New Developments 2005

Evidence for Discrete Cosmological Self-Similarity in Delta Scuti Variable Stars [December 2005]

I. Introduction

This “New Development” builds upon the results obtained in its predecessor, which discussed the discrete fractal behavior of RR Lyrae variables.  For that class of stars, it was demonstrated that the masses, radii and oscillation periods of RR Lyraes have a discrete self-similar relationship to the masses, radii and transition periods of their Atomic Scale counterparts: helium atoms in moderately excited Rydberg states undergoing single-level transitions.  The techniques that were used to achieve this unique result are now applied to a distinctly different class of variables: δ Scuti stars.  The new results contain some interesting surprises, but they are in general agreement with the discrete fractal paradigm and offer additional supportive evidence for the principle of discrete cosmological self-similarity.

The presentation given here assumes a basic understanding of the SSCP (see Paper #1 in the “Selected Papers” section of this website) and a familiarity with the previous “New Development” (see “RR Lyrae Stars, Helium Atoms And Energy State Transitions” – Sept 2005).

At critical points in this presentation we will rely on our trusty discrete scale transformation equations:

                                    R = Λr           ,                         (1)

                                   P = Λp           and                    (2)

                                   M = Λ^D m      ,                         (3)

where R, P and M are radii, periods and masses of Stellar Scale systems and r, p and m are the equivalent parameters for their Atomic Scale counterparts.  As before, Λ and D are dimensionless self-similar scaling constants with values of ≈ 5.2 x 10¹⁷ and ≈ 3.174, respectively.  The numerical value of Λ^D is ≈ 1.70 x 10⁵⁶.

II. Delta Scuti Variable Stars

Compared with RR Lyrae stars, δ Scutis are a somewhat unruly lot.  For example, the amplitudes of their oscillation periods can vary radically.  As Breger and Pamyatnykh¹ point out: “A star may change its pulsation spectrum to such an extent as to appear as a different star at different times”, although “modes do not completely disappear, but are still present at small amplitudes.”  Here we will exclusively work with high amplitude δ Scuti stars (HADS) because their pulsation behavior is simpler (less multi-periodic) and more regular than low amplitude δ Scutis (LADS), and because they are thought to pulsate mainly in radial modes,² which helps in identifying specific energy level transitions.  Delta Scutis have spectral classifications of A to F and can be designated dwarf or subgiant stars.³  Most importantly, their masses⁴ typically range from 1.5 M_¤ to 2.5 M_¤.  This is a major change from the situation with the RR Lyrae class wherein the lower, narrower and better-defined mass range was 0.4 M_ to 0.6 M_¤.

As of 2004 roughly 400 δ Scutis were known.  A typical oscillation period would be 0.1 day, and the cutoffs for this class occur at roughly 0.04 day and 0.2 day.  In Figure 1 we show a representative histogram of oscillation periods for 193 HADS.² Superimposed upon this distribution are lines corresponding to typical oscillation periods for Stellar Scale n values, which will be derived and explained below.

Figure 1.  Histogram of periods for 193 high amplitude δ Scuti stars.2  Superimposed upon the period distribution are lines representing the scaled oscillation periods for Rydberg atoms excited to various n values.

In this particular sample of 193 HADS stars, there were 40 double-mode pulsators and twelve stars that simultaneously pulsed at 3 or more periods.  Unfortunately, data on the radii of HADS stars appear to be very limited; this causes a minor problem with regard to identifying relevant energy levels, but the problem can be circumvented.

III. Atomic Scale Counterparts of δ Scuti Stars

Given Eq. (3) and the mass of the proton (m_p), we can calculate that the Stellar Scale proton analogue will have a mass of about 0.145 M_¤.  The mass range (1.5 M_¤ – 2.5 M_¤) for δ Scuti stars would correspond to an Atomic Scale mass range of approximately 10 m_p to 17 m_p, or about 10 – 17 atomic mass units (amu).  The atoms that dominate this mass range are Boron (11 amu), Carbon (12 amu), Nitrogen (14 amu) and Oxygen (16 amu).  Therefore the majority of δ Scuti stars are hypothesized to be analogues of these atoms.  Because reliable radius data are not available, an alternative method must be derived for determining the approximate energy levels of the relevant Atomic Scale transitions.  There is a general relationship between the radial quantum numbers (n) for Rydberg atoms and their oscillation periods of the form:

                                                p_n ≈ n³p_o              ,                                   (4)

where p_o is the classical orbital period of the groundstate H atom:≈ 1.5 x 10^-16 sec.  The Stellar Scale equivalent to this relationship would be:

                                                P_n ≈ n³P_o          ,                                   (5)

where P_o ≈ Λp_o ≈ (5.2 x 10¹⁷)(1.5 x 10^-16 sec) ≈ 78 sec.  Using Eq. (5) we can generate the lines plotted in Fig. 1 for the different oscillation periods associated with Stellar Scale n values.  These results tell us that if the discrete fractal paradigm is correct, and if we are dealing with analogues to Rydberg atoms undergoing transitions with Δn≈ 1, then the relevant range of n values for δ Scuti variables is predominantly 3 ≤ n ≤ 6, with n = 5 being the most probable radial quantum number for this class of stars.

