Obliquity, inclination and eccentricity of Earth – a model: Part 1

Earth's Axial Tilt, or Obliquity [Credit: Wikipedia]

First let's get the approximate target numbers for the model.

'The inclination of Earth's orbit varies with respect to the solar system's invariant plane with a period of roughly 71000 years.
. . .
Taken in conjunction with the 26000-year spin-axis precession, the 71000-year orbit precession causes a 41000-year oscillation in the tilt of the earth's axis, about plus or minus 1.3 degrees from its average value of 23.3 degrees. This number is not absolutely stable – it depends on the combined positions of all the planets through time.'

Astronomy: precession of Earth (Washington State University)
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Origin of the 100 kyr Glacial Cycle: eccentricity or orbital inclination?

'Spectral analysis of climate data shows a strong narrow peak with period ~ 100 kyr, attributed by the Milankovitch theory to changes in the eccentricity of the earth's orbit. The narrowness of the peak does suggest an astronomical origin; however the shape of the peak is incompatible with both linear and nonlinear models that attribute the cycle to eccentricity or (equivalently) to the envelope of the precession. In contrast, the orbital inclination parameter gives a good match to both the spectrum and bispectrum of the climate data.'

Richard A. Muller — University of California, Berkeley and
Gordon J. MacDonald — University of California, San Diego

Let obliquity (tilt) = letter 'O'

Model assumption:
Inclination (estimated as 71000 years) = Geometric mean of 1*O and 3*O
Eccentricity (estimated as 100000 years) = Geometric mean of 2*O and 3*O

For two numbers, geometric mean = the square root of their multiplied value:
Square root of 1 * 3 = 1.7320508
Square root of 2 * 3 = 2.4494897

If O = 41000 years:
Inclination = O * 1.7320508 = 71014 years (~99.98% match to target)
Eccentricity = O * 2.4494897 = 100429 years (~99.57% match to target)
– – –
So that's the basic idea with Part 1 this model: inclination and eccentricity can be derived from obliquity, according to these results.

In Part 2 the orbital factor comes in, with relevant supporting numbers. It will also show where the idea of the obliquity multiples (2 and 3) originates from.

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