Why Phi? – a unified precession model

‘Because of apsidal precession the Earth’s argument of periapsis slowly increases; it takes about 112000 years for the ellipse to revolve once relative to the fixed stars. The Earth’s polar axis, and hence the solstices and equinoxes, precess with a period of about 26000 years in relation to the fixed stars. These two forms of ‘precession’ combine so that it takes about 21000 years for the ellipse to revolve once relative to the vernal equinox, that is, for the perihelion to return to the same date (given a calendar that tracks the seasons perfectly).’ – Wikipedia

Here we’ll fit the three precession cycles into one model and briefly examine its workings.

In an earlier Talkshop post we discussed the quarter ‘axial precession’ cycle (sometimes called ‘precession of the equinoxes’) and produced a number chart showing the events that go to make a period of 6441 tropical years (TY), or 339 Metonic cycles of 19 TY.

Explanation of the graphic : it uses the 6441 TY = quarter axial precession as a basis for deriving all the precession periods shown. The numbers of precessions of each type are in Fibonacci proportions (numbers of cycles or sub-cycles: 2,3,8, and 13)

Also: 6441 TY = 6440.75 sidereal years (SY), meaning that the axial precession period is 4 times these numbers i.e. 25764 TY = 25763 SY = a difference of 1.

Noting that TY – AY (anomalistic years) = 1 is the definition of the combined precession cycle (Wikipedia calls it ‘This interaction between the anomalistic and tropical cycle’), we find:
20933.25 TY – 20932.25 AY = a difference of 1

At four times these periods we get whole numbers:
83733 TY = 83729 AY
83733 / 6441 = 13
so
83733 TY = 4 (83733 – 83729) combined precession cycles = 83733/4 = 20933.25 TY

Half the combined precession period is 20933.25 TY / 2 = 10466.625 TY
10466.625 TY / 6441 = 1.625 = 13/8
so
the ratio of half the axial precession to the combined precession is 13:8
(13 and 8 are consecutive Fibonacci numbers with a ratio close to phi)

83733 TY = 13 x 113 x 3 Metonic cycles (4407)
334932 TY / 3 = 111644 TY = apsidal precession cycle in tropical years
111644 TY = 13 x 113 x 4 Metonic cycles (5876)

Summary of the model

The apsidal precession cycle of the perihelion is 111,644 tropical years
The axial precession cycle is 25,764 years
The combined precession cycle is half the harmonic mean, or axial period, of those two cycles:
111644 x 25764 / (111644 + 25764) = 20,933.25 TY

Axial period: (20933.25 / 111644) + (20933.25 / 25764) = 0.1875 + 0.8125 = 1
Ratio of 0.8125:0.1875 = 13:3 (see chart)

Note also that the combined precession cycle consists of 2,366 lunar apsidal cycles which is 7 x 13².
7 apsidal cycles is 766 lunar synodic months and 13 is a Fibonacci number.
The 6441 TY period = 766 x 13 x 8 synodic months.

Reference: Sidereal, tropical, and anomalistic years