The above considerations lead us to conclude that δ Scuti variables correspond to a very heterogeneous set of B, C, N and O atoms excited to Rydberg states with n approximately varying between 3 and 6.  The next step is to test this hypothesis by comparing the oscillation periods of δ Scuti stars with specific oscillation periods of their predicted Atomic Scale counterparts.  However, there is a serious problem with a straightforward comparison between Stellar Scale and Atomic Scale frequency/period spectra, as was possible in the case of the RR Lyrae class.  The main source of this problem is the heterogeneity of the δ Scuti class of stars.  We are dealing with analogues of at least four different atoms, each of which can be in several different isotopic configurations.  Moreover, each of those four species of atom can be in several different ionization states: -1, neutral, +1, +2, etc.  A further complication is that each of the atoms can have an entirely different set of energy levels for each of the singlet, doublet, triplet, etc., spin-related designations that apply.  Also, we are less confident about using the Δn ≈ 1 and l ≤ 1 restrictions that were so helpful in the RR Lyrae case.  Therefore, we are faced with a very large number of potential Atomic Scale transition periods to compare with the δ Scuti period distributions.  A meaningful test requires that the number of Stellar Scale periods should not be outnumbered by the experimental Atomic Scale periods with which they are to be compared.  In the case of the δ Scuti class of stars, the number of empirical comparison periods would be in the hundreds, exceeding the number of observed HADS periods!  Compounding this general lack of Atomic Scale specificity are the usual uncontrollable physical factors that can result in additional shifting of Stellar Scale energy levels away from unperturbed values: ambient pressures, temperatures, electric fields and magnetic fields.

Fortunately, there is a way to circumvent the alarming specificity problems discussed above.  If we have enough accurate information for an individual δ Scuti star, then we can use the data to identify the specific Atomic Scale analogue of the star, to restrict the number of possible energy level transitions for that specific atom, and to construct a valid test between a uniquely predicted oscillation period and a reasonably restricted number of Atomic Scale comparison periods.  Very recently, the δ Scuti star GSC 00144–03031 was analyzed in detail by Poretti et al⁵ and this system has certain characteristics that make it an excellent test star for our purposes.  First and foremost, its mass has been determined with reasonable accuracy and is approximately 1.75 M_.  Using our knowledge that 0.145 M_¤ ≈ 1 SMU (Stellar Mass Unit) ≈ Λ^D amu (atomic mass units) ≈ Λ^Dm_p, we can determine that 1.75 M_¤ corresponds to 12 SMU and therefore GSC 00144–03031 can be identified with a high degree of confidence as an analogue of a ¹²C atom.  Other advantages of using this star as a test system are that it is a classic HADS system (regular, high amplitude pulsations), that it has a dominant fundamental mode that is highly radial in character (which helps in narrowing energy level possibilities), and finally that it is a pure double-mode pulsator (providing us with an second test period).  The fundamental radial mode has a period of about 0.058 day (≈ 83.6 min or ≈ 5017.42 sec) and its amplitude is 4 times greater than the secondary pulsation which has a period of ≈ 3872.70 sec.

Table 1.  Physical Properties of GSC 00144–03031⁵

IV. Test of the Discrete Self-Similarity Principle

If the principle of discrete cosmological self-similarity is correct, then we should be able to use the oscillation periods of GSC 00144-03031 to derive predicted oscillation periods for comparison with empirical oscillation periods of its Atomic Scale analogue undergoing corresponding transitions.  We have identified the Atomic Scale analogue as a ¹²C atom and the transition has a strong radial mode (l = 0) character.  From Figure 1, we can determine that the position of the dominant period for GSC 0144–03031 falls between the n ≈ 5 and the n ≈ 4 lines and so we anticipate a n = 5 → 4 (low l) transition.  Using Eq. (2) we can calculate the predicted Atomic Scale period:

                        p  ≈  P ÷ Λ  ≈  5017.42 sec ÷ 5.2 x 10¹⁷ ≈  9.65 x 10^-15 sec.

