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

Posted: February 2, 2019 by oldbrew in Celestial Mechanics, solar system dynamics

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)
– – –
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.

  1. donald penman says:

    Well imo which hemisphere gets less sunlight during the summer months maybe not so important as many people think in starting glaciations after all glaciations occur at the same time in both hemisphere we do not have a northern hemisphere ice age and a southern hemisphere ice age. the minimum arctic sea ice extent could be more important than the level of solar radiation at an arbitary lattitude in the northern hemisphere in allowing ice sheets to form in the high lattitude land areas, my idea would be that if the arctic ocean fills up with sea ice and does not melt back in the summer then these land masses will not be able to warm up during the summer months but will still get colder during the winter months.
    The idea that solar radiation must reach a certain low level in the northern hemishere summer before ice sheets form is i think behind the idea that the next ice age is a long way in the future but a high sea ice minimum would produce the same low summer temperatures , the mini ice age and 1940-1970 cooling are examples.

  2. oldbrew says:

    Spectrum of 100-kyr glacial cycle: Orbital inclination, not eccentricity [1997 paper]
    Richard A. Muller* and Gordon J. MacDonald

    In this paper we note five sets of observations which conflict with the suggestion that insolation variations associated with eccentricity are responsible for the dominant 100,000-year cycle.

    – – –
    They seemed keen on this idea 20+ years ago. But if both inclination and eccentricity derive from the period of the obliquity/tilt cycle, is the point moot?
    = = =
    Another point of view…
    Obliquity pacing of the late Pleistocene glacial terminations [2005]
    Huybers P, Wunsch C.

    Here we present a statistical test of the orbital forcing hypothesis, focusing on the rapid deglaciation events known as terminations. According to our analysis, the null hypothesis that glacial terminations are independent of obliquity can be rejected at the 5% significance level, whereas the corresponding null hypotheses for eccentricity and precession cannot be rejected. The simplest inference consistent with the test results is that the ice sheets terminated every second or third obliquity cycle at times of high obliquity, similar to the original proposal by Milankovitch. We also present simple stochastic and deterministic models that describe the timing of the late-Pleistocene glacial terminations purely in terms of obliquity forcing. [bold added]

    – – –
    Of course some scientists/commentators say eccentricity alone couldn’t cause glaciations, even if the timescales look very similar.

  3. J Martin says:

    The BRI work would suggest that the 70,000 cycle is not relevant. That precession is not relevant and that only obliquity and eccentricity play a role.

    It would be interesting if eccentricity is a function of obliquity.

  4. dscott says:

    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.

    uhm, no, linear suggests proportionality, a sin wave varies non linearly in proportion to either a progressively increasing or decreasing response. It’s more of a cascade response to a series of variables. If you don’t account for the major variables, you can’t accurately account for the response.

    Photosynthesis responses by plant life to insolation changes may not be proportional. Plant density and color and variety are all variables that affect the ecosystem and in turn climate. Savanahs and forest cover affect temperature and therefore the mirco-climate of the area. Add up all the mircro-climates and you end up with a composite of the whole. Nothing is done in a vacuum.

  5. stpaulchuck says:

    more orbital mechanics, after Scafetta and others, this is not a surprise but a welcome addition to actual science and not more BS incompetent computer programs. I understand they are creating computer models too, but it would appear theirs are based on empirical data from the real world with no magic “forcings” numbers or universal variable constants added for effect.

    I thank all these folks for their contributions to our knowledge base. I am so worn out of the modern Piltdown Man scam called AGW, or nowadays – climate extremes.

  6. Chaeremon says:

    The orbital elements of mathematical Milankovitch cycles have been modeled in NASA/JPL DE421: there are four immense peaks represented by equinox (solstice) and perihelion (aphelion) falling together, in that model, at circa 4027bce, 1169ce and 6374ce.
    Since we have no record from neanderthals (or what they left for us is always academically gobbledegooked — with sensational help from fake media), nobody can say that such cycles have serious meaning or authenticity. And those who corroborate with confirmation bias only create vicious circles in reasoning.
    How can we get out of that.

  7. oldbrew says:

    Since the ~71kyr and ~100kyr periods are derived from the same shorter period with factors of √ 3 and √ 6, their ratio to each other in the model is 1:√ 2. (Geometric note: so in a diagram of a square, the 71kyr period would be any side and 100kyr would be any diagonal).

  8. oldmanK says:

    This looks like becoming an interesting thread. My 2c worth; but I’ll add something else later when prepared.

