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| Astronomy Quiz | Appendix 1 | Appendix 2 | Appendix 3 | Appendix 4 | Appendix 5 |

Chapter 8
OUR MATHEMATICAL MOON

It's a question of balance

Let us try to envisage the delicate balances that are involved in our Moon's orbit and position. It will help in appreciating the Moon's deep connection with Earth, to ponder the enigmas of its motion; what follows is a mathematical look at patterns it generates in space and time.

Firstly, the size of the Moon; being one quarter of the Earth's diameter it is too large to be an ordinary satellite, yet rather too small to be a planet. Other planets have satellites far smaller in relation to their size, so one quarter of a diameter is rather large, and yet the Moon is smaller than Mars or Venus in its size. Astronomers are unsure whether to regard it as a satellite of Earth or a companion planet. Further, there is only one of it. Planets further away from the Sun have several moons, whereas planets nearer to the Sun have none. The latter would be unable to hold on to a moon, because the Sun's gravity pull is too strong.

Next, there is the Moon's distance from Earth, which is 30 Earth- diameters. It is so far away that it is puzzling how it every got into such a position, and yet orbit in the nearly circular orbit that it does. All other satellites are held within the gravitational pull of the planet which they orbit - that is to say they are pulled much more strongly by it than by anything else. One would naturally assume that satellites are held within the gravitational pull of their parent planet - the Moon, though, defies this simple rationale. It does so by standing well outside the Earth's gravitational domain. How far outside? It is 2.2 times more strongly pulled by the Sun that by the Earth: always twice as strongly attracted towards the Sun than the Earth. So although it appears to be going in a nearly circular path, as if just orbiting round Earth, if one looks at the solar system as a whole, it is orbiting primarily around the Sun, the Sun's pull is stronger.

If the Moon is so far away from us, why then does it have so nearly circular an orbit? Why is it not some widely elliptical orbit like other satellites that are far away from their parent planet? Astronomers have a problem trying to answer this one.

		Showing Jupiter's moons Io, Europa, Ganymede and Callisto, and Saturn's moon Titan (Martineau)

If, as we have seen, the Moon is pulled much more strongly by the Sun, why should it be facing Earth? The Moon's own rotation is locked into its orbit round the Earth, whereby it unfailingly shows the same side of itself to us. Surprisingly, the two sides of the Moon have turned out to be quite different. Huge seas of frozen, once-molten lava make up the familiar, charming features of its face, of a charcoal-dark hue owing to the presence of certain heavy metals. These characteristics are not seen on the opposite side. The long-hidden, back of the Moon turned out to be full of nondescript mountain ranges. This asymmetry of the Moon strongly suggests that it was formed in relation to Earth.

		Sea-shell of 'Nautilus'

The Moon has a quite balanced orbit, and stays pointing Earthwards. The nearest astronomers have come to explaining this is to say that once it was very close to Earth. The gravity pull was then much stronger, as a result of which it lost any original motion it once had and ended up always pointing towards Earth. Gradually, it spun further and further away, still keeping, however, its position in relation to Earth. You may wonder whether there is any trace of evidence to support this theory. Well, there is one piece of supporting evidence. Astro-physicists have become interested in a humble sea creature living at the bottom of the ocean! The fossil shells of 'Nautilus' have an odd calendar. Its spiral is said to have a lunar-monthly calendar record in it. Once a month its shell grows a new wall, or 'septum'. It seems that this happens once per lunar month, though whether this occurs at each Full or New Moon, or whenever, nobody seems to know. It isn't easy to investigate a creature living on the ocean bed without disturbing its habits. The chambers thereby constructed enable Nautilus to float up and down, by filling them with gas - as its ancestors have been doing for half a billion years. Every day the shell wall is marked with a line of growth, giving 30 or so growth lines in between each of these septa. Going back to the much earlier fossil shells, the numbers decrease through the ages down to just 12 or 15. So, there is this one piece of evidence, from the family tree of Nautilus, to support the theory that the Moon was once close to Earth. However, this is but one piece of evidence amidst a sea of speculation.

Sea-shell of 'Nautilus' with the chambers it grows, once a month

The Moon's orbit presently moves out from Earth at the rate of three centimetres a year. If we extrapolate this back to when it was close to Earth, it seems to concur with the dating of the Moon rock, which is around 4 billion years. There tends to be a discrepancy between these two approaches of one or two billion years, but it seems that such mis-dating is not regarded as too bad.

