va41_4_543

Pergamon

Vistas in Asrronomy Vol. 41, No. 4, 543-571, 1998 pp. @ 1998 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0083-6656/97 $15.00 + 0.00

PI I: SOOS3-6656(98)00004-X

THE CYCLES OF SELENE

B.A. STEVES

Department of Mathematics, Glasgow Caledonian University, Glasgow, G4 OBA, UK

Abstract- The discovery and use of the Saros, a h&r cycle of 18 years and 10 or 11 days, is reviewed from its earliest origins two millennia ago to the present day, when it is known with precision and enables the accurate prediction of both time and type of solar and lunar eclipses. The theoretical basis for the Saros is discussed, along with other historically known lunar cycles. The geometry of the Sun-Moon-Earth system is found to repeat itself after one Saros, not only at eclipses but also at any phase of the cycle, indicat- ing that the Moon moves in a nearly periodic orbit. The search for periodic orbits using the Saros has led to the discovery of a set of eight periodic orbits of period equal to one Saros whose time evolutions closely resemble that of the real Moon. Finally, the potential of the Saros in studying the dynamics and stability of the Earth-Moon system is examined and the existence of other Saros-like cycles of longer periods in the present, past arid future of the Earth-Moon-Sun system is explored. @ 1998 Elsevier Science Ltd. All rights reserved.

1. PROLOGUE

The old seer carefully moved his aching legs up the time-worn stairs of the stone watch-tower as he had done countless nights for so many years. The sky was lit up with the cold brilliance of thousands of stars, awesome in their beauty. He caught his breath. Its icy tendrils swirled into the night. The beauty - the immensity of the sky. It cut him like a knife. It neverfailed to. It was so huge. He was nothing. How dare he follow the gods, like a thie$ through the mighty skies.

But like familiar fiends, they appeared out of the vastness. There was Ishtal; star of both evening and morning, enticing in herfluctuating brightness. She was the goddess of youth, beauty and love, but she could also be a fierce Lady in battle riding astride a great lion. Ma&k, the king of gods, was also visible with a strong steadfast glow. He could bring cataclysm and storms, but at a whim he couldalso reveal a regal graciousness to his subjects. The malevolent red gleam of Nergal, god of war; shone down on the seer: Nergal could herald death and destruction. And there was the faint light of Ninurta, the god of time. He moved so slowly he was still in the same constellation after many years, while Nebu, the messenger, raced off through the skies to play his tricks.

Their movements ofen puzzled the seer: Sometimes the gods would stride ahead purposely, then stop in the vicinity of a constellation only to reverse and move backwards for a while. They

544 B.A. Steves

were the bibbus, ‘the wild goats’, but they heM human destiny in their hands and as such the seer and many before him hadfaithfully recorded their movements throughout the years.

He fondly watched them as they marched across the sky. He was excited. Would it happen? Would the gods show theiFfaVOUF to him? While studying the clay tablets that recorded the details of one hundred years of the travels of Selene, the Moon, he had discovered that she appeared to pass through the same pattern of eclipses every 18 years and 10 days. If he was right, she would meet with an eclipse tonight.

And it came to pass. With intense excitement he watched the crescent of darkness hide more and more of the golden face of Selene until like a phantom barely seen, her face shone faintly, glowing like the last embers of afire. He continued to watch until she reappeared in herfill glory once more.

What power his discovery gave him! He was now able to predict the actions of the gods. The priests would be able to use his knowledge to increase their hold over the people and the King! The seer shivered. He was old. He was tired. Let younger men do what they would. There was power here, but also what beautiful symmetry the gods had shown him.

2. THE DISCOVERY OF THE SAROS

It is not known who the Chaldean genius was who first discovered the Saros, the most famous cycle of Selene, the Moon. A period of 6585.32 days (i.e. approximately 18 years and 10 or 11 days depending on the number of leap years in the interval), it is the time that elapses between successive repetitions of a particular sequence or “family” of solar and lunar eclipses. ‘Saros’, in fact is the Greek word for ‘repetition’. After one Saros period, a sequence of eclipses will recur with approximately the same order of type of eclipses, the same duration of the eclipses and the same time intervals between the eclipses, as the previous sequence of eclipses. In other words, after one Saros period a large partial lunar eclipse will be followed by a large partial lunar eclipse; a total solar eclipse of short duration will be followed by a total solar eclipse of short duration; an annular solar eclipse will be followed by an annular solar eclipse and so on. Thus, the solar and lunar eclipse pattern which occurred in the previous Saros period can be used to predict the solar and lunar eclipses in the next Saros period.

The Saros cycle was most probably first discovered by the Babylonians from their records of eclipses extending back over many centuries. In his book, A History of Astronomy, A. Pannekoek [ 171 gives an excellent, well documented history of the development of Babylonian astronomy. In summary, the Babylonians lived on the plains between the Euphrates and the Tigris rivers from some time before 3000 BC. The whole of Mesopotamia was first united as early as 2500 BC. Babylon became the capital and grew into a great commercial and cultural centre circa 2OOfl BC under the reign of Hammurabi.

The Babylonians began making regular observations of the Moon in order to be able to predict when the different seasons were approaching and therefore to know when to start planting and harvesting crops. This led to a belief that the heavenly bodies were related to the Gods and Goddesses, and that the movements of these bodies held great significance to changes in the lives of people on Earth.

The Assyrians, a military people living on the northern part of the Tigris river, rose to be- come the most powerful state in near-Asia around 800 BC, and eventually destroyed Babylon in 689 BC. Although the Assyrians conquered the Babylonians, they adopted much of their culture and it became very important for the Assyrians to know what was happening in the sky in order to interpret its meaning and relate it to the success of any large-scale military enterprise or the

The cycles of Selene 545

well-being of the king and country. Eclipses were, in particular, very significant omens. The exact month, day, time of day, and

place in the sky for every eclipse observed was recorded. A different interpretation of the eclipse was made, depending on the exact time it occurred. Thus, for example, we find recorded such predictions as this (see Ref. [ 171, but originally published in R.C. Thompson’s book The Reports of the Magicians and Astrologers of Ninevah and Babylon, 1900, page 27 1):

“An eclipse in the morning-watch means disease.. . The morning-watch is Elam, the 14th day is Elam, Simannu is Amurru, the second side is Akkad.. . When an eclipse happens in the morning-watch and it completes the watch, a north wind blowing, the sick in Akkad will recover: When an eclipse begins on thefirst side and stands on the second side, there will be slaughter of Elam; Guti will not approach Akkad.. . When an eclipse happens and stands on the second side, the gods will have mercy on the land. When the Moon is dark in Simannu, after a year Ramanu [the storm-god] will inundate. When the Moon is eclipsed in Simannu, there will beflood and the produce of the waters of the land will be abundant.. . ”

From these accurate observations of eclipses, the Babylonian-Assyrians were able to identify regular patterns. Their reports then began to show that they expected eclipses to occur, by an- nouncing the consequences of such events before they occurred. For example, Thompson (1900, pages 273-274) also records the following prophecies made by court astrologers:

“On the 14th an eclipse will take place; it is evil for Elam and Amurru, lucky for the king, my lord; let the king, my lord, rest happy. It will be seen without Venus. To the king, my lord, I say; there will be an eclipse. From Irasshi-ilu, the king’s servant.”

“To the king of countries, my lord thy servant Bil-usur May Bel, Nebo and Shamash be gra- cious to the king, my lord. An eclipse has happened but it was not visible in the capital. As that eclipse approached, at the capital where the king dwells, behold, the clouds were everywhere, and whether the eclipse took place or did not take place we don’t know. Let the lord of kings send to Ashul; to all cities, to Babylon, Nippul; Uruk and Borsippa; whatever has been seen in those cities the king will hear for certain. . . The great gods who dwell in the city of the king, my lord, overcast the sky and did not permit to see the eclipse. So let the king know that this eclipse is not directed against the king, my lord, nor his land. Let the king rejoice. . . ”

At this point in time, they were probably using the simple regularity (see Section 3) that once an eclipse series began, there would be five or six lunar eclipses each separated by approximately six months. Thus, when a new series of eclipses began one or two years later, they could predict eclipses again.

The Assyrian empire, weakened by wars against barbarian tribes from Europe, eventually crumbled under the combined attack of the Babylonians and the Medes. In 606 BC, Nineveh, the Assyrian capital, was ruined and Babylon became the capital once again. The new Babylonian empire was extended by Nebuchadnezar (604-561 BC) to include all of near-Asia. During his time, the Babylonian priesthood held great power arising from its ability to predict the fortunes of people and countries by reading the signs laid out in the sky. In 539 BC, the Persians conquered Babylon and eventually Babylon was reduced to the status of other Persian capitals.

The Persian kings had no use for omens of good luck or evil from foreign gods; nevertheless, the priests maintained their power by changing their role from that of court astrologer to that of the elite group of people who knew the ways of the gods by knowing their movements in the heavens. In order to prove this knowledge, their observations of the sky became more accurate and detailed. It was probably during this interval that they discovered the Saros period.

The well-known ‘Saros-Canon’ tablet studied by J.N. Strassmaier and J. Epping is a fragment of a list of eclipse months which extend from 373 to 277 BC. Because each column consists of 38 lines spanning 223 months, and each eclipse series is clearly demarcated by horizontal

546 B.A. Steves

Sun

: :

:

Distant Stars :- : completed : here

Fig. 1. Solar and lunar eclipses can only occur when the Earth, Moon and Sun are collinear, i.e. when the Moon is new or full.

lines and a total eclipse month is located in the centre of each series, it is believed that this list of eclipse months was used as a means of predicting future eclipse months by applying the repetitive properties of the Saros cycle. Certainly, the Babylonians knew of and used the Saros period in later centuries because Babylonian ‘Auxiliary Tables’ have been discovered where it is obvious that eclipse times have been calculated from data taken 18 years earlier.

