Then I simplified my program and tried to plot occurences of great tides . As the terrestrial great tides occur when the Moon and the Sun are in geocentric syzygies, solar ones occur on heliocentric syzygies of two or more of the tidal planets . The tidal coefficients of all planets, which give the amplitude of the tide that they bring on the Sun's surface, are related to the mass divided by the cube of the mean distance to the Sun (radius). Taking as units the earth mass and astronomic unit (Sun-Earth radius), it comes :
The two main effects are due to Venus and Jupiter, and two smaller effects are due to Mercury and Earth. I tried to search great tides in finding syzygies or oppositions of Jupiter and Venus. This event occurs every 119 days. Quantitative correlation studies between solar data (Wolf number, magnetic data, etc...) could possibly show the existence of this cycle. I decided however to search for the more evident of the solar cycles : the 11 years sunspot cycle. I tried to find a cycle of occurence of 3 or 4 planets syzygies.
Its is clear that the amplitude of the tide given by 3 planets at the syzygie is greater if the syzygie is perfect (the 3 planets aligned with the Sun), than if the syzygie is approximative (the 3 planets passing in an angular sector of n degrees). I plotted on a diagram the quality of the syzygies ( linked to the amplitude of the tide), putting a vertical bar centered on the date where the angular sector is minimum, and an height which is a linear decreasing function of this angular sector in degrees. If "n" is the minimum value of the angular sector aperture at the syzygie, "m" a fixed maximum value, and "K" a fixed coefficient useful to plot the quality signal on the same diagram as the sunspot number, the quality signal is given by :
q = k * (m - n)
m is the angular sector limit : no syzygie if the angular sector in which the planets gather is greater than m degrees.
This approach assumes that the rather short tidal effect given at the syzygy could be plotted as a Dirac impulse with an height representing the energy involved.
Plotting this diagram for the Venus Earth Jupiter syzygie, hereafter named the VeEaJu syzygie,
I found that the maxima of the quality signal occured on the maxima of the Wolf number curve
and that the syzygies occured in bursts, separated each by 594 days.
This value is equal to the
heliacal cycle of Venus (584 days) plus a phase shift of 10 days. It is an
interesting result due to the fact that the syzygie cycle of Venus and Jupiter
is 238 days, so that the half cycle (delay between a syzygie and an opposition)
is 119 days, and that 5 times this value equals 595 days.
I found also that two separate and distinct bursts were interleaved. Syzygies were caracterized by the fact that Venus and Earth were in true syzygie (even sunspot cycles) in the first burst, or in opposition (odd cycles) in the following one. I decided to plot positively the quality of the syzygies if Venus and Earth were in syzygie, and negatively if they were in opposition (fig2), giving the following quality signal :
q = + k * (m- n) if VeEa syzygie q = - k * (m- n)) if VeEa opposition
Varying the m parameter between 0 and 45 degrees, I found that, for m = 15 degrees, the two successive (even-odd) bursts were separated and that the quality curve given by linking the top of the bars was a quasi-sinusoidal wave, in phase with the Wolf number curve. This plot can be seen, on K = 10 (Fig. 3), under the signed Wolf number curve. The two signals are in phase, for three centuries, exept some periods where phase shifts of 30 to 80 degrees occur.
Examining this diagram, it can be seen that the number of syzygies in a burst and their amplitude have long term variations. It seems that the global amplitude of the Wolf number is linked to the number and amplitude of the syzygies in the corresponding burst. I decided to filter the syzygies quality signal (pulses) by a centered finite impulse response filter ( Hanning window) with a horizon of one half of a sunspot cycle (70 month).
The result (Fig. 4), reshaped at the same scale, shows clearly the 22 years sinusoidal signal, modulated by a 160 years waveform, which could be related to the 157 years waveform found by Bracewell.
The curve of Fig. 5 shows the same filtered syzygies signal for the 1700-2000 period, along with the signed Wolf number curve.
It can be seen that the phasing of the sunspot cycle seems to change slightly, depending on the amplitude of the modulation function. Sunspots seem to come in advance when the amplitude is great, and in lag when the amplitude is small.