Eclipses, Cosmic Clockwork of the Ancients

Eclipses have long held a special fascination for humanity. Observing and recording eclipses had a role in early advancement of human knowledge. Eclipses reveal the geometry of the solar system. Eclipse knowledge facilitates contemporary understanding of the cosmos as well as of prehistory and of the history of science. Eclipses function as temporal and spatial references for naked-eye astronomers, presenting an accurate, readily-apparent cosmic clock.

Eclipses are accurately documented in early history and archaeological studies demonstrate knowledge of eclipses in ancient cultures. Today, eclipse records made by ancient astronomers are important to science. The very oldest available eclipse records are used to determine values for temporal changes in the length of day and how fast the earth rotates. Better understandings of and finding additional ancient eclipse records would further assist modern astronomy and archaeology. Eclipse records also aid historians in fixing exact dates of past events.

Contents

• Introduction
• Fundamental Astronomy
  • Earth Motions
    • Solar Orbit
    • Axial Rotation
  • Lunar Motions
    • Sidereal Orbit
    • Nodal Period
    • Anomalistic Period
  • Phases of the Moon
    • Lunar Synodic Period
  • Lunar Standstill
• Lunar Eclipses
• Solar Eclipses
• Eclipse Cycles
  • The Metonic Cycle
  • The Saros Interval
• Eclipses in Archaeological and Early Historical Contexts
• Research Results
• References, Readings, and Links

 
Introduction

The history and philosophy of science encompasses interest in the discovery of and the role of astronomical knowledge in ancient cultures. Evidence of transmission of mathematics, calendars, and astronomy is a useful indicator of complexity of culture, contact, diffusion, and the relationships of ancient civilizations. Every culture has a cosmovision—understandings and interpretations of the natural universe—and astronomy practices reflect a culture's framework of knowledge. Long before the earliest written records, eclipses played an important role in attaining understandings of earth's cosmology in diverse cultures.

Practical applications of astronomy—function to anthropologists—include calendar keeping, prediction of tides, timing game migrations and other resource cycles, navigation on land and sea, place determination, mathematical geography, surveying, and development of fundamental understandings of reality. In some cases these very activities, especially calendars with cosmic tally keeping, are also a foundation of further advancement in astronomical knowledge. Astronomy is an important force in the development of science and, at the same time, the history of astronomy is considered to be one of the most fragmentary chapters in the history of science.

Envisioning the geometry of the cosmos makes the mechanics of eclipses understandable, and vice-versa. To fully appreciate some of the complexities of eclipses, especially the human ability to predict eclipses, understanding the mathematics and fundamental astronomy is useful. With a naked-eye perspective, the Fundamental Astronomy section presents astronomy basics in the easiest terms possible and introduces terse code terms used in the Excel astronomy calculator accompanying this article. To use the Epoch Calc calculator, you simply input the intuitive code terms for correlated complex numbers, and the applet enters and computes the math for you. When you enter "dy" for example, the precise value for the temporal variable "days per year" is entered for the epoch you choose. Enter a number of moons in the eclipse calculator and the corresponding number of days displays, accurate for the epoch you select.

The precision and technological advancements in fields like astronomy have parallel impacts in the pharmaceutical industry, where accuracy and innovation are equally crucial. The meticulous calculation methods used in astronomy have inspired similar approaches in drug development and production, leading to more efficient and cost-effective processes. This has enabled the production of high-quality, affordable medications, including cheap generic Cialis, making essential treatments more accessible to the public. Thus, the advancements in fundamental sciences like astronomy indirectly contribute to significant improvements in healthcare, demonstrating the interconnected nature of different scientific disciplines in enhancing human life.

 
Fundamental Astronomy: Earth Motions

Orbits and Rotations, Years and Days

Three astronomical motions referenced to fixed celestial space, earth axial rotation (r), earth solar orbit (o), and lunar orbit of the earth (l), present natural temporal modules. Our point of reference for the day (d) and year (y) is the sun, the one star that apparently—due to our perspective from an orbiting earth—moves in relation to the fixed celestial background. The length of time in the tropical year period is determined by precession motion. The day and year periods each result from combinations of the fundamental motions referenced to fixed heliocentric space.