3. ECLIPSE PREDICTION USING THE SAROS

A solar or lunar eclipse occurs whenever the Earth, the Moon and the Sun lie in approximately a straight line. If the Moon’s orbital plane were exactly the same as the ecliptic plane, a solar eclipse would occur at every new moon when the Moon is at conjunction, and a lunar eclipse would occur at every full moon when the Moon is at opposition. See Fig. 1. However, since the two planes are inclined at an angle of about 5” to each other, eclipses can only be possible when the Sun is also near a node and is therefore located in approximately the same plane as the Barth-Moon system. It is obvious that (Fig. 2), if the Moon is near the same node as the Sun, a solar eclipse will occur, while, if the Moon is near the opposite node, a lunar eclipse will occur.

The ecliptic limit, i.e. the maximum angular distance that the new or full moon can be from the nodes in order for an eclipse to occur, is about 10” to 12” for a partial eclipse and 5” to 6” for a total eclipse, the particular value depending on the exact geometry.

Consecutive lunar eclipses occur approximately six months apart, when the Sun returns to a position near one of the Moon’s nodes. After six synodic months (i.e. 177.18 days), the Sun and hence the positions of the full or new moon have moved, on average, by about (177.18 days/36524 days) x 360’ = 174.64’ in longitude. However, in this time, because the nodes of


Q_.u,r orbital plane _

Ecliptic

Y

plane

Sun

Line of nodes 1997

Line of nodes 1999

Fig. 2. Solar and lunar eclipses can only occur when the Moon is located near the intersection of the lunar orbital plane and the ecliptic plane, i.e. when the Moon lies near its line of nodes.

No eclipse Partial Total? TOtal Total Partial Partial? No eclipse

13.67’ 9.65’ 5.62 ’ I .59 * -2.43’ -6.45’ - 10.48’ -14.50’

Fig. 3. A possible series of eclipses which occur approximately every six months. The large and small circles denote the Earth’s and Moon’s shadows respectively. The scale indicates the angular separation (in degrees) between the Moon’s position and the Moon’s orbital nodes. Diagram and data derived from Ref. [ 171.

t.be lunar orbit are moving at a rate of -O.O52954”/day, the nodes have receded by about 177.18 days x(-O.O52954’/day) = 9.38”. The opposite node is located at a longitude of 180”-9.38” = 170.62”. Therefore, the full moon has moved relative to the node by about 174.64” - 170.62” = 4.02” [ 171.

An eclipse will thus occur as long as the Moon is located within approximately f 12” of a node. If the Moon progresses about 4” relative to the node every time it comes back to a position where both it and the Sun are near a node, then approximately five or six consecutive lunar eclipses separated by about six months are possible before the Moon moves beyond the ecliptic limits. Fig. 3 portrays a possible series of eclipses. Note that the type of eclipse depends on the angular distance from the Moon to its node.

A new series of eclipses then begins when the positions of the Sun, the full moon and the nodes coincide again. This occurs about 47 months after the first series began. The new series,

548 B.A. Steves

like the old, contains five or six lunar eclipses each approximately six months apart; however, the characteristics of the new series, namely the eclipse type and duration, are not the same as that of the old series.

The type of eclipse which occurs depends not only on the angular separation between the Moon and its node, but also on the distance between the Earth and the Moon, and on the position of the Earth observer. Although the Moon is almost four hundred times smaller than the Sun, it is also approximately four hundred times closer to the Earth than the Sun. Thus, having almost the same apparent angular size as the Sun, the Moon can totally obscure the Sun. If the Moon is far enough away, it appears smaller than the Sun. In this case, a ring of light encircles the Moon as the Moon eclipses the Sun. This type of eclipse is called an annular eclipse.

If the Moon is close enough to the Earth, the Moon’s apparent diameter is greater than that of the Sun and the umbra or dark cone-shaped inner region of the Moon’s shadow just barely reaches the Earth’s surface. Any observer on Earth positioned along the path taken by the umbra will experience a total solar eclipse. Any Earth observer located in the outer lighter region or penumbra of the Moon’s shadow will see a partial solar eclipse. The extent of the penumbra and umbra shadow regions, and therefore likewise the types of both lunar and solar eclipses observed depend on the distance between the Moon and the Earth at that time.

The Earth-Moon separation also affects the duration of an eclipse. The length of time that the Moon and the Sun remain in an eclipse configuration is governed by the Moon’s velocity, which varies according to its position in its elliptical orbit, and hence varies according to the distance between the Earth and the Moon.

Therefore, because the eclipse characteristics such as duration and type depend on the Earth- Moon distance, the first repetition of an eclipse series where the eclipse characteristics are also repeated, occurs when not only the position of the Sun, the new or full moon and the nodes lie in approximately a straight line again, but when this event also coincides with the Moon and the Sun lying in the same positions relative to their pericentres. This occurs approximately after one Saros period or about five series of eclipses.

4. THR REPETITION OF THE RELATIVE GEOMETRY OF THR EARTH-MOONSUN SYSTEM OVER ONE SAROS

The ease with which people can use records of eclipses that occurred in the previous Saros period to predict future solar and lunar eclipses suggests how closely the geometry of the Earth- Moon-Sun system must be repeated every Saros period. Table 1, for example, shows the values of the semi-diameters of the Moon and the Sun during four eclipses, which occurred at approximately Saros intervals in the years 1898,1916,1934,1952 and 1970. The data is taken from the relevant Astronomical Ephemerides [22,25,19].

Even without allowing for the fact that over a span of seventy-two years, methods for calculat- ing the ephemerides have changed (i.e. orbital constants of the Earth, Moon and Sun have been improved, and more accurate and constant time reference frames have been implemented), the semi-diameters of the Moon and the Sun vary only slightly from eclipse to eclipse over a single Saros period. Because the Saros cycle is not exactly periodic, in other words the relative geometry of the Earth-Moon-Sun system is not repeated exactly over one Saros period, a comparison of the relative geometry over many Saros periods will start to show discrepancies. However, a comparison over even four Saros periods still shows a close agreement between the lunar and the solar semi-diameters at the beginning and the end of the four Saros periods.

This is particularly remarkable, given that the total range within which the semi-diameters of

Table 1


Semi-diameters (in arc-minutes) for the Sun and Moon during eclipses which occurred at Saros intervals. The data below are taken from the appropriate Nautical Alamanacs and Astronomical Ephemerides [22]

Eclipse type

1898 1916

Year

1934 1952 1970

A=Partial Lunar eclipse Moon Sun

B = Total Solar eclipse Moon Sun

C = Partial Lunar eclipse Moon Sun

D = Annular Solar eclipse Moon Sun

Jan. 7 Jan. 19 Jan. 30 Feb. 10-l 1 Feb. 21

14.867’ 14.830’ 14.808’ 14.788’ 14.780’ 16.265’ 16.255’ 16.235’ 16.207’ 16.172’

Jan. 21 Feb. 3 Feb. 13-14 Feb. 25 Mar. 7

16.405 16.423 16.455 16.487 16.527 16.247 16.225 16.193 16.157 16.113

Jul. 3 Jul. 14 Jul. 26 Aug. 5 Aug. 17

16.722 16.715 16.718 16.720 16.732 15.731 15.735 15.748 15.770 15.798

Jul. 18 Jul. 29 Aug. 10 Aug. 20 Aug. 3 1Sep. 1

14.765 14.733 14.720 14.708 14.710 15.739 15.755 15.780 15.810 15.847

Table 2 Comparison of the differences in the lunar and solar semi-diameters (in arc minutes) over 1 and 4 Saros periods for the eclipses in Table 1

Eclipse type Average absolute differences in % differences in semi-diameters semi-diameter (in arc-minutes) relative to total possible differences

Over 1 Saros Over 4 Saros’ Over 1 Saros Over 4 Saros’

A Moon 0.022’ 0.087’ 1.08% Sun 0.023 0.093 4.18

B Moon 0.03 1 0.122 1.53 Sun 0.034 0.134 6.18

C Moon 0.006 0.010 0.30 Sun 0.017 0.067 3.09

D Moon 0.015 0.055 0.74 Sun 0.027 0.108 4.91

4.29%

16.91

6.01 24.36

0.49 12.18

2.71 19.64

the Sun and the Moon can vary over time is considerably larger than the small differences in their semi-diameters over one Saros period. For example, Table 2 compares the sizes of the average absolute differences in semi-diameters for the Sun and the Moon over one Saros period with the total possible range in semi-diameters for the Sun and the Moon. The lunar differences over one Saros period are about 1% of the total differences possible, while the solar differences are about 3-696 of the total,

The solar semi-diameter and hence the Sun’s geocentric distance will not be as closely repeated as the lunar semi-diameter because, after one Saros period, the Sun has not returned to exactly the same position relative to its apse line. Instead of revolving through 18 complete cycles of its orbit, the Sun has gone through 18.03 cycles. Of course, the apse line of the Sun has also moved

550

Table 3

B.A. Steves

Comparison of the relative position and velocity coordinates for the Moon and the Sun during a partial lunar eclipse which occurred on Feb. 11.1952 and then recurred one Saros period later on Feb. 21,197O [22]

Eclipse date Positional coordinates

Feb. lO-11,1952 Feb. 21,197O

179.91Y -0.8473” 16.207’ 14.788’ 179.912 -0.8615 16.172 14.780

Eclipse date Velocity coordinates

Feb. lO-11,1952 Feb. 21.1970

- 10.944”lday -10.915

65.348’lday 65.305

-0.18”lday -0.22

3.84”lday 3.49

in those 18 years and 10 or 11 days. However, since these changes in the Sun’s orbital geometry are small over one Saros period, the Sun’s semi-diameters are still very closely repeated.

A repetition of the configuration and characteristics of an eclipse must include a repetition of the relative velocity vectors, as well as the relative radius vectors of the Moon and the Sun. In other words, the complete relative dynamical geometry of the Earth-Moon-Sun system must be repeated. If we now look at the full set of relative position and velocity coordinates for the Earth-Moon-Sun system at a particular eclipse epoch and compare these quantities with the same eclipse event one Saros period later, we find that, in general, all of them are very closely repeated.

In Table 3 the semi-diameters of the Sun and the Moon, os and a,,,, the differences between the Sun and the Moon’s geocentric ecliptic longitudes )cs - &, and latitudes Bs - /I,,,, and the daily rates of change of these coordinates, are seen to return to much the same values after one Saros period. The data in Table 3 describe a partial eclipse of the Moon which occurred on February 10-l 1, 1952 and recurred one Saros period later on February 21, 1970 [22,25,19].

