSCP analysis of ZZ Ceti phenomena is summarized in Table 3.
Table 3. SSCP Analysis of the ZZ Ceti Class
Because the SSCP analysis of ZZ Ceti variable stars is quite speculative and requires the crucial hypothesis that the low-amplitude multi-mode oscillation periods of ZZ Cetis are directly related to the discrete transition periods of their Atomic Scale counterparts, it is highly desirable to test the SSCP analysis in a quantitative manner. A relevant test has been identified and conducted, but before presenting the results it is prudent to briefly review the physical factors that can interfere with a straightforward determination of discrete self-similarity between Atomic Scale and Stellar Scale analogues. These primary physical considerations are listed in Table 4 below.
Table 4. Factors Influencing Self-Similarity Tests for Atomic/Stellar Systems
Ambient
Galactic Scale parameters that can cause shifts in stellar oscillation
periods |
Factors
that generate additional or extraneous oscillation periods |
Electric fields |
Isotope heterogeneity |
Magnetic fields |
Analogue heterogeneity |
Temperature |
Singlet, doublet, … multiplicity |
Pressure |
Ionic state heterogeneity |
Radiation fields |
Harmonics, aliases, combin./diff. periods |
The ZZ Ceti sample of Bergeron et al2 has a considerable range of masses, as shown in Table 5.
Table 5. Mass2 and Dominant Periods3 for a sample of ZZ Ceti Variables
If our preliminary analysis of ZZ Ceti phenomena is on the right track, then the lowest mass stars are 4He+ analogues and their dominant oscillation periods will match up with scaled transition periods for 4He+. On the other hand, the high mass stars from the sample should not be 4He+ analogues and their dominant periods should not match up well with the scaled 4He+ transition periods. For our 4He+ analogue subsample we choose the four lowest mass systems (0.53 M¤ ≤ M ≤ 0.55 M¤) and concern ourselves only with the dominant oscillation period of each star. As shown in Figure 3, the results of this test agree with our expectations: each of the four dominant periods lines up close to one of the scaled transition periods of 4He+.
Figure 3. Superposition of the dominant periods
of the low mass ZZ Ceti subsample and the scaled 4He+ transition
periods.
It is interesting that in each case the observed period is slightly shorter than the matching predicted period, suggesting the possibility of a small systematic shift such as might be caused by an ambient magnetic field in the Galactic disk. The quantitative data for this test are given in Table 6.
Table 6. Quantitative Data Shown in Figure 3Star
ID |
Mass (M¤) |
Dominant
Period (sec) |
Closest
Scaled 4He+ Period (sec) |
Error
(%) |
0.53 |
209.2 |
210.6 |
0.7 |
|
HL Tau 76 |
0.55 |
541 |
555.3 |
2.6 |
G38 – 29 |
0.55 |
<924.2> |
938.2 |
1.5 |
G238 - 53 |
0.55 |
206.2 |
210.6 |
2.1 |
As a check on the uniqueness of the agreement between the observed periods of the low mass subsample and the predicted set of Stellar Scale periods, we can repeat the test for the high mass subset of ZZ Cetis listed in Table 5. Our second subsample consists of five stars with masses ranging from 0.75 M¤ to 1.10 M¤, and the dominant modes for these stars are compared with the scaled 4He+ transition periods in Figure 4. For this subsample of high mass ZZ Cetis we see very little agreement between the observed and predicted periods, as expected.
This result supports the contention that the agreement found in the low mass ZZ Ceti case was unique rather than coincidental. Quantitative data for this second part of our uniqueness test of the SSCP analysis are presented in Table 7.
Table 7. Quantitative Data Shown in Figure 4Star
ID |
Mass (M¤) |
Dominant
Period (sec) |
Closest
Scaled 4He+ Period (sec) |
Error
(%) |
BPM 30551 |
0.75 |
<742.5> |
8.6 |
|
HE 0532-5605 |
0.92 |
688.8 |
812.4 |
15.2 |
LTT 4816 |
1.10 |
192 |
188.1 |
2.7 |
G226-29 |
0.79 |
109.5 |
168.5 |
35.0 |
G207-9 |
0.83 |
318.0 |
284.3 |
11.9 |
In conclusion, the SSCP interprets the overwhelming majority of ZZ Ceti stars as discrete self-similar analogues of He+, Li+ and Li++ ions. These systems are highly perturbed and are hypothesized to be oscillating at discrete transition periods, but their low-amplitude and multi-mode characteristics seem to argue against an analogy to true energy level transitions.
The helium ion analogues, which should outnumber the lithium ion analogues, have relevant n values of 2 ≤ n ≤ 8 and relevant Δn values of 1 ≤ Δn ≤ 6. Preliminary results indicate that the ranges of n and Δn are very similar for the lithium ion analogues, but an analysis of this smaller, more complicated, subset of analogues must remain a project for the future. The helium analogues appear to be oscillating at specific sets of transition periods associated with transitions to or from n = 2, n = 3 and n = 4.
A more complete understanding of ZZ Ceti stars
will require inclusion of scaled frequency spectra for the more complicated
Li ions and heavier ions. Accurate mass and radius data
for individual stars would greatly reduce the complexity of the research
effort by putting limits on relevant Atomic Scale analogues, their ionic
states and their energy levels.