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Solar eclipses and the Saros cycle
- pre-Christian predictions -

Tsuneaki Daishido
Professor, Faculty of Education and Integrated Arts and Sciences, Waseda University

With the end of the rainy season, I have high hopes for good weather in the Tukhuar Islands and the Bonin Islands, which are expected to have views of a total eclipse of the sun. When I was a junior high school student, a large partial eclipse was witnessed in Tokyo, and on Hachijo Island, an annular eclipse was visible. As schools and teachers were more flexible in those days, and because the event occurred on a Saturday, students at some schools were specially advised to go home early to observe the eclipse.

When the sun started being obscured, the process was the same as a partial eclipse that I had already seen, so it seemed that there was nothing special. We were observing the sun through a glass blackened with candle fire to protect our eyes while chatting to one another about "the eclipse getting started."

As the sun was further obscured, however, the sky changed colors drastically and a cool wind began blowing, which produced an extraordinary atmosphere. If it were a total eclipse, we would have experienced an even more dramatic scene. Nevertheless, we enjoyed a large partial eclipse in which around 80% of the sun was obscured.

Though I have never seen a total eclipse of the sun, it is said that an annular eclipse is 100 times as impressive as a partial eclipse, and further that a total eclipse is 100 times as impressive as an annular eclipse. We can see the profound impressions that total eclipses make reflected in many art works including "St. