Lunar connection: the Saros, nodal and apsidal cycles

Posted: September 8, 2019 by oldbrew in Cycles, Maths, moon, solar system dynamics
Tags: ,

Image credit: naturalnavigator.com


The contention here is that in the time taken for 14 lunar nodal cycles, the difference between the number of Saros eclipse cycles and lunar apsidal cycles (i.e the number of ‘beats’ of those two periods) is exactly 15.

Since 15-14 = 1, this period of 260.585 tropical years might itself be considered a cycle. It is just over 9 Inex eclipse cycles (260.5 years) of 358 synodic months each, by definition.

Although it’s hard to find references to ~260 years as a possible climate and/or planetary period, there are a few for the half period i.e. 130 years, for example here.

Calculations:
Nodal cycle = 6798.33 days
Apsidal cycle = 3231.49 days
Saros = 6585.3213 days (223 synodic months, by definition)

6798.33 / 3231.49 = 2.1037756
6798.33 / 6585.3213 = 1.0323459
2.1037756 / 1.0323459 = 1.0714297
1.0714297 * 14 = 15.000015

Referring back to charts from an earlier post: Two long-term models of lunar cycles, the 14:15 ratio can be confirmed.

Lunar chart 1

In the first chart (shown, right) 106 nodal cycles = 223 apsidal cycles.

The second chart (not shown here), which is 7 times the period of the first chart, equates to 766 Saros – because the number of synodic months(SM) is shown as 766 * 223, and 223 SM = 1 Saros, as mentioned above and in the linked post.

7 * 106 = 742 nodal
7 * 223 = 1561 apsidal
1561 – 766 = 795 (beats of apsidal and Saros)
Ratio of 742:795 (dividing by 53) is 14:15.
(795 – 742 = 53)

Comments
  1. oldbrew says:

    The ratio of Jupiter’s orbit period to the mean ’11-year’ (more like 11.07y) solar cycle could also be 14:15.
    – – –
    The 18.61 year lunar nodal, or precession, cycle.

  2. Chaeremon says:

    @oldbrew, 9 Inex is not only in the other half of the year but also at the other node. Odd Inex rule.

    The Mayas had a 260 figure; I’ll look at what 9 Inex minus the previous eclipse season (same equinox of the year and same node) comes out.

  3. oldbrew says:

    I have made this 9:14 Inex:LNC link before, although it’s not quite 100% precise. But the other figures here should be accurate…

    18:28 = 9:14 ratio

    From this post: Why Phi? – the Inex eclipse cycle, part 2
    https://tallbloke.wordpress.com/2015/03/20/why-phi-the-inex-eclipse-cycle-part-2/

    It’s not flagged in this chart, but 199 and 521 are Lucas numbers (the only L no. in between is 322).
    18 (Inex) is also a Lucas number.

    Re. the other numbers:
    Draconic months = 777 * 9
    Synodic months = 716 * 9 (shown as 537 * 12)
    Draconic years = 61 * 9 (61 = 777 – 716)

    Lunar evections in latitude = 716 – 61 = 655 (not shown).
    Of course 716 / 2 = 358 SM = 1 Inex, by definition.

  4. Ian Wilson says:

    oldbrew,

    This plot shows that a strong 130-year cycle appears in the amplitude spectrum of the North American (proxy) air temperatures.

  5. Ian Wilson says:

    My post has been sent to Purgatory again because of some Godforsaken wordpress rule

    [mod] unknown issue, sorry – retrieved now

  6. oldbrew says:

    IW – so the ‘strong 130-year cycle’ looks like 7 lunar nodal.

    Update 3:40 PM – the 133 years on the Fourier spectrum graphic above is 7 Metonic cycles.
    In that period there is an exact number of solar (1914) and Carrington (1781) rotations, difference = 133.

  7. Chaeremon says:

    @oldbrew, 23 Inex – 22.5 Saros is 260.0~ years. From 1600 to 2100, some 66.~% eclipses (same node) pair @ 3216.5 moons. Incidently the timeframe has 365.~ Tzolkʼin as well (the other figure in the Maya 260 thing).

