The Saros cycle by numbers

Posted: April 14, 2020 by oldbrew in Analysis, Cycles, data, moon

The basis for discussion is the abstract of the paper below. Instead of their ‘high-integer near commensurabilities among lunar months’ we’ll just say ‘numbers’ and try to make everything as straightforward as possible. This will expand on a previous Talkshop post on much the same topic.

Hunting for Periodic Orbits Close to that of the Moon in the Restricted Circular Three-Body Problem (1995)
Authors: G. B. Valsecchi, E. PerozziA, E. Roy, A. Steves

The role of high-integer near commensurabilities among lunar months — like the long known Saros cycle — in the dynamics of the Moon has been examined in previous papers (Perozzi et al., 1991; Roy et al., 1991; Steves et al., 1993). A by-product of this study has been the discovery that the lunar orbit is very close to a set of 8 long-period periodic orbits of the restricted circular 3-dimensional Sun-Earth-Moon problem in which also the secular motion of the argument of perigee ω is involved (Valsecchi et al., 1993a). In each of these periodic orbits 223 synodic months are equal to 239 anomalistic and 242 nodical ones, a relationship that approximately holds in the case of the observed Saros cycle, and the various orbits differ from each other for the initial phases. Note that these integer ratios imply that, in one cycle of the periodic orbit, the argument of perigee ω makes exactly 3 revolutions, i.e. the difference between the 242 nodical and the 239 anomalistic months (these two months differ from each other just for the prograde rotation of ω).
[bold added]

To start with we can create a model that pretends the ‘high-integer near commensurabilities’ really are whole numbers, then break down the logic of the result to see what’s going in with the Moon at the period of one Saros cycle.

This diagram conforms to the statement in the abstract above:
In each of these periodic orbits 223 synodic months are equal to 239 anomalistic and 242 nodical ones.

Reading from left to right and comparing the anomalistic and draconic numbers, we see they all have a difference of 3 (207-204, 19-16, 242-239). As the abstract says:
Note that these integer ratios imply that, in one cycle of the periodic orbit, the argument of perigee ω makes exactly 3 revolutions.

We’ll come back to the revolutions shortly.

Reading top and bottom numbers:
207+239 = 204+242 = 446 = 223*2

The evections are the counterparts, or ‘mirrors’ of the anomalistic and draconic month, so where their numbers are greater than the 223 synodic months of the Saros, the evections are correspondingly less.
223-16(FMC) = 207 LE1 (evection-in-longitude)
223-19(DY) = 204 LE2 (evection-in-latitude)

In reality the numbers, apart from the Saros itself, are as the abstract says *near* commensurabilities. But when summed to 446 as shown, the result is exactly 223*2 synodic months.

The number of full moon cycles is the difference between the number of anomalistic months and 223 (shown as 16, but in reality very slightly less).

The number of draconic years is the difference between the number of draconic months and 223 (shown as 19, but in reality very slightly less).

It’s also true that the number of:
LE1 minus LE2 = DY minus FMC = AM minus DM = ~3 (in fact: 3.00652)

The reason behind that is that these pairs all have the same beat period, which is ‘the revolution of the argument of perigee’, as the abstract calls it. This is the period in which the sum of the number of lunar apsidal and nodal cycles is 1, and is just under 6 years:

18.030012 (Saros in tropical years) divided by 3.00652 = 5.99697~ TY (about 2190.35 days).

Astronomer Willy de Rop’s short analysis of the quasi 6-year period is here (see Fig. 3 showing the half period, i.e. ~3 years).

  1. Chaeremon says:

    Re: Willy de Rop manuscript, 1971
    Is there at least one exemplary interval [teh thing named reality] with Rop’s 300 p (close to his wished ~1800 years), by explicit start date and explicit end date?
    The Saros series ends after ~72 members (~1298 years) because after that, line-of-nodes and line-of-apsides have diverged soo much they cannot result in Rope’s figure 3 at interval end.
    From the other direction: one sixth of Rop’s 300 p can be approximated with 6×Inex + 7×Saros (has ~12 members, short of [closest to] ~300 years), but again that cannot match Rop’s wished ~1800 years at interval end.
    Any other resort?

