Recursive Integer Sequences, Detected in Solar-Cycle Periodicities Measured in Numbers of Rigid Rotations of the Sun

Abstract

Consecutive integers in the recursive number sequences, the Fibonacci sequence (F_n) and the Lucas sequence (L_n), are detected in the lengths of solar-activity variations from ≈ 1 yr to ≈ 12 yr, measured in rigid rotations of the Sun at the helioseismologically determined frequency in the radiative zone, $433 \pm 3$ nHz. One rotation is denoted by the symbol $\Omega$. Firstly, in the new international sunspot-number record (Ri) the most frequent (modal) sunspot-cycle length, which is also the period defined by autocorrelation for the recurrence of sunspot cycles, has been $144 \pm \approx 2~\Omega$ ($\mbox{F}_{12} = 144$). The most frequent length for a descending leg of the cycle has been 89 ± 2 $\Omega$ (F$_{11} = 89$), and for an ascending leg 55 ± 1 $\Omega$ (F$_{10} = 55$). Secondly, there is some observational evidence of Ri spectral peaks at the consecutive L_n numbers of $\Omega$: 18 $\Omega$ (≈ 1.3 yr), 29 $\Omega$ (≈ 2.1 yr), 47 $\Omega$ (≈ 3.4 yr), and 76 $\Omega$ (≈ 5.6 yr), which are harmonics of the 144 $\Omega$ period divided by the first four F$_{{n}} > 1$: 2, 3, 5, and 8. The numbers of $\Omega$: 144, 89, and 55 may be kinematical thresholds in the dynamo process starting at sunspot maximum, when the poles change polarity and the process is re-set. The ratio of two consecutive F_n or L_n converges to $\frac{1+ \sqrt{5}}{2}$, hence it is suggested that this proportion plays a role in solar behavior over time, described numerically. The length ratio $\frac{1+ \sqrt{5}}{2}$ also is characteristic of fivefold symmetry in space. Since the icosahedral group is the link between numerical and spatial expressions of fivefold symmetry, it is proposed that the presence of icosahedral symmetry in the large-scale geometry of the Sun could also be tested.

Appendices

Appendix A

1.1 Interpolated Sunspot-Minimum Epochs

1.1.1 Epochs Obtained from Two Months Verifying the Minimal 13-Month Smoothed Monthly Sunspot Number Ri

1996.498 is the mean of 1996.373 (Ri = 11.2) and 1996.624 (Ri = 11.2).

1923.580 is the mean of 1923.538 (Ri = 9.4) and 1923.623 (Ri = 9.4).

1755.163 is the mean of 1755.123 (Ri = 14.0) and 1755.204 (Ri = 14.0).

1.1.2 Epochs Obtained from More than Two Months Verifying the Minimal 13-Month Smoothed Monthly Sunspot Number Ri

1823.288 is the center of three consecutive months (1823.204, 1823.288, and 1823.371), each verifying Ri = 0.2.

1810.580 is the central date in a sequence of ten consecutive months (1810.204 to 1810.958), each verifying Ri = 0.0, when this central date is taken either (i) as the mean of the epochs of the fifth and sixth months, i.e. of the two central months in the cited sequence: 1810.538 and 1810.623, or (ii) as the mean epoch of the first and final months, or (iii) as the mean of the epochs of the ten months in the sequence.

Appendix B

2.1 Range of Angular Velocities for the Rigid Rotation of the Sun: 430–436 nHz

2.1.1 Definition of Rigid Rotation in the Sun

The Sun is helioseismologically known to rotate rigidly, i.e. at the same angular velocity irrespective of heliolatitude, only in a certain spherical shell deeper than ≈ 0.7 R_SUN. The latitude-independent rotation is perhaps constant, irrespective of depth, only in a narrower shell. The shell for which there is evidence of uniform rotation is given, e.g., as 0.4 < R_SUN < 0.7 in Eff-Darwich and Korzennik (2012). Inward of 0.4 R_SUN evidence of the solar rotation profile is more limited. In the Sun’s energy-generating core ($<\,\approx$ 0.3 R_SUN) the solar rotation profile is most uncertain.

2.1.2 Lower Bound for the Angular Velocity of This Rigid Rotation $\Omega$

The observed value of latitude-independent rigid angular velocity in the Sun was initially estimated below the tachocline as “about 430 nHz” by Schou et al. (1998) based on the first 144 days of MDI observations. This estimate is still widely retained, e.g. in the reference book Priest (2014) or as an assumption in a model: e.g., in Lund, Miesch, and Christensen-Dalsgaard (2014). In those of the most recently published results which do not state an error bar ≥± 10 nHz for this quantity, the value of 430 nHz is confirmed (Eff-Darwich et al. 2008) or is slightly revised upward at 431 nHz (Eff-Darwich and Korzennik 2012). The latter wrote that their results are compatible “with a radiative zone rotating rigidly at a rate of approximately 431 nHz”. Thus, the value of about 430 nHz apparently withstood the test of time, now being based on an “entire solar cycle” (Eff-Darwich and Korzennik 2012) or on the joint consideration of data sets from three distinct instruments: MDI, GONG, and GOLF (Eff-Darwich et al. 2008). The latter authors wrote: “In the outer radiative zone (0.4 > r/R < 0.7) (…) the rotation rate is almost constant”, and “decreasing below 0.2 R”. According to some authors, instead, angular velocity is suggested to increase in the most inward part of the radiative zone and in the solar core (see below). Since 2000, no estimate < 430 nHz, for the Sun’s rigid rotation rate in the radiative zone, could be found in the literature.

2.1.3 Higher Bound for the Angular Velocity of This Rigid Rotation ${\Omega}$

Most recently, rigid rotation with characteristic angular velocity ≥ 433 nHz was suggested. Based on $g$-modes, Fossat et al. (2017) estimated a rigid rotation rate of 433 nHz. These authors, however, stated that this value is “less accurately” measured, compared to estimates “directly” based on $p$-modes. They suggested that published results are “compatible with a radiative zone that rotates rigidly at a mean rate of $433 \pm 10$ nHz”, i.e. possibly up to 443 nHz. The so-defined central value of 433 nHz is, at present, retained as an assumption in many simulations, e.g. in several papers by Garcia and his collaborators, lastly in Garcia (2015). Bertello et al. (2000), based on their MDI rotational splittings and on a rotation profile derived from “medium-$\ell$ MDI data”, suggested a still faster characteristic rotation with a central-value “rate of about 435 nHz” for the “uniform rotation of the solar core”. For the rotational splittings calculated in Bertello et al. (2000), based both on MDI and GOLF observations, Gelly et al. (2002) (Table 3) gave a $\sigma$-weighted value of 436 nHz for $\ell = 1$.

2.1.4 Retained Range for the Sun’s Rigid Rotation $\Omega$, as Defined Here

For the purposes of this paper, a relevant characteristic range for ${\Omega}$, the value of solar rigid rotation in the Sun must be selected. The range is retained here, for the radiative zone of the Sun including the solar core, as in the latest estimate cited above ($433 \pm 10$ nHz). However, the range is narrowed-down to only ± 3 nHz. This range is defined by the two extreme estimated individual values (430 nHz and 436 nHz) cited in this brief review.

References include references for Supplementary Material.