Download I. ASYMMETRY OF ECLIPSES. CALENDAR CYCLES

I. ASYMMETRY OF ECLIPSES. CALENDAR CYCLES
Igor Taganov & Ville-V.E. Saari
1.1 Metaphysics of solar eclipses
1.2 Calendar cycles of solar eclipses
Literature
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p. 20
p. 26
To describe the two main types of solar eclipses in modern astronomy the old Latin terms – umbra,
antumbra and penumbra are still used (Fig. 1.1). A “partial eclipse” (c. 35 %) occurs when the Sun and
Moon are not exactly in line and the Moon only partially obscures the Sun. The term “central eclipse” (c.
65 %) is often used as a generic term for eclipses when the Sun and Moon are exactly in line. The strict
definition of a central eclipse is an eclipse, during which the central line of the Moon’s umbra touches the
Earth’s surface. However, extremely rare the part of the Moon’s umbra intersects with Earth, producing
an annular or total eclipse, but not its central line. Such event is called a “non-central” total or annular
eclipse [2].
Fig. 1.1. Main types of solar eclipses
The central solar eclipses are subdivided into three main groups: a “total eclipse” (c. 27 %) occurs
when the dark silhouette of the Moon completely obscures the Sun; an “annular eclipse” (c. 33 %) occurs
when the Sun and Moon are exactly in line, but the apparent size of the Moon is smaller than that of the
Sun; a “hybrid eclipse” or annular/total eclipse (c. 5 %) at certain sites on the Earth’s surface appears as a
total eclipse, whereas at other sites it looks as annular. There are more annular solar eclipses than total
because on average the Moon moves too far from the Earth to cover the Sun completely.
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From ancient times several picturesque effects during solar eclipses are known, for example,
“Diamond rosary” (Fig 1.2-1), “Diamond ring” (Fig 1.2-2) and “Wedding ring” (Fig 1.2-3). The
“Diamond rosary” and “Diamond ring” now astronomers name the “Baily’s beads” in honor of English
astronomer Francis Baily (1774–1844) who first published an explanation of these phenomena – the
bursts of light appear around the lunar silhouette when during total sun eclipse the sunlight shines through
rugged lunar limb in some places.
Fig. 1.2. “Diamond rosary” (1) and “Diamond ring” (2) during total sun eclipse on November 25,
2011 [http://www.spacetribe.com/]. “Wedding ring” (3) – the annular sun eclipde on May 20, 2012
[/wikipedia.org/].
A lunar eclipse occurs when the Moon passes behind the Earth into its shadow (umbra). Lunar
eclipses occur when the Sun, Earth and Moon are aligned in syzygy (from the Ancient Greek σύζυγος,
suzugos, meaning “yoked together”) – a straight-line configuration of all three celestial bodies (Fig. 1.3).
The Moon crosses ecliptic – the plane of the Earth’s orbit around the Sun at positions called “nodes”
twice every month and when the Full Moon occurs in the same position at the node, a lunar eclipse can
occur. These two nodes allow two to five eclipses per year, parted by approximately six months. An
eclipse of the Moon can only take place at Full Moon, and only if the Moon passes through some portion
of Earth’s shadow, which composed of two cone-shaped parts, one nested inside the other.
Fig. 1.3. The geometry of lunar eclipse. Right part: the lunar eclipse on October 27, 2004 (Eclipse
Predictions by Fred Espenak, NASA’s GSFC; http://eclipse.gsfc.nasa.gov/. For earthly observer the
Moon daily crosses the sky from east to west, however, with respect to the Earth’s shadow cone and the
stars it moves from west to east.
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When the Moon enters into the Earth’s penumbra a partial penumbral eclipse (Fig. 1.4-1) starts with a
subtle darkening of the Moon’s surface. When the Moon moves exclusively within the Earth’s penumbra
a total penumbral eclipse (Fig. 1.4-2) is visible. Total penumbral eclipses are rare, and when these occur,
that area of the Moon which is closest to the umbra can appear somewhat darker than the rest of the
Moon. If the entire Moon passes through the umbral shadow, then a total eclipse (Fig. 1.4-3) of the Moon
occurs. The type and length of a lunar eclipse depend upon the Moon’s location relative to its orbital
nodes.
Fig. 1.4. Lunar eclipses: partial (1), total penumbral eclipse (2) and total eclipse (3; the “Blood
Moon”).
The Moon’s speed through the Earth’s shadow is about one kilometer per second, and total eclipse
may last up to more than 100 minutes. However, the total time between the Moon’s first and last contact
with the Earth’s shadow is much longer, and could last up to 4 hours. In contrast to a solar eclipse, which
can only be viewed from a certain relatively small area of the world, a lunar eclipse may be viewed from
anywhere on the night side of the Earth. A lunar eclipse lasts for a few hours, whereas a total solar eclipse
lasts for only a few minutes at any given place, due to the smaller size of the Moon’s shadow. A totally
eclipsed Moon occurring near Moon’s apogee where its orbital speed is the slowest will lengthen the
duration of total eclipse.
The inner shadow (umbra) is a region where Earth blocks all direct sunlight from reaching the Moon.
The outer shadow (penumbra) is a zone where Earth blocks only part of the Sun’s rays. Though within the
umbra the Moon is totally shielded from direct illumination by the Sun, the Moon does not completely
disappears as it passes through the umbra because of the refraction of sunlight by the Earth’s atmosphere
into the shadow cone. The amount of scattered and refracted light depends on the amount of dust and
clouds in the Earth’s atmosphere that influences the color of eclipsed Moon. During an eclipse, the
Moon’s disk can take on a dramatically colorful appearance from dark gray or bright orange to copperyred or dark brown. To describe the color of eclipsed Moon the scale proposed in 1930s by French
astronomer André-Louis Danjon (1890–1967) is sometimes used: L=0: Very dark eclipse with Moon
almost invisible; L=1: Dark gray or brownish eclipse; L=2: Deep red or rust-colored eclipse; L=3: Brickred eclipse with the umbral shadow having a bright rim; L=4: Intense copper-red or orange lunar disc
having a bright rim and the bluish umbral shadow.
A “selenelion” or “horizontal eclipse” occurs when both the Sun and the eclipsed Moon can be
observed at the same time, which happens just before sunset or just after sunrise and both bodies will
appear just above the horizon at nearly opposite points in the sky. There are a number of high ridges
undergoing sunrise or sunset that can see it. Although the Moon is in the Earth’s umbra, the Sun and the
eclipsed Moon can both be seen at the same time because the refraction of light through the Earth’s
atmosphere causes each of them to appear higher in the sky than their actual geometric positions.
The red coloring of eclipsed Moon arises because sunlight reaching the Moon must pass through a
long and dense layer of the Earth’s atmosphere, where shorter wavelengths are more likely to be scattered
by the air and dust particles, and so by the time the light has passed through the atmosphere, the longer
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wavelengths dominate. Such scattered light we see as red hues during sunsets and sunrises and it often
paints the eclipsed Moon by a reddish color being a cause of the “Blood Moon” phenomenon.
Eclipse season is the only time during which the Sun is close enough to one of the Moon’s nodes to
allow for an eclipse to occur (Fig. 1.7). During the Eclipse season, whenever there is a Full Moon a lunar
eclipse will occur and whenever there is a New Moon a solar eclipse will occur. Each season lasts from
31 to 37 days, returning about every 6 months (173.31 days – half of an Eclipse year). At least two – one
solar and one lunar, in any order, and at most three eclipses – solar, lunar, then solar again, or vice versa,
will occur during every Eclipse season. Since it is about 15 days between Full Moon and New Moon, if
there is an eclipse at the very beginning of the Eclipse season, then there is enough time for two more
eclipses. If the last eclipse of an Eclipse season occurs at the very beginning of a calendar year, it is
possible for seven eclipses to occur since there is still time before the end of the calendar year for two full
Eclipse seasons, each having up to three eclipses.
Occasionally 4 total lunar eclipses occur in a sequence with intervals of 6 lunations (the average time
for one lunar phase cycle i.e., the synodic period of the Moon, or the period from one New Moon to the
next). Such remarkable lunar eclipse series is called the “Tetrad”. Italian astronomer and science historian
Giovanni Schiaparelli (1835–1910) noticed that there are eras when Tetrads occur comparatively
frequently, interrupted by eras when they are rare and later Dutch astronomer Antonie Pannekoek (1873–
1960) explained Tetrad phenomenon and found a period of 591 years. Recently Tudor Hughes explained
the Tetrad period variation by secular changes in the eccentricity of the Earth’s orbit and estimated the
current period as about 565 years. To describe the regular appearance of Tetrads Belgian astronomer Jean
Meeus proposed special astronomical period – the “Tetradia”, which consists of 6984 synodic months and
595 eclipse years (see details in [6]).
