Orbital resonance and the celestial origins of Earth’s climatic changes – Why Phi?

Posted: October 30, 2021 by tallbloke in Analysis, Astrophysics, Celestial Mechanics, climate, COP26, Cycles, Ice ages, modelling, moon, Natural Variation, Phi, research, Solar physics, solar system dynamics
Tags: ,

A year after I wrote the original ‘Why Phi’ post explaining my discovery of the Fibonacci sequence links between solar system orbits and planetary synodic periods here at the Talkshop in 2013, my time and effort got diverted into politics. The majority of ongoing research into this important topic has been furthered by my co-blogger Stuart ‘Oldbrew’ Graham. Over the last eight years he has published many articles here using the ‘Why Phi’ tag looking at various subsystems of planetary and solar interaction periodicities, resonances, and their relationships with well known climatic periodicities such as the De Vries, Hallstatt, Hale and Jose cycles, as well as exoplanetary systems exhibiting the same Fibonacci-resonant arrangements.

Recently, Stuart contacted me with news of a major breakthrough in his investigations. In the space of a few hours spent making his calculator hot, major pieces of the giant jigsaw had all come together and brought ‘the big picture’ into focus. In fact, so much progress has been made that we’re not going to try to put it all into a single post. Instead, we’ll provide an overview here, and follow it up with further articles getting into greater detail.

One of the longest known climatic periods is the ~413,000 year cycle in the eccentricity of Earth’s orbit. This period has been found in various types of core sample data and discussed in many paleoclimatic science papers, along with cyclicities around 95, 112 and 124kyr, and shorter periods such as Earth’s obliquity variation, ~41Kyr and Earth’s equinoctial-precession periods of ~19 and ~23kyr. Stuart has discovered how all of these periods are related to each other and to the planetary orbits and their synodic conjunctions.

We’ve also been able to link these Earth Orientation Parameters and climatic periodicities to the planetary orbital and synodic conjunction periods which we believe are key to modulating solar activity. The basis for these were laid out in my 2011 post on Jupiter and Saturn’s motion and further developed with the valuable input of many Talkshop contributors, culminating in the solar variation models published by Rick Salvador and Ian Wilson in the 2013 special issue of Pattern Recognition in Physics.

Solar Total Solar Irradiance (TSI) prediction model hindcast created by Rick Salvador using planetary periods discussed at the Talkshop in 2013

Figure 1 below scratches the surface of what we have discovered. These relationships are all precise whole number ratios, not approximations. The red ‘Graham Cycle’ is a novel addition to previously known cyclic periods which connects the three areas of the figure; Solar-Planetary at the top, climatic periods bottom left, and Earth Orientation Parameters bottom right. Of note, are the ratios between the 60kyr Graham Cycle period and the periods in the three groups. They are mostly ratios of Fibonacci numbers or combinations of them. We know from a previous investigation that Fibonacci and phi (Golden Section) related periodicities tend to be stable and minimally resonant. It could be that the reason the 60kyr period hasn’t been found previously is due to it not showing up strongly in periodograms and other spectral analyses. Nonetheless, it’s an important period for our ‘Why Phi’ investigation and has a lot more connections than we wanted to clutter up Figure 1 with, as it already looks pretty busy!

Figure 1. Spatio-temporal diagram showing solar system dynamical arrangement with particular reference to solar modulation, climatic periods and Earth Orientation Parameters affecting Milankovitch cycles.

Solar cycles

Starting with the upper ‘Solar planetary’ section of figure 1, Ian Wilson’s 2013 PRP paper noted that the Hale cycle and Jupiter-Saturn synodic (J-S) have a 193 year beat period, which is evident in Oxygen18 isotope data as well as Group Sunspot Numbers and 10Be ice core data. This was picked up by the Helmholtz Institute research lab and covered in our earlier post on the Solar Magnetic cycle. What they didn’t pick up on is the fact that the same 193year beat period can also be derived from the 178.8yr Jose cycle and the 2403yr Solar Inertial Motion (SIM) period.

This second route to the 193 year solar magnetic cycle is a novel result revealed in this post. Using the beat period formula of (A*B)/(A-B) = period, the solar inertial motion cycle (A) proposed by Charvatova of ~2403 tropical years and the Jose cycle (B) produces the same 193 year result. It was then possible to tie all this together in the 60 kyr cycle shown in the diagram.

There are 336 Jose and 25 SIM in 60 kyr which means the beat period produces 336-25 = 311 solar magnetic cycles of 193 years each. The number of Hale cycles in 60 kyr is given by the number of J-S minus the number of solar magnetic cycles. i.e. 3024-311 = 2713. It’s notable that 311 and 2713 are both prime numbers. Coupled with the fact that the number of J-S in 60Kyr is the Fibonacci multiple 144×21, we think this is a strong indicator that both 193yr and 60kyr periods are significant solar-planetary cyclic periods.

Support for the 60kyr period comes from Russia, where in 2017 A. S. Perminov and E. D. Kuznetsov produced a paper at at Ural Federal University, Yekaterinburg, entitled ‘Orbital Evolution of the Sun–Jupiter–Saturn–Uranus–Neptune Four-Planet System on Long-Time Scales’. This paper shows inter-related variations in the orbital parameters of the gas giants including antiphase changes in the eccentricities and orbital inclinations of Jupiter and Saturn at ~60kyr and in-phase changes in those parameters at ~400kyr, antiphase to Uranus. These ~400kyr variations are likely to be drivers of Earth’s 413kyr eccentricity cycle.

ISSN 0038-0946, Solar System Research, 2018, Vol. 52, No. 3, pp. 241–259. © Pleiades Publishing, Inc., 2018.
Original Russian Text © A.S. Perminov, E.D. Kuznetsov, 2018, published in Astronomicheskii Vestnik, 2018, Vol. 52, No. 3, pp. 239–259
.

Planetary-climatic cycles

Moving on to the lower left ‘climatic and planetary cycles’ section of Figure 1,

The de Vries cycle is half of 21 J-S and is a prominent climatic cycle. It also links to other cycles through resonant harmonics: Hallstatt = 11 de Vries, J-S synodic precession cycle = 12 de Vries. 6 de Vries is 7 Jose cycles. 33 de Vries is 7 Eddy cycles. See also Why Phi? – Jupiter, Saturn and the de Vries cycle.

The lunar-terrestrial year (L-T) is 13 lunar months. Earth’s tropical year is used throughout this post. Whole numbers of both occur at 353 tropical years and 363 lunar years, forming 10 beats (363-353) of 35.3 years. An important period is 13 L-T, which is 2 Hallstatts and 11 de Vries cycle pairs (22 de Vries). This is 1/9th of the obliquity cycle. It is also 3x7x11 J-S. It follows that the 41kyr obliquity cycle is 3x7x11 Jose cycles, because the Jose cycle is 9 J-S. 3,7 and 11 are all Lucas numbers. We will post a separate article on the inter-relation of the Fibonacci and Lucas series, as they relate to orbital resonance. See also Sidorenkov and the lunar or tidal year (2016)

An explanation for the effect of the motion of the gas giants on these and other climatic periods is found in Nicola Scafetta’s 2020 paper ‘Solar Oscillations and the Orbital Invariant Inequalities of the Solar System’ discussed here at the talkshop.

EOP

At the lower right of Figure 1 we find Earth orientation parameters and associated cycles. To understand how these link to planetary periods we need to look at the motions of Jupiter and Saturn in particular. Kepler gives us this useful graphic in his book De Stella Nova (1606).

Kepler’s trigon showing the ~60 year cycle in the longitude of the Jupiter-Saturn synodic conjunctions. This may be linked to the ~60yr cycle of Earth’s major oceans, giving rise to the 30 year global cooling scare 1960-90 and the subsequent global warming scare 1990-2020.

From an earlier post: ‘As successive great conjunctions occur nearly 120° apart, their appearances form a triangular pattern. In a series every fourth conjunction returns after some 59.8 years to the vicinity of the first. These returns are observed to be shifted by some 7–8°’. Wikipedia. [2019 version]. After 3 J-S the conjunctions have nearly described an exact triangle, but the start position has moved (precessed) slightly, by 60/7 degrees of precession of the J-S conjunction axis. It takes 42 of those (42*3 J-S) to complete the precession cycle in 2503 years. (41×61.051 y = 41×360 degrees movement of the axis).

The 413kyr eccentricity cycle is equivalent to 55*3 of these J-S synodic precession periods, and 6765 or 55×123 (Fibonacci and Lucas numbers) of the 61.051 360 degree periods. Additionally 413 kyr = 10 obliquity periods.

In the brown triangle: the 19 kyr and 23 kyr periods have a beat period of the 112kyr perihelion precession.
23 kyr is 10 Hallstatt cycles.

In the blue triangle: the 95 kyr (5×19 kyr) and 124 kyr (3 obliquities) have a beat period of 413 kyr i.e. Earth’s eccentricity cycle (mentioned in various research papers). Since our 95 kyr = 353×270 and our 124 kyr = 353×351, we find: (351×270) / (351-270) = 1170, and 1170*353 = 413010 years (the obliquity period).

Discussion

The 95 and 124kyr eccentricity cycles are linked with glacial periods. From Park and Maarsch (1993) paper ‘Plio—Pleistocene time evolution of the 100-kyr cycle in marine paleoclimate records’: “The DSDP 607 time scale is more favorable to an abrupt jump in amplitude for the 95-kyr δ18O envelope, but not in the 124-kyr envelope. Rather, long-period δ18O fluctuations appear phase-locked with the 124-kyr eccentricity cycle some 300-400 kyr prior to its growth in amplitude and phase-lock with the 95-kyr eccentricity cycle in the late Pleistocene.” Because the 124kyr period is 3x41kyr (obliquity period), this may help explain the change from glacial periods around 41kyr to around 100kyr.

The bi-modality of glacial cycles and the 95 and 124kyr cycles is one of the modes of variation mirrored between celestial cyclic motion and Earth climatic events. There are also many periods which are ‘quasi-cyclic’ and vary in length within bounds whose attractor nodes fit our phi-Fibonacci scheme. We are not claiming to have elucidated a deterministic and predictable system with our precise whole-number orbitally resonant ratios. We are offering this scheme as a potentially useful roadmap for further investigations into the intriguing numerical links between planetary orbits, synodic timings, planar inclinations, eccentricities, energy transfers and other celestial mechanical and orientation data.

As an example of how our scheme links shorter to longer term cycles, there are exactly 9 Jupiter Saturn conjunctions in the period of the Jose cycle of 178.8 years. There are 55x21x2 Jose cycles in the 413kyr eccentricity period. Experienced researchers like Paul Vaughan will immediately see that this product of multiple Fibonacci numbers resolves to the product of the first 6 prime numbers 1,2,3,5,7,11.

The solar system is organised by the forces of gravity and electro-magnetism into a log-normal distribution of which the Fibonacci series and Lucas series are examples which maintain the stability of the system. Resonance is minimised, but also utilised to transfer energy between orbits in order to resolve inequalities through resonance-forced changes to the eccentricity and inclination of orbits. These changes give rise to the cyclic changes in climatic factors on Earth observed at all timescales from the ~22yr Hale and ~60yr J-S trigon to the ~100kyr and 413kyr glaciation in core sample data and other indices.

Data sources and acknowledgements

Planetary data used is from NASA JPL which gives the Seidelmann values for orbital periods. Our thanks to Paul Vaughan for insisting on their use.

The periods we have calculated can all be reproduced using the ratios we have provided on Figure 1 and the NASA JPL values for the Jupiter, Saturn and Uranus orbital periods.

Comments
  1. Paul Vaughan says:

    “[…] long-periodic astronomical phenomena whose records we succeeded to find in the Dresden Codex – a synchrony of Venusian heliacal risings with solar eclipses (interval of about 132 years) […]” — 2013 dating of Mayan Calendar using long-periodic astronomical phenomena in Dresden Codex

    132 year review

  2. Paul Vaughan says:

    climate exploration nostalgia: “line-of-nodes and line-of-apsides coincide every six years,
    the points in the lunar orbit at which these coincidences occur are not necessarily aligned with the
    positions of the Full Moon and New Moon […] point of alignment slowly rotates with
    respect the New Moon/Full Moon line, only producing a grand alignment once every 66 years”

    (p.22)

  3. Paul Vaughan says:

    crude (can be refined) preliminary check on Horizons:

    2319.0467418165 = 1/(1/11.8618572921452-3/29.4567725360326+1/84.02044892998+1/164.753642595886)

    planning further diagnostics

  4. Paul Vaughan says:

    phase (in pi radians) and cosine

  5. Paul Vaughan says:

    CO[U-N?]T
    sea cure IT?
    in sol ace sh!one:
    Capt. Dove 8˚North
    with allure misst knew miracle myth OD

    No.SSTallGaia Vlad dove US talk:
    “just tag URL in the” (no doubt!!! laugh-T) “www!hiRled”

    rush in 60 seconds = 1 minute peace folk us putting_6 dovetail write side [“dull Mn“] up “like 36” for goal post-h***stat*

    =
    Horizons On-Line System News
    January 26, 2022
    — Fourteen Saturnian satellites were updated to the new SAT441L solution
    from R. Jacobson (JPL): 601-609, 612-614, 632, and 634, along with
    planet-center 699.
    =

    “T weed dull dumb there snow CO[]MPairus[]UN” mathematic ally know doubt reference frame sporadic(except sh!U-N)ally

    171.406220601552 = beat(164.791315640078,84.016845922161)
    16.9122914926352 = harmean(29.4474984673838,11.8626151546089)

    18.7636626447678 = beat(171.406220601552,16.9122914926352)
    338.432743555958 = slip(19.8650360864628,18.7636626447678)

    4622.9545388013 = slip(338.432743555958,9.93251804323141)
    2311.47726940065 = slip(338.432743555958,4.9662590216157) —— ll.07 = 166/2/07 ?

    178.006192198129 = beat(4622.9545388013,171.406220601552)

    4622.52465229533 = (e^√(19*2π) – ⌊√(e^√(19*2π))⌉^2) / (⌊√(e^√(19*π))⌉^2 – e^√(19π))
    2311.26232614767 = 4622.52465229533 / 2 ———————— no doubt

    178.000120709909 = beat(4622.52465229533,171.4) —————– B’S at turn
    49000.79211442 = beat(208.886643858907,207.999955892563)—-(in!)000MOC˚K˚CRown
    more ID?
    y’know 11over
    turn in “sure!”cue11aceU-N

  6. Paul Vaughan says:

    sew sea & iceburg in NO?
    SAM muck ably (in can’TA11key)

    11.071

    22.142
    44.284; 284 = s(220)

  7. Paul Vaughan says:

    “on average” ? …..know doubt:

  8. Paul Vaughan says:

    11.0693… = 33.208 / 3
    (0+11.86+11.86+11.86+11.86+11.86+11.86+11.86+11.86+11.86+11.86+11.86+11.86+11.86+11.86)/15

  9. Paul Vaughan says:

    Suggest Dove 0.0744 Puzzle Peace

    2545.40891524211 = √2*harmean(1470,5*(11.86-(19+43+67+163+28)*(0.07)^2/2)+191*11.86)
    1799.87590486039 = harmean(1470,5*(11.86-(19+43+67+163+28)*(0.07)^2/2)+191*11.86)
    2320.64 = 5*(11.86-(19+43+67+163+28)*(0.07)^2/2)+191*11.86
    11.84 = (5*(11.86-(19+43+67+163+28)*(0.07)^2/2)+191*11.86)/(5+191)
    s(1210) = 1184 — nos.
    s(1184) = 1210 — amicable
    11^2 = 121
    76 = 19 mod 24 + 43 mod 24 + 67 mod 24 + 163 mod 24

  10. Paul Vaughan says:

    Seidelmann (1992) mean elements from tropical (mislabelled sidereal) & synodic (Table 15.6).

    25407.7660910075 = beat(0.240846733026329,0.24084445)
    25729.7982184441 = beat(0.615197278962733,0.61518257)
    25684.2432466089 = beat(1.00001755422766,0.99997862) —- sidereal est. from harmean of others
    25760.4321570756 = beat(1.88084836649705,1.88071105)
    25691.0410103856 = beat(11.8619993833167,11.85652502)
    25754.8216760853 = beat(29.4571726091513,29.42351935)
    25699.0740448472 = beat(84.021212742844,83.74740682)
    25757.04141984 = beat(164.770559417647,163.7232045)
    25677.6970524456 = beat(250.439797262174,248.0208)

    Earth is left blank in the synodic column, but diagnostics suggest the EMB Keplerian sidereal period 1.00001743371442 (Table 5.8.1) was used to balance k ~= 25685. Treat 1.00001755422766 as a question inviting deep scrutiny.

  11. Paul Vaughan says:

    25721.8900031954 = 360*60*60/50.3851

    perusing consensus history

  12. Paul Vaughan says:

    before-adjustment value for comparison:
    25771.4533429313 = 360*60*60/50.2882

    from old reference frame pub.

    Earth-Venus
    241.155400410678 = 88082.01 / 365.25

    Jupiter-Saturn
    853.993867214237 = 311921.26 / 365.25
    336.051413467044 = beat(853.993867214237,241.155400410678)
    84.012853366761 = 336.051413467044 / 4

    854.011006160164 = 311927.52 / 365.25
    336.048759632077 = beat(854.011006160164,241.155400410678)
    84.0121899080192 = 336.048759632077 / 4
    84.0120465434634 = 1/U —– NASA “factsheet”

    never saw that one pointed out before

  13. Paul Vaughan says:

    compare-and-contrast (as a botanisst wood) supplementary

    k = “p_A” in those tables

    can’t just trust astronomy summaries — e.g. notation isn’t defined, models are mixed, etc.
    need to check
    found they mixed ~Williams J & S mean elements with ~Seidelmann Keplerian E & V

    19.8588772513307 = beat(29.4571389459274,11.8619822039699)
    60.9470469878813 = slip(29.4571389459274,11.8619822039699)
    883.192112166325 = slip(60.9470469878813,19.8588772513307)

    853.83263208956 = axial(25685,883.192112166325)
    853.873341278472 = axial(25721.8900031954,883.192112166325)
    853.92785846215 = axial(25771.4533429313,883.192112166325)

    0.995849084939309 = beat(0.615197263396975/8,1.00001743371442/12)
    241.147989898212 = beat(0.999978614647502,0.995849084939309)

    usefully suggestive but underscores:
    authors aren’t clear on (sometimes mixed) definitions & models
    sew: carefully diagnose, sort, classify shape-shifting morphology comparatively

  14. Paul Vaughan says:

    notation link — & a link from there:

    Park, Ryan S.; Folkner, William M.; Williams, James G.; Boggs, Dale H. (2021). “The JPL Planetary and Lunar Ephemerides DE440 and DE441“. The Astronomical Journal. 161 (3): 105.

  15. Paul Vaughan says:

    I used to think it was ridiculous to take measurements of model features to deduce mean model parameters. However, a lot of forensic diagnostic trouble is resulting from other authors’ summaries.

    I have a big study in mind — measuring with generalized wavelets in diverse reference frames to explore why mainstream authors are missing simple, key constraints when estimating central limits.

    Generalized wavelets — to be neither confused nor conflated with conventional wavelets — subsume other period estimation methods.

  16. Paul Vaughan says:

    some quick notes
    1/E sidereal
    365.256 days Williams — diagnostics are making it look “too rounded-off”
    365.256367664193 Seidelmann

    25763.987503107 = beat(1.00001743371442,0.99997862)
    compare: fraction-of-a-year difference at link vs. 50 year JPL k correction (LNC,LLR) above

    review: replace φ with 240 (E8) in Schneider’s classic equation:
    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)
    Let us find peace and tranquility.

  17. Paul Vaughan says:

    ~midpoint dates of “solar derailment” from “jupiter groove”
    2nd column = step size since last
    1BC moved to 0

    -9129.36111111111	
    -8950.44444444445	178.916666666666
    -7621.36111111111	1329.08333333333
    -7442.69444444444	178.666666666667
    -7263.86111111111	178.833333333333
    -7084.94444444444	178.916666666667
    -6945.02777777778	139.916666666667
    -6906.11111111111	38.9166666666661
    -6766.27777777778	139.833333333334
    -6587.44444444444	178.833333333333
    -6408.52777777778	178.916666666667
    -4681.36111111111	1727.16666666667
    -4502.44444444444	178.916666666667
    -4323.52777777778	178.916666666667
    -2994.36111111111	1329.16666666667
    -2815.69444444444	178.666666666667
    -2636.86111111111	178.833333333333
    -2457.94444444444	178.916666666667
    -2318.02777777778	139.916666666667
    -2278.94444444444	39.0833333333335
    -2139.27777777778	139.666666666667
    -1960.44444444444	178.833333333333
    -233.194444444444	1727.25
    -54.3611111111111	178.833333333333
    124.555555555556	178.916666666667
    303.472222222222	178.916666666667
    482.305555555556	178.833333333333
    1632.63888888889	1150.33333333333
    1811.30555555556	178.666666666667
    1990.22222222222	178.916666666667
    2169.22222222222	179
    2308.97222222222	139.75
    2487.80555555556	178.833333333333
    2666.63888888889	178.833333333333
    4393.88888888889	1727.25
    4572.72222222222	178.833333333334
    4751.63888888889	178.916666666666
    4930.55555555556	178.916666666667
    5109.38888888889	178.833333333333
    5249.30555555556	139.916666666667
    6259.80555555556	1010.5
    6438.55555555556	178.75
    6617.47222222222	178.916666666667
    6796.55555555556	179.083333333333
    6936.13888888889	139.583333333333
    7114.97222222222	178.833333333334
    9021.05555555555	1906.08333333333
    9199.88888888889	178.833333333334
    9378.72222222222	178.833333333334
    9557.63888888889	178.916666666666
    9697.55555555555	139.916666666666
    9876.38888888889	178.833333333334
    

    gap between 2 noteworthy pairs of events spaced ~39 years apart:
    4627.16666666667 = (-2278.94444444444) – (-6906.11111111111)

  18. Paul Vaughan says:

    ~1727.2 review

  19. Paul Vaughan says:

    a bit choppy with monthly output (horizons ICRF jovian barycenters & SSB)

  20. Paul Vaughan says:

    column:variable
    A:date
    B:x_J
    C:y_J
    D:x_sun
    E:y_sun

    F:
    leave F1 blank
    enter in cell F2
    =MOD(AVERAGE(MOD(ATAN2(B1,C1)/PI()-ATAN2(B2,C2)/PI()+1,2)-1,MOD(-ATAN2(D1,E1)/PI()+ATAN2(D2,E2)/PI()+1,2)-1)+1,2)-1
    copy/paste down column F

    G:integrate column F
    H:round off column G

    dates of discrete steps (of 1) in column H give summary above

  21. Paul Vaughan says:

    correction
    B:x_sun
    C:y_sun
    D:x_J
    E:y_J

  22. Paul Vaughan says:

    typo above:
    ” 4622.52465229533 = (e^√(19*2π) – ⌊√(e^√(19*2π))⌉^2) / (⌊√(e^√(19*π))⌉^2 – e^√(19π)) “

    should read:
    4622.52465229533 = (e^√(19*2)π – ⌊√(e^√(19*2)π)⌉^2) / (⌊√(e^√(19)π)⌉^2 – e^√(19)π)
    4622.52465229533=(exp(1)^((19*2)^(1/2)*pi())-round((exp(1)^((19*2)^(1/2)*pi()))^(1/2),0)^2)/(round((exp(1)^((19)^(1/2)*pi()))^(1/2),0)^2-exp(1)^((19)^(1/2)*pi()))

  23. Paul Vaughan says:

    Long Jump Track

    typo nostalgia no doubt peacefully writes too vague SST left know how?

    2311.26232614767 = 4622.52465229533 / 2
    2313.52777777778 = 4627.05555555556 / 2

    2360310.64278926 = beat(2313.52777777778,2311.26232614767)
    2360489.30437277 = 2/(1/(φ^22+1/11)^(e/11+1/22)-1/ln(163*67*43*19*11*7*3*2*1))
    405375.147994516 = 1/((1/(φ^22+1/11)^(e/11+1/22)-1/ln(163*67*43*19*11*7*3*2*1))/2+1/271/43/7/3/2)

    2313.52760614138 = beat(2360489.30437277,2311.26232614767)
    4627.05521228277 = 2313.52760614138 / 2

    -7621.36111111111	-0.0214855354269731
    -6945.02777777778	0.00937170630674525
    -2994.36111111111	0.118785930341576
    -2318.02777777778	0.150143934714772
    1632.63888888889	0.258513139461028
    2308.97222222222	0.289510321287933
    6259.80555555556	0.424222798309845
    6936.13888888889	0.458736536733029
    

    visually notice ~62k longitude drift cycle

    4627 = (-7621.36111111111) – (-2994.36111111111)
    4627 = (-6945.02777777778) – (-2318.02777777778)
    4627 = (-2994.36111111111) – (1632.63888888889)
    4627 = (-2318.02777777778) – (2308.97222222222)
    4627.16666666667 = (1632.63888888889) – (6259.80555555556)
    4627.16666666667 = (2308.97222222222) – (6936.13888888889)
    4627.05555555556 = average (with monthly resolution model sampling)

    2403.15249392055 = beat(62033.6579963371,2313.52760614138)
    2403.15253519605 = beat(62033.7538930916,2313.52777777778)

    5000.00255456061 = beat(62033.6579963371,4627.05521228277)
    5000.00233239882 = beat(62033.7538930916,4627.05555555556)

  24. Paul Vaughan says:

    ˚Ti.e.Gores tribe swan N/A f(11)ear US?

    refininin’ “math or myth?” crude mystery USing˚T(able) :

    65972.0774235673 = 4627/average(0.118785930341576,-(-0.0214855354269731))
    65737.3979559194 = 4627/average(0.150143934714772,-0.00937170630674525)
    66229.0477160309 = 4627/average(0.258513139461028,-0.118785930341576)
    66400.5161326476 = 4627/average(0.289510321287933,-0.150143934714772)
    55846.6742229939 = 4627.16666666667/average(0.424222798309845,-0.258513139461028)
    54686.1685052325 = 4627.16666666667/average(0.458736536733029,-0.289510321287933)
    62033.7538930916 = harmean

    less sun?
    off ice yell parameters
    Mask wear?
    as model snow doubt “amiMa[˚K]in’MI’$elf…C11ear?”