We assume that the ¹²C atom is most likely to be in a neutral, rather than an ionized, state.  Therefore the quantitative test is whether a neutral ¹²C atom has a radial mode transition between the n = 5 and n = 4 levels that involves an oscillation period of about 9.65 x 10^-15 sec.  We use a standard source for atomic energy level data⁶ and find that the n = 5 energy level with the least non-radial character is the 1s²2s²2p5p(J=0) ¹S singlet level with an energy of 85625.18 cm^-1.  The n = 4 energy level with the least non-radial character is the 1s²2s²2p4p(J=0) ¹S singlet level with an energy of 82251.71 cm^-1.  Subtracting the energies for these neighboring energy levels gives 3373.47 cm^-1 as the transition energy (ν) for the n = 5(J=0) → n = 4(J=0) transition.  We can calculate the oscillation period for the transition by using the relation ν = νc for electromagnetic radiation, and we find that ν = (3373.47 cm^-1)(2.99 x 10¹⁰ cm/sec) = 1.01 x 10¹⁴ sec^-1.  Since p = 1/ν, the period for the transition is 9.88 x 10^-15 sec.  This value is higher than the predicted value of 9.65 x 10^-15 sec by a factor of 0.024, but considering the numerous sources of small uncertainties that are involved in this test, and the uncontrollable physical factors that can shift the Stellar Scale oscillation period, the agreement between the predicted and experimental values is quite good.  Table 2 summarizes the discrete self-similarity between GSC 00144–03031 and ¹²C [1s²2s²2p5p → 4p, (J=0), ¹S].

Table 2.  Comparison of the Fundamental Mode Properties of GSC 00144–03031 and Its ¹²C Analogue Undergoing a [1s²2s²2p5p(J=0) → 4p(J=0), ¹S] Transition

To check on the uniqueness of the discrete self-similar relationship between the specific oscillation periods of GSC 00144–03031 and ¹²C [5p(J=0) → 4p(J=0), ¹S], the oscillation periods of other transitions in the 3 ≤ n ≤ 5 range were checked.  The closest alternative match occurred for the [5p(J=2) → 4p(J=2), ¹D] transition.  However, its oscillation period (9.18 x 10^-15 sec) is about 5% low and the transition is not similar to a fundamental radial mode oscillation.  Oscillation periods for other ¹²C transitions [3 ≤ n ≤ 5; ¹S and ³S] differed from the predicted period of 9.65 x 10^-15 sec by 10% or more.  Given the good quantitative match between the fundamental period of GSC 00144–03031 and the single uniquely specified transition period of ¹²C, it seems fair to conclude that discrete cosmological self-similarity has been shown to apply in this case.

To further demonstrate the uniqueness of our result, we can repeat the same analysis for other atoms such as H, He, Li, Be, B and N.  The results of these calculations are summarized in Table 3.

Table 3.  Oscillation Periods [5(l ≈ 0) → 4(l ≈ 0)] for Atoms Other Than ¹²C

The oscillation periods for the most radial transitions [5(l=0) → 4(l=0)] of H, He, Be, B and N are definitely not in good agreement with our predicted test period.  The closest alternative match is the 5s → 4s transition for Li with an oscillation period of 1.02 x 10^-14 sec, which is higher than our predicted period by about 5.4%, and would require an unreasonable 42% error in the mass estimate of GSC 00144–03031.

V. The Secondary Period of GSC 00144–03031

As an additional check on the uniqueness of the above results, we now explore the secondary oscillation period of GSC 00144–03031, which is 3872.70 sec.  At this point in the development of the SSCP, we are still trying to fully understand single-mode pulsators, although research on double-mode pulsation is on-going and seems promising.  That preliminary work suggests that both oscillation periods of a double-mode pulsator come from the discrete spectrum of allowed transition periods for that system.  Therefore, we can predict that for a ¹²C atom undergoing transitions with 3 ≤ n ≤ 5, there will be a transition period very close to (3873 sec)(1/Λ).  Actually, when this prediction is tested we find three candidates.  The [5p(J=1) → 4p(J=1), ¹P] transition comes within 4.4% of the predicted period of 7.45 x 10^-15 sec, but the [(J=1), ¹P] nature of this transition does not have much radial character and we expect the match between predicted period and comparison period to be at the 3% level, or better.  The triplet version of ¹²C has a [1s²2s²2p(²P^o)5s(J=2) → 1s²2s²2p3d(J=1), ³P^o] transition with a period of 7.48 x10^-15 sec (only 0.4% high), and the (J=0) version of that transition has a period of 7.55 x 10^-15 sec (1.3% high).  These two intimately related transitions offer very close quantitative matches, but the acceptability of having simultaneous transitions involving both singlet and triplet configurations remains to be more fully explored.  The third possible match, and possibly the most interesting, occurs with the [1s²2s²2p4p(J=0), ¹S → 1s²2s²2p3d(J=2), ¹D^o] transition.  This transition has an oscillation period of 7.32 x 10^-15 sec, which is quite close to the predicted period of 7.45 x 10^-15 sec (1.8% low).  It also has the unique feature that it is directly linked to the fundamental pulsation period since both share the [1s²2s²2p4p(J=0), ¹S] energy level.  Whereas the previous candidates would seem to require some sort of “superposition” of possibly competing transitions, the third candidate suggests an alternative qualitative explanation for double-mode pulsation, wherein the system is undergoing a succession of two separate, but related, transitions and the second oscillation begins before the first oscillation has completely finished.