  2. Paul Vaughan says:

    Event series need not be considered to review the “1800 year” attractor:

    5.99685290323073 = (18.6129709123853)*(8.84735293159855) / (18.6129709123853 + 8.84735293159855)

    5.99685290323073 = (0.0745030006844627)*(0.0754402464065708) / (0.0745030006844627 – 0.0754402464065708)

    ⌊ 5.99685290323073 / 1.00002638193018 ⌉ = ⌊5.99669469884984⌉ = 6
    5.99685290323073 / 6 = 0.999475483871788
    i.e. harmonic of 5.99685290323073 nearest 1.00002638193018 is 5.99685290323073 / 6 = 0.999475483871788
    1814.31362251033 = (1.00002638193018)*(0.999475483871788) / (1.00002638193018 – 0.999475483871788)

  3. oldbrew says:

    Chaeremon – there’s this…

    The 1,800-year oceanic tidal cycle: A possible cause of rapid climate change
    Charles D. Keeling and Timothy P. Whorf
    PNAS April 11, 2000 97 (8) 3814-3819;

    A well defined 1,800-year tidal cycle is associated with gradually shifting lunar declination from one episode of maximum tidal forcing on the centennial time-scale to the next. An amplitude modulation of this cycle occurs with an average period of about 5,000 years, associated with gradually shifting separation-intervals between perihelion and syzygy at maxima of the 1,800-year cycle.
    – – –
    de Rop doesn’t mention the Saros at all, but says:
    Other less intensive maxima are superposed
    on the period of 1800 years. They are caused
    by a coincidence of the line of nodes and the
    line perihelion-Sun as well as a coincidence of
    the line of apsides and the line perihelion-Sun

  4. Chaeremon says:

    @Paul Vaughan, thanks, appreciate your event series not needed view, gives limits for pondering.

    @oldbrew, Re: intervals … at maxima of the 1,800-year cycle. I define maxima on my bank account, without ever having to show any start date and end date. Not really expected from professional astronomy; thanks. Is there any other source|paper, I mean with testable dates?

  5. oldbrew says:

    De Rop gives a date in the first sentence of his paper.

    See also:

    Both mention 1433.

  6. Chaeremon says:

    @oldbrew, I’m trying to check against Ian Wilson: A Severe Case of Cognitive Dissonance, so where are “all” the maxima?

    Can we agree that Ian’s 31 (62) years and Rop’s 1800 years don’t fit together \(0.0)/

  7. oldbrew says:

    Chaeremon – my understanding is that the 31 and/or 62 years period is anomalistic i.e. relates to apsidal and full moon cycles. However once you get to 3*62 that’s nearly 10 nodal cycles. Note 3*62 is a multiple of 6 years…

    De Rop is talking about the period of both apsidal and nodal cycles relative to each other, which is just under 6 years. In that time the sum of their occurrences is 1 (approx. 0.678 to 0.322 ratio). So for every 300 of those, a total of 1 anomalistic year is ‘lost’, giving his 1799 anom. year cycle as he explains.

    De Rop: So, when the perigee of the Moon’s orbit coincides with the ascending node, then this situation repeats after 2190.340565 days.

  8. Chaeremon says:

    @oldbrew, Re: 3*62 is a multiple of 6 years

    Okay, that’s 2078 synodic, 2255 nodal, 2246.0~ sidereal (therefore 9 nodal cycles, not 10).
    Since that has eclipse distance (2×Unidos + 2×Metonic) we can already judge from the eclipse characteristics that 2078 synodic overshoots* the year’s bound (and any multiple thereof can not do better). Overshoot* is good, if needed for the precession of perihelion, but not for year’s bound.
    We’re still not in a position to finding 1 case for figure 3 of Rope in the ephemerides data, and by that I mean 1 interval. Why I’m asking?: nodical & apsidal falling together (and/or on constant° offset) are easy to examine, in the eclipse canon.