    If IT could IT wood “bury center” …sew let US find peace and tranquility instead ˚f(Be˚in˚) divided and conquereadbuyMSM PR˚script sh!…

  25. Paul Vaughan says:

    More orders available for central limit leveraging (beyond the 8 points that first caught attention).

    4626.91666666667 = (-4502.44444444444) – (-9129.36111111111)
    4626.91666666667 = (-4323.52777777778) – (-8950.44444444445)
    4627 = (-2994.36111111111) – (-7621.36111111111)
    4627 = (-2815.69444444444) – (-7442.69444444444)
    4627 = (-2636.86111111111) – (-7263.86111111111)
    4627 = (-2457.94444444444) – (-7084.94444444444)
    4627 = (-2318.02777777778) – (-6945.02777777778)
    4627.16666666667 = (-2278.94444444444) – (-6906.11111111111)
    4627 = (-2139.27777777778) – (-6766.27777777778)
    4627 = (-1960.44444444444) – (-6587.44444444444)
    4627 = (-54.3611111111111) – (-4681.36111111111)
    4627 = (124.555555555556) – (-4502.44444444444)
    4627 = (303.472222222222) – (-4323.52777777778)
    4627 = (1632.63888888889) – (-2994.36111111111)
    4627 = (1811.30555555556) – (-2815.69444444444)
    4627.08333333333 = (1990.22222222222) – (-2636.86111111111)
    4627.16666666667 = (2169.22222222222) – (-2457.94444444444)
    4627 = (2308.97222222222) – (-2318.02777777778)
    4627.08333333333 = (2487.80555555556) – (-2139.27777777778)
    4627.08333333333 = (2666.63888888889) – (-1960.44444444444)
    4627.08333333333 = (4393.88888888889) – (-233.194444444444)
    4627.08333333333 = (4572.72222222222) – (-54.3611111111111)
    4627.08333333333 = (4751.63888888889) – (124.555555555556)
    4627.08333333333 = (4930.55555555556) – (303.472222222222)
    4627.08333333333 = (5109.38888888889) – (482.305555555556)
    4627.16666666667 = (6259.80555555556) – (1632.63888888889)
    4627.25 = (6438.55555555556) – (1811.30555555556)
    4627.25 = (6617.47222222222) – (1990.22222222222)
    4627.33333333333 = (6796.55555555556) – (2169.22222222222)
    4627.16666666667 = (6936.13888888889) – (2308.97222222222)
    4627.16666666667 = (7114.97222222222) – (2487.80555555556)
    4627.16666666667 = (9021.05555555555) – (4393.88888888889)
    4627.16666666667 = (9199.88888888889) – (4572.72222222222)
    4627.08333333333 = (9378.72222222222) – (4751.63888888889)
    4627.08333333333 = (9557.63888888889) – (4930.55555555556)
    4627.08333333333 = (9876.38888888889) – (5249.30555555556)

    4627.0763870233 = harmean

    69319.8319294751 = 4626.91666666667 / (mod(average(-0.156033953247042,-(-0.289528698917344))+1,2)-1)
    72522.1226808762 = 4626.91666666667 / (mod(average(0.00957376131761016,-(-0.118026383179726))+1,2)-1)
    65972.0774235673 = 4627 / (mod(average(0.118785930341576,-(-0.0214855354269731))+1,2)-1)
    62642.0400575176 = 4627 / (mod(average(0.246762713723871,-(0.0990344470278467))+1,2)-1)
    58162.8085052871 = 4627 / (mod(average(0.409765504860617,-(0.25066039563676))+1,2)-1)
    57218.4124896072 = 4627 / (mod(average(0.590350933719956,-(0.428619777657546))+1,2)-1)
    65737.3979559193 = 4627 / (mod(average(0.150143934714772,-(0.00937170630674525))+1,2)-1)
    52288.3862005778 = 4627.16666666667 / (mod(average(0.7748405198333,-(0.597854117961309))+1,2)-1)
    62133.8886342867 = 4627 / (mod(average(0.298120241269768,-(0.149183804111166))+1,2)-1)
    58031.3186779579 = 4627 / (mod(average(0.46630071916349,-(0.30683510282355))+1,2)-1)
    64055.5936384889 = 4627 / (mod(average(-0.184800789851074,-(-0.329269047412262))+1,2)-1)
    66553.5604114132 = 4627 / (mod(average(-0.0169880488236772,-(-0.156033953247042))+1,2)-1)
    66164.790281135 = 4627 / (mod(average(0.149436669681404,-(0.00957376131761016))+1,2)-1)
    66229.0477160309 = 4627 / (mod(average(0.258513139461028,-(0.118785930341576))+1,2)-1)
    62379.7218720458 = 4627 / (mod(average(0.395112204909195,-(0.246762713723871))+1,2)-1)
    54893.8699843549 = 4627.08333333333 / (mod(average(0.57834838432064,-(0.409765504860617))+1,2)-1)
    54004.0629503972 = 4627.16666666667 / (mod(average(0.761714583559292,-(0.590350933719956))+1,2)-1)
    66400.5161326476 = 4627 / (mod(average(0.289510321287933,-(0.150143934714772))+1,2)-1)
    56620.6567661775 = 4627.08333333333 / (mod(average(0.461561769418814,-(0.298120241269768))+1,2)-1)
    56387.6155673988 = 4627.08333333333 / (mod(average(0.630417725593836,-(0.46630071916349))+1,2)-1)
    65548.762952841 = 4627.08333333333 / (mod(average(-0.207172348220246,-(-0.348352230948537))+1,2)-1)
    68686.7571285196 = 4627.08333333333 / (mod(average(-0.0500707916164622,-(-0.184800789851074))+1,2)-1)
    68877.3083573421 = 4627.08333333333 / (mod(average(0.117369214654959,-(-0.0169880488236772))+1,2)-1)
    64173.0306806299 = 4627.08333333333 / (mod(average(0.293643146584791,-(0.149436669681404))+1,2)-1)
    60459.3548952395 = 4627.08333333333 / (mod(average(0.465014848671486,-(0.311950584543929))+1,2)-1)
    55846.6742229939 = 4627.16666666667 / (mod(average(0.424222798309845,-(0.258513139461028))+1,2)-1)
    49572.7606882443 = 4627.25 / (mod(average(0.581797389908275,-(0.395112204909195))+1,2)-1)
    52466.6053025123 = 4627.25 / (mod(average(0.754736773594929,-(0.57834838432064))+1,2)-1)
    54816.663980246 = 4627.33333333333 / (mod(average(0.930544023271306,-(0.761714583559292))+1,2)-1)
    54686.1685052325 = 4627.16666666667 / (mod(average(0.458736536733029,-(0.289510321287933))+1,2)-1)
    54999.6938581224 = 4627.16666666667 / (mod(average(0.629823312059504,-(0.461561769418814))+1,2)-1)
    63490.6451187531 = 4627.16666666667 / (mod(average(-0.061413342054471,-(-0.207172348220246))+1,2)-1)
    64141.1102649451 = 4627.16666666667 / (mod(average(0.0942100494090025,-(-0.0500707916164622))+1,2)-1)
    66149.9993768388 = 4627.08333333333 / (mod(average(0.257265915393354,-(0.117369214654959))+1,2)-1)
    60895.1252476894 = 4627.08333333333 / (mod(average(0.445612070682173,-(0.293643146584791))+1,2)-1)
    69993.9063542172 = 4627.08333333333 / (mod(average(0.153726074736133,-(0.0215121842892383))+1,2)-1)
    

    60891.850424131 = harmean

    To refine beyond crude suggestions, try sampling the NASA JPL model more frequently than monthly near the critical dates (21st of months used here).

  26. Paul Vaughan says:

    refined crude summary

    2313.53819351165 = 4627.0763870233 / 2

    2404.91090073075 = beat(60891.850424131,2313.53819351165)
    2404.89946055601 = beat(60891.850424131,2313.52760614138)

    5007.59574851986 = beat(60891.850424131,4627.0763870233)
    5007.57094786665 = beat(60891.850424131,4627.05521228277)
    2503.79787425993 = 5007.59574851986 / 2
    2503.78547393333 = 5007.57094786665 / 2

    60891.8504241311 = beat(2503.79787425993,2404.91090073075)
    60891.850424131 = beat(2503.78547393333,2404.89946055601)

    Recognize this?
    8600.60617512082 = harmean(60891.850424131,4627.0763870233)
    8600.56959601988 = harmean(60891.850424131,4627.05521228277)

  27. Paul Vaughan says:

    “8600” …with “6441”

    Seidelmann 1992
    tropical
    19.8588720868409 = beat(29.42351935,11.85652502)
    61.0914225103732 = slip(29.42351935,11.85652502)
    800.898956784996 = slip(61.0914225103732,19.8588720868409)

    sidereal from synodic (NOT Keplerian)
    19.8589101021728 = beat(29.4571726091513,11.8619993833167)
    60.9472122984759 = slip(29.4571726091513,11.8619993833167)
    883.152947004205 = slip(60.9472122984759,19.8589101021728)

    8599.1727824054 = beat(883.152947004205,800.898956784996)

    substituting Williams’ tropical “means” :
    19.8588772513307 = beat(29.4571389459274,11.8619822039699)
    60.9470469878813 = slip(29.4571389459274,11.8619822039699)
    883.192112166325 = slip(60.9470469878813,19.8588772513307)

    8600.85489073177 = beat(883.152947004205,800.91354556689)

    with Williams sidereal & tropical:
    19.8588772513307 = beat(29.4571389459274,11.8619822039699)
    60.9470469878813 = slip(29.4571389459274,11.8619822039699)
    883.192112166325 = slip(60.9470469878813,19.8588772513307)

    8597.14206514499 = beat(883.192112166325,800.91354556689)
    and using that some might see where some other ideas came from, k?…
    6441.18426216303 = axial(25685,8597.14206514499) ……….no doubt!

    Note weather myth and math, together comparatively.

  28. Paul Vaughan says:

    Instead of savagely, relentlessly, & viciously harassing curious, innocent, nature-loving explorers a decade ago, “experts” could have simply said:

    mod(average(mod(atan2(B1,C1)/pi()-atan2(B2,C2)/pi()+1,2)-1,mod(-atan2(D1,E1)/pi()+atan2(D2,E2)/pi()+1,2)-1)+1,2)-1

  29. Paul Vaughan says:

    peace together clear aggregation criteria

    no matter the aggregation criteria, they can be stated mathematically
    same for relations between different aggregation criteria
    compare and contrast diversity of models
    sort and classify features comparatively (like botanist developing taxonomic key)

    discourse stalled decades by
    narrative spin agents opaquely mixing different models & aggregation criteria

    left with feasible impression for right team with sufficient resources:
    automation of precise ephemeride model intercomparison diagnostics (no doubt has 405 applications…)

  30. Paul Vaughan says:

    Seidelmann (1992) tropical with LLR-based k:

    11.8619928147296 = beat(25721.8900031954,11.85652502)
    29.4572157446799 = beat(25721.8900031954,29.42351935)

    19.8588720868409 = beat(29.4572157446799,11.8619928147296)
    60.9466695881755 = slip(29.4572157446799,11.8619928147296)
    883.419711511591 = slip(60.9466695881755,19.8588720868409)
    883.419711511591 = harmean(883.710029359836,883.129584352078)

    19.8588720868409 = beat(29.42351935,11.85652502)
    61.0914225103732 = slip(29.42351935,11.85652502)
    800.898956784996 = slip(61.0914225103732,19.8588720868409)

    441.709855755795 = 883.419711511591 / 2
    883.710029359836 = beat(883.129584352078,441.709855755795)

    2313.58289306322 = 883.710029359836 * φφ
    4627.16578612644 = 2313.58289306322 * 2

  31. Paul Vaughan says:

    crude mnemonic trick to assist recall (φ,2,42,43,71,84,100) :

    441.564792176039 = beat(8600,420) ———————— 8600 = 43*(378-178)
    883.129584352078 = 2 * 441.564792176039
    800.886917960089 = harmean(8600,420)
    8600 = beat(883.129584352078,800.886917960089)
    420 = axial(883.129584352078,800.886917960089)
    840 = harmean(883.129584352078,800.886917960089)
    1420000 = 2 * 71 * 100^2
    883.679164079454 = beat(1420000,883.129584352078)
    2313.50208671011 = 883.679164079454 * φφ
    4627.00417342021 = 2313.50208671011 * 2

  32. Paul Vaughan says:

    tricky thing about last 2 comments?
    mixed aggregation criteria (user ‘nevermind IT’ hazard so rightly-left best as side)

  33. Paul Vaughan says:

    Select Barycentric Detail

  34. Paul Vaughan says:

    4627.26 = 209*22.14
    4627.0962773767 = 209*φ(φφ)^e
    4627.09938493153 = 209*(φ^22+1/11)^(e/11+1/22)
    4627.18618248315 = 209*ln(163*67*43*19*11*7*3*2*1)

  35. Paul Vaughan says:

    according to Williams’ “factsheet” :
    4627.20054512577 = 209 * φφ / (J+S) = 209 * 22.1397155269176= 209 * φφ * 8.45661883002872
    4627.13699402865 = 209 / (3V-5E+2J) = 209 * 22.1394114546826

  36. Paul Vaughan says:

    64k answer no doubt “too reference frame”

    according to Williams’ “factsheet” :
    22.1394114546826 — jev-
    18.0042929108891 — sev-
    10.0369907682676 — uev-
    8.98466169603683 — nev-
    6.39987970183041 — jsev+

    A: 4627.20054512577 / each of these
    B: round-off each
    B/A for each
    harmean = 0.999978565343834 = 365.242170991835 days — compare with:
    tropical = 0.999978614647502 = 365.242189 / 365.25 (Meeus)

    Repeat the calculations with all the same values, except substitute “Seidelmann (1992) sidereal“:

    22.1369448465182
    18.0059244891283
    10.0374978119762
    8.98506798919295
    6.40008584740693

    0.999999991983752 = 365.249997072065 days = harmean of B/A list
    1 julian year = 365.25 days

    explore tie-in with longitude drift (blue dots graph) above:

    60876.0000002838 = beat(1.00001642710472,1)

    36163.6011172769 = harmean(60876.0000002838,25721.8900031954) —- k from LLR
    36127.1256108909 = harmean(60876.0000002838,25685) ——— k
    36133.4834429326 = beat(29.4474984673838,29.42351935) —– Seidelmann (1992) tropical
    18066.7417214663 = 36133.4834429326 / 2
    25691.4278186716 = beat(60876.0000002838,18066.7417214663) — within bounds & near k

    8600.66250323489 = harmean(60876.0000002838,4627.20054512577)
    800.944087641983 = axial(8600.66250323489,883.192112166325) —- compare with:
    800.91354556689 —- Seidelmann (1992) tropical

    interpretation:
    CLT retrieves k balance constraint from “mean” (not truly mean) “expert” aggregation

  37. Paul Vaughan says:

    typo: “0.999999991983752 = 365.249997072065 days = harmean of B/A list”
    it’s A/B for this case
    improved documentation on the NASA “factsheet” pages (including sources) could potentially eliminate some of the diagnostic & interpretive tedium — but even if it appears I’m inclining towards careful model exploration with generalized wavelet (&/or angle differintegrals) to work around ambiguities (absolutely shouldn’t be necessary given vast time & resource differential)

  38. Paul Vaughan says:

    flipping aggregation around the other way (i.e. B/A) :
    1.00000000369856 = 365.2500013509 days
    it would certainly be helpful to have the source for “365.256” (to clarify aggregation & truncation methods, rather than having to do tedious forensic diagnostics to work backwards to them)

  39. Paul Vaughan says:

    that’s with harmean, so equal with average
    no doubt such a boring (but necessary) analysis i’m falling asleep

  40. Paul Vaughan says:

    Mayan Credit No Doubt

    “Serpent” no. = 15009.1608487337

    25673.4038500704 = beat(36133.4834429326,15009.1608487337)
    20986.6183290126 = beat(25673.4038500704,11547.3140098413)
    60977.4364403431 = beat(25673.4038500704,18066.7417214663)
    (blue dot longitude drift)

    32000.1001653059 = beat(60977.4364403431,20986.6183290126)
    review links: _1_, _2_

    clear after a quick review compiling from different sources:
    20978.8759312685 = beat(25685,11547.3140098413)
    20954.32984611 = beat(25721.8900031954,11547.3140098413)
    20934.9357937308 = beat(1.00002638193018,0.999978614647502)
    20937.2820046388 = beat(1.00002638193018,0.99997862)
    20710.5713193904 = beat(1.00002638193018,0.999978097193703)

    60912.1198244773 = beat(25685,18066.7417214663)
    60705.6488226157 = beat(25721.8900031954,18066.7417214663)

    32000.1001653058 = beat(60912.1198244773,20978.8759312685)
    32000.1001653059 = beat(60705.6488226157,20954.32984611)

    J direction’s opposite (compare frequency coefficient sign pattern)
    to S,U,N in XEV relations where X=J,S,U,N (stuff we covered long, long ago)

    Refinement’s no doubt possible, but to first order that further clarifies ingredients needed to translate Seidelmann’s (1992) Keplerian framing to Williams’ “factsheets”, Horizons output, & Seidelmann’s (1992) “synodic”.

    supplementary
    1.00002623256874 = beat(20978.8759312685,0.999978565343834)
    1.00002628840921 = beat(20954.32984611,0.999978565343834)
    1.00002633262181 = beat(20934.9357937308,0.999978565343834)
    1.0000263272688 = beat(20937.2820046388,0.999978565343834)
    1.00002685012553 = beat(20710.5713193904,0.999978565343834)
    1.00002638193018 —- anomalistic year period (from 1 source)

    answer to 64k question:
    Mayan wheel reinvention no doubt

  41. Paul Vaughan says:

    Hale
    Stat
    Ice Sun how http://www..

    Recall 2545 ≠ 2318 ~= 2320 motivated this tedious Halstatt diagnostic side-trail.

    Side-benefit of the exercise: clearly noted 4627 events distinct from 2318 ~= 2320 orbitally invariant “means”. (still need superior “means” to diagnose around 2318 ~= 2320)

    From different aggregation criteria arise summaries with different mathematical properties. It is necessary to develop understanding of the distinctions to make sense of conflicting quantities countless commentators have mixed into the discourse for decades — mixing different models and different reference frames into the discourse weather ignorantly, deceptively, and/or otherwise, with apparent inattention to nuanced interpretive distinctions.

    The NASA rep to whom I wrote (with all necessary details) apparently failed to appreciate (or showed no indication of recognizing) the 64k question (rightly left veiled).

    “[…] claimed U.S. President Dwight D. Eisenhower himself did not want to be disturbed while the show was on […]”

    My suggestion is educational unity (weather myth or math) with developing hindsight as follows:
    Surrender exclusive framing to promote democratic inclusivity. Certainly NASA (or some other well-funded agency) has the “means” to support more efficient (& surely harassment-free) discourse.

    white-collar Discourse Crystallization framing above ^
    –hybrid–
    hears the blue-CO[II]air Fram U-N v

    folk cuss no doubt Diss Curse
    C R US Ta11 ice-ace sh!own
    22 need all timin’docs MET
    trick “believe” IT
    or[well k]not observes
    209 moment” of sc11U-N*zz…
    (oversatiatation no doubt; video CHALKED FULL of carefully time-indexed mnemonic orbital#theory)

    White AND blue collar folk US invariant peace (2320~=2318) * 2 weather left and write 4627.

  42. Paul Vaughan says:

    As noted above:

    25721.8900031954 = 360*60*60/50.3851

    before-adjustment value for comparison:
    25771.4533429313 = 360*60*60/50.2882 ———————————————————— NOTICE

    diagnosing (a) Seidelmann Keplerian vs. (b) Williams “factsheet” “means”

    222314.53951386 = beat(11.8626151546089,11.8619822039699)
    25771.8797028462 = beat(222314.53951386,23094.6280196825) ————————– J
    20921.2705663812 = axial(222314.53951386,23094.6280196825)
    89978.8376726197 = beat(29.4571389459274,29.4474984673838)
    60381.2322858743 = beat(89978.8376726197,36133.4834429326)
    25780.5804579468 = axial(89978.8376726197,36133.4834429326) ———————— S
    25776.2293461625 = harmean(25780.5804579468,25771.8797028462) ———————– J&S

    thus deduce conversions from (a) to (b) …

    222346.270902445 = beat(25771.4533429313,23094.6280196825)
    11.8619822942945 = axial(222346.270902445,11.8626151546089)
    11.8619822039699

    89867.7550325788 = beat(36133.4834429326,25771.4533429313)
    29.4571508661187 = beat(89867.7550325788,29.4474984673838)
    29.4571389459274

    … and conversions vice versa :

    11.8626150642747 = beat(222346.270902445,11.8619822039699)
    11.8626151546089

    29.4474865550032 = axial(89867.7550325788,29.4571389459274)
    29.4474984673838

  43. Paul Vaughan says:

    ??sh!in!Clue sieve SOSlieIT?

    Next compare Seidelmann Keplerian with Seidelmann “synodic”.

    228517.20342556 = beat(11.8626151546089,11.8619993833167)
    25691.0410103856 = beat(228517.20342556,23094.6280196825)
    20974.8475614111 = axial(228517.20342556,23094.6280196825)
    89665.8397310115 = beat(29.4571726091513,29.4474984673838)
    60523.006270857 = beat(89665.8397310115,36133.4834429326)
    25754.8216760853 = axial(89665.8397310115,36133.4834429326)
    25722.8918067878 = harmean(25754.8216760853,25691.0410103856)

    Recognize ~LNC-adjusted k (~25722) based on LLR.

    Note well that this is becoming a true peace of work with no doubt.

  44. Paul Vaughan says:

    ˚T UN in’ IT in with queen11y nos.

    Seidelmann’92 tropical & Keplerian sidereal

    26114.2236547808 = beat(84.016845922161,83.74740682)
    25259.6956047041 = beat(164.791315640078,163.7232045)
    771931.758604905 = beat(26114.2236547808,25259.6956047041)
    25679.8527338808 = harmean(26114.2236547808,25259.6956047041)

    23094.6280196825 = beat(11.8626151546089,11.85652502)
    36133.4834429326 = beat(29.4474984673838,29.42351935)
    64000.2003306117 = beat(36133.4834429326,23094.6280196825)

    59100.2493255391 = axial(771931.758604905,64000.2003306117)
    59100 = 64k – 70^2
    70 = √4900 = √(64k – 59100)

    64000.2 = 64000 + 1/5
    59100.25 = (64000 + 1/5) – (70^2 – 1/20) = 64000 – 70^2 + 1/4

    Why sing? Peace inClue sov. 11y.

  45. Paul Vaughan says:

    Parameter set mismatches may seem puzzling initially, but careful diagnostics clarify relations.
    Some examples have been noted above.
    Seidelmann’s tropical relates to Keplerian & synodic sidereal as follows:

    11.85652502
    11.8565248071438 = 1/(+0.5/64000.2+1/25721.8900031954+0.5/11.8626151546089-0.5/29.4474984673838+0.5/11.8619993833167+0.5/29.4571726091513)

    29.42351935
    29.4235180391966 = 1/(-0.5/64000.2+1/25721.8900031954-0.5/11.8626151546089+0.5/29.4474984673838+0.5/11.8619993833167+0.5/29.4571726091513)

    83.74740682
    83.7473417981136 = 1/(+0.5/64000.2-0.5/59100.25+1/25721.8900031954+0.5/84.016845922161-0.5/164.791315640078+0.5/84.021212742844+0.5/164.770559417647)

    163.7232045
    163.722956001282 = 1/(-0.5/64000.2+0.5/59100.25+1/25721.8900031954-0.5/84.016845922161+0.5/164.791315640078+0.5/84.021212742844+0.5/164.770559417647)

  46. Paul Vaughan says:

    thought this was already noted:

    25699.0740448472 = beat(84.021212742844,83.74740682)
    25757.04141984 = beat(164.770559417647,163.7232045)
    25728.0250810588 = harmean(25757.04141984,25699.0740448472)

    25771 vs. 25721 sorts by context

  47. Paul Vaughan says:

    translating (b) to (a) gives some noteworthy tweaks:
    835.499521684417 = slip(61.0465822533736,19.8650412541591)
    2938.33144181534 = slip(504.264086710815,131.717210113846)
    2938.33144181488 = slip(356.573966040843,50.0711015696908)
    1469.16572090767 = slip(504.264086710815,65.8586050569228)
    979.443813938687 = slip(304.366068147897,131.717210113846)
    supplementary:
    divisors of 1671: 1, 3, 557, 1671
    1671 = 2 * 835.5

  48. Paul Vaughan says:

    EUrope
    LoSST
    ..and found:

    2545=ΣΣδ(220)’samicably “heavy ‘pettin'” Ha!stat˚T˚˚missh__˚˚

    “The
    SSTorrery
    of US:
    ITa11waySSTarts the same” – lg

  49. Paul Vaughan says:

    Found
    Peace

    2 equations
    ..and too UN owns

    CO[www]buoy surely roads weather
    ferry yore? we11bridge

    11.8626151546089
    11.8626147284818 = 1/(-1/64000.2+2/25721.8900031954+1/29.4474984673838-2/29.42351935+1/11.8619993833167+1/29.4571726091513)

    29.4474984673838
    29.4474958413624 = 1/(+1/64000.2+2/25721.8900031954+1/11.8626151546089-2/11.85652502+1/11.8619993833167+1/29.4571726091513)

    axial frame
    sh! own long ago
    0.000000000000 = % error

    ???0LA!SSTurgeUN0we[]D-out

  50. Paul Vaughan says:

    11.8626131311487 = 1/(-1/64000.2+2/25771.4533429313+1/29.4474984673838-2/29.42351935+1/11.8619822039699+1/29.4571389459274)

    11.8626135250709 = 1/(-1/64000.2+2/25771.4533429313+1/29.4474984673838-2/29.4235181382615+1/11.8619822039699+1/29.4571389459274)

    11.8626151546089

    29.447485998276 = 1/(+1/64000.2+2/25771.4533429313+1/11.8626151546089-2/11.85652502+1/11.8619822039699+1/29.4571389459274)

    29.4474984673838

    29.4475117889182 = 1/(+1/64000.2+2/25771.4533429313+1/11.8626151546089-2/11.8565229295003+1/11.8619822039699+1/29.4571389459274)

  51. Paul Vaughan says:

    Reference Frame Know Doubt

    Williams “factsheet” tropical & sidereal:
    25762.0064305964 = beat(11.8619822039699,11.8565229295003)
    25779.6502299353 = beat(29.4571389459274,29.4235181382615)
    25770.825310343 = harmean(25779.6502299353,25762.0064305964)

    Pay attention to 25721 vs. 25771.

