The Metonic cycle in long-term lunar harmonics

Posted: August 30, 2019 by oldbrew in Maths, moon, solar system dynamics
Tags: , ,

Image credit: interactivestars.com


In 2015 this post discussed long-term lunar precession from an apsidal, or anomalistic, standpoint.

We saw that all the numbers related to an exact number (339) of Metonic cycles (19 tropical years each, as discussed below).

Here we show the equivalent from a nodal, or draconic, standpoint.

Again, all the numbers relate to an exact number (337 this time) of Metonic cycles.

6747 draconic years (DY) = 6403 tropical years (TY) = 344 lunar nodal cycles (LNC)
(6747 – 6403 = 344).
6747 * 346.62008 days = 2,338,645.68
6403 * 365.24219 days = 2,338,645.74

[click on chart to enlarge]


The original chart had anomalistic months (AM); this one has draconic months (DM).

Together these posts put the Metonic cycle at the heart of the long-term harmonics of lunar motion, as related to the Earth.

NB – Wikipedia says:
Nineteen tropical years are about two hours shorter than 235 synodic months. The Metonic cycle’s error is, therefore, one full day every 219 years, or 12.4 parts per million.

19 tropical years = 6,939.602 days (12 × 354-day years + 7 × 384-day years + 3.6 days).
235 synodic months (lunar phases) = 6,939.688 days (Metonic period by definition).
254 sidereal months (lunar orbits) = 6,939.702 days (19 + 235 = 254).
255 draconic months (lunar nodes) = 6,939.1161 days.

The contention here is that there is no ‘error’ in the Metonic cycle – it is 19 tropical years. What happens is that after every 6400 years or so, one synodic month and one tropical month have been ‘lost’ (compared to multiples of 235 and 254 respectively), a type of precession.

Comments
  1. oldbrew says:

    The commonly used solar numbers, solar sidereal rotation and Carrington rotation, line up at seven Metonic cycles.

    1914 SSR * 25.38 = 48577.32 days
    1781 CR * 27.2753 = 48577.309 days
    1914 – 1781 = 133 tropical years
    133 * 365.24219 = 48577.211 days

    133 / 19 = 7 Metonic

    This can be taken further – maybe tomorrow 😋

  2. oldbrew says:

    Re earlier comment: what do 1914 solar and 1781 Carrington rotations tell us? To find out, we can refer to the post.

    ‘235 synodic months (lunar phases) = 6,939.688 days’ and
    ‘254 sidereal months (lunar orbits) = 6,939.702 days (19 + 235 = 254)’

    In 7 Metonic cycles:
    1781 = 254 * 7, +3
    1914 = 273 * 7, +3

    273 = 254 + 19

    The solar rotations have a lunar equivalent, which is the axial period i.e. the time taken for the number of tropical years and the number of tropical months to sum to 1:
    (TY * TM) / (TY + TM) = 25.42006 days

    133 TY / 25.42006 = 1910.9793 days — just over 3 less than the solar rotations.

    In 337 Metonic cycles (see chart above) these will sum to 85597 + 6403 = 92000.

  3. E.M.Smith says:

    This is all fun numerology, but is it important in some way?

    Does it mean anything or influence or change anything?

    Does setting out some ratio of things tell us anything other than the ratio?

  4. oldbrew says:

    Another confirmation of the numbers can be found in the lunar evections in latitude (called LE2 here: https://tallbloke.wordpress.com/2018/12/02/why-phi-a-lunar-evection-model-part-2/)’

    From the current post: 6747 * 346.62008 days = 2,338,645.68
    2,338,645.74 / 32.280777 (LE2) = 72447.004
    79194 SM – 72447 LE2 = 6747 DM (see chart above)

    72447 is divisible by 19, result 3813 (3*31*41).
    So 337 tropical years = exactly 3813 lunar evections in latitude.
    (337 is a prime number – again, see chart above).

  5. oldbrew says:

    Sorry EMS – WordPress sent your comment to the bin, just found it.

    I like to record these obvious matches of different planetary or lunar phenomena when they show up. If no immediate usefulness is apparent, it may happen later.

    In any case it looks as if the Metonic cycle is a bit underrated in the Earth-Moon-Sun dynamics. As I mentioned above, this post is the nodal/draconic counterpart to the 2015 apsidal/anomalistic post.