Lunar evections and the Saros cycle

Posted: May 7, 2019 by oldbrew in Maths, moon, solar system dynamics

Credit: Matthew Zimmerman @ English Wikipedia

The Saros cycle can be used to predict eclipses of the Sun and Moon, and is usually defined as 223 lunar synodic months, or about 11 days over 18 years.

But there are a few other lunar-related periods which can used to arrive at 223.

One Saros cycle can be said to be the difference between the number of:
— anomalistic months and full moon cycles (239 – 16)
— draconic months and draconic years (242 – 19)
— tropical months and tropical years (241 – 18)

That may be fairly well known, but then there are the lunar evections.

One Saros cycle can also be the sum of the number of:
— draconic years and lunar evections in latitude (204 + 19)
— full moon cycles and lunar evections in longitude (207 + 16)

In each case (except tropical months/years) both numbers are marginally lower than the whole number, but the difference is still exactly 223.

With tropical months/years the evections are marginally higher than the whole number, but the sum is still exactly 223.

To convert the ‘marginally’ inexact numbers to whole ones we need to multiply by 766, but that’s another story.

  1. tallbloke says:

    How often is there a triple conjunction of Earth-Moon, Jupiter and Venus?

  2. oldbrew says:

    Need some numbers to work with, TB.

    Another way to look at the 223 synodic months:
    204 evections (lat.) + 242 DM = 223 * 2
    207 evections (long.) + 239 AM = 223 * 2

  3. tallbloke says:

    Eyeballing the solar sym I get

    dec 6 1858
    mar 6 1862
    dec 18 1882
    mar 11 1886
    sep 17 1903
    sep 19 1927
    mar 23 1945
    jun 20 1948
    mar 23 1969
    jun 23 1972
    dec 31 1989
    Apr 2 1993
    Jan 8 2014

    diffs of whole years


    Average: 13.
    Earth-Venus conjunction cycle is 8, so we have adjacent fibonacci numbers there.

  4. oldbrew says:

    These dates match up to 110.3~ years (170 J-V or 101 J-E or 69 V-E):

    dec 6 1858 – mar 23 1969
    mar 6 1862 – jun 23 1972
    dec 18 1882 – apr 2 1993
    sep 17 1903 – Jan 8 2014

    No Moon link though?
    Ratio of 110.3~ year JEV period to the Jose cycle is about 1.621:1

  5. Chaeremon says:

    @oldbrew Re: Moon link

    e.g. 1858/12/06 smallest |abs| declination (turning point) during +/- 6 months, climbed to -27.99330°

    e.g. 1969/03/25 largest |abs| declination (turning point) during +/- 6 months, climbed to 28.72559°

    Didn’t check the other ones; who’s looking at +/- max |abs| declination, anyways, (except for draconic related).

    Solex data points say the avg temp dist is 1703.75 moons, 50312.7 days.

  6. Chaeremon says:

    Re: [above] May 7, 2019 at 4:02 pm
    Sorry, the last line (days, moons) has copy/paste errors; must tame my new XPS ubuntu 18.4 …

  7. tallbloke says:

    OB: No Moon link though?

    That’s what I was fishing for. Both V and J have similar magnitude grav effects on E-M system.
    Your 18 year 11 day period sits between Nodal and 2# precession lunar cycles.

  8. oldbrew says:

    5 nodal cycles = 7.845158 Jupiter orbits = 3 * 2.6150527 [34/13 = 2.6153846]
    5 nodal cycles is also exactly 13*81 evections in latitude.

    742 nodal cycles = 766 Saros (see link at end of blog post).

  9. oldbrew says:

    The beat period of the rotations of Mercury and Venus is 77.300419 days (BPMV).
    So (BPMV / Me(r)) – (BPMV / Ve(r)) = 1.

    77.300419 / 29.530589 (lunar synodic month) = 2.6176389 [89/34 = 2.617647].
    189 BPMV (21*9) = 40 tropical years.

  10. oldbrew says:

    TB – re: Jan 8 2014

    2013–14 United Kingdom winter floods
    Date: 5 December 2013 – 25 February 2014

    The Met Office reported the storms were responsible for the wettest December to January period since 1876.

    JEV in Jan. 2014 – Jupiter-Sun line close to SS barycentre.

  11. tallbloke says:

    Hmm. Interesting OB.