  52. Paul Vaughan says:

    Terminology Know Doubt

    Williams “factsheet” tropical & synodic

    21867.405834306 = beat(11.8629550321199,11.8565229295003)
    23741.9817459875 = beat(29.4600280504908,29.4235181382615)
    22766.170636964 = harmean(23741.9817459875,21867.405834306) ~= 23k

    37012.3865742555 = beat(83.9373297002712,83.7474058863792)
    34730.9484755915 = beat(164.498657117277,163.723203285421)
    35835.3926584539 = harmean(37012.3865742555,34730.9484755915) ~= 36k

    62424.1184948485 = beat(35835.3926584539,22766.170636964)

    a variation on 64k but manifest from different combinations

  53. Paul Vaughan says:

    Seidelmann tropical & synodic presents interesting puzzle
    coarse balance is clear near ~2722 (LNC,LLR adjusted k)
    but clustered imbalance provokes deeper aggregation scrutiny

    J,U
    25691.0410103856 = beat(11.8619993833167,11.85652502)
    25695.0568997754 = harmean(25699.0740448472,25691.0410103856)
    25699.0740448472 = beat(84.021212742844,83.74740682)
    —–

    J,S,U,N
    25722.8918067878 = harmean(25754.8216760853,25691.0410103856)
    25723.9988806482 = harmean(25757.04141984,25691.0410103856)
    25725.4581878491 = harmean(25755.9315001361,25695.0568997754)
    25726.9176606306 = harmean(25754.8216760853,25699.0740448472)
    25728.0250810588 = harmean(25757.04141984,25699.0740448472)

    —–
    S,N
    25754.8216760853 = beat(29.4571726091513,29.42351935)
    25755.9315001361 = harmean(25757.04141984,25754.8216760853)
    25757.04141984 = beat(164.770559417647,163.7232045)

  54. Paul Vaughan says:

    inclination node out:
    18978.2646244704 = beat(11.863936923446,11.85652502)
    25728.2492709741 = beat(72337.575351641,18978.2646244704)
    25725.2841498062 = beat(72361.0252351259,18978.2646244704)

  55. Paul Vaughan says:

    “In fact, the very definitions of these planes are problematic for high-precision work.”

    no doubt

  56. Paul Vaughan says:

    2319.71393669362 = 1/(1/11.861990807677-3/29.4571309198874+1/84.01495797691+1/164.786005834669)

    table 2a

  57. Paul Vaughan says:

    2534.00777150548 = 1/(1/11.86260678-3/29.4480064+1/84.0175262+1/164.7903053)

    table 1 at same link
    I’m sure some can see what’s going on with the shorter model-duration biases (e.g. Seidelmann 1992 Keplerian model was fit to just 66 years-worth of observations)

    multidisciplinary teams could be kept busy for years exploring systematic sampling & aggregation biases

    they don’t leverage CLT (central limit theorem) — very curious

    some of the summaries suggest an “intermediate” (deformed) coordinate system (based on number theory) is used in some of the “more special” models (raises some of the most interesting curiosities about knew miracle methods “weather trade” secret or ignorance)

    one of the things I started doing was putting together all possible combinations of crosses of different model parameter-sets to sort and categorize reference-frame & definition ambiguities, offsets, & biases — rapidly clarifies the main structures …even if the detective knows nothing about what’s being modelled — quick avenue to comparative sense

    I’m sure I could develop really simple methods to precisely sharpen “mean” elements and organize a vastly more intuitive model structure that would facilitate use and interpretive ease for a multidisciplinary audience …but I almost certainly won’t do it without stable, secure, long-term funding and a diverse, talented, subordinate team (meaning no internal politics).

    problem: I’ve identified far too many avenues needing exploration right when my time & focus for this has been cut by an order of magnitude; probably select a small, special subset for now (picking stuff no one else would, no doubt…)

  58. Paul Vaughan says:

    “expert” knew miracle myth OD

    pop quiz (taken silently withOUTtakin’ D-bait) :
    what organization’s sampling (& 1911-1977 training) interval found QUITE OBVIOUS bias (without simply saying so UPFRONT in 2008)????

  59. Paul Vaughan says:

    Williams “factsheet” sidereal
    2317.51251648274 = 1/(1/11.8619822039699-3/29.4571389459274+1/84.0120465434634+1/164.788501026694)

    Seidelmann synodic
    2320.97373292688 = 1/(1/11.8619993833167-3/29.4571726091513+1/84.021212742844+1/164.770559417647)

    Seidelmann tropical (mislabelled as sidereal in pub.)
    2320.01916295313 = 1/(1/11.85652502-3/29.42351935+1/83.74740682+1/163.7232045)

    Williams “factsheet” tropical
    2319.96076275948 = 1/(1/11.8565229295003-3/29.4235181382615+1/83.7474058863792+1/163.723203285421)

    Standish sidereal Keplerian 3000BC-3000AD (unadjusted)
    2319.71393669197 = 1/(1/11.861990807677-3/29.4571309198874+1/84.01495797691+1/164.786005834669)

    orbital invariance in (well overdue) hindsight:
    initial “expert” guidance? no. k
    (too) costly delays without especially deep, early focus on reference frames

    USinilliberal climbIT
    Diss Cursin’ list of crucial “expert failed guide dense” :

    This one at least makes the top 10 – probably the top 5.

  60. Paul Vaughan says:

    html-typo (hidden) in last comment:
    bold-off html-tag after “IT” should be link-off i.e. /a not /b


    further diagnostics forthcoming (at a very slow pace)

  61. Paul Vaughan says:

    high frequency stuff here relates to “long.peri. […] deg/Cy” Table 2a = Table 8.10.3

    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)

    936.955612197409 = 1/(-2/11.8627021700857+5/29.4701958106261)

  62. Paul Vaughan says:

    Conclusion

    The parameter bias in the short-duration models is systematic and diagnosable with (a) traverse-closing methods from a basic surveying course and (b) central limit theorem (intro stats course).

  63. Paul Vaughan says:

    I’ve also identified the source of asymmetry here. I’m honestly starting to wonder if all this stuff was planted purposefully.

  64. Paul Vaughan says:

    interpret carefully:

    835.26379569011 = beat(936.955612197409,441.596056083158)
    835.028692086849 = beat(936.955612197409,441.530332692567)

    883.350168557103 = harmean(936.955612197409,835.546575435631)

    936.600040514322 = beat(835.546575435631,441.596056083158)
    936.304439796406 = beat(835.546575435631,441.530332692567)

    441.5 = 883 / 2

  65. Paul Vaughan says:

    25731.3307999186 = beat(11.861990807677,11.85652502)
    25721.4885660692 = beat(11.861990807677,11.8565229295003)

    25786.7295255651 = beat(29.4571309198874,29.42351935)
    25785.7988537611 = beat(29.4571309198874,29.4235181382615)

    25759.0003768883 = harmean(25786.7295255651,25731.3307999186)
    25753.6035620709 = harmean(25785.7988537611,25721.4885660692)

    4267.83999767789 = slip(164.786005834669,84.01495797691)
    19.8589050137632 = beat(29.4571309198874,11.861990807677)
    60.9473428358019 = slip(29.4571309198874,11.861990807677)
    883.060665385134 = slip(60.9473428358019,19.8589050137632)
    25557.2802223533 = slip(4267.83999767789,883.060665385134) ————————————-
    36003.3715298079 = harmean(60891.850424131,25557.2802223533)
    25688.6057708122 = beat(60891.850424131,18066.7417214663)
    36129.9162647266 = harmean(60891.850424131,25685)
    60912.1198244773 = beat(25685,18066.7417214663)
    1806.17637816608 = slip(883.060665385133,60.9473428358019)

    36133.4834429326 = beat(29.4474984673838,29.42351935)
    36131.65611118 = beat(29.4474984673838,29.4235181382615)


    compare with
    4278.72826993103 = slip(164.788501026694,84.0120465434634)
    1798.25390532451 = slip(883.192112166317,60.9470469878813)

    23126.4013526327 = beat(11.8626067830889,11.85652502)
    23118.4507121144 = beat(11.8626067830889,11.8565229295003)

    35384.5860615844 = beat(29.4480063959839,29.42351935)
    35382.8336890746 = beat(29.4480063959839,29.4235181382615)

    66756.8778237927 = beat(35384.5860615844,23126.4013526327)
    66696.8976941191 = beat(35382.8336890746,23118.4507121144)

    26048.6678680319 = beat(84.0175261973943,83.74740682)
    26048.5775452477 = beat(84.0175261973943,83.7474058863792)

    25283.456332553 = beat(164.790305314929,163.7232045)
    25283.4273672742 = beat(164.790305314929,163.723203285421)

    860677.506841794 = beat(26048.6678680319,25283.456332553)
    860742.554004726 = beat(26048.5775452477,25283.4273672742)

    72370.1350864148 = beat(860677.506841794,66756.8778237927)
    72299.19036667 = beat(860742.554004726,66696.8976941191)
    72334.6453311711 = harmean(72370.1350864148,72299.19036667)
    compare
    72337.575351641
    72337.575351641
    72361.0252351259

  66. Paul Vaughan says:

    26k ~= slip(4270,883)

    pentagonal pyramidal numbers:
    936, 1470, 1800, 4200

  67. Paul Vaughan says:

    178.942598954722 = ΦΦ * 936.955612197409 / 2

  68. Paul Vaughan says:

    468.300020257161 = 936.600040514322 / 2
    468.152219898203 = 936.304439796406 / 2
    600.203243993438 = harmean(835.546575435631,468.300020257161)
    600.081837182902 = harmean(835.546575435631,468.152219898203)

    600.276235692238 = harmean(936.955612197409,441.596056083158)
    600.215511380798 = harmean(936.955612197409,441.530332692567)

    pentagonal pyramidal number:
    7200

  69. Paul Vaughan says:

    lunisolar review link

  70. Paul Vaughan says:

    0.0748026830551271 = axial(1.00002638193018,0.0808503463381246)

    18.5964370693548 = beat(0.0748026830551271,0.0745030006844627)
    8.85109350901809 = beat(0.0754402464065708,0.0748026830551271)

    184.063510192393 = slip(18.5964370693548,8.85109350901809)
    600.241396282931 = slip(184.063510192393,5.99685290323073)

    compare

  71. Paul Vaughan says:

    22.1391843145908 = 1/(3/0.615197265341166-5/1.0000174322536+2/11.8626067830889)
    22.123085762185 = 1/(3/0.615197293550571-5/1.00002641247066+2/11.8634375929488)
    30424.2928774432 = beat(22.1391843145908,22.123085762185)
    304242.928774432 = 2 * 5 * 30424.2928774432
    173801.88427185 = axial(405378.494928687,304242.928774432)
    173801.269042893 = axial(405375.147994516,304242.928774432)
    compare
    173913.043478261
    173889.708842077
    173804.240903943
    close enough to suspect 5:1 DH resonance

  72. Paul Vaughan says:

    note well that perihelion rates in Standish’s long model match systematically-biased mean longitude rates in Seidelmann’s short model (another way to understand 64k)

    the “expert guidance” we had in “climate discourse” more than a decade ago was grossly and reprehensibly insufficient

  73. Paul Vaughan says:

    936 = beat(883,600)/2

  74. Paul Vaughan says:

    Decisions made by officials hundreds of years obstruct efficient climate exploration in the present.

    A few talkshop readers may recall that years ago I wrote to NASA about a discrete discontinuity in Horizons output. It was coincident with a major historical calendar switch. The problem was readily acknowledged by an official.

    I’m now finding a serious discontinuity obstructing refined measurement of mean elements from fresh Horizons output (to be neither confused nor conflated with Horizons output from many years ago). Upon first glance it looks related to the same calendar issue. Tedious diagnostics will be necessary.

    It is very disappointing to discover that it’s necessary to do such time-consuming, careful checks on such a well-resourced organization.

    I’m refining a clearer sense of the issues obstructing Halstatt communication. I’m increasingly aware of how the specialists could better serve the broader climate discussion community to help remove sources of persistent ambiguity, thus facilitating efficient, independent learning and understanding.

  75. Paul Vaughan says:

    typo above — should read:
    “Decisions made by officials hundreds of years ago obstruct efficient climate exploration in the present.” (church then and government now carrying forward mess created then)

    ridiculously anomalous longitude rates (& a step change) at this date:
    1582.805556 (“the moon is glued to a picture of heaven and…” -SG)

    The step changes critically interfere with Halstatt estimation (how I ended up on this totally-unwanted tangent).

    old comment from way back:
    “I was ready to expand my investigation to include Mercury, but I hit a glitch in the NASA Horizons online software. I sent [multiple] requests to the system that tripped it up, causing staff to discover a bug in the program around the time where they have to switch from one calendar system to another”

    =
    Nov 07, 2009:
    — Version 3.35a
    Fixed a bug that caused no output when using calendrical output
    stepping to step by month into a non-existent range of date labels
    (i.e., the Gregorian calendar switch-over point in October 1582).
    =

    This is where it would be helpful to have a budget and a diverse, multidisciplinary team of technicians. Fussing with a neverending stream of tedious technical hurdles is an unacceptable waste of an explorer’s time and resources.

    An option to have Horizons output everything in a Julian frame (with no weird gaps & steps) would (quite obviously & simply) be helpful.

  76. Paul Vaughan says:

    All of the mean longitude rates after 1582 are contaminated. That’s 42% of the record. This is totally ridiculous. Correcting for the contamination appears simple in central limit but not otherwise (more tedious diagnostics needed for event-level precision). I may stop when I’m satisfied with central limit estimates (and let these well-resourced organizations spend their own time & resources on the event-level clean-up post-1582).

  77. Paul Vaughan says:

    figured out: superior method for estimating mean elements

    s(4370) = 4270

    report on Halstatt?

    936.17955717643 = 3 * φ * s(4370) / 22.14

    …soon

  78. Paul Vaughan says:

    Past 378

    Perihelion rates systematically bias sidereal longitude rate estimates based on short sampling intervals.

    There’s a simple way to adjust for this, but be well-aware that it isn’t noted on the webpages climate explorers might use as sources upon which to found their explorations.

    Bias structure from Table 2a:

    11.8627021700857 = 360*100/(3034.90371757-0.18199196)
    29.4701958106261 = 360*100/(1222.11494724-0.54179478)
    84.0331316671926 = 360*100/(428.49512595-0.09266985)
    164.793624044745 = 360*100/(218.46515314-0.01009938)

    936.955612197409 = 1/(-2/11.8627021700857+5/29.4701958106261)
    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)

    string of related notes (mnemonically interspersed within clue$ave links)
    (March 8, 2022 at 9:28 pm to March 9, 2022 at 10:23 pm)

    be aware of a few typos above
    (obvious to anyone reviewing carefully enough, even where not pointed out explicitly)

  79. Paul Vaughan says:

    Top 4˚C

    brief:
    1st marvelled rightly
    at˚Knew miracle methods

    ace:
    parsimonious statistical model
    with systematic physical bias

    wonderous miss 51 lb:
    why wood ~1934 decoys
    be left buy commerce?

    “now i no. why” – lb

  80. Paul Vaughan says:

    Finding Peace

    Paddling inlet.
    Between mountains.
    Left steep cliff. Headed right towards island pair.
    Lightning track approached (not forecast). Wide spinning cloud descended.
    Trees swayed violently. Some snapped. Hat, hood suddenly thrown with 2 circling eagles edging only as necessary — and always gracefully — on intense wind bursts.
    Anticlimatically, sea-chop damped within 15 min.

    (1929.22222222222)
    94874.0989279822 = slip(15403.6217843836,4870.92829844298)

    (1930.13888888889)
    19.8588778386491 = beat(29.4571541785584,11.8619848835796)
    60.9469873120748 = slip(29.4571541785584,11.8619848835796)
    883.230870074906 = slip(60.9469873120748,19.8588778386491)
    2320.22316001908 = 1/(1/11.8619848835796-3/29.4571541785584+1/84.0206328418372+1/164.770073333031)
    15403.4824158992 = slip(936.955612197409,883.230870074906)
    4870.91188428463 = slip(2320.22316001908,936.955612197409)
    94883.7340415051 = slip(15403.4824158992,4870.91188428463)

    94885.6349884336 = beat(304407.424910486,72337.575351641) ~= 95ka
    94885.8433982369 = beat(304405.279928371,72337.575351641) ~= 95ka

    (1933.47222222222)
    94918.7878447315 = slip(15402.9756291991,4870.85219470599)

    94926.7650262257 = beat(304399.417131486,72361.0252351259) ~= 95ka

    ((1934.47222222222))
    19.8588781067985 = beat(29.4571537687132,11.8619849127921)
    60.9469915921757 = slip(29.4571537687132,11.8619849127921)
    883.228703878724 = slip(60.9469915921757,19.8588781067985)
    2320.23196371032 = 1/(1/11.8619849127921-3/29.4571537687132+1/84.0206333236655+1/164.77007177171)
    15402.8235915673 = slip(936.955612197409,883.228703878724)
    4870.83428686792 = slip(2320.23196371032,936.955612197409)
    94929.3094879479 = slip(15402.8235915673,4870.83428686792)

    Thank God 4 gracious guidance.

  81. Paul Vaughan says:

    sidereal orbital invariance with anomalistic bias

    (1930.13888888889)
    4932.39309395826 = slip(2362.15308834917,936.955612197409)
    9658.89570899433 = slip(4270.51884168654,936.955612197409)
    231392.295261542 = slip(9658.89570899433,4932.39309395826)
    compare with g_1 & s_1 (mercury)

    (1934.47222222222)
    4932.23055879401 = slip(2362.14066213527,936.955612197409)
    231750.569835749 = slip(9658.89570899433,4932.23055879401)

    supplementary:
    7255.79048704034 = slip(2362.15308834917,883.230870074906)
    7255.58494846013 = slip(2362.14066213527,883.228703878724)

  82. Paul Vaughan says:

    Linguistic Functional Numeracy (aka dumb muck .4 blue COIIair ‘less Sun’)

    11 = 55 / 5 ———- nasa horizons measure mint
    55 = 28^2 – 27^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2

    55 = 2*harmean(2320,2362) – 4627
    5 = 2*harmean(2320,2362) / 936

    crude state mean˚T:
    42 = 2362 – 2320 ~= harmean(u,n/2)
    21 = average(-2320,2362)
    1930˚Figure$D-pressUN$script
    editor[We11]inkin’to 836 (lowest un˚touchable weird no.)
    with˚C air fully-refined folk us ‘ape beat’ note$131k:
    2320.2, 2362; 41.8, 20.9
    check date $0 EU Don˚T .4 get_mnemonically_5,4,3,2,1 LB gravity (where 51 is Shorthand for S’un’)
    2320.4, 2362; 41.6, 20.8
    hedgeUNno.ME?go bull?USmmm? if $0: know doubt

  83. Paul Vaughan says:

    muck clock clue in side real possess un? 95ka dates refined:

    ((1930.30555555556))
    94885.4859731875 = 1/(4/11.8619848847033-9/29.4571541627973-2/84.0206328603223-2/164.770073273138-8/11.8627021700857+20/29.4701958106261)
    94885.6349884336 = beat(304407.424910486,72337.575351641)
    94885.8433982369 = beat(304405.279928371,72337.575351641)

    ((1934.22222222222))
    94926.6787401574 = 1/(4/11.8619849111069-9/29.4571537923607-2/84.0206332957994-2/164.770071862025-8/11.8627021700857+20/29.4701958106261)
    94926.7650262257 = beat(304399.417131486,72361.0252351259)

    11.8627021700857 & 29.4701958106261 are anomalistic periods from Standish (1992) Table 2a
    (anomalistic tell-tale clue: 936 js & 4270 jsun are stable & clear across entire nasa horizons model range)

    $era-wise the grave IT model tuned 2*ear11y˚30s date$sew the mystery turned up hears

    plea$11et?us˚Find˚Fin’n’cia11peace0vvheat with no doubt
    ((˚Fin’ridiculous “leadership” from ALL the big countries))

  84. Paul Vaughan says:

    refined dates most closely matching mercury (at monthly resolution) :

    (1925.55555555556)
    231016.042780749 = s_1 (La2011 Table 6 La2010a)
    231014.076532225 = 1/(6/11.8619848526737-15/29.4571546119796-9/84.0206323349967+18/164.770074974996+7/11.8627021700857-20/29.4701958106261+5/84.0331316671926+5/164.793624044745)

    (1935.55555555556)
    231842.576028623 = g_1 (La2011 Table 6 La2004a & La2011 Table 6 La2010a)
    231842.576028623 = s_1 (La2011 Table 6 La2004a)
    231840.243340341 = 1/(6/11.8619849200943-15/29.4571536662419-9/84.0206334445196+18/164.770071380022+7/11.8627021700857-20/29.4701958106261+5/84.0331316671926+5/164.793624044745)

    (1940.88888888889)
    232283.957100594 = g_1 (La2011 Table 5)
    232282.319418881 = 1/(6/11.861984956037-15/29.4571531617084-9/84.0206340418668+18/164.770069443753+7/11.8627021700857-20/29.4701958106261+5/84.0331316671926+5/164.793624044745)

    The measurements are simply a (gaussian) function of the nasa horizons model parameterization.

    The mystery that arose:
    Why are 95ka, 131ka, & ~231ka all tuned to ~early 1930s (tightly-clustered despite 20ka span)?

    not necessarily trying to answer the question
    but reporting 1 more might˚T˚Cur(i0)us IT

  85. Paul Vaughan says:

    95 & 131 with 405 & 2360 (metronome)

    124 = axial(2360,131)
    124 = beat(405,95)
    99 = beat(2360,95)
    99 = axial(405,131)
    77 = axial(405,95)
    55 = axial(131,95)

    in kiloyears

    95 = axial(405,124)
    95 = axial(2360,99)
    131 = beat(405,99)
    131 = beat(2360,124)

    compare:
    107 = harmean(2360,55) = harmean(2360,axial(131,95))
    110 = harmean(131,95)

    98 = axial(1180,107)
    110 = axial(1180,121)
    121 = beat(1180,110)
    121 = axial(405,173)
    173 = beat(405,121)

    Mercury: plea$$T˚ache˚C˚Eur.alarm˚F***the˚T˚i.e.[won??N/A!!!]g_0R.

    73 = axial(232,107)
    73 = axial(231,107)
    69 = axial(232,98)
    69 = axial(231,98)

  86. Paul Vaughan says:

    Tropical Review
    (now with double-hindsight)

    The incorrect tropical “correction” resulted in a systematically biased decoy.

    NASA Horizons is close to the following:

    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)
    25763.8618611039 = beat(1.00001743390371,0.99997862)

    0.240846701471137 = beat(25763.8618611039,0.24084445)
    0.615197259514841 = beat(25763.8618611039,0.61518257)
    1.00001743390371 = beat(25763.8618611039,0.99997862)
    1.88084834821604 = beat(25763.8618611039,1.88071105)
    11.8619839030546 = beat(25763.8618611039,11.85652502)
    29.457160787184 = beat(25763.8618611039,29.42351935)
    84.0205219636868 = beat(25763.8618611039,83.74740682)
    164.770280379209 = beat(25763.8618611039,163.7232045)
    250.431628515718 = beat(25763.8618611039,248.0208)

    130762.093818245 =1/(3/11.8619839030546-8/29.457160787184-2/84.0205219636868+7/164.770280379209)

    mnemonic supplement to go with last comment
    (˚Knew miracle methods of ((no doubt)) integration have limits) :

    2360489.30437277 = 2/(1/(φ^22+1/11)^(e/11+1/22)-1/ln(163*67*43*19*11*7*3*2*1))
    1180244.65218639 = 1/(1/(φ^22+1/11)^(e/11+1/22)-1/ln(163*67*43*19*11*7*3*2*1))
    405375.147994516 = 1/((1/(φ^22+1/11)^(e/11+1/22)-1/ln(163*67*43*19*11*7*3*2*1))/2+1/271/43/7/3/2)

    lim x→∞
    (φ^x+2/x)^(2e/x+1/x)
    = φ(φφ)^e

  87. Paul Vaughan says:

    supplementary to last comment:

    those are conversions of Seidelmann (1992) tropical to sidereal — the better option they had – but did not use – for the “correction”

    lol at the layers of corrections, disjoint calendars, systematic biases, & forensic diagnostics these folks have left a lone, displaced ecologist with no funding to sort out for them (a large, well-resourced organization) —- ˚F˚in’ ridiculous western inequality

    be aware that Williams’ “factsheet” sidereal for U & N are nowhere near NASA Horizons (J & S are fine)

    in this ongoing review, so far Standish (1992) gives the anomalistic jsun periods best-matching Horizons

    what a piece of (should be unnecessary) diagnostic work this crew has left us

    I now have an extensively long list of further things to check. Need 10 years of secure, stable funding and talented team to do it comprehensively. I’ve dug out some old prototypes (combinations of gaussian & generalized wavelets) and furthered them. Under better circumstances it could go a very long way. The algorithms take to way too long to run. Team: My life history leaves me effective but not efficient at the task. Get me funding and I’ll shepherd and delegate the development. I know no one else will do it right.