Although at present we do not have enough information to definitively choose between the three candidate matches for the secondary oscillation of GSC 00144–03031, we can safely say that this additional check on the uniqueness of the primary results for the dominant oscillation period yields confirmatory results.  Had we not found any period matches at the ≤ 5% level, then that might have indicated a serious problem with the analysis, or possibly with the whole concept of discrete cosmological self-similarity.

VI. Conclusions

The class of δ Scuti variable stars is a much more heterogeneous class than the RR Lyrae class, corresponding to a collection of Atomic Scale systems with masses in the 10-17 amu range.  The high amplitude δ Scuti stars appear to be limited to low Δn transitions (n_i – n_f ≈ 1) primarily within the range 3 ≤ n ≤ 6.  The substantial heterogeneity of this class interferes with a simple comparison of sizeable samples of δ Scuti oscillation periods with predicted periods derived from Atomic Scale data, although this is possible in principle.  However, we have achieved the specificity required for a meaningful test of discrete cosmological self-similarity by focusing on an individual, well-characterized δ Scuti star.  The success in predicting: (1) a specific Atomic Scale analogue (¹²C), (2) a most likely transition (1s²2s²2p5p → 4p, J=0, ¹S) and (3) a correct oscillation period (to < 3%), based purely on physical data for GSC 00144–03031, combined with the previous successful demonstration of discrete self-similarity between RR Lyrae stars and He atoms undergoing Dn = 1 transitions between Rydberg states, lends further support to our contention that discrete cosmological self-similarity is a fundamental property of nature.  These successes also lead us to predict that the same methods that have been applied here, and in the case of the RR Lyraes, can be successfully applied to other δ Scuti stars if the following criteria are met.  The stellar mass must be known to an accuracy of ≤ 0.05 M_¤, so that the correct Atomic Scale analogue can be identified.  Ideally the star should pulsate in a single dominant oscillation mode, although double-mode pulsators can also be analyzed by our methods.  Multi-mode pulsators with three or more low amplitude periods appear to be analogous to excited, highly perturbed, atomic systems that are oscillating at several potential transition periods, but are not yet undergoing single specific transitions between energy levels, as will be discussed in the next “New Development” on the class of ZZ Ceti variable stars.  At any rate, the higher the amplitude of the dominant oscillation period of the δ Scuti star, the more likely it is that we are observing an event that is self-similar to a full-fledged transition between discrete energy levels.  Although we may be getting a bit ahead of ourselves here, it is conceivable that a typical single-mode HADS star evolves from a multi-mode LADS star when the latter absorbs a sufficient amount of energy at an appropriate frequency in order to trigger a genuine energy level transition.

If you like solving puzzles, you might enjoy the challenge of identifying a well-studied δ Scuti variable and using the above methods to determine its self-similar counterpart on the Atomic Scale.

REFERENCES

1. Breger, M. and Pamyatnykh, A.A., preprint, arXiv:astro-ph/0509666 v1, available at    http://www.arXiv.org, 2005

2. Pugulski, A., et al., preprint, arXiv:astro-ph/0509523 v1, available at   http://www.arXiv.org, 2005.

3. Alcock, C., et al., Astrophys. J. 536, 798-815, 2000.

4. Fox Machado, L., et al., preprint, arXiv:astro-ph/0510484 v1, submitted to Astron. and Astrophys., available at http://www.arXiv.org, 2005.

5. Poretti, E., et al., preprint, arXiv:astro-ph/0506266 v1, submitted to Astron. and Astrophys., available at http://www.arXiv.org, 2005.

6. Bashkin, S. and Stoner, Jr., J.O., Atomic Energy Levels and Grotrian Diagrams (Vol. I. Hydrogen I – Phosphorus XV), North-Holland Pub. Co., Amsterdam; American Elsevier Pub. Co., Inc., New York, 1975.

RR Lyrae Stars, Helium Atoms and Energy State Transitions [September 2005]

Magnetic Activity Cycles in Stars and Electron Spin Transitions in Atoms [July 2005]

Introduction

One prediction mentioned in Paper #2 (see the “Selected Papers” section of this website) asserted that a specific Atomic Scale analogue to the Sun's 22-year magnetic activity cycle would eventually be discovered.  The morphologies and motions of the structures involved in the magnetic field flipping should be similar for the analogues on both Scales, and their timescales should be related be the usual SSCP scaling equations.  In this case,

P_stellar » LP_atomic » (5.2 x 10¹⁷)(P_atomic).

As discussed below, it appears that an excellent candidate for the Atomic Scale analogue to the Sun's 22-year magnetic reversal cycle, and related activity cycles in other stars, may have been identified.

