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SOME MOON RHYTHMS

By Josephine Coffey jocoffey@worldnet.att.net
Copyright 2000 Josephine Coffey - All rights reserved
Updated December 11, 2000


Acknowledgement

Many thanks to Victor Reijs for his helpful comments on an earlier draft of this paper. Naturally, this does not mean that Victor should be held responsible for anything said here.


INTRODUCTION

I first met Martin Byrne in 1999, around Samhain. He introduced me to the ceremonial complex centered on Knocknarea, the Hill of the Moon. And he said he thought the great astronomical ceremonial sites in Ireland – LoughCrew on the Sliabh na Caillighe, and the Brugh na Bóinne: Dowth, Knowth and Newgrange on the River Boyne, as well as the ones we were visiting that day there within sight of the Bay of Donegal – were as much about the moon as they are about the sun. He took me to Cairn G in Carrowkeel, a small passage cairn with a roof box located near a summit in the Bricklieve mountains. The roof box opens to the northwest, too far north for the setting sun, but not too far for the northernmost reaches of the moon, where the moon goes once every 18 to 19 years

About 6 months later Victor Reijs started the Irish stones group www.egroups.com/group/irish-stones. Early on, there was a lot of discussion about the moon. We came to the realization that there was a good chance that the 2001, 2002 and 2003 summer solstice full moons would shine into Newgrange. It was mentioned a few times in the course of this that Martin Brennan had said that Knowth, which is aligned (roughly) to the solar equinoxes, was about the moon. Brennan’s intuitions about these ceremonial sites, by now 20 years old, have proved to be worth checking out. Also pointed out during the discussion was Gillies MacBain’s observation that the great mound at Knowth has a kerb of 127 boulders. 127 is half the number of tropical moons in the Metonic cycle. Yet another authority mentioned was John North, who said (in an appendix to his Stonehenge: a new interpretation of man and the cosmos, 1996) that the northern reach of the moon is always at an equinox. That would be a connection with Knowth. It was not obvious why the southernmost (and northernmost) [1] moon should always be around equinox - in fact, if the period of the nodes is 18.6 years, it would seem that it couldn’t always be at the equinox. From one cycle to another, the southernmost moon should float, eventually hitting every month in the year. But particularly since it would justify a strong connection between Knowth and the moon, it seemed worth spending some time charting the position of the southernmost moon over time. Fortunately, and this is one of the miracles of the Internet, the data to do such an analysis is easily available. Victor introduced us to the JPL web site http://ssd.jpl.nasa.gov, which provides via e-mail detailed data about numerous celestial objects, including positions of the moon for 3000 BCE to 3000 CE. JPL works for NASA, the American space program, and its practical experience locating objects in the solar system lends confidence in its data (2). It is difficult to overestimate the importance of this data. The motions of the moon are so difficult to figure out that I certainly wouldn’t have been able to derive correct positions. I might have tried, but I wouldn’t have succeeded. But the data is there, so all I had to do was figure out how to ask for it, and then run some simple analytical tools against it. This paper is a report on those analyses. I’ll give the plot away. North was right. The southernmost (and northernmost) moon is always in the spring or the fall, near an equinox. This supports the belief that the builders and users of Knowth were consciously focused on the moon. Also, once you look at the data, it makes sense why the users of Knowth might have been interested in the tropical moon and the Metonic cycle.

The neolithic stone monuments have had a life in the communities in which they have existed for all these centuries. The local people know about them, have great stories about them, and in some cases, attribute a connection between the monuments and the sun, moon and stars. For example, the large unopened passage cairn near Sligo sits atop Knocknarea, the Hill of the Moon; before its excavation, local people near Newgrange told stories about the sun entering the chamber at mid-winter. Outside these local communities, an awareness of the astronomic significance of some of these places has been growing since the 19th century. This awareness got a big boost in the mid 20th century due to the work of two men, one an astronomer, Gerald Hawkins, the other an engineer, Alexander Thom. Hawkins (1965) used a computer, a fairly new tool at the time, to show that Stonehenge was built and used to mark extreme positions reached by the sun and moon during their major cycles - the annual cycle of the sun and the 18.6 year cycle of the moon. Beginning earlier than that, Alexander Thom had been investigating hundreds of stone circles, mostly in Scotland and England, analyzing them for their geometric and construction properties, and also for their possible astronomic alignments. He too found evidence of a deliberate intent to mark the extreme positions of the sun and the moon. The work of Thom in particular inspired many researchers in the new interdisciplinary field of archaeoastronomy in Europe.

One of Thom’s interests in the 19 year lunar nodal cycle had been the 9 minute wobble or perturbation in the lunar orbit, a perturbation with an average period of 173 days that causes the mean declination of the moon to be increased or decreased by 9 minutes, or 15% of a degree. Thom believed that neolithic builders of the stone circles were aware of this perturbation, and that some of the stones in neolithic monuments were deliberately aligned to measure it. (See, for instance, Thom, 1967, chapters 10 and 11). This has tended to frame the interest of later researchers in the lunar nodal cycle. Insofar as this cycle is concerned, they have focused their attention on testing the reasonability of Thom’s belief that the builders of some stone circles were attempting to measure the 9 minute wobble. An excellent and very thorough example of this can be found in Ruggles, 1999. (See also Heggie, 1981).

The analysis in this paper takes a slightly different tack. As mentioned earlier, my chief motivation was to check North’s statement (1996) that the 9 minute wobble would cause the moon to reach an extreme position only when the sun was at an equinox. Because of the availability of the JPL data, I was able to avoid having to theoretically determine the positions the moon might reach at standstills. There is an excellent theoretical discussion of this wobble on Victor Reijs website: www.geniet.demon.nl/eng/moonfluct.htm. The availability of JPL data also permitted me to easily look at data over more than one cycle. This provided a real bonus, the discovery that the distance from one major standstill to another is often a Metonic cycle. One could have deduced this from the fact that the standstills happen only at an equinox - it has to be at half year boundaries, and 18.5 and 19 are the choices for a cycle that’s 18.6 years - but I hadn’t realized it until I saw the data.