Therefore at the end of a Saros period, the Earth, the Sun, the Moon, the nodes of the Moon’s orbit and the pericentre and apocentre of the Moon’s orbit have all returned to approximately the same relative positions that they held at the beginning of the Saros period.

5. EXPLANATION OF THE NEAR REPETITION OF ECLIPSES EVERY SAROS PERIOD

The near repetition of eclipses is a consequence of the set of high-integer near commensurabilities which exist between the Moon’s synodic period, its anomalistic period and its nodical period. The lunar synodic period is the time taken for the Earth-Moon-Sun system to move from one conjunction to the next, i.e. for the Moon to pass from full moon to full moon (Fig. 1). The lunar anomalistic period is the time taken for the Moon to move through one complete cycle of its orbit relative to its line of apses. Finally, the lunar draconic or nodical period is the time taken for the Moon to move through one complete cycle of its orbit relative to its line of nodes.

The mean values of these periods are listed in the 1988 Nautical Almanac as


Synodic period Ts = 29.530589 days, Anomalistic period TA = 27.554550 days, Nodicai period TN = 27.212221 days.

Although the actual values of these different lunar months can vary quite extensively from one revolution of the Moon to another because of the solar perturbations in the Moon’s orbit, these mean values remain constant to within one second over many centuries.

The commensurable set of integers which makes up the Saros period is then

223Ts = 6585.3213 days, 239TA = 6585.5375 days, 242T~ = 6585.3575 days.

In order for an eclipse to occur, the Earth, the Sun and the Moon must lie in approximately a straight line. This configuration can occur only when the Moon is at conjunction or opposition, and the Moon and the Sun are located near the Moon’s orbital nodes. At this point, a series of five or six lunar eclipses will occur about every six months before the required configuration is disrupted once again. A conjunction of the Moon and Sun occurs every lunar synodic period, while the Moon passes its ascending node every lunar nodical month. Therefore, conditions for the occurrence of an eclipse series must result whenever an integer multiple of the synodic month is approximately equal to an integer multiple of the nodical month. The lowest integer multiple set which meets these requirements is

47Ts = 1387.937683 days, 5lT~ = 1387.823271 days.

In this, we have the reason why a new eclipse series was observed to begin approximately every 47 lunar months.

To obtain a repetition of the characteristics of the eclipse series as well, the Earth-Moon and the Earth-Sun distances should also be repeated. A repetition of the Earth-Moon distance occurs once every lunar anomalistic month. A repetition of eclipse series characteristics would therefore occur over a period which equaled integer multiples of the synodic, nodical and the anomalistic months. The first set of integers to meet these requirements approximately, is the set which makes up the Saros period.

It might be thought that a repetition of the characteristics of an eclipse series would also re- quire a commensurability with the Sun’s anomalistic period. This does not occur within the Saros cycle. However, because the Saros period is only N 10 days longer than 18 solar years and because the Sun’s orbit is almost circular with an eccentricity of only 0.017, the Sun’s geocentric radius vector is still repeated approximately over one Saros period, despite its lack of commensurability with the solar anomalistic period. After one Saros period, the Sun’s geocentric radius vector is only about (10 days/365 days) x 360” w 10’ from its former position. Using familiar elementary properties of elliptical orbital motion [22], a displacement of 10’ in the true anomaly changes the Sun’s radius vector, velocity vector and the angle between them by at most 0.3%. A commensurability with the Sun’s anomalistic period does not therefore seem to be so crucial for a close repetition of the relative geometry of the Earth-Moon-Sun system.

The near commensurabilities between Ts, TA and TN, suffice to ensure that the mean relative geometry of the Earth-MoonSun system at the beginning of a Saros cycle is almost exactly repeated at the end of a Saros cycle. The whole system is simply rotated about 10” from its previous position.

552 B.A. Steves

6. OTHER CYCLES OF SELENE

Greek study of the heavens was possibly partially motivated by a need to improve the naviga- tion of their ships and to enable accurate calendar timekeeping. Thales of Miletus (624-547 BC) was accorded by Herodotus with having predicted a solar eclipse. However, Herodotus only states that Thales foretold that an eclipse would occur within the year. Because of the vagueness of Thales’ prediction, Pannekoek [ 171 surmises that Thales was probably unaware of the use of the Saros cycle as a means of predicting eclipses.

Another Greek, Meton (c. 433 BC) discovered the 19 year Metonic cycle within which an integer multiple of lunar months approximately equals an integer multiple of solar tropical years. A tropical year is the time interval between two successive passages of the Sun through the vernal equinox, in other words, the solar nodical period. Since the seasons recur every tropical year, the tropical year is the ideal average length for the calendar year.

Given that the mean solar tropical year is TY = 365.24220 days, the metonic cycle consists of

235Ts = 6939.69 days, 19T, = 6939.60days

and allows time-keeping by the lunar month to be incorporated into a solar calendar which keeps a measure of the changing seasons. In other words, for twelve calendar years out of the nineteen, the year would consist of twelve lunar months, while for seven calendar years a thirteenth month would be intercalated to make up for the fact that twelve lunar months are only 354.37 days and not the 365.24 days of the solar year. The Babylonians also knew of this cycle and it is not known whether Meton discovered it independently or borrowed it from them.

Hipparchus made careful observations of eclipses between 146 and 135 BC and compared these results with earlier Babylonian eclipse records, not to predict future eclipses, but in order to get more accurate values of the mean synodic and nodical lunar months. According to Deslambre [4, p. 1441, Hipparchus not only used the Saros cycle, but he also used a 345 year cycle of

4267Ts = 126007.02 days, 4573T~ = 126006.96 days, 4612Tsi = 126007.51 days,

3451; = 126008.56 days,

where Tsi = 27.321662 days is the mean lunar sidereal period. The lunar sidereal period is the time taken for the Moon to move through one complete cycle of its orbit relative to the distant stars (see Fig. 2). When the above cycle period is doubled, a commensurability with TN results:

8534Ts = 252014.05 days, 9261TN = 252012.38 days, 9146TA = 252013.91 days, 9224Tsi = 252015.01 days,

69OT, = 252017.12 days.

In addition, Hipparchus used a cycle of about 441 years and 103 days

5458Ts = 161177.95 days, 5923TN = 161177.99 days

and a cycle of 20 years and approximately 107 days given by


Table 4 Summary of historically known lunar cycles. E = lunar and solar eclipses; m = apparent sernidiameter of the moon; S = apparent semi-diameter of the sun; C = calendar date; L = location on earth for viewing solar eclipse

Cycle Commensurability in Repeatability of

Name Duration T, Ts TN TA Tsi EmSCL

saros - 400 BC

Meton c. 433 BC

Hipparchus 146135 BC 1. Short

2. Medium

3.1 Long

4. Long

Exelignos Triple Saros

Perchot 1894

Stockwell 1901

Crommelin 1910 1. Megalosaros

2. 18xStockwell

18.030 y 6585.321 d

19.000 6939.69

20 294 7412.178

441291 161177.95

3449% 126007.02

689 992 252014.05

54090 19 755.964

168011 61364,564

28.945 10571.951

1805.023 659270.399

521.011 190 295.116

223

235

251

5458

4267

8534

669

2078

358

22325

6444 6993.002

241.999

-

5 922 999

-

9261061

725 996

2255.037

388.500

24 226.997

238.992

269 000

4573 002

9146005

716976

2227.021

23 926.009

YYY--

- - Y Y -

- -y__-

- y - - - -.

4611982 - Y Y - Y

9223965 Y Y Y Y

- YYY--

2246004 Y Y Y Y

Y-Y--

YYY--

- Y-Y--

25lTs = 7412.1778 days, 2693~ = 7412.1740 days.

These lunar cycles and others are summarised in Table 4. Edmond Halley was the first astronomer in recent history to use the Saros to predict eclipses,

being generally credited by his contemporaries with having discovered it [14]. Halley, in fact, gave the Saros its name. He also seems to be the first astronomer to realise that detailed predictions of the Moon’s circumstances during an eclipse and not just general predictions of when an eclipse might occur could be made using the Saros [I]. In 173 1, Halley proposed a new method of determining the longitude of the Moon at any given time using the Saros cycle to improve the predictive accuracy of Newton’s Lunar Theory of 1702 [8].

He hoped to predict the lunar longitude with enough accuracy that comparison with the moon’s current position would give a universal time, which could then be compared with the local time to give the longitude of the observer. The problem of how to determine one’s longitude at sea was of great importance to the powerful naval nations of the time and a prize of 20 000 pounds (equivalent to 2 million pounds today) was offered in 1714 by the British parliament for the first person to solve it.

By comparing his observations with those of the first Astronomer Royal, John FQunsteed made

554 B.A. Steves

18 and 36 years before, Halley established that the errors between the observed lunar longitudes and those predicted by Newton’s procedure were repeated after 1 Saros period [8]. Halley made over 2200 observations of the lunar longitude positions during one Saros period from 1722 to 1740. This enabled him to empirically predict corrections to Newton’s Lunar Theory, which could then be included to improve the lunar theory. Kollerstrom [ 1 l] evaluated Halley’s method by repeating Halley’s calculations over three successive Saros cycles for the years 1690, 1708 and 1726, confirming that Halley was able to predict the lunar longitude to within approximately 2 arcminutes.

Unfortunately, Halley never published his data during his lifetime and so none of his contemporaries were able to check his new method. When his data was finally published in 1749 [9], his method was already becoming redundant with the development of new lunar theories based on the theory of gravity and calculus. Little attention, thereafter, was paid to his use of the Saros to predict the lunar longitude.

More recently, various commentators on the Saros cycle have taken it to refer principally to the commensurability existing between the lunar synodic period and the lunar nodical period because a good commensurability between these two values is the minimum requirement for eclipse prediction. Consequently, they have often ignored the fact that the Saros cycle also contains a commensurability with the lunar anomalistic period and have searched, instead, for more accurate metonic-like cycles (i.e. cycles produced by commensurabilities between only two periods) which consist of integer multiples of the synodic period approximately equalling integer multiples of the nodical period.