    Re the two links in the subject matter:
    The WSU show obliquity curve for more than 600k yrs, but do not prove its provenance, or/and its validity. J Bronowski ” they are not here to worship what is known, but to question it.” ― Jacob Bronowski, The Ascent of Man ”

    The Muller UC: quote in summary ” However we know of no model, linear or nonlinear, that can reconcile the of eccentricity with the observed narrow 100 kyr glacial cycle. This is a severe and possibly fatal problem for the conventional Milankovitch mechanism of insolation, ——- may have serious systematic errors.”

    Chaeremon says (an invite? 🙂 ) : February 3, 2019 at 7:42 am ; quote “Since we have no record from neanderthals,–“. Not from neanderthals, since they were brighter than us. They have left us a record/method of how to predict the day and the hour of the solstice, days in advance. Inadvertently they also left a record of obliquity values over a span of time, which should put paid to the Stockwell/Newcomb assertion that there is nothing beyond the secular changes to obliquity.

    D P Rubincam in “Has climate changed the Earth’s tilt?” https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/95PA00578 This paper proposes a cyclic mechanism whereby a freely rotating body with varying M of I in the xyz planes can change its orientation. (it is precisely what the ‘neanderthal’ record shows, but in much shorter time-spans. Those events have left their mark all over the various proxies. But that is like reading Sumerian at a time when Sumerian did not exist).

    Once the obliquity of a rotating sphere in ‘free’ space is seen to be subject to both secular and abrupt adjustments, then the matter completely changes. The question is: What and Where is the evidence for that? The not so neanderthal ‘neanderthals’, and what the classic writes have left us (Plato Herodotus–) is food for thought.

  9. oldbrew says:

    Mars currently has a similar tilt to Earth.

    Ancient Impact Craters Reveal Mars’ First Equator
    – – –
    More Evidence for an Ancient Ocean on Mars

    Just like the Earth, Mars is wider around its equator than it is from pole to pole. At some point, several billion years ago, when the planet still had an ocean, it experienced a redistribution in its mass. Perhaps this came from a series of volcanic eruptions, such as the ones that created the Tharsis Bulge and Olympus Mons. This additional mass caused the planet to rebalance its spin, shifting its pole 50 degrees away from its position.


    Of course that was a very long time in the past, even if true.

  10. Chaeremon says:

    @oldmanK: shoot 🙂

  11. oldmanK says:

    @ Chaeremon: thanks for the push.

    The below was discovered a 3wks ago. The implications are multi-fold. Tectonically; astronomically; human achievement and interaction; anthropology; etc


    Re this thread, the question of earth tilt and its possible abrupt shifts needs reconsideration. Much of what is generally discussed, such as insolation/latitude; volcanism; changes of ocean currents; land areas – emerged or submerged; all are collateral of a possible abrupt tilt change. The evidence there is points to such. The opposite seems to be inadequate mathematical analysis.

  12. oldbrew says:

    Reminder: the time taken to go from min-max-min obliquity (currently ~41 kyr) is the basis of this thread 🙂

  13. Chaeremon says:

    @oldmanK, interesting find, unenvious respect. But the interpretation makes forefathers of old way too stupid (so that wannabee literary scholars can pass their exam by copy & paste).
    1. megalithic buildings are not there for just reminding that a year has another season; and they are only needed for educating the offspring, in epochs to come, by referring to a common base (skills, crafts, knowledge of what was and may be coming again).
    2. the last astronomical year which had Venus transits in weeks short before equinox was 6995bce.
    3. the first astronomical year which had all Venus transits in weeks after equinox was 2 times 13 lunestice epochs later.
    4. this began the epoch where the orientation of and graffiti on buildings became outdated (luminous example: Göbekli Tepe).
    5. the forefathers’ method for epochal time-keeping was the lunestice, it takes 13 lunestices until the epoch with another Venus transit of similar (trustworthily comparable …) characteristics.
    6. lunestices occur at the solstice ecliptic position, AFAIK this is an astronomical constant which hasn’t changed over time.
    Please: restrict to new arguments, avoid reiterations, TIA.

  14. oldbrew says:

    Chaeremon says:
    DE421: there are four immense peaks represented by equinox (solstice) and perihelion (aphelion) falling together, in that model, at circa 4027bce, 1169ce and 6374ce.
    Since we have no record from neanderthals (or what they left for us is always academically gobbledegooked — with sensational help from fake media), nobody can say that such cycles have serious meaning or authenticity.

    Those periods of ~5200 years would fit quite well into this.

    On both Earth and Mars, these two precessions are in opposite directions, and therefore add, to make the precession cycle between the tropical and anomalistic years 21,000 years on Earth and 56,600 years on Mars.