BRIEF ASTRONOMICAL BACKGROUND

This will be a very brief discussion of observational astronomy for the sun, earth and moon. For further background, there are excellent introductions to observational astronomy. I particularly like Chapter III of Anthony Aveni’’s Skywatchers of Ancient Mexico (1980) and Joachim Schultz’s Movements and Rhythms of the Stars (1987). Alexander Thom also gives an introduction to astronomy in Chapter 3 of his Megalithic Sites in Britain (1967), but I found that a bit too terse.

Basically, all motions of the sun and moon seen from the earth are of a body (the moon) rotating around itself and revolving around the earth, of a body (the earth) rotating around itself, and a system of bodies (the earth and moon) revolving around the sun. I suppose that’s the way we would actually see it if we had greatly expanded vision and were perched in space someplace looking at the inner solar system. But we’re on the earth, and to us, it looks as if the moon and the sun and the entire dome of the heavens revolve around us from east to west in a day. In addition, it looks as if the moon and the sun travel eastward through the dome of the heavens, making a complete circuit in a month (for the moon), or in a year (for the sun). Each body also shifts its rising position north or south along the horizon every day, making a complete north/south circuit in the course of one cycle

These observational phenomena give rise to the picture astronomers make to initially orient themselves in the heavens.




Figure 1. Celestial sphere, celestial equator and ecliptic

The large circle in the figure represents the the sky, the dome of the heavens. It is conceived of as a sphere, and all the heavenly bodies are conceived of as being located on the inner surface of that sphere. Just as mapmakers locate a given point on the earth with a coordinate system (latitude and longitude), astronomic map-makers have devised a coordinate system to locate points on the celestial sphere. One coordinate is measured from the celestial equator. That is a great circle extended from the earth’s own equator. It is perpendicular to the earth’s poles, the axes around which the earth turns, that is, it is 90 degrees away from the celestial north (or south) pole, the point in the sky around which the dome of the heavens appears to turn. This coordinate is known as declination. It is similar to latitude (3) on an earth map. The other coordinate is taken from one of the points where the celestial equator intersects another great circle, the ecliptic. The ecliptic is the apparent path of the sun across the celestial sphere in the course of the year. It is called ecliptic because an eclipse can happen only when the moon and the sun are on this path. Notice that the ecliptic is at an angle to the celestial equator. It is about 23.5 degrees, which is how far the sun is away from the celestial equator at the winter and summer solstices. The ecliptic intersects the equator at two points, the vernal equinox and the autumnal equinox. The second coordinate is measured from the vernal equinox, and is called right ascension. Since right ascension is in the direction that the sphere turns in one day, you sometimes see it given in hours, though it is also given in degrees, as is declination. Right ascension measures all the way around the great circle of the celestial equator, so it goes from 0 to 360 degrees; declination, which measures from the south pole through the celestial equator to the north pole, goes from -90 degrees at the south pole, through 0 degrees at the equator, to 90 degrees at the north pole. Just as latitude and longitude will give you an exact location on the surface of the earth, declination and right ascension will give an exact location on the surface of the celestial sphere.

There is one more measurement that we’re interested in, and that is the position of a heavenly body as it crosses the horizon at rising or setting. The horizon is also conceived of as a great circle and it too ranges from zero to 360 degrees. The zero point is due north. Measurement is clockwise, so east is 90 degrees, south is 180 degrees, west is 270 degrees. This measurement is called the azimuth. The azimuth depends on declination and right ascension, but it also depends on the (earthly) latitude of the observer. The declination and the right ascension of the sun, say, at a given point of time, is the same all over the earth, but the azimuth may change. As you move north, a difference in declination will result in a greater difference in azimuth.


Figure 2. Celestial sphere, ecliptic and lunar orbit

The ecliptic is the apparent path of the sun. The moon has a path that is close to, but not identical with, the ecliptic. It is inclined at an average of 5.15 degrees to the ecliptic. The relationship between the lunar orbit and the ecliptic is similar to that between the ecliptic and the celestial equator. They are both great circles on the celestial sphere, and they cross one another at two points. Astronomers call the two points the nodes, the ascending node where the moon’s orbit crosses the ecliptic on its way north, the descending node where it crosses on its way south. I didn’t mention this when talking about the ecliptic, but I might as well say it now. The points where the ecliptic crosses the celestial equator (the equinox points) do not stand still. They slowly shift westward along the equator, taking about 26,000 years to complete a circuit. This is known as the precession of the equinoxes. In just the same way, the lunar nodes shift westward. The lunar nodes move much faster than the equinox points - it takes the nodes on average 18.6 years to complete a circuit. It is this cycle that is being examined in this paper. The moon will reach its maximum southernmost position when the ascending node is at the vernal equinox, because at that point, you get full effect of both inclinations - the sun’s -23.5 degrees plus the moon’s -5.15 degrees for a total of -28.65 degrees. Half a cycle later, when the descending node is at the vernal equinox, there is another extreme position, but this time, the lunar orbit is within the ecliptic, so the sun’s -5.15 degrees is subtracted from the sun’s -23.5 degrees for a total of -18.35 degrees. These two positions, the southernmost (and northernmost) position, and, 9 or 9.5 years later, the moon’s minimum southern (and northern) position have been called lunistices (See Aveni, 1997) because of their similarity to the sun’s extreme positions, the solstices. Thom called these the major and minor standstills respectively. (See Figures 5 and 6)