  9. oldbrew says:

    therefore 9 nodal cycles, not 10

    Maths problem there. 62*3 = 186, nodal cycle is 18.6~ years.

    Tide Gauge Records Show That the 18.61‐Year Nodal Tidal Cycle Can Change High Water Levels by up to 30 cm

  10. Chaeremon says:

    @oldbrew, Re: April 15, 2020 at 2:23 pm

    Yes I know “the maths problem”. Just take one lunestice date and show the previous and/or next lunestice date at statistical 18.6 distance: you will report none, otherwise I buy you a [big!] beer.

  11. oldbrew says:

    They don’t mention any ‘lunestice’ here.

    But I noticed this:
    The equation of the ellipse yields an eccentricity of 0.0549, and, perigee and apogee distances of 362,600 km and 405,400 km respectively (a difference of 12%).

    405400 / 362600 = 1.1180364 = half the square root of 5 (close enough to 1.118034)
    Add 0.5 and it’s Phi, or subtract 0.5 for 1/Phi.
    – – –
    Update: found it…

    A lunar standstill is the gradually varying range between the northern and the southern limits of the Moon’s declination, or the lunistices, over the course of one-half a sidereal month (about two weeks), or 13.66 days. (Declination is a celestial coordinate measured as the angle from the celestial equator, analogous to latitude.) One major, or one minor, lunar standstill occurs every 18.6 years due to the precessional cycle of the lunar nodes at that rate.

  12. Chaeremon says:

    Re: lunestices

    I compute their communicable tally like all other cycles, as diff between x and sidereal, err observation 😉 This is not the same as statistics with their cherry-picked start date / end date, and the latter is misleading [often aca-fool-dense, Akadämlich, like in CO2 iceheat trends].

    In the recent Metonic thread I reported the braided nodal cycles. Empiric data, from ephemerides, show that lunestice can only occur during equinox month (moon); therefore statistical 18.6 is over and out in reality.

    P.S. good find Phi in anomaly, perhaps this helps with detecting intervals for the 1800 years thing in ephemerides.

  13. oldbrew says:

    But ‘Tide Gauge Records Show That the 18.61‐Year Nodal Tidal Cycle Can Change High Water Levels by up to 30 cm’

    Sounds like reality? Btw the Saros doesn’t seem to relate directly to the nodal cycle, but the difference between (e.g.) the number of full moon cycles and draconic years in one Saros *is* exactly the number of 2190.34~ day periods — which are determined by the interplay of nodal and apsidal cycles, as de Rop diagrammed and explained.

  14. Chaeremon says:

    @oldbrew, I leave you alone with your not showing dates; the aca-fool-dense designations are not my métier.

  15. oldbrew says:

    Remember de Rop never mentioned the Saros.

  16. oldbrew says:

    Nasa: Five Millennium Canon of Solar Eclipses:
    –1999 to +3000 (2000 BCE to 3000 CE)

    October 2006
    Fred Espenak and Jean Meeus [60 pp.]

    4.2 Saros Series

    The periodicity and recurrence of eclipses can be investigated using the Saros cycle, a period of approximately 6,585.32 days (~18 years 11 days 8 hours). It was known to the Chaldeans as an interval when lunar eclipses appeared to repeat, but the Saros is applicable to solar eclipses as well.

    The Saros arises from a harmonic between three of the Moon’s orbital cycles. All three periods are subject to slow variations over long time scales, but their current values (2000 CE) are:
    Synodic Month (New Moon to New Moon) = 29.530589 days = 29d 12h 44m
    Draconic Month (node to node) = 27.212221 days = 27d 05h 06m
    Anomalistic Month (perigee to perigee) = 27.554550 days = 27d 13h 19m

    One Saros is equal to 223 synodic months, however, 242 draconic months and 239 anomalistic months are also equal (within a few hours) to this same period:
    223 Synodic Months = 6585.3223 days = 6585d 07h 43m
    242 Draconic Months = 6585.3575 days = 6585d 08h 35m
    239 Anomalistic Months = 6585.5375 days = 6585d 12h 54m

    Any two eclipses separated by one Saros cycle share similar characteristics. They occur at the same node with the Moon at nearly the same distance from Earth and at the same time of year.