Let us return to the present position the Moon, 30 Earth-diameters away, and look at the coincidences which it generates. To start with, it is able exactly to eclipse the Sun: the Sun is 400 times larger than the Moon in diameter, and 400 times further away, which by scaling gives the same angular size. In fact during a total eclipse an exact fit can give 'Bailey's beads,' when the Sun's corona is seen to glitter through the mountains of the Moon.

		'Baily's beads,' glittering through the lunar mountains during a total eclipse

As the Moon moves from its furthest to its nearest approach each month, apogee to perigee, it draws 10% nearer to the Earth. As a result of this, tides on average grow 30% larger. This increase in the tidal height shows how the tides grow much larger as the Moon comes a small amount closer to Earth. It is here that once again we see the delicate balance of the Moon, for were it just 5% closer to us, huge areas of land would regularly become flooded. Were the Moon a fraction nearer, or if it weren't of such light density, half that of earth, the tides would be unduly large.

It would seem that with the Moon we come up against a number of 'coincidences'; its convenient distance from Earth is just one of them; its low density is another (it couldn't have any lower density for its size). It appears to be 'balanced' in every respect.

Earthrise above the Moon photographed by the Clementine spacecraft orbiting the Moon in 1994

The Moon makes a circuit of the Earth 12 times in one year, giving the all-important number 12. The 'calendar' which reflects the motion of the Moon, comes from the word 'Calare' which means to 'cry out', which is what the High Priest did from his temple or Ziggurat when he saw the first thin crescent of the New Moon appearing. The division of time obtained by reconciling the lunar motion to the solar year gives the 12-fold division of the zodiac. This is the division of the course of the year in time, and also of the Sun's course in space. The Moon's motion fits into the solar year 12 1/3rd times. Were it a mere 2-3% closer to Earth, there would be approximately 12 2/3rd revolutions in a year, so instead of having 12 months in one year, and occasionally 13, there would be 13 months in a year and sometimes 12. With 13 months in the average year, things would be unimaginably different - no longer the four seasons of the year, or the division of the circle of 360 degrees, or the month of 30 days. (The Babylonian division of the circle through 12 x 30 = 360 degrees, is an expression in space of its calendar system.). In the same way that one cannot envisage anything other than 12 months in the year, and 12 divisions in space, so it is difficult for us to imagine the Moon differing in apparent size from the Sun. It is difficult to regard these ratio as deriving from chance since so much depends upon them. At this point let us turn to a number I have become quietly obsessed by, that is 27.3. 27.3 is the number of days the Moon takes to go round in space, against the fixed stars; also it orbits once on its own axis, every 27.3 days, two different cycles which coincide. Scientists measure various effects on Earth according to 27.3 day rhythms, for example, doctors observe some types of illness varying according to the 27.3 day pattern2; or, habits of some rhodents were observed to follow a 27.3 day cycle3. There is said to be a 27.3 day rainfall pattern.4 In actual fact these 27.3 day rhythms observed are solar patterns, and not lunar ones! What is the Sun doing with the Moon's period? The 27.3-day solar activity cycle shows up in the sunspot patterns. The most frequently quoted mean value of the sunspot cycle is 27.3 days, though one English meteorologist takes 27.5 days as his estimate of the mean solar sunspot pattern. It would be oversimplifying to say that this is the period the Sun takes to rotate, since it is a gaseous body. It rotates about once in 25 days at the equator, and once in about 30 days at the poles. It is the mean latitude of sunspots that determines their 27.3 day mean period of rotation. Any study of monthly rhythms in organic life needs to take into account these two different cycles; one 29.5 day rhythm, the lunar influence, and one 27.3 day rhythm, which is solar. For these two different patterns suitable statistics are used to separate the cycles. Then the mean diameter of the Moon is 27.3% that of the Earth. The Earth's diameter varies slightly depending on where it is measured, and one might get 27.4%, but basically it's something rather similar.

