## The Coincidences Between The Months

The Metonic cycle represents a coincidence between the synodic month and the solar year. There are three other definitions of the month we have met (the sidereal, anomalistic, and nodical months), each of them lasting for 27 days plus some fraction. In discussing eclipses we are not much worried about the stars, and so the sidereal month can be laid aside. But consider the mean lengths of the other three types of month:

Synodic month: S = 29.53059 days

(full moon to full moon) Anomalistic month: A = 27.55455 days

(perigee to perigee) Nodical month: N = 27.21222 days

(node to node)

Using those figures we can explore various matters of interest. For example, during a single lunation it is brightest at full moon, but not all full moons are equally bright: if opposition occurs near apogee then the full moon will be dimmer than during an opposi tion near perigee, because that orb is farther from us. We could ask then: How long is the period between those ultra-bright full moons near perigee? The answer is given by multiplying the synodic month by the anomalistic month and dividing by their difference [(S X A) / (S — A)], the result being about 412 days. That value is 13.94 times S: 13 complete synodic months plus about 94 percent of such a month, or 0.06 months (actually 1.64 days) short of the next full moon. Thus starting with a full moon at perigee, the fourteenth full moon will occur about a day and a half after perigee, and there will be a long-term cycle in full moon brightness.

One could take the broad question further. The brightness of full moon will depend upon how far above or below the ecliptic the Moon happens to be at opposition. One might imagine that brighter full moons occur when the Moon is at a node at opposition. In fact that would be the dimmest possible full moon, because that is when a lunar eclipse takes place. (Nevertheless, the brightest the Moon ever gets to be occurs just before a lunar eclipse, because then it is the nearest it ever comes to being precisely opposite the Sun in the sky, and that favors back-scattering of sunlight, plus the bonus of being closest if at perigee.)

Eclipses are what we are interested in here, and in this respect the month lengths we have labeled S, A, and N above have some remarkable relationships. We shall now examine just what sorts of cycles exist by doing a little numerical manipulation.

Full moon occurs near perigee about every 412 days, but over longer intervals there are cycles that are much more precise. Try doing the following sums on your calculator (the justification for them will soon become apparent):

This means that after 223 synodic months the Moon has returned to close to the same node as at the start of that sequence. The difference amounts to merely 51 minutes.

We are also interested in when perigee occurs, so consider the anomalistic month:

That is only about five hours longer than the canonical 223 synodic months above.

Shortly we will see the interval of 6,585.32 days to be extremely significant, but first we must learn about yet another type of year: the eclipse year.

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