  88. Paul Vaughan says:

    Commerce Dove:
    Mint Heart DEEP

    1st note hear$ whirl D???why???D lower & middle ˚Class˚C?buy done˚$ 1930$ purge yuan plan IT (no doubt hear$us?hint:too UNaffordable lie ability)

    (1929.22222222222)
    130697.862041146 = beat(173913.043478261,74619.9907876555) (La2011 Table 6 La2004a)
    130697.499887022 = 1/(3/11.861984877399-8/29.4571542652499-2/84.0206327402408+7/164.770073662204)

    (1930.13888888889)
    130711.04387292 = beat(22379.3746087485,19107.8673985264) (La2011 Table 6 La2010a)
    130711.463657429 = 1/(3/11.8619848835796-8/29.4571541785584-2/84.0206328418372+7/164.770073333031)

    (1934.47222222222)
    130777.916695678 = beat(173804.240903943,74626.0277273697) (La2011 Table 5)
    130777.4870868 = 1/(3/11.8619849127921-8/29.4571537687132-2/84.0206333236655+7/164.77007177171)

    (1933.47222222222)
    130762.248768164 = 1/(3/11.8619849060515-8/29.4571538632991-2/84.020633212248+7/164.77007213279)

    Seidelmann (1992) tropical
    130762.093818127 = 1/(3/11.85652502-8/29.42351935-2/83.74740682+7/163.7232045)

    (sarc) Great! (/sarc)
    D-press 0˚D-on˚T no? side R-eel

    Seidelmann (1992) tropical converted to sidereal
    130762.093818245 =1/(3/11.8619839030546-8/29.457160787184-2/84.0205219636868+7/164.770280379209)

    Weather east AND west,
    let us pray for peace (financial, political, and otherwise).

  89. Paul Vaughan says:

    last comment
    130711.04387292 = beat(22379.3746087485,19107.8673985264)
    true – but this perspective was in mind
    130711.04387292 = beat(173889.708842077,74619.9907876555) (La2011 Table 6 La2010a)

  90. Paul Vaughan says:

    Read Together:
    “the date witch figures a line meant”

    quickly peace together (sensibly & pragmatically)

    25899.2748828505 = slip(4270.51884168654,883.230870074906)
    72333.0975802726 = slip(25899.2748828505,936.955612197409)
    400501.909620738 = slip(25899.2748828505,4270.51884168654)

    crude suggestion
    25899.1238467892 = 6 * beat(405000,4271)
    4270.51802039155 = 5 * axial(25899.1238467892,883.230870074906)

    no. doubt sew EU can refine “on Eur. own”?
    4 equations
    4 unknowns

    there was another album before that too (best ˚Figure$ away too avoid IT entirely)

  91. Paul Vaughan says:

    anomalistic orbital invariant
    2790.83682251396 = 1/(-2/11.8627021700857+5/29.4701958106261+3/84.0331316671926-6/164.793624044745)

    130548.596389613 = slip(2790.83682251396,936.955612197409)

  92. Paul Vaughan says:

    supplementary

    where ~72k is g_4 from sources noted in brackets below

    924.974835143217 = axial(72337.575351641,936.955612197409) (La2011 Table 6 La2004a)
    924.974835143217 = axial(72337.575351641,936.955612197409) (La2011 Table 6 La2010a)
    924.978668097723 = axial(72361.0252351259,936.955612197409) (La2011 Table 5)

    25899.2953840101 = 28 * 924.974835143217
    25899.2953840101 = 28 * 924.974835143217
    25899.4027067362 = 28 * 924.978668097723

    orbital invariant clue in standish (1992) biased sidereal U & N:
    2328.32483734181 = 1/(-2/11.861990807677+5/29.4571309198874+3/84.01495797691-6/164.786005834669)
    2135.25942084327 = 1/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)
    25750.7410386977 = beat(2328.32483733907,2135.25942084327)

  93. Paul Vaughan says:

    Horizons DATA (to be neither confused nor conflated with MODELS) :
    https://ssd.jpl.nasa.gov/planets/obs_data.html

  94. Paul Vaughan says:

    “About
    […]
    This viewer was implemented using 2-body methods, and hence should not be used for determining accurate long-term trajectories (over several years or decades) or planetary encounter circumstances. For accurate long-term ephemerides, please instead use our Horizons system.

    Version 1.0.0 (2018 March)”

    “Limitations

    The position of the small body is computed using so-called two-body equations: only the gravitational force of the Sun is considered in the viewer. If the small body makes a close approach to the Earth or a planet, its position as shown in this viewer may become inaccurate. You are especially cautioned against using this viewer to make predictions of the small body’s position a long time in the future or past.

    Accurate positions of small bodies can be obtained from our Horizons ephemeris system which uses a numerically integrated high fidelity model which includes gravitational perturbations by the Sun, all the planets, and some of the largest asteroids.”

  95. Paul Vaughan says:

    Take 58

    weathered myth, math, observation, & model:
    246671132.298687 = beat(4627.18618248315,4627.09938493153)

    405375.730975656 = 246671132.298687 / 608.5
    1180244.65214683 = 246671132.298687 / 209
    2360489.30429365 = 246671132.298687 / 104.5

    220 = s(284); 284 = s(220) ————- lowest amicable pair
    504 = 284 + 220 = 608.5 – 104.5
    s(496) = 496 = s(4*163) = s(s(608)) —- perfect no doubt

    supplementary
    4627.09501221791 = 365.25 / 365.2425 * 4627
    236 = 047 + 59 + 59 + 071

  96. Paul Vaughan says:

    1/(1/6.57-1/8.45) = 29.53

  97. Paul Vaughan says:

    Misnomer Alert

    unhelpful conventions are used in astronomy field (or at least Horizons) where simpler, more intuitive forms of model organization will -no doubt- be quickly obvious to (some) outsiders looking in

    =
    [Horizons] Output type : [geometric] osculating elements
    JDTDB Julian Day Number, Barycentric Dynamical Time
    [Calendar Date (TDB) before 1582-Oct-15 Julian calendar system; later calendar dates Gregorian]
    EC Eccentricity, e
    QR Periapsis distance, q (au)
    IN Inclination w.r.t X-Y plane, i (degrees)
    OM Longitude of Ascending Node, [omega], (degrees)
    W Argument of Perifocus, w (degrees)
    Tp Time of periapsis (Julian Day Number)
    N Mean motion, n (degrees/day)
    MA Mean anomaly, M (degrees)
    TA True anomaly, nu (degrees)
    A Semi-major axis, a (au)
    AD Apoapsis distance (au)
    PR Sidereal orbit period (day)
    =

    ♈︎ : reference direction
    Ω = ascending node longitude
    ϖ = periapsis longitude
    ϖ ≡ Ω + ω
    ω = periapsis argument
    ν = true anomaly
    L = true longitude
    L = ϖ + ν
    L = Ω + ω + ν
    M = mean anomaly
    l = mean longitude
    l = ϖ + M
    l = Ω + ω + M
    _

    mean not meant when “mean” said (“freedom of choice is words that they will bend — metallica)

    potential for (innocent perhaps) cross-disciplinary miscommunication:
    “mean” as defined by this astronomy field convention (counterintuitive historical legacy no doubt)
    is not in accord with intuition, convention, and perspective brought from other fields.

    effect: builds delays into what might otherwise be efficient, harmonious cross-disciplinary exchange

  98. Paul Vaughan says:

    Myth 0 “logical math”? Horizons
    with crude means (refinable)

    302998.544820593 = beat(11.8624525330717,11.8619881339068)
    302998.544575869 = beat(11.8623051105517,11.8618407229289)

    56498.324215687 = beat(29.4726630438482,29.4572964823535)
    56498.3242128238 = beat(29.4711887578906,29.4558237332929)

    395868.561366945 = beat(84.036052500683,84.0182168855969)
    395868.561421726 = beat(84.0343524991551,84.0165176055963)

    644223.254745067 = beat(164.797156009437,164.755010438071)
    644223.254719654 = beat(164.791273817488,164.74913125433)

    95238.1591975846 = harmean(302998.544820593,56498.324215687)
    412149.950131739 = harmean(644223.254745067,302998.544820593)
    98883.9415073487 = harmean(395868.561366945,56498.324215687)
    171631.483256873 = axial(395868.561366945,302998.544820593)

    130097.139297195 = beat(412149.950131739,98883.9415073487)
    123859.079046001 = beat(412149.950131739,95238.1591975846)

    109971.337347519 = harmean(130097.139297195,95238.1591975846)
    109971.337347519 = harmean(123859.079046001,98883.9415073487)

    121171.903078096 = axial(412149.950131739,171631.483256873)

    1189710.99772792 = beat(121171.903078096,109971.337347519)
    2379421.99545584 = 2 * 1189710.99772792
    98580.8900682825 = axial(1189710.99772792,107487.428178114)
    107487.428178114 = harmean(2379421.99545584,54985.6686737593)

    that’s without even using gaussian

    a key to sorting out disorienting “clerical (?) errors” contaminating various “official” summaries:
    reference frame diagnostics expose puzzle pieces not sharing a common geometry

    common frames used by different models (not always made explicit but detectable with diagnostics)
    • sidereal
    • solar system invariant (i.e. relative to J node)
    • ecliptic at epoch (often incorrectly noted as “ecliptic” but diagnosable as “at epoch”)

    The calculations above are with Jupiter node drift subtracted from horizons osculating output:
    • beat(M,l)
    • beat(ν,L) — (notation)

    It’s obvious at this point that a stable, compact model spanning millions of years is feasible.

  99. Paul Vaughan says:

    recall:
    anomalistic 4270 (jsun) & 936 (js) stable over entire 20ka Horizons span

    Standish’s f in Table 2b
    (used in M = mean anomaly estimation formula)

    U & N
    4693.45849222646 = 360 * 100 / 7.67025
    / 5 =
    938.691698445292 = 360 * 100 / 38.35125
    J & S

    note:
    47478.8397233984 = harmean(49188.090971097,45884.3897482377) —- Saturn s & g
    4692.59798767156 = beat(47478.8397233984,4270.51884168654)
    / 5 =
    938.519597534312

    answers & raises questions

  100. Paul Vaughan says:

    Past Decoys

    “Such elements are not intended to represent any sort of mean; they are simply the result of being adjusted for a best fit.” — Standish (1992)

    That may be — and no doubt true means can be found with more care.

    130762.093818127 = beat(2361.92512664079,2320.01916295311) ~= 131k —- orbital invariants
    2340.78460414247 = harmean(2361.92512664079,2320.01916295311) ———- arose from pairs
    1170.39230207124 = axial(2361.92512664079,2320.01916295311)

    1199.97261013651 = beat(47478.8397233984,1170.39230207124)
    936.754162989874 = axial(4270.51884168654,1199.97261013651) ~= axial(4271,1200)

  101. Paul Vaughan says:

    Horizons (in hindsight)

    shows up in Ω:
    883.178329571106 = harmean(936,836)
    = harmean(836+average(-178,378),836)

    shows up in ω & ϖ = Ω + ω :
    922.210960518562 = harmean(936,908.822299651568)
    = harmean(836+average(-178,378),harmean(836+average(-178,378),harmean(836+average(-178,378),836)))
    where
    908.822299651568 = harmean(936,883.178329571106)
    = harmean(836+average(-178,378),harmean(836+average(-178,378),836))

    supplementary
    600.073619631902 = harmean(936,441.589164785553)
    = harmean(836+average(-178,378),axial(836+average(-178,378),836))

  102. Paul Vaughan says:

    “Steep is the mountain which we climb” — Metallica

    2400 2341 59
    2362 2320 42
    38 80
    76 160
    152 320 = 163+67+43+19+28
    304 640
    608 1280

    2*(163+67+43+19)
    2400 / 365.25

  103. Paul Vaughan says:

    Standish’s T^2 term:
    it’s not really time

    after accounting for regular oscillations (in Horizons 20ka span) :
    mean longitude rates drift linearly
    mean anomaly rates drift as a quadratic of that (r^2 very nearly 1 for both)

    compact closed form equations look feasible

    clean factor of 3 appears in temporal scaling (geometric basis)
    systematically unbiasing biased estimates appears feasible

  104. Paul Vaughan says:

    Horizons Anomalistic Tunin’

    197703.858840665 = harmean(405568.048748278,130711.04387292)
    11.8627025553864 = beat(197703.858840665,11.861990807677)
    29.4700074925316 = axial(936,11.8627025553864/2)

  105. Paul Vaughan says:

    A rose myth (or math)?

    mimi˚C˚Kin’SSTand!sh more precisely:
    29.4701937613057 = 1/(-2/405568.048748278-2/11.861990807677+1/29.4571309198874+2/11.8627025553864)

    405568.048749313 = 1/(-1/11.861990807677+0.5/29.4571309198874+1/11.8627025553864-0.5/29.4701937613057)

    Standish (1992) :
    404924.648581828 = 1/(-1/11.861990807677+0.5/29.4571309198874+1/11.8627021700857-0.5/29.4701958106261)

    ‘n’ ice !
    98851.9294203326 = beat(304405.279928371,74619.9907876555)
    197703.858840665 = 2 * 98851.9294203326

  106. Paul Vaughan says:

    0 know doubt weather memorable myth or math

    22.1396468061395 = ln(163*67*43*19*11*7*3*2*1)

    with Standish (1992) anomalistic periods
    22.1205725094214 = 1/(3/0.615197860179071-5/1.0000262476142+2/11.8627021700857)

    25675.4767814772 = beat(22.1396468061395,22.1205725094214)
    compare Mayan

    and:
    25679.8527338808 = UN harmean
    25679.08722502 = beat(22.1396468061395,22.1205751889305)
    22.1205751889305 = 1/(3/0.615197860179071-5/1.0000262476142+2/11.8627025553864)

  107. Paul Vaughan says:

    Jupiter-Earth-Venus with Standish (1992) anomalistic

    1.59867106414771 = beat(1.0000262476142,0.615197860179071)

    0.761769224080824 = harmean(1.0000262476142,0.615197860179071)
    0.380884612040412 = axial(1.0000262476142,0.615197860179071)
    0.190442306020206 = 0.380884612040412 / 2

    0.814043420635227 = beat(11.8627021700857,0.761769224080824)
    0.715803548953639 = axial(11.8627021700857,0.761769224080824)

    44.2411450188424 = slip(1.59867106414771,0.814043420635227)
    6.84967828238651 = slip(1.59867106414771,0.715803548953639)

    835.563824740778 = slip(6.84967828238651,0.761769224080824)
    417.781912370389 = slip(6.84967828238651,0.380884612040412)
    208.890956185194 = slip(6.84967828238651,0.190442306020206)

    149 = 298 / 2
    163
    331
    353 (orbital invariant 130672)
    883

    +101 (lowest odd prime Mertens 0-crossing)
    883.185646071486 = harmean(936.563824740778,835.563824740778)
    909.091872159363 = harmean(936.563824740778,883.185646071486)
    922.62339315001 = harmean(936.563824740778,909.091872159363)

    with Williams’ “factsheet” values:
    60.9470469878813 = slip(29.4571389459274,11.8619822039699)
    883.192112166325 = slip(60.9470469878813,19.8588772513307)

    with Seidelmann decoy bias (short-duration model) no doubt clear in hindsight:
    61.0464822565173 = slip(29.4474984673838,11.8626151546089)
    835.546575435631 = slip(61.0464822565173,19.8650360864628)

  108. Paul Vaughan says:

    71, 171, 271

    review
    1180244.65218282 = beat(22.1396468061395,22.1392315068494)
    2360489.30436563 = beat(44.2792936122789,44.2784630136989)

    with Standish (1992) anomalistic
    171.471519050756 = beat(164.793624044745,84.0331316671926)
    4231.48507417337 = slip(164.793624044745,84.0331316671926)
    13120.0921835138 = slip(4231.48507417337,171.471519050756)
    26240.1843670276 = 2 * 13120.0921835138
    22.1392386512889 = beat(26240.1843670276,22.1205751889305)
    22.1392359672564 = beat(26240.1843670276,22.1205725094214)
    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    22.1205680565315 = axial(26240.1843670276,22.1392315068494)

    26246.4517581191 = beat(22.1392315068494,22.1205725094214)
    26250.2245780852 = beat(22.1392315068494,22.1205751889305)

    wonderous weather decoy
    26253.7412640768 = 262537412640768744 / 10^13
    22.1205776858368 = axial(26253.7412640768,22.1392315068494)
    in mnemonic hindsight

    405375.147994305 = 1 /(1/2360489.30436563+1/271/43/7/3/2)

  109. Paul Vaughan says:

    Great Dough
    Press $0 Don˚T !
    No. 1930 Why $

    130711.04387292 = beat(173889.708842077,74619.9907876555)
    94885.8433982369 = beat(304405.279928371,72337.575351641)
    109953.889784291 = harmean(130711.04387292,94885.8433982369)

    For˚Tallies hear?

    109954.696553367 = 2/(7/11.8619848847033-17/29.4571541627973-4/84.0206328603223+5/164.770073273138-8/11.8627021700857+20/29.4701958106261)

    hot pole-boydone$in sh! IT O˚C he

    egg rush
    in Zest˚T
    tie sh! rum
    more $isle 0we
    sci. b a n j o u r 0 ˚ K
    i . e . ˚C air folly read 2˚C a rose

    get away from $8
    in SST 0˚C mark at e^C˚0˚Know ME plea$
    “thought” $0˚C old “global leader” $$T 00P aid
    for$sober second

    5 lb decoy$ in clue nos. 1
    T˚ache 58˚F* linguist “˚C ˚K people”; 104 “love IT when EU can˚Tri”

    109942.314217849 = beat(183569.40509915,68760.6112054329)

  110. Paul Vaughan says:

    Point of Clarification

    s(4370) – 2 * 19 = 4270 – 38 = 4232

    Seidelmann sidereal short (bias = decoy) model
    4270.09258127429 = slip(164.791315640078,84.016845922161)

    Standish sidereal long model
    4267.83999767789 = slip(164.786005834669,84.01495797691)

    Standish anomalistic long model
    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)
    4231.48507417337 = slip(164.793624044745,84.0331316671926)

    Seidelmann tropical converted to sidereal (not biased)
    4232.67340063361 = slip(164.770275240329,84.0205206274557)

    Horizons ~1930 sidereal (not biased)
    4232.12222054646 = slip(164.770073333031,84.0206328418372)

    2 equations
    2 unknowns

  111. Paul Vaughan says:

    Notes (anomalistic)

    anomalistic estimates via Seidelmann tropical-converted-to-sidereal
    84.032929278878 = 3/(4/4270.51884168654-2/11.8627025553864+6/29.4701937613057+1/84.0205206274557)
    164.794135784188 = 3/(2/4270.51884168654-1/11.8627025553864+3/29.4701937613057-1/84.0205206274557)

    anomalistic estimates via Horizons ~1930 sidereal
    84.0330017055757 = 3/(4/4270.51884168654-2/11.8627025553864+6/29.4701937613057+1/84.0206328418372)
    164.793857247713 = 3/(2/4270.51884168654-1/11.8627025553864+3/29.4701937613057-1/84.0206328418372)

    latter fit better (compare)
    leaves a few questions on J & S refinement but crude outline’s complete

    reviewing perspective (always aware jovians shape EMB orbit envelope),
    how did this line of exploration arise? ———————————————————— question

    with widening gaussian filter on Horizons rates (of several different metrics), 936 (js) & 4270 (jsun) are the last waves to drop out; their periods are stationary (a definitive clue)

    100ka sidereal-anomalistic cycles turned up in this exploration:

    16.9132450828034 = harmean(29.4571309198874,11.861990807677)
    19.8589050137632 = beat(29.4571309198874,11.861990807677)

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)

    99476.8155050052 = beat(16.9161211952138,16.9132450828034)
    99764.9717395856 = harmean(100054.802238465,99476.8155050052)
    100054.802238465 = beat(19.8589050137632,19.8549641949401)

    These are really basic and simply-physical cycles.
    Curiosity: can’t recall anyone (“expert” or otherwise) pointing them out.
    Raises question: Have some records been misinterpreted? (always a naive question)
    _
    also noting 40ka (diagnostically not a function of k)
    60.9473428358019 = slip(29.4571309198874,11.861990807677)
    60.8544553085225 = slip(29.4701958106261,11.8627021700857)
    39929.1213729784 = beat(60.9473428358019,60.8544553085225)

  112. Paul Vaughan says:

    quick note on Standish nodes:
    19.8630730877524 = beat(29.4511026866654,11.862499899747)
    61.0124738503575 = slip(29.4511026866654,11.862499899747)
    851.495746676794 = slip(61.0124738503575,19.8630730877524)
    review link for comparison

  113. Paul Vaughan says:

    also compare figures at last review link with Standish (1992)

    sidereal
    1.59868960462765 = beat(1.0000174152119,0.615197263077614)
    2.67043880509308 = slip(1.0000174152119,0.615197263077614)
    8.10187019969398 = slip(2.67043880509308,1.59868960462765)
    238.924841351755 = slip(8.10187019969398,2.67043880509308)

    anomalistic
    1.59867106414771 = beat(1.0000262476142,0.615197860179071)
    2.67055352572244 = slip(1.0000262476142,0.615197860179071)
    8.10445902963256 = slip(2.67055352572244,1.59867106414771)
    233.230092320364 = slip(8.10445902963256,2.67055352572244)
    with hindsight: important detail missing in past (lunisolar) discussion with Ian Wilson

    nodes
    1.59868736807262 = beat(1.00001071395229,0.615194395759546)
    2.67039725923815 = slip(1.00001071395229,0.615194395759546)
    8.10116286399172 = slip(2.67039725923815,1.59868736807262)
    240.447503790019 = slip(8.10116286399172,2.67039725923815)

  114. Paul Vaughan says:

    405 & 100
    can agree on won thing in mediation mathematical
    i.e. equivalent “too what?”$in site bull-locked already note$too simply above ad UN clarity

    133k = beat(405k,100k) 131k

    197810.936263778 = beat(11.8627021700857,11.861990807677)
    132891.645799905 = axial(404924.648582574,197810.936263778)
    132891.645799828 = beat(404924.648582574,100054.802238465)
    66445.8228999138 = 132891.645799828 / 2
    66445.8228999526 = beat(29.4701958106261,29.4571309198874)

    UNode doubtsstand!sh
    4232.95187099857 = slip(164.7814326883,84.0262133014925)

    witch “experts” know doubt?
    promise to tell the truth
    the whole truth? ————————————————— where “experts” failed
    & nothing but the truth — (at times counterproductive) populace response to “expert” failure observed

    “gods” rich expectation know doubt:
    repentance weather classified or not

    maybe the populist response “too expert” aggression reminds us quite humbly:
    “Thou shalt not take false gods before me.”

    Still it isn’t entirely clear how God would like us to respond when “experts” fail to convey wholly truth and we instinctively deeply sense darkness weather by ignorance and/or deception. Keep shining light is no doubt part of the answer. Does God expect us “too humbly” submit “too expert” darkness and secrecy 0˚F*˚Tech˚Know log_cal elite D-$eat? I don’t think $0. Stable, secure financial equality would no doubt help eliminate gaps weather logical or other why$

    SST and! shout’n’ode
    $0 far left IT isn’t write 2D-bait in dumb mock crazy?
    inequality how sing dove[11a Davo$$T˚C˚K?]hide’n’concur? a better way can be found no doubt

    Pragmatic ally-why$sensibly negotiate$set11mean˚T?
    IT’s C11ear there will never B ag[D]ream mint.