Fig. 1.5. Correlation of 2014-2015 Tetrad with the Tetrad, which accompanied the dramatic Gospel
events and Jesus Christos crucifixion [1] .
Since the usual red coloring of totally eclipsed Moon the Tetrad of 4 total lunar eclipses sometimes is
named the Blood Moon Tetrad. From time to time the Blood Moon Tetrads coincides with feasts of
ancient calendars, which are also based on the lunar phase cycles. One of such coincidence is the
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correlation of 2014-2015 Tetrad (on Gregorian calendar) with the Tetrad, which accompanied the
dramatic Gospel events and Jesus Christos crucifixion (Fig. 1.5).
The first Blood Moon of 2014 is total lunar eclipse on April 15, which was visible across most of
North America, Latin America and the Caribbean, coincides with the first day of Jewish Passover 2014.
The second Blood Moon on October 8 coincides with the 1st day of Sukkoth (Tabernacles) 2014, the
third on April 4, 2015 coincides with the 1st day of Passover 2015 and the fourth on September 28, 2015
coincides with the 1st day of Sukkoth 2015. In between all these events is a total eclipse of the Sun on
March 20, 2015 – the First day of the Jewish month of Nissan, the “leading month” in the Jewish
calendar, the month when Israelites were freed from slavery in ancient Egypt. Astronomers found that
there was seven Blood Moons on the first day of Jewish Passover and the first day of Sukkoth since
beginning of the Christian era and the 2014-2015 Tetrad is the eighth coincidence.
The second Astronomer Royal in Britain Edmond Halley (1656–1742) succeeding the first
Astronomer Royal John Flamsteed is one of the founders of modern eclipse astronomy. Halley spent most
of his time on lunar and comet observations, but was also interested in the problems of gravity,
considering the possible derivations of Kepler’s laws of planetary motion. In August 1684, during
discussion in Cambridge he persuaded Sir Isaac Newton to publish his calculations based on the inverse
square law of gravitation. The first of the three constituent books of Newton’s Philosophiæ Naturalis
Principia Mathematica was sent to Halley in spring 1686 and first 300 copies published by Halley at his
own financial risk appeared in July 1687.
Fig. 1.6. The Astronomer Royal in Britain Edmond Halley.
Edmond Halley firstly worked out the historical astronomy methods and started to use them in
calculations with comets, in particular, for the orbit of Kirch Comet. In 1705, applying his historical
astronomy methods, Halley published Synopsis Astronomia Cometicae, which stated his belief that the
comet observations of 1456, 1531, 1607, and 1682 related to the same comet, which he predicted would
return in 1758. Halley did not live to witness the comet’s return, but when it did, the comet became
generally known as the “Halley’s Comet”.
By 1706 Halley had learned Arabic and completed the translation started by Edward Bernard of
Books V-VII of Apollonius’s Conics from copies found at Leiden and the Bodleian Library at Oxford. He
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also completed a new translation of the first four books from the original Greek that had been started by
David Gregory. He published these books along with his own reconstruction of Book VIII in the first
complete Latin edition in 1710. In 1718, he discovered the proper motion of the “fixed” stars by
comparing his astrometric measurements with those given in Ptolemy’s Almagest.
The base of the sun eclipse phenomenology is the “Saros cycle”, which has been used from the epoch
of ancient Babylonian astronomy to predict eclipses of the Sun and the Moon. This cycle was named the
“Saros” (Greek: σάρος) in 1691 by Edmond Halley, who took this word from the “Suda”, a Byzantine
lexicon of the 11th century. Though the word “saros” comes from Sumerian number sar (3600) and does
not mean “cycle” or “period”, as thought by Halley, it is nonetheless used in modern astronomy.
Lunar and sun eclipses could take place when the Earth and the Moon are aligned with the Sun, and
the shadow of one body cast by the Sun falls on the other. At Full Moon, when the Moon is in opposition
to the Sun, the Moon may pass through the shadow of the Earth, and a lunar eclipse is visible from the
night half of the Earth. At New Moon, when the Moon is in conjunction with the Sun, the Moon may pass
in front of the Sun as seen from a narrow region on the surface of the Earth and cause a solar eclipse. A
sun eclipse does not happen at every New Moon, because the plane of the Moon’s orbit around the Earth
is tilted with respect to ecliptic. A lunar eclipse can only occur when the Moon is close to the plane of
ecliptic near one of the two its nodes and the Sun is near the opposite lunar node (Fig. 1.7).
Fig. 1.7. Eclipses in the Sun-Earth-Moon planetary system
The inclination of lunar orbit (about 5.15°) is much larger than the average apparent diameter of the
Sun (0.53°), much larger than the Moon diameter, as seen from the surface of the Earth (0.525°), and
much larger than the shadow of the Earth at the average lunar distance (1.4°). Therefore, at most New
Moons the Earth passes too far north or south of the lunar shadow, and at most solar eclipses the apparent
angular diameter of the Moon is insufficient to fully obscure the solar disc, unless the Moon is close to
perigee.
If a solar eclipse occurs at some New Moon, which must be close to a node, then at the next Full
Moon the Moon is already more than a day past its opposite node, and may or may not miss the Earth’s
shadow. By the next New Moon it is even further ahead of the node, so it is less probable that there will
be a solar eclipse. However, about 5 or 6 lunations later (half of an Eclipse year) the New Moon will be
close to the opposite node and the Sun will also move to the opposite node and therefore the geometry of
Sun-Earth-Moon system will again be appropriate for one or more eclipses.
The time it takes for the Moon to return to a node – the draconic month (27.21 days) is less than the
time it takes for the Moon to return to the same ecliptic longitude as the Sun – the synodic month (29.53
days) because during the time that the Moon has completed an orbit around the Earth, the Earth-Moon
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system have completed about 1 13 of its orbit around the Sun and the Moon has to make up for this in
order to come again into conjunction or opposition with the Sun.
The orbital nodes of the Moon precess westward in ecliptic longitude, completing a full circle in about
18.5 years, so a draconic month is shorter than a sidereal month (27.32 days) – the time it takes the Moon
to return to a given position among the stars. As seen from the Earth, the Sun passes both nodes as it
moves along its ecliptic path and the period for the Sun to return to a node is called the “Eclipse year”
(346.62 days), which is shorter than a sidereal year (365.24 days) – the time taken by the Earth to orbit
the Sun with respect to the fixed stars, because of the precession of the nodes.
Any two sun eclipses separated by one Saros consisting of 223 synodic months (18 years 11 days and
8 hours) have very similar characteristics – they occur at the same node with the Moon at nearly the same
distance from Earth and at the same time of the year. However, even though the geometry of the SunEarth-Moon system will be almost identical after a Saros, the Moon will be in a different position with
respect to the fixed stars due to the Moon’s orbit precession. Since the Saros cycle is not equal to a whole
number of days, the extra 8 hours shift each succeeding eclipse path on the Earth’s surface by nearly 120
degrees westward. Saros eclipse series returns to the same geographic region every three Saros periods
(the “Exeligmos” – 54 years and 34 days).
The interrelations between lunar and sun eclipses are determined by the “Sar” cycle – a half of the
Saros cycle. After a certain lunar or solar eclipse, after 9 years and 5.5 days (a Sar cycle) an eclipse will
occur that is lunar instead of solar, or vice versa, with similar properties. For example, 9 years and 5.5
days after a total solar eclipse a total lunar eclipse will occur.
Table 1.1. Main characteristic periods (approximate), which are used in eclipse astronomy
synodic month
draconic month
sidereal month
anomalistic month
Eclipse year
lunar year
tropical year
Sar
Saros
Metonic cycle
Inex
Exeligmos
Callippic cycle
Hipparchic cycle
Great Babylonian cycle
Tetradia
solar days
29.53
27.21
27.32
27.55
346.62
354.37
365.24
3292.66
6585.32
6939.69
10571.95
19755.96
27758.75
126007
161177.95
206241.63
synodic month
1
0.92
0.925
0.93
11.74
12
12.37
111.5
223
235
358
669
940
4267
5458
6984
Eclipse years
tropical years
1
1.02
1.05
9.5
19
20
30.50
57
80
363.53
465
595
0.95
0.97
1
9.015
18.03
19
28.945
54.09
76
345
441.29
564.67
In eclipse astronomy and calendar study, besides Sar, Saros and Exeligmos several other periods with
historical names are used. These periods contain the integer numbers of synodic months. The “Metonic
cycle” or “Enneadecaeteris” (from Ancient Greek “nineteen years”) was introduced by Greek astronomer
Meton of Athens (5th century BCE) as a period of very close to 19 years, which is remarkable for being
nearly an integer number of the solar years, Eclipse years and the synodic months. The “Callippic cycle”
proposed by Greek astronomer Callippus (c. 370–c. 300 BCE) is an approximate of 76 years and the
integer number of synodic months. The Greek astronomer Hipparchus of Nicaea (c. 190–c. 120 BCE)
introduced the eclipse cycle that have been named after him in later literature, which closely matches an
integer number of synodic months (4267), anomalistic months (4573), years (345), and days (126007). By
comparing his own eclipse observations with Babylonian records from 345 years earlier, Hipparchus also
verified the accuracy of various eclipse periods that the Chaldean astronomers used.