    Since the eclipses occur regular, this could have been the pre-historic civilization’s astron-observational beat — but years instead of know-nothing-for-all academicelized “sacred” days.

  8. oldbrew says:

    Re 766 Saros (see post):
    766*223 = 170818 synodic months (SM)

    Add one SM = 170819 = 3223 * 53
    9*358 SM (Inex) = 3222

    So every 14 nodal cycles take about one SM longer than 9 Inex, less one SM for every 53 sets of 14 – which then lines up with the Saros figure.

  9. oldbrew says:

    For more re 133 year cycle in IW’s graphics, see:

    Comment – oldbrew says: August 31, 2019 at 12:24 pm
    https://tallbloke.wordpress.com/2019/08/30/the-metonic-cycle-in-long-term-lunar-harmonics/

  10. Ian Wilson says:

    Sidereal year = 365.256363 days
    Tropical year = 365.242188 days
    Synodic period = 29.530588861 days
    Tropical month = 27.3215823 days
    Sidereal month = 27.3216616 days
    Draconic Month = 27.2122208 days

    (Tropical) Nodal cycle = (27.3215823 x 27.2122208) / (27.3215823 — 27.2122208)
    _________________= 6798.379047 days = 18.61335 tropical years

    (Sidereal) Nodal Cycle = 27.3216616 X 27.2122208) / (27.3216616 — 27.2122208)
    __________________= 6793.472709 days = 18.59919 sidreal years

    [Note: The difference between the Sidereal and Tropical frames-of-reference comes about because of the 25,700-year precession of the Earth’s rotation axis with respect to the stars.]

    Metonic Cycle = 235 Synodic cycles
    19.0 sidereal years = 6939.870897 days
    235 Synodic period = .6939.688382 days = 18.999500 Sidreal years = 19.000238 Tropical years
    19.0 tropical years = .6939.60157 days

    oldbrew,

    I assume that you are highlighting the difference between:

    seven X Metonic cycle = 7 x 19.000238 Tropical years = 133.0017 years

    and seven x Nodal cycle = 7 x 18.61335 tropical years = 130.2935 tropical years?

    This might explain why the 133-year cycle appears in the spectrum of the solar activity index, while the ~ 130-year cycle appears in the proxy temperature record.

    The 19-year Metonic cycle of the Moon (a.k.a 133-year cycle) connects the lunar orbit to a whole multiple of the Earth’s orbit about the Sun. Hence, this is connected to the level of solar activity via the linkages between the lunar orbit periods and the orbital periods of Jupiter and Venus.

    The 18.61-year Nodal cycle should influence the Earth’s climate through the Moon’s effects upon the Earth’s atmospheric and oceanic tides.

  11. oldbrew says:

    IW – re. Metonic Cycle = 235 Synodic cycles

    I suggest it loses one synodic month every 339 Metonic cycles (of 19 tropical years each).


    https://tallbloke.wordpress.com/2017/10/15/lunar-precession-update/

    235 synodic months and 254 tropical months are not exactly the same number of days, although the difference is very small. But multiplying that by 339 gives 2.2374 days, whereas using the numbers in the chart gives zero difference – on a pocket calculator at least!

  12. Chaeremon says:

    @oldbrew, from astron. -2443/06/16 to 3998/06/23 = 79665 SM.

    Are you using wiki-pravda [etc] base numbers? suggest the use of R. H. van Gent’s calc, his base numbers are concerted for centuries and millennia.

  13. oldbrew says:

    Chaeremon – 339 * 235 = 79665.

    I am saying the definition of a Metonic cycle is 19 tropical years, not 235 synodic months. It can’t be both as there’s a known slight mis-match which also applies to 254 tropical months. In the chart it balances out.

    79664 = 766 * 104. This gives 7 * 104 apsidal cycles, making 766 SM = 7 apsidal.
    All the other (non-bracketed) figures in the chart are 7 less than a multiple of 104.

    An oddity – perhaps – is that 235+104 = 339.

  14. Chaeremon says:

    @oldbrew, after 4 Metonic (235 SM) the equinox (Earth) and synodic (Moon) diverge, sometimes also near to 4 Metonic [IMHO depends on what phase you measure, and phase = Moon + Sun, inseparable].