    10 – or 20? – Times Richer

    6.56993811757712 = axial(14.7255513433327,11.862499899747)
    13.1398762351542 = harmean(14.7255513433327,11.862499899747)
    16.912768715208 = harmean(29.4511026866654,11.862499899747)
    19.8630730877524 = beat(29.4511026866654,11.862499899747)
    61.0124738503575 = slip(29.4511026866654,11.862499899747)
    851.495746676794 = slip(61.0124738503575,19.8630730877524)

    17.2555048518984 = beat(851.495746676794,16.912768715208)
    16.5833825558784 = axial(851.495746676794,16.912768715208)

    65.7216461925254 = slip(19.8630730877524,8.6277524259492)
    131.443292385051 = slip(19.8630730877524,17.2555048518984)

    casually exploring further
    too ponder more
    questions arose

    171.807870026969 = slip(65.7216461925254,9.93153654387618)

    50.2176861731761 = slip(19.8630730877524,8.29169127793919)
    95.0744667586459 = slip(19.8630730877524,4.1458456389696)
    100.435372346352 = slip(19.8630730877524,16.5833825558784)
    251.161498722936 = slip(95.0744667586459,16.912768715208)
    172616.932121584 = slip(251.161498722936,50.217686173176)

    good methods with nodecoy models

  115. Paul Vaughan says:

    from review link:

    109830.508474576 = beat(183829.787234043,68753.3156498674)
    109942.314217849 = beat(183569.40509915,68760.6112054329)

    109886.382906563 = harmean(109942.314217849,109830.508474576)

    _
    100046.317739694 = harmean(183569.40509915,68760.6112054329)
    100077.22007722 = harmean(183829.787234043,68753.3156498674)

    100061.766522545 = harmean(100077.22007722,100046.317739694)

    _
    219772.765813125 = 2 * 109886.382906563

    183699.503897945 = beat(219772.765813125,100061.766522545)
    68756.9632341238 = axial(219772.765813125,100061.766522545)

  116. Paul Vaughan says:

    Distinction

    130711.04387292 = 1/(-1/173889.708842077+1/74619.9907876555-0/72337.575351641-0/304405.279928371)

    132807.5411729 = 1/(0.5/183569.40509915+0.5/68760.6112054329-1/173889.708842077-0/74619.9907876555-0/72337.575351641+1/304405.279928371)

    raises diagnostic questions about Halstatt & Seidelmann (1992)

    tropical:
    130762.093818127 = 1/(3/11.85652502-8/29.42351935-2/83.74740682+7/163.7232045)
    130762.093817963 = 1/(1/2320.01916295313-1/2361.92512664087)
    (recall: same for tropical-converted-to-sidereal because orbitally invariant k-balance)

    synodic:
    132447.316536213 = 1/(3/11.8619993833167-8/29.4571726091513-2/84.021212742844+7/164.770559417647)
    132447.316535802 = 1/(1/2320.97373292688-1/2362.37133892198)

    synodic suggests trivial jsun anomalistic solution

    supplementary
    109942.314217849 = 1/(1/68760.6112054329-1/183569.40509915)
    109953.889784291 = 2/(1/74619.9907876555+1/72337.575351641-1/173889.708842077-1/304405.279928371)

  117. Paul Vaughan says:

    Seidelmann (1992) synodic with Standish (1992) nodes

    281135.559177404 = beat(11.862499899747,11.8619993833167)
    110491.22840888 = harmean(281135.559177404,68756.9632341238)

    142925.41912705 = beat(29.4571726091513,29.4511026866654)
    132497.268142232 = beat(142925.41912705,68756.9632341238) =======

    132464.669333742 = harmean(133828.914107874,131127.95796173) =====

    1411839.124377 = beat(84.0262133014925,84.021212742844)
    131127.95796173 = harmean(1411839.124377,68756.9632341238) ++++++++++

    2497052.60854115 = beat(164.7814326883,164.770559417647)
    133828.914107874 = harmean(2497052.60854115,68756.9632341238) ————

    426807.179318294 = beat(130704.452624679,100061.766522545)
    200224.836371855 = beat(11.8627021700857,11.8619993833167)

    6484482.69562381 = beat(426807.179318294,400449.672743711)

    133831.621979313 = beat(12968965.3912476,132464.669333742) —————-
    131125.358392832 = axial(12968965.3912476,132464.669333742) ++++++++++

  118. Paul Vaughan says:

    Weather “60k mysteriously” or “60072 mathematically”, sh!op nos…

    60077.769536193 = axial(132447.316536213,109951.487723802)
    60058.3268992756 = axial(132447.316536213,109886.382906563)
    60078.4866779602 = axial(132447.316536213,109953.889784291)
    60075.0306299611 = axial(132447.316536213,109942.314217849)
    60077.0524115463 = axial(132447.316536213,109949.085768262)
    60041.6324548796 = axial(132447.316536213,109830.508474576)

    60081.3396304648 = axial(132464.669333742,109951.487723802)
    60061.8946831129 = axial(132464.669333742,109886.382906563)
    60082.0568574667 = axial(132464.669333742,109953.889784291)
    60078.6003987141 = axial(132464.669333742,109942.314217849)
    60080.6224205865 = axial(132464.669333742,109949.085768262)
    60045.1982554562 = axial(132464.669333742,109830.508474576)

    comparative diagnostics ID ~common model features & reference frames

    according to Standish (1992)
    197810.936263778 = beat(11.8627021700857,11.861990807677)
    695709.040566689 = beat(11.8627021700857,11.862499899747)
    308037.559876028 = harmean(695709.040566689,197810.936263778)
    308043 = g_5 according to Berger 1988 Table 4 (based on Berger 1978)

  119. Paul Vaughan says:

    1/s_7 ~= 433k is key to conversion between Standish & Horizons’ early 1930s mean longitude rates
    100061.766522545 = harmean(183699.503897945,68756.9632341238)
    100402.649875387 = axial(433078.965717205,130704.452624679)
    100745.863762348 = beat(100061.766522545,50201.3249376937)
    100752.924584289 = beat(100054.802238478,50201.3249376937)

  120. Paul Vaughan says:

    g_7 & s_7 harmean
    100032.902990717 = axial(426282.531723861,130704.452624679)
    113330.117656829 = harmean(130704.452624679,100032.902990717)
    1.0000262486283 = beat(113330.117656829,1.00001742446316)
    1.0000262476142 Standish for comparison

  121. Paul Vaughan says:

    Basics — Part VI

    La2011 Table 6 La2010a
    419696.458422524 = 1 / g_7
    433078.972953216 = 1 / s_7
    1925651.43034595 = 1 / g_8
    1873536.29976581 = 1 / s_8

    La2011 Table 6 La2004a
    419695.778851413 = 1 / g_7
    433078.958481195 = 1 / s_7
    1925645.70793482 = 1 / g_8
    1873547.13358854 = 1 / s_8

    harmean(2004,2010)
    419696.118636694 = 1 / g_7
    426282.531723861 = harmean(433078.965717205,419696.118636694)
    433078.965717205 = 1 / s_7
    1925648.56913613 = 1 / g_8
    1899237.81359039 = harmean(1925648.56913613,1873541.71666151)
    1873541.71666151 = 1 / s_8

    Berger 1988 Table 4 (based on Berger 1978)
    422814 = 1 / g_7
    432023 = 1 / s_7
    1940518 = 1 / g_8
    1874374 = 1 / s_8

  122. Paul Vaughan says:

    UN certain bias

    Puzzle pieces organize neatly with a list of anomalistic & nodal periods and unambiguous determination of reference frame conventions — in particular the role of s_3, as with k before.

    Crossing combinations of periods from different lists & sources turns out to be a crucial diagnostic where (sometimes biased) lists are contaminated by “clerical errors” & omissions (e.g. undefined or incompletely defined terms, mixed types of estimates in the same list, mislabelled estimates, drifting estimates (reflecting different or evolving sampling &/or numerical methods)).

    With improved organization it’s becoming clear these pieces fit together quite neatly in several ways not noted by source authors (raises questions). Above are a few rough notes on U. Below on N.

    review —– Seidelmann short-duration sidereal with tropical
    23094.6280196825 = beat(11.8626151546089,11.85652502)
    36133.4834429326 = beat(29.4474984673838,29.42351935)
    64000.2003306117 = beat(36133.4834429326,23094.6280196825)
    extension
    66232.0778884627 = beat(1899237.81359039,64000.2003306117)
    132464.155776925 = 2 * 66232.0778884627 —- compare:
    132464.669333742 compare and carefully notice systematic bias

  123. Paul Vaughan says:

    aka “60k” or “60072”

    19.8549641949401 = beat(29.4701958106261,11.8627021700857) — Seidelmann (1992) synodic
    19.8589101021728 = beat(29.4571726091513,11.8619993833167) — Standish (1992) anomalistic

    99925.8030607636 = beat(19.8589101021728,19.8549641949401)

    30036.625757983 = beat(99925.8030609588,23094.6280196825)
    60073.251515966 = 2 * 30036.625757983

    mnemonically

    178k ~= beat(100k,64k)
    100k ~= beat(178k,64k)
    78k ~= harmean(100k,64k)

    99929.8245614035 = beat(178k,64k)

    30036.2624186769 = beat(99929.8245614035,23094.6280196825)
    60072.5248373539 = 2 * 30036.2624186769

  124. Paul Vaughan says:

    anecdote

    with Berger 1988 Table 4 (based on Berger 1978)
    95212.6482654869 = beat(308043,72732)
    99590.2129742594 = beat(308043,75259)
    123749.898156006 = beat(176420,72732)
    131248.13693024 = beat(176420,75259)

    110363.325696666 = harmean
    55181.6628483332 = harmean / 2

    400478.81907478 = beat(64000.2003306117,55181.6628483332)
    110363.325696666 = harmean(400478.81907478,64000.2003306117)
    25899.2909772319 = 6 * beat(400478.81907478,4270.51884168654)
    883.230851357431 = beat ( 25899.2909772319, 4270.51884168654 / 5 )

    883.230870074906 = slip(60.9469873120748,19.8588778386491) — with ~1930
    60.9469873120748 = slip(29.4571541785584,11.8619848835796) — Horizons
    19.8588778386491 = beat(29.4571541785584,11.8619848835796) — output

    25899.2909772318 = slip(4270.51884168654,883.230851357431)
    400478.819074885 = slip(25899.2909772318,4270.51884168654)

    even though this is not 405k, ~this (noticed in wide gaussian smooths of horizons output) prompted the successful search for 405k (which is simpler, noted above)

  125. Paul Vaughan says:

    weather too confuse(or,clarify?)
    U,N bias with the center of the sample

    66445.8228999526 = beat(29.4701958106261,29.4571309198874)
    142925.41912705 = beat(29.4571726091513,29.4511026866654)
    142889.793300912 = beat(1899237.81359039,132891.645799905)

    132428.181070613 = beat(123859.129354423,64000.2003306117)
    132421.415512283 = beat(123865.048265316,64000.2003306117)
    132434.947320298 = beat(123853.211009174,64000.2003306117)
    132553.277441197 = beat(123749.898156006,64000.2003306117)

    123745.929709558 = axial(1873541.71666151,132497.268142232)
    123856.611088565 = axial(1899237.81359039,132497.268142232)

    1898438.644812 = beat(132497.268142232,123853.211009174)
    1874451.80258414 = beat(132497.268142232,123749.898156006)

    426193.700655367 = beat(130762.09381872,100061.766522545)
    426282.531723861 = harmean(433078.965717205,419696.118636694)
    426310.538349634 = beat(130777.4870868,100077.22007722)

    427158.866183256 = beat(130697.862041146,100077.22007722)
    427368.896577944 Berger

    113224.424017561 = beat(1.0000262476142,1.0000174152119)
    113356.074521123 = harmean(130697.862041146,100077.22007722)
    113370.308271798 = harmean(130762.09381872,100061.766522545)
    113386.012502077 = harmean(130777.4870868,100077.22007722)
    113462.111826241 = beat(1.0000262476142,1.00001743371442)

    special k˚s_3-point interpretation
    JS: mirror
    U : VE
    N : VMa

  126. Paul Vaughan says:

    1800 lunisolar with Standish (1992) anomalistic

    0.0748026823036094 = axial(1.0000262476142,0.0808503463381246)
    18.5964835171849 = beat(0.0748026823036094,0.0745030006844627)

    8.85108298702051 = beat(0.0754402464065708,0.0748026823036094)
    16.8899183125299 = beat(18.5964835171849,8.85108298702051)

    184.049860297634 = beat(18.5964835171849,16.8899183125299)

    5.99685290323073 = beat(0.0754402464065708,0.0745030006844627)
    5.99685290323073 = axial(18.5964835171849,8.85108298702051)

    1814.75583949423 = slip(5.99685290323073,1.0000262476142)
    1787.32381267794 = slip(184.049860297634,16.8899183125299)

    1800.93537030388 = harmean(1814.75583949423,1787.32381267794)

    68961.1021012713 = slip(1800.93537030388,0.999978614647502)
    68935.6558846914 = slip(0.99997862,0.999964114549628)

  127. Paul Vaughan says:

    Diagnostics suggest numerical methods, limits, and tuning.

    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)

    0.0374013164945983 = axial(0.500008716951856,0.0404251731690623)

    9.29976596105445 = beat(0.0374013164945983,0.0372515003422313)
    4.42519629511916 = beat(0.0377201232032854,0.0374013164945983)

    2.99842645161536 = beat(0.0377201232032854,0.0372515003422313)
    2.99842645161536 = axial(9.29976596105445,4.42519629511916)
    8.4424457329891 = beat(9.29976596105445,4.42519629511916)
    91.5792802799901 = beat(9.29976596105445,8.4424457329891)
    922.126441327183 = slip(2.99842645161536,0.500008716951856)
    600.450003140476 = slip(91.5792802799901,8.4424457329891)

    2607982.70831762 = slip(1800.93537030388,600.450003140476)
    2607198.33309978 = 1/(3/11.8619822039699-8/29.4571389459274-2/84.0120465434634+7/164.788501026694)
    2607198.3328459 = beat(2317.51251648274,2319.57436292966)
    sidereal orbital invariant with Williams’ “factsheet”

    0.0748024157783867 = axial(0.999978614647502,0.0808503463381246)

    8.84735293159855 = beat(0.0754402464065708,0.0748024157783867)
    18.6129709123853 = beat(0.0748024157783867,0.0745030006844627)
    5.99685290323073 = beat(0.0754402464065708,0.0745030006844627)
    5.99685290323073 = axial(18.6129709123853,8.84735293159855)

    1986.46983643213 = slip(5.99685290323073,0.999978614647502)

    with Seidelmann (1992)
    1.00001743371442 = 1/E
    600.450003140476 becomes
    600.445772016063

    6442.77629910435 = slip(1986.46983643213,600.445772016063)
    25771.1051964174 = 4 * 6442.77629910435

    2313.55148640991 = slip(600.445772016063,18.6129709123853)
    4627.10297281983 = 2 * 2313.55148640991

    2499.39163457868 = slip(1200.90000628095,18.6129709123853)
    2402.88367676045 = harmean(2499.39163457868,2313.55148640991)

  128. Paul Vaughan says:

    correction last 2 rows sidereal consistency
    2497.01127082478 = slip(1200.89154403213,18.6129709123853)
    2401.78308806424 = harmean(2497.01127082478,2313.55148640991)

    insight pouring faster than recording speed now (omission necessary)
    Standish (1992) was missing link
    conventional approach to hierarchical bias correction clarifying through diagnostic exploration

  129. Paul Vaughan says:

    Reference Frames:
    It’s the things they don’t tell you.

    66445.8228999526 = beat(29.4701958106261,29.4571309198874)
    132669.109137331 = harmean(132891.645799905,132447.316536213)
    132670.71628812 = beat(405629.613215262,99972.3915877777)

  130. Paul Vaughan says:

    reference frame exploration with
    beats of synodic Jupiter-Saturn beats & harmonic means (~100ka)
    Earth sidereal year length
    405ka

    Review:
    66445.8228999526 = beat(29.4701958106261,29.4571309198874) — Standish (1992)
    132891.645799905 = 2 * 66445.8228999526

    La2011 Table 6 La2010a
    405568.048748278 = beat(304405.279928371,173889.708842077)
    harmean(2004,2010)
    405629.613215262 = beat(304406.35241565,173901.37537739)
    La2011 Table 6 La2004a
    405691.196375825 = beat(304407.424910486,173913.043478261)

    Notice how the preceding bracket some of the following.
    Each calculation chain first specifies Earth sidereal used (“supplementary” below gives detail).

    _
    1.00001743371442 —- Seidelmann (1992) short-duration Keplerian model

    99925.8030609588 = 1/(-1/11.8619993833167+1/29.4571726091513+1/11.8627021700857-1/29.4701958106261)
    100019.023577014 = 1/(0.5/11.8619993833167+0.5/29.4571726091513-0.5/11.8627021700857-0.5/29.4701958106261)
    99972.3915878471 = 1/(-0.25/11.8619993833167+0.75/29.4571726091513+0.25/11.8627021700857-0.75/29.4701958106261)
    132447.316536213 = 1/(3/11.8619993833167-8/29.4571726091513-2/84.021212742844+7/164.770559417647)
    132669.109137331 = harmean(132891.645799905,132447.316536213)
    405644.637269099 = beat(132669.109137331,99972.3915877777)

    _
    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6) ——– Schneider$Fi[N]e!SST!m[U]te

    99925.8030609588 = 1/(-1/11.8619994099501+1/29.4571727733966+1/11.8627021700857-1/29.4701958106261)
    100020.917159549 = 1/(0.5/11.8619994099501+0.5/29.4571727733966-0.5/11.8627021700857-0.5/29.4701958106261)
    99973.3374875151 = 1/(-0.25/11.8619994099501+0.75/29.4571727733966+0.25/11.8627021700857-0.75/29.4701958106261)
    132447.31653597 = 1/(3/11.8619994099501-8/29.4571727733966-2/84.021214079097+7/164.770564556546)
    132669.109137209 = harmean(132891.645799905,132447.31653597)
    405660.210867366 = beat(132669.109137209,99973.3374874111)

    _
    1.0000174152119 — Standish (1992)

    99925.8030608895 = 1/(-1/11.8619967799723+1/29.4571565546135+1/11.8627021700857-1/29.4701958106261)
    99834.2762309764 = 1/(0.5/11.8619967799723+0.5/29.4571565546135-0.5/11.8627021700857-0.5/29.4701958106261)
    99880.0186778954 = 1/(-0.25/11.8619967799723+0.75/29.4571565546135+0.25/11.8627021700857-0.75/29.4701958106261)
    132447.316536822 = 1/(3/11.8619967799723-8/29.4571565546135-2/84.0210821278182+7/164.770057105325)
    132669.109137637 = harmean(132891.645799905,132447.316536822)
    404128.108246229 = beat(132669.109137637,99880.0186778262)

    _
    1.00001642710472 — Williams’ NASA “factsheet” (“too rounded off” for Earth)

    99925.8030608895 = 1/(-1/11.861857752743+1/29.4562992042035+1/11.8627021700857-1/29.4701958106261)
    90870.4823490607 = 1/(0.5/11.861857752743+0.5/29.4562992042035-0.5/11.8627021700857-0.5/29.4701958106261)
    95183.2568831366 = 1/(-0.25/11.861857752743+0.75/29.4562992042035+0.25/11.8627021700857-0.75/29.4701958106261)
    132447.3165367 = 1/(3/11.861857752743-8/29.4562992042035-2/84.0141073585951+7/164.743236088451)
    132669.109137576 = harmean(132891.645799905,132447.3165367)
    336870.502763842 = beat(132669.109137576,95183.2568831052)

    _____________
    supplementary

    Seidelmann (1992) Table 15.6 synodic column p.367(352)

    0.3172553045859 = 115.8775 / 365.25
    1.59868966461328 = 583.9214 / 365.25

    2.13534866529774 = 779.9361 / 365.25
    1.0920848733744 = 398.884 / 365.25
    1.03515920602327 = 378.0919 / 365.25
    1.0120629705681 = 369.656 / 365.25
    1.00612375085558 = 367.4867 / 365.25
    1.00402655715264 = 366.7207 / 365.25

    conversions with Schneider

    0.240846733037309 = axial(1.00001743390371,0.3172553045859)
    0.615197279034371 = axial(1.59868966461328,1.00001743390371)

    1.88084836716665 = beat(2.13534866529774,1.00001743390371)
    11.8619994099501 = beat(1.0920848733744,1.00001743390371)
    29.4571727733966 = beat(1.03515920602327,1.00001743390371)
    84.021214079097 = beat(1.0120629705681,1.00001743390371)
    164.770564556546 = beat(1.00612375085558,1.00001743390371)
    250.439809134 = beat(1.00402655715264,1.00001743390371)

  131. Paul Vaughan says:

    “Be Cause”…

    compare ~132447 with ~132464 (from previous notes) :

    132677.81404965 = harmean(132891.645799905,132464.669333742)
    132677.55644373 = harmean(132891.645799905,132464.155776925)

    1.00001743371442:
    405563.279197667 = beat(132677.81404965,99972.3915877777)
    405565.686218759 = beat(132677.55644373,99972.3915877777)

    405568.048748278 La2011 Table 6 La2010a — bracketed by last & next —————–

    1.00001743390371 (Schneider240) :
    405578.846548265 = beat(132677.81404965,99973.3374874111)
    405581.253754146 = beat(132677.55644373,99973.3374874111)

    1.0000174152119:
    404047.357304897 = beat(132677.81404965,99880.0186778262)
    404049.74636555 = beat(132677.55644373,99880.0186778262)

    1.00001642710472 (NASA “factsheet” Earth sidereal lacked precision at retrieval date) :
    336814.391520401 = beat(132677.81404965,95183.2568831052)
    336816.051655712 = beat(132677.55644373,95183.2568831052)

    many distinctions to be made when comparing periods with different properties

    for example: diagnostics immediately above reveal that Standish’s (1992) Keplerian Earth sidereal year length isn’t the Earth sidereal year length used to tune Standish’s (1992) Keplerian Saturn sidereal & anomalistic year lengths

    careful study of boundary condition aggregation criteria (and knew miracle methods)

    prerequisite for “physics” pea & shell games, where pieces of many models may be mixed into shared belief systems based on favorite orreries

    big, clear questions arising about lunar ephemerides
    see above and note glaring inconsistencies with tuning of longer ephemeris
    can 25771 vs. 25721 be diagnosed without LLR? no doubt (“could be nothing at all” L&R)

    goal isn’t necessarily to finally pick best but to first at least be aware of recipes

  132. Paul Vaughan says:

    La2011 Table 5 (1 more thing to compare)
    405113.811661464 = beat(304399.417131486,173804.240903943)

    Combination best compared with (Schneider240) last comment:
    405578.45627326 = beat(304399.417131486,173889.708842077)
    ______________
    supplementary (focusing 100ka intuition)

    19.8549641949401 = beat(29.4701958106261,11.8627021700857) — Standish (1992) anomalistic

    1.00001743371442:
    19.8589101021728 = beat(29.4571726091513,11.8619993833167) — Seidelmann (1992) synodic
    99925.8030607636 = beat(19.8589101021728,19.8549641949401)
    100019.023576957 = beat(16.9161211952138,16.9132606717144)
    99972.391587704 = harmean(100019.023576957,99925.8030607636)

    1.00001743390371:
    19.8589101021728 = beat(29.4571727733966,11.8619994099501) — Seidelmann (1992) synodic
    99925.8030609435 = beat(19.8589101021728,19.8549641949401)
    100020.917159414 = beat(16.9161211952138,16.9132607258603)
    99973.3374874229 = harmean(100020.917159414,99925.8030609435)

  133. Paul Vaughan says:

    different way of looking at the above (supplementary – nothing new)

    25772.7186475618 = beat(1.0000174152119,0.999978614647502)

    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)

    1.00001743390557 = 62/(42/0.0745030006844627+20/0.0754402464065708-62/0.0808503463381246+3/1800.93537030388-1/2607198.33309978)

  134. Paul Vaughan says:

    58, 59 mnemonic

    19.8589101021728 = beat(29.4571726091513,11.8619993833167) — synodic
    19.8549641949401 = beat(29.4701958106261,11.8627021700857) — anomalistic
    99925.8030607636 = beat(19.8589101021728,19.8549641949401)
    3422 = beat(59,58)
    58.4957264957265 = harmean(59,58)
    136880 = 40 * 3422
    5845232.44570826 = 58.4957264957265 * 99925.8030607636
    133747.973551142 = axial(5845232.44570826,136880)
    13374797.3551142 = 100 * 133747.973551142
    25721.8906850397 = axial(13374797.3551142,25771.4533429313)
    25721.8900031954
    0.000002650833 = % error

  135. Paul Vaughan says:

    concise notes dialing in Seidelmann synodic 132447 orbital invariant
    (also recording a few notes along the way)

    with harmean(2004,2010)
    1182481.75182481 = beat(73001.7461837436,68756.9632341238)
    2364963.50364963 = beat(74619.9907876555,72337.575351641)
    143000.04832872 = axial(2364963.50364963,152203.176207825)
    29.4511058537984 = axial(143000.04832872,29.4571726091513)
    29.4511026866654 nodal Standish (1992)

    11.8624616355381 = beat(304406.35241565,11.8619993833167)
    11.862499899747 nodal Standish (1992)

    132433.196342637 = beat(143000.04832872,68756.9632341238)
    132448.674250615 = harmean(132464.155776925,132433.196342637)

  136. Paul Vaughan says:

    few more notes
    132669.790266541 = harmean(132891.645799905,132448.674250615)
    197152.808533573 = beat(405629.613215262,132669.790266541)
    11.8627045448705 = beat(197152.808533573,11.861990807677)
    11.8627021700857 Standish (1992) anomalistic

  137. Paul Vaughan says:

    goes with last
    99971.8657741115 = axial(405629.613215262,132669.790266541)

    noting now (maybe revise/tighten later)
    133805.278809805 = beat(13216090.441739,132464.155776925)
    2480700.64426686 = beat(68756.9632341238,66902.6394049025)
    164.781504366091 = beat(2480700.64426686,164.770559417647)
    164.7814326883

    131149.649967513 = axial(13216090.441739,132464.155776925)
    1416885.58942143 = beat(68756.9632341238,65574.8249837565)
    84.0261954901455 = beat(1416885.58942143,84.021212742844)
    84.0262133014925

    subtleties between different authors and types of summaries keep clarifying even with minor effort since missing link found

    apologies for lack of explanation — available time got reduced by another order of magnitude

    those looking independently: be tediously careful differentiating Keplerian sidereal from synodic sidereal — may look quite similar initially, but subtle systematic differences clarify upon increasingly careful study

  138. Paul Vaughan says:

    173362.29482452 = harmean(173901.37537739,172826.54615749)
    110456.356060155 = axial(304406.35241565,173362.29482452)
    280684.610556601 = beat(68756.9632341238,55228.1780300774)
    11.8625007039119 = beat(280684.610556601,11.8619993833167)
    11.862499899747
    lot of things near-equal
    continually invites better balance
    leaving it there for now

  139. Paul Vaughan says:

    reorganizing some of preceding, found easy way to remember synodic-anomalistic relations

    360 = σ(323)

    29.4571726091513
    29.4571726172074 = 5/(2/11.8619993833167+29/3/2/2/11.8627021700857-29/2/2/29.4701958106261+29/3/2/2/84.0331316671926+29/3/2/2/164.793624044745+1/3/2/323000)

    11.8619993833167
    11.8619993800508 = 2/(5/29.4571726091513-29/3/2/2/11.8627021700857+29/2/2/29.4701958106261-29/3/2/2/84.0331316671926-29/3/2/2/164.793624044745-1/3/2/323000)

    so that’s down to fine diagnostics on sidereal earth year-length

  140. Paul Vaughan says:

    politics apparently transformed into weird, unhelpful noise
    numbers still a casual hobby — exploratory meandering (no rush & nothing to do with politics)
    _
    short-duration model bias is systematic
    translations between models are feasible

    orbital invariant matched with combinations of neptune offsets (g_ & s_8) from different authors (~1.9 Ma)

    132464.155776897 = 1/(-0.5/11.8626151546089+0.5/29.4474984673838+0.5/11.85652502-0.5/29.42351935-0.5/1899237.81359039)

    132454.908763513 = 1/(-0.5/11.8626151546089+0.5/29.4474984673838+0.5/11.85652502-0.5/29.42351935-0.5/1903047.54215808)

    132445.663041064 = 1/(-0.5/11.8626151546089+0.5/29.4474984673838+0.5/11.85652502-0.5/29.42351935-0.5/1906872.58550543)