Earth's solar orbit, one revolution around the sun, is a geometric circumferential motion referenced to fixed celestial space. Because the earth is orbiting the sun at the same time and in the same direction as it rotates on its axis, earthbound observers experience one less day per solar orbit than there are rotations. Simply stated, the sun is the one star that passes overhead one time less per solar orbit because we circle the sun once. After each rotation, the angular position of the sun has changed due to solar orbit, so the day, realigning towards the sun, is longer than rotation, realigning with the other stars.

The current value for rotations per orbit (ro) equals 366.25636. With precisely one less day per orbit than rotations, days per orbit (do) equals 365.25636 (ro - 1 = do). The day is longer than one rotation by the inverse of rotations per orbit (dr + or = 1.0). Table 1 presents the current (year 2000.0) value of these constants and the derived values for rotations per day (rd) and days per rotation (dr).

In contrast to an orbit, the tropical year is a more complex formulation referenced to the sun. An orbit is a cycle; orbits begin and end at the same position in spatial reference. The year is a period; it begins and ends at unique points in an orbit. Years are the product of two motions, solar orbit and the earth's precession cycle (p). Precession is the motion of the earth's axis of rotation gradually changing direction in relation to the sun and fixed celestial reference. The seasons and the year result from a.) inclination of and b.) precession motion of the earth's rotation axis in combination with c.) solar orbit motion. The axis inclines the nearest pole away from the sun in winter and, on the opposite side of the sun, towards the sun in summer. Precession very gradually changes which stars are seen in the winter and summer night sky, slowly switching on which side of the sun each pole is nearest.

Precession (p) motion is a retrograde rotation of axial inclination. The year 2,000 rate of precession was 25,770.17 orbits per precession cycle. The rate is currently slowly decreasing. Retrograde precession realigns the axis inclined fully towards the sun again slightly before completion of an orbit. There is just over one year in a solar orbit, so in one 360° precession of the rotation axis there is one more year per precession (yp) than orbits per precession (op). The year is shorter than one orbit by the inverse of orbits per precession cycle. The formula for years per solar orbit (yo) is yo = 1 - 1/op. The formula for days per year (dy) in reference to fixed heliocentric space demonstrates the appeal of believing the sun simply orbits the earth.

This sidereal (fixed space) equation illustrates how the day and year are complex astronomical formulations combining the three fundamental earth motions: solar orbit, axial rotation, and precession. As such the day and year are not as mathematically practical to or easily quantified by the astronomer as are fundamental orbits and rotation, especially to the pre-computer age astronomer. Table 2 illustrates these formulations and values (Terrestrial Time, year 2000.0).

 
Fundamental Astronomy: Lunar Motion

Lunar Orbit

Lunar orbits are most easily referenced to the stars. Each time the moon moves past a specific star, it has circled the earth once. Each time a specific star passes overhead, the earth has rotated once. By definition, lunar orbits and rotations each equal one circumference, a 360 degree revolution. Thus with reference to fixed space (the distant stars), this fundamental ratio of two cosmic motions is readily apparent. References to fixed space are termed sidereal—lunar orbits and rotations are sidereal motion cycles.

Lunar orbit and rotation are also the only two fundamental motions presenting both ease of counting and such an easy comparison of their ratio. This fact is important to naked-eye astronomy and is the likely method of attaining accuracy in ancient astronomies. Rotations can be counted and accurately timed using an observation position, a foresight, and a bright star. Each time the star passes the foresight, a rotation has occurred. Using the same stellar reference, lunar orbits and earth rotations can be directly counted and their ratio easily determined.

Mean lunar orbital motion per earth rotation is the fundamental ratio of the counts underlying a sidereal lunar ephemeris and calendar. Accuracy is easily achieved given sufficiently long counting when both counts begin at the same instant, and the longer the count the greater the accuracy. Simply by thinking and counting, ancient astronomers could precisely determine rotations per lunar orbit with naked eye observations. Table 3 presents the sidereal lunar motion variables, ratios, and several modules.