The use of one such period of 29 years minus 20 days where

358Ts = 10571.95 1 days, 388.5T~ = 10571.948 days,

but 383.673T~ = 10571.95 days

was advocated by Stockwell [31]. He felt that the 29 year cycle was a vast improvement on the Saros cycle traditionally used for eclipse prediction because it was more accurate, and therefore slower to change from cycle to cycle. It was also longer than the Saros cycle and therefore contained a larger series of eclipses for prediction. At the same time, it was not too long that use of it for eclipse predictions became cumbersome.

Crommelin [2] and Newcomb [ 151 point out that cycles like Stockwell’s cycle, while they may have very close commensurabilities between the synodic and nodical periods, and therefore can be used to predict with great accuracy when eclipses will occur, are of no use for predicting the type of successive eclipses or the location of these eclipses. Without the added commensurability with the anomalistic period which the Saros cycle contains, the characteristics of an eclipse cannot be repeated every cycle because the Moon’s true anomaly has not returned to its original value at the beginning of the cycle. In addition, the Saros cycle brings the Sun’s position back to within 10” of its former true anomaly. This means that, unlike the other cycles so far described, the Saros cycle enables the circumstances of an eclipse at its beginning to be almost totally duplicated at its end or in other words, the relative geometry of the Earth-Moon-Sun system to be almost exactly repeated.

Crommelin [2] mentions a few other cycles of interest which do maintain some of the important characteristics of the Saros cycle. He describes the triple Saros of 54 years and 33 days:

3 x 223Ts = 19755.964 days, 3 x 239T~ = 19756.612 days, 3 x 242T~ = 19756.072days,


which was known to the Creeks as the ‘Exelignos’ cycle. It has the advantage of having a period almost equal to a whole number of days. ‘Ihis means that after one ‘Exelignos’ cycle, the Earth has rotated back to its original position and that therefore the eclipse track on the Earth will also be found in almost the same position. This is helpful for predicting the locations on Earth where the solar eclipses can be seen. However after one Exelignos cycle, the Sun has moved about 30” from its former position and the Sun’s geocentric distance is not quite repeated.

He also studied a cycle of about 1805 years minus 6 days tlrst discovered by hf. Oppert, which Crommelin calls the Megalosaros:

22325Ts = 659270.40 days,

23926TA = 659270.16 days,

24227T~ = 659270.48 days.

The Megalosaros is just slightly greater than 100 times the Saros cycle. The commensurabilities over one Megalosaros cycle are closer to being exact than the commensurabilities for the Saros cycle are over the same time span. However, the relative geometry of the Earth-Moon-Sun system is still repeated more accurately over one Saros period than over one Megalosaros period.

Finally, Crommelin mentions a 521 year cycle which is 18 Stockwell cycles:

18 x (358)Ts = 6444Ts = 190295.12days, 18 x (388.5)T~ = 6993TN = 190295.06&ys,

6906TA = 190291.72 days, 521T, = 190292.19 days.

Again it is not as accurate as the Saros cycle. All of these cycles show close commensurabilities between Ts, TN and TA, but because their

large sixes make them a bit unmanageable for eclipse prediction and because they are less accurate, the use of the Saros cycle has remained the most popular method of predicting eclipses.

7. A MODERN USE OF THE SAROS AS AN INDICATION OF THE NEAR-PRRIODICITY OF THE EARTH-MOONSUN DYNAMICAL SYSTEM

So far, well-known ground has been covered in the present paper with respect to cycles involving the Sun, Moon and Earth used for calendar and eclipse predictions. Modem predictions of eclipses use sophisticated computer programs and complex lunar theories. It might be thought that such metonic and Saros type cycles are therefore of purely historical interest and of no importance to modem celestial mechanics research into the dynamics of the Earth-Moon system. Recent research by Perozxi, Roy, Steves and Valsecchi [29,19,25,32,33,30,34] has, however, made use of the existence of the Saros to investigate the dynamics of the Earth-Moon system. Their interest began with the property that the Saros has, of almost repeating exactly the relative geometry of the Earth-Moon-Sun system.

Crommelin [2] states that it is well-known that the Saros cycle “reproduces the distances, diameters and rates of motion of Sun and Moon with very considerable accuracy”. Yet no one previously seems to have considered that the existence of the Saros cycle for at least 2300 years, i.e. more than 130 Saros cycles, implies that the Earth-Moon system perturbed by the Sun is moving in a nearly periodic orbit of one Saros period.

So far, by focusing on the mean motions of the Moon and the Sun through the use of the mean periods Ts, TN and TA, we have been studying a fictitious dynamical system. Yet, despite the

556 B.A. Steves

fact that the Saros cycle is a relationship involving these mean motions, the closeness with which eclipses can be predicted in the real Earth-Moon-Sun system suggests that not only is the mean relative geometry repeated over one Saros period, but so is the real relative geometry.

This result is surprising, given that the eccentricities of the solar and lunar orbits may cause the Sun and the Moon to be up to f2” or f5” respectively from their mean positions. Their relative positions can, as a result, vary by as much as 7” from their mean relative positions at any time. Newcomb [ 151 mentions that such a large variation could change the eclipse time by half a day, and the distance of the Sun and the Moon from the nodes by about 2”. The combination of the above two effects could cause a recurring eclipse to be almost a day late or early. The character of the eclipse could change from a total to a partial eclipse, or a partial eclipse to one that fails to occur at all, or the reverse. Because none of these possible scenarios actually occur, the real relative geometry of the Earth-Moon-Sun system, as well as the mean relative geometry, must be nearly repeated every Saros period.

More importantly, this repetition of the relative geometry over one Saros period does not just occur at eclipse times, but at any time. After all, there is nothing unique about the geometry of an eclipse except that its occurrence is visible from the Earth. If the relative geometry of an eclipse is nearly repeated every Saros period, then so must the relative geometry of the Earth-Moon-Sun system in general also be repeated.

In order to test this hypothesis Perozzi, Roy, Steves and Valsecchi [29,19,25] adopted the following procedure. An epoch tl was chosen at random, but avoiding the time of a solar or lunar eclipse. Then, using the JPL high-precision numerically integrated planetary and lunar ephemerides [28], the relative position and velocity coordinates of the Moon and the Sun were found at this epoch tl. Similarly to the case described in Table 3 where the epoch is taken to be the time of an eclipse, they found the geocentric distance of the Moon r,,,, the differences between the Sun and the Moon’s geocentric ecliptic longitudes (& - hm) and latitudes (/ls - #Im) and the daily rates of change of these coordinates im, (i, - i&, (& - bm>. The Sun’s geocentric distance and rate of change were not considered as they would introduce a long-term libration with a period of the order 18 x 360/10 = 650 years (see Section 5).

Taking the value of the Saros period to be approximately T = 6585.3 days, the ephemeris was then searched for the time t2 near tl + T which minimized Q, the sum of the squares of the normalised differences between the relative position and velocity coordinates of the two epochs tl and t2. Perozzi et al. [19] describe the derivation of the expression for Q in more detail. T* = t2 - tl was then defined as the “osculating” value of the Saros period for that particular epoch tt . The procedure was repeated for 100 values of t1 spanning two Saros periods from 1952 to 1988.

The latter rows of Table 5 show a comparison of the relative positional and velocity coordinates for the two epochs tl and t2 for a sample set of 5 values of tl . The results confirm that as expected, regardless of the initial time chosen, the relative positions and velocities of the Moon and Sun are best repeated after an interval of time close to the classical Saros period, i.e. T* w Tsaros = 223Ts. The last two columns of Table 5 indicate the accuracy of the repetition, with Q being the minimum sum of the differences and P being the percentage relative discrepancy obtained by dividing Q by the total number of coordinates taken into consideration. From the sample of 100 epochs, the average difference P of a relative coordinate of the Earth-Moon-Sun system over T* x one Saros period is 0.5%. The average period T* for the sample was 6585.320 days, with a standard deviation of 0.03 days.

Traditionally the Saros period is taken to be 6585.321 days, which involves only the synodic month. By adopting the idea of minimizing the expression Q to find numerically the Saros period, Perozzi et al. [19] changed its definition; however, the average Saros period found in this manner is remarkably close to the traditional result, suggesting that the synodic month plays the dominant

1 Be cycles oj Selene 557

Table 5 Comparison of the relative position and velocity coordinates for the Moon and the Sun over one Saros period beginning at epochs chosen far from eclipses. For comparison, the first row corresponds to the partial lunar eclipse which occurred on Feb. 11.1954 [ 191

Saros period Beginning %)

rm ls-& is-i, Bs-& js-A Q P

(days) End (km/day) (“) (‘/day) (“) (‘/day)

year

6585.32696 1952 (eclipse) 1970

6585.32751 1958 1976

6585.27911 1964 1982

6585.36501 1974 1992

6585.36290 1979 1997

6585.32358 1988 2006

---_---- mth day

2 11.02729 403657.9 -1749.7 179.925 -10.942 -0.848 1.091 0.015 0.19 2 21.35425 404162.9 -1591.9 179.924 -10.916 -0.864 1.087

10 20.89109 392136.2 4829.7 2S4.564 -11.633 -4.980 0.401 0.023 0.29 10 31.21860 391372.4 4877.1 254.561 -11.677 -4.990 0.397

7 15.63149 390917.8 5398.4 279.984 -11.755 -5.163 0.227 0.032 0.40 7 26.91061 389942.7 5522.6 279.978 -11.817 -5.184 0.210

2 4.86549 359560.4 -2023.8 209.769 -13.991 1.620 1.281 0.004 0.05 2 16.23050 359473.7 -2206.4 209.769 -14.001 1.611 1.282

10 31.60589 364882.5 -1481.2 229.510 -13.543 1.011 1.272 0.009 0.11 11 10.96879 364603.2 -1727.4 229.510 -13.559 0.992 1.276

6 7.71649 370856.4 1595.6 84.094 -13.086 -0.380 -1.234 0.014 0.17 6 19.04007 370542.8 1308.2 84.093 -13.112 -0.305 -1.237

role in driving the system towards the repetition of any particular configuration. It should also be noted that this confirmation of the value for the Saros period to this accuracy

is obtained using only the criterion that the relative geometry of the Earth-Moon-Sun system is repeated after one Saros period. The Saros period is therefore evident in the JPL Ephemeris despite the fact that the JPL Ephemeris, in its calculations of the positions of the Earth, Moon and the Sun at any time, incorporates not only the effects of the gravitational interaction between the three bodies, but also the effects of the figures of the Earth and the Moon, the effects of the tides raised on the Earth by the Moon, and the effects of the point mass gravitational interactions of the other planets and the five largest asteroids. This underlines the fact that at any given time, these other perturbations on the Moon’s orbit are very small, being several orders of magnitude smaller than the gravitational perturbations of the Sun.