At this point I began to wonder about the number 27.3, it being configured four times in relation to the Moon. Dare one say it, 27 is the third power of 3 and 9 is the second power of three. Nine is a highly lunar number, being the mean number of months in conception. If 266 days is the mean duration between conception and birth, then there are nine Moons between conception and birth.
        29.53 x 9 = 265.8
The menstrual cycle is often quoted as 28 days. Various surveys have looked at this, and a couple of very large surveys have found that the period decreased with age6. The childbearing years, between twenty and thirty, give a mean value between 29 and 30, while 28 days was the mean length for women aged thirty-five while it was longer, 30 days or more, for teenage girls. Thus during childbearing years, the period duration is indistinguishable from that of the lunar month.

The tilt of the lunar orbit to the ecliptic affects how often eclipses occur. This tilt angle of 5° - 6°, is a thing not predicted on any theory that astronomers have for the Moon's origin. The two main theories used to be capture and fission. On the capture theory a body in the solar system became caught by the Earth's gravitation. It would be expected to end up orbiting in the same plane as the rest of the solar system. The fact that the Moon is not in the same plane puts the odds against this theory. The theory also predicts a highly elliptical orbit, which the Moon doesn't have. The other theory, that it was torn off from the Earth, predicts a Moon orbiting in the plane of the Earth's equator, i.e. at 23° to the ecliptic. In general scientists group all satellites into one of these two categories: those formed by fission from the planet - which orbit in the equatorial plane - and those which were captured, which orbit much further away, and have elliptical orbits in the plane of the ecliptic. The Moon, like some riddle of the Sphinx, belongs to neither of these categories, but has her own plane at 5 or 6 degrees to the ecliptic.

		Earthrise on the Moon

To get round the impossible difficulties which both the fission and capture theories presented, the accretion theory was developed. According to this theory, a large amount of rubble circulating around Earth somehow coalesced to form the Moon. The problem with the accretion theory was that, if some early moon had been continually impacting on rubble, it would gradually lose its kinetic energy, and thereby drift continually nearer to Earth, whereas the trick was to have it move out, far away from Earth. Another problem, mentioned earlier, is that the Moon is clearly facing towards Earth. So they grew tired of the accretion theory, and now have a 'sudden impect' model, whereby Earth suffered a cataclysmic encounter, as supposedly produced Luna.

Turning to longer-term cycles, the Saros gives the pattern by which eclipses repeat. Several cycles have to come together for a repetition of the same eclipse. The phase of the Moon has to be the same, that is either full or new, and the nodes have to coincide with the phase. Thus the primary condition for an eclipse to happen is the alignment of the phase and node cycles. To get an eclipse which looks the same, the Moon must be the same distance away from Earth; because if it's further from Earth - near apogee - it will appear smaller in relation to the Sun, and won't cover the whole solar disc. To get the same sort of eclipse the apogee/perigee cycle has to be the same. The Sun also changes its apparent size with the season of the year, because in summer and winter it's further or nearer as Earth goes round it in an ellipse. It would seem, then, that there are four cycles which have to come together to get the same eclipse happening again. How often would these four cycles come together, that is three monthly cycles and one yearly? A thousand years, maybe? It turns out to be only eighteen years, quite a 'coincidence.' Mathematically, it requires six -figure accuracy to express this:

A Saros 'family' of Eclipse paths. A 'chain' of eclipses are here shown, at intervals of 18 years, 11 days: a Saros family starts off near the North Pole and gradually winds its way due South7.

		Paths of a Saros 'family' of eclipses, at intervals of 18 years, 11 days

The three monthly cycles of lunar phase, node and apogee-perigee all come into synchrony with the Saros: the phase and nodal cycles to within an hour and the apogee-perigee within seven hours. Also, the interval of 11 days means that the season of the year is much the same, so the fourth cycle is also satisfied. It also happens, although this is not particularly relevant, that the sidereal cycle is much the same, that is to say it brings one back to much the same point against the stars. Altogether this means that, to quote from an astronomy book, '....though it seemed that a similar eclipse would take place only after an extremely long interval, two prodigious coincidences bring the period to less than twenty years, and make the Saros Cycle a cycle of considerable interest, although defying all the laws of logic.'8,9 The pattern of eclipses does not come back geographically to the same place, because of the extra third of a day; for the eclipses to reappear at a similar geographic position, it is necessary to count three of them - that is, 54 years.

