Tropical timings – Earth’s obliquity and more

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

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Obliquity, or tilt, is the third of Earth’s three main orbital variations.

From the two previous posts on precession (here) and eccentricity (here), a diagram (here) was produced showing relationships of these orbital parameters of the Earth in numerical terms, using simple ratios. A Great Precession Cycle (GPC) was also identified and defined.

To add obliquity, which is often given in science papers as ’41 kyr’, we propose 10 per long eccentricity cycle (LEC):
412344 / 10 = 41234.4 tropical years (TY). The GPC is then equivalent to 10 obliquities + 10 axial precessions:
(41234.4 + 25771.5) * 10 = 670059 TY (1079*621).

This gives obliquity a 10:16 (= 5:8) ratio with the axial precession (AP) cycle in the period of one LEC, so the 5:8 ratio repeats every half LEC (207*996 TY). See also Conclusion below, with its diagram of ratios relating to obliquity (click on image to enlarge).

23 kyr
The period of the 3 (8-5) harmonics of obliquity and AP is:
(41234.4*5)/3 = 68724 TY or 69*996 TY. This can be split into 3 parts of ’23 kyr’ each, i.e. 23*996 = 22098 TY. Therefore 5 obliquity = 8 axial precessions = 9*23 kyr in half LEC (meaning 23 kyr = LEC/18).
Note: the sum of PP and AP = 6*23 kyr exactly = LEC/3.

19 kyr
This period derives from perihelion precession (PP) and 23 kyr.
PP = 111676.5 TY (see here)
23 kyr = 22908 TY
(PP*23 kyr) / (PP+23 kyr) = 19008.765 TY (19 kyr)
Ratio PP:23kyr:19kyr = 8:39(3*13):47(8+39)
It follows: (23 kyr*19 kyr) / (23 kyr+19 kyr) = PP

124 kyr
This period, sometimes called 123 or 125 kyr, is often quoted in science papers discussing Earth’s orbital parameters, e.g. J.Rial (see eccentricity post) says: ‘(Fig. 1C) shows evidence of frequency modulation as the observed 75- and 123-ky peaks coincide with the predicted sidebands of a 95-ky carrier frequency-modulated by a 413-ky signal.’

The two eccentricity cycles have a ratio of 3:13. The sub-harmonic is 0.3 LEC = 1.3 SEC with a difference of 1 in 1/10th (13-3=10) of the full period.
1/10th of 3 LEC = 3 obliquities = 41234.4*3 = 123703.2 TY (aka 124 kyr).

Conclusion
If obliquity:axial precession ratio is 5:8, that’s 3 (8-5) beats per 13 (8+5) axial periods = 3:13 beat:axial ratio. The periods of each are 1/6 LEC for beats and 1/6 SEC for axials, and LEC:SEC = 3:13. The beat periods are 3*23 kyr each. The ratio of the two precessions, perihelion and axial, is also 3:13.

Of course for the 3:13 ratios of both the precession and eccentricity pairs it’s the other way round…10 (13-3) beats and 16 (13+3) axial is a ratio of 10:16 = 5:8 = obliquity:axial precession.

Note: 3,5,8 and 13 are Fibonacci numbers, ‘a sequence in which each element is the sum of the two elements that precede it’ and which is ‘strongly related to the golden ratio’ (Wikipedia). Some of the periods are variable within limits, so the diagrams are intended to relate to mean values.
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Image: Earth’s Axial Tilt, or Obliquity [credit: Wikipedia]