It was therefore likely that the main characteristics of the Saros cycle could also be investi- gated in the much simplified model of the elliptic restricted three-body problem of the Earth- Moon system disturbed only by the Sun. Perozzi et al. [ 191 then numerically integrated a system consisting of the Sun, the Earth moving in a fixed Keplerian ellipse with its mass augmented by that of the real Moon, and a massless Moon perturbed by the Earth and Sun. The initial positions and velocities of the Earth and Moon were taken from the JPL Ephemeris DE-l 18 at JD 2434000.5. The equations of motion were integrated in Cartesian coordinates using the RADAU integrator [5] ’ to 15th order to give the time evolutions of the Moon’s semimajor axis, eccentricity, inclination, argument of perigee and longitude of the nodes over two Saros periods. Here the Saros period was taken to be 223 synodic months, where the synodic month had to be calculated using Delaunay’s formulae [3] for the rates of change in the longitude of the nodes and pericentre as functions of v = nl /n, e, and i, because the initial osculating semimajor axis and the initial osculating eccentricity chosen for the Earth from the JPL Ephemeris are constants in

’ See “Explanatory supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Al- manac (1961) H.M.S.O.”

558 B.A. Steves

the restricted elliptical problem and these values were not the mean values for the Earth which give the current synodic lunar month. All the orbital parameters of the Moon with the exception of longitude of the node were shown to reproduce themselves very well after one Saros period, thus confirming that the Saros cycle is essentially a three-body phenomenon. The longitude of the node behaves differently because its revolution period is 18.6 years, slightly longer than that of the Saros cycle. The net result is that eclipses, which repeat after one Saros period, will occur with the three bodies aligned in a different direction with respect to a fixed reference frame than before. All other characteristics of the Earth-Moon-Sun configuration will be repeated.

Generally, it may be concluded that the relative dynamical geometry of the Earth-Moon-Sun system over one Saros period is repeated at any “osculating” phase of the period, and not just in the mean geometry reference frame nor simply at the occurrence of certain particular events such as eclipses. In other words the perturbations of the Sun on the Earth-Moon system, particularly the large disturbances in the Moon’s semi-major axis, eccentricity and inclination, are almost completely cancelled out by each other over any Saros period started at any time. This suggests that the Saros period could have relevance to any question of the stability of the Earth-Moon system against solar perturbations.

8. MIRROR CONFIGURATIONS IN THE SAROS CYCLE

How is it that the solar perturbations acting on the Earth-Moon system are nearly cancelled over one Saros period to such an extent that no matter when the Saros begins, the dynamical geometry of the system is very nearly repeated at its end?

A well-known theorem by Roy and Ovenden [24] states that if at a certain epoch a system of n gravitating point masses (n I 2) is aligned in such a way that every radius vector relative to the system’s centre of mass is perpendicular to every mutual velocity vector, the behaviour of the system after that epoch will be a mirror image of its behaviour before that epoch. There exist only two such types of possible configurations, one where the bodies are collinear with their velocity vectors perpendicular to the line (hereafter called collinear MC’s) and one where the bodies are coplanar with their velocity vectors lying perpendicular to that plane (hereafter called coplanar MC’s). Those solar and lunar eclipse configurations in which the Moon is at perigee or apogee are examples of collinear MC’s. Because of the relatively large value of the inclination of the lunar orbit, coplanar MC’s do not correspond to eclipse configurations.

A corollary to the mirror theorem states that if a dynamical system passes through two such mirror configurations, then the system is periodic, with a period equal to twice the time interval between successive mirror configurations.

Even if only a near mirror configuration occurs, the occurrence of a second near mirror configuration can be enough to ensure that the dynamical system moves in a quasi-periodic orbit for a finite length of time dependent on the deviation of the bodies’ configuration from exact mirror configurations. Since the Saros cycle has been known to exist for at least 2300 years, it would appear that the Earth-Moon-Sun system must pass through a minimum of two near mirror configurations every Saros period.

This can be easily proven in the case of a hypothetical perfect Saros cycle where the Sun’s geocentric orbit is assumed to be essentially circular. Let the Saros period be Tsaros = 6585.3 days and the Saros cycle be exactly

223Ts = 239T~ = 242T~ = Tsms,

If the Barth, Moon and Sun are now taken to be collinear - the Moon to be located at its pericentre, and the Moon’s line of nodes to be collinear with the line formed by the Earth, Moon


(01 (bl

Fig. 4. ‘hvo successive collinear mirror configuration events of the Earth (E)-Moon (M)-Sun (S) system, separated by one-half Saros period, and occurring at a conjunction (a) and opposition (b) respectively. The location of the ascending node of the Moon’s orbit is indicated by 52 [ 191.

and Sun - then at that epoch, the mutual velocities of the three bodies are perpendicular to their radius vectors and an exact mirror configuration exists. Suppose the Earth-Moon-Sun system is also in conjunction and the Sun is located in the direction of the descending node v (See Fig. 4a).

Then one half Saros period later, when Tsaros/2 = 115.5T, = 119.5T~ = 12lT~ months have elapsed, the Earth, Moon and Sun are once again collinear but in opposition. The Moon is now at its apocentre and the Sun is located in the direction of the ascending node Q (See Fig. 4b). Hence, a second mirror configuration is formed which will reverse the solar perturbations built up in the first half of the Saros period to return the system to its original mirror configuration, by the end of the Saros period.

The Saros cycle is, however, not perfect and the Sun’s geocentric orbit is not circular. If the observed mean motions of the Moon, the Sun, the Moon’s pericentre and the Moon’s ascending node are then studied, it can still be shown that even in the actual Earth-Moon-Sun mean dynamical system, any mean near mirror configuration of the Earth, Moon and Sun is followed one half Saros period later by a second near mirror configuration.

The observed mean longitudes of the Moon km, the Moon’s pericentre m, the Moon’s ascending node s2 and the Sun )Ls are given by the following formulae taken from The Explanatory Supplement to the Astronomical Ephemeris (1961, pp. 98 and 107):

a ,,, = 270.434164 + 13.1763965268 d”,

m = 334.329556 + 0.1114040803 d”,

Q = 259.183275 - 0.0529539222 d”,

hs = 279.696678 + 0.9856473354d”,

d being the number of Julian days that have elapsed from the epoch 241 5020.0 JD or 1900 January 0.5 ephemeris time. Higher order terms of d are neglected since they are very small for times of the order of one Saros period.

The Moon’s mean position with respect to the Sun, with respect to the lunar pericentre and with respect to the lunar ascending node are therefore given by Eqs. (1) to (3), respectively,

AA = AC, - )Ls = 350.737486 + 12.1907491914d”,

Am = A,,, - w = 296.104608 + 13.0649924465 da,

AQ = &, - s2 = 11.250889 + 13.2293504490d”.

(1)

(2)

(3)

560 B.A. Steves

Let us study the values of AA, Am and AQ at a time d and at a time d + tl later, where tl is 6585.3/2 = 3292.65 days or half a Saros period. If we take the differences between the values of A&Am and ASZ at these two times, we obtain

Ak(d + tl) - Al(d) = 40140.479” = 180.479”,

Aw(d + tl) - Am(d) = 179.101”,

A52 (d + tl) - A52 (d) = 0.282”.

Thus the results show that if the Earth-MoonSun’s mean dynamical system is at, or near, a mirror configuration at time d, a similar configuration will occur at time d + tl , one half Saros period later.

Perozzi et al. [ 191 studied the occurrence of mirror configurations in the case of a coplanar Earth-Moon-Sun system. This is not an unreasonable simplification as the Moon’s inclination is small. Using current values of the Sun’s mean motion n 1 and the rate of change of the Moon’s line of apses w, they discovered that the time taken for an opposition of the Ear&Moon-Sun system to move from the farthest possible angle from the line of nodes to the line of nodes and hence form a mirror configuration is of the order of 5.08 years, less than one half Saros period. After 5 years the opposition will, at best, be exactly aligned with the apse or, at worst, the opposition on one side of the moving apse and the succeeding one on the other side of the moving apse wiIl be equidistant from it at an angle no bigger than 0.36”. Therefore even in the ‘worst’-case scenario, the best near mirror configuration that occurs within one half a Saros period deviates only slightly from a perfect mirror configuration.

By the arguments given at the beginning of this section, once the first good near mirror configuration occurs within a half Saros period, a second near mirror configuration will follow one half Saros period later. The occurrence of two good near mirror configurations in any Saros period implies that the Earth-Moon-Sun dynamical system is moving in a nearly periodic orbit of period equal to one Saros period.

9. PERIODIC ORBITS IN THE MAIN LUNAR PROBLEM

A periodic orbit of period one Saros might therefore exist in an Earth-Moon-Sun system whose lunar mean semi-major axis, eccentricity and inclination were close in value to those of the Moon. A periodic orbit of period one Saros cannot exist in the elliptic restricted three- dimensional three-body model of the Earth-Moon-Sun system because the geocentric distance of the Sun is not repeated after one Saros period, due to the fact that 223 synodic months do not correspond to an integral number of years but instead to 18 years 10 days. A circular restricted model, however, does not have this problem. Since the Earth’s eccentricity is small and the difference between the Saros period and the duration of the year is only 10 days, the behaviour of a periodic orbit found in the circular problem should still be very similar to that of the real lunar orbit.

