*It turns out that the previous post was only one half of the lunar evection story, so this post is the other half.*

There are two variations to lunar evection, namely evection in longitude (the subject of the previous post) and evection in latitude, which ‘generates a perturbation in the lunar ecliptic latitude’ (source).

It’s found that the first is tied to the full moon cycle and the second to the draconic year.

The lunar evection in latitude is the beat period of the synodic month and the draconic year. The result is that it should average about 32.280777 days (46484.32 minutes).

Comparing synodic months (SM), draconic months (AM), and lunar evections in latitude (LE2) with the draconic year (DY) we find:

1 DY = 11.737662 SM

1 DY = 11.737662 + 1 = 12.737662 DM

1 DY = 11.737662 – 1 = 10.737662 LE2

Since 0.737662 * 385 = 283.99987 (284), we can say for our model:

3850 + 284 = 4134 LE2 (55*75, +9)

4235 + 284 = 4519 SM (55*82, +9)

4620 + 284 = 4904 DM (55*89, +9)

385 DY = 55*7 and 89-82 = 7 and 82-75 = 7

55 and 89 are Fibonacci numbers.

A very near-equivalent is: 61 DY = 655 LE2 = 716 SM = 777 DM.

That should be accurate enough (> 99.999%) for most purposes.

**Analysis**

The lunar evection in latitude is the beat period of the draconic year and the synodic month.

(Alternatively, DY = the beat period of LE2 and SM).

The difference between the the number of evections in longitude (LE) and evections in latitude in any given period = the difference between the number of draconic years and full moon cycles in the same period i.e.:

(Number of) LE – LE2 = (Number of) DY – FMC

The key period is the ‘lunar wobble’, approx. 2190.35 days, where both differences = 1.

This is the ‘axial’ period of the lunar nodal and apsidal cycles, i.e. the time taken for the sum of their occurrences to equal 1 (about 0.3222 nodal plus 0.6778 apsidal).

This model from an earlier post shows how some of these lunar periods fit together.

Note: ‘lunar wobble’ is shown as RLA in the chart.

The numbers of DY, FMC and RLA are divisible by 7, so:

297 DY – 250 FMC = 47 RLA.

LE – LE2 will also equal 47 in that period, as will draconic minus anomalistic months.

Source: Two long-term models of lunar cycles

**Update 15/12/18**

The ‘lunar wobble’ is a period where the number of:

Draconic months minus anomalistic months = 1

Draconic years minus full moon cycles = 1

Evections in longitude minus evections in latitude = 1

Apsidal cycle (portion of 1) plus nodal cycle (portion of 1) = 1

Apsidal: Anomalistic months, full moon cycle, evections in longitude, apsidal cycle

Nodal: Draconic months, draconic years, evections in latitude, nodal cycle

The Saros eclipse cycle = 6585.3212 days.

After one saros, the Moon will have completed roughly an integer number of synodic, draconic, and anomalistic periods (223, 242, and 239) and the Earth-Sun-Moon geometry will be nearly identical: the Moon will have the same phase and be at the same node and the same distance from the Earth.https://en.wikipedia.org/wiki/Saros_(astronomy)

204 lunar evections in latitude = 6585.2785 days.

The difference from the Saros is just over an hour.

5 lunar nodal cycles = 1053 evections (LE2) = 13 * 81 (81 = 3^4th power).

3, 5, and 13 are Fibonacci numbers.

The ratio of 81 evections (LE2) to half the lunar apsidal cycle (perigee to apogee, or vice versa) is:

2614.7429 days (LE2) : 1615.7452 days = 1.6182891:1 (99.984% match to Phi, aka the golden ratio).

oldbrew,

Here are some re-statements of what you have posted plus a few additions of my own.

Let:

SYM = Synodic month = 29.53 05889 days

AM = Anomalistic month = 27.554550 days

DM = Draconic month = 27.212220 days

DY = Draconic year = 346.620075883 days

FMC = Full Moon Cycle = 411.78443029 days

LE1 = Lunar evection longitude cycle = 31.811941 days

LE2 = Lunar evection latitude cycle = 32.280777 days

++++++++++++++

1 / (SYM/2) — 1 / DM = 1 / LE2_____________________(1)

__1 / (SYM) — 1 / DY = 1 / LE2_____________________(2)

Hence by eliminating LE2,

____________1 / DY = 1 / DM — 1 / SYM_____________(3)

++++++++++++++

1 / (SYM/2) — 1 / AM = 1 / LE1_____________________(4)

__1 / (SYM) — 1 / FMC = 1 / LE1_____________________(5)

Hence by eliminating LE1,

____________1 / FMC = 1 / AM — 1 / SYM_____________(6)

++++++++++++++

From (2) and (5)

____ 1 / FMC = 1 / SYM — 1 / LE1____________________(7)

______ 1 / DY = 1 / SYM — 1 / LE2___________________(8)

Hence by elimination SYM,

_____ 1 / LE1 — 1 / LE2 = 1 / DY — 1 / FMC = 2190.34838058 days ____(9)

___________________________________ = 5.99674257 sidreal years

i.e. the number of (LE1 — LE2) = the number of (DY — FMC)

+++++++++++++

Let :

Sidereal year = 365.256363 days

Venus Sidereal year = 224.70069 days

SVE = Synodic Period of Venus-Earth 583.9206726 days

LE1 = Lunar evection longitude cycle = 31.811941 days

LE2 = Lunar evection latitude cycle = 32.280777 days

Finally,

I have shown in a previous post that:

___ 1 / (2*DY) + 1 / (9*FMC ) = 1 / SVE_________________(10)

[N.B: This gives an SVE of 583.99961228 days which is only slightly off the nominal value – the difference could be explained by the relative drifts of the Perihelia of the orbits of Venus and the Earth.]

This means that:

___ (11 / SYM) — (9 / LE2) — (2 / LE1) = (18 / SVE) __________(11)

Equations (10) and (11) both indicate that there is a strong connection between the changing tilt and shape of the lunar orbit and the synodic period of Venus and the Earth.

Equations (10) and (11) both indicate that there is a strong connection between the changing tilt and shape of the lunar orbit and the synodic period of Venus and the Earth.4 lunar wobbles = 5 V-E * 3 (within about 2.25 days in just under 24 years).

The wobble is the key period here. It’s when the sum of nodal and apsidal cycles is 1.

Add to earlier list:

LE1 – LE2 = LNC + LAC (sum of lunar nodal and apsidal cycles) in any period.

E.g. in one tropical year both would be 0.1667~