    The 7 apsidal thing is interesting indeed.

  15. Ian Wilson says:

    Modified post to get around WordPress bug

    oldbrew,

    Here is some info about the Metonic cycle – explaining why it associated with the alignment of the lunar Spring tides (that occur at New and Full Moon) with the annual seasonal cycle.

    Higher than normal spring tides occur once every semi-synodic month (Msf), whenever the Sun, Earth, and Moon are co-aligned at either New or Full Moon.

    It turns out that 12.5 synodic months are 3.890171 days longer than one tropical year (= 365.2421897 days (J2000) (McCarthy and Seidelmann 2009).

    Hence, if a spring tide occurs on a given day of the year, 3.796 tropical years will pass before another spring tide occurs on the same day of the year.

    This occurs because: (0.5 synodic months) / (12.5 synodic months – tropical year) = (14.7652944 days / 3.890171 days) = 3.796 years.

    In addition, it can be shown that multiples of half of the lunar synodic cycle (Msf) are almost exactly equal to whole multiples of a year, at:

    4.0 years,
    4.0 + 4.0 = 8.0 years,
    4.0 + 4.0 + 3.0 = 11.0 years,
    4.0 + 4.0 + 3.0 + 4.0 = 15.0 years, and
    4.0 + 4.0 + 3.0 + 4.0 + 4.0 = 19.0 years.

    Hence, Spring tides that occur on roughly the same day of the year (i.e the same point in the seasonal cycle) follow a 4:4:3:4:4-year spacing pattern, with an average spacing of:

    (4 + 4 + 3 + 4 + 4)/5 = 3.8 years

    with the pattern repeating itself after a period of almost exactly 19 years.

    The 19.0 year period is known as the Metonic cycle. This cycle results from the fact that 235 Synodic months = 6939.688381 days = 19.000238 Tropical years.

    McCarthy DD, Seidelmann PK. Time from earth rotation to atomic physics. Weinhein: WileyVCH Verlag GmbH & Co KGaA 2009

  16. Ian Wilson says:

    Chaeremon – apologies for using mean value approximations for the lunar orbital parameters. I am just doing a rough calculation at this point, which should eventually be done using an accurate lunar ephemeris.

    oldbrew,

    You posted the statement:

    “I suggest it loses one synodic month every 339 Metonic cycles (of 19 tropical years each).”

    Here is my interpretation of what you are saying:

    1.0 tropical year = 365.2421897 days
    1.0 Synodic month = 29.530588853 days
    235 Synodic months = 19.000238 tropical years

    339 x 235 Synodic months = 6441.080682 tropical years
    339 x 19 years__________= 6441.000000 tropical years

    which differ by 0.00682 tropical years = 29.468470 days ~ 29.47 days.

    This is not far off the nominal 29.53 day period for the Synodic month.

    &&&&&&&&&&&&&&&&&&&

    The Sidereal year (= 365.256363 days) is the time taken for the Earth to complete one revolution of its orbit, as measured with respect to the stars.

    The mean tropical year is defined as the period of time for the mean ecliptic longitude of the Sun to increase by 360 degrees. Since the Sun’s ecliptic longitude is measured with respect to the equinox, the tropical year comprises a complete cycle of the seasons.

    Because of the 25,772-year period for the precession of the Earth’s axis, the tropical year is about 20 minutes shorter than the sidereal year. The mean tropical year is approximately 365 days, 5 hours, 48 minutes, 45 seconds, using the modern definition (= 365.24219 days of 86400 SI seconds).

    &&&&&&&&&&&&&&&&&&&&

    I assume that you realize that:

    4 x 6,441 tropical years = 25,764 tropical years

    which is not dar off of the nominal 25,772-year period for the precession of the Earth’s orbit.

    So the question becomes why do we have a roughly 4 synodic months difference between:

    4 x 339 x 235 Synodic months = 25,764.322728 tropical years
    4 x 339 x 19.0 tropical years__= 25,764.000000 tropical years

    which differ by 0.322728 tropical years = 117.873881 days ~ 117.87 days

    This is not far off the nominal period for 4.0 x Synodic months = 118.122355 days ~ 118.12 days

    Doesn’t this suggest that the Synodic month of the lunar orbit is somehow locked into some sort of resonance with the period of precession of the Earth’s rotation axis?