The cycle of the earth’s revolution around the sun is the year. The moon’s circuit around its orbit is the month. But life is more complicated for the moon than the sun. There are two monthly lunar orbital cycles that we are interested in here. The first is the cycle we are most familiar with, the cycle of the phases from full moon to full moon. This is called the synodic moon, and it is about 29.5 days. The second is the cycle from (solar) equinox point to equinox point. This is the tropical moon. Its period is about 27.3 days. The synodic period is longer than the tropical period because after the moon has left its position opposite the sun (full moon), it has to travel more than a complete orbit to be opposite the sun again (full moon), because the earth-moon system has moved a bit along its orbit in the meantime. The tropical period is extremely close to another lunar cycle, the sidereal period (See Appendix B) which is the time it takes the moon to return to the same position among the fixed stars. The two periods are different because of the precession of the equinox. The equinox points are slowly moving westward, so the moon reaches its starting point relative to the equinox a bit before it reaches its starting point relative to the fixed stars. It really is just a bit. The difference between the two periods amounts to 6.8 seconds. So, normally it doesn’t matter which of the two cycles you refer to, and they are frequently used interchangeably. But if you’re using solar measurements based on the tropical year - and normally you are - it’s more consistent to refer the tropical moon



Figure 3. Phases of the Moon



Figure 4. Cycle of Moon’s Declinations

These 2 charts illustrate the difference between the synodic and tropical moon. They show the rise, transit and set positions for the moon from January 1 through February 28, 2001. The Y Axis in each chart is different, and the difference reflects the way we ‘measure’ both moons. The first chart shows percent illumination from new moon at zero to full moon at 100. The synodic moon is the moon of the phases. As you can see, the two months I chose cover not quite two cycles. The little tail at the right hand side (on February 28) doesn’t quite reach the percent illumination it started with on January 1.

The tropical moon, which is measured by how far north and south (declination) the moon goes, is shown on the second chart, and notice that the moon does move through the entire declination range in the course of a month. The tropical moon covers a bit more than two cycles in January and February. The lunar cycle that is the focus of this paper affects the maximum declinations reached during the tropical month.

The two periods - synodic and tropical - come together in some interesting ways. Perhaps the best known is the Metonic (4) cycle. This is a cycle of 19 years that brings together the synodic month with the tropical solar year, and also with the tropical solar month. There is less than a day’s difference between 19 years, 254 tropical moons and 235 synodic moons. Actually, there are some very beautiful harmonics that are formed by the rhythms of the day, the synodic moon, the tropical moon, and the year, and I’ve described some of them in Appendix C.


METHOD

You can ask for the JPL data in an input text file where the options are set with key words. There’s a sample input file in Appendix A. Basically, I asked for rise, transit, set data for the moon for Knocknarea for (1) a complete 19 year cycle for 1987-2006, (2) 10 sets of 3 years data around the southern standstills from 1837 to 2006, and (3) 10 sets of 3 years data from 2983 BCE to 2816 BCE (5). Rise, transit, set gives you three positions for each day, two of them as the body crosses the observer’s horizon. This will not necessarily yield the actual southernmost position of the moon since that might occur between those times, but the builders and users of the monuments were almost certainly taking a fix on the moon at moonrise or moonset, so it would seem safest to look for horizon events. In any event, the day the moon hits her extreme rising (or setting) point is probably the day, or very near the day when she reaches her true extreme. As to the place, I chose Knocknarea because I feel particularly attached to it. The readings are not going to be all that different for the other astronomical monuments in Ireland

Once JPL sent me the data, I passed it into some BASIC (Future Basic 3) routines that selected out records for the specific analysis I wanted . For example, one routine selected the rising records with the maximum declination reached for each period. Another selected the rising records with the maximum phase (i.e., the full moons) for each period. Yet another translated the Julian day number into the Gregorian date. This last was necessary because for dates prior to the beginning of the Gregorian calendar (Oct 15, 1582), JPL uses Julian dates (6). In particular, it uses Julian dates for the BCE data. I share Pope Gregory’s preference for having the vernal equinox be at or near March 21st, so I translated all dates for this analysis into Gregorian.

I then passed that output to a spreadsheet (APPLE WORKS 6) to actually draw the charts.


ANALYSIS


Figure 5. Declinations for a Metonic Cycle

This chart shows the extreme declinations (north and south) for the rising moon each sidereal month for a Metonic cycle from October 2 1987 to October 2 2006.



Figure 6. Azimuths for a Metonic Cycle

And this chart shows the azimuths for the same period. Notice that the maximum azimuth range at the latitude of Knocknarea - a bit less than 115 degrees - is almost double the maximum declination range - a bit less than 60 degrees. Also notice that the moon’s positions are shifted a bit to the south (south is at the top of the azimuth chart, the bottom of the declination chart.) I don't understand why this should be so, but I suspect astronomers do. For me, this is an issue for future study.

As the charts indicate, I’ve chosen a cycle so that the end points occur when the moon is in the region of her extreme southern (and northern) positions in her 18.6 year nodal cycle. When I first started this investigation I believed that the two cycles - the Metonic and the cycle of the nodes - were completely separate. However, during the course of the investigation, I realized that, since the extreme southernmost positions of the moon happen in the spring or the fall, the length of time between two major standstills can not be 18.6 years. I could see in the data that it switches back and forth between 18.5 years and 19 years. This raises the possibility that the two cycles could have been observed simultaneously, observations of one helping with the tracking of the other..



Figure 7. Azimuths at Four Major Standstills - Present Time

This may become clearer by looking closely at the behavior of the moon at the time of her extreme positions. This chart shows, for the current and three most recent nodal cycles, the maximum southern azimuth position of the rising moon reached during each month for approximately three years. The data shown are the maximum rising moons for the periods Mar 1, 1949 - Mar 31, 1952; Sep 1, 1967 - Sep 30, 1970; Mar 1, 1986 - Mar 31, 1989; Sep 1, 2004 - Sep 30, 2007. The moons for the dates in between these maximum periods are not charted. I’ve labeled the date of the maximum moon of each period. Notice that, in two periods, 1950 and 2006, the extreme moons are in the fall, near the autumnal equinox, and in two periods, 1969 and 1988, they are in the spring, near the vernal equinox. The cycle from 1969 to 1988 is a Metonic cycle. The month and day are exactly the same, and, as can be seen in Table 1, the phase of the moon is almost exactly the same, as is the rising time.