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The Moon’s orbit is not a perfect circle but approximates an ellipse and the lunar distance from Earth
varies throughout the lunar cycle. The varying distance from the Moon to Earth changes the apparent
diameter of the Moon, and therefore influences the probability, duration, and type of an eclipse. The
orientation of Moon’s orbit is not fixed and, in particular, the position of the “extreme points line” – the
line of the apsides: perigee and apogee, makes a full circle (lunar precession) in about 3233 days (8.85
years). It takes the Moon the anomalistic month (27.55 days) to return to the same apsis. The Moon
moves faster when it is closer to the Earth (near perigee) and slower when it is near apogee (furthest
distance), thus periodically changing the timing of syzygies by up to ±14 hours (relative to their mean
timing), and changing the apparent lunar angular diameter by about ±6%. An eclipse cycle must comprise
close to an integer number of anomalistic months in order to perform well in predicting eclipses.
During a central sun eclipse, the Moon’s umbra (or antumbra, in the case of an annular eclipse) moves
in a roughly west-east direction across the Earth’s surface at the speed of the Moon’s orbital velocity
minus the Earth’s rotational velocity. The width of the path of a central eclipse varies depending on the
relative apparent diameters of the Sun and Moon. When a total eclipse occurs very close to perigee, the
path can be over 250 km wide and the duration of eclipse may be over 7 minutes.
Though the length of eclipse path is several thousand kilometers, and an eclipse is visible on the Earth
during several hours, there are certain coordinates and time of the “Greatest Eclipse”, which define the
site and the instant when the axis of the Moon’s shadow cone passes closest to Earth’s center. Although
the Greatest Eclipse differs slightly from the instant of the greatest magnitude and the Greatest Duration
for sun eclipses, these differences are small. For lunar eclipses, Greatest Eclipse is defined as the instant
when the Moon passes closest to the axis of Earth’s shadow. The coordinates and time of the Greatest
Eclipse are individual characteristics of each eclipse and they are used to identify eclipses in databases.
Fig. 1.8. The sun eclipse track and Greatest Eclipse on March 29, 2006 (Fred Espenak and Jay
Anderson, NASA/TP-2004-212762 “Total Solar Eclipse of 2006 March 29”).
For the Greatest Eclipse the main eclipse characteristics – eclipse Magnitude and Gamma can be
evaluated. Eclipse magnitude is the fraction of the Sun’s diameter occulted by the Moon. It is a ratio of
diameters and should not be confused with eclipse obscuration, which is a measure of the Sun’s surface
area occulted by the Moon. Eclipse Magnitude may be expressed as either a percentage or a decimal
fraction (e.g., 80% or 0.80). Gamma is the distance of the Moon’s shadow axis from Earth’s center in
units of equatorial Earth radii and at the instant of Greatest Eclipse its absolute value is at a minimum. For
lunar eclipses, the Gamma is the Moon’s minimum distance measured with respect to the axis of Earth’s
shadow in units of equatorial Earth radii.
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As an example we can consider the sun eclipse on March 29, 2006, belonging to Saros 139 (Fig. 1.8).
The narrow band of total sun eclipse began in the Eastern Brazil (“first contact”) then crossed the Atlantic
to Africa, where the Moon’s shadow travelled over the northern part of the continent. It went on to cross
the Mediterranean Sea to Turkey, after which it passed through part of central Asia to end in Mongolia
(“fourth contact”). Partial phases of the solar eclipse were observed by the whole of Europe. Only the
south-eastern parts of Africa missed the partial eclipse. At the extreme parts of the eclipse path of totality
the duration of “totality” – the maximum phase of a total eclipse during which the Moon’s disk
completely covers the Sun, was less than 2 minutes, but it was as long as 4 minutes and 7 seconds in
Libya at 9:18 UT (Universal Time), at the moment of Greatest Eclipse. At this moment, the path of
totality of the solar eclipse was about 180 kilometers wide.
The known history of solar eclipse maps begins in the 17th century, when a path of total solar eclipse
crossed northern Europe on August 12, 1654 and German astronomer Erhard Weigel created the earliest
known solar eclipse map on the day before the eclipse. In 1676, Johann Christoph Sturm produced an
almanac of eclipses, which included a map of two eclipses, one total and one annular, for that year. In
1700 the leading astronomer of the Paris Observatory Giovanni-Dominique Cassini created an eclipse
map of the total solar eclipse of 1699.
In 1715, Edmond Halley using the Isaac Newton’s theory of gravitation created several eclipse maps
of very high accuracy. Later, Halley analyzed observational reports from the 1715 eclipse and produced
another map with a refinement of the path of the 1715 eclipse along with the forthcoming 1724 eclipse.
Fig. 1.9. Edmund Halley’s sun eclipse map (1715). Right: the modern eclipse map of predicted sun
eclipse (Saros 145) on August 21, 2017 with up to 2.7 minutes of totality. Light blue lines parallel to the
dark blue track of totality indicate the degree of partial eclipse to be seen (http://eclipse.gsfc.nasa.gov/
“Eclipse Predictions by Fred Espenak (NASA's GSFC)”).
By the 20th century, the publication of eclipse maps became routine in most of the major almanacs of
the world and continues to this day. In the United States, the U.S. Naval Observatory began publishing
eclipse maps in USNO Eclipse Circulars in 1949.
In 1970s, started the international program of Lunar Laser Ranging Experiment and “Lunokhod-1”
(1970) was the first of two unmanned lunar rovers landed on the Moon by the Soviet Union as part of its
Lunokhod program. Lunochods carried small light reflecting arrays and reflected signals were initially
received from Lunokhod-1, but no return signals were detected from 1971 until 2010, due to significant
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uncertainty in Lunochod locations on the Moon. In 2010, Lunochods were found in Lunar
Reconnaissance Orbiter photographs and the retro-reflectors have been used again.
Astronauts on the Apollo 11, 14, and 15 missions left more perfect retro-reflectors on the Moon as
part of the Lunar Laser Ranging Experiment. The Lunar Ranging Retro-Reflector (LRRR) of Apollo 11
consisted of an array of 100 fused silica cubes, arranged to reflect a beam of light back on a parallel path
to its origin. The Apollo 15 reflector had 300 silica cubes. The LRRRs placed on the Moon are aligned
precisely so that they face the Earth and scientists from around the world can direct laser beams at the
instruments, which reflect beams back to Earth. Such laser locations of the Moon allow precise
measurements of distances between the Earth and Moon and the procession of measurement sets gives the
accurate estimations of main characteristics of the Moon’s motion in orbit.
Fig. 1.10. The ranging retro-reflector on the Moon’s surface.
Starting in the 1970s, the laser locations of the Moon and modern computer technologies were applied
to very precise calculations and new maps of solar eclipses. In the 1980’s, the task of publishing detailed
eclipse reports and maps in USA was transferred to NASA under the leadership of Fred Espenak of the
NASA/Goddard Space Flight Center and Jay Anderson of the Royal Astronomical Society of Canada.
Modern theories of the Earth and Moon orbital motions with the value of the Moon’s secular
acceleration ( 25.858 arc-sec/cy2) deduced from the Lunar Laser Ranging Experiment allow calculating
the eclipse characteristics for the periods of several millennia. In this book we use the “Five Millenniums
Canon of Solar Eclipses: –1999 to +3000 (2000 BCE to 3000 CE)” [4], which we henceforth will refer to
as “Canon”. An example of information about main characteristics of solar and lunar eclipses from Canon
presented in Tables 1.2 and 1.3.
Table 1.2. Main characteristics of 2013-2015 solar eclipses [4] and (http://eclipse.gsfc.nasa.gov/
“Eclipse Predictions by Fred Espenak (NASA's GSFC)”.
Date
5/10/ 2013
11/3/2013
4/29/2014
10/23/2014
3/20/2015
9/13/2015
Time of
Greatest
Eclipse
(UTC)
00:26:20
12:47:36
06:04:33
21:45:39
09:46:47
06:55:19
Saros
Type
Gamma
Magnitude
Central
duration
Location of
Greatest
Eclipse
Path
width
(km)
138
143
148
153
120
125
Annular
Hybrid
Annular
Partial
Total
Partial
-0.2694
0.3272
-1.0000
1.0908
0.9454
-1.1004
0.954
1.016
0.987
0.811
1.045
0.788
6m 3s
1m 40s
—
—
2m 47s
—
2.2N;175.5E
3.5N;11.7W
70.6S;131.3E
71.2N;97.2W
64.4N;6.6W
72.1S;2.3W
173
58
—
—
463
—
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Table 1.3. Main characteristics of 2014-2015 lunar eclipse Tetrad (http://eclipse.gsfc.nasa.gov/
“Eclipse Predictions by Fred Espenak (NASA's GSFC)” [5].