The Solar Magnetic Reversal Cycle

The Sun's magnetic activity cycle is truly a remarkable physical process that is periodic and global, but enigmatic.  The basic properties of the cycle have been observed and described with increasing sophistication since the time of Galileo, but a rigorous understanding of the physics that underlies the activity cycle has eluded scientists. 

Starting at a time of minimum activity, which is characterized by low numbers of sunspots, new sunspots begin to form at mid-latitudes in both hemispheres.  As the cycle proceeds the sunspot activity migrates toward the Sun's equator.  The sunspots tend to come in pairs that have opposite polarities.  During each half-cycle the leading spots in the pairs will tend to have one polarity in the Sun's northern hemisphere and the opposite polarity in the southern hemisphere.  The polarities of the sunspot pairs reverse during the next half-cycle.  The number of sunspots taking part in this migration from middle to low latitudes increases rapidly and remains at an elevated level until the Sun approaches a new activity minimum roughly 11 years later.  At that time the Sun's global dipole magnetic field will have its “north” and “south” polarities reversed from what they were at the previous minimum.  Two 11-year half-cycles are required for the system to return to its initial state, so the full cycle has a period of about 22 years.

When the solar activity cycle is studied over hundreds of years, it is found that half-cycle periods can vary in length from 8 years to 15 years with an average period of 11.1 years, that the amplitude of the sunspot numbers can vary by ≤ 3, that half-cycles can overlap by 2 to 3 years, that during the Maunder Minimum (1654 to 1715) there were very few sunspots at all, and that the establishment of the reversed magnetic fields at the North and South poles of the Sun can lag behind one another by up to 9 months.  Despite the somewhat erratic nature of this phenomenon, which reminds one of a slightly cantankerous clock, the Sun's activity cycle has persisted for centuries with an average full-cycle period of 22.2 years.

An Atomic Scale Analogue?

The Sun's activity cycle is thought to take place primarily in a thick (» 0.3 R_) outer layer of the Sun referred to as the convection zone.  According to the SSCP, this outer spherical shell of charged matter in a plasma state corresponds to an S-state electron wavefunction representing the outermost electron of the inner core of a highly excited Rydberg atom.  Such atoms, with their compact inner cores and nearly classical Rydberg electron orbiting very far from the core, are often called “planetary atoms” for obvious reasons.

We then ask whether electrons undergo processes associated with flipping of their intrinsic magnetic fields, and are led to the process of hyperfine electron spin transitions as the most promising category of potential analogues to what we are observing in the solar magnetic activity cycle. The simplest and best-known example of magnetic spin transitions on the Atomic Scale occurs in the ground state hydrogen atom.  The proton and the electron of H both have spins of  ½ and the total number of spin states is 2S + 1 = 2, representing spins parallel (both “up”) and spins anti-parallel (1 “up”, 1 “down”).  In an environment that is cold enough to prevent excitation of the atom to n ³ 2 states, and where strong magnetic fields are absent, the H atom can undergo a transition from the higher energy parallel state to the lower energy anti-parallel state, emitting electromagnetic radiation with a frequency (n) of 1420 MHz.  The period of this oscillation is 1/n = 0.704 x 10^-9 sec.  To convert this period into an analogous Stellar Scale period we multiply by the usual scaling factor of L (»5.2 x 10¹⁷) for lengths and times: (0.704 x 10^-9 sec)(5.2 x 10¹⁷) ¸ (3.17 x 10⁷ sec/yr) = 11.62 years.  It is encouraging that our initial test of the potential analogy gives results that are of the correct order of magnitude, and in fact are within a factor of 2.

 As has been demonstrated elsewhere at this website, the mass of the Sun suggests that it is an analogue of Li rather than H.  The unpaired outer electron of the ground state Li atom has a hyperfine spin transition with a frequency of about 803.5 MHz, which corresponds to an Atomic Scale period of 1.24 x 10^-9 sec and an analogous Stellar Scale period of 20.5 years.  That is rather close to 22.2 years!  In the case of electron spin transitions involving spherical S-states, which would seem appropriate for the proposed analogy, electric dipole radiation is strictly forbidden and so the transition must proceed via magnetic dipole radiation.  An oscillating magnetic dipole would seem to required in order to generate this type of electromagnetic radiation.  Therefore we have identified potentially analogous processes on the Stellar and Atomic Scales that share some fairly unique qualitative properties and that appear to have periods that are reasonably well correlated by the SSCP's discrete self-similar transformation equations.  Below we will explore this potential example of cosmological self-similarity in more detail.