Figure 8. Phases of the Moon during 1988 Major Standstill

To get a sense of the phases of the moon during a major standstill, take a look at this chart which depicts the moon phases during the standstill of 1988. (7)

Around spring equinox, the moon reaches her southernmost point in that part of the sky that the sun had been three months earlier, at winter solstice when the sun was at his extreme southernmost point. A spring equinox major standstill will have the picture shown here: extreme moon at third quarter. The picture of an autumnal equinox major standstill will look the opposite. The extreme southern moon will be where the sun will be three months ahead, again at winter solstice, and this moon will be at first quarter.


Figure 9. Azimuths at Four Major Standstills - Third Millennium BCE

This next chart shows a similar set of data to Figure 7, this time three millennia before Christ. This is not as early as the building of some of the neolithic monuments, but JPL provides moon data only as far back as 3000 BCE. This is within 500 years of the building of Knowth and Newgrange. The data shown are the maximum moons for the periods Aug 23, 2929 BCE - Sep 4, 2926 BCE; Feb 15, 2910 BCE - Feb 24, 2907 BCE; Feb 15, 2891 BCE - Feb 24, 2888; Sep 15, 2873 BCE - Sep 30, 2870 BCE. The charts are very similar to the ones shown above for current time - for example, like the current set of cycles, there’s a Metonic cycle here - Sep 22 2909 BCE to Sep 24 2890 BCE (8), but there are some interesting differences. For one thing, the moon’s orbit was farther from the celestial equator in the old days. There’s an almost 1.5 degree difference (in azimuth) between then and now.


Greg Date

Time

Illm

Azimuth

Julian Day

Cycle length

Sep 26 1838

15:34

47.7

145.7216

2392644

 

Mar 19 1857

4:17

46.9

145.7638

2399392

6748

Mar 18 1876

4:06

48.8

145.6561

2406331

6939

Oct 5 1894

14:55

39.3

145.7482

2413107

6776

Mar 29 1913

3:46

53.7

145.6206

2419855

6748

Mar 1 1932

5:15

34

145.8074

2426767

6912

Oct 16 1950

14:26

32.9

145.7244

2433571

6804

Mar 12 1969

4:50

39.8

145.7574

2440292

6721

Mar 12 1988

4:40

41.7

145.7232

2447232

6940

Sep 2 2006

17:15

68.5

145.6262

2453981

6749

 

 

 

 

Average length

6815.22222

Feb 25 2983 B

5:52

28.1

147.1502

631959

 

Sep 13 2965 B

16:35

58.6

147.0068

638734

6775

Sep 14 2946 B

16:29

57.5

147.1271

645673

6939

Mar 6 2927 B

5:18

34.8

147.1582

652422

6749

Sep 22 2909 B

15:51

49

147.1207

659197

6775

Sep 24 2890 B

15:51

48.8

147.3001

666137

6940

Feb 17 2871 B

6:14

23.9

147.1725

672858

6721

Oct 3 2853 B

15:20

41.8

147.2344

679662

6804

Mar 27 2834 B

3:45

54.1

147.3351

686411

6749

Sep 17 2816 B

16:21

55.4

147.2397

693159

6748

 

 

 

 

Average length

6800


Table 1. Extreme Southern Moons

This table summarizes the position of the extreme southern rising moon for 10 cycles in current time, and 10 cycles in the third millennium BCE. Note that in the modern period, there are three Metonic cycles, two nearly perfect cycles, and one, Mar 29, 1913 - Mar 1, 1932, that is off by just one tropical month. In the third millennium BCE, there are 2 Metonic cycles. Not every cycle is a Metonic cycle, of course, and there are never two Metonic cycles in a row. This is impossible given the length of the nodal cycle (18.6 years) and the length of the Metonic cycle (19 years). However, there is this interesting relationship between nodal cycles and Metonic cycles - 5 nodal cycles is roughly 93 years; 5 Metonic cycles is 95 years. You could use the Metonic cycle to anticipate extreme southern positions so long as you subtracted 2 years every 5 cycles, on average. This would be the reverse of the familiar calendric technique of intercalating - adding a month or day; here you would do the reverse, and it’s something like a reverse leap year.

Notice the time of rising and the percent illumination. The spring moons always rise in the early morning; they are always near third quarter. And the fall moons are the opposite; they’re near first quarter moons rising in the afternoon.

I’m puzzled at the difference between the average cycle length in modern times, and that in ancient times. There’s roughly a half moon difference. Perhaps that would even out with more data, or perhaps it indicates something that should be investigated.

In this analysis, I’ve been focusing on the moon’s extreme position. I haven’t given much attention to the new and full moon, which we can expect would have been important phenomena. The full moon is worth noticing in its own right, and new and full moons are associated with eclipses, which surely would have been important. So that’s an entire line of research that’s worth pursuing. For now, the following chart will give some indication of how the full moon operates around the moon’s extreme positions.



Figure 10. Full Moon Azimuths at Major Standstill

This chart shows the azimuths of the full moons during the next extreme season - September, 2004 through September, 2007. Notice that the southernmost full moon is near summer solstice just prior to the extreme southernmost moon of the standstill on September 2, 2006. To get an idea of how ‘unextreme’ this full moon is, the azimuth of the summer solstice full moon is 144.3258, some 1.3 degrees, or between 2 and 3 moon diameters less southerly than the azimuth of the first quarter moon in September. The time of the northernmost moon is interesting - December 15, 2005 - a full three quarters of a year before the moon’s extreme. Why isn’t the extreme northern moon the following year? Probably a couple of reasons. If we look back at the chart of major standstills, September 2, 2006 looks like a late peak. It’s probably a tad past the conjunction of the nodes with the equinox points. Also December 15 is closer to winter solstice than is January 3, which is the next solstice full moon after September 2, 2006, and the closer the moon is to the winter solstice, the nearer it will be to its own northernmost orbital point.