    Further on:
    Saros series do not last indefinitely because the synodic, draconic, and anomalistic months are not perfectly commensurate with one another.

    But the post explains that, hopefully.

  17. oldbrew says:

    Re. the ‘braided nodal cycles’: that looks like when the tropical and draconic years match in whole numbers, or differ by a multiple of 0.5, e.g. 242 TY = ~255 DY =~13 nodal cycles (i.e. 255-242).
    – – –
    Possible forcing of global temperature by the oceanic tides
    Charles D. Keeling and Timothy P. Whorf
    PNAS August 5, 1997

    An even longer perspective of strong tidal forcing is gained from a study of the timing of astronomical alignments by Cartwright (43), who showed that the greatest tide raising forces in the past millennium occurred between A.D. 1340 and 1619, at 93-year intervals, when the perigean eclipse cycle was almost optimally timed with respect to perihelion.

    By his calculation, tides of such great magnitude will not occur again until A.D. 3182, an interval corresponding approximately to the near 1,800-year return period of optimal timing of perigean eclipses with perihelion discussed above in Section 4. According to the calculations of Otto Pettersson made many years ago, tides of great magnitude conforming roughly to this return period also occurred near 3500 B.C., 1900 B.C., and 200 B.C., as well as in A.D. 1433 (ref. 6, pp. 220–222). [bold added]

    Re. 1433: de Rop quoted Petterson, and the ‘near 1800-year period’ is de Rop’s 1799 anomalistic years.
    Re. 1340 and 93 year intervals: 1340 + 93 = 1433.

    Keeling & Whorf seem unaware of de Rop’s paper. They also say:
    The line of apsides and line of nodes come into mutual alignment on average twice every 5.997 years. This time interval, the perigean eclipse cycle, is also the beat period between the anomalistic and nodical months.
    — i.e. 2190.34~ days.

    Lots of actual dates in that paper too. Not academic mumbo-jumbo at all.

  18. Paul Vaughan says:

    You don’t need to know anything about event series to deduce a central limit.

    It remains at best curious — and at worst suspicious — that some are UNable/UNwilling to acknowledge a simple mathematical fact: I limit my public participation.

  19. oldbrew says:

    The linked images can be magnified:

    Historic Tidal and Cool Events

    Lunisolar Tidal Events
    (B to C is 1881-1974: 93 years – see K&W study, below)

    From: Keeling&Whorf 1997
    Possible forcing of global temperature by the oceanic tides

    K&W: The shortest period in which syzygy, perigee, an eclipse, and perihelion nearly coincide is 18.030 tropical years (18.029 anomalistic years), consisting nominally of 223 synodic, 239 anomalistic, and 242 nodical months (see ​Table1).

    As described in this post: the 223 is exact, the 239 and 242 match with the two evection types to sum to 223*2 (446) synodic (18.030 tropical years is the Saros).

    Valsecchi noted the 446 in the Discussion section of another paper:

    • the variation, a term depending on 2Δλ, an angle that makes 446 revolutions in the periodic orbits associated to the Saros;
    • the parallactic inequality, depending on Δλ, that makes 223 revolutions;
    • the evection, to which are due a perturbation in e and ω, depending on 2λ′−2ω˜, that makes 32 revolutions, and another perturbation in λ, depending on λ−2λ′+ω˜, that makes 207 revolutions;
    • the principal perturbation in latitude, depending on λ − 2λ′+Ω, that makes 204 revolutions.

    [bold added]

    None of the 204, 207, 239, or 242 is exact on its own but they do sum to 446:
    ~207+~239 = ~204+~242 = 446 = 223*2
    And the difference between them gives twice the exact number of full moon cycles (~239 minus ~207) or draconic years (~242 minus ~204) per Saros – see post.