Valsecchi et al. [32] searched for a periodic orbit of period one Saros in the restricted circular three-dimensional three-body Earth-Moon-Sun problem with mean semimajor axis Q, eccentricity e and inclination i close to that of the Moon. They discovered, numerically, not just one, but eight related periodic orbits based on the 8 possible pairs of the 16 different mirror configurations that exist in the problem.

In the restricted circular three-dimensional three-body problem, the 16 MC’s can be uniquely identified by the values of three angles: the argument of perigee o, the mean anomaly M of


Table 6 The 16 possible MC’s of the circular restricted three-dimensional three-body problem [32]

MC 0 M A1

ooo O0

002 O0 020 0" 022 0”

100 9o” 102 9o” 120 90”

122 9o”

200 180”

202 180”

220 180” 222 180” 300 270”

302 270’

320 210”

322 270°

O0 O0

180’ 180°

O0 O0

180”

180” O0 O0

180’ 180”

0” 0”

180” 180’

O0 180”

O0 180”

O0 180”

O0

180” O0

180’ 0”

180” O0

180’ O0

180’

the Moon and the difference between the mean geocentric longitudes of the Moon and the Sun A)c = )L - Al. Eight collinear MC’s are possible when w, M and A)c are either 0” or 180”, while eight coplanar MC’s are formed when u is either 90” or 270” and M and A1 are either 0” or 180“. Table 6 unambiguously identifies the 16 MC’s with a three-digit code representing w, M and A)c as multiples of 90”.

The revolution periods of the angles AA and M are the synodic month Ts and the anomalistic month TA respectively, while the revolution period of the angle w + A4 is the nodical month TN. In the Saros, the three months are related by

223Ts = 6585d.32d 21239T~ 2: 242T~. (4)

Since the difference of duration between TA and TN is due to the secular rotation of w, Eq. (4) can also be written as

223Ts N- 239T~ N 3T,, (5)

where To is the period of revolution of w. In periodic orbits of period one Saros = 223Ts, relation (5) must hold exactly.

In general, for any periodic orbit based on Saros-type commensurabilities between the synodic, anomalistic and nodical months (or the revolution periods of the angles o, M and AA), the following relation must hold exactly:

NAA Ts = NM TA = NoTo = 1 &OS period,

where NAP, NM and No are suitable integers.

(6)

Hereafter ‘a Saros period’ will denote the period of a periodic orbit which results from the commensurabilities given in Eq. (6) where Ts, TA and T, are variables, while ‘the Saros period’ will refer to the period of the commensurability given in Eq. (5) where Ts is that of the current Moon. From Eq. (6), the pairs of MC’s which exist in each of the 8 periodic orbits are then uniquely given by the values N,, N,u and NAA. If we start from a MC, then half a Saros period later the second MC will be reached where w, M and AA will differ from the initial values by

562

Table I

B.A. States

The seven possible sets of pairs of MC’s contained in periodic orbits of the circular restricted 3- dimensional three- body problem. The coefficients No, NM, NAJ. are, respectively, even, even, odd for EEO, even, odd, even for EOE, etc. [32]

EEO EOE EOO OEE OEO OOE 000

ooo 000 000 ooo ooo 000 ooo 002 020 022 200 202 220 222

020 002 002 002 002 002 002 022 022 020 202 200 222 220

200 200 200 020 020 020 020 202 220 222 220 222 200 202

220 202 202 022 022 022 022 222 222 220 222 220 202 200

100 100 100 100 100 100 100 102 120 122 300 302 320 322

120 102 102 102 102 102 102 122 122 120 302 300 322 320

300 300 300 120 120 120 120 302 320 322 320 322 300 302

320 302 302 122 122 122 122 322 322 320 322 320 302 300

(N,/2)360”, (N~/2)360” and (N~~/2)360”, respectively. Table 7 lists the 8 pairs that one can have, given the different possible parities of N,, NM and NAP, where E = even and 0 = odd numbers. For the Saros, the parity for N,, NM and NAA is therefore denoted at 000.

Four of the orbits contain collinear MC’s, four contain coplanar MC’s and all of the orbits correspond to the same mean values of a, e and i, for a given set of N,, NM and NAP. The three coefficients N,, NM and NAA must not be divisible by a common multiple or a shorter periodic orbit would exist. Hence the case EEE is not allowable.

A restricted circular three-dimensional three-body model with a massless Moon and an Earth whose mass was augmented by that of the real Moon was used to find the 8 periodic orbits numerically. The Moon was placed at one of the mirror configurations by specifying the appropriate values for w, M and A)c. A first guess at the starting geocentric orbital elements for the Moon ao. eg and io was made and the problem was then numerically integrated for one half Saros period, using Everhart’s [5] RADAU integrator taken to 15th order. After one half Saros period, the Moon’s coordinates X, y, z, i, Jo, i were examined in a sidereal ecliptic frame where the Sun- Earth direction returns to the x-axis every half Saros period. If the orbit is periodic, a second MC should exist at this time. For collinear MC’s, lunar coordinates y, z and i should be zero. For coplanar MC’s, y, i and i should be zero. The initial guesses for ao, eo and io were then adapted until the relevant lunar coordinates were zeroed at the half Saros period stage.

Table 8 gives the resulting osculating values for a, e and i at each of the 16 MC’s in the 8 periodic orbits. Fig. 5 shows the time evolution of the Moon’s orbital elements a, e, i and o over one year for: (1) the periodic orbit passing through the MC’s coded as 200 and 022 in Table 8 and (2) the ‘real’ Moon as given by the JPL DE 102 [16] starting at a time when the Moon was very nearly in a MC of type 200. Although the periodic orbit was developed from values of the mean semi-major axis, eccentricity and inclination close to those of the Moon, the resulting time evolutions of the osculating orbital elements bear a remarkable similarity to the

Table 8


The osculating elements a, e and i of the Moon for the 16 possible MC’s which form the 8 periodic orbits associated with the Saros in the circular restricted three-dimensional three-body problem [32]

MC a (AU) e i 0

000 222

200 022

002 220

202 020

100 322

300 122

102 320

0.00258099716 0.083356670 0.00259063207 0.056488496

0.00258099716 0.083356670 0.00259063207 0.056488496

0.00258082066 0.083864066 0.00259085405 0.056953569

0.00258082066 0.00259085405

0.00258081914 0.00259043342

0.00258081914 0.00259043342

0.00258064647 0.00259065 120

0.083864066 0.056953569

0.08393625 1 0.057165911

0.083936251 0.057165911

0.084452672 0.057640009

5.6908146 5.6881713

5.6908146 5.6881713

5.6915135 5.6887261

5.6915135 5.6887261

5.3658476 5.3481729

5.3658476 5.3481729

5.3655959 5.3475655

302 0.00258064647 0.084452672 5.3655959 120 0.00259065120 0.057640009 5.3475655

‘real’ osculating orbital elements of the Moon. The time evolutions of the semi-major axis are practically indistinguishable, while most of the differences in the evolutions of the eccentricity and the inclination involve a systematic shift arising from the difference between the values of e and i for the real Moon’s orbit and the periodic orbit. Valsecchi et al. [32] show that the close similarity between the osculating orbital elements continues throughout at least one Saros period.

10. THE SAROS CYCLE AS A POSSIBLE STABILISING MECHANISM FOR THE MOON’S ORBIT

Periodicity, in itself, does not guarantee the stability of a system against perturbations by external forces. A system can be periodic and still be unstable, in the sense that a slight disturbance by a small external force could send the system into a non-periodic orbit which could result in a marked irreversible change from the original orbit.

Roy and Ovenden [24] argued that although periodicity does not guarantee stability, periodic orbits which pass through mirror configurations are more stable, if the time intervals between successive mirror configurations are shorter. As long as a second mirror configuration is able to occur before disturbances have enough time to accumulate to the point of causing irreversible change to the orbit, the system will return to the neighbourhood of its original orbit, with the second mirror configuration reversing the perturbations caused by the small external force.

While Roy and Ovenden [24] stated that they did not treat the question of stability in an exhaustive and rigorous manner, they did show that their hypothesis is confirmed by observations made in the solar system. They proved that two orbiting masses having nearly commensurable mean motions, and certain values for either of the orbital parameters ti or !& depending on whether eccentricities or inclinations dominate the system, will produce frequent occurrences of mirror configurations. They then studied the nearly commensurable systems of low integer

B.A. Steves 564

a(AU)

0.00258

0.00256

0.00254

I e a

i / 237!900 237’4100 t’“:‘]

5.6”: w : 5.20: w 1 4.8’.

b

I. I I I

2371900 2372100 t( JD) 2371900 2372100 t( JD)

Fig. 5. (a) Time evolution of the osculating semi-major axis u of the Moon for the JPL Ephemeris DE 102 and two periodic orbits described in the text over approximately one year starting with an MC. Although practically indistinguishable, tbere are actually three curves in the figure. (b) The equivalent of (a) for the osculating eccentricity e. The upper curve refers to the periodic orbit associated with the Saros, the lower curves to the “long” periodic orbit (see Section 12) and to the DE 102 orbit. (c) The equivalent of (a) for the osculating inclination i. The upper awe refers to the periodic orbit associated with the Saros, the central one to the DE 102 orbit, and the lowest curve to the “long” periodic orbit (see Section 12). (d) The equivalent of (a) for the osculating argument of perigee o. The individual curves are difficult to distinguish, but the curve with the smaller amplitude refers to the DE 102 orbit and the “long” periodic orbit [34].

values in the solar system and verified that these systems are indeed arranged in such a way as to allow the most frequent occurrences of mirror configurations. The greater stability of systems passing through frequent mirror configurations is then suggested to be the reason why more commensurable systems exist in the solar system than are expected by mere chance [23].

In actual fact, the orbital mean motions do not necessarily have to be nearly commensurable in order for frequent mirror configurations to occur. That they are generally commensurable for most of the three-body systems in the solar system exhibiting frequent occurrence of mirror configurations is simply the result of the fact that most of the systems have values of ti or ti which are much smaller than their mean motions.