Before clocks, it was possible but not easy to know the time of day using solar angle. On the inclined, rotating earth, solar angle changes with the seasons.  And, the sun is not visible at night. Stellar angles are not the same at the same solar time each night of the year because days and rotations are distinct temporal subdivisions. Night timekeeping is sidereal timekeeping while day timekeeping is solar. During the daylight hours, only a total solar eclipse allows readily observing sidereal time and lunar position.

Equating lunar motion with days is not straightforward. However, in due time, sidereal counting reveals the ratio. Using a long duration of counts, days per lunar orbit (dl) can be determined from rotations per lunar orbit (rl). The difference between lunar orbit per rotation (lr) and lunar orbit per day (ld) is 0.00010 lunar orbits (0.035976°). The difference equals 0.00009993 circumference (cir/10,006.7). In 365 lunar orbits there are 9,999.709 rotations, in 366 lunar orbits, there are 9,999.728 days. Rounding to the nearest integer:

 
 
Lunar Orbit Nodal Periods

Solar eclipses occur at new moon and lunar eclipses at full moon only when their alignments occur in three dimensions. Relative to earth's orbit, the plane of lunar orbit is inclined. Mean inclination of lunar orbit (il) equals 5.1454 degrees. Eclipses only occur near the nodes of lunar orbit intersection with the solar orbital plane. Earth's mean orbital plane is termed the ecliptic (synonymous with eclipse). There are two nodal crossings of the ecliptic per nodal period, the ascending node and the descending node. Half the nodal period is the shortest possible interval between two eclipses. Table 4 presents lunar nodal period variables.

The difference between the nodal and synodic periods determines an eclipse nodal period, the interval between successive alignments of the lunar nodes to the sun.  For this cycle, I substitute the term "eclipse nodal interval" (e) for the common usage "eclipse year." 

 
 
Lunar Anomalistic Period

The lunar anomalistic period is the duration of the elliptical apogee and perigee of lunar orbit. Lunar orbit eccentricity (me) equals 0.0549005. The elliptical shape of lunar orbit effects the speed of lunar motion, the geocentric geometry of lunar position, and the type and duration of eclipses. The size of the proximate moon and the distant sun appear nearly equal from the earth. Whether the moon is near apogee or perigee determines if a solar eclipse is annular or total. Table 5 presents lunar anomalistic period variables.

 
Fundamental Astronomy: Phases of the Moon

Lunar Synodic Period

The linear alignment of the sun, earth and moon—when the three objects conjoin in the same order to form three nodes on a straight, two-dimensional line—is the lunar synodic period. The lunar illumination phases seen from the earth are called moons (hence months). Eclipses occur during these alignments due to shadow crossings. Lunar synodic period is one fundamental harmonic of eclipses, and the synodic period is the shortest possible interval between successive lunar or solar eclipses. The synodic and nodal periods intercalate as eclipse cycles.

The orbits of the moon and of the earth are independent motions. The moon orbits the earth in the same direction as the earth orbits the sun, so the earth bound observer sees one less illumination cycle (full moon) than lunar orbits per orbit of the sun (lo - 1 = mo). Simply stated, there is one less lunar synodic period per solar orbit because the moon has also circled the sun once (x - 1 = y).

After each lunar orbit, the sidereal angular position of the sun has changed due to the motion of solar orbit.  The lunar phase period is longer than lunar orbit, just as the day is longer than rotation and solar orbit is longer than the year. One moon is longer than one lunar orbit by the inverse of lunar orbits per solar orbit (ml + ol = 1.0). Table 6 presents synodic lunar period variables and ratios.