Date
4/15/2014
10/8/2014
4/4/2015
9/28/2015
Time of Greatest Eclipse
(UTC)
07:46:48
10:55:44
12:01:24
02:48:17
Saros
Type
122
127
132
137
Total
Total
Total
Total
Umbral
Magnitude
1.291
1.166
1.001
1.276
Partial/Total Duration
03h35m/01h18m
03h20m/00h59m
03h29m/00h05m
03h20m/01h12m
Dutch Astronomer George van den Bergh (1890–1961) in his book Periodicity and Variation of Solar
(and Lunar) Eclipses introduced the numbering system that is often used for the Saros series. All 8000
solar eclipses from Theodor von Oppolzer’s Canon der Finsternisse (1887) are placed into a twodimensional matrix, where each Saros series is arranged as a separate column with the eclipses in
chronological order. The Saros series columns are staggered so that the interval between any two eclipses
in adjacent columns is the “Inex” cycle – 358 synodic months (29 years minus 20 days). Each eclipse has
two numbers determining its position in Saros-Inex Panorama (Fig. 1.11-1) and Inex cycle marks the time
interval between consecutively numbered Saros series.
Within each column of Saros-Inex Panorama is a complete Saros series, which progresses from partial
eclipses into total eclipses and back into partials. Each Inex series extends as a graph row, but it gradually
bends due to long term period variations as can be seen at Panorama for 61775 solar eclipses, which has
been produced by astronomers Luca Quaglia and John Tilley (Fig. 1.11-2).
Fig. 1.11. 1. The fragment of Saros-Inex Panorama. 2. Saros-Inex Panorama for 61775 solar eclipses.
3. Several central solar eclipses from Saros 145 that began with a partial eclipse near the North Pole in
1639 and will terminate with the last eclipse near the South Pole in 3009 (http://eclipse.gsfc.nasa.gov/
“Eclipse Predictions by Fred Espenak (NASA's GSFC)”. The black circles are locations of Greatest
Eclipses.
The Saros numbering sequence does not depend on when a series begins or ends and the numbering
tends to follow the order in which each series “peaks” – the peak of a series occurs when the umbral
shadow axis passes closest to the center of Earth. Since the duration of each series varies up to several
11
hundred years, the first eclipse of a series, which peaks later, can actually precede the first eclipse of a
series that peaks earlier. There are approximately forty different Saros series in progress at any one time –
as old series terminate, new ones are beginning and take their places. The Saros numbering of lunar
eclipses is opposite to that for solar eclipses. Lunar eclipses occurring near the Moon’s ascending node
have even Saros numbers and correspondingly, lunar eclipses occurring near the Moon’s descending node
have odd Saros numbers.
1.1 Metaphysics of solar eclipses
A very rare phenomenon in the planetary mechanics – a total eclipse of the central star in satelliteplanet structure in the Sun-Earth-Moon system is determined by its special geometry, which provides a
close match of apparent angular sizes  of the Moon and the Sun. Modern astronomers explain the
possibility of a total solar eclipse by the fact that the ratio of the Sun diameter to the Moon diameter:
A1  dS d M  400.5 is almost the same as the ratio of the average half-axis of the Earth’s orbit to the
average half-axis of the lunar orbit: A2  aE aM  389.3 (Fig. 1.12). The difference between these ratios
is only 2.9% and hence:

tg
dM aM
dS aE
0.53 (1.1)
However, in astrology and metaphysics of the East a possibility of total solar eclipses is explained
otherwise – by the declaration that the solar eclipse as well as the whole geometry of the Universe
determines the unique sacred number 108. Indeed, the ratio of average half-axis of the lunar orbit to the
diameter of the Moon: M1  aM d M  110.6 is very close to the ratio of the Sun diameter to the Earth
diameter: M 2  dS d E  109.1 . The difference between these ratios even less than the difference of A1
and A2 – only 1.4%, and the ratios M 1 , M 2 within the accuracy of the ancient astronomical observations
could well be estimated by the sacred number 108:
aM dM
dS dE
108 (1.2)
Figure 1.12 visually illustrates that the close values of ratios: A1 A2 provide an approximate
similarity of the trapeziums that makes possible the total solar eclipses. However, it is difficult to explain
what relation to the geometry of total solar eclipses could have the Earth’s diameter d E , which is included
in the ratio M2 and then into relation (1.2).
Fig. 1.12. The geometry of solar eclipse.
In Eastern world, the number 108 is considered sacred by several religions, including Hinduism,
Jainism and Buddhism. In Sanskrit alphabet, which is used in sacred Vedic texts there are 54 letters and
each letter has masculine and feminine (Shiva and Shakti) symbolic significances. Therefore, the letters
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of Sanskrit alphabet have totally 54 × 2 = 108 symbolic meanings. The Upanishads – philosophical texts
completing the Vedic corpus are classically counted as 108 and are divided into four categories according
to the particular Veda to which each of them belong.
Some Hinduism traditions assert that there are 108 deities each having 108 names and there are 108
paths to God. The well known bas-relief carving at the famous Angkor Wat temple in Cambodia relates
the Hindu story of a serpent being pulled back and forth by 108 gods and asuras (demons) – 54 gods
pulling one way, and 54 asuras pulling the other, to churn the Ocean of milk in order to produce Amrita –
the elixir of immortality. In front of Angkor Wat temple gate stand 108 statues of 54 gods and 54 demons,
which represent the sacred churning of the Ocean.
Many Hinduism traditions use the Sri Yantra formed by nine interlocking triangles that surround and
radiate out from the central point (bindu), the junction point between the physical universe and its
unmanifest source (Fig. 1.13). Four triangles with the apices upwards represent Shiva and the Masculine
principle. Five triangles with the apices downward, symbolize female embodiment Shakti and therefore
Sri Yantra represents the union of Masculine and Feminine Divine. The nine triangles of various sizes
intersect with one another and on the Sri Yantra there are 54 marmas where three lines intersect. Each
intersection has masculine and feminine (Shiva and Shakti) qualities and therefore there are totally 54 × 2
= 108 special points of the symbolic Universe. According to Ayurveda, these 108 points correspond to
pressure points in the body, where consciousness and flesh intersect to give life to the living being. The
Sri Yantra also represents the Heart Chakra, where 108 body energy lines intersect.
Fig. 1.13. The Sri Yantra with 54 marmas.
The Lankavatara Sutra has a section where the Bodhisattva Mahamati asks Buddha 108 questions and
another section where Buddha lists 108 statements of negation. The Sanskrit word translated as
“statement” is pada, which also means “foot-step” and therefore many Buddhist temples have footways
with 108 steps. In Tibet, China and Japan, at the end of the year, a bell is chimed 108 times in many
Buddhist temples to finish the old year and welcome the new one with each ring representing one of 108
earthly temptations a Buddhist must overcome to achieve nirvana.
The Hinduist and Buddist ritual rosaries (mala) have 108 beads and the Meru – a larger bead that is
not tied in the sequence of the other beads. It is the guiding bead, the one that marks the beginning and
end of the mala. The Chinese Buddhists and Taoists use a 108 bead rosary (su-chu), which has three
dividing beads, so the rosary is divided into three parts of 36 each. The Tibetan Buddhist malas include
the Guru-bead and 108 beads symbolizing the most important words of the Buddha writen in the sacred
texts of Kangyur consisting of 108 volumes. Traditionally, yoga students stop at the 109th Meru-bead,
flip the mala around in their hand, and continue reciting their mantra as they move backward through the
beads. The Meru-bead represents the summer and winter solstices, when the Sun appears to stop in its
course and reverse directions. In the yoga tradition using a 108-beads mala is a symbolic way of
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connecting ourselves with the cosmic cycles governing our Universe. If the yogin is able to be so calm in
meditation as to have only 108 breaths in a day, the Enlightenment will come.
In modern number theory, the number 108 is an abundant semi-perfect number, which divides the
twin primes: 107,108,109 and has 12 positive divisors {1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108}. It is a
refactorable number (tau-number) since it is divisible by the total number of its devisers: 108 : 12 = 9.
