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My comments on these results :

From these qualitative remarks, and assuming that this nice effect is not an artefact, as was the correlation observed decades ago between sunspot cycles and the level of the Victoria lake, I can tempt some conjectures, recognizing that I have no theoretical support to prove them, and that I only express a systems engineer intuitive feeling of the situation :

It seems that Mercury and its syzygies with Venus and Jupiter have no effect. Perhaps the quality of a syzygie is also linked to the time it lasts (say the number of days during which the angular sector is smaller than 15 degrees), or to the the time integral of the tidal effect. The impulse of tidal energy looks like a triangle rather than like a Dirac impulse, the time integral of the energy being lower, for the same amplitude, if the basis of the triangle is shorter). Due to the higher angular speed of Mercury, the syzygie time is smaller for syzygies including this planet.

Solar cycle 22 and 23 occur on the maximum of the modulation : they should be of greater amplitude than the preceding cycles. Syzygies of the 4 planets ( including Mercury) occur during cycle 22, and a great syzygie including all planets from Mercury to Neptune (March excepted) occurs in January 1990. A high rate of birth of active centers should be expected around this date (as established by Trellis). The delayed effect due to the migration of these active centers towards the solar equator and the fact that a strong MeVeEaJu syzygie will appear in 1991 could give a stronger than expected sunspot activity from 1990 to 1993. These events could force cycle 22 to have an abnormally great amplitude (Wolf number greater than 200), culminating during an unusually long time.

Remark (october 1995) : the above paragraph was written in 1989. In fact, cycle 22 proved to be of the same amplitude than cycle 21. Could cycle 23 be of great amplitude ? It seems already that cycle 22 ends in advance, which could indicate a phase shift. Cycle 23 could start also in advance, and like for the situations encounterd for cycles 3 or 19 (fig 1), (fig 5), be of great amplitude. This effect could be reinforced by the fact that, since half a century (as Bracewell found it), "negative" cycles have a greater amplitude than "positive" cycles.

The bimodal distribution of the sunspot cycle period could be due to the fact than either 6 or 7 syzygies occur in a burst (fig 2). As going from an even cycle burst to an odd one (or vice versa) implies passing from syzygies to oppositions, syzygies cycles being 19.5 month, and going from the last syzygie to the first opposition (or vice versa) taking 231 days (7,5 month) we find that syzygies cycles have a duration equal to one of the following values :

6 x 19,5 + 7,5 = 124,5 month
7 x 19,5 + 7,5  = 144 month

This is to compare with the values found by Wilson (122 and 140 month [17]. It could be assumed that the impulsive nature of the excitation signal gives a bifurcation type response in the syzygie-opposition transition, or on the contrary, that the bimodal distribution is due only to the fact that an integer number of convection rolls, migrating from polar to equatorial regions, can establish during a cycle, so that the above numerical correlation would be an artefact. Both of these hypotheses could also be linked together, assuming that the VeEaJu syzygy tidal energy impulse gives birth to a new convection roll.

The VeEaJu syzygies gives tidal waves, or more precisely tidal acoustic constraints, not only in the outer part of the Sun, but also deeply into the convective zone, and the interaction between these waves, the convection rolls mechanism, and the deep internal core, where complex nuclear reactions and magnetohydrodynamic processes occur [7], forces the Sun's core to oscillate at the frequency of the tidal excitation waveform. It could be assumed that the Sun has a natural tendency to oscillate on a period close to 22 years, and that the oscillator is captured by a close excitation frequency. The influence of the external excitation terms in the state equation could however be strong, if the bimodal mode of the sunspot signal was due to the numerical distribution of syzygies in the bursts.

The fact which seems to contradict these conjectures is that a maximum of the modulation function occured around 1650. Normally, the Maunder minimum should have occured on the next minimum, in 1730. Perhaps we could solve this contradiction, in the following way. Let us suppose that the Sun's internal oscillator is not linear, but that the oscillation period decreases with the amplitude of the oscillations. This would be the case, for example, with an oscillator having a state equation of the form :

X" = - K * X3

The VeEaJu tidal excitation being periodic, the more the amplitude of the oscillation increases, the more the oscillation comes in advance upon the excitation. Situations can arise, on an overshoot of the oscillator, where excitation becomes opposed to the oscillation, and a fast and complete 180 or 360 degrees phase shift of the oscillator occurs.

After a few periods, it starts again in phase whith the tidal excitation signal, and the process goes on. In such a non linear oscillator, energy involved in the oscillation process increases very fastly with the amplitude of the oscillations : it is possible that the occurence of a strong maximum of oscillation implies that a threshold is overpassed in the system, breaking the oscillation and pushing the system into an other mode of evolution.

If this last hypothesis was good, such a phase shift could have been provoqued by the maximum of the modulation function around 1640, and the oscillator could have taken again its normal mode in 1700. Occurence of regular 160 years period fluctuation, and sporadic phase shifts like the Maunder minimum could explain long terme solar activity records such those reported by Beer et al. [1] from ice drills analyses, or by Schove [14] from auroral historical data and radiocarbon production in tree rings. Brutal changes of internal magnetic and hydrodynamics conditions during these phase shifts, changes of configuration of the radiative and convective processes under the surface of the Sun [13] could explain it's diameter's growth and it's slower external rotation during theses crises , as supposed by Ribes et al. [11, 12] from old observation data analyses.

Observing the amplitude and phase shifts of the sunspot cycle regarding to the syzygies cycle seems also confirm that there are two different cycles which interact together : an hydroacoustic cycle with a period of 11 years, always in phase with the VeEaJu syzygie cycle, and a magnetic cycle with a period of 22 years, as suggested by Ribes et al. [12]. It could be assumed, regarding the dynamical evolutions of the system, that the free magnetic oscillation period is 18 or 20 years rather than 22 when the amplitude increases, and that the magnetic oscillator has a tendancy to escape from the regular excitation, to take some cycles of advance.

Fig. 7 shows an extension of the signals for the years before 1700. It is a romantic reconstruction based on some sunspots descriptions during the 1620-1700 period. It is likely that the 11 years cycle was still present during the sunspots absence, as the few ones described were seen at the right times, in phase with the VeEaJu syzygies cycles ... exept for the 1655 sunspot which is aberrant. I have drawn (blue dotted line) on the same diagram an hypothetic magnetic cycle curve between 1645 and 1700 showing a 360 degrees phase shift in advance.

It is even possible to think that the magnetic cycle has reversed during the Maunder minimum, with a 180 degrees phase shift instead of 360 degrees : the corresponding magnetic cycle curve, if plotted, would have the look of the nodes found by Williams and Sonett in the Elatina formation. Knowing if this phase shift was 180 or 360 degrees would be of great interest, to know if there is a direct relation between the type of VeEa syzygie and the polarity of the poloidal field oscillation. If such a relation was found, we could infer that the interaction between the Sun and the planets creating the magnetic solar cycles is not only gravitational but also magnetic, or that the tidal effects have a different interaction with the magnetic core's oscillation depending on the relative situation of Earth and Venus (syzygie or opposition) in the VeEeJu syzygie burst.

What about the future ?

It is interesting to note that we are in 1989-1993 in the same conditions than in 1640, with a strong maximum of the modulation function, and a sunspot cycle which starts in advance : a phase shift could occur in the coming years ( 1995-2040), with all the perturbations of the Sun and of the climates found during the Maunder minimum. If this hypothesis was verified, the climatic effects of this new little ice age could be minimized by the carbon dioxid diffusion in the atmosphere and the greenhouse effect...

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