Obliquity, inclination and eccentricity of Earth – a model: Part 2

Kepler's trigon – the orientation of consecutive Jupiter-Saturn synodic periods, showing the repeating triangular shape (trigon).

This of course follows on from the very recent Part 1 of the model. Since Jupiter and Saturn are the dominant planets in our solar system, we can speculate that they may have a significant effect on the obliquity of smaller bodies. Or they may not – no-one knows, but we can look at possible evidence.
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Precession of the Jupiter-Saturn conjunction (J-S) was worked out by Kepler centuries ago, as shown in his diagram to the right.

'As successive great conjunctions occur nearly 120° apart, their appearances form a triangular pattern. In a series every fourth conjunction returns after some 60 years in the vicinity of the first. These returns are observed to be shifted by some 7–8°' – Wikipedia.

The conjunction periods in the Kepler diagram have been rounded up to 20 years each, but in reality the mean period of 9 such conjunctions (aka the Jose cycle or solar inertial motion cycle ) is 178.78532 years, rather than 9 * 20 = 180 years.

To get an accurate period for the return of the conjunction to its exact original position in the 360°circle, we start with the degrees of movement in one conjunction, i.e. 117.14701°. To cut the story short, we find a match at 2012 revolutions:
2012 * 360 = 724320 degrees
724320 / 117.14701 = 6183.0003 J-S

6183 / 9 = 687 Jose cycles = 10354 Jupiter orbits = 4171 Saturn orbits

The point of this is: 687 * 178.78532 y = 122825.51 years
10354 Jupiter orbits @ 11.862615 y = 122825.51 years
4171 Saturn orbits @ 29.447498 y = 122825.51 years
Data source: JPL/NASA
(Note: 2012 = 6183 – 4171)

Comparing to the estimated 41,000 year obliquity cycle:
3 * 41000 = 123000
122825.51 / 123000 = 0.9985813 (i.e. a ~99.86% match).

Since 687 is divisible by 3 (result = prime number 229), the pattern should repeat 3 times in the period, i.e. once per 120° of the full 360° cycle:
122825.51 / 3 = 40941.836 years = obliquity period (according to this model).

This is where the idea of the obliquity multiples of 2 and 3, as discussed in Part 1 of the model, comes from.

Note also that Mars, with a similar mean tilt angle (~25°) to that of Earth (~23.3°), has an estimated obliquity of 124,000 years (source – Wikipedia), i.e. ~3 times that of Earth and closely matching the J-S period above (122,825 years).
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Summary – the model proposes that:
Ratio of Obliquity cycle to Inclination cycle is 1:√3
Ratio of Obliquity cycle to Eccentricity cycle is 1:√6
therefore
Ratio of Inclination cycle to Eccentricity cycle is 1:√2.

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