Impediments To A More Quantitative Test Of The Analogy

If we wanted to do a more rigorous test of the proposed analogy between the solar magnetic cycle and a lithium atom undergoing a specific S-state hyperfine magnetic spin transition, we would have to know the internal configuration of the solar electron analogues.  We know (Papers #1 and #2) that the outermost electron analogue for the solar system has a principal quantum number n of about 168, and an orbital quantum number l  that is nearly equal to n.  Given the radius of the Sun's convection envelope and the 160 minute oscillations of the Sun's surface layers, it has been previously concluded that the outer electron analogue of the system's core has n = 5 and is in an S-state with l = 0.  The problem is that a neutral lithium atom analogue would have one more electron analogue well inside the core, below the convective layer.  Its n value could be any number between 1 and 4.  Even more problematic is the fact that the Sun could also be analogous to the core of a Li⁺ ion with no interior electron analogue, or it could be analogous to a Li^- ion with 2 internal electron analogues in unknown states.  (Note: As if all these uncertainties were not enough, there is also the possibility that the correct spin transition analogy involves electron spin resonance wherein external magnetic fields and electromagnetic radiation drive the system at a specific oscillation frequency.)  

It is conceivable that with a lot of astute physics, and working on both the atomic and stellar ends of the puzzle, one might be able to use a chain of reasoning to narrow down the set of possible electron configurations that apply in the proposed analogy.  Detailed studies of this type will have to remain a project for the future.  What can be done at present is to take a broader look at magnetic activity cycles in other stars to see if they exist and, if so, how their cycle periods compare with the distribution of periods for hyperfine transitions in atoms.

Magnetic Activity Cycles In Other Stars

If our analogy to hyperfine spin transitions is valid, then we would expect that some other stars have activity cycles similar to the Sun's and others do not.  Atoms must have the right configuration in order for hyperfine transitions to occur.  In atoms with filled electron shells and no unpaired electrons, the hyperfine transitions cannot occur.  The ground state of helium is one example and a Li⁺ ion with both electrons in the same S-shell is another.  Since hyperfine spin transitions are fairly common, our analogy would be considerably weakened if other stars did not have magnetic activity cycles.  On the other hand, if all stars had them, then the analogy would also be weakened.

Fortunately for the proposed analogy the observations verify the expected dichotomy: some stars have activity cycles and some apparently do not.  The most extensive study of magnetic activity cycles in stars has been carried out at the Mt. Wilson Observatory.  Since the program began in 1966, over 100 stars of spectral types F, G, K and M have been monitored for indications of cyclical magnetic activity.  In 1995 the results for 112 stars, including the Sun, were reported (Baliunas, et al, 1995).  Thirty-one stars were observed to have flat or linear baselines, suggesting no cyclic activity or extremely long timescales.  Twenty-nine stars had irregular variations in their magnetic activities but lacked periodic cycles.  Fifty-two stars were observed to manifest periodic magnetic activity cycles with estimated half-cycle periods ranging from 2.5 years to 25 years.  The quality of the observational estimates for the cycle periods were rated either excellent, good, fair or poor.  Below is a table of the magnetic cycling stars in the good and excellent categories.

*  Here we presume that the observed cycle is a half-cycle, as is known to be the case for the Sun.

**  Longer term averages that take all aspects of the cycle into account are 11.1 years for the half-cycle and 22.2 years for the full cycle.

From the stellar data presented in the table above we can see that the periods of the presumed full cycles range from 14 years to 42 years, with an average of about 24.3 years.  Shorter periods are favored in the analysis so the apparent cutoff at about 14 years, or more conservatively on the order of 10 years, can be considered potentially important and diagnostic.  The available data strongly select against long-period cycles, so they are probably under-represented.  In the case of very long periods, say 100 years or more, no star other than the Sun has been studied for a sufficiently long time to properly define these periods.  The Sun is believed to have long-term amplitude variations in its cycle with periods of about 80-90 years (the Gleissberg cycle) and about 150-210 years (the Suess cycle) (Bonev, et al, 2004).

Hyperfine Spin Transitions

For comparison we will look at some typical Atomic Scale hyperfine spin transition periods for low-mass atoms in states with low n and l values, since the relevant stars in the comparison are relatively small, low-mass systems.  The table below also shows predicted periods for Stellar Scale analogues of the specific transition periods.

When we compare the scaled electron spin transition periods with the presumed full cycle periods of the stars, we see that the agreement is reasonably good at the short-period end of the ranges but not at the long-period end.  As mentioned above, in addition to its primary 22.2-year cycle period the Sun has secondary cycling oscillations at about 80-90 years and about 150-210 years.  In the case of other stars, however, astronomers currently do not have the capability to test for magnetic activity cycles with periods on the order of 10² years to 10⁴ years.

A serious concern is that the number of stars in the Sun's neighborhood that are undergoing magnetic activity cycles appears to be on the order of 50%.  Given that hyperfine transitions tend to have low (and sometimes very low) probabilities, 50% is a very high number.  This might suggest that these magnetic activity cycles are driven by resonant forcing in analogy to the phenomenon of electron spin resonance, rather than our original analogy to spontaneous spin transitions.

