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

Posted: February 4, 2019 by oldbrew in Celestial Mechanics, solar system dynamics
Tags:

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.
– – –
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 José 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 José 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).
– – –
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.

Comments
  1. Stephen Richards says:

    But why did the iceages once follow the 41000 year cycle ?

  2. oldbrew says:

    Stephen R: the moon on a stick costs extra 😎

    https://en.wikipedia.org/wiki/Milankovitch_cycles#Transition_problem

    Update: this attempts to explain it…

    Earth’s orbital variations and sea ice synch glacial periods
    January 26, 2017
    https://m.phys.org/news/2017-01-earth-orbital-variations-sea-ice.html

  3. tallbloke says:

    Good work OB!
    Three obliquity cycles during the precession cycle and also three grand cycles of J-S conjunctions.
    Looks like a strong possibility of a link there to me.

  4. oldbrew says:

    Thanks TB. I forgot to mention the Neptune link…

    Take the 122825.51 year J-S period (~3 obliquity cycles) and multiply by 3.
    We now have : 2061 Jose cycles (687*3) = 2059 * 14 Jupiter-Neptune conjunctions.
    It’s also 2236 Neptune orbits.

    This is an outcome of these numbers:
    9 J-S = 178.78532 y (Jose cycle)
    14 J-N = 178.959 y

    PS Re ‘Three obliquity cycles during the precession cycle’ – no, see summary.

  5. oldbrew says:

    Re obliquity:
    Saturn’s [largest] moon Titan has a cycle of approximately 60,000 years…
    Neptune’s [largest] moon Triton has a variation similar to Titan’s
    .

    https://en.wikipedia.org/wiki/Milankovitch_cycles#Outer_planets

    So approx. half of Mars and 3:2 with Earth.

    One estimate sets the present Martian obliquity period at 120,000 years (Perron and Huybers, 2009).
    https://www.sciencedirect.com/topics/earth-and-planetary-sciences/obliquity

  6. tallbloke says:

    Right you are.
    Maybe I was thinking about a multiple of the axial precession period. 128860yrs is 5 x 25772yrs

  7. oldbrew says:

    Re: We now have : 2061 Jose cycles (687*3) = 2059 * 14 Jupiter-Neptune conjunctions.
    It’s also 2236 Neptune orbits.

    So nine obliquity cycles (3*3) looks like another Grand Synod.
    2236 N = 43 * 52
    52 Neptune = 102 Uranus = 291 Saturn

    Just multiply those numbers by 43.
    We already have a whole number of Jupiter orbits (10354 * 3).
    In fact Uranus is about 1/4 of an orbit ‘off’, but in a total of 4386 (43*102) that’s a very small difference.

    And 9*3 = 27 obliquity cycles = 752 Dansgaard-Oeschger cycles.
    27 * 40941.836 = 1,105,429.5 years
    1,105,429.5 y / 752 = 1469.986 y

    Dansgaard–Oeschger events are rapid climate fluctuations that occurred 25 times during the last glacial period. Some scientists say that the events occur quasi-periodically with a recurrence time being a multiple of 1,470 years – Wikipedia

    An obliquity/inclination/eccentricity factor in D-O patterns? Looks like it…

    Time-frequency analysis shows that in GRIP the DO mode undergoes a frequency modulation that is phase-locked with the Earth’s obliquity cycle
    https://www.sciencedirect.com/science/article/abs/pii/S0277379101001408
    GRIP: https://en.wikipedia.org/wiki/Greenland_ice_core_project
    – – –
    IIRC somebody referred to a 1.1 million year Jupiter-Neptune period in one of the old Talkshop brainstorming posts.

