Movements of the Moon

MoonThe Metonic Cycle - Dozens and ThirteensMoon

AN EASY QUESTION FOR ASTRONOMERS?

Here's a very interesting question which many astronomers would initially imagine is easy to answer:

"If you saw the Full Moon above Orion tonight, when could you expect to see another Full Moon in EXACTLY the same position among the stars again??"

If that's got you thinking, don't be surprised. The answer will not roll of your tongue, even if you are an astronomer! The first time I was asked that question, I was at a loss. You see, the Moon's movements through the sky are not straightforward, like those of its companion, the Sun. With the Sun, we know it takes 365 and a quarter days to make a full journey through the Zodiac and return to the same position in the sky again. It follows the same imaginary line (Ecliptic) every time it does this journey. It's regular and easy to follow.

The picture on right shows a Full Moon above Orion on a sample date, December 18, 2021, at 23:57. When will the Full Moon return to exactly that position between Taurus' horns?

The Moon above Orion


TRYING TO WORK IT OUT
The Moon makes its full journey through the sky in, roughly speaking, 27 days (the exact decimal figure is 27.322 days). That's called the Siderial Lunar Month, or the Tropical Lunar Month. Simple enough, one would think. So the Moon returns to the same background stars every 27.3 days. There's the answer to the question. But wait - the question is "If you saw the Full Moon above Orion tonight, when could you expect to see another Full Moon in EXACTLY the same position among the stars again??"

And herein lies the problem. While the Moon takes 27.3 days (Tropical Month) to return to the same background stars, it does not return to the same phase until two days after. In other words, the time it takes the Moon to return from one Full phase to the next is, roughly speaking, 29 and a half days. (Actual period expressed as a decimal is 29.5306 days). That period is called a Synodic Lunar Month.

So, we have a Full Moon above Orion's outstretched hand. We know the Moon will come back to this position in 27.3 days (Tropical Month), but it won't be full until 29.5 days (Synodic Month). So when can we expect to see it Full again, and back in the same position between the tips of Taurus' horns? We will need to count the number of same phases (Synodic Months) and the number of returns Orion's hand (Tropical Months) to find out.

THE LUNAR YEAR
12 synodic months, or 12 returns of the Moon to the same phase, forms the period of time known as a lunar year. If you start from the Full Moon closest to the time of Winter Solstice (Dec. 21st), and count how many times the Moon returns to this position and how many times it returns to the same phase, you will find that in the time it has returned to Full Moon 12 times, it will have passed Orion's hand 13 times. This is one lunar year. 12 same shapes, 13 returns to the same stars. This lunar year is exactly 354.372 days long, which is a whole 11 days shorter than a Solar tropical year.

So, could that be the answer? Is 12 returns to the same shape, one Lunar Year, the time it takes the Moon to return to exactly the same background stars?

Let's try it out. Using an astronomy program such as SkyMap Pro, go forward 354 days from our sample date, Dec. 18 2021, and we get December 7, 2022. This is the first time since our start date that the Full Moon is visible in the horns of Taurus, so it's pretty close. But it's not bang on. Remember, we're looking for the Full Moon in exactly the same position.

If we wait another lunar year, another 12 returns to the same phase, 13 returns to the same shape, we see the Full Moon in this region of the sky again, but its position is under the Pleiades, a bit west from the original position. So we have seen 24 Full Moons and 26 returns to this part of the sky. This is two pure lunar years. To get a more accurate return of the Full Moon to the horns of Taurus, wait another Synodic Month. This takes us to December 26, five days after Winter Solstice, the year 2023. We have seen 25 Full Moons and 27 returns to Orion's hand.

One year later

In order to keep the lunar periods attached to the solar year (remember we are watching for the Full Moon closest to Winter Solstice), we have added 1 "pure" lunar year containing 12 synodic months, 13 tropical months, with a period we will call the lunar "leap" year - 13 synodic months, 14 tropical months. This is a very valuable 'first lesson' in learning the Metonic cycle - "Dozens and Thirteens". We can express these periods in an easy-to-remember fashion as follows:

12,I

12 synodic months ending 11 days before 1 tropical year

II,25

25 synodic months ending 8 days after 2 tropical years

In this notation, developed by Charlie Scribner, the 12 comes before the I because the 12 synodic months ends 11 days BEFORE 1 year. In the second period, the 25 follows the II because the 25 sm ends 8 days AFTER 2 years.

We use the period counts of same Moon phases and returns to the same stars, called the Synodic month and Tropical month, to warn us when to pay close attention to what the Sun is doing and to better manage time. If we continue our series, we add another "pure lunar year". This time, we will see the Full Moon for the 37th time, and we've seen it pass Orion's hand 40 times. It's now December 14, 2024 and the Full Moon is this time located just above the upper horn of Taurus. This gives us the third Metonic interval:

37,III

37 synodic months ending 3 days before 3 tropical years

The numbers of tropical years in our evolving series have an interesting quality. They are equal to the numbers of tropical months (in the latest instance 40) minus the numbers of synodic months (37). So 3ty = 40tm - 37sm.

Remember the formula: TY = TM - SM

The number of tropical years equals the number of tropical months minus the number of synodic months.