    21000 = 5250 * 4

  15. Chaeremon says:

    @oldbrew, the picture has basis same mathematical astronomy, sure good fit. The authors could have indicated the perihelion (aphelion) movement a bit more direct; one can guess that from the difference in area / size above and below their horizontal line (Keplerian areas expressing orbital “Jan 4th” speed).

  16. oldbrew says:

    As on Earth, Mars experiences Milankovitch cycles that cause its axial tilt (obliquity) and orbital eccentricity to vary over long periods of time, which has long-term effects on its climate. The variation of Mars’s axial tilt is much larger than for Earth because it lacks the stabilizing influence of a large moon like Earth’s moon. Mars has a 124,000-year obliquity cycle compared to 41,000 years for Earth.


    That looks a lot like 3 * Earth’s obliquity cycle : 41,000 * 3 = 123,000 years
    We’ll come to that in Part 2.

  17. oldmanK says:

    oldbrew: reminder noted; seems I got carried away.

    Still I would recommend in parallel, checking the foundations of this extensive subject. links here put it in perspective.

    @ Chaeremon: limiting myself here since off-subject. You seem to be missing the fundamental issue. It is not reckoning time from the divination of graffiti and such. It is simple technique and basic mathematics. Explained here: https://www.facebook.com/melitamegalithic/videos/458039251394832/ Note that structure is still fully functional, but originally designed on much lower obliquity.

  18. oldmanK says:

    Links in above appear non functional but did open before. They are

    Title: The evolution of adopted values for precession
    Authors: Lieske, J. H.
    Journal: Celestial Mechanics (ISSN 0008-8714), vol. 37, Nov. 1985, p. 209-238.
    Bibliographic Code: 1985CeMec..37..209L

    Title: Progress Report of the International Astronomical Union Division I Working Group on Precession and the Ecliptic
    Authors: Hilton, J. L.
    Affiliation: AA(U.S. Naval Observatory)
    Publication: American Astronomical Society, DDA meeting #36, id.05.02; Bulletin of the American Astronomical Society, Vol. 37, p.524
    Publication Date: 05/2005
    Origin: AAS
    Bibliographic Code: 2005DDA….36.0502H

  19. Chaeremon says:

    @oldmanK, I’m not missing your issue or something except a not-so-small extension of just 5 minutes, to understand your find (i.o.w. more time). Thanks for the vid.
    Next time you travel, and crossing or pass by Lower Saxony, we have to sit together, with a wine glas [me: white] on a terrace in mild weather, and talk through the possibilities.
    Yet without elimination of the impossible (from a list big as possible), I won’t ratify your interpretation. Please you begin and email me how the 3195bce dating was done and how they cross-checked it with geochronological method/s (surface exposure dating).

  20. nickreality65 says:

    Just in case you missed earlier missives, allow me to summarize this science yet again.

    1) 288 K – 255 K = 33 C warmer with the atmosphere is rubbish. 288 K is a WAG pulled from WMO’s butt. NOAA/Trenberth use 289 K. The 255 K is a theoretical S-B temperature calculation for a 240 W/m^2 ToA (w/ atmosphere!!) ASR/OLR balance (1,368/4 *.7) based on a 30% albedo.

    By definition no atmosphere includes no clouds, no water vapor, no oceans, no vegetation, no ice, no snow an albedo perhaps much like the moon’s 0.15. 70% of the lit side would always be above freezing, 100 % for weeks due to the seasonal tilt, not that it matters since there would be no water to freeze.

    Without the atmosphere the earth will get 20% to 40% more kJ/h depending on its naked albedo. That means a solar wind 20 to 30 C hotter w/o an atmosphere not 33 C colder. The atmosphere is like that reflective panel behind a car’s windshield.

    2) The 396 W/m^2 upwelling ideal BB LWIR that powers the RGHE is, as demonstrated by experiment, not possible. If this upwelling energy does not work – none of RGHE works.

    3) The 333 W/m^2 up/down/”back” GHG energy loop is thermodynamic nonsense, i.e. it’s calculated energy appearing out of nowhere, a 100% efficient perpetual energy loop, energy from cold to hot without work. (396 – 333 = 63) “net” radiation is thermodynamic nonsense.

    4) 1) + 2) + 3) = 0 RGHE & 0 GHG warming & 0 man caused climate change.

    I’ve got the science. If you have some anti-science, BRING IT!!

    Nick Schroeder, BSME CU ’78, CO PE 22774

    P.S. According to NOAA the current rate of sea level rise is 3 mm/y. That’s not even a foot per century.

    P.P.S According to JAXA and DMI the sea ice and ice cap volumes have not deviated significantly from decades of natural variability.