The digital representation of number 108 depends on the used counting base:
108(10)  423(5)  154(8)  90(12)  30(36) . In decimal counting system the number 108 is a Harshad number
(Niven number), which is an integer divisible by the sum of its digits: 108/(1+0+8=9)=12, and has a rare
feature to keep constant the sum of digits in three successive divisions by 2:
108 (1+0+8 = 9) : 2 = 54 (5+4 = 9) : 2 = 27 (2+7=9) : 2 = 13.5 (1+3+5 = 9)
(1.3)
However, the number theory cannot explain the mysterious role of quantity 108 in the geometry of
Sun-Earth-Moon system governing the solar eclipses and to clear up the old puzzle we shall consider the
ancient metaphysics of the East.
The Shulba Sutras (from Sanskrit sulba – “cord, rope”) that are considered to be appendices to the
Vedas contain geometry related to fire-altar construction and are the most important sources of
knowledge of Indian mathematics from the Vedic period (I mill. BCE). The different fire-altar shapes
were associated with different gifts from the gods: “He who desires heaven is to construct a fire-altar in
the form of a falcon… those who wish to destroy existing and future enemies should construct a fire-altar
in the form of a rhombus”. The need to manipulate different fire-altar shapes led to the creation of the
pertinent Vedic mathematics. For example, the “Baudhayana Shulba Sutra” (c. 800 BCE) describes the
construction of geometric shapes such as squares and rectangles and the geometric area-preserving
transformations: a square into a rectangle, an isosceles trapezium, an isosceles triangle, a rhombus, and a
circle, and transforming a circle into a square.
The altar construction also led to an estimation of the square root of 2, using of Pythagorean triples
and knowledge of irrationality and irrational numbers. The religious Indian texts of the Vedic period
provide evidence for the use of large numbers from 102 (sata) to 1012 (paradha). The mystical properties
of numbers were considered spiritually powerful that consequently led to their incorporation into religious
texts. Treatises of Jain1 mathematicians of the second half of the first millennia BCE are the links
between the mathematics of the Vedic period and that of the Classical period. Jain mathematicians
investigated the first powers of numbers like squares and cubes, which enabled them to define simple
algebraic equations, and started to use the concept of zero. Probably, Jain mathematicians firstly found
the expressive “magic” representations of ancient sacred numbers 9, 12, 108 as the hyperfactorials of
metaphysical Ultimate Symbols 1, 2, 3:
9  32
1
12  13  22  31 108  11  22  33 (1.4)
The so called Classical period (400–1600) is the golden age of Indian mathematics when mathematics
was included in the “astral science” (Jyotiḥsastra), which consisted of three sub-disciplines: mathematical
sciences (ganita), horoscope astrology (hora) and divination (samhita). During this period the
achievements of Indian mathematicians had spread to the all East Asia, the Middle East, and eventually to
Europe.
The astronomical symbolism of the sacred number 108 started to spread in the East after the works of
prominent Indian astronomer and mathematician Aryabhata (476–550) and his followers. The
mathematical part of Aryabhata’s main work named by later commentators the Aryabhatiya covers
arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions,
1
Jainism is a religion and philosophy that predates its most famous exponent Mahavira (6th century BCE) who was a
contemporary or predecessor of Gautama Buddha.
14
quadratic equations, sums of power series, and a table of sines with the word kha (“emptiness”) to mark
“zero” in tabular arrangements of digits. In the first chapter of his book Aryabhata described a system of
numerals based on Sanskrit phonemes, which attributes a numerical value to each syllable possible in
Sanskrit phonology – from ka = 1 up to hau = 1018. Probably, he also firstly used the astrological
interpretation of the sacred number 108 – the total quantity of planet positions in all Signs of zodiac:
108  9 12 (1.5)
The followers of Aryabhata often referred to his book as Arya-shatas-aShTa – literally, “Aryabhata’s
108”, because there are 108 verses in the text written in the style typical of sutra literature, in which each
line is an aid to memory for a complex system.
The numerological methods of Indian astrology, which are abundant contrasting the European
tradition, are based on the interpretation of the symbolic formula (1.5) and its graphical representation
(Fig. 1.14).
Fig. 1.14. The magic square of 9 planets at the center of 12 zodiacal Signs.
Figure 1.14 shows the traditional zodiacal and lunar House coordinate systems of Indian astrology
with nine planets placed in the central “magic square”. The positions of planets in square grid correspond
to the Hindu customs with the Navagraha typically placed in a single square with the Sun (Surya) in the
center and the other planet deities surrounding Surya. In central magic square, red numbers corresponding
to planets, form an arrangement, where the numbers in each row, and in each column, and the numbers in
the forward and backward main diagonals, all add up to the same number – the “magic constant” equal to
15. The 3×3 magic square has been a part of sacred rituals in India since Vedic times and still is today, for
example, the well-known “Ganesh yantra” and the 10th-century 4×4 magic square in the Parshvanath Jain
temple in Khajuraho.
The authority of number 108 became highest when it was discovered that this number is tightly
connected with the Golden ratio and defines the geometry of the simplest polygon, which can exist as a
regular star – the pentagram of Pythagorean metaphysics. The interior angle of a convex regular n-gon is
 n  180  360 n that gives for pentagon ( n  5 ) the sacred number  n  108 degrees.
The number 108 is not a member of classical Fibonacci series of numbers that starts with two
predetermined terms and each term afterwards is the sum of the preceding two terms:
[0, 1] 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377…
15
However, the number 108 is a member of so-called “tetranachi” series of numbers that starts with four
predetermined terms, each term afterwards being the sum of the preceding four terms:
[0, 0, 0, 1] 1, 2, 4, 8, 15, 29, 56, 108, 208, 401…
The “tetranacci constant” – the ratio toward which adjacent tetranacci numbers tend is 1.92756…that
differs from the Fibonacci constant (Golden ratio)   1.61803… Nevertheless, the tight connection of
the number 108 with Golden ratio demonstrates the analysis of pentagon geometry.
As firstly demonstrated Euclid in his Elements (c. 300 BCE) a regular pentagon is constructible using
a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge.
In a regular pentagon, all sides are equal in length and each interior angle is 108°. A regular pentagon has
central angles of 72° that is a half of Fibonacci number 144. The diagonals of a regular pentagon d are in
Golden ratio to its sides s: d s    1,61803….(Fig. 1.24-2)
Fig. 1.15. 1. The Golden Triangle (in red) in the proportions of Leonardo da Vinci. 2. Geometric
parameters of a pentagon.
Angle determining a length of pentagon side is   2 5  72 , and angle defining pentagon diagonal
is: 2  4 5  144 . Side length of a pentagon is: s  2sin( 2)  2sin( 5) and the diagonal length is:
d  2sin( )  2sin( 5) . The ratio of the diagonal to the side for a pentagon is: d s  sin  sin( 2) .
This formula can be transformed by using the formula for a double angle: sin   2sin( 2) cos( 2) .
The transformation yields: d s  2cos( 2) and after substitution of   2 5 , we get for a pentagon:
d s  2cos( 5)  1,61803...   .
Formula   2cos( 5) and its inverse:   5arccos( 2) determine the exact relation for two
irrational numbers  and  . Using the formula: 2cos   ei  ei , we can find the exact relation for
three irrational numbers  ,  and e:
  2cos( 5)  ei 5  ei 5 (1.6)
The metaphysics traditionally considered numbers  and  not as just numbers but as symbols of the
rational order of the Universe, an imminent natural law, a life giving force behind all things, the universal
structure governing and permeating the world. From old times mathematicians were looking for the
laconic fractions that represent these irrational numbers with the maximal exactness, for example:
16
  22 7  3.14285714 (Archimedes, 3 century BCE);   377 120  3.14116667 (Aryabhata, 5 century);
  355 113  3.14159292 (Zu Chong Zhi, 5 century). Following this tradition we can transform the
relations (1.6) into the laconic approximate formula for  ,  and e, which gives a correct value for its
first five decimal digits:
(e)   7 5  1.40001384… (1.7)
The answer to the question formulated at the beginning of this chapter – “What is the connection of
the sacred number 108 and the diameter of the Earth with a solar eclipse?” we can find in the ancient
Indian Shulba Sutras, dealing with the sacred geometry of Hindu temples. In these treatises one can read:
“Take a pole, mark its height, and then remove it to a place 108 times its height. The pole will look
exactly of the same size as the Moon and the Sun”.
Fig. 1.16. Number 108 in the geometry of Sun-Earth-Moon system.
And indeed the angular size of a pole at a distance of 108 of its height is:
 arctg (1/108) 9.26 103 rad 0.53 , that coincides with the mean angular size of the Sun and
Moon in the sky (Fig. 1.25). This means that in the geometry of the Sun-Earth-Moon planetary system the
number 108 approximately determines not even two, but three relations of characteristics (see Table.
1.4). Analysis of relations (1.1) and (1.2) shows that they are both consequences of the formula (1.8).