DISCUSSION

One might ask why we should think that the ancient builders and users of the passage mounds were interested in the moon. In trying to answer that, we have to make some reasonable guesses as to what these people were doing when they built and used these mounds. One of the things that early anthropologists noticed about the ‘primitive’ people they studied - people, that is, who had not been touched by western civilization - is that they saw nature as alive. The anthropologists even coined a term for that - animism. They needed the term because the anthropologists themselves ‘knew’ that a stone or the sun or a lake were not alive. As Aristotle had pointed out long ago, only plants, animals and humans were alive. Further, only animals and humans could sense things, and only humans were truly conscious. Western civilization has been deeply committed to these categorizations. But ‘primitive’ people are not. They see wise spirits in plants, they call the ocean “grandmother”, they are alive to spiritual presence in all of nature. I think it’s reasonable to assume that the builders of the passage mounds were more like the ‘primitives’ than they were like the anthropologists. And if they were looking at the sky - and by now, hardly anyone doubts that they were looking at the sky at least some of the time - they cannot have avoided seeing the moon. Even we, their over-rationalized descendants, can appreciate how powerful the spiritual presence of the moon might have been. Once several years ago, I was a passenger in a car driving south along the east side of San Francisco Bay. It was fall, and the sun was setting to my right, and the full moon was rising to my left. For several minutes, I was completely absorbed in watching first one, then the other. I was literally awe-struck, and dumb-struck. No words came then or now to adequately describe what happened. I don’t think this is uncommon; I’ll bet most people can remember having some experience like that, particularly with the moon. I think such experiences are of the nature of the way ‘animistic’ people perceive the world around them, and as such, they provide a gateway for appreciating what people were doing when they built and used these ritual centers.

But even if we can agree that they were interested in the moon, why should we think they were interested in this particular 18/19 year cycle? Why not in the synodic monthly cycle, or an eclipse cycle? Well, they probably were interested in both of those cycles, but we have good reasons for thinking they were also interested in the cycle where the moon reaches her extreme southern position.

The first reason is historical.

“Now for our part, since we have seen fit to make mention of the regions of Asia which lie to the north, we feel that it will not be foreign to our purpose to discuss the legendary accounts of the Hyporboreans. Of those who have written about the ancient myths, Hecataeus and certain others say that in the regions beyond the land of the Celts there lies in the ocean an island no smaller than Sicily. This island, the account continues, is situated in the north and is inhabited by the Hyperboreans, who are called by that name because their home is beyond the point whence the north wind (Boreas) blows; and the island is both fertile and productive of every crop, and since it has an unusually temperate climate it produces two harvests each year. Moreover, the following legend is told concerning it: Leto was born on this island, and for that reason Apollo is honoured among them above all other gods; and the inhabitants are looked upon as priests of Apollo, after a manner, since daily they praise this god continuously in song and honour him exceedingly. And there is also on the island both a magnificent sacred precinct of Apollo and a notable temple which is adorned with many votive offerings and is spherical in shape. Furthermore, a city is there which is sacred to this god, and the majority of its inhabitants are players on the cithara; and these continually play on this instrument in the temple and sing hymns of praise to the god, glorifying his deeds

The Hyporboreans also have a language, we are informed, which is peculiar to them, and are most friendly disposed towards the Greeks, and especially towards the Athenians and the Delians, who have inherited this good-will from ancient times. The myth also relates that certain Greeks visited the Hyperboreans and left behind them there costly votive offerings bearing inscriptions in Greek letters. And in the same way Abaris, a Hyperborean, came to Greece in ancient times and renewed the good-will and kinship of his people to the Delians. They say also that the moon, as viewed from this island, appears to be but a little distance from the earth and to have upon it prominences, like those of the earth, which are visible to the eye. The account is also given that the god visits the island every nineteen years, the period in which the return of the stars to the same place in the heavens is accomplished; and for this reason the nineteen-year period is called by the Greeks the ‘year of Meton.’ At the time of this appearance of the god he both plays on the cithara and dances continuously the night through from the vernal equinox until the rising of the Pleiades, expressing in this manner his delight in his successes. And the kings of this city and the supervisors of the sacred precinct are called Boreadae, since they are descendants of Boreas, and the succession to these positions is always kept in their family.”

– Diodorus of Sicily, 1935 / 1st century BCE).

This is a very interesting text from a first century BCE Roman historian, referring to legends told by a Greek named Hecataeus and others. Hecataeus is from the 6th century BCE. The text is usually taken to refer to Britain and Stonehenge. The edition from which I have the quotation cites an article by R. Hennig (1928) who makes that identification, and most writers who refer to this quotation seem to have followed him. But I think the place sounds more like Ireland and Brugh na Bóinne. For one thing, ‘spherical in shape’ better describes a cairn than it does a stone circle like Stonehenge. And, as Ellen Evert Hopman points out, Ireland is more the size of Sicily than Britain. Further, the backstone in Knowth’s eastern passage is decorated with art that has been identified as a map of the moon. P.J. Stooke (1994), who does astronomic maps for NASA, regards this stone as the earliest moon map. He too cites this passage of Diodorus, with its mention of the visible prominences of the moon. (Stooke’s paper is at http://publish.uwo.ca/~pjstooke/knowth.htm.) We don’t know whether Hennig or those who followed in thinking Diodorus was talking about Stonehenge even considered Brugh na Bóinne. It’s possible they didn’t know of it, or didn’t realize that it is as old as it is. Of course, even if the citation does refer to Brugh na Bóinne in the 6th century BCE, that does not necessarily tell us what the builders of those monuments were doing two millennia earlier. And the citation might not refer to Brugh na Bóinne at all. But whether it does or not, it certainly does provide legendary evidence that there were rituals in the area of Britain celebrating the moon and a 19 year cycle at least as early as the sixth century BCE. I think that is enough reason for keeping 19 year lunar cycles in mind when looking at these monuments, when trying to interpret what the monuments themselves can tell us in their architecture and in their art.