    132447.316536213 = 1/(3/11.8619993833167-8/29.4571726091513-2/84.021212742844+7/164.770559417647)

    132446.730558588 = beat(143000.04832872,68760.6112054329)
    where
    143000.04832872 = axial(2364963.50364963,152203.176207825)
    2364963.50364963 = beat(74619.9907876555,72337.575351641)
    152203.176207825 = 304406.35241565 / 2

    _
    concise outline of comparable U offset (g_ & s_7) translations (~427 ka)

    in the following
    100061.766522545 = harmean(183699.503897945,68756.9632341238)
    130704.452624679 = beat(173901.37537739,74619.9907876555)

    68749.8116120017 = 1/(-1/84.021212742844+1/84.0262133014925+2/132454.908763513+0.5/11.8619993833167-0.5/11.8627021700857-1/426825.022891826)

    68750.2745749087 = 1/(-1/84.021212742844+1/84.0262133014925-1/100061.766522545+1/130704.452624679+2/132454.908763513+0.5/11.8619993833167-0.5/11.8627021700857)

    68753.5961356997 = 1/(-1/164.770559417647+1/164.7814326883+2/132454.908763513-0.5/11.8619993833167+0.5/11.8627021700857+1/426825.022891826)

    68753.1331280572 = 1/(-1/164.770559417647+1/164.7814326883+1/100061.766522545-1/130704.452624679+2/132454.908763513-0.5/11.8619993833167+0.5/11.8627021700857)

    _
    intermodel comparatison diagnostic measures
    harmonic mean of g_2, g_3, g_4, g_5, s_2, s_3, s_4
    100078.929091917 — La2011 Table 6 La2004a
    100076.152524661 — harmean(2004,2010)
    100073.376111466 — La2011 Table 6 La2010a
    100743.899044297 — Berger 1988 Table 4 (based on Berger 1978) —————- note difference
    _
    latter & following suggest features to explore comparatively
    _
    221349.897911753 = harmean(304406.35241565,173901.37537739)
    112568.239609489 = beat(221349.897911753,74619.9907876555)
    111613.565931354 = harmean(221349.897911753,74619.9907876555)
    1.00002639378076 = beat(111613.565931354,1.00001743390371)
    1.00002638193018 — anomalistic (source trail)
    vs.
    224351.275783703 = harmean(308043,176420)
    113248.339496145 = beat(224351.275783703,75259)
    112709.436417295 = harmean(224351.275783703,75259)
    1.00002624574898 = beat(113248.339496145,1.0000174152119) — Standish (1992) sidereal
    1.0000262476142 — Standish (1992) anomalistic
    comparisons focus questions
    haven’t yet encountered clear, concise materials bypassing slow process of exploring and figuring it out independently without sufficient preparation and support
    _
    systematic bias note:
    keplerian models (whether short- or long-duration) have ~4270 year UN sidereal slip cycle ~matching length of ~4270 year JSUN anomalistic orbital invariant, whereas UN slip cycle for everything else is ~4232 (anomalistic, sidereal, & nodal)

  141. Paul Vaughan says:

    132461.310003517 = 1/(1/173901.37537739/3+1/74619.9907876555/6+1/72337.575351641/6+1/304406.35241565/3)
    60853.6256423468 = axial(132461.310003517,112568.239609489)
    25695.4149624003 = beat(60853.6256423468,18066.7417214663)
    review: 25722 (“2722” is typo at link)
    sources of bias clarifying

  142. Paul Vaughan says:

    further diagnostics: harmonic mean of sidereal-anomalistic J-S and (J+S)/2 explains refinement of ~100.7ka to ~100.0ka (with k adjustment)

  143. Paul Vaughan says:

    completes puzzle
    rearrange, solve — connects old notes (turns out it wasn’t a decoy after all)

  144. Paul Vaughan says:

    arrive back at same conclusion

  145. Paul Vaughan says:

    11.8626151546089
    11.8626150543774 = 1/(-1/164.770559417647+1/164.7814326883-0.5/11.8619993833167+0.5/11.8627021700857+1/426825.022891826-1/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1903047.54215808)

    29.4474984673838
    29.4474990850308 = 1/(1/11.8626151546089-1/11.85652502+1/29.42351935+1/1903047.54215808+1/164.770559417647-1/164.7814326883+0.5/11.8619993833167-0.5/11.8627021700857-1/426825.022891826+1/68756.9632341238)

  146. Paul Vaughan says:

    The Seidelmann (1992) short-duration model is systematically biased by precession of nodes & perihelion of U & N, diagnosable by simply noting 36ka ≠ 26ka ≠ 23ka (———-).

    11.8626151546089
    11.8626150900371 = 1/(-1/84.021212742844+1/84.0262133014925-1/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1899237.81359039+0.5/11.8619993833167-0.5/11.8627021700857-1/426825.022891826)
    11.8626149417082 = 1/(-1/84.021212742844+1/84.0262133014925-1/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1903047.54215808+0.5/11.8619993833167-0.5/11.8627021700857-1/426825.022891826)

    29.4474984673838
    29.4474988652882 = 1/(1/84.021212742844-1/84.0262133014925+1/68756.9632341238+1/11.8626151546089-1/11.85652502+1/29.42351935+1/1899237.81359039-0.5/11.8619993833167+0.5/11.8627021700857+1/426825.022891826)
    29.4474997793212 = 1/(1/84.021212742844-1/84.0262133014925+1/68756.9632341238+1/11.8626151546089-1/11.85652502+1/29.42351935+1/1903047.54215808-0.5/11.8619993833167+0.5/11.8627021700857+1/426825.022891826)

  147. Paul Vaughan says:

    inner system reflects this in terms of E & V nodes & perihelion:

    11.8626151546089
    11.8626151889228 = 1/(-1/164.770559417647+1/164.7814326883-0.5/11.8619993833167+0.5/11.8627021700857+1/100061.766522545-1/130704.452624679-1/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1899237.81359039)
    11.8626150405938 = 1/(-1/164.770559417647+1/164.7814326883-0.5/11.8619993833167+0.5/11.8627021700857+1/100061.766522545-1/130704.452624679-1/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1903047.54215808)

    29.4474984673838
    29.4474982559348 = 1/(1/11.8626151546089-1/11.85652502+1/29.42351935+1/1899237.81359039+1/164.770559417647-1/164.7814326883+0.5/11.8619993833167-0.5/11.8627021700857-1/100061.766522545+1/130704.452624679+1/68756.9632341238)
    29.4474991699678 = 1/(1/11.8626151546089-1/11.85652502+1/29.42351935+1/1903047.54215808+1/164.770559417647-1/164.7814326883+0.5/11.8619993833167-0.5/11.8627021700857-1/100061.766522545+1/130704.452624679+1/68756.9632341238)

    where
    130704.452624679 = beat(173901.37537739,74619.9907876555)
    100061.766522545 = harmean(183699.503897945,68756.9632341238)

  148. Paul Vaughan says:

    stated alternately:

    11.8626151546089
    11.8626151889228 = 1/(-1/164.770559417647+1/164.7814326883-0.5/11.8619993833167+0.5/11.8627021700857+0.5/183699.503897945+1/173901.37537739-1/74619.9907876555-0.5/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1899237.81359039)
    11.8626150405938 = 1/(-1/164.770559417647+1/164.7814326883-0.5/11.8619993833167+0.5/11.8627021700857+0.5/183699.503897945+1/173901.37537739-1/74619.9907876555-0.5/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1903047.54215808)

    29.4474984673838
    29.4474982559348 = 1/(1/11.8626151546089-1/11.85652502+1/29.42351935+1/1899237.81359039+1/164.770559417647-1/164.7814326883+0.5/11.8619993833167-0.5/11.8627021700857-0.5/183699.503897945-1/173901.37537739+1/74619.9907876555+0.5/68756.9632341238)
    29.4474991699678 = 1/(1/11.8626151546089-1/11.85652502+1/29.42351935+1/1903047.54215808+1/164.770559417647-1/164.7814326883+0.5/11.8619993833167-0.5/11.8627021700857-0.5/183699.503897945-1/173901.37537739+1/74619.9907876555+0.5/68756.9632341238)

  149. Paul Vaughan says:

    further comparisons for more thorough consideration (not really necessary at this point) :

    11.8626151546089
    11.8626152027063 = 1/(-1/164.770559417647+1/164.7814326883-0.5/11.8619993833167+0.5/11.8627021700857+1/426825.022891826-1/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1899237.81359039)
    11.8626151038207 = 1/(-1/84.021212742844+1/84.0262133014925-1.5/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1899237.81359039+0.5/11.8619993833167-0.5/11.8627021700857-0.5/183699.503897945-1/173901.37537739+1/74619.9907876555)
    11.8626151038207 = 1/(-1/84.021212742844+1/84.0262133014925-1/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1899237.81359039+0.5/11.8619993833167-0.5/11.8627021700857-1/100061.766522545+1/130704.452624679)
    11.8626149554917 = 1/(-1/84.021212742844+1/84.0262133014925-1.5/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1903047.54215808+0.5/11.8619993833167-0.5/11.8627021700857-0.5/183699.503897945-1/173901.37537739+1/74619.9907876555)
    11.8626149554917 = 1/(-1/84.021212742844+1/84.0262133014925-1/68756.9632341238+1/29.4474984673838+1/11.85652502-1/29.42351935-1/1903047.54215808+0.5/11.8619993833167-0.5/11.8627021700857-1/100061.766522545+1/130704.452624679)

    29.4474984673838
    29.4474981709978 = 1/(1/11.8626151546089-1/11.85652502+1/29.42351935+1/1899237.81359039+1/164.770559417647-1/164.7814326883+0.5/11.8619993833167-0.5/11.8627021700857-1/426825.022891826+1/68756.9632341238)
    29.4474987803512 = 1/(1/84.021212742844-1/84.0262133014925+1.5/68756.9632341238+1/11.8626151546089-1/11.85652502+1/29.42351935+1/1899237.81359039-0.5/11.8619993833167+0.5/11.8627021700857+0.5/183699.503897945+1/173901.37537739-1/74619.9907876555)
    29.4474987803512 = 1/(1/84.021212742844-1/84.0262133014925+1/68756.9632341238+1/11.8626151546089-1/11.85652502+1/29.42351935+1/1899237.81359039-0.5/11.8619993833167+0.5/11.8627021700857+1/100061.766522545-1/130704.452624679)
    29.4474996943842 = 1/(1/84.021212742844-1/84.0262133014925+1.5/68756.9632341238+1/11.8626151546089-1/11.85652502+1/29.42351935+1/1903047.54215808-0.5/11.8619993833167+0.5/11.8627021700857+0.5/183699.503897945+1/173901.37537739-1/74619.9907876555)
    29.4474996943842 = 1/(1/84.021212742844-1/84.0262133014925+1/68756.9632341238+1/11.8626151546089-1/11.85652502+1/29.42351935+1/1903047.54215808-0.5/11.8619993833167+0.5/11.8627021700857+1/100061.766522545-1/130704.452624679)

  150. Paul Vaughan says:

    review

    132465.224494761 = 1/(0.5/29.4571726091513-0.5/29.4511026866654+0.5/74619.9907876555-0.5/72337.575351641-1/304406.35241565+1/68756.9632341238)

    132465.224494761 = 1/(0.5/29.4571726091513-0.5/29.4511026866654+0.75/68756.9632341238+0.25/73001.7461837436-1/304406.35241565)

  151. Paul Vaughan says:

    1899237.81359039 = harmean(1925648.56913613,1873541.71666151)
    123859.129354423 = beat(173901.37537739,72337.575351641)
    132500.150033928 = beat(1899237.81359039,123859.129354423)
    142922.065898762 = beat(132500.150033928,68756.9632341238)
    29.4511025442826 = axial(142922.065898762,29.4571726091513)
    29.4511026866654 saturn node Standish (1992)
    29.4511025442826 = 1/(1/29.4571726091513+0.5/1925648.56913613+0.5/1873541.71666151+1/173901.37537739-1/72337.575351641+1/68756.9632341238)
    29.4511025442826 = 1/(1/29.4571726091513+1/1899237.81359039+1/173901.37537739-1/72337.575351641+1/68756.9632341238)

  152. Paul Vaughan says:

    132466.664728053 = 1/(0.5/29.4571726091513-0.5/29.4511025442826+0.75/68756.9632341238+0.25/73001.7461837436-1/304406.35241565)

    68756.1458829955 = 1/(-1/84.021212742844+1/84.0262133014925+2/132466.664728053+0.5/11.8619993833167-0.5/11.8627021700857-1/426825.022891826)

    68759.4680111832 = 1/(-1/164.770559417647+1/164.7814326883+1/100061.766522545-1/130704.452624679+2/132466.664728053-0.5/11.8619993833167+0.5/11.8627021700857)

  153. Paul Vaughan says:

    68759.9311041526 = 1/(-1/164.770559417647+1/164.7814326883+2/132466.664728053-0.5/11.8619993833167+0.5/11.8627021700857+1/426825.022891826)

    68756.608931217 = 1/(-1/84.021212742844+1/84.0262133014925-1/100061.766522545+1/130704.452624679+2/132466.664728053+0.5/11.8619993833167-0.5/11.8627021700857)

  154. Paul Vaughan says:

    harmonic mean (La2011 Table 6 La2004a, La2011 Table 6 La2010a)
    110478.869637491 = 8/(1/173901.37537739+1/74619.9907876555+1/72337.575351641+1/304406.35241565+1/183699.503897945+1/68756.9632341238+1/73001.7461837436+1/405629.613215262)
    110478.869637491 = 8/(2/173901.37537739+1/74619.9907876555+1/72337.575351641+1/183699.503897945+1/68756.9632341238+1/73001.7461837436)

    55239.4348187456 = 110478.869637491 / 2
    280975.609756098 = beat(68756.9632341238,55239.4348187456)
    11.8625001846852 = beat(280975.609756098,11.8619993833167)
    11.862499899747
    __________________________

    La2011 Table 6 La2004a
    110482.401457754 = 8/(1/173913.043478261+1/74619.9907876555+1/72337.575351641+1/304407.424910486+1/183829.787234043+1/68753.3156498674+1/72993.5229512813+1/405691.196375825)
    110482.401457754 = 8/(2/173913.043478261+1/74619.9907876555+1/72337.575351641+1/183829.787234043+1/68753.3156498674+1/72993.5229512813)

    55241.200728877 = 110482.401457754 / 2
    281082.253429486 = beat(68753.3156498674,55241.200728877)
    11.8624999946712 = beat(281082.253429486,11.8619993833167)
    11.862499899747 Standish (1992) Jupiter node

  155. Paul Vaughan says:

    2.4 Ma U-N
    171.446863617875 = beat(164.770559417647,84.021212742844)
    171.471519050756 = beat(164.793624044745,84.0331316671926)
    1192364.14480033 = beat(171.471519050756,171.446863617875)
    2384728.28960065 = 2 * 1192364.14480033
    2384110.34604552 = beat(74626.0277273697,72361.0252351259) ————— La2011 Table 5

  156. Paul Vaughan says:

    2.4 Ma U-N metronome – more

    1177263.23982246 = beat(164.793624044745,164.770559417647)
    592382.787802241 = beat(84.0331316671926,84.021212742844)
    2354526.47964493 = 2 * 1177263.23982246
    2369531.15120897 = harmean(2384728.28959931,2354526.47964493)
    2369531.15120897 = 4 * 592382.787802241
    1184765.57560448 = axial(2384728.28959931,2354526.47964493)
    1184765.57560448 = 2369531.15120897 / 2 = 2 * 592382.787802241

    thus ~conversions:

    164.770559417647
    164.770705450344 = axial(1184765.5756047,164.793624044745)
    84.021212742844 = axial(592382.787802352,84.0331316671926)
    84.021212742844

    164.793624044745
    164.793477971421 = beat(1184765.5756047,164.770559417647)
    84.0331316671926 = beat(592382.787802352,84.021212742844)
    84.0331316671926

    compare:
    2364963.50364963 = beat(74619.9907876555,72337.575351641)
    La2011 Table 6 La2004a & La2011 Table 6 La2010a

  157. Paul Vaughan says:

    2.4 Ma metronome part 3

    La2011 Table 6 La2004a , La2011 Table 6 La2010a & La2011 Table 5 all together

    2364963.50364963 = beat(74619.9907876555,72337.575351641)
    1182481.75182481 = beat(73001.7461837436,68756.9632341238)
    2384110.34604552 = beat(74626.0277273697,72361.0252351259)
    1173060.8742562 = beat(1192055.17302276,591240.875912407)

    164.793624044745
    164.793706682921 = beat(1173060.8742562,164.770559417647)
    84.0331546904585 = beat(591240.875912407,84.021212742844)
    84.0331316671926

    164.770559417647
    164.770476802685 = axial(1173060.8742562,164.793624044745)
    84.0211897261213 = axial(591240.875912407,84.0331316671926)
    84.021212742844

    with attention to orbital invariants
    small J offset attributable to different (& systematically biased) sidereal reference frame

    395218.586983767 = axial(1192055.17302276,591240.875912407)
    11.8619900818382 = axial(197609.293491884,11.8627021700857)
    11.861990807677 Standish (1992) sidereal

    11.8627028960116 = beat(197609.293491884,11.861990807677)
    11.8627021700857 Standish (1992) anomalistic

  158. Paul Vaughan says:

    inclination

    mercury

    110992.163747698 = harmean(231428.571428571,73001.7461837436)
    106640.335719575 = beat(231428.571428571,73001.7461837436)
    106016.605996155 = harmean(231428.571428571,68756.9632341238)
    97818.7032983621 = beat(231428.571428571,68756.9632341238)

    review graph panel (D)

    shaped by S, U, & N nodes & perihelion

    55490.5640640029 = beat(433078.965717205,49188.0779029847) ; * 2 = 110981.128128006
    53429.7226968636 = beat(426282.531723861,47478.7901510379) ; * 2 = 106859.445393727
    48696.1383511397 = beat(1899237.81359039,47478.7901510379) ; * 2 = 97392.2767022795
    47004.3265016268 = beat(1925648.56913613,45884.3085224084) ; * 2 = 94008.6530032537

    50011.488147239 = axial(106859.445393727,94008.6530032537)
    100022.976294478 = harmean(106859.445393727,94008.6530032537)

    fits J & S

    8.45663033585722 = axial(29.4571726091513,11.8619993833167)
    8.45806059760692 = axial(29.4701958106261,11.8627021700857)

    16.9132606717144 = harmean(29.4571726091513,11.8619993833167)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)

    50009.5117884787 = beat(8.45806059760692,8.45663033585722)
    100019.023576957 = beat(16.9161211952138,16.9132606717144)

  159. Paul Vaughan says:

    earth anomalistic year
    in terms of JSUN synodic-anomalistic beats & axial
    _________
    111743.763579682 = 1/(0.25/11.8627021700857-0.75/29.4701958106261+0.75/84.0331316671926-0.25/164.793624044745-0.25/11.8619993833167+0.75/29.4571726091513-0.75/84.021212742844+0.25/164.770559417647)

    1.00002638193018
    1.00002638315183 = 1/(-0.25/11.8627021700857+0.75/29.4701958106261-0.75/84.0331316671926+0.25/164.793624044745+0.25/11.8619993833167-0.75/29.4571726091513+0.75/84.021212742844-0.25/164.770559417647+1/1.00001743371442)
    _________
    111743.763579642 = 1/(0.25/11.8627021700857-0.75/29.4701958106261+0.75/84.0331316671926-0.25/164.793624044745-0.25/11.8619994099501+0.75/29.4571727733966-0.75/84.021214079097+0.25/164.770564556546)

    1.00002638193018
    1.00002638334112 = 1/(-0.25/11.8627021700857+0.75/29.4701958106261-0.75/84.0331316671926+0.25/164.793624044745+0.25/11.8619994099501-0.75/29.4571727733966+0.75/84.021214079097-0.25/164.770564556546+1/1.00001743390371)
    _________
    forensic diagnostics clarify reference frames used (and not used) by authors
    the following clarify what was not used:
    _________
    111743.763579604 = 1/(0.25/11.8627021700857-0.75/29.4701958106261+0.75/84.0331316671926-0.25/164.793624044745-0.25/11.8619967799723+0.75/29.4571565546135-0.75/84.0210821278182+0.25/164.770057105325)

    1.0000262476142
    1.00002638193018
    1.00002636464897 = 1/(-0.25/11.8627021700857+0.75/29.4701958106261-0.75/84.0331316671926+0.25/164.793624044745+0.25/11.8619967799723-0.75/29.4571565546135+0.75/84.0210821278182-0.25/164.770057105325+1/1.0000174152119)
    _________
    111743.763579645 = 1/(0.25/11.8627021700857-0.75/29.4701958106261+0.75/84.0331316671926-0.25/164.793624044745-0.25/11.861857752743+0.75/29.4562992042035-0.75/84.0141073585951+0.25/164.743236088451)

    1.0000262476142
    1.00002638193018
    1.00002537652411 = 1/(-0.25/11.8627021700857+0.75/29.4701958106261-0.75/84.0331316671926+0.25/164.793624044745+0.25/11.861857752743-0.75/29.4562992042035+0.75/84.0141073585951-0.25/164.743236088451+1/1.00001642710472)

  160. Paul Vaughan says:

    alternate perspective on first calculation in last comment:

    19.8589101021728 = beat(29.4571726091513,11.8619993833167)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    99925.8030607636 = beat(19.8589101021728,19.8549641949401)

    171.446863617875 = beat(164.770559417647,84.021212742844)
    171.471519050756 = beat(164.793624044745,84.0331316671926)
    1192364.14480033 = beat(171.471519050756,171.446863617875)

    109066.059069573 = beat(1192364.14480033,99925.8030607636)

    16.9132606717144 = harmean(29.4571726091513,11.8619993833167)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    100019.023576957 = beat(16.9161211952138,16.9132606717144)

    111.291640445866 = harmean(164.770559417647,84.021212742844)
    111.307357343015 = harmean(164.793624044745,84.0331316671926)
    788169.463380577 = beat(111.307357343015,111.291640445866)

    114556.259185123 = beat(788169.463380577,100019.023576957)

    111743.763579479 = harmean(114556.259185123,109066.059069573)
    1.00002638315183 = beat(111743.763579479,1.00001743371442)
    1.00002638193018
    111759.01908408 = beat(1.00002638193018,1.00001743371442)

    1.0000174324928 = axial(111743.763579479,1.00002638193018)
    1.00001743371442

  161. Paul Vaughan says:

    earth tropical year
    elaboration on 100.1ka (typo at link: “100.0ka”)

    quote from another comment above (with another typo corrected)
    :
    intermodel comparison diagnostic measures
    harmonic mean of g_2, g_3, g_4, g_5, s_2, s_3, s_4
    100743.899044297 — Berger 1988 Table 4 (based on Berger 1978) ———— note large difference
    100078.929091917 — La2011 Table 6 La2004a
    100076.152524661 — harmean(2004,2010)
    100073.376111466 — La2011 Table 6 La2010a
    :

    measure k

    anomalistic inputs from Standish (1992)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)

    tropical baseline from Seidelmann (1992)
    19.8588720868409 = beat(29.42351935,11.85652502)

    100897.671749148 = beat(19.8588720868409,19.8549641949401)
    _____
    I.
    100743.899044297 — Berger 1988 Table 4 (based on Berger 1978)
    50371.9495221485 = 100743.899044297 / 2
    100590.59433958 = beat(100897.671749148,50371.9495221485)

    16.9132769229027 = axial(100590.59433958,16.9161211952138); / 2 = 8.45663846145137
    11.8620005953416 = harmean(19.8588720868409,8.45663846145137)
    29.457263727112 = 2 * beat(19.8588720868409,8.45663846145137)

    25685.3568935629 = beat(11.8620005953416,11.85652502)
    25685.3568935608 = beat(29.457263727112,29.42351935)

    0.999978501184162 = axial(25685.3568935608,1.00001743371442)
    _____
    II.
    100076.152524661 — harmonic mean of La2011 Table 6 La2004a & La2011 Table 6 La2010a
    50038.0762623304 = 100076.152524661 / 2
    99267.9030446487 = beat(100897.671749148,50038.0762623304)

    16.9132390309646 = axial(99267.9030446487,16.9161211952138); / 2 = 8.4566195154823
    11.8619819569911 = harmean(19.8588720868409,8.4566195154823)
    29.4571487860026 = 2 * beat(19.8588720868409,8.4566195154823)

    25773.0455911765 = beat(11.8619819569911,11.85652502)
    25773.0455911756 = beat(29.4571487860026,29.42351935)
    0.999978633640778 = axial(25773.0455911756,1.00001743371442)
    _____

    0.999978501184162 = 365.242147557515 / 365.25 — I
    0.999978614647502 = 365.242189623 / 365.25 — Meeus & Savoie (1992)
    0.99997862 = 365.242190955 / 365.25 — Seidelmann (1992)
    0.999978633640778 = 365.242195937294 / 365.25 — II

    365.242147557515 = 0.999978501184162 * 365.25 — I
    365.242189623 = 0.999978614647502 * 365.25 — Meeus & Savoie (1992)
    365.242190955 = 0.99997862 * 365.25 — Seidelmann (1992)
    365.242195937294 = 0.999978633640778 * 365.25 — II

  162. Paul Vaughan says:

    alert: Meeus & Savoie (1992) typos near end of last comment (corrections below)
    _____
    with Standish (1992) sidereal earth (1.0000174152119) :
    _____
    III.
    0.999978482683075 = axial(25685.3568935608,1.0000174152119)
    _____
    IV.
    0.999978615139687 = axial(25773.0455911756,1.0000174152119)
    _____
    typo correction, clarification, expanded perspective