Table 1.4. Unusual concidences in the geometry of the Sun-Earth-Moon system (millions km)
Sun (S)
a (orbit half-axis)
d (diameter)
1.392
aE d S

dS aE
dM aM
Earth (E)
149.6
0.0127
Moon (M)
0.384
0.0035
107.5
aM d M
dE dS
1/108
dS dE
109.6
109.7
3
9.26 10 rad
0.53 (1.8)
Of course, the estimations of visible angular sizes of the Sun and Moon by (1.8) are only approximate,
because the angular dimensions of these celestial bodies are changing due to the ellipticity of the Earth
and Moon orbits: S  0.533  0.009 and M  0.525  0.035 . Moreover, the ratio aM d M 109.7 is
valid only for a relatively short historical period. While the distance between the Sun and its planets is
fairly stable, the distance between the Earth and the Moon is subject to steady and ultimately very sizable
changes. The distance from the Earth to the Moon due to energy losses in tidal friction increases by about
3.8 cm per year. This means that billions of years ago the Moon was much closer to the Earth and its
apparent angular size was larger.
The modern favored scientific hypothesis for the formation of the Moon is the “giant impact
hypothesis” (Big Splash), which suggests that the Moon was formed out of the debris left over from a
collision between young Earth and some other heavenly body. One scenario is the collision between the
Earth and a planet the size of Mars (George Darwin, 1898; Daly, 1946; Hartmann & Davis, 1975).
17
Another scenario proposes that the Moon and the Earth were created together in a giant collision of two
planetary bodies that were each five times the size of Mars (NASA Lunar Science Institute, Robin M.
Canup, 2012). The Big Splash hypothesis explains the Earth’s spin and Moon’s orbit having similar
orientations; the Moon samples indicating that the surface of the Moon was once molten and the Moon’s
relatively small iron core; lower Moon density compared to the Earth and the identical stable isotope
ratios of lunar and terrestrial rocks.
The Big Splash could initially form a triple system – the Earth with a twin pair of moons, surrounded
by a common silicate vapor atmosphere, and the Earth-Moons system could became homogenized by
convective stirring. The second smaller moon could be formed in a Lagrange point of the large moon and
after tens of millions of years, as the two moons migrated outward from the Earth, solar tidal effects
would have made the Lagrange orbit unstable, resulting in a slow-velocity collision of both moons that
explains the thickened crust of the Moon’s Far side. The resulting lunar mass irregularities could
subsequently produce a gravity gradient that resulted in “tidal locking” of the Moon so that today, only
the Moon’s Near side remains visible from Earth.
Several billion years ago the rotation rates of the Earth and Moon were larger than today and the
Moon was closer to Earth. The mathematical models of the Earth’s rotation estimate the LOD (Length of
a Day) as about 6-8 hours at a time 4 billions years ago. The gravitational interaction between Sun, Earth
and Moon accompanied by the tidal energy dissipation steadily decreases rotation of the Earth and
increases the semi-axis of the Moon’s orbit. In our time the moon and sun’s gravity add about 1.7
milliseconds to the length of a day each century. It appeared that the rate of earth’s rotation in the distant
past can be estimated by paleontological studies2.
Periodic growth structures – lines, bands, and rings are preserved in the skeletons or hard parts of
some organisms – stromatolites, fossil bivalves and corals. These growth indicators show the number of
days in a year at the time when these organisms lived. Measurements of daily layers and yearly patterns in
fossil corals from 180 to 400 million years ago show year lengths from 381 to 410 days (Wells, 1963;
Scrutton, 1970; Eicher, 1976). The oldest radiologically dated material with well-defined fossil corals
revealed that 600 million years ago the day would have been 3.3 hours shorter, corresponding to 424 days
in a year.
Fig. 1.17. 1. Proterozoic stromatolites from Bolivia. 2. Nonlinear dependence of LOD (hours) on time.
(“Ma” for megaannus – a unit of time equal to one million years; blue circles for fossil coral estimations,
red circles for stromatolites estimations).
Pannella, G.,. Paleontological evidence on the Earth’s rotational history since the early Precambrian // Astrophysics and
Space Science (1972) 16; 212-237. Rosenberg, G. D. and S. K. Runcorn (eds.),. Growth Rhythms and the History of the Earth's
Rotation. New York, 1975. Schopf, J.W. (ed.),. Earth's Earliest Biosphere. Its Origin and Evolution. Princeton, New Jersey:
Princeton University Press, 1983.
2
18
Stromatolites, which form yearly patterns much like rings in trees, are other indicators of the Length
of Day. These layered bio-chemical structures were formed in shallow water by microorganisms,
especially by blue-green cyanobacteria that use water, carbon dioxide and sunlight to create their food.
Due to intense ultra violet radiation during the day, the stromatalitic bacteria restrict their activity to night
time. Fine grain sediments are bound together by excretions adhered atop one another by the bacteria’s
flagella, forming layers that correspond to periods of high activity. The investigation of fossil
stromatolites from China showed that one billion years ago, a year was composed of about 516 days and
each day would have been on the order of 17 hours long (Pannella et al., 1968; Mohr, 1975; Williams,
1997). Figure 1.17 demonstrates the paleontological estimations of the LOD for past epochs.
At the current rate of retreating of the Moon from the Earth about 3.8 cm per year the increase of
semi-axis of the lunar orbit is approximately: aM 0.384  3.8 105  T (million km), and the increase of
apparent angular size of the Moon is determined by the approximate formula:

dM aM
0.0035 (0.384  3.8 105  T ) (1.9)
In this formula, T (Ma – million years) is negative for the past epochs of Earth’s history.
Calculations by formula (1.9) show that, for example, two billion years ago, the apparent size of the
Moon was noticeably larger than the Sun ( M 0.65  S 0.53 ), and from Earth surface could be
observed only total or partial solar eclipses. In the next two billion years, the Moon’s disk will be
noticeably smaller than the Sun ( M 0.43  S 0.53 ), and terrestrial observatories will not be able
to observe total eclipses of the Sun.
Fig. 1.18. The Sun and the Moon – the ancient Biblical symbols. 1. Hartmann Schedel, Liber
chronicarum. Nuremberg: Koberger, 1493. 2. From Sphæra Mundi; Joannes Sacro Bosco, Venice, 1482.
Eastern sacred number 108 does not play noticeable role in Christian religious philosophy; however
European medieval astrologers found another riddle that still amazes modern astronomers. The Sun and
the Moon are ancient Biblical symbols, and in the Middle Ages astronomers and astrologists asserted that
the days of a week are marked by special astrological signs to remind the sacred days of Creation. In
particular, they claimed that on Wednesdays (the middle day of Creation) there are much more sun
eclipses, than on Saturdays (the day of God’s rest).
During long time, this claim of medieval astronomers was considered as an astrological folklore.
However, in 2000s a Belgian astronomer Jean Meeus analyzed the statistics of more than 3500 predicted
sun eclipses visible in Berlin, Madrid and Moscow for the years 1CE–3000CE [7], and found the
19
expressive calendar cycles. The probabilities of solar eclipses visible from these European capitals
dramatically differ from the random probability 1 7 0.143 resulting in the set –
(0,17;0,1;0,19;0,1;0,18;0,12;0,14) for days of a week starting from the Monday. Figure 1.28
demonstrates this astonishing statistics.
Fig. 1.19. Probabilities of sun eclipses on different days of a week in Berlin, Madrid and Moscow, which
dramatically differ from the expected random probability of 1/7.
These varying probabilities show that at least in Western Europe a largest number of solar eclipses
seem to occur on a Wednesday though Monday and Friday appear to have eclipse probabilities nearly as
high. Modern astronomy proposing arbitrary grouping of days into seven day weeks have no idea what
causes the long-standing “Wednesday paradox” with its strange distribution of solar eclipses as the week
cycles.
1.2 The calendar cycles of solar eclipses
The second half of first millennium of our era was the golden age of Eastern astronomy. Aryabhata
(476–550), which we already mentioned, was the first in the row of great Indian scientists of the Classical
age3. He was born around 476 in central India and probably became the head of the newborn Nalanda
University and its observatory. Nalanda University that is about 100 kilometers south-east of Patna
remained a leading centre of learning in India from the fifth century to twelfth century.
Aryabhata is the author of several treatises on mathematics and astronomy, many of which are lost.
His major work, Aryabhatiya, a compendium of mathematics and astronomy was extensively referred to
in the Eastern literature and has survived to modern times. In the astronomical part of his treatise
Aryabhata asserted that the Earth spins on its axis, the Earth moves around the Sun and the Moon rotates
round the Earth. He believed that the Moon and planets shine by reflected sunlight, insightfully assuming
that the orbits of planets are ellipses. Aryabhata described a geocentric model of the Solar system, in
which the Sun and Moon are each carried by epicycles – the motions of the planets are governed by two
epicycles, a smaller manda (slow) and a larger sighra (fast).