One of the things that Knowth tells us is that it has a kerb of 127 stones (Eogan, 1986). That’s half the number of tropical moons in the Metonic cycle. It seems strange that people would keep track of the tropical moon. The synodic moon - the full moon - is so much more obvious. But if you wanted to prepare for the god’s 19 year visit, you might track how far south she went each month, and then you’d be tracking the tropical moon.

Diodorus refers to the 19 year return of the god, and then calls it the Metonic cycle. He would have known about Meton independently of the legends about the Hyperboreans, as the text itself suggests. His description of the cycle is a bit confusing - “the period in which the return of the stars to the same place in the heavens is accomplished” - that sounds more like the precession of the equinoxes, but there is a return of sorts in the Metonic cycle. It’s the time that the same phase of the moon returns to the same place in the sky at the same time of year. But this doesn’t sound much like a visit, because it doesn't specify a particular place. What sounds like a visit is the moon reaching an extreme position on the horizon, and with that we’re dealing with the nodal cycle, the 18 to 19 year cycle where the moon reaches her extreme southern position (9) - her major standstill. As we’ve seen, although the nodal cycle (at 18.6 years) is different from the Metonic cycle (at 19 years), because the major standstills happen when the sun is at an equinox position, not infrequently one extreme moon is a Metonic cycle away from the next. In fact, one extreme moon is always either 18.5 or 19 years away from the last. 18.6 is the length of the nodal cycle, but it is not the length of the major standstill cycle.

Another clue that the ritual is dealing with the nodal rather than the Metonic cycle is the mention of the vernal equinox. There’s no preferred time of year for the Metonic cycle to start or end - the Metonic cycle is really a kind of yardstick; you can start it anytime - but as we’ve seen, the extreme moons in the nodal cycle are always near either the vernal or autumnal equinox. Diodorus’ text says vernal; it doesn’t also say autumnal. But there’s a reason why people might have chosen to celebrate only the vernal equinox. As Table 1 shows, the spring extreme moon always rises in the early morning - in the dark. But the autumn extreme moon always rises in the afternoon, when it would be more difficult to see. As Figures 7 and 8 show, there’s only a few tenths of a degree difference - half a moon diameter or less - between the southernmost moon and the moon five to seven tropical months earlier, and the moon five to seven tropical months later. So there’s not that much difference between a vernal extreme moon and its neighboring autumnal extreme moons when the moon is at her major standstill. If you were celebrating the return of the moon to her southernmost position, you could safely plan to have your celebrations at the vernal equinox. You could use the Metonic cycle (19 years) to predict the next return of the moon to her southernmost position so long as in three out of five cycles on average, you subtracted one year from the cycle length of 19 years, and expected her in 18 years. Five nodal cycles is extremely close to 93 years; five Metonic cycles is 95 years.

So I’m suggesting, from the evidence of the legends Diodorus recounts and from the 127 kerb stones at Knowth, that the people who put those stones there may have used them as part of a system to track the timing of a ritual celebrating the 19 (sometimes 18) year return of the moon in the spring, when the sun reaches the vernal equinox. I’m not suggesting that that’s all they used the stones for, but this may have been one use.

But there’s yet another reason for thinking that the builders of Brugh na Bóinne were interested in the cycle of the lunar nodes. It is a cycle that is similar to the solar year in that it has standing positions - lunistices - and points halfway in between. And, as it reaches those positions, the moon can be received and honored by the same structures that are used to honor the sun, but in a complementary way.




This diagram shows the solar year with the Brugh na Bóinne structures that receive the rising sun at or near the solstices and equinoxes (10)




Here’s the same diagram, but this time showing the lunar nodal cycle, and the Brugh na Bóinne structures that are associated with the moon at or near the major standstill and the mid points. Notice that the structures have switched positions. Newgrange, which receives the sun at winter solstice, receives the full moon at summer solstice in between the moon’s minor and major standstills. The next such moon will be near summer solstice, 2001. And Knowth, which, for the sun, is associated with the equinoxes, for the moon, is associated with her major standstill. I personally find the parallelism of the two cycles, and this same but different way the two cycles interact with the Brugh na Bóinne structures a powerful argument that the builders of those structures were consciously thinking of both the sun and the moon, and of these solar and lunar cycles.


CONCLUSION

I hope that it’s become clearer what the 18.6 year cycle of the lunar nodes looks like in our time and in ancient times, and I hope this can provide guidance in the interpretation of sites that seem to have an astronomical intent.

There’s a lot of lunar data that may be important at Knowth and elsewhere that I haven’t talked about here. I haven’t talked about the minor standstill in the nodal cycle. I haven’t talked about eclipses at all, and I haven’t talked very much about the full moon. As Martin Byrne and others have been saying, these people were celebrating the interaction between the sun and the moon, and that means definitely eclipses, perhaps rising full moon, setting sun (or rising sun, setting full moon) phenomena. The astronomical possibilities there need to be explored. There’s a lot of interesting work ahead..