    Yes – it was Ray Tomes:
    For example alignments of the 4 gas giants are much more likely to have effects than of say Mercury, Mars and Pluto. Long term solar system calculations show for example a 1.1 million year energy exchange between Jupiter and Neptune.

    https://tallbloke.wordpress.com/2011/06/30/further-terrestrial-evidence-of-planetary-cycles-affecting-climate/#comment-7392

    27 * 41000 = 1,107,000

    Do I feel a Part 3 coming on – or did I just write it? 😃

  8. oldbrew says:

    Another interesting-looking paper…

    Obliquity‐driven expansion of North Atlantic sea ice during the last glacial

    Chris S. M. Turney Zoë A. Thomas David K. Hutchinson Corey J. A. Bradshaw Barry W. Brook Matthew H. England Christopher J. Fogwill
    First published: 24 November 2015

    4.1 Mechanisms of Change: Sea Ice and Obliquity
    The trends in autocorrelation and variance in both records, and resultant timing of the climate shift, parallel the decrease in the Earth’s axial tilt (obliquity) [Berger, 1978] (Figure 1), in marked contrast to summer insolation at 65°N, which rises and then falls over the same period. Low obliquity is known to drive high sea ice extent in the Arctic due to decreased seasonality [Mantsis et al., 2011; Tuenter et al., 2005].
    . . .
    4.2 Wider Implications
    . . .
    Regardless of the origin of D‐O events, however, our results suggest that obliquity tipped the North Atlantic into a more frozen state, driving synchronous hemispheric climate change that led to the establishment of the Last Glacial Maximum.

    https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2015GL066344
    – – –
    Yes, ‘ship of fools’ Chris Turney…

    https://tallbloke.wordpress.com/2014/01/06/nature-takes-its-turney-excuse-pill/

    Ship of Fools in the Antarctic
    By Jack Kelly
    January 12, 2014

    So when the research ship Akademik Shokalskiy got stuck in the ice about 40 miles from Antarctica, some who knew the purpose of its voyage dubbed it the “Ship of Fools.”

    “It would take a heart of stone not to laugh,” said British journalist Leo McKinstry.

    Aboard the vessel were 22 scientists headed by Chris Turney, a professor of climate change at the University of New South Wales, four journalists and 26 tourists.

    By comparing their measurements with those taken by Australian explorer Sir Douglas Mawson in 1913, they hoped “to prove the East Antarctic ice sheet is melting,” noted the Australian, a newspaper in Sydney.

    It was “a pseudo-scientific expedition,” the director of the French Polar Institute told Agence France Presse.

    “The debacle in the Antarctic ice is probably the largest setback for global warming campaigners since the Climategate scandal in 2009,” said the (London) Financial Times.

    https://www.realclearpolitics.com/articles/2014/01/12/ship_of_fools_in_the_antarctic_121200.html

    ‘MV Akademik Shokalskiy is an Akademik Shuleykin-class ice-strengthened ship’ – Wikipedia

  9. oldmanK says:

    Picking up a clue from Malaga Bay for March 4th 1345ce, a near peak on the Eddy cycle corresponding to some astronomical event in the MWP (a trip point?) the planetary setup on Solar Simulator was interesting. Something similar an ~~Eddy cycle before (RWP) produced somewhat similar arrangement.

    Eddy cycle is given by 976+/- 53. A modulated cycle by planet positions ?????

  10. oldbrew says:

    16 J-S rotations of 360 degrees = 976.7 years.

    980 years is exactly 2/3rds of the Dansgaard-Oeschger cycle of 1470 years where:
    41 S-N = 74 J-S = 115 J-N (41+74) [Saturn, Jupiter, Neptune)

    Re D-O see comment: February 5, 2019 at 9:07 am

    Also – Earth, Venus and Mercury line up at 980 years (but not at the same time as J-S-N).

  11. oldmanK says:

    My point was that March 4th 1345ce had Earth by itself on one side of sun, and all others in two groups opposite of sun. Somewhat similar March 411ce

  12. oldbrew says:

    oldmanK – our solar system has 2 groups of planets, the fast and slow orbiters.

    All four slow orbiters can be ‘on one side of the Sun’ for 5-6 years at a time i.e. half a Jupiter orbit, longer for the other three giant planets – up to 80+ years.

    The four fast orbiters meanwhile are going round every year or less, apart from Mars (1.88 years). Is it all that unusual for one of the fast ones to be briefly on one side of the sun at a given snapshot in time, while the other three are briefly on the other side?

    Simulator example: late April 435.