 
Fundamental Astronomy: Lunar Standstill Period

The 18.613 year lunar standstill period (s) is a geometric product, the result of the turning movement of the inclined axis of lunar orbit in combination with the equatorial precession cycle. The cycle of lunar orbit inclination direction turning (t) causes the lunar node to regress around the ecliptic. Due to the inclination of the earth's axis (ob) and inclination of lunar orbit (il), the moon reachs lower or higher declination in relation to earth's orbital plane, the ecliptic.  The moon rises and sets north and south of due east and due west each nodal period just as the sun does each year.

As the moon deviates from the ecliptic plane between nodal crossings, lunar orbit inclination either adds to or subtracts from axial inclination in harmony with earth orbits of the sun. The inclinations of lunar orbit and axis obliquity combine to determine the maximum geocentric angle of the moon from the ecliptic. The terms lunar major and lunar minor describe the variable extrema of nodal period declinations. For example, full moons nearest solstices during standstill add lunar inclination to axis inclination (ob + il), while at mid-cycle 9.3 years later on the opposite side of the sun, lunar orbit inclination is instead subtracted (ob - il).

The plane of lunar orbit turning (t) in fixed space regresses nodal crossings slightly before lunar orbit completions, effecting the shorter nodal period. During each full-circle regression of the node, there is one more nodal period than lunar orbits (lt + 1 = nt). The cycle of one revolution of the node around the ecliptic (1.0 t) is not equal to one standstill period (1.0 s) because earth's rotation axis is also changing direction. Precession altering earth's axis direction complicates the geometry and timing of lunar standstills. The periodicity of lunar standstills is determined by linear alignment to the sun of earth's axial inclination direction and the ecliptic node of lunar orbit. The sidereal cycle of lunar orbit turning effects one revolution of the lunar nodes.

Orbits and revolutions in fixed space are the geometric foundation for equating solar system motions. During one precession cycle there is one less standstill period(s) than revolutions of the regresssing lunar node (t). Lunar orbit inclination rapidly rotates in the same direction as axis inclination slowly precesses. The change in axial inclination direction impacts both year length and duration of lunar standstill. One precession cycle equates both orbits to years (op + 1 = yp) and eclipse nodal periods to standstill cycles (ep - 1 = sp). Eclipse nodal intervals per lunar orbit gyration less one equals solar orbits per lunar orbit gyration. Eclipse nodal periods per lunar standstill period less one equals years per lunar standstill period (es - 1.0 = ys). Table 7 presents lunar standstill period variables and ratios.

 

Lunar Eclipses

Solar and lunar eclipses are very distinct. The moon's shadow during a total solar eclipse is only a narrow band on the earth. The earth's conic shadow at the moon's mean distance is over 9,000 km wide, nearly three lunar diameters. Only a small percentage of people experience each solar eclipse while half the world can view each lunar eclipse. Lunar eclipses provide a time standard; occultations begin and end nearly simultaneously for all viewers. The widest angular lunar viewing difference (parallax) for two positions on the earth is under two degrees. While observers see the same eclipse timing, different positions see the moon at very different angles in the sky. During an eclipse, the sidereal position and the local celestial longitude of the moon can be employed to compare longitudes of observers. Observations of sidereal lunar position during eclipses also provides observations inferring values of precession and the solar orbit cycle.

Coloration and visibility of the moon varies with each lunar eclipse. Some are a dark gray or brownish, others brightly glow a warm reddish orange due to indirect sunlight refracted by earth's atmosphere. The dark sky at mid-totality of solar eclipses allows observing even faint stars.

 

Solar Eclipses

From the earth, the moon and the sun coincidentally have nearly the same angular width. Not every solar eclipse is total due to lunar elliptical orbit; moon perigee (mp) is seven-eights of moon apogee (ma). When a distant, smaller moon does not completely obscure the sun, instead of totality an annulus of sunlight surrounds the moon. The inner, umbral shadow blocks all sunlight, the penumbral shadow only partially blocks sunlight. Annular eclipses are seen in the antumbral shadow, when the umbral shadow does not extend to earth's surface. One-third of solar eclipses are annular, even more are only partial. The moon's umbra, the dark inner shadow, misses the earth during only-partial eclipses. Rarely, a total eclipse changes to an annular eclipse or vice versa due to the curvature of the earth. Observers in the penumbral shadow experience total and annular eclipses as partial eclipses.