    Why 4.0 Synodic months?

  17. oldbrew says:

    Re: 235 Synodic months = 6939.688381 days = 19.000238 Tropical years

    19.000238 – 19 = 0.000238 TY = 0.0869276 days
    0.0869276 * 339 = 29.468456 days = 99.79% of a synodic month

    That’s why the chart has 79664 SM and not 79665 (= 339 * 235).
    86105 tropical months = 79664 synodic months (within 0.1 day)
    Add 1 to each and the difference becomes 2.2~ days.

  18. Ian Wilson says:

    olbrew,

    An important clue might come from the connection between the Synodic month, the sidereal month, and the sidereal year.

    (1 / (Synodic month)) = (1 / (sidereal month)) — (1 / (sidereal year))

  19. oldbrew says:

    IW – I’m not disputing that 🙂
    – – –
    Doesn’t this suggest that the Synodic month of the lunar orbit is somehow locked into some sort of resonance with the period of precession of the Earth’s rotation axis?

    From the chart: 4 * 6441 = 25764 tropical years = 25763 sidereal years.
    This is very much like the axial precession period* and has been discussed in earlier posts, e.g.:
    https://tallbloke.wordpress.com/2016/02/01/why-phi-a-unified-precession-model/

    (*Wiki currently quotes figures around 25770y).

  20. Ian Wilson says:

    Oldbrew,

    Sorry, there was a typo in my post of September 11, 2019 at 6:27 pm.

    Everywhere I have put 238 Synodic months it should have been 235 Synodic months. However,
    since I was working in tropical years, and I was using 235 Synodic months = 19.000238 tropical years, my calculations are still correct.

    Yes, 339 x 235 SM = 79665 SM

    So, why is this slippage of 4 SM occurring every 25,764 years?

  21. oldbrew says:

    Synodic and tropical months both slip by 1 in the ‘quarter precession’ period of 339 Metonic cycles, so that looks like a precession in its own right since they aren’t 1:1 periods.

    In 25,764 years they repeat the cycle 4 times.

  22. Ian Wilson says:

    oldbrew,

    From my September 11, 2019 at 6:39 pm post you can see that the synodic month (SM)
    is [by defintion] set by the lengths of the sidereal month and the sidereal year.

    Hence, you would expect that if you were investigating multiples of the lunar tropical month (TM), they should drift away from multiples of the SM at a rate that is determined by the drift between the tropical frame of reference (i.e the Equinox) and the sidereal frame of reference (i.e the stars). This by definition must be related to the 25772-year precession of the rotation axis of the Earth with respect to the stars.

    So we are still left with the question, why is this slippage of almost exactly 4 SMs occurring every 25,764 years?

  23. Ian Wilson says:

    oldbrew,

    Just a hunch but it may have something to do with the mystery I proposed at:

    http://astroclimateconnection.blogspot.com/2017/04/i-need-help-to-solve-lunar-puzzle.html

    I Need Some Help to Solve an Interesting Lunar Puzzle

  24. oldbrew says:

    OK the question is there, but so is the observation. Does this help at all?

    NB needs to be seen in the context of the post linked a few comments back.
    83733 = 6441 * 13

  25. oldbrew says:

    IW – this turned up, may be a red herring but anyway…

    My chart has 766*104 synodic months. Call it 7660 * 10.4 SM instead.
    1 full moon cycle / 10.4 SM = 411.78443 / 307.11812 = 1.3408014

    Your blog post refers to 1.340930~ degrees.
    5713 FMC (from my chart) * 1.3408014 = 7660.73

    Maybe this is all circular, but there it is.

  26. Paul Vaughan says:

    If you study the works of Ramanujan you’ll better appreciate that Mayan astronomers were well aware that 73 is the smallest prime congruent to 1 modulo 24. 24 the smallest 5-hemiperfect number. 232 is the discriminant in Ramanujan’s calculation of 208.

