![Combined precession cycle [credit: wikipedia]](https://tallbloke.wordpress.com/wp-content/uploads/2016/02/precess2.jpg)
Combined precession cycle [credit: wikipedia]
‘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.
Three linked precessions
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 (4407) Metonic cycles
334932 TY / 3 = 111644 TY = apsidal precession cycle in tropical years
111644 TY = 13 x 113 x 4 (5876) Metonic cycles
Summary of the model
The apsidal precession cycle of the perihelion is 111,644 tropical years
The axial precession cycle is 25,764 tropical 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 tropical years
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 x 2.
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 = 7 x 13 x 8 lunar apsidal cycles.
Reference: Sidereal, tropical, and anomalistic years
Tallbloke's Talkshop 
If the numbers look a bit hard to digest, try the summary section only.
Remarkable. Nice work OB.
So you’ve reconciled apsidal and equinoctal precession using Fibonacci/phi ratios.
This on top of the amazing results Paul Vaughan has been getting with the solar-planetary theory this last week.
We’re on a roll!
Dr Bob put me on to it…
His comment starts: ‘Your logic is correct’
http://answers.yahoo.com/question/index?qid=20080122112444AAz6n6y
TB says: ‘We’re on a roll!’
And there’s a rumour we may have something on solar rotation up our sleeves 😉
Make that Rock and Roll then.
I am a bit wary of that argument. Question: how sensitive is it to a varying day length?
I have indicated elsewhere that there is evidence that I suspect indicates a varying day-length between 5000 to 2345 bce, maybe of considerable amount. As also explained here by Richard Heath https://www.academia.edu/8379226/A_Proposed_Itinerary_for_Megalithic_Astronomy pg3 from year to year the sun is seen to creep and then come back after the fourth day. Evidence seem to indicate that that creep varied, four spots for a quarter day extra, three for a third of a day, five… and one for a year with a whole number of days.
Now that would happen if earth inertia along ‘z’ axis changes due to polar ice loading fluctuations (termed as climate feedback). If obliquity changes too (as argued elsewhere that it has) that is an added complication because precession cycle duration may also change.
oldmanK: as I understand it precession works on the equatorial bulge, not on the axis.
I may be wrong but my understanding is that precession is the dynamic behavior of a spinning gyro (sort of). The influence of the planets on the equatorial bulge is what makes the tilt angle (obliquity) oscillate. A spinning top still precesses but its tilt angle remains the same (at least that’s what I understood in maths, and maths was not my forte’, so do correct me here).
Varying Iz with respect to Ix(=Iy) varies the tilt.
Hi OldmanK,
You’re right that the changing disposition of other celestial bodies in the Z axis will make a difference to obliquity via a quadrupole moment on the equatorial bulge. The precession of the line of apsides (where perihelion/aphelion occurs) is also an effect of other celestial bodies, which is reasonably well theorised. Compare the theoretical and observed/projected apsidal precession rates of the planets here:
http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node115.html
I thought it was mostly the Sun and the Moon acting on the bulge?
Wikipedia says: ‘There are a variety of factors which can lead to periastron precession, such as general relativity, stellar quadrupole moments, mutual star–planet tidal deformations, and perturbations from other planets.’
The Moon is the nearby celestial body with the biggest variable effect on the Earth tidally, but doesn’t affect apsidal precession. See the link I gave for the effects of the Sun and other planets on apsidal precession in the equations. Don’t forget the Earth’s apsidal precession is more properly the apsidal precession of the Earth-Moon system as a whole.
There was something else I left out (in haste) the relevant point to the subject matter.
if inertia in z axis changes, say increases, and assuming constant rotational energy, then speed of rotation decreases, making day length longer, and number of days per orbit lower (orbit time remaining constant –?).
It may also effect speed of precession (rate of precession -requiring a revised formula? ) , but I am even less sure here.
Orbit time depends only on distance from the Sun.
http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_law
If the Earth wants to change its behaviour it has to know a response from the Moon is possible 😉
TB says: ‘The Moon is the nearby celestial body with the biggest variable effect on the Earth tidally, but doesn’t affect apsidal precession.’
Not directly, but the orientation of the equatorial bulge changes as the tilt changes, which might have a non-trivial effect?
We could go the whole hog here, so to speak, and propose that the eccentricity cycle is 12/13ths of the apsidal (perihelion) precession, and that the obliquity cycle is 2/5ths of the eccentricity cycle.
Eccentricity: 103056 TY [usually estimated to be about 100,000 years]
Obliquity: 41222.4 TY [usually estimated to be about 41,000 years]
It then turns out that the axial precession of 25764 TY would be 5/8ths of the obliquity cycle, and 1/4 of the eccentricity cycle (25764 x 4 = 103056).
Since all these periods relate to the Earth-Moon system, it’s not unthinkable that they could also relate to each other. Of course we’re calling it a model, so if the governing period were to vary, the other cycles would be expected to adjust to that, one way or another.
NASA says:
‘What impact do the varying length of the anomalistic month and the direct (eastward) rotation of the Moon’s elliptical orbit have on the length of the lunation? To answer this, one must first consider Earth’s elliptical orbit around the Sun, which has a mean eccentricity of 0.0167. The center-to-center distance between Earth and the Sun varies with mean values of 147,098,074 km at perihelion to 152,097,701 km at aphelion. The direction of Earth’s orbital line of apsides also changes but at a rate far slower than the Moon’s. Having a direct (eastward) shift with a mean value of 0.0172° per year, it takes about 20,500 years for Earth’s major axis to make one complete revolution.’
360 / 0.0172 (deg. per year) = 20930.23~ years
Figures used in this post: 20933.25 TY – 20932.25 AY = a difference of 1
Thumbs up from NASA for our calc. 🙂
I noticed that also !
We can fit these cycles with Fibonacci sequence !
During 17 apsidal precession cycles there will be 72 precession of the equinoxes cycles and so 89 climatic precession cycles (17 + 72 = 89).
17 = 34 / 2 and 72 = 55 + 17 = 55 + 34 / 2.
34 + 55 = 89 is a step of the Fibonacci sequence as 89 / 34 = ~2.618 and 89 / 55 =~ 55 / 34 = ~ 1.618
89 / 17 =~ 3 + square root of 5
89 / 72 =~ -1 + square root of 5
72 / 17 =~ 2 + square root of 5
Etc… 🙂