Aryabhata probably recognized the general metaphysical Principle of relativity, which asserts that the
essence of physical laws does not depend on the employed coordinate system. He not only uses
geocentric model but writes about the position of a planet in relation to its movement around the Sun. He
estimates radiuses of the planetary orbits in terms of the radius of the Earth orbit around the Sun
evaluating planetary periods of rotation around the Sun.
India’s first satellite “Aryabhata” launched by Russian space rocket on April 19, 1975 and the lunar crater Aryabhata are
named in Aryabhata’s honor. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the
Aryabhata Research Institute of Observational Sciences (ARIES) near Nainital, India.
3
20
Instead of the astrological tradition in which eclipses were caused by demons Rahu and Ketu,
Aryabhata firstly explained eclipses in terms of shadows cast by and falling on Earth. He considered the
role of moving lunar nodes and estimated the size and extent of the Earth’s shadow. Aryabhata’s
computational methods improved by his followers was so accurate that 18th-century scientist Guillaume
Le Gentil, during a visit to Pondicherry in India, found that the Indian computations of the duration of the
lunar eclipse of 30 August 1765 to be short only by 41 seconds whereas his own charts were long by 68
seconds. Aryabhata estimated the Earth’s circumference with the error of only 0.2 % and the length of the
sidereal year with the error less than one second.
Fig. 1.20. Indian mathematician and astronomer Aryabhata (476–550). 2. Chinese astronomer and
engineer Yi Xing (683–727).
The astronomical works of Aryabhata and his followers significantly influenced the Eastern
astronomy and in particular were commented and included in the Chinese astrology encyclopedia the
Treatise on Astrology of the Kaiyuan Era (Da Tang kai yuan zhan jing) compiled by the head editor
Gautama Siddha, astronomer and astrologer of Indian descent and numerous scholars from 714 to 724
during the Kaiyuan era of Tang Dynasty. The book, also known as the Kaiyuan Star Observations,
contains approximately 600 000 words in 120 chapters (juan). By the tenth century, copies of the
manuscript began to disappear and it was said that only in the 17th century Buddhist scholar Cheng Ming
Shan found a copy of the original work in the stomach of an old Buddha sculpture [3].
This monumental treatise begins with a discussion of the structure of Heaven and Earth, discusses
divinations relating to Heaven, Earth, Sun, Moon, the “five stars”, 28 constellations; meteors, “parasitic”
stars, nova and comets, “guest” stars and “shooting” stars. Most importantly, the work collected the
results of the observations, theories, and prognostications of earlier astronomers and astrologers prior to
the Tang dynasty.
Among such diverse astronomical information in the book there is a section describing works of
Chinese astronomer, the Buddhist scholar Yi Xing (born Zhang Sui, 683–727), which is known as an
author of astronomical celestial globe featured a clockwork escapement mechanism, the first in a long
tradition of Chinese astronomical clockworks. The celestial globe was made in the image of the round
heavens and on it were shown the lunar mansions in their order, the equator and the degrees of the
heavenly circumference. In the early 8th century, Yi Xing beside of his engineering work was in charge
of a terrestrial-astronomical survey established by the Tang court in order to obtain new astronomical data
that would aid in the prediction of solar eclipses.
In addition to the well-known Saros cycle in the Treatise on Astrology of the Kaiyuan Era one can
find a hint at the irregular seasonal and geographical frequency of solar eclipses. Commenting the eclipse
studies of Aryabhata, Yi Xing wrote the following:
21
“The Moon consists of water, the Sun of fire, the Earth of earth, and the Earth’s shadow of darkness.
The merciful Moon shadows the fierce Sun more often in summer and in southern lands”.
In 2006, F. Espenac and J. Meeus confirmed the seasonal irregularity of sun eclipses by statistical
analysis [4]. However, this unusual type of sun eclipse variability was not investigated in detail by
planetary astronomy until recently (Taganov&Saari, 2014 [8, 9]).
Using the dates and coordinates of the Greatest Eclipses in database of Canon one can evaluate the
probabilities of all types of eclipses for all calendar months: pei ( N )  nei ( N ) nei and for different
geographical sites: pel  nel ne . Here and henceforth, the index “e” denotes the evaluations based on the
Canon database; N is the calendar number of a month; nei ( N ) is the quantity of i-type eclipses in the Nth month; nei is the total quantity of i-type eclipses during considered epoch; nel is the quantity of central
eclipses with the Greatest Eclipse at latitude l for total set of eclipses ne .
As demonstrates the statistical analysis of sun eclipses in Canon the calendar distributions (amid
months) of all eclipses, – central, partial and hybrid eclipses are accidental. The relative frequencies or
probabilities of these eclipses for any month on average correspond to: p  1 12 0.0833 . More exactly
these probabilities correspond to:
p( N )  (0.0849;0.0821;0.0773) (1.10)
This formula accounts for 31, 30 and 28.25 days in a month. For example, for the February:
p(2)  28.25 365.25  0.0773 . All statistically estimated probabilities at Figs. 1.21, 1.23 are normalized
(  pei  1 ).
(i )
Figure 1.21 demonstrates the close coincidence of the evaluated probabilities peC ( N ) of all central
eclipses (filled diamonds) with theoretical probabilities p( N ) (broken green curve 1 near straight dotted
line p  1 12 0.0833 ).
In contrast to this almost uniform eclipse distribution the probabilities of total and annular eclipses
demonstrate the expressive calendar cycles (Fig. 1.30). In spring-summer half year (March–August) the
cycles predict 58 % of all total and 42 % of all annular eclipses with the vice versa eclipse proportion in
the autumn-winter half year.
To make the quantitative description of eclipse calendar cycles it is necessary to consider the dynamics
of “Saros series”. The position of the Sun and the Moon conjunction shifts by about 0.48º with respect to
the Moon’s nodes every Saros, and this gives rise to a series of eclipses (Saros series) that may last 1226
to 1550 years and is comprised of 69 to 87 eclipses, of which about 40 to 60 are central (i.e., total, hybrid
or annular).
If the Moon were in a circular orbit moving a little closer to the Earth, and in the same orbital plane,
there would be total solar eclipses every single month, corresponding to the uniform calendar eclipse
distribution. However, the elliptical orbit of the Moon around Earth is inclined about 5.1º to Earth’s
elliptical orbit around the Sun, and the Moon’s orbit crosses the ecliptic at two nodes that are 180º apart
and slowly regress by 19.35º per year with complete cycle 18.6 years. The solar eclipse will be visible
from some place on the Earth only if New Moon takes place within the range 15.39º–18.59º (about 10º–
12º for central eclipses) of a node, which happens at two Eclipse seasons approximately six months
(173.3 days) apart. These orbital characteristics define the dynamics of Saros series.
22
Fig. 1.21. Calendar cycles of total (filled circles) and annular (contoured circles) sun eclipses for
epoch of 3500 years: 2000 BCE–1500 CE with 4887 central eclipses. Broken green line 1 is the
probability estimations by (1.10); blue curve 2 is the probability estimations by (1.15); red curve is the
probability estimations by (1.14). Here pi is the eclipse probability and N is the standard calendar
number of a month except December with N  0 .
The Saros series begins, for example, when the New Moon appears 17º–18º east of the node. The
Moon’s shadow passes about 3500 km south of Earth and a partial eclipse with small magnitude will be
visible from high southern latitudes. One Saros later, the Moon’s shadow passes around 250 km closer to
the Earth’s geometric center and a partial eclipse of larger magnitude will result. This is the start of the
initial phase of the Saros series, which lasts about 2.5 centuries with 6 to 25 partial eclipses. The peak
phase of a Saros series lasts for 7 to 10 centuries with 39 to 59 central eclipses visible almost every Saros
and moving gradually northward. In the middle of the peak phase, central eclipses of long duration occur
near the equator.
The last central eclipse of the peak phase takes place at high northern latitudes. The final phase of
Saros series lasts about 2.5 centuries with 6 to 24 partial eclipses at high northern latitudes with
successive smaller amplitude. The considered Saros series ends near North Pole 12 to 15 centuries after it
began near the opposite South Pole.
Fig. 1.22. Sun eclipse paths of Saros 136 for 1937–2081 years (http://eclipse.gsfc.nasa.gov/ “Eclipse
Predictions by Fred Espenak (NASA's GSFC)”.
23
A good example is the eclipse series of Saros 136 that is currently producing the longest total solar
eclipses of the 20th and 21st centuries and totally will produce 71 eclipses over 1262 years in the
following order: 8 partial, 6 annular, 6 hybrid, 44 total, and 7 partial. Figure 1.31 shows nine solar
eclipses from Saros 136 for the years 1937 through 2081. The westward shift about 120° of each eclipse
path is a consequence of the extra 8 hours in the length of the Saros period. The northward shift of each
path is due to the progressive motion of the Moon with respect to its descending node at each eclipse.