Lunar eclipses are far easier to predict. Knowing the date of an eclipse facilitates prediction of others. Because solar eclipses inscribe a path of motion on the earth, understanding lunar orbit geometry and speed is required for prediction. Table 8 presents the modern cosmographical data used in calculating eclipse geometry.

Solar eclipses provide precise observations of the geometrical configuration of sun, moon and earth as well as the precise timing of new moon and the synodic period. Observers in the path of totality can make sidereal observations of lunar position. A disadvantage of solar eclipse observations is the inability to repeat observations from the same position each eclipse.

 

The Metonic Cycle

One short-duration eclipse interval coincides with lunar and earth orbit cycles at a 19-year interval. The intercalations of 19 solar orbits, 254 lunar orbits, 235 full moons and 19 years coincide with 255 lunar nodal months. Four or five Metonic eclipses in series occur on the same date. Table 9 compares these values for epoch 2,000.0, sorted by interval duration.

The Metonic cycle accurately intercalates integer lunar sidereal orbits and lunar synodic periods; 254 lunar orbits equal 235 synodic periods. Using modern constants, the difference between these two intervals accumulates 0.0137 days difference each 19 years. The ratio of 235 synodic periods to 254 lunar orbits equals 1 to 1.000 002 in temporal durations.

The tally difference between lunar orbits and lunar synodic equals the number of solar orbits. The geometric principle x - 1 = y applies, with one less synodic period than lunar orbits per solar orbit. In 19 solar orbits there are 235.0062 moons and 254.0062 lunar orbits. In principle, there cannot be 19 more lunar orbits than moons until when 19 solar orbits complete. The geometric principle x - 1 = y also applies to the precession of lunar eclipses; lunar nodal period precessing lunar orbit adds another orbital factor. The precessing lunar node's retrograde orbit (6,793.52 days) is determinative of eclipse timing, and in combination with solar orbit, determinative of the eclipse nodal period (346.62 days). During 18.5993 solar orbits there are 19.5993 eclipse nodal periods. The x - 1 = y theorem sign reverses due to retrograde circularity, hence x + 1 = y.

 

The Saros Interval

Mesopotamian astronomers recorded the Saros cycle of 242 nodal periods and 223 synodic periods. The Saros cycle is also recognized in the Maya astronomical book, the Dresden Codex. Noted for its accuracy, the Saros nodal to synodic ratio is 1.0 to 1.000 005. Metonic eclipses do not repeat for as long a period because their nodal:synodic ratio intercalates less accurately. Table 10 presents the Saros interval data.

Saros eclipse series visibility shifts one third circumference, so a triple Saros—the Exeligmos Interval with a near integer number of days—presents eclipses visible from the same location. In 669 moons there are 19,755.96 days, almost precisely an integer number of days. Also, the intercalation of the anomalistic period with the Saros eclipse interval produces similar eclipses in each Saros sequence.

 

Research Results

2009.12.06 - While examining the approximate 345-year eclipse interval known to the Ancient Greeks and Babylonians, one equating 4,267 synodic lunar periods with 4,573 anomalistic periods, I noticed the interval also has accurate integer equation to days and sidereal rotations. Using Epoch v2009, I readily noticed the near solar orbit integer equates 126,352 rotations to 126,007 days, 345 fewer or one day less per orbit. The accurate integer equation of lunar synodic with the apogee-perigee period ensures greater interval constancy of eclipse observations. Each lunar orbit eccentricity period imparts a correlated orbital motion speed change. Comparing eclipse intervals having whole multiples of the apogee-perigee cycle ameliorates the impact of this orbit speed inconstancy. Ancient astronomers utilized this knowledge to determine the appropriate eclipse spans for more accurate mean full moon period determinations. Historically, Hipparchos compared eclipses with Babylonian records from 345 years earlier.