A Very Speculative Possible Solution

From the discussion above, we have seen that the Sun's magnetic activity cycle appears to have four relevant periods: 11.1 years, 22.2 years, 80-90 years and 150-210 years.  Consider the following table of data.

Is it coincidental that the orbital periods of the Solar System's four major planets closely parallel the four main periods of the solar cycle?  Perhaps not.  Within the conceptual framework of the SSCP, the planetary system constitutes an analogue to the wavefunction for one electron in a very high-n Rydberg state (n » 168 and l » n).  It is conceivable that for a lithium atom with the unusual configuration described in this study, the Rydberg electron can resonantly drive the spin transition of the n = 5, l = 0 outer core electron to resonate at the orbital frequencies of the Rydberg electron's wavefunction.  By analogy, the Sun's magnetic activity cycle might not be self-similar to a spontaneous spin transition, but rather to resonant spin transitions regulated by interactions with the planetary system.  This is an admittedly wild idea, but an interesting one no less.  If other stars with well-documented activity cycles were found to have planetary systems with similar correspondences between orbital and cycle periods, then this wild idea might be worthy of very serious consideration.  A preliminary but hopeful sign is that the stars HD 179949 and n And have chromospheric activity cycles that appear to be coupled to the orbital periods of close-in Jupiter-mass planets (Shkolnik et al, 2005).  These authors note that since the enhanced activity occurs once per orbit, the interaction is probably electromagnetic in origin, rather than tidal.

Final Thoughts

The present work on the potential self-similarity between electron spin transitions and magnetic activity cycles in stars can only be viewed as a rudimentary beginning.  It remains to be seen whether more thorough and sophisticated research will support the proposed self-similarities, or falsify them.

REFERENCES

Active Galaxy Oscillation Periods Agree with SSCP Prediction of » 10⁷ Years [June 2005]

The SSCP predicts that ultracompact objects with extremely high densities comprise the overwhelming majority of the mass/energy found on each of the three observable cosmological Scales (Atomic, Stellar and Galactic).  Atomic nuclei play this dominant role on the Atomic Scale, while the analogous class of self-similar nuclear objects on the Stellar Scale includes neutron stars (observed) and predicted stellar-mass black holes (possibly detected in dark matter microlensing experiments).  Based on their sizes, shapes and central singularities, the SSCP identifies galaxies as the Galactic Scale counterparts to Atomic Scale particles and nuclei under high energy and density conditions.

Rotation and Oscillation Periods for Atomic, Stellar and Galactic Scale "Nuclei"
	Typical Rotation Period	Typical Oscillation Period
Atomic Nuclei	5 x 10-20 sec	1 x 10-21 sec
Neutron Stars	3 x 10-2 sec	5 x 10-4 sec
Active Galaxies	1 x 1016 sec  (3 x 108 yr)	2.5 x 1014 sec  (8 x 106 yr)

Two reasonably fundamental physical properties of these ultracompact objects on the three different Scales are their typical rotational and oscillatory periods, since these periods are closely related to the spin and mass/energy of the objects, respectively. If the SSCP is a useful paradigm and the correct self-similar analogues have been chosen, then the rotation and oscillation periods for these analogues should obey the basic self-similar scaling relation that governs temporal phenomena: 

P_N » LP_N-1   ,

where P is any periodic phenomenon, N and N-1 indicate neighboring Scales of the cosmological hierarchy and L is the dimensionless scaling constant for lengths and times, which » 5.2 x 10¹⁷.

The Table above shows that our expectations for the relationships among the rotational and oscillatory periods for the proposed Atomic, Stellar and Galactic Scale analogues are met.  At the time that these relationships were initially tested (papers #1 and #2), the typical oscillation period of » 10⁷ years for active galaxies could only be predicted, since galactic oscillations had not been observed yet.  In the last 10 years this situation has changed and it appears that another SSCP prediction may have been vindicated.

In 1995 Walter Moody of Brigham Young University and colleagues reported observing periodic ejections from the galaxy M101.  The estimated period was 1.1 x 10⁷ years (J. Hecht, New Scientist, 2 Dec 1995).  In 2000 A. Schoenmakers and colleagues published a paper in the 21 Jun 2000 issue of Monthly Notices of the Royal Astronomical Society that identified repeated and symmetrically opposed ejections from a radio galaxy with a periodicity of roughly 3 x 10⁷ years.  They noted that at least 8 other examples of radio galaxies undergoing similar events had been identified at that time (G. Schilling, Science, 289, 28, 2000).   More recently, in 2004, William Forman and colleagues reported periodic outbursts from the active galaxy M87.  Again the timescale for those oscillations was on the order of 10⁷ years [arXiv:astro-ph/0312576 v2].

Bearing in mind that only a subset of galaxies would be undergoing global oscillations at any given epoch, it appears that there is now a growing body of evidence supporting the idea that massive active galaxies can eject energy and matter in periodic pulsations with timescales typically on the order of 10⁷ years, as predicted by the SSCP.