  20. Paul Vaughan says:

    Others please carry on expressing attention to events while understanding my intention is to establish that civil servants are unable to infer the implications of central limits. The goal is to lay the foundations for a textbook example of government failure.

  21. oldbrew says:

    Millennial Repetitions.

    The perigean eclipse cycle also influences tidal events on the millennial time scale because the return time for near coincidence of events of this cycle with perihelion is approximately 1,800 years. We propose that the repeat time of millennial extremes in tide raising forces, discussed below, relates to this return time, although the actual timing of such millennial events must be irregular, being sensitive to the exact time of syzygy (ref. 23, pp. 201–249). The near coincidence of perihelion with this cycle in the present millennium occurred near the time of a climactic tidal event in A.D. 1433 (ref. 6, p. 220).
    [bold added]

    Keeling & Whorf

    De Rop also referred to 1433.
    – – –
    Ref. 6:
    Every winter from 1433/4 to 1437/8 described as severe. Lamb
    [see Section 2 – Reference material]

    Click to access long-slow-thaw-supplementary-information.pdf

  22. Paul Vaughan says:

    We went over all that many years ago.
    It terrifies the Dutch — understandably.

    Here’s the aggregate proof review:
    Keep in mind that lunisolar only accounts for 15-17% of the variation in SST.

    It’s a civil service technician’s job to work out the details (which we KNOW exist from the clear and simple aggregate). They are well-paid. They have comfortable lives and the needed resources. Some (e.g. at NASA JPL) have the ability (probably LOTS already known and classified there but few of today’s climate scientists would understand it).

    They could contract me to teach them how to generalize wavelet methods (that subsume conventional methods as a single case in an infinite field). Regrettably, I think it will be decades before anyone builds the generalized wavelet methods needed to simplify analyses by orders of magnitude. I don’t have the needed resources — only the prototyped and successfully tested concepts. For me this further underscores that our society’s institutions are built fundamentally wrong. They obstruct efficiency and build in delay. The supports do not exist where they are needed.

    Thus, I choose when to deliberately boycott (e.g. event series) in order to focus productively elsewhere — e.g. aggregates that a few rare luminaries can recognize as strictly logically inferring cognitive gaps in conventional measurement and modeling.

    Let’s not forget that 83-84% of the SST variance is solar …and it’s NOT the integral of ENSO.

    The DO-timescale records are misinterpreted. On Suggestions-42 I’ve brought up Rial’s work again for review. I’ll be bringing up 27.03 days again soon too.

  23. Paul Vaughan says:

    Here’s the aggregate numeric summary:
    Today I reviewed the images and animations that taught me that.
    This is crystal clear, really easy stuff.
    Civil servants should be to be the ones who publicly present any grand outline of the intricate detail that summarizes succinctly and clearly in aggregate as I effortlessly showed years ago.
    They have to end the lockdowns and do this to restore trust. It’s entirely doable. We’ll watch and see what they do. Because of how savage the lockdowns have been, I advise that this be their last chance. Existing in a deep state of mistrust is UNtenable. Terrorizing the public is unethical.

  24. Paul Vaughan says:

    Mode 1:
    Mode 2:

    Ignoring thermal tides — the heat engines (shaping ~85% of SST variance) — while pointing only to gravitational tides (shaping ~15% of SST variance) is a dirty trick to fool innocent members of the public.

    Duke cousins: Watch out for the “31/62/93” trick. It’s designed to fool you into surrendering your freedom. You’ll lose the farm and sleep on the the street in -30 with no reliable food supply. Gravitational tides play a role in the leftover ~15% and can be paid respect in that context, but modes 1 & 2 are thermal — NOT gravitational — and this should be just plain obvious intuition for anyone with lights on upstairs.

    Be aware that a very nasty campaign of vandalism was waged to attack the SST record:

    When corrections still had not been made after patiently waiting a full year, I initiated a set of strict, permanent boycotts to limit my public participation, knowing the public was being herded towards Orwellian dystopia.

    Shortly thereafter I started developing communication protocols for the sci11UNs censor ship of battles tar gel act to cull, bearing in Mayan-D IT’s AIweis data that saves US from borg assimilation.

