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 Λ (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.