Lunar cycles, the ‘square year’ and the 1:81 ratio

This post looks at some published data and combines it with results from a previous Talkshop post. Will the number 81 appear again?

The ‘square year’ is a term for 385 draconic years, also known as eclipse years. It equates to 365.37~ Earth years, close to 365.24219 days (one tropical year) squared. [Source – Wikipedia]

The second column from the right (below) from the same source confirms that the square year is 4519 synodic months (SM) = 385 draconic years (aka eclipse cycles, shown as ‘full EY’).

Lunar chart 1

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Lunar chart 1

Now we refer back to the 2015 Talkshop post featuring the chart on the right.

This shows 2079 draconic years (DY) as the period for all those numbers. The square year is 385 DY:
2079 / 385 = 5.4
To convert to a whole number of square years:
5.4 * 5 = 27
27*385 or 5*2079 = 10395 DY.
All the other numbers in the chart also increase by a factor of 5.

Returning to the square year, the 4519 synodic months tells us two things about the 385 DY period:
Anomalistic months = 4519 + 385 = 4904
Evections in latitude = 4519 – 385 = 4134

Evections aren’t often mentioned anywhere, but there’s a description here: ‘Evection in latitude generates a perturbation in the lunar ecliptic latitude that oscillates sinusoidally with a period of 32.3 days’.

One way to think of it is that evections in latitude are nodal (or draconic), and evections in longitude – not relevant to this post – are apsidal (or anomalistic).

Anyway, the number per 385 DY can be shown like this:
4134 = 3*13*106
Checking the 2015 chart above, there’s a clue: 106 lunar nodal cycles (LNC), tagged LP in the chart.
Therefore, multiplying those chart numbers by 5 to give 27*385 DY must mean:
Evections in latitude in the *5 period = 27*3*13*106, and also 5*106 LNC.
The evections can be re-stated as 13*81*106, which in turn means the number of evections in latitude in 5 lunar nodal cycles is 13*81.

Summary:
— In every 5 lunar nodal cycles a sequence of 13 evections in latitude will occur 81 times.
— In a square year a sequence of 13 evections in latitude will occur 318 times.
— 13 evections in latitude = 419.6501 days, or 385/318 eclipse cycles.
* * *
Next time we’ll look at the expanded version of the lunar chart above, bringing all the relevant numbers together. We’ll see if or how the Chandler wobble might fit into the 81 scheme, plus one or two other details not covered so far.