When the synodic month and tropical month come back into phase with one another, when same shapes return to the same stars, the synodic and tropical months also come back into phase with the sun and his seasons. The numbers of tropical years are equal to the numbers of tropical months plus the number of synodic months. The three different periods form what we now call an harmonic. Adding another pure lunar year takes us to:

49,IV,53

49 synodic months ending 14 days before 4 tropical years (53 tropical lunar months)

If we add a second lunar leap year to the series, we arrive at another Metonic interval:

V,62,67

62 synodic months ending 5 days after 5 tropical years (67 tropical lunar months)

The first lesson, "Dozens and Thirteens", continues:

74,VI,80

74 synodic months ending 6 days before 6 tropical years (80 tropical lunar months)

VIII,87,94

87 synodic months ending 13 days after 7 tropical years (94 tropical lunar months)

VIII,99,107

99 synodic months ending 2 days after 8 tropical years (107 tropical lunar months)

Metonic interval V62
Metonic interval VIII,99

The Full Moon back in Orion's hand after Metonic Interval V,62 - five days after 5 tropical years, date: 23 Dec. 2026.

The Full Moon in Orion's hand again, this time at Metonic Interval VIII,99 - 2 days after 8 tropical years, date: 20 Dec. 2029.

This latest Metonic interval, VIII,99, brings the Full Moon in Orion's hand to within just two days of the date of the same Moon we saw eight years back. The original observation was made on December 18th (2021), with the current observation on December 20th (2029) and since our very first Full Moon eight years ago we have seen 99 Full Moons, and a whopping 107 returns of the Moon to Orion's hand. That's a lot of moon watching!

Here's an interesting fact: This VIII,99 Metonic subunit which brings the same phase of the Moon back to the same part of the sky two days after eight solar years, is actually made up of two of the smaller intervals. You can add them up yourself to see how it works:

37,III,40

37 synodic months ending 3 days before 3 tropical years

V,62,67

62 synodic months ending 5 days after 5 tropical years
VIII,99,107 99 synodic months ending 2 days after 8 tropical years

 

37,III,40 + V,62,67 = VIII,99,107

Adding 37,III, the 3 days before, to VIII,99, the 2 days after, finds the even stronger tie:

136,XI,147

136 synodic months ending about a day before 11 tropical years

Add VIII,99 to 136,XI and find the answer to the question!:

XIX,235,254

235 synodic months ending at the same time as 19 tropical years or 254 tropical lunar months.

This is the Metonic Cycle, and it brings the Full Moon back to where we first observed it, between the horns of Taurus all those long 19 years ago.

The Full Moon back above Orion at the 11-year Metonic interval of 136 synodic months. This interval brings the same phase to within one day of its original date.

Finally, the Full Moon returns to the exact position where we saw it 19 years before. Compare this with the very first image at the top of the page.

It's most incredible. If you see the Moon tonight, watch closely its position and phase, because you won't see it returning to that exact position and phase for another 19 years, or 235 synodic months, 254 tropical lunar months. You might not even be alive the next time it happens. Try it with a computer program like SkyMap. Just pick a date and look at the phase and position of the Moon and add 19 years. Here's how it works out in terms of actual days:

365.24 days (solar tropical year) x 19 = 6939.56 days

29.5306 days (lunar synodic month) x 235 = 6939.691 days

27.322 days (lunar tropical month) x 254= 6939.788 days

But remember, you DO NOT have to know the day counts in order to see the Metonic Cycle in action. It's the whole period counts which give us the intervals. We don't think of 12,I -11 as being 354 days. We think of it as being 12 returns of the Moon to the same shape, in this case, Full Moon , and that this is 11 days before the Winter Solstice sunrise. Try it with another example, this time the Full Moon on Spring Equinox, 2000, March 20, the old pagan Easter, with the Moon under Denebola, the tail of Leo the Lion, in the stars of Virgo.

Full Moon at Spring Equinox, 2000
Full Moon at Spring Equinox 2019

Full Moon on March 20, 2000, the Spring Equinox, under Denebola in the stars of Virgo.

Full Moon 19 years later, March 20, 2019, in exactly the same location under Leo's tail.

You can try some of the other intervals too, and watch how the Full Moon returns to this part of the sky. But remember, think of whole period counts instead of big numbers of days. Add 12 synodic months, one solar year minus 11 days.

If you can look out a window and see a moon among the stars right now, you will see this Moon return to the same shape and passing the same stars in 19 tropical years, 235 synodic months, 254 tropical months. We have seen how this can be uncovered visually, without the need for complex mathematics and astronomical instrumentation, and also how we do not need to know the actual day counts because we can record the cycle with period counts - synodic months, tropical months and tropical years. We don't even need to know about fractions. This "Metonic Cycle" is named after a Greek, called Meton, who lived in the 5th Century BC, and who claimed he discovered the cycle on his own. It seems that simple visual observations are all that's needed to see the cycle . . . and there's plenty of evidence it was known and recorded long before Meton ever existed.

If you have SkyMap, you can download the starter maps which I have used for the examples on this page and open them with SkyMap (after you save the file, go to SkyMap, click on 'File' -> 'Open' and locate the file). Here are the files:

Full Moon in Orion's hand, 18th Dec. 2021

Full Moon on Spring Equinox, 2000

PAGES OF INTEREST:
Calendarstone: See how the Metonic interval V,62,67 was recorded on a 5,000-year-old stone at Knowth.
Lunar stone: More lunar calculations at Knowth.