Appendix A - Sample JPL Input


!$$SOF (ssd) JPL/Horizons Execution Control VARLIST
!
!*** Find rise, transit, set data several months 1838 major standstill

EMAIL_ADDR = ’ your e-mail address here’

!Looking for moon
COMMAND = ’301’

OBJ_DATA = ’YES’
MAKE_EPHEM = ’YES’
TABLE_TYPE = ’OBS’
CENTER = ’coord’
COORD_TYPE = ’GEODETIC’

! Looking for Knocknarea
!54;15.155 8;34.440
SITE_COORD = ’-8.574,54.2526,0’
START_TIME = ’1837-mar-01 00:00 UT’
STOP_TIME = ’1840-mar-31’
R_T_S_ONLY = ’YES’
STEP_SIZE = ’1m TVH’
QUANTITIES = ’2,4,10’
CAL_FORMAT = ’BOTH’
APPARENT = ’REFRACTED’
TIME_DIGITS = ’MIN’
TIME_ZONE = ’+00:00’
ELEV_CUT = ’-90’
SKIP_DAYLT = ’NO’
SOLAR_ELONG= ’0,180’
AIRMASS = ’38.0’
EXTRA_PREC = ’NO’
CSV_FORMAT = ’YES’
);




Appendix B - Some Heavenly Rhythms

Solar
Tropical Year 365.242303 days (equinox to equinox)
Sidereal Year 365.25636050 days (fixed star to fixed star)

Lunar
Synodic 29.530.5881 days (new moon to new moon)
Tropical 27.3215816 days (equinox to equinox)
Sidereal 27.3216609 days (fixed star to fixed star)
Nodal 27.2122178 days (node to node)

Metonic cycle
(235 synodic months) 6939.68819 days
(254 sidereal months)
(19 tropical years)

1 nodal cycle 6793.39 days
(18.5997 years)

(After Schultz, 1986 and Deprit, 1971
)




Appendix C - Some Harmonics among the Synodic moon, the Tropical Moon and the Year


You’ll notice that 19, the number of years in the Metonic cycle, is the difference between the number of synodic months and number of tropical months. This does not hold just for 19 years. For any year, if you subtract the nearest whole number of synodic moons in that year from the nearest whole number of tropical moons in that year, you obtain the number of years. So, for example, the nearest whole number of synodic moons in a year is 12; the nearest whole number of tropical moons is 13; difference is 1, same as the number of years. The nearest whole number of synodic moons in 5 years is 62 (62 moons is actually 4 days more than 5 years, but it is the nearest whole number of synodic moons to 5 years). There’s 67 tropical moons in that same time; difference 5, same as the number of years. Try it. I haven’t found this not to work.

The Metonic cycle is a really important beat in the rhythms of the earth, moon and sun. There’s less than 1 day, less than 3/4 of a day between 19 years and 235 synodic months. This cycle repeats faithfully every nineteen years.




Figure C1. Number of Days Difference between Synodic ‘Years’ and Solar Years in One 19 Year Cycle




Figure C2. Number of Days Difference between Synodic ‘Years’ and Tropical ’Years’ in One 19 Year Cycle


These two charts go a little further into the interactions between synodic moon, tropical moon and year (11). They show, for 19 years, the difference in number of days between, in the first chart, the number of whole synodic moons in the year and the year and in the second, the number of whole synodic moons in the year and the number of whole tropical moons in the year.

Here’s the math for these charts (and if the sight of equations gives you hives, just skip them, but be sure to pick up right after the equations because there’s some really interesting stuff here):

a = Number of days in tropical solar year = 365.2422034
b = Number of days in synodic month = 29.5305881
c = Number of days in tropical month = 27.3215816
n = Number of years
x = Number of days in solar years = a*n
y = Number of days in synodic ’year’ = INT((a*n)/b)* b
z = Number of days in tropical ’year’ = INT((a*n/c)*c

Note: INT means round up, but if one of the lunar lengths has a decimal slightly above .5, and the other has one slightly below, don’t round one up and the down. In such a case, don’t round at all, but just take the integer portion of the number.

Data points in first chart = (y - x) as n varies from 1 to 19
Data points in second chart = (y - z) as n varies from 1 to 19

For example, for five years, the values of x, y and z are:
x = 1826.211017 days
y = 1830.8964622 days
z = 1830.5459672 days

y - x = 4.6854452 days.
y - z = 0.350495 days.

So, in the fifth year of a Metonic cycle, the synodic ‘year’ is out of phase with the solar year by a little over 4 days, but the synodic ‘year’ is out of phase with the tropical ‘year’ by less than half a day.

One of the very striking things about these two charts is that they look so much alike. The amplitude of the curve - the spread from high to low - is much greater in the chart that compares the synodic ‘year’ with the solar year, but the form of the two charts look just about identical. And they are indeed very close. Furthermore, the ratio of the data points in the first chart and the data points in the second chart turns out to be extremely close to the real number of tropical moons in one year (the real number is the number with the decimal part included). In other words, in our example for 5 years, if you take 4.6854452 days and divide that by .350495 days, you get 13.36808. That is almost exactly what you get when you divide the real number of days in one year (365.2422034) by the real number of days in one tropical month (27.3215816), which is 13.3682672. And I didn’t just pick a good number of years. This works for any number of years. Interestingly, our old friend the Metonic cycle shows up here. The ratio is least good for multiples of 19 years, but even there, it’s identical up to the second decimal place (for 19 years, the ratio is 13.32986). There’s one more harmonic, and I don’t show this in the charts: if you reverse the role of the tropical moon and synodic moon in the above equations, the ratio between the differences in number of days turns out to be extremely close to 12.26835094, the real number of synodic months in one year.

This is a really extraordinary and totally unexpected set of harmonics between the year, the synodic month and the tropical month. And this is not just an artifact of the equations. It’s not like the game that went around the playground when I was a kid, where you did a series of additions and multiplications, at times using any number you want and always wound up with 14 as the result. I tried this set of equations with other values for the lengths, and didn’t come up with charts that looked like this. For instance, using the nodal cycle length (27.2122178) instead of the tropical cycle length gives an entirely different chart - second chart; the first remains the same, of course. It’s even noticeably different if you use the sidereal cycle length in place of the tropical cycle length. Try it in your spreadsheet. This math may look a little complicated because of the number of equations, and because the numbers are big, but it’s just addition, subtraction, multiplication and division. No trig, no calculus, no statistics.