Revised Stellar Mass And Radius Limits Agree With SSCP Predictions [May, 2005]

Smallest Red Dwarfs

Papers #1 and #2 in the “Papers” section show how to calculate the Stellar Scale equivalent of the Bohr radius (a₀ = 0.53 x 10^-8 cm), which is identified as A₀ and is » (a₀)(L) » (0.53 x 10^-8 cm)(5.2 x 10¹⁷) » 2.76 x 10⁹ cm.

In the Atomic Scale case, the electron’s wave function for the hydrogen atom peaks at a₀, but extends beyond that radius to approximately 3a₀. A simple test of the SSCP is whether or not low-mass (M £ 0.2 M_¤) red dwarf stars have a minimum radius of about 8.3 x 10⁹ cm ( » 3A₀). When papers #1 and #2 were written in 1988, the minimum radius for a red dwarf star was estimated to be about 8.7 x 10⁹ cm, which was in reasonably good agreement with the predicted value. In 2005 a European team analyzing transiting star data from the Optical Gravitational Lensing Experiment reported a new observational estimate for the smallest red dwarf radius of 8.35 x 10⁹ cm (Pont, F., et al, Astron.& Astrophys. 433, L21-L24, 2005), which is quite close to the predicted value. This measurement is a “direct radius determination” that involves fewer assumptions and uncertainties than previous methods.

The wave function cutoff at 3a₀ is a fairly crude approximation, but the peak at a₀ is a better-defined physical parameter. At some point in the future it may be possible to infer the inner mass distribution of the lowest-mass red dwarf stars. The SSCP uniquely predicts a distinct and dominant peak in the mass distribution of the star’s outer layers at A₀ ( » 2.8 x 10⁹ cm) for these smallest red dwarf stars.

Largest Star

Atoms in very high Rydberg states can be relatively gigantic, literally comparable to the size of a small microbe. In 1988 (paper #1) the largest atoms commonly observed had a principle quantum number of » 110, which translates into a radius of 12,100 a₀ or nearly 0.001 millimeters! At that time the largest observed stars had radii on the order of 12,140 A₀ or about 3.4 x 10¹³ cm. In 2004 the star VY CMa set a new radius record at a humongous 3020 R_¤, or about 2.1 x 10¹⁴ cm, which is approximately 76,594 A₀ (Monnier, J.D., et al, Astrophys. J. 605, 436, 2004). According to the SSCP this would correspond to a Rydberg atom with a principal quantum number of at least 195 if the outer radius is about 2 times the radius of the mass density peak, and could be as high as 276 in the case of a very thin outer envelope. Although the critical distance and luminosity estimates for VY CMa are “fairly certain” and “well-established”, respectively, it has been noted that “none of [VY CMa’s] stellar properties are in accord with stellar evolution theory” (Levesque, E.M., et al, Astrophys. J., in press, 2005, [arXiv:astro-ph/054337]). The SSCP predicts that the observed sizes of the largest atoms and the largest stars will continue to be of the same order of magnitude when scaled by the R_star » LR_atom relation.

Stellar Mass Limits

A much more important, and potentially more rigorous, test of the Self-Similar Cosmological Paradigm concerns mass limits for stars. At the start of the 21st century the most widely accepted theories for the formation and evolution of stars put no well-defined upper limit on the mass of a star. Supermassive stars of over 300 M_¤ were considered to be possible. Conversely, the SSCP predicted that single stars, like atoms, have an upper mass limit due to strict stability constraints. For the Atomic Scale, the most massive naturally occurring atoms have masses of about 240 times the mass of the proton. Since we have already determined that the Stellar Scale equivalent of the proton mass is » 0.145 M_¤, we can predict that the largest masses for single stars will be on the order of (240)(0.145 M_¤), or roughly 35 M_¤. Observationally we find that stars with masses greater than 20 M_¤ are quite rare, less than one per thousand stars (Reid, I.N., et al, Astron. J. 124, 2721-2738, 2002). Very recently it was reported (Figer, D.F., Nature 434, 192-194, 2005) that for the Arches cluster, an environment favoring the formation of very massive stars, the most massive stars were estimated to have masses of about 130 M_¤. This figure is about 3.7 times larger than the SSCP prediction, but the values are of the same order of magnitude, and one must bear in mind that mass estimates for stars at the upper and lower ends of the stellar mass scale are based on numerous theoretical assumptions, making significant systematic errors a definite possibility. The main message is that astrophysicists now suspect that stars, like atoms, appear to have an upper mass limit. A mass cutoff below 150 M_¤ has no clear explanation within previous stellar formation/evolution theories, but was predicted by the SSCP. We may further predict that the observed upper mass limit will gradually decrease and eventually agree with the SSCP prediction for the scaled upper mass limit for atoms.