    Watch for 27.03 day geometric insight *left on Suggestions-42.
    We’re not homeless yet and if we play our cards *right it will stay that way.

  25. Chaeremon says:

    Found a typo in lunations calc (yielded wrong #), sorry, better approach with 2×Unidos +3×Saros +1×(Meton/10), sum is 3×Inex +5½×Saros (eclipse series 14 members expectable), still shows that (off by months) statistical 10×LNC is not astronomical 186 yrs (base as suggested 3× of Ian’s 62 yrs).
    P.S. winters of 1433/4 & 1437/8 is Not + Never anything about 1800 yrs Interval (a date ≠ an interval); have a great time with 1×Date = 1×Interval thing (pull the other one, is that Englisch?).
    P.P.S. following the JPL/ephemerides data, one can expect perihelion + perigee events, if during ±5×LNC then, close to 1169AD (overlaps the phantom dark ages of H. Illig).

  26. oldbrew says:

    I forgot about this post…

    Lunar connection: the Saros, nodal and apsidal cycles
    Posted: September 8, 2019
    – – –
    Keeling & Whorf say:

    Millennial Repetitions.
    The perigean eclipse cycle also influences tidal events on the millennial time scale because the return time for near coincidence of events of this cycle with perihelion is approximately 1,800 years. We propose that the repeat time of millennial extremes in tide raising forces, discussed below, relates to this return time, although the actual timing of such millennial events must be irregular, being sensitive to the exact time of syzygy (ref. 23, pp. 201–249). The near coincidence of perihelion with this cycle in the present millennium occurred near the time of a climactic tidal event in A.D. 1433 (ref. 6, p. 220).


    Worth a look. They accept there are issues:

    The motions of the Earth and Moon, although periodic, do not produce truly periodic strong tidal events, because these events require the near coincidence of four incommensurate recurring astronomical relationships, namely syzygy, perigee, eclipse, and perihelion. This circumstance, although adding complexity to the analysis, may, however, be an asset in proving a connection between tides and temperature, because interrupted or transient tidal periodicities should produce characteristic signatures of tidal forcing in temperature records.

  27. Chaeremon says:

    Re: near coincidence of four incommensurate recurring astronomical relationships, namely syzygy, perigee, eclipse, and perihelion

    Willkommen an Board, new passengers 😉

    The LAC draws pattern (form on x/y chart: lid of eye) when filtered for apsis nearest to equinox, look for 1666ad March and then 241 yrs in both directions … yes I work with two dates per interval — ahoy statistical voodoo correlations.
    The LNC draws also pattern (linked above), choose the 242 yrs pattern — ahoy again, statistical voodoo correlations.

    Going back + forth one single step finds that the overlap of both intervals shrinks / widens by 1 (one) year.

    Previous stop: 1907ad, next stop: 2148ad, you might want to check earlier dates for closeness to 1433.

    Dunno how long the series, JPL/ephemerides dilutes towards the past.

  28. oldbrew says:

    The lunar nodal cycle can be described as the beat period of the draconic and tropical years.

    After 14 nodal cycles the (unnamed?) beat period of the Saros and apsidal cycles has occurred 15 times, i.e. exactly once more than the nodal cycle occurrences.

  29. Chaeremon says:

    Re: Saros and 14 nodal cycles

    The interval of 14 nodal cycles does not divide even (or ½) into Saros 223 lunations.

    But as ballpark figure it is 260.~5~ years (“sacred” Maya 260 figure), therefore: looking into eclipse canon for Equinox 2015ad and 1755ad, the pair at interval’s ends is 2015 Sept and 1755 Mar, good shot.

    Now this is 4×Unidos (260.0~ yrs) +1×Semester lun. (6 lunations), hence 9×Inex, again not Saros. Here the Solar anomaly is ~3° short of 180°, the Lunar anomaly is that of 3×3Inex = 3×Triad (can, as 2015 Sept shows, converge to perigee, but must not).