    179.323556519781 = (232.356682331997)*(146.000401793801)/((232.356682331997+146.000401793801)/2)

    ⌊ 18.6129709123853 / 8.84735293159855 ⌉ = ⌊2.10378980654299⌉ = 2
    18.6129709123853 / 2 = 9.30648545619264

    harmonic of 18.6129709123853 nearest 8.84735293159855 is 18.6129709123853 / 2 = 9.30648545619264

    179.333323110834 = (9.30648545619264)*(8.84735293159855) / (9.30648545619264 – 8.84735293159855)

  27. Paul Vaughan says:

    WordPress bins another calculation.

  28. oldbrew says:

    IW – re. So we are still left with the question, why is this slippage of almost exactly 4 SMs occurring every 25,764 years?

    Is there a ‘slippage’? That relies on the idea that 19 tropical years ‘should’ be 235 SM, but in the 25764 TY period (25763 sidereal years) the number of lunar sidereal months is exactly 1 less than the tropical months i.e.:

    86105*4 = 344420 TM (*27.321582 = 9410099.2 d)
    25763*365.25636 / 27.321662 = 344419 Sid.M
    344420 -344419 = 1

    Therefore the difference between SM and TM is reduced by 1, giving one less sidereal than tropical years):
    79664*4 = 318656 SM
    344419 – 318656 = 25763

    25763*365.25636 = 9410099.6 d
    25764*365.24219 = 9410099.7 d

    Also: 6441 TY = 728 apsidal cycles (tropical perspective)
    4 * 728 = 2912
    2912 – 1 = 2911
    9410099.6 / 2911 = 3232.6003 apsidal cycles (sidereal)

  29. oldbrew says:

    Younger-Dryas Period–Mystery Solved?

    The precession of the equinoxes is the observable phenomena of the rotation of the heavens around the Earth–a cycle that is said to span a period of (approximately) 25,920 years (Platonic year).

    The cause of the precession of the equinoxes remains a hotly debated topic. At the heart of the debate is the source of the underlying motion that cause the equinoxes to precess. I believe that motion is a cycle of 80-years and that apsidal precession is the phenomena that produces the 25,920-year precession cycle.[bold added]

    https://ancient-astronomer.com/tag/apsidal-precession/
    – – –
    We know the period of the precession cycle varies and is declining in the current era, but some interesting graphs and discussion at the link.
    – – –
    Going back to the ‘Lunar chart’ [oldbrew says: September 10, 2019 at 1:42 pm ]…

    Apsidal cycles and synodic months are multiples of 104 = 13*8.
    All other non-bracketed numbers (trop. years, full moon cycles etc.) are 7 short of a multiple of 104, i.e. 1 short (each) per 104 apsidal cycles.

    So for example:
    62 – ‘7/104ths’ tropical years = 55 – ‘7/104ths’ full moon cycles = 7 apsidal cycles (TY-FMC).

    BTW the evections in longitude go the other way, i.e. 7 more than a multiple of 104.
    79664 (SM) – 5713 (FMC) = 73951 Ev. long. = 711*104, +7.
    Ev. long. + AM = SM*2

  30. Paul Vaughan says:

    JEV-Lunar

    Ramanujan, Mayan, & JEV slip cycle review:

    https://tallbloke.wordpress.com/2019/09/05/wild-geese-take-climate-action/comment-page-1/#comment-151719

    https://tallbloke.wordpress.com/suggestions-39/comment-page-1/#comment-150260

    232 = 4*(⌊1/JEV⌋+⌊1/SEV⌋+⌊1/UEV⌋+⌊1/NEV⌋) = discriminant
    232 = 4*58 = 232
    396 = 11*5256/146

    11 = 396/36
    22 = 396/18
    44 = 396/9

    1 = 1^2 = (1#)^2
    4 = 2^2 = (2#)^2
    36 = 6^2 = (3#)^2
    # = primorial

    1, 4 & 36 are the only square highly composite numbers.

