The Saros cycle by numbers

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

Abstract
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
223-19(DY) = 204 LE2

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).