Why Phi? – modelling the solar cycle

We’re familiar with the idea of the solar cycle, e.g.:
‘The solar cycle or solar magnetic activity cycle is the nearly periodic 11-year change in the Sun’s activity (including changes in the levels of solar radiation and ejection of solar material) and appearance (changes in the number of sunspots, flares, and other manifestations).

They have been observed (by changes in the sun’s appearance and by changes seen on Earth, such as auroras) for centuries.’
http://en.wikipedia.org/wiki/Solar_cycle

Here we’ll try a bit of pattern-hunting, so to speak.

This post stems from comments on an earlier Talkshop thread e.g.:
https://tallbloke.wordpress.com/2015/12/21/why-phi-an-orbital-parameters-test/comment-page-2/#comment-113202
https://tallbloke.wordpress.com/2015/12/21/why-phi-an-orbital-parameters-test/comment-page-2/#comment-113371
https://tallbloke.wordpress.com/2015/12/21/why-phi-an-orbital-parameters-test/comment-page-2/#comment-113885
H/T Paul Vaughan for the Hale cycle/Jupiter-Saturn/phi connection.

Theory: an average solar (Hale) cycle is about 22.14 years.
It consists of one North and one South ’11 year’ magnetic cycle, due to magnetic reversal.
Paper: Synchronized Helicity Oscillations: A Link Between Planetary Tides and the Solar Cycle? [Aug.2016]
Quote: ‘we recover a 22.14-year cycle of the solar dynamo.’
http://arxiv.org/pdf/1511.09335.pdf

Jupiter-Saturn axial period is 8.456146 years.
That’s when the sum of J and S orbital movement in the conjunction period = 1, as shown here:
J-S conjunction period = 19.865036 years
(19.865036 / J orbit period) + (19.865036 / S orbit period) = 2.3491832
19.865036 / 2.3491832 = 8.456146 years
(8.456146 / J orbit period) + (8.456146 / S orbit period) = 1
Data (Sidereal Orbit Period) : http://ssd.jpl.nasa.gov/?planet_phys_par

If we assume two axial periods plus a remainder period:
22.14 – (2 x 8.456146y) = 5.227~y (= remainder)
8.456146 / 5.227 = 1.61778~ = ~Phi
22.14 / 8.456146 = 2.6182~ = ~Phi²
22.14 / 5.227 = 4.2357~ = ~Phi³

If we break the 22.14 years up into those parts like this:
8.456146 + 5.227 + 8.456146 = ~22.14y
it is equivalent to saying, in terms of a model:
1/Phi²(0.382) + 1/Phi³(0.236) + 1/Phi²(0.382) = 1 (Hale cycle)

Timo Niroma’s solar cycle analysis shows the min./max. periods of solar cycles.

From: Timo Niroma and Ray Tomes on solar cycle lengths and planetary alignments
https://tallbloke.wordpress.com/2010/07/20/timo-niroma-on-solar-cycle-lengths/
Sunspot cycles: Min. = 8.7 years, max. = 13.9 years
8.7 is about 1/4 of a year more than 8.456146y
13.9 is about 1/4 of a year more than 8.456146 + 5.227y (= 13.683y)
13.9 – 8.7 = 5.2, matches to 5.227

Very close to the phi pattern (above).

The variability of a single solar cycle lies within the 5.2 year period (1/Phi³) in the middle of the Hale cycle. The question as always is: what lies behind the observed variability?
—
Note: J-S axial period / Hale cycle = 1/Phi²