    11 = 11*1 = 11*(1#)^2
    44 = 11*4 = 11*(2#)^2
    396 = 11*36 = 11*(3#)^2

    36 = 1+(3*5*7*11*13)/11/(3+5+7+11+13)
    396 = 11+(3*5*7*11*13)/(3+5+7+11+13)

    18 = 11+(3*5*7)/(3+5+7)
    36 = 2*(11+(3*5*7)/(3+5+7))
    396 = 2*11*(11+(3*5*7)/(3+5+7))

    all calculations in julian years = 365.25 days

    51.9443628734759 = 207.777451493904 / 4
    58.0891705829991 = (491.049532942454)*(51.9443628734759) / (491.049532942454 – 51.9443628734759)
    232.356682331997 = 4*58.0891705829991
    179.323556519781 = (232.356682331997)*(146.000401793801)/((232.356682331997+146.000401793801)/2)

    ⌊ 18.6129709123853 / 8.84735293159855 ⌉ = ⌊2.10378980654299⌉ = 2
    18.6129709123853 / 2 = 9.30648545619264

    harmonic of 18.6129709123853 nearest 8.84735293159855 is 18.6129709123853 / 2 = 9.30648545619264

    179.333323110834 = (9.30648545619264)*(8.84735293159855) / (9.30648545619264 – 8.84735293159855)

    6.40217174588736 = 2*(3*5*7)^(1/4)
    8.84758375100207 = 2*(Φ√5)*(3*5*7)^(1/4)
    8.84735293159855 = observed

    6.4099785937056 = (5256.63940169013)*(6.40217174588736) / (5256.63940169013 – 6.40217174588736)
    6.40939079526111 = observed

    tropical year harmonic nearest draconic month
    0.999978614647502 / 27 = 0.0370362449869445

    0.0372514804884878 = (6.4099785937056)*(0.0370362449869445) / (6.4099785937056 – 0.0370362449869445)
    0.0745029609769756 = 2*0.0372514804884878
    27.2122064968403 d = 365.25*0.0745029609769756
    27.212221 d = observed

    9.30676715942977 = (179.323556519781)*(8.84758375100207) / (179.323556519781 – 8.84758375100207)
    18.6135343188595 = 2*9.30676715942977
    18.6129709123853 = observed

    0.0748023666518021 = (18.6135343188595)*(0.0745029609769756) / (18.6135343188595 – 0.0745029609769756)
    27.3215644195707 d = 365.25*0.0748023666518021
    27.3215823630557 d = observed

    0.0754401796567736 = (8.84758375100207)*(0.0748023666518021) / (8.84758375100207 – 0.0748023666518021)
    27.5545256196366 d = 365.25*0.0754401796567736
    27.55455 d = observed

  31. Paul Vaughan says:

    another calculation vanished

  32. Paul Vaughan says:

    also: the decoy model vanished on Glasgow — need that one to illustrate a point coming in a series of 4 more posts

  33. Paul Vaughan says:

    minimization on a matrix of polynomials
    next level is weighted matrix to discern drift of local solutions relative to global for different processes sharing a common timing framework
    tight first-order fit’s easy even for left (artistically primorial) parameterizations
    Ramanujan’s almost-integer right-parameterization undersores center of 208 E-V drift past J-S
    Mayan & Seidelmann models exactly match Ramanujan integer floor

    recall:

    1 / 52.0073396621029 = (Φ/1)(J-S)(Φ/1)
    1 / 104.014679324206 = (Φ/√2)(J-S)(Φ/√2)
    1 / 208.029358648411 = (Φ/2)(J-S)(Φ/2)
    1 / 260.036698310514 = (Φ/√5)(J-S)(Φ/√5)

  34. oldbrew says:

    This chart derives (see the ‘*4’ numbers) from the 6441-year chart shown earlier, also here:
    ‘https://tallbloke.files.wordpress.com/2015/11/lunar_ty.jpg’

    One less lunar month because 27.321662 (sidereal) is very slightly greater than 27.321582 (tropical).

    The rest follows because this changes some of the differences by 1.
    Not shown: 6441*4 = 25764 tropical years, less 1 = 25763 sidereal years.

    Synodic months = 766*416 (=318656).

  35. oldbrew says:

    Since 15-14 = 1, this period of 260.585 tropical years might itself be considered a cycle.

    163 Venus-Earth conjunctions = 260.582 years.

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