    0.999978482683075 = 365.242140799993 / 365.25 — III
    0.999978501184162 = 365.242147557515 / 365.25 — I
    0.999978614647502 = 365.242189 / 365.25 — Meeus & Savoie (1992) (with common rounding)
    0.999978615139687 = 365.242189179771 / 365.25 — IV – – – – – – – – – – – – – – – – – – – – – – – – –
    0.999978616353183 = 365.242189623 / 365.25 — Meeus & Savoie (1992)
    0.99997862 = 365.242190955 / 365.25 — Seidelmann (1992)
    0.999978633640778 = 365.242195937294 / 365.25 — II

    365.242140799993 = 0.999978482683075 * 365.25 — III
    365.242147557515 = 0.999978501184162 * 365.25 — I
    365.242189 = 0.999978614647502 * 365.25 — Meeus & Savoie (1992) (with common rounding)
    365.242189179771 = 0.999978615139687 * 365.25 — IV – – – – – – – – – – – – – – – – – – – – – – – – –
    365.242189623 = 0.999978616353183 * 365.25 — Meeus & Savoie (1992)
    365.242190955 = 0.99997862 * 365.25 — Seidelmann (1992)
    365.242195937294 = 0.999978633640778 * 365.25 — II
    _____

    subtle diagnostics (compare here & above there)
    distinction between ~131k vs. ~132k invariants from Seidelmann (1992) clarifying

  163. Paul Vaughan says:

    2.4 Ma U-N Metronome – more detail

    171.446863617875 = beat(164.770559417647,84.021212742844)
    171.471519050756 = beat(164.793624044745,84.0331316671926)
    1192364.14480033 = beat(171.471519050756,171.446863617875)

    111.291640445866 = harmean(164.770559417647,84.021212742844)
    111.307357343015 = harmean(164.793624044745,84.0331316671926)
    788169.463380577 = beat(111.307357343015,111.291640445866)

    394084.731690211 = axial(1177263.23982246,592382.787802241)
    592382.787802241 = beat(84.0331316671926,84.021212742844)
    788169.463380422 = harmean(1177263.23982246,592382.787802241)
    1177263.23982246 = beat(164.793624044745,164.770559417647)
    1192364.14479966 = beat(1177263.23982246,592382.787802241)
    _____
    exploring 1-size-fits-all:

    2369531.15120897 = 4 * 592382.787802241; 592068.36536811 = harmean / 4
    2364508.39014127 = 3 * 788169.463380422; 789424.48715748 = harmean / 3
    2354526.47964493 = 2 * 1177263.23982246; 1184136.73073622 = harmean / 2
    2384728.28960065 = 2 * 1192364.14480033; 1184136.73073622 = harmean / 2
    2368273.46147244 = harmean

    compare harmean(2004,2010) = harmonic mean of La2011 Table 6 La2004a & La2011 Table 6 La2010a
    2364963.50364963 = beat(74619.9907876555,72337.575351641); look 4 lines down (exactly equal)
    / 2 = 1182481.75182481
    / 3 = 788321.167883209
    / 4 = 591240.875912407
    1182481.75182481 = beat(73001.7461837436,68756.9632341238); * 2 = 2364963.50364963 (look 4 lines up)

    compare La2011 Table 5
    2384110.34604552 = beat(74626.0277273697,72361.0252351259)
    / 2 = 1192055.17302276
    / 3 = 794703.448681839
    / 4 = 596027.586511379

    84.0331379977269 = beat(592068.36536811,84.021212742844)
    84.0331316671926

    164.793490144188 = beat(1184136.73073622,164.770559417647)
    164.793624044745

    171.471690382177 = beat(164.793490144188,84.0331379977269) = beat(1184136.73073622,171.446863617875)
    171.471519050756
    _____
    beyond one-size-fits-all (above) explore (below) 500x/(g_2-g_5) fine-tuning where x=1,11/2

    500 ~= 499.998175691058 = 202814066.613901 / 405629.613215262
    202814066.613901 = beat(1184136.73073606,1177263.23982246)
    202814806.607631 = 500 * 405629.613215262

    2750 ~= 2749.98996630082 = 1115477366.37645 / 405629.613215262
    1115477366.37645 = beat(592382.787802241,592068.365368028)
    1115481436.34197 = 2750 * 405629.613215262

    11 / 2 = 2750 / 500
    = 2749.98996630082 / 499.998175691058
    = 1115477366.37645 / 202814066.613901
    = 1115481436.34197 / 202814806.607631

    1192364.1145724 = beat(1177263.26475563,592382.786654425)
    1177263.26475563 = axial(202814806.607631,1184136.73073606)
    788169.467952245 = harmean(1177263.26475563,592382.786654425)
    592382.786654425 = beat(1115481436.34197,592068.365368028)
    394084.733976122 = axial(1177263.26475563,592382.786654425)

    synodic-anomalistic conversions thus:

    84.0331316672157 = beat(592382.786654425,84.021212742844)
    84.0331316671926

    164.793624044256 = beat(1177263.26475563,164.770559417647)
    164.793624044745

    171.471519051381 = beat(164.793624044256,84.0331316672157) = beat(1192364.1145724,171.446863617875)
    171.471519050756

    orbital invariants clarified reference frames with links to Standish (1992)

  164. Paul Vaughan says:

    estimating earth tropical year length using
    La2011 Table 6 La2010a
    Seidelmann (1992) tropical
    Standish (1992) anomalistic

    100073.376111466 = harmean(173889.7088,74619.99079,72337.57535,304405.2799,183569.4051,68760.61121,73009.97127)
    = 7/(1/173889.708842077+1/74619.9907876555+1/72337.575351641+1/304405.279928371+1/183569.40509915+1/68760.6112054329+1/73009.9712692243)

    25773.4138919541 = 1/(-2/100073.376111466-0.5/11.85652502+1.5/29.42351935+0.5/11.8627021700857-1.5/29.4701958106261)
    = 1/(-2/7/173889.708842077-2/7/74619.9907876555-2/7/72337.575351641-2/7/304405.279928371-2/7/183569.40509915-2/7/68760.6112054329-2/7/73009.9712692243-0.5/11.85652502+1.5/29.42351935+0.5/11.8627021700857-1.5/29.4701958106261)

    0.999978615694116 = 1/(-2/100073.376111466-0.5/11.85652502+1.5/29.42351935+0.5/11.8627021700857-1.5/29.4701958106261+1/1.0000174152119)
    = 1/(-2/7/173889.708842077-2/7/74619.9907876555-2/7/72337.575351641-2/7/304405.279928371-2/7/183569.40509915-2/7/68760.6112054329-2/7/73009.9712692243-0.5/11.85652502+1.5/29.42351935+0.5/11.8627021700857-1.5/29.4701958106261+1/1.0000174152119)

    0.999978615694116 = 365.242189382276 / 365.25
    365.242189382276 = 0.999978615694116 * 365.25
    compare

    _____
    supplementary
    Berger 1988 Table 4 (based on Berger 1978)

    100743.899044297 = 7/(1/176420+1/75259+1/72576+1/308043+1/191404+1/68829+1/72732)

    25685.356893565 = 1/(-2/100743.899044297-0.5/11.85652502+1.5/29.42351935+0.5/11.8627021700857-1.5/29.4701958106261)

    _____
    not necessarily picking best but learning to recognize model combinations underlying narratives

  165. Paul Vaughan says:

    earth anomalistic year supplementary

    1.00002636464897 = beat(111743.763579479,1.0000174152119)
    1.00002638169098 = beat(111743.763579479,1.0000174322536)
    1.00002638193018
    1.00002638315183 = beat(111743.763579479,1.00001743371442)
    1.00002638334112 = beat(111743.763579479,1.00001743390371)

    365.259629688035 — Standish (1992) short-duration model
    365.259635912629 — Standish (1992) long-duration model
    365.259636
    365.259636446204 — Seidelmann (1992) short-duration model
    365.259636515344; 1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)

    Standish (1992) short-duration model
    113224.424017561 = beat(1.0000262476142,1.0000174152119)

    Standish (1992) long-duration model
    111360.765447049 = beat(1.00002641247066,1.0000174322536)

  166. Paul Vaughan says:

    significant typos in Table 2
    some values in table are inconsistent with figures below it
    typo exploration:
    36750.3190131859 = axial(74615.6946283609,72418.4175234689)
    73500.6380263718 = harmean(74615.6946283609,72418.4175234689)

  167. Paul Vaughan says:

    for comparison

    Zeebe 2017 Table 4:

    232170.688450583 = 1 / g_1 = 360*60*60/5.5821
    173822.073793908 = 1 / g_2 = 360*60*60/7.4559
    74613.546734218 = 1 / g_3 = 360*60*60/17.3695
    72327.8864184302 = 1 / g_4 = 360*60*60/17.9184
    304403.992953611 = 1 / g_5 = 360*60*60/4.2575
    45883.9023975755 = 1 / g_6 = 360*60*60/28.2452
    419716.302869357 = 1 / g_7 = 360*60*60/3.0878
    1923990.49881235 = 1 / g_8 = 360*60*60/0.6736
    3709215.79851173 = 1 / g_9 = 360*60*60/-0.3494

    230826.773055961 = 1 / s_1 = 360*60*60/-5.6146
    183494.032196407 = 1 / s_2 = 360*60*60/-7.0629
    68762.0705023451 = 1 / s_3 = 360*60*60/-18.8476
    73017.3754310053 = 1 / s_4 = 360*60*60/-17.7492

    49188.1675130371 = 1 / s_6 = 360*60*60/-26.3478
    433068.234979616 = 1 / s_7 = 360*60*60/-2.9926
    1872561.76853056 = 1 / s_8 = 360*60*60/-0.6921
    3691256.05240672 = 1 / s_9 = 360*60*60/-0.3511

    94869.298508883 = beat(304403.992953611,72327.8864184302)
    98840.7565588774 = beat(304403.992953611,74613.546734218)
    123870.967741935 = beat(173822.073793908,72327.8864184302)
    130729.5029051 = beat(173822.073793908,74613.546734218)

    405202.60130065 = beat(304403.992953611,173822.073793908)
    2361085.80797959 = beat(74613.546734218,72327.8864184302)

    97936.975742462 = beat(230826.773055961,68762.0705023451)
    106802.037149968 = beat(230826.773055961,73017.3754310053)
    109973.100715334 = beat(183494.032196407,68762.0705023451)
    121276.774936133 = beat(183494.032196407,73017.3754310053)
    172795.392122877 = beat(68762.0705023451,49188.1675130371)

    1179898.03350328 = beat(73017.3754310053,68762.0705023451)

  168. Paul Vaughan says:

    significant discrepancies:
    do careful comparison and find several figures inconsistent with tabulated values

    also — more benign, easy to spot:
    number of table values off by power of 10 — some *10, some /10

    searched for errata publication (not found) in top search results for article title

    suggestion:
    compare and contrast table 2 from that article with table 4 from a different article

  169. Paul Vaughan says:

    2.4 Ma UN concisely

    171.446863617875 = beat(164.770559417647,84.021212742844)
    171.471519050756 = beat(164.793624044745,84.0331316671926)
    111.291640445866 = harmean(164.770559417647,84.021212742844)
    111.307357343015 = harmean(164.793624044745,84.0331316671926)

    592382.787802241 = beat(84.0331316671926,84.021212742844)
    788169.463380422 = harmean(1177263.23982246,592382.787802241)
    788169.463380577 = beat(111.307357343015,111.291640445866)
    1177263.23982246 = beat(164.793624044745,164.770559417647)
    1192364.14479966 = beat(1177263.23982246,592382.787802241)
    1192364.14480033 = beat(171.471519050756,171.446863617875)

    1-size-fits-all model
    2368273.46147211 = harmean
    2369531.15120897 = 4 * 592382.787802241; 592068.365368028 = 2368273.46147211 / 4
    2364508.39014127 = 3 * 788169.463380422; 789424.48715737 = 2368273.46147211 / 3
    2354526.47964493 = 2 * 1177263.23982246; 1184136.73073606 = 2368273.46147211 / 2
    2384728.28959931 = 2 * 1192364.14479966; 1184136.73073606 = 2368273.46147211 / 2

    anomalistic estimates (from synodic) with 1-size-fits-all model
    84.0331379977269 = beat(592068.365368028,84.021212742844)
    164.793490144188 = beat(1184136.73073606,164.770559417647)
    171.471690382177 = beat(1184136.73073606,171.446863617875)
    111.307332352834 = beat(789424.48715737,111.291640445866)

    _____

    100 ka JS analogously

    19.8589101021728 = beat(29.4571726091513,11.8619993833167)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    16.9132606717144 = harmean(29.4571726091513,11.8619993833167)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)

    66658.6205013893 = beat(29.4701958106261,29.4571726091513)
    99925.8030609183 = beat(200224.836371855,66658.6205013893)
    99925.8030607636 = beat(19.8589101021728,19.8549641949401)
    100019.023577046 = harmean(200224.836371855,66658.6205013893)
    100019.023576957 = beat(16.9161211952138,16.9132606717144)
    200224.836371855 = beat(11.8627021700857,11.8619993833167)

    1-size-fits-all model
    200022.497116091 = harmean
    200224.836371855 = 1 * 200224.836371855; 200022.497116091 = 200022.497116091 / 1
    200038.047154092 = 2 * 100019.023577046; 100011.248558045 = 200022.497116091 / 2
    199975.861504168 = 3 * 66658.6205013893; 66674.1657053636 = 200022.497116091 / 3
    199851.606121837 = 2 * 99925.8030609183; 100011.248558045 = 200022.497116091 / 2

    anomalistic estimates (from synodic) with 1-size-fits-all model
    11.8627028810547 = beat(200022.497116091,11.8619993833167)
    29.4701927729007 = beat(66674.1657053636,29.4571726091513)
    19.8549675654917 = axial(100011.248558045,19.8589101021728)
    16.9161214176327 = beat(100011.248558045,16.9132606717144)

    ____
    orbital solutions piece together preceding

  170. Paul Vaughan says:

    amicable structure

    200022.497116091 (JS)
    2368273.46147211 (UN)
    2368266.36585452 = 11.84 * 200022.497116091

    2392283.52682109 = 4 * 598070.881705272 (JU analogy to JS & UN above)
    2396408.70582883 = 4 * 599102.176457208 (SU analogy)

    2394344.33952442 = harmean(2396408.70582883,2392283.52682109)
    2394 = 1210 + 1184

    supplementary
    126173.107872115 = harmean(1177263.23982246,66658.6205013893)
    342242.293975111 = harmean(1177263.23982246,200224.836371855)

    252 = average(220,284) = 2 * 126
    342 = Σδ(220)

  171. Paul Vaughan says:

    104429.377596496 = beat(2368273.46147211,100019.023577046)
    104429.806007131 = harmean(173901.37537739,74619.9907876555)

    100112.418185928 = 200224.836371855 / 2
    104531.194491385 = beat(2368273.46147211,100112.418185928)
    104480.261238591 = harmean(104531.194491385,104429.377596496)
    104482.425024186 = harmean(183699.503897945,73001.7461837436)

    enough to solve

    68760.5409705648 = axial(1184000,73000)

  172. Paul Vaughan says:

    quick note (more details at another time)

    orbital solutions sorting & classification:
    group 1: Berger, La2021
    group 2: La2004, La2010, Zebe 2017
    initial diagnostics reveal fundamentally contrasting architecture

  173. Paul Vaughan says:

    carefully (subtle enough?) compare 2020 models with 2021 model

    fascinating: these models (and narratives based on them) are not consistent with one another

    supplementary: 200ka

  174. Paul Vaughan says:

    amicable 304 ka heuristic

    initial conditions guidance
    173849.206949429 = 2 * beat(130762.093818127,104429.377596496/2)
    74629.1626477617 = harmean(130762.093818127,104429.377596496/2)
    72349.0326948921 = axial(2368000,74629.1626477617)
    183697.183506996 = beat(73000,104480.261238591/2)
    68760.5409705648 = axial(1184000,73000)

    304000.007496581 = 2/7/(-2/7/173849.206949429-2/7/74629.1626477617-2/7/72349.0326948921-2/7/183697.183506996-2/7/68760.5409705648-2/7/73000-1/2/11.85652502+3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261-1/G6091)

    304000.007494335 = 1/(-1/173849.206949429-1/74629.1626477617-1/72349.0326948921-1/183697.183506996-1/68760.5409705648-1/73000-7/4/11.85652502+21/4/29.42351935+7/4/11.8627021700857-21/4/29.4701958106261-7/2/25773.41)

  175. Paul Vaughan says:

    60072 precisely from
    Mayan serpent series
    lunisolar precession
    100ka differentiation

    25771.4533429313 = 360*60*60/50.2882
    25721.8900031954 = 360*60*60/50.3851

    13374613.0030966 = beat(25771.4533429313,25721.8900031954)
    25672.5169367299 = axial(13374613.0030966,25721.8900031954)
    15009.1608487337 = 5482096 / 365.25
    36135.2404360745 = beat(25672.5169367299,15009.1608487337)
    23095.3457556756 = axial(64000.2003306117,36135.2404360745)

    30030 = 13*11*7*5*3*2 = 13#
    30030.1274606336 = beat(100011.248558045,23095.3457556756)

    30030:
    46197.4595666555 = harmean(100076.152524661,30030)
    60071.6981458392 = beat(200022.497116091,46197.4595666555)

    30030.1274606336:
    46197.6103909058 = harmean(100076.152524661,30030.1274606336)
    60071.9531664205 = beat(200022.497116091,46197.6103909058)

  176. Paul Vaughan says:

    27.5 Ma fits amicably

    23.6800407305146 = 1/(3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261)
    23.6800709481954 = 2368273.46147211 / 100011.248558045

    13767156.3895453 = beat(23.6800407305146,23.68) = beat(18556852.4806421,7903570.7243091)
    7903570.7243091 = beat(23.6800709481954,23.68)
    18556852.4806421 = beat(23.6800709481954,23.6800407305146)

    27534312.7790905 = 2 * 13767156.3895453
    27.5 million years

  177. Paul Vaughan says:

    Orbital Solutions Sorting & Classification

    25772.7186475618 = beat(1.0000174152119,0.999978614647502)
    25773.8517155112 = beat(1.0000174152119,0.999978616353183)

    Zeebe 2017
    100065.077596762 = harmean(173822.073793908,74613.546734218,72327.8864184302,304403.992953611,183494.032196407,68762.0705023451,73017.3754310053)
    25774.5149030924 = 1/(-2/100065.077596762-1/2/11.85652502+3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261)

    La2011 Table 6 La2010a
    100073.376111466 = harmean(173889.708842077,74619.9907876555,72337.575351641,304405.279928371,183569.40509915,68760.6112054329,73009.9712692243)
    25773.4138919219 = 1/(-2/100073.376111466-1/2/11.85652502+3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261)

    harmean(2004,2010) = harmonic mean of La2011 Table 6 La2004a & La2011 Table 6 La2010a
    (recall ~2.4 Ma metronome property for this combo: 2/(s_4-s_3) = 1/(g_4-g_3) exactly equal)
    100076.152524661 = harmean(173901.37537739,74619.9907876555,72337.575351641,304406.35241565,183699.503897945,68756.9632341238,73001.7461837436)
    25773.0455911473 = 1/(-2/100076.152524661-1/2/11.85652502+3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261)

    La2011 Table 6 La2004a
    100078.929091917 = harmean(173913.043478261,74619.9907876555,72337.575351641,304407.424910486,183829.787234043,68753.3156498674,72993.5229512813)
    25772.6773008985 = 1/(-2/100078.929091917-1/2/11.85652502+3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261)

    _____
    architecturally different group of orbital solutions

    La2021
    100732.896221555 = harmean(174006.444683136,75045.60062538,72418.4175234689,304407.281910738,193173.349232374,69035.3166782081,73191.3932343141)
    25686.7875634114 = 1/(-2/100732.896221555-1/2/11.85652502+3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261)

    Berger 1988 Table 4 (based on Berger 1978)
    100743.899044297 = harmean(176420,75259,72576,308043,191404,68829,72732)
    25685.3568935329 = 1/(-2/100743.899044297-1/2/11.85652502+3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261)

  178. Paul Vaughan says:

    typo exploration” above (i.e. change table 2 g_3 rate 17.269 to 17.369 (closer to estimate of other authors)) corresponds with libration state: 3rd panel top row figure 7 — a detail washed out by the tabulated long-run convergence —- study the figures in this paper very carefully, comparing each distribution mode with values tabulated by these authors & others

  179. Paul Vaughan says:

    author comments on increasing skewness over time
    growing skewness reflects variable wait-time until state-jump
    see figure 9
    compare with figure 7 (link in last comment)
    mean converges but (decreasing number of) individual cases still locked in (or recently drifting from) less-dominant libration state, as modeled marginally (not jointly) and statistically computed (MH MCMC)

    authors clearly state they ignored joint distributions
    table 2 may be a useful indicator of extreme marginal averages
    thus practical to consider combinatorically all possible crosses of marginal modes fully realizing observed state is (curiously) marginally disfavored by long-run MH MCMC configured to ignore joint distribution

    ~25685 & ~25770 are at the extremes (governed by assumptions)
    precession rate’s variable including systematic variation and sampling (sampling bias correctable with sufficient attention & care)

    fig. 13
    g_4 estimate fine but g_3 way off (curious, arouses suspicion)

    section 6.2 & fig. 12
    highly curious errors — very suspicious —- scrutinize every detail (start with the location of the vertical lines)

  180. Paul Vaughan says:

    La2021

    225039.069282862 = 1 / g_1 = 360*60*60/5.759
    174006.444683136 = 1 / g_2 = 360*60*60/7.448
    75047.773466906 = 1 / g_3 = 360*60*60/17.269; [74615.6946283609 = 360*60*60/17.369]
    72418.4175234689 = 1 / g_4 = 360*60*60/17.896
    304407.281910738 = 1 / g_5 = 360*60*60/4.257454
    45883.8536630787 = 1 / g_6 = 360*60*60/28.24523
    419694.963368985 = 1 / g_7 = 360*60*60/3.087957
    1925637.98273374 = 1 / g_8 = 360*60*60/0.6730237

    229299.363057325 = 1 / s_1 = 360*60*60/-5.652
    193173.349232374 = 1 / s_2 = 360*60*60/-6.709
    69035.3166782081 = 1 / s_3 = 360*60*60/-18.773
    73191.3932343141 = 1 / s_4 = 360*60*60/-17.707

    49188.0368318198 = 1 / s_6 = 360*60*60/-26.34787
    433078.799289029 = 1 / s_7 = 360*60*60/-2.992527
    1873545.50850714 = 1 / s_8 = 360*60*60/-0.6917366

    95024.7922322511 = beat(304407.281910738,72418.4175234689)
    99600.0167844473 = beat(304407.281910738,75045.60062538)
    124042.879019908 = beat(174006.444683136,72418.4175234689)
    131955.403960698 = beat(174006.444683136,75045.60062538)

    406200.0673239 = beat(304407.281910738,174006.444683136)
    2068635.27533919 = beat(75045.60062538,72418.4175234689)
    [—
    2459203.03605313 = beat(74615.6946283609,72418.4175234689) —— [typo]
    36750.3190131859 = axial(74615.6946283609,72418.4175234689) —– [exploration]
    73500.6380263718 = harmean(74615.6946283609,72418.4175234689) — [aside]
    —]
    98772.9593780962 = beat(229299.363057325,69035.3166782081)
    107507.258399005 = beat(229299.363057325,73191.3932343141)
    107427.055702918 = beat(193173.349232374,69035.3166782081)
    117839.607201309 = beat(193173.349232374,73191.3932343141)
    171092.045144009 = beat(69035.3166782081,49188.0368318198)

    1215759.84990619 = beat(73191.3932343141,69035.3166782081)

  181. Paul Vaughan says:

    La(2004a,2010a)average (Table 1 in La2020)

    231842.576028623 = 1 / g_1 = 360*60*60/5.59
    173901.37537739 = 1 / g_2 = 360*60*60/7.4525
    74619.9907876555 = 1 / g_3 = 360*60*60/17.368
    72337.575351641 = 1 / g_4 = 360*60*60/17.916
    304406.35241565 = 1 / g_5 = 360*60*60/4.257467
    45884.3085224084 = 1 / g_6 = 360*60*60/28.24495
    419696.118636694 = 1 / g_7 = 360*60*60/3.0879485
    1925648.56913613 = 1 / g_8 = 360*60*60/0.67302
    3702804.24565363 = 1 / g_9 = 360*60*60/-0.350005

    231428.571428571 = 1 / s_1 = 360*60*60/-5.6
    183699.503897945 = 1 / s_2 = 360*60*60/-7.055
    68756.9632341238 = 1 / s_3 = 360*60*60/-18.849
    73001.7461837436 = 1 / s_4 = 360*60*60/-17.753

    49188.0779029847 = 1 / s_6 = 360*60*60/-26.347848
    433078.965717205 = 1 / s_7 = 360*60*60/-2.99252585
    1873541.71666151 = 1 / s_8 = 360*60*60/-0.691738
    3702962.94179834 = 1 / s_9 = 360*60*60/-0.34999

    94885.7391932208 = beat(304406.35241565,72337.575351641)
    98851.8163220366 = beat(304406.35241565,74619.9907876555)
    123859.129354423 = beat(173901.37537739,72337.575351641)
    130704.452624679 = beat(173901.37537739,74619.9907876555)

    405629.613215262 = beat(304406.35241565,173901.37537739)
    2364963.50364963 = beat(74619.9907876555,72337.575351641) ———————- “2.4 Ma”
    2364963.50364964 = 2 * beat(73001.7461837436,68756.9632341238) —————– “2.4 Ma”

    97818.7032983621 = beat(231428.571428571,68756.9632341238)
    106640.335719575 = beat(231428.571428571,73001.7461837436)
    109886.382906563 = beat(183699.503897945,68756.9632341238)
    121144.139091419 = beat(183699.503897945,73001.7461837436)
    172826.54615749 = beat(68756.9632341238,49188.0779029847)

    1182481.75182482 = beat(73001.7461837436,68756.9632341238) ——————– “1.2 Ma”
    1182481.75182481 = beat(74619.9907876555,72337.575351641) / 2 —————– “1.2 Ma”

  182. Paul Vaughan says:

    20212020 = 1 suggestion:
    Srutinize curiously suspicious La2021 errors — e.g. incorrectly-placed lines on graphs and concomitant misleading claims in the text.