For the collinear-type mirror configurations, the shortest time between successive perfect mirror configurations in a three-dimensional system occurs when the bodies are in conjunction or opposition every time the bodies are positioned along their mutual line of nodes and their orbital apsidal lines. For example, if we take a case similar to the Ear&Moon-Sun system where the Sun’s orbit is taken to be circular, frequent occurrences of mirror configurations will occur if


the satellite, the Sun, the satellite’s line of nodes and the satellite’s pericentre or apocentre are aligned as often as possible, i.e.

AiTs = A~TA = A~TN, (7)

where Ai for i = 1,2,3 are small integers. In other words, frequent occurrence of mirror configurations occurs when the system contains a perfect Saros cycle.

Eq. (7) can be rewritten in terms of the mean motions of the satellite n, the Sun n 1, the satellite’s pericentre Tb and the satellite’s line of nodes fi, i.e.

At 2n-

A2

n - nl =27t7 =2n& (8)

n--m

and hence can be rearranged in the more familiar form describing the conditions that will produce resonances in the circular three-body problem, i.e.

Bi(n - nt) = Bz(n - ti) = B3(TZ - Cl),

where Bi for i = 1,2,3 are integers.

(9)

For example, in the case of periodic orbits associated with the Saros, where

223Ts = 2393~ = 242T~,

we could write

239(n - nt) = 223(n - tb) and 242T~ = 223(n - S?Z),

which leads to

16n - 239n1 + 223ti = 0 and 19n - 242n1 + 223fi = 0.

In general, Saros-like periodic orbits are equivalent to e-i resonances of the form

iln - iznl + i3ti = 0 and i4n - ipI+ i3d = 0. (10)

Thus, Saros-like periodic orbits are not necessarily close, in general to mean motion resonances. Only for those systems where rir and b are small relative to their mean motions n, would

the mean motions be nearly commensurable and frequent occurrences of mirror configurations occur, viz.

Bl(n-n1)2:B2nCBqz or irk Bl B1 -2:-. B1 - B2 B1 - B3

The Metonic cycle of 19n 2: 254nl is such a relation in the Earth-Moon-Sun system, but in- spection of the lunar orbital behaviour in the IPL Ephemeris shows that the lunar orbit is not nearly periodic over this cycle’s period.

In general, frequent occurrences of mirror configurations in the circular three-body problem result when the dynamical system contains a set of commensurabilities like those of Eq. (7) or, in other words, when the system contains a Saros cycle. Extension of the solution to that of the general three-body problem simply requires the addition of commensurabilities with the Sun’s anomalistic and nodical period, to enable the frequent alignments of the Sun’s line of nodes and apses with a conjunction or opposition, as well as with the Moon’s line of nodes and apse. Then for example,

566 B.A. Steves

Bl(n - nl) = Bz(n - ti) = Bg(n - h) = B4(nl - zirl) = B5(n1 - !&). (11)

In the simpler case, where the Sun is taken to move in an elliptical orbit with small rates of change of pericentre and node relative to its mean motion, the following set of commensurabilities is sufficient for frequent occurrences of mirror configurations:

Bl(n - nl) 2: Bz(n - rb) ” B3(n - ti) 2: B4(nl) = Bs(nl).

or in terms of the mean periods

AITS 2: A~TA 2: A~TN = AdTo,

where Bi and Ai are integers and To is the Sun’s sidereal period. The Hipparchus cycle of - 690 years and the Perchot cycle [ 18,261 of

2078Ts 21 2227T~ N 225ST~ 2: 168To

so that

149n - 2227n1+ 2078ti = 0;

177n - 2255nl+ 20786 = 0; and

2078n - 2246nt = 0

are examples of the above type of resonance in which there is a possibility that periodic orbits associated with it may be found in the elliptic problem.

Eqs. (9) and (11) are recognisable as the conditions which will produce resonances in the circular three-body problem and general three-body problem, respectively. Thus, the statement that a three-body system moves through two mirror configurations which form a periodic orbit, is equivalent to the statement that the system is moving in a resonance. The existence of resonances within a planetary system is known to be closely related to the stability of that system [7,6]. Perhaps the mirror theorem can be used as a simple way of understanding physically how resonances can act to stabilise a system.

If the hypothesis, that “periodic orbits are more stable against external perturbations, the shorter the time interval between successive mirror configurations” is true, then circular three- body dynamical systems containing perfect Saros cycles, which we have already seen move through two mirror configurations, will be more stable the smaller the Saros period. Saros cycles whose periods are too large may not have enough time to return the dynamical system to its original geometry before solar perturbations grow to the point where the system’s orbital elements are changed irreversibly.

If the Saros cycle is not perfect, as in the case of the Earth-Moon-Sun system, its duration will also depend on how closely the near Saros cycle approaches a perfect Saros cycle. Saros cycles which are too inaccurate can not successfully cancel solar perturbations and therefore will not last very long.

Also, if a near Saros cycle can help the system to endure long enough to reach a Saros cycle that is closer to a perfect Saros cycle but of much longer period, then the system will again last longer than if it was only able to use the solar perturbation cancelling properties of the first Saros cycle.

In summary, it seems that a circular three-body system will be more stable: (1) the shorter the Saros period; (2) the closer the near Saros cycle is to a perfect Saros cycle; (3) the greater the number of near Saros cycles operating at any given time.


11. THE EFFECT OF TIDAL EVOLUTION ON THE SAROS

If the Saros cycle does act as a stabilising mechanism, then of importance is the question of how probable it is that such cycles comprised of commensurabilities between synodic, anomalistic and nodical periods should exist, especially given the knowledge that tidal evolution has changed these periods in the Moon’s past and is continuing to change them today.

Authors such as Mitchell [ 131 and Crommelin [2] merely note that the anomalistic period plays a very convenient role in helping to predict eclipse characteristics. Newcomb [ 151 in his paper on “The Recurrence of Solar Eclipses”, talks of “two remarkable chance relations connected with the Saros.. . without which the period would not have served the purpose of foreseeing eclipses so well as it actually does.” He is referring to the fact that after one Saros period, the mean anomalies of the Moon and the Sun have returned to within less than 3” and 12” respectively of their original positions. He goes on to say, “There is no a priori reason that this should be the case: it arises only from the fact that 18 years is a close multiple, not only of the times of revolution of the Sun and the Moon, but also of the times of revolution of the Moon’s node and perigee”.

Many people have remarked on the fortuitous existence of the near commensurabilities form- ing the Saros cycle, but to our knowledge no one has asked the question, “Just how much of a coincidence is the existence of a Saros-type cycle within the orbital dynamics of the Earth- Moon-Sun system?” If the existence of a Saros-type cycle is improbable, what implications does this have for the Moon’s orbital evolution when in the past or the future, the Moon may not have had, or may not have, a Saros-type cycle to use as a stabilising mechanism against solar perturbations?

With regards to the past history of the Moon’s orbit, Sonett et al. [27] have estimated the duration of half a synodic lunar month to be 0.038095 f 0.00003 yr in the late Precambrian era approximately 6.8 x lo* years ago. They determine this value under the assumption that lunar and solar tides were the source of the laminae of the Elatina Formation in South Australia.

If we use Delaunay’s expression for the rate of change of the lunar perigee and node as functions of a, e, i, el and assuming that the mean eccentricity and inclination of the Moon have not changed very much from present values as a result of tidal evolution [ 121, it is found that another Saros-like relationship exists in the Precambrian era of the form

512Ts = 1424gd.4 N 547T~ ” 553T~ (12)

Valsecchi et al. [32] showed that a set of 8 periodic orbits in which Eq. (12) holds in its exact form can be found in the restricted circular three-dimensional three-body problem. The mean lunar eccentricity and inclination for these periodic orbits are even closer to the current ones than those of the periodic orbits found for the Saros. Their period of 39.01 yr is very close to an integral number of years suggesting the possibility that a periodic orbit may also exist nearby in the mean a, e, i and ei phase space in the elliptic restricted three-dimensional three-body problem.

Can, therefore, a Saros-like periodic orbit close to the ‘real’ lunar orbit be found for any value of the Moon’s semi-major axis that may exist during the Moon’s evolution? Recently Steves, Valsecchi, Perozzi and Roy [29,30] showed that this is not the case. The Earth-Moon-Sun dynamical system exhibits near Saros-like cycles for only 25 to 30% of the time interval between -5 x IO7 to +5 x lo7 years given an upper limit of 1000 on the size of integers allowed in the commensurable relations and a maximum error of 1% of the synodic period allowed from exact equalities in the commensurabilities. In addition, if the variations of the Earth’s mean eccentric-

568 B.A. Steves

ity are taken into account, a Saros-like cycle which appears to exist for a constant value of et in fact exists only intermittently throughout its calculated duration.

Curiously, the present Saros is found to have the smallest period of the Saros-like cycles during this time. It also endures the longest and has a period closest to an integral number of anomalistic years, making it one of the most efficient Saros-like cycles for reversing solar perturbations in the main lunar problem.

What happens to the Moon’s orbit during epochs when tidal friction and/or the change in eccentricity of the Earth’s orbit take the Moon into situations where no short period Saros-like cycles exist in its orbital dynamics is still unknown. Possibly, solar perturbations may grow unchecked to the point where the Moon’s orbital elements are changed irreversibly before another Saros-like cycle of short period is found.

12. THE ARRANGEMENT OF LONGER SAROS-LIKE CYCLES IN MEAN ELEMENT SPACE

Periodic orbits derived from Saros-like cycles (SLC’s) of much longer durations are, however, very abundant in the Earth-Moon-Sun dynamical model [33]. This is not surprising given Poincare’s famous conjecture [20]:

“There is one thing I was not able to demonstrate rigorously, but that I think is very likely. That given a system of canonical equations (he refers to the N-body equations of motion) and one of its particular solutions, it must be always possible to find a periodic solution (possibly of a very long period), so that the difference between the two solutions is arbitrarily small for an arbitrary long time span”.

In other words, Poincare suggested that for every bounded trajectory in phase space there should exist a periodic orbit arbitrarily close to that trajectory but possibly of very long period.