  13. tallbloke says:

    ~980years was identified by Semi as a key period of solar system motion in his analysis too.
    https://tallbloke.wordpress.com/2009/12/30/meet-the-new-kepler-p-a-semi/

  14. […] Obliquity, inclination and eccentricity of Earth – a model: Part 2 […]

  15. oldbrew says:

    From Part 1 of the obliquity model:

    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

    – – –
    What has been overlooked so far is: 3 is the geometric mean of 1 and 9.

    1 = J-S conjunction and Obliquity period (1*O)
    3 = J-S Trigon and J-S precession cycle (~123 kyr = 3*O)
    9 = Jose cycle and J-S-U-N Grand Synod (3² *O)

    In other words:
    — The trigon is the geometric mean of the J-S conjunction and the Jose (or solar inertial motion) cycle
    — The J-S precession cycle is the geometric mean of the Obliquity period and the J-S-U-N Grand Synod

  16. tallbloke says:

    61.5kyr seems to be key, because it’s the a*b/a-b result from 123*100/(123-100) and also 123*41/(123-41)

    61.5kyr is near the Titan and Triton cycles and half the Mars cycle you mentioned.

    It’s also half of the 123kyr itself!

    So 61.5kyr is in a ~2:3 with the Obliquity cycle, a ~3:2 with the glacial cycle, and a 2:1 with the Mars period and ~matches the Titan and Triton periods.

    Interesting!

  17. oldbrew says:

    No, geo mean is the ‘n’th root of n numbers, so for two numbers it’s just sqrt(a*b).
    Sqrt(123*41) = 71

    For 1 and 9 it’s sqrt(1*9) = 3.

    So it’s a number where dividing by it gives you the lower number, and multiplying gives the higher number:
    71 kyr / sqrt(3) = obliquity
    71 kyr * sqrt(3) = 3 * obliquity

  18. tallbloke says:

    Right you are. I realised that, and edited my comment. Have another look. 🙂

  19. oldbrew says:

    61.5kyr seems to be key, because it’s the a*b/a-b result from 123*100/(123-100) and also 123*41/(123-41)

    The first part isn’t right : 123*100/(123-100) = 534.78

    61.5 kyr is 3/2 obliquities, as you said.

  20. tallbloke says:

    Serves me right for trying to take short cuts with the calculator.

  21. tallbloke says:

    Thinking about the harmonic of the 1*O and 3*O periods:
    122825.51 * 41000 / (122825.51 – 41000) = 61543yrs

    That’s pretty close to a phi ratio with the 100kyr glacial cycle isn’t it? 🙂

  22. oldbrew says:

    Yes – and/or with the 100kyr eccentricity cycle.

    It amounts to 1/4 of the square root of 6 = 0.61237

  23. tallbloke says:

    Well the geometric mean of Phi and 0.61237 is 0.61519 so we’ve cracked it. 🙂

  24. oldbrew says:

    If you say so 🤔

    According to Steve Holmes, the Venus ‘precession rate is surprisingly rapid – 44.74 arc-sec/yr.’
    https://www.drivehq.com/web/FreeHostSpace/steveholmes/dwrntm/obliq/obliq.htm

    That’s about 8:9 with Earth, and returns 28967~ years (162 José = 28963y).
    28967 * √2 = 40966y i.e. almost the same as our Earth obliquity number (40941.8)

    Not found anything on a Venus obliquity period though, apart from this:

    Obliquity Variability of a Potentially Habitable Early Venus [2016]

    The orbital variations of Venus and Earth involve some changes in the orbital inclination of the two planets, represented by the distance of the lines in Fig. 2 from the origin. The primary effect, though, is counterclockwise near-circular changes that correspond to the precession of the orbit through space. We call that effect nodal precession, as it drives monotonic increases in the element known as the orbit’s ascending node, the angle at which the planet comes up through the reference plane from below. We show the effective period of the nodal precession of the orbits of Venus and Earth in Fig. 3. Although the precession rate changes as the orbital inclinations of each planet vary, the long-term average precession rate for both planets is in the vicinity of ~70,000 years. [bold added]

    https://webpages.uidaho.edu/mhedman/papers_published/Quarles_Venus_AstBio_2016.pdf

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