    Apr 2 1993
    The Great Flood of 1993 (or “Great Mississippi and Missouri Rivers Flood of 1993”) was a flood that occurred in the Midwestern United States, along the Mississippi and Missouri rivers and their tributaries, from April to October 1993. The flood was among the most costly and devastating to ever occur in the United States, with $15 billion in damages.

    dec 31 1989

    jun 23 1972

    One of the worst rainstorms in decades slammed up the Eastern Seaboard yesterday, dropping a torrent of water that knocked out transportation and communications, sent tens of thousands fleeing from their homes and unleashed deadly flash floods. The death toll from the tropical storm known as Agnes, as it moved slowly in a 250mile front from the Carolinas and Virginia into Pennsylvania and New York, was at least 26, with many persons listed as missing. And authorities said it was certain to rise as rescue workers reached submerged vehicles. Earlier, the storm left 16 dead in Cuba and Florida. It was one of the most widespread flood disaster situations in the nation’s history, officials at American National Red Cross headquarters in Washington said.

    mar 23 1969
    According to hydrologist Guido van Langenhove in the Ministry of Agriculture, the highest record for the Zambezi was 8,16m in 1969.
    “Reports from Zambia are that water levels there are also the highest on record and that the flood will still be increasing, although at least rainfall would be less in the coming days,”

  12. oldbrew says:

    Maybe the JEV triple can act as a trigger of some kind, leading to weather disturbances if initial conditions allow?

  13. tallbloke says:

    Well there are floods somewhere in the world on any given date. But some of these seem to have been particularly severe ones. I don’t have the time or patience for a serious statistical study.

  14. waterside4 says:

    Please Tallbloke, I am an old wrinklie and you nearly gave me a heart attack. I read Soros cycle above!
    I am really in dread of old Gregor Sorearse and all the devastation he is wreaking on the world.

  15. oldbrew says:

    Re – oldbrew says: May 7, 2019 at 11:15 am

    What it boils down to is:

    Synodic month / anomalistic month, plus synodic month / evection in longitude = 2
    Synodic month / draconic month, plus synodic month / evection in latitude = 2

    Beat period of anomalistic month and evection in longitude = half the full moon cycle
    Beat period of draconic month and evection in latitude = half the draconic year

    In any period the sum of each pair must be the same, and be twice the number of synodic months.

  16. oldbrew,

    What you have found is a direct result of the very definitions of lunar evection by longitude and latitude.

    Lunar evection in longitude is the periodic oscillation in (ecliptic) longitude of the position of the real Moon about the position of the (fictitious) Mean Moon by +/- 1.274 degrees once every Full Moon Cycle (= 411.87443 days) or evective year.

    In other words, the amount of lunar evection in longitude is simply a function of the angle between the line-of-syzygies (i.e. the Earth-Sun line) and the line-of-Apsides.


    AM = anomalistic month = 27.554549878 days
    SYM = Synodic month = 29.530588853 days
    FMC = Full Moon Cycle = 411.78443025 days
    EVL = Evective (in longitude) month = 31.8119411 days

    Since there are two ends to the line-of-Apsides, this means that:

    (1/AM) — (1/EVL) = 1 / (FMC/2) = 2 / FMC

    and given that by definition:

    (1/AM) – (1/SYM) = 1/FMC


    (2/FMC) = (1/AM) — (1/EVL) = (2/AM) — (2/SYM)

    which can re-arranged to give:

    (2/SYM) = (1/AM) + (1/EVL)


    Similarly, Lunar evection in latitude is the periodic oscillation in ecliptic) latitude of the position of the real Moon about the position of the (fictitious) Mean Moon once every Draconic Year (= 411.87443 days) or evective year.

    In other words, the amount of lunar evection in latitude is simply a function of the angle between the line-of-syzygies (i.e. the Earth-Sun line) and the line-of-nodes.


    DM = Draconic month = 27.21222082 days
    SYM = Synodic month = 29.530588853 days
    DY = Draconic Year = 346.62007644 days
    EVLat = Evective (in latitude) month ~ 32.2807767 days

    Since there are two ends to the line-of-Apsides line-of-Nodes, this means that:

    (1/DM) — (1/EVLat) = 1 / (DY/2) = 2 / DY

    and given that by definition:

    (1/DM) – (1/SYM) = 1/DY


    (2/DY) = (1/DM) — (1/EVLat) = (2/DM) — (2/SYM)

    which can re-arranged to give:

    (2/SYM) = (1/DM) + (1/EVLat)


  17. Error: The eighth line from the end of the last post should read:

    “Since there are two ends to the line-of-nodes, this means that:”

  18. oldbrew says:

    Saros long-term cycle (1 Saros = 223 SM)

    182467 DM = 168142 SM = 153817 LE2 = 754 Saros

    182467 – 168142 } = 14325 Draconic Years
    168142 – 153817 } (= 754*19, -1 DY)

    DM = Draconic Month
    SM = Synodic Month
    LE2 = Lunar Evection in Latitude

    754 = 377*2 (Fibonacci numbers)
    204 = 34*3*2 (Fibonacci numbers)
    223 = 204+19
    242 = 204+19+19

    The cycle is complete when both 1 extra evection and one less draconic month than an exact multiple of 754 have been reached (at the same time), i.e. after 754 Saros.
    (Sum of LE2 and DM = 754*223*2).

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