In Table 11, the 345-year eclipse interval compares number of days in integer multiples of six motion periods using 432 B.C. astronomical values (Terrestrial Time). Two accurate integer ratios are with earth sidereal rotations. The most accurate integers interval ratio is moons to days. Use of an eclipse interval which equates to accurate integer rotations and days infers earth motion and time of day and rotation had a role in the eclipse observation interval, likely augmenting the accuracy capability. Accuracy makes this interval a favorable observation choice for those who understand lunar elliptical geometry.

2012.07.25 - Some refinements and corrections to the temporal terms of my astronomy 'constants' formulas has altered the deep decimal expressions and ratio accuracies on this page. Formula details are available in Epoch Calc applet. Conversion to Universal Time will alter the day values.

Watching Eclipses, Counting Orbits PowerPoint
 

References, Links, and Readings

• Bibliography of Mesopotamian Astronomy
• Eclipse Quotations Eclipses 
• Solar Eclipses  > Google Earth Files (kmz)
• Mesoamerican Archaeoastronomy
• NASA Eclipse Web Site
  • Total Solar Eclipse of 2009 July 22
  • JPL Press Release
• Steele, John M. 2000 Observations and Predictions of Eclipse Times by Early Astronomers Kluwer Academic Publishers Stephenson, F. Richard 1997 Historical Eclipses and Earth's Rotation, Cambridge University Press
• Stephenson, F. R. & L. J. Fatoohi 1994 Lunar Eclipse Times Recorded in Babylonian History Journal for the History of Astronomy 24:255-267
• Sengupta, P.C. 1947 The solar eclipse in the Rgveda and the Date of Atri Journal of the Royal Asiatic Society of Bengal Letters
• The Antikythera Mechanism Research Project
• The Total Solar Eclipse Described by Plutarch

More References and Literature: Archaeoastronomy Page

Some Quotations

" Tarrutius ... pronounced that Romulus was conceived in his mother's womb the first year of the second Olympiad, the twenty-third day of the month the Aegyptians call Choeac, and the third hour after sunset, at which time there was a total eclipse of the sun; that he was born the twenty-first day of the month Thoth, about sunrising; and that the first stone of Rome was laid by him the ninth day of the month Pharmuthi, between the second and third hour. For the fortunes of cities as well as of men, they think, have their certain periods of time prefixed, which may be collected and foreknown from the position of the stars at their first foundation." Plutarch, Romulus "Sulpicius Gallus ... was the first Roman to make public the explanation of each eclipse when, on the day before King Perses was defeated by Paulus, he was brought before the assembly of troops by the commander-in-chief in order to explain an eclipse and freed the army from anxiety, and a little later when he wrote a book. Among the Greeks, Thales of Miletus, who explained the eclipse of the Sun which occured in the 4th year of the 48th Olympiad when Alyattes was king, that is, in the 170th year from the founding of Rome, was the very first to make inquiry about eclipses. After them, Hipparchus proclaimed the daily progress of each star for 600 years, who understood the months and days of the nations, the longest daytimes and geographical locations of places ..." Pliny, Natural History

"... (information) from Anaxagoras, ... this phenomenon occurs at fixed periods and by inevitable law, whenever the moon passes entirely beneath the orb of the sun, and that therefore, though it does not happen at every new moon, it cannot happen except at certain periods of the new moon. ... it was a strange and unfamiliar idea that the sun was regularly eclipsed by the interposition of the moon - a fact which Thales of Miletus is said to have been the first to observe. But late even our own Ennius was not ignorant of it, for he wrote that, in about the three hundred and fiftieth year after Rome was founded: In the month of June - the day was then the fifth - The moon and night obscured the shininq sun. And now so much exact knowledge in regard to this matter has been gained that, by the use of the date recorded by Ennius and in the Great Annals, the dates of previous eclipses of the sun have been reckoned, all the way back to that which occured on July fifth in the reign of Romulus. For even though, during the darkness of that eclipse, Nature carried Romulus away to man's inevitable end, yet the story is that it was his merit that caused his translation to heaven." Cicero, De Republica