    B.t.w. 9×Inex (3×Triad) is 242 years pattern, from LNC pattern above, plus 229 lunations (1×Saros +1×Semester lun.); confirms ballpark of 14 nodal intervals at eclipse distance.

  30. Chaeremon says:

    Woops, line missing in the previous: anomaly (3×Triad)/15 ≅ nodal (3×Triad)/14, numerically.
    Also please note that 3×Triad (uneven Inex) is always alternate node at interval’s ends.

  31. oldbrew says:

    Chaeremon: The interval of 14 nodal cycles does not divide even (or ½) into Saros 223 lunations.

    True, but that’s not unusual for a beat period. See the blog post I linked to – the whole numbers are in there, at a much longer period.

    Those numbers show that the nodal cycle-Saros beat period (575.35~ years) occurs 8 times for every 13*21 nodal-apsidal beat periods (16.86~ years). I hadn’t noticed that before – 8,13,21 are Fibonacci numbers.

    Btw 575.35 years is about 26 Hale cycles, and is a period that appeared in another context in one of the PRP papers. A subject for a future blog post, delayed by the ‘lockdown’ 😎

  32. Chaeremon says:

    Re: 16.86~ years

    Take that ×24 then there is eclipse distance (9×Inex +8×Saros), members 11 expectable, almost same apsis (but LAC not # whole), declination of equinox at one end & declination of solstice at other end.
    So, equinox (years) is not whole & LAC is not whole. Did you try times 2×24?

  33. oldbrew says:

    742 nodal = 766 Saros = 1561 apsidal

    Those are the numbers here:

    Then: 1561 = 223*7 (7 apsidal = 766 synodic months (SM))
    1 Saros = 223 SM

    1561-766 = 53*15
    742 = 53*14
    Ratio = 14:15

    1561-742 = 819 = 3*13*21
    766-742 = 24 = 3*8

    3 factor cancels, leaving ratio of 8:(13*21)
    That’s the model.

  34. Chaeremon says:

    Re: 7 apsidal

    This is 766.06 synodic, says JPL ephemerides for 2299ad to 1583ad; I don’t multiply with such big slippage — better expose dates of interval’s ends.

    In my work I search for equinoctial bounds in multiples of anomalistic (easy for nodical, see link to LNC pattern).

  35. oldbrew says:

    What answer do you get from:
    (Anomalistic Month * Tropical Month) / (Anomalistic Month – Tropical Month) — multiplied by 7 and converted to Synodic Months?

  36. Chaeremon says:

    @oldbrew, Re: what answer

    Try your formular with 565.98~ yrs (7000 lunations, 354.0~ synodic Venus), this I can lookup in JPL/ephemerides.

  37. oldbrew says:

    Q: (Anomalistic Month * Tropical Month) / (Anomalistic Month – Tropical Month) — multiplied by 7 and converted to Synodic Months?

    A: (27.554549*27.321582) / (27.554549-27.321582) = 3231.5042 days

    Alternative using full moon cycle and tropical year:
    (411.78443*365.24219) / (411.78443-365.24219) = 3231.4955 days

    766 Synodic months / 7 = 3231.4901 days = 99.9997% match to 3231.5 days.
    – – –
    From the recent thread: 728 (7*104) apsidal = 6441 trop. years (= 766*104 synodic months)
    6441*365.24219 / 728 = 3231.4902 days

    The sum of the number of full moon cycles and apsidal cycles in a period should equal the number of tropical years, by definition. Or to put it another way, since TY and FMC have published values:
    TY no. minus FMC no. = Aps. no.

  38. oldbrew says:

    Re 7000 lunations:

    7000*29.5305889 = 206714.12 days
    206714.12 days / 365.24219 = 565.9645 trop. yrs.
    206714.12 days / 411.78443 = 501.99595 full moon cycles
    565.9645 – 501.99595 = 63.96855 apsidal cycles

    206714.12 days / 63.96855 = 3231.4961 days per apsidal
    (3231.4961*7) / 29.5305889 = 766.00141

    Looks a lot like 766 there.

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