		The lunar nodes

There are two other 18-19 year cycles often confused with the Saros, the Metonic cycle and the 'nutation' cycle. The Saros cycle gives the eclipse pattern, while the Metonic cycle is the calendar computer. The latter has nothing to do with eclipses, but is used for drawing-up calendars, as it says where the Full Moons fall each year: it happens by another most remarkable coincidence that the lunar period of 29.5 days goes very exactly into 19 years. Every 19 years it comes back to the same position. Diaries tell what part of the Metonic Cycle we're in, giving the Metonic number between 1 and 19 (which the Jews still use for calculating their religious calendar). This is so accurate that its predictions only require one day's correction every three centuries.

The lunar nodes rotate against the stars once in 18.6 years, the nutation period. The Saros and Metonic cycles are not lunar cycles as such, rather they were convenient ratios between other cycles, but nutation is a true lunar cycle. Neither the Saros nor Metonic cycles have got any business existing: they are merely high-precision coincidences which happen to be found to be there, in the machinery of things10. In 18.6 years the lunar nodes move around the zodiac, so that the eclipse seasons of the year move through the four seasons. These are presently (2004) in October and April, when eclipses happen.

I hope that this has given some idea of the unique nature of Luna, how it is quite different from any other known satellite, how its origin remains shrouded in mystery, and how its position so far away, yet pointing towards us continues to baffle those who study it. The Moon is so much a part of nature that one cannot really imagine life without it, and yet when one tries to put together an explanation as to how it got there, it never seems to fit.

		Moon over the ocean

Lunar Quiz

1. Is the first thin crescent of the New Moon visible at dusk or at dawn?

2. In what month is the Harvest Moon?

3. Name the line joining the apogee and perigee positions.

4. At what time of year does the full Moon rise highest?

5. Name the line joining the full and new Moon positions.

6. What cycle has a 27.3 day period?

7. What was the legal distinction between lunacy and madness in the 18th century?

8. Name a well-known murderer, about whom a story was written, who pleaded that he was not feeling himself at full moon.

9. Does the Moon spend more time in the day or night sky?

10. If you were the keeper of the calendar, what would the ratio 7:19 mean to you?

1) P.G. Kahn and S.M. Pompea, Nature, 1978, 19 October, p.606
2) "The method of the 27-day solar calendar for treatment of experimental data has been known for a long time .... some neuropsychic and somatic diseases show a 27-day recurrence. A 27-day period has also been observed in the course of biological processes and physicochemical reactions," owing to "the 27-day recurrence of geomagnetic storms." from A. Dubrov, "The Geomagnetic Field and Life" Plenum Press, NY 1978 (translated from the Russian).
3) See, e.g., J.R. Brown, 'Living Clocks,' Science, 1939, 130, p.1535.
4) J.W. King, "Sun-Weather Relationships," Astronauts and Aeronautics, April 1975, p.14.
5) J.W. King took 27.5 days as his average value over the ten-year period 1963-1972, Journal of Atmospheric and Terrestrial Physics 1977, p.39, p.1357.
6) A. Theloar et al., "Variation of the Human Menstrual Cycle through Reproductive Life," International Journal of Fertility, 1967, 12, p.77.
7) Bernadette Brady's 'Predictive Astrology: The Eagle and the Lark' 1999 is the astrologers' favourite on how Saros-families work.
8) Vincent de Callatay Atlas of the Moon p.56.
9) Edmond Halley was the first known astronomer to use the Saros for eclipse prediction, and he named it. The Saros interval was known to the ancients as a synchrony, being in Ptolemy's Almagest, but it was not stated as being relevant for eclipses. The Saros cannot be used for predicting eclipses at a given locality, it only works globally.
10) The Metonic cycle, used for constructing the calendar, also works to this same, six-figure accuracy. In nineteen years, there are 365.242 x 19 = 6939.60 days, which contain a whole number of lunar months 29.5306 x 235 = 6939.69 days (where 235 = 12x12 + 13x7): these nineteen-year periods coincide within an hour or two! The Metonic cycle means that, on the 19th birthday, and multiples thereof, the luminaries arrive at the same positions in the zodiac as they occupied at birth, within about one degree. The 'sacred' numbers twelve and seven here appear, with seven years counting thirteen lunar months and twelve with only twelve.

		Edmond Halley, first known astronomer to use the Saros