During one Saros on average 40 solar and 29 lunar eclipses can be observed. These eclipses belong to
19–21 Saros series, moving from the South Pole, and to 19–21 Saros series, moving from the North Pole.
These series are at different phases of their evolution, but majority of series (about 65 %) is at the peak
phase, producing central solar eclipses.
The distribution of Greatest Eclipses over the Earth’s surface is an important geographical
characteristic of sun eclipses, which was not investigated in planetary astronomy until now. Statistical
analysis of the Canon reveals the expressive dependence of the probabilities of latitudinal coordinates of
the Greatest central eclipses on the latitude (Fig. 1.23). Almost 40 % of the Greatest central eclipses are
positioned in the tropics (23.5ºS–23.5ºN) and more than 70 % in the area between 45ºS and 45ºN.
Fig. 1.23. The dependence of latitudinal coordinate probabilities of the Greatest central eclipses pel
(filled circles) on latitude l (degrees with negative values for southern hemisphere) for the epoch of 2000
years: 2000 BCE–1 CE with 2915 central eclipses. Solid curve is the theoretical probability estimations
by (1.11). 2. The insolation distribution on the Earth’s surface.
The place of the possible Greatest sun eclipse is always positioned near the intersection of the Earth’s
surface with the line, connecting the centers of the Earth and the Sun. Projections of the possible Greatest
Eclipse are distributed randomly on the frontal plane that is perpendicular to the line, connecting the
centers of the Earth and Sun, and corresponding uniform statistical density grows with the increase of
considered epoch. The Earth’s rotation averages the longitudes of possible greatest sun eclipses forming
uniform longitudinal distribution of sun eclipses.
However, the projection of the uniform Greatest Eclipse density in the frontal plane onto the Earth’s
curved surface is heterogeneous. As the angle increases between the direction connecting the centers of
the Earth and Sun and the coordinate vector of considered meridian, the average Greatest Eclipse
density in the frontal plane is reduced in proportion to the cosine of the latitude l. This is the same
effect as the decrease of average insolation4 with the growth of the incidence angle. Therefore, the
“Insolation” (solar irradiation) is the total amount of solar radiation energy received on a given surface area during a
given time.
4
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average latitudinal density of central eclipses is nearly proportional to the average insolation, and can be
described by the following relation:
pl  0.194  cos l
(1.11)
The amplitude in (1.11) was estimated by the least square method from the database of Canon, and this
relation estimates the central eclipse probabilities with relative error less than 2.5 %.
Figure 1.24, which shows the paths of total and annular solar eclipses in the period 2000–2020.,
visually demonstrates a greater frequency of eclipses in the equatorial and sub-equatorial latitudes.
Fig. 1.24. The paths of total, annular and hybrid solar eclipses in the period 2000–2020. Blue paths – total
eclipses, red paths – annular and hybrid eclipses (http://eclipse.gsfc.nasa.gov/ “Eclipse Predictions by
Fred Espenak (NASA’s GSFC)”.
The movement of the Earth and the Moon in elliptical orbits cyclically changes the apparent sizes of
the Sun and the Moon. The Magnitude of an eclipse that is the ratio of the apparent size of the Moon to
the apparent size of the Sun is maximal (total eclipse) when the Moon is near its perigee and when the
Earth approaches its farthest distance from the Sun in July. Quite the opposite, the magnitude of the
eclipse is minimal (annular eclipse) when the Moon is near its apogee and when the Earth approaches its
closest distance to the Sun in January. Therefore, the relative frequency of total eclipses is larger in
summer months and the relative frequency of annular eclipses is larger in winter months.
The calendar cycles of total and annular eclipses represented in Fig. 1.30 are results of statistical
interference of many different Saros series with raised probabilities of total eclipses in summer months
and raised probabilities of annular eclipses in winter months. This qualitative analysis is confirmed by the
fact that probability differences peT ( N )  peT ( N )  p( N ) and peA  peA ( N )  p( N ) appear to be
proportional to the sun declination  that depends on a year’s season:
peT ( N )  aT  ( N )
paA  aA ( N ) (1.12)
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These proportionalities can be used to obtain the semi-empirical quantitative description of considered
calendar cycles. Since the eccentricity of the Earth orbit is quite low, it can be approximated to a circle,
and  is approximately given by the following expression [2]:     sin[2 365  (284  n)] . In this
formula  is the maximal declination on summer solstice in the northern hemisphere:   23.45
( 0.41 rad.) and n is the day number of a year starting from n  1 for January 1. We can use this
formula with the more convenient form, defining the day number n  365 12  N by the calendar number
N of a month:
 ( N )    sin(2 365  C   6  N ) (1.13)
In this formula N is the standard calendar number of a month except December with N  0 .
Using the least square method, we can estimate for database of Canon the optimal values of
parameters C, aT , aA and then from (1.12, 1.13) we can derive the following semi-empirical formulae:
pT ( N )  p( N )  0.018  sin(2 365  C   6  N ) (1.14)
pA ( N )  p( N )  0.013  sin(2 365  C   6  N ) (1.15)
In this formulae as in (1.13) N is the standard calendar number of a month except December with
N  0.
As a measure of divergence of theoretical estimates pi by (1.14, 1.15) and Canon evaluations pei we
can use r.m.s. (root mean square) value of the relative differences: si  [( pei  pi ) pei ] 100(%) . In the
Canon for epoch of 1200 years: 1601 CE–2800 CE with 1682 central eclipses the Gregorian calendar is
used, and the optimal value of calendar parameter in (1.14, 1.15) is C  240 that provides: sT  1.08% ;
sA  2.54% .
For epoch of 3500 years: 2000 BCE–1500 CE (Fig. 1.30) with 4887 central eclipses, where only
Julian calendar is used for eclipse dates, the optimal value of calendar parameter in (1.14, 1.15) is
C  297 , and sT  1% ; sA  0.45% .
In Canon, the Julian calendar is used for eclipse dates before October 15, 1582 and then onwards the
eclipse dates correspond to Gregorian calendar. For such mixed use of calendars the optimal value of
calendar parameter in (1.14, 1.15) is C  284 , and for the eclipse statistics for the epoch of 5000 years:
2000 BCE–3000 CE with 7129 central eclipses the divergences of theoretical estimates (1.14, 1.15) and
the Canon evaluations are: sT  0.74% ; sA  0.51% .
Calendar cycles of total and annular sun eclipses are results of statistical interference, summation of
manifestations of the three main cyclic movements – the Earth and the Moon movements in elliptical
orbits and the slow regress of lunar nodes by 19.35º per year with complete cycle 18.6 years. The
medieval Indian and Chinese astronomers were right in their intuitive guesses – the total eclipse
probabilities are larger in summer months (Fig. 1.21) and in southern regions (Fig. 1.23).
This short section of our book does not pretend to the complete quantitative description of considered
phenomena. In particular, the next step could be the representation of empirical amplitudes in (1.11, 1.14,
1.15) as functions of the Earth’s and the Moon’s orbital parameters.
Literature
1. Bible (English Standard Version).Crossway Bibles, 2011. http://www.biblegateway.com/
2. Danby, J.M.A. Fundamentals of Celestial Mechanics. Willmann-Bell, 1992.
26
3. Da Tang kai yuan zhan jing (开元占经 (開元占經) is included in the Siku quanshu. In 1989 the
Zhonghua shuju has published a separate facsimile of this edition. A separate edition was printed during
the Daoguang reign (1821–1850) by the Hengde Hall. http://www.wdl.org/en/item/11423/.
4. Espenac, F., Meeus, J. Five Millenniums Canon of Solar Eclipses: - 1999 to + 3000 (2000 BCE to
3000 CE). NASA/TP-2006-214141.
5. Espenak, F., Fifty Year Canon of Lunar Eclipses: 1986-2035, Sky Publishing Corp., Cambridge, MA,
1989.
6. Meeus, J. Mathematical Astronomy Morsels III. Willmann-Bell, Richmond VA, 2004. ISBN 0-94339681-6.
7. Meeus, J. More mathematical astronomy morsels. Willmann-Bell, Inc., 2002.
8. Igor Taganov & Ville Saari. Antike Rätsel der Sonnenfinsternisse. Die Kalendarzyklen // Nachrichten
der Olbers-Gesellschaft Bremen, 244, Januar 2014. 4-10.
9. Таганов И.Н., Саари Вилле. Древняя загадка солнечных затмений – календарные циклы //
Система «Планета Земля»: XX лет Семинару «Система “Планета Земля”». М.: ЛЕНАНД, 2014.
ISBN 978-5-9710-0929-0. (стр. 317-325).
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