Valsecchi et al. [33] used the general relationship for a Saros-like cycle given by Eq. (6) to explore the mean lunar eccentricity and inclination phase space for other SLC’s (i.e. other periodic orbits). Keeping the synodic period constant at its present value, the other lunar periods were varied until integer multiples NAA, NM and NW were found which satisfied Eq. (6) exactly. The resulting three lunar periods were then used in Equations (8) to find the mean motion of perigee and node. The mean motion of perigee and node and the present values for v = nl/n and er, were then substituted into Delaunay’s equations (see Ref. [30]) which could then be solved numerically for the corresponding mean lunar eccentricity and inclination.

Fig. 6 shows the arrangement of the resulting SLC’s in e-i phase space for values of No limited to 10,20,40 and 80 in turn. Each point or SLC corresponds to a set of 8 periodic orbits which can be found numerically using the procedure in Section 9. The curious figure resembles a city map, with “squares”, large “avenues” and an intricate, self-similar network of “narrower streets”. The “squares” are situated at the crossings of “avenues” and have at their centres a point representing a set of periodic orbits whose NAP is much smaller than those of the surrounding orbits. The “streets” get narrower and the “squares” get smaller as values of No get larger.

The point representing the Saros is located near the centre of the figure at the centre of the largest square (hereafter called Saros square), while the point representing the location of the Moon’s present orbit in e-i phase space is, interestingly enough, also close to Saros square. Approximately 10000 SLC’s were discovered in e-i phase space within the ranges of 0 5 e i 0.1 and 0 5 i 5 lo”, when a maximum of 115 revolutions of w was allowed.

Valsecchi, Perozzi, Roy and Steves [34] then searched the arrangement of SLC’s for periodic orbits of longer durations whose e and i were much closer to those of the ‘real’ Moon than the e


Fig. 6. Mean eccentricities and inclinations of periodic orbits found in the restricted circular three-dimensional three-body problem close to the orbit of the Moon, for a number of revolutions of the argument of perigee o of up to 10 (upper left), 20 (upper right), 40 (lower left) and 80 (lower right) [33].

and i of the periodic orbit associated with the Saros. The time evolutions of the osculating lunar orbital elements of one such periodic orbit of period 7571 Ts (called the ‘long’ periodic orbit) is shown in Fig. 5. Although difficult to distinguish because of the scale, the ‘long’ periodic orbit is indeed an improvement in matching the eccentricity, inclination, semi-major axis and argument of perigee of the Moon’s orbit. This feature seems to be supporting evidence for the truth of Poincare’s conjecture. However, in the case of the real Moon’s orbit, a preliminary analysis suggests that even over a short time span of one year the residuals from the much longer periodic orbit show a significant trend which is probably due to the neglecting of the Earth’s eccentricity. Thus, for continued improvement, it would be better to upgrade to a more general model such as the elliptic restricted three-dimensional three-body model.

13. CONCLUSIONS

It appears that the Saros cycle has been used for little in the past but prediction of eclipses. It is therefore curious that a feature known for over two thousand years, particularly to all those who have worked on Lunar theories, should have something new to say to us. Not only does it lead to the existence and determination of three-dimensional periodic orbits of long duration, but it also provides supporting evidence for the conjecture made by Poincare over a hundred years ago.

But Selene has yet more secrets. Tidal friction will push the Moon out until the day equals the month, both Earth and Moon turning the same faces towards each other. What we have not yet learned is whether, in those far-future times, the Moon will still have Saros-type cycles as

570 B.A. Steves

a defence against the enhanced gravitational disturbances of the Sun or whether the Moon will escape from the Earth to lead to an independent existence as a planet.

ACKNOWLEDGEMENTS

The author gratefully acknowledges her indebtedness to her fellow collaborators in Project POINCARE (Periodic Orbits Involving Numerous Circuits: Assessment, Research, and Explo- ration): Dr E. Perozzi (Nuova Telespazio, Rome, Italy), Dr G. Valsecchi (CNR, Riparto Plane- tologia, Rome, Italy) and Professor A.E. Roy (Dept. of Physics and Astronomy, Glasgow Uni- versity, Glasgow, UK).

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[l] A. Cook, Halley and the Saros, Q.J.R. Astl: Sot. 37 (1996) 349-353. [2] A.C.D. Crommelin, The 29-Year eclipse cycle, The Observatory 24 (1901) 379. [3] Ch. Delaunay, Note sur les mouvements du perigee et du noeud de la Lune, Compt. rend.

hebdom. Acad. Sci. 74 (1872) 17. [4] J.B.J. Deslambre, Histoire de1 1’Astronomie Ancienne II, Paris (1817). [5] E. Everhart (1985) An efficient integrator that uses Gauss-Radau spacings. In Dynamics of

Comets: their Origin and Evolution, A. Carusi, G.B. Valsecchi, Eds. (Reidel, Dordrecht, 1985) p. 185.

[6] P. Goldreich, An explanation of the frequent occurrence of commensurable mean motions in the Solar System, Mon. Not. Roy. Ast,: Sot. 130 (1965) 159.

[7] R. Greenberg, Orbital resonances among Saturn’s satellites. In Saturn, T. Gehrels, M.S. Mathews, Eds. (Univ. of Arizona Press, Tucson, 1984).

[8] E. Halley, A Proposal of a method for finding the longitude at sea within a degree or twenty leagues by Dr Edmund Halley Astr. Reg. Vice-President of the Royal Society. With an account of the progress he bath made therein by a continued series of accurate observations of the Moon, taken by himself at the Royal Observatory at Greenwich, Philos. Trans. 37, No. 421 (1731) 185-195.

[9] E. Halley, Edmundi Halleii Astronomi dum viveret Regii Tabulae Astronomicae accedunt de usu Tabularum Praecepta, J. Bevis, Ed., London (1749);

[lo] E. Halley, Astronomical tables with Precepts both in English and in Latin for computing the places of the Sun, Moon, &c. (translations of Halley, 1749), London (1752).

[ 1 l] N. Kollerstrom, A reintroduction of epicycles: Newton’s 1702 Lunar Theory and Halley’s Saros correction, Q.J.R. Ask Sot. 36 (1995) 357-368.

[ 121 F. Mignard, The lunar orbit revisited, III, Moon & Planets 24 (198 1) 189-207. [ 131 S.A. Mitchell, Eclipses of the Sun, 5th ed. (Columbia University press, New York, 195 1). [14] 0. Neugebauer, The Exact Sciences in Antiquity (Brown Univ. Press, 1957) pp. 141-142. [ 151 S. Newcomb, The recurrence of solar eclipses, Astron. Papers of the Amel: Ephemeris 1

(1822) 7. [ 161 X.X. Newhall, E.M. Standish, J.G. Williams, DE 102: A numerically integrated ephemeris

of the Moon and planets spanning forty-four centuries. Astron. Astrophys. 125 (1983) 150. [17] A. Pannekoek, A History of Astronomy, (George Allen & Unwin, London, 1961). [ 181 J. Perchot, Bull. Astron. 11 (1894) 21.


[19] E. Perozzi, A.E. Roy, B.A. Steves, G.B. Valsecchi, Significant high number commen surabilities in the main lunar problem. I: The Saros as a near periodicity of the Moon’s orbit, Celest. Mech & Dynam. Astron. 52 (1991) 241-261.

[20] H. Poincare, Les Methodes Nouvelles de la Mecanique Celeste (Gauthier-Villars, Paris, 1893).

[21] A.E. Roy (1983) Asymptotic approach to mirror conditions as a trapping mechanism in N-Body hierarchical dynamical systems. In Dynamical trapping and evolution in the Solar System, V.V. Markellos, Y. Kozai, Eds. (Reidel, Dordrecht, 1983) pp. 227.

[22] A.E. Roy, Orbital Motion, 3rd ed. (Adam Hilger, Bristol, 1988). [23] A.E. Roy, M.W. Ovenden, Mon. Not. Roy. Astron. Sot. 114 (1954) 232. [24] A.E. Roy, M.W. Ovenden, On the occurrence of commensurable mean motions in the solar

system II. The mirror theorem, Mon. Not. Roy. Astron. Sot. 115 (1955) 296-309. [25] A.E. Roy, B.A. Steves, G.B. Valsecchi, E. Perozzi, Significant high number commen

surabilities in the Main Lunar Problem: a postscript to a discovery of the ancient Chaldeans. In Predictability, Stability and Chaos in N-Body Dynamical Systems, A.E. Roy, Ed. (Plenum, London, 1991) pp. 273-282.

[26] K. Schwarzschild, Astron. Nuchl: 147 (1898) 289. [27] C.P Sonett, S.A. Finney C.R. Williams, The lunar orbit in the late Precambrian and the

Elatina sandstone laminae, Nature 335 (1988) 806-808. [28] E.M. Standish, M.S.W. Keesey, X.X. Newhall, JPL Development Ephemeris Number 96’,

NASA Technical Report 32 (1976) 1603. [29] B.A. Steves, Finite-Time Stability Criteria for Sun-Perturbed Planetary Satellites, PhD

Thesis, Univ. Glasgow (1990). [30] B.A. Steves, G.B. Valsecchi, E. Perozzi, A.E. Roy, Significant high number commen

surabilities in the Main Lunar Problem. II: The occurrence of Saros-like near-periodicities, Celest. Mech & Dynam. Astron. 57 (1993) 341.

[31] J.N. Stockwell, (1901) Eclipse-cycles,Astmn. J. 21(1901) 185. [32] G.B. Valsecchi, E. Perozzi, A.E. Roy, B.A. Steves, Periodic orbits close to that of the Moon,

Astnm. Astrophys. 271(1993) 308. [33] G.B. Valsecchi, E. Perozzi, A.E. Roy, B.A. Steves, The arrangement in mean elements’

space of the periodic orbits close to that of the Moon, Celest. Mech. & Dynam. Astrvn. 56 (1993) 373.

[34] G.B. Valsecchi, E. Perozzi, A.E. Roy, B.A. Steves, Hunting for periodic orbits close to that of the Moon in the restricted circular three-body problem, in: From Newton to Chaos, A.E. Roy, B.A. Steves, Eds. (Plenum, New York 1995) p. 231.

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