Bibliography


Aveni, Anthony. 1980. Skywatchers of ancient Mexico. Austin, Texas: University of Texas Press.

Aveni, Anthony. 1997. Stairways to the stars. New York: John Wiley & Sons.

Brennan, Martin. (1984). The Stars and the stones: ancient art and astronomy in Ireland. New York: Thames & Hudson.

Deprit, André. 1971. “The Motions of the moon in space.” in Physics and Astronomy of the Moon 2d ed. ed Kopal, Zdenek. New York: Academic Press.

Diodorus of Sicily. 1935. Library of history. tr. C. H. Oldfather: Cambridge, Massachusetts: Harvard University Press. Diodorus lived in the 1st century BCE.

Eogan, George. 1986. Knowth and the Passage-Tombs of Ireland. London: Thames and Hudson.

Hawkins, Gerald. 1965. Stonehenge Decoded. New York: Dell.

Heggie, Douglas. 1981. Megalithic science: ancient mathematics and astronomy in North-West Europe. London: Thames and Hudson.

Hennig, R. 1928. “Die Anfange des kulturellen und Handelsverkehr in der Mittelmeerwelt.” Historische Zeitschrift, 139.

JPL documentation is at http://ssd.jpl.nasa.gov/iau-comm4/README

North, John. 1996. Stonehenge: a new interpretation of man and the cosmos. New York: The Free Press.

Ruggles, Clive. 1999. Astronomy in prehistoric Britain and Ireland. London: Yale University Press.

Schultz, Joachim. 1986. Movement and rhythms of the stars. tr. John Meeks. Spring Valley NY: Anthroposophic Press. Original work published in German in 1963.

Stooke, P.J. 1994. Neolithic Lunar Maps at Knowth and Baltinglass, Ireland, Journal for the History of Astronomy, XXV: 39-55.

Thom, Alexander. 1967. Megalithic sites in Britain. Oxford: Clarendon Press.




Footnotes


1. North spoke of the northernmost moon. More often than not, the northernmost and southernmost moons happen in the same month. For reasons that should become clearer later, I think the stone age and bronze age people that used the monuments were more focused on the southernmost moon, so that’s the direction I work with in this paper - mostly.

2. Leaving aside their embarrassing confusion between meters and feet on the recent Martian mission. But, though it was JPL that managed that project, the actual mistake was made at Lockheed.

3. But don’t call declination ‘latitude’. Astronomers use the terms latitude and longitude, but they use them for a different coordinate system, a system based on the ecliptic rather than the celestial equator.

4. Named for the 5th century BCE Greek astronomer, Meton. As will be discussed later, I believe the cycle was known in Ireland earlier than that, perhaps millennia earlier.

5. I had asked for refracted apparent azimuth and elevation. The algorithm JPL uses to derive these returns a slightly different elevation each day. For example, for the 4 major standstill seasons shown in Figure 7, the average elevation is -.1788 and the standard deviation is .0253. This variability in the elevation might be a problem in a study trying to determine site lines, but that’s not a concern in this study. Here, what’s of interest is when in the cycle the major standstill happens.

6. And this Julian has nothing to do with the Julian day number mentioned previously. Western civilization has bequeathed us a particularly cumbersome set of techniques for keeping a calendar over a long period of time. The Julian day number is a sequential numbering of days starting with a day in the 5th millennium BCE. The Julian date is the month/day/year in the calendar introduced by Julius Caesar. This is the calendar, modified slightly by Pope Gregory XIII in the 16th century, that we use today. The Julian and Gregorian dates may differ, and the number of days of difference depends on the distance between the date and Jan 1, 1 CE.

7. The symbols don’t tell an exact story. The actual percent illumination of the moon for the dates shown here are:

1987-Sep-29   41.3%
1987-Oct-27   26.9%
1987-Nov-23    6.9%
1987-Dec-20   0.4%
1988-Jan-17   5.4%
1988-Feb-13   25.9%
1988-Mar-12   41.7%
1988-Apr-08    68.2%
1988-May-05    89.3%
1988-Jun-01   96.7%
1988-Jun-28   98.6%
1988-Jul-26   92.4%
1988-Aug-22   71.7%
1988-Sep-19   56.2%

8. I puzzled over the two day difference in these dates - for a Metonic cycle, they should be at most a day apart - until I realized that it’s an artifact of the Gregorian calendar. The year 2900, which falls between these two dates, is not a leap year, so there’s one fewer February 29th in these 18 years than there is normally, thus adding 1 to the second date. If you look at Table 1, you will see the this cycle has the same number of days as the cycle between Mar 12, 1969 and Mar 12, 1988.

9. It is because the southernmost moon appears close to the earth throughout her period of visibility - she not only rises and sets south, she’s very low in the sky throughout the night - that I think these rituals honor the southern moon rather than the northern moon. The northernmost moon does not appear closer to the earth; she actually looks farther away. This also may be why we have a structure aligned to the southernmost position of the sun (winter solstice), and not to the northernmost position (summer solstice) in Brugh na Bóinne. Perhaps it is when they appear closest to the earth that the gods, the children of Leto (as the Greeks knew her), are visiting.

10. I realize there is some question about the way Knowth might receive the sun because the alignment of the eastern passage is not due east, and the western passage is not straight. I am leaping over that question for the time being, and just making the point that there does seem to be some solar equinox connection at Knowth.

11. The data for these charts is derived from equations, not from JPL data, so it is very definitely a theorical model, and does not reflect the vagaries of reality.

RESEARCH PAPERS





The URL of this page is: www.astroarchaeology.org/research/papers/sunmoon
Updated 29 January 2001
For more information contact Michael O'Callaghan at moc@global-vision.org

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