  183. Paul Vaughan says:

    Period (a) – Frequency (”/a) Conversions

    _____
    Zeebe 2017

    94869.298508883 = beat(304403.992953611,72327.8864184302) = 360*60*60 / 13.6609
    98840.7565588774 = beat(304403.992953611,74613.546734218) = 360*60*60 / 13.112
    123870.967741935 = beat(173822.073793908,72327.8864184302) = 360*60*60 / 10.4625
    130729.5029051 = beat(173822.073793908,74613.546734218) = 360*60*60 / 9.9136

    405202.60130065 = beat(304403.992953611,173822.073793908) = 360*60*60 / 3.1984
    2361085.80797959 = beat(74613.546734218,72327.8864184302) = 360*60*60 / 0.5489
    1179898.03350328 = beat(73017.3754310053,68762.0705023451) = 360*60*60 / 1.0984

    97936.975742462 = beat(230826.773055961,68762.0705023451) = 360*60*60 / 13.233
    106802.037149968 = beat(230826.773055961,73017.3754310053) = 360*60*60 / 12.1346
    109973.100715334 = beat(183494.032196407,68762.0705023451) = 360*60*60 / 11.7847
    121276.774936133 = beat(183494.032196407,73017.3754310053) = 360*60*60 / 10.6863
    172795.392122877 = beat(68762.0705023451,49188.1675130371) = 360*60*60 / 7.5002

    _____
    La(2004a,2010a)average
    94885.7391932208 = beat(304406.35241565,72337.575351641) = 360*60*60 / 13.658533
    98851.8163220366 = beat(304406.35241565,74619.9907876555) = 360*60*60 / 13.110533
    123859.129354423 = beat(173901.37537739,72337.575351641) = 360*60*60 / 10.4635
    130704.452624679 = beat(173901.37537739,74619.9907876555) = 360*60*60 / 9.9155

    405629.613215262 = beat(304406.35241565,173901.37537739) = 360*60*60 / 3.195033
    2364963.50364963 = beat(74619.9907876555,72337.575351641) = 360*60*60 / 0.5480
    1182481.75182482 = beat(73001.7461837436,68756.9632341238) = 360*60*60 / 1.096

    97818.7032983621 = beat(231428.571428571,68756.9632341238) = 360*60*60 / 13.249
    106640.335719575 = beat(231428.571428571,73001.7461837436) = 360*60*60 / 12.153
    109886.382906563 = beat(183699.503897945,68756.9632341238) = 360*60*60 / 11.794
    121144.139091419 = beat(183699.503897945,73001.7461837436) = 360*60*60 / 10.698
    172826.54615749 = beat(68756.9632341238,49188.0779029847) = 360*60*60 / 7.498848

    _____
    La2021

    95024.7922322511 = beat(304407.281910738,72418.4175234689) = 360*60*60 / 13.638546
    99600.0167844473 = beat(304407.281910738,75045.60062538) = 360*60*60 / 13.012046
    124042.879019908 = beat(174006.444683136,72418.4175234689) = 360*60*60 / 10.448
    131955.403960698 = beat(174006.444683136,75045.60062538) = 360*60*60 / 9.8215

    406200.0673239 = beat(304407.281910738,174006.444683136) = 360*60*60 / 3.190546
    2068635.27533919 = beat(75045.60062538,72418.4175234689) = 360*60*60 / 0.6265
    1215759.84990619 = beat(73191.3932343141,69035.3166782081) = 360*60*60 / 1.066

    98772.9593780962 = beat(229299.363057325,69035.3166782081) = 360*60*60 / 13.121
    107507.258399005 = beat(229299.363057325,73191.3932343141) = 360*60*60 / 12.055
    107427.055702918 = beat(193173.349232374,69035.3166782081) = 360*60*60 / 12.064
    117839.607201309 = beat(193173.349232374,73191.3932343141) = 360*60*60 / 10.998
    171092.045144009 = beat(69035.3166782081,49188.0368318198) = 360*60*60 / 7.57487

    _____
    La2011 Table 5 (has only perihelia — no nodes)

    94926.7650262257 = beat(304399.417131486,72361.0252351259) = 360*60*60 / 13.65263
    98863.1425160258 = beat(304399.417131486,74626.0277273697) = 360*60*60 / 13.109031
    123977.271216256 = beat(173804.240903943,72361.0252351259) = 360*60*60 / 10.453529
    130777.916695678 = beat(173804.240903943,74626.0277273697) = 360*60*60 / 9.90993

    405113.811661464 = beat(304399.417131486,173804.240903943) = 360*60*60 / 3.199101
    2384110.34604552 = beat(74626.0277273697,72361.0252351259) = 360*60*60 / 0.543599

  184. Paul Vaughan says:

    note 171 ≠ 173

    2.4 Ma (an important one) : tediously scrutinize every detail of column 1 row 3 graph ( g_4 – g_3 ) ; then read the claims about fig. 13 (in section 6.2)

    to ease comparisons

    probably enough said (0.6265 ≠ 0.5480) on this

  185. Paul Vaughan says:

    from Hinnov 2013
    general (~25770) = planetary + lunisolar (~25720) precession
    25685 ~= 360*60*60 / 50.4576

    Contribution	Precession rate (''/a)	cumulative	precession (a)	cumulative (a)	(factor*	sun)
    Sun first order	15.94887	15.94887	81259.67545	81259.67545	1	81259.67545
    Moon first order	34.457698	50.406568	37611.33434	25710.93513	2	162519.3509
    Second order	-0.000468	50.4061	-2769230769	25711.17385	4	325038.7018
    J_4	0.000026	50.406126	49846153846	25711.16058	8	650077.4036
    Tilt effects	-0.002686	50.40344	-482501861.5	25712.53073	16	1300154.807
    Direct planetary	0.000318	50.403758	4075471698	25712.36851	32	2600309.614
    Geodesic precession	-0.019194	50.384564	-67521100.34	25722.16364	64	5200619.229
    -	-	-	-	-	128	10401238.46
    Total space motion	50.384565	50.384565	25722.16313	25722.16313	256	20802476.92
    Ecliptic motion	-0.096865	50.2877	-13379445.62	25771.70958	512	41604953.83
    General motion	50.2877	-	25771.70958	-	1024	83209907.66
    ^ "RATE K, GIVEN H= 0.0032737634 AND E 23.43926° AT J2000 (FROM WILLIAMS, 1994)" https://d3i71xaburhd42.cloudfront.net/ae8889e07e04bebace9efc958147817a5a361139/7-Table2-1.png						
    v "*Assuming a mean precession rate k = 50.4576 arcsec/yr and ellipticity H = 0.00328005 [...]" https://d3i71xaburhd42.cloudfront.net/ae8889e07e04bebace9efc958147817a5a361139/7-Table1-1.png						
    mean prec. rate	50.4576	-	25684.93151	~= 25685	https://www.semanticscholar.org/paper/Cyclostratigraphy-and-its-revolutionizing-in-the-Hinnov/ae8889e07e04bebace9efc958147817a5a361139	
    
  186. Paul Vaughan says:

    2310 = 11*7*5*3*2 = 11#

    45884.4 ~= 1 / g_6 ~= harmean(2311/(Ja-Sa),2310.5/(Jy-Sy)) —– +1 & +1/2 from primorial 11#

    19.8549641949401 = beat(29.4701958106261,11.8627021700857) — anomalistic Standish (1992)
    19.8589101021728 = beat(29.4571726091513,11.8619993833167) — synodic Seidelmann (1992)

    45884.8222545066 = 2311 * 19.8549641949401
    45884.4170192096 = harmean(45884.8222545066,45884.0117910702) —————————–
    45884.0117910702 = 2310.5 * 19.8589101021728

    45884.3897482377 = 1 / g_6 = 360*60*60/28.2449 — La2011 Table 6 La2010a
    45884.3085224084 = 1 / g_6 = 360*60*60/28.24495 — La(2004a,2010a)average
    45884.2272968667 = 1 / g_6 = 360*60*60/28.245 — La2011 Table 6 La2004a
    45883.9023975755 = 1 / g_6 = 360*60*60/28.2452 — Zeebe 2017 Table 4
    45883.8536630787 = 1 / g_6 = 360*60*60/28.24523 — La2021 Table 2

  187. Paul Vaughan says:

    with diversified perspective
    36750 = 73500 / 2 review

    8 / (g_2 + 5*(g_3 + g_4) + g_6 + s_2 + s_3 + s_4 + s_6)
    harmonic mean

    36746.6589355023 — Zeebe 2017 Table 4
    36749.7014379182 — La2011 Table 6 La2010a
    36750.0196670027 — La(2004a,2010a)average
    36750.3190131859 = 1/(g_3+g_4) — La2021 Table 2 with exploratory adjustment
    36750.3379015986 — La2011 Table 6 La2004a

    36889.5636478399 — La2021 Table 2
    37209.6176613942 — Berger 1988 Table 4 (based on Berger 1978)

    supplementary

    36746.6589355023 = harmean(173822.073793908,183494.032196407,74613.546734218/5,68762.0705023451/5,72327.8864184302,73017.3754310053,45883.9023975755,49188.1675130371)

    36749.7014379182 = harmean(173889.708842077,183569.40509915,74619.9907876555/5,68760.6112054329/5,72337.575351641,73009.9712692243,45884.3897482377,49188.090971097)

    36750.0196670027 = harmean(173901.37537739,183699.503897945,74619.9907876555/5,68756.9632341238/5,72337.575351641,73001.7461837436,45884.3085224084,49188.0779029847)

    36750.3379015986 = harmean(173913.043478261,183829.787234043,74619.9907876555/5,68753.3156498674/5,72337.575351641,72993.5229512813,45884.2272968667,49188.0648348793)

    36889.5636478399 = harmean(174006.444683136,193173.349232374,75047.773466906/5,69035.3166782081/5,72418.4175234689,73191.3932343141,45883.8536630787,49188.0368318198)

    37209.6176613942 = harmean(176420,191404,75259/5,68829/5,72576,72732,49434,49339)

  188. Paul Vaughan says:

    1536 obliquely bonds with 1470, 6000, etc.

    Capitaine 2003 (Table 5) precession in obliquity alongside Standish long-duration model anomalistic orbital invariants (936 & 4270) to realign perspective

    1535.88262671198 = 360*60*60/843.81448
    1535.88333657714 = 360*60*60/843.81409
    1535.88339118218 = 360*60*60/843.81406

    1200.30319518678 = beat(4270.51884168654,936.955612197409)
    1198.77624475105 = beat(4270,936)

    1536.74746987137 = harmean(4270.51884168654,936.955612197409)
    1535.42835190165 = harmean(4270,936)

    883.178329571106 = harmean(936,836)

    3001.42588931156 = harmean(64000.2,1536.74746987137)
    2999.77634751171 = harmean(64000.2,1535.88262671198)
    2999.77770147709 = harmean(64000.2,1535.88333657714)
    2999.77780562832 = harmean(64000.2,1535.88339118218)
    2998.90987783669 = harmean(64000.2,1535.42835190165)

    65008.4580351334 = beat(1535.42835190165,1500); 8126.05725439168, 16252.1145087834, 32504.2290175667, 65008.4580351334, 130016.916070267, 260033.832140534, 520067.664281067, 1040135.32856213, 2080270.65712427, 4160541.31424854, 8321082.62849708 — compare “(factor* sun)

    2400.60639037357 = beat(1536.74746987137,936.955612197409)
    2402.7198838921 = beat(1535.88262671198,936.955612197409)
    2402.71814662908 = beat(1535.88333657714,936.955612197409)
    2402.71801299356 = beat(1535.88339118218,936.955612197409)
    2403.83248224843 = beat(1535.42835190165,936.955612197409)

    2400.60639037357 = beat(4270.51884168654,1536.74746987137)
    2398.49661175262 = beat(4270.51884168654,1535.88262671198)
    2398.49834291632 = beat(4270.51884168654,1535.88333657714)
    2398.49847608287 = beat(4270.51884168654,1535.88339118218)
    2397.38894612274 = beat(4270.51884168654,1535.42835190165)

    68756.9632341238 = 1 / s_3 — La(average(2004a,2010a))
    34378.4816170619 = 68756.9632341238 / 2

    1470.99283468751 = axial(34378.4816170619,1536.74746987137)
    1470.20039920479 = axial(34378.4816170619,1535.88262671198)
    1470.20104965322 = axial(34378.4816170619,1535.88333657714)
    1470.20109968773 = axial(34378.4816170619,1535.88339118218)
    1469.78414257271 = axial(34378.4816170619,1535.42835190165)

    only one (“It looks like there aren’t many great matches for your search”) search result (from google) for
    “1536 years” obliquity

  189. Paul Vaughan says:

    weatheringssteadysstudiesof(oblique’solution’symmetry)

    130704.452624679 = beat(173901.37537739,74619.9907876555)
    433078.965717205 = 1 / s_7
    100402.649875387 = axial(433078.965717205,130704.452624679)
    50201.3249376937 = 100402.649875387 / 2
    100731.284587288 = beat(100076.152524661,50201.3249376937)

    25722.6032216452 = 1/(-2/100731.284587288-1/2/11.85652502+100011.248558045/2368273.46147211)

    68760.5409705648 = axial(1184000,73000)
    41096.3025926102 = beat(68760.5409705648,25722.6032216452)
    20548.1512963051 = 41096.3025926102 / 2
    11423.0240729734 = axial(25722.6032216452,20548.1512963051)

    1535.88296112698 = 3 * beat(11423.0240729734,490)
    843.814296272316 = 360*60*60 / 1535.88296112698

    843.81448 — L77
    843.81448 — IAU2000
    843.81409 — W94
    843.81406 — P03

    843.8142775 = average
    1535.88299529573 = 360*60*60/843.8142775

    Slower methods do no better than symplectic integrators.
    Results of both depend on underlying ephemerides.
    Therefore problem reduces to comparative ephemerides, with combinatoric attention to assumptions.

    Conjecture: All “orbital solutions” map exactly to other orbital solutions with sufficient comparative attention to underlying epheride assumptions (parameters in the translation).

    Slight change in assumptions (doesn’t necessarily but) can lead to different convergence either side of a symmetry point in a landscape of nonlinear assumptions.

    Recommending: tree level attention in the forest of matrices & computing for generators of “120000 solutions”.

    1353.85090672047 = axial(11423.0240729734,1535.88296112698) — “1350”
    980 = harmean(1353.85090672047,767.941480563488) ————— 1536, 768

  190. Paul Vaughan says:

    UK SST ink sh!…

    25722.6032216452 = 1/(-2/100731.284587288-1/2/11.85652502+100011.248558045/2368273.46147211)

    25770.753387539 = 1/(-2/100731.284587288-1/2/11.85652502+100011.248558045/2368273.46147211+3/2/29.42351935+1/2/11.8627021700857-3/2/29.4701958106261-1/23.68)

    13767156.3909689 = beat(25770.753387539,25722.6032216452)
    27534312.7819379 = 2 * 13767156.3909689
    27.5 million ~= 2 * beat(25770.753387539,25722.6032216452)

  191. Paul Vaughan says:

    25770.0359146014 = 360*60*60/50.290966

    25720.5198783896 = 360*60*60/50.387784 — IAU1976 (L77)
    25722.0354624594 = 360*60*60/50.38481507 — P03
    25722.0495372735 = 360*60*60/50.3847875 — P03pre1
    25722.0495372735 = 360*60*60/50.3847875 — B03
    25722.0495372735 = 360*60*60/50.3847875 — MHB
    25722.0495372735 = 360*60*60/50.3847875 — IAU2000
    25722.0526360831 = 360*60*60/50.38478143 — F03
    25722.1631216381 = 360*60*60/50.38456501 — W94

    23739.3950755189 = axial(49188.0779029847,45884.3085224084)
    25747.3224920285 = beat(304406.35241565,23739.3950755189) ———————
    25746.6557925767 = harmean(25770.753387539,25722.6032216452) —————
    27534312.7819354 = beat(25746.6557925767,25722.6032216452)
    27534312.7819403 = beat(25770.753387539,25746.6557925767)

    27.5 = average( -(27^2) , 28^2 ) = average( 1^2 + 3^2 + 5^2 , 2^2 + 4^2 )
    55 = 28^2 – 27^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2

  192. Paul Vaughan says:

    007won weather heuristic or mnemonic with SSTand!sh anomalistic

    memorable precession
    constantly rating obliquity

    1473.04563141139 = beat(4627,1800) / 2
    1504.22242843624 = beat(71071.71,1473.04563141139)
    1536.74746553612 = beat(71071.71,1504.22242843624)
    1536.74746987137 = harmean(4270.51884168654,936.955612197409)

    review (table 2a)
    936.955612197409 = 1/(-2/11.8627021700857+5/29.4701958106261)
    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)

  193. Paul Vaughan says:

    quote from above:
    architecturally different group of orbital solutions”

    _____

    map from La2021 to La2010a “405ka”

    La2021
    100732.896221555 = harmean(174006.444683136,75045.60062538,72418.4175234689,304407.281910738,193173.349232374,69035.3166782081,73191.3932343141)

    433078.958481195 = 1 / s_7 = 360*60*60/2.9925259 — La2004
    130711.04387292 = beat(173889.708842077,74619.9907876555) = 360*60*60 / 9.915 — La2010a
    405568.069599057 =2/(3/2/100732.896221555-1/433078.958481195-1/130711.04387292)
    405568.048748278 = beat(304405.279928371,173889.708842077) = 360*60*60 / 3.195518 — La2010a

    _____

    map to La2021 “406ka” (curious outlier attracting inquiry) from Berger

    Berger 1988 Table 4 (based on Berger 1978)
    100743.899044297 = harmean(176420,75259,72576,308043,191404,68829,72732)

    432023 = 1 / s_7 = 360*60*60/2.99984028628105 — Berger
    130704.452624679 = beat(173901.37537739,74619.9907876555) = 360*60*60 / 9.9155 — La(2004a,2010a)avg
    406200.879541285 =2/(3/2/100743.899044297-1/432023-1/130704)
    406200.0673239 = beat(304407.281910738,174006.444683136) = 360*60*60 / 3.190546 — La2021

    _____
    the critical diagnostic:
    Berger is the s_7 outlier

    432023 — Berger 1988 Table 4 (based on Berger 1978)
    433068.234979616 — Zeebe2017
    433078.799289029 — La2021
    433078.958481195 — La2011 Table 6 La2004a
    433078.965717205 — La(2004a,2010a)average
    433078.972953216 — La2011 Table 6 La2010a
    = 1/s_7

  194. Paul Vaughan says:

    The 2 architecturally different groups of “orbital solutions” estimate different (and systematically related) physical quantities. To underscore the point: These are NOT just different measurements of the same phenomena. They characterize different — but related — physical phenomena. Transformation to a common framework with phenomenological accord is both feasible and prerequisite to meaningful comparison.

  195. Paul Vaughan says:

    5 Times the Symmetry

    19.8549641949401 = beat(29.4701958106261,11.8627021700857) —— Standish (1992)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857) — long-model anomalistic

    Horizons insight: La2021 mirrors La(2004a,2010a)average
    100750 = beat(19.858877815863,19.8549641949401)
    100071 = beat(16.9161211952138,16.9132621572062)
    symmetry date = 1929+(9+20/30)/12

    “Dewey first became interested in cycles while Chief Economic Analyst of the Department of Commerce in 1930 or 1931 because President Hoover wanted to know the cause of the Great Depression. Dewey reported that each economist to whom he spoke gave him a different answer and he lost faith in the current economic methods. He received and took advice to study how business behaviour occurred rather than why. Therefore, his views are generally regarded as inconsistent with mainstream economics.”

    “When transforming between the underlying ICRF reference frame, Horizons uses the IAU76/80 fixed obliquity of 84381.448 arcsec at the J2000.0 standard epoch, and an associated time-varying model for “of-date” ecliptic.”

    1535.88262671198 = 360*60*60 / 843.81448

    “When transforming between FK4/B1950, a fixed obliquity of 84404.8362512 arcseconds is used at the standard epoch, with an associated time-varying model for other instants.”

    1535.45703962144 = 360*60*60 / 844.048362512

  196. Paul Vaughan says:

    counter-mythology (what truly accounts for coincidence of dates?)
    relationship parameter uncertainty minimized at multivariate sample center

  197. Paul Vaughan says:

    different bias with different framing:

    2166101.14285714 — B
    2159260.67718384 — Seid trop

    2384728.28959929 — seid synodic
    2384110.34604552 — T5

    2604030.68238338 = beat(2166101.14285714,1182481.75182481)
    2607198.33309978 — W

    shift ~early 1930s (sample center) “95ka” & “131ka” (previously noted with entertainment links from way back in the old days before entertainment was abandoned) by “2.4Ma”: gives “405ka” in tight accord with convention

    mystery vanishing

  198. Paul Vaughan says:

    the view has become orders of magnitude more clear than I ever imagined it would

    the authors just don’t have the presentation organized as a botanist would design a taxonomic key (comparing & contrasting reference frames, parameter values, etc. to simply ID reliably different species & varieties)

  199. Paul Vaughan says:

    supplementary

    25747.7997010794 = harmean(25773.0455911473,25722.6032216452) — La(2004a,2010a)avg
    25747.3224920285 (SSTartin’ too˚C weather peace’s_f(IT) buck$in˚K sh!yen/ruble/yuan?)
    25747.3026929123 = harmean(25808.1036291693,25686.7875634114) — La2021 with below
    25747.43606059 = harmean(25808.3716271706,25686.7875634114)
    25746.5839635401 = harmean(25808.1036291693,25685.3568935329) — Berger with below
    25746.717323772 = harmean(25808.3716271706,25685.3568935329)

    5.99685290323073 = beat(0.0754402464065708,0.0745030006844627)
    179.333323110834 = slip(18.6129709123853,8.84735293159855)
    1986.46983643213 = slip(5.99685290323073,0.999978614647502)
    25808.3716271706 = slip(1986.46983643213,179.333323110834)

    25808.1036291684 = beat(4231.48507417337,3635.42278750964)
    4231.48507417337 = slip(164.793624044745,84.0331316671926) —- Standish anomalistic
    3635.42278750964 = slip(163.7232045,83.74740682) ——————— Seidelmann tropical

    keplerian models with hindsight:
    long-duration models good for anomalistic estimates but sidereal biases to 4270 = s(4370)

  200. Paul Vaughan says:

    Confirmation: Mapping from any parameter list to any other is feasible with a little diagnostic work.

    With the forest in focus the list of noteworthy trees is long —
    too long to report; some selective highlights maybe.

    Meanwhile almost nothing of interest comes up in “climate discussion” anymore.
    Found an exception and scrolled through it in under a minute:

    2022 Climate effects on archaic human habitats and species successions

  201. oldmanK says:

    Another paper on human civilisation collapse, this time blaming volcanoes.
    “https://eos.org/articles/did-volcanoes-accelerate-the-fall-of-chinese-dynasties” Ming dynasty collapse in 1644ce.
    Or “One Drought and One Volcanic Eruption Influenced the History of China: The Late Ming Dynasty Mega-drought” link https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2020GL088124

    1644ce is a precise Eddy cycle root. The one single common element in multi civilisations collapse in the past 8k2 years.

  202. Paul Vaughan says:

    Seidelmann (1992) short-model sidereal
    Standish (1992) long-model anomalistic

    171.406220601552 = beat(164.791315640078,84.016845922161)
    171.471519050756 = beat(164.793624044745,84.0331316671926)

    111.292543528394 = harmean(164.791315640078,84.016845922161)
    111.307357343015 = harmean(164.793624044745,84.0331316671926)

    450106.937906397 = beat(171.471519050756,171.406220601552)
    836224.782693008 = beat(111.307357343015,111.292543528394)
    585215.415762275 = harmean(836224.782693008,450106.937906397)

    84.020635232164 = 7/(2/585215.415762189-6/836224.782693201+7/84.016845922161)
    84.0206328002627

    164.770100737597 = 7/(6/585215.415762189-4/836224.782693201+7/164.791315640078)
    164.770073467735

  203. Paul Vaughan says:

    last comment underscores sample center — background:

    171.444974999322 = beat(164.770073482697,84.0206327956442)
    111.291020849569 = harmean(164.770073482697,84.0206327956442)

    758281.302167317 = beat(171.444974999322,171.406220601552)
    8134256.92902343 = beat(111.292543528394,111.291020849569)

    836235.807754755 = beat(8134256.92902343,758281.302167317)

    4232.12253521941 = slip(164.770073482697,84.0206327956442)
    4270.09258127429 = slip(164.791315640078,84.016845922161)

    475942.4050886 = beat(4270.09258127429,4232.12253521941)

    584818.172857209 = harmean(758281.302167317,475942.4050886)

  204. Paul Vaughan says:

    15268.3462104784 = 3 * beat(791.282376817764,936.955612197409) — stand a short
    15351.8779640976 = beat(883.060665385146,936.955612197409) — standish s
    15379.8163558059 = beat(883.152947004212,936.955612197409) — seid synodic
    15391.7026544853 = beat(883.192112166333,936.955612197409) — W’factsheet’s
    15403.482415902 = beat(883.230870074916,936.955612197409) — 1930.13888888889
    15403.6217843793 = beat(883.231328292231,936.955612197409) — 1929.22222222222
    15407.6872446168 = beat(883.244691321446,936.955612197409) — 25771.4533429313
    15415.6993862807 = beat(883.271007507227,936.955612197409) — 25763.987503107
    15439.8479288541 = 2 * beat(835.546575435627,936.955612197409) — seid s short
    15461.1213414356 = beat(883.419711511583,936.955612197409) — 25721.8900031954
    15559.2279828429 = 5/3 * beat(851.49574667679,936.955612197409) — stand n
    15811.8617894692 = 2 * beat(837.679677971629,936.955612197409) — stand s short
    16546.1976870195 = 3 * beat(800.898956785007,936.955612197409) — Seid trop
    16858.3647714886 = 3 * beat(803.058289202714,936.955612197409) — stand n short

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