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## Long term modulation of the solar cycle

I modified the "plots.pas" program to allow the
plotting of the syzygies (raw or filtered), either without sign (usual Wolf numbers
convention), either signed (Bracewell's convention). I gave the choice, for the
filtered signal, to widen the Hanning window to 500 month (about 4 sunspots cycles).
If it is asked to plot from -300 to 2100 (now possible without problem), all values
positive (Wolf numbers convention) and widen to 500 years the Hanning window,
here is what can be seen :

There is a periodic modulation, giving 13 cycles on 2165 years, i.e. a period of
166.5 years. This value is to compare with the 157 years modulation found by
Bracewell on a Laplace analysis of Wolf numbers. More interesting, this curve looks
exactly like the curve found by Williams and Sonett (Nature Vol 318 12dec85, p.525,
fig. 3b) : The main period (166,5 years) of the actual syzygies signal would
be to relate to the main period (314 years) of the amplitude curve on the quoted
figure.

Looking for the 314 years modulation found by Williams and Sonett, I tried to
modify the angular speeds of Earth and Venus, as I did for finding the good
correlation before 260 AD for the Schove's dates. So I added the two following
lines at the beginning of the "syzygie.pas" program :

with planete [rang_venus] do LP := LP * 0.999855 ;
with planete [rang_terre] do LP := LP * 1.000145 ;

LP being the angular velocity (degrees / days) on the orbit,
this means that I slow down a little Venus of minus 1.45E-4 and that I accelerate a
little Earth of plus 1.45E-4, Jupiter being supposed to stay regular.
here is what can be seen :

There is still a periodic modulation, giving 6 cycles on about 1880-1890 years, i.e. a period of
approximately 313-315 years. This value is to compare with the 314 years modulation
quoted above.

It is curious to see that such small variation of the planets orbital parameters,
possible by secular interaction beetween them, gives so great waveforms
modifications. It must be reminded that this enveloppe amplitude modulation is
a beat beetween two frequencies : this amazing effect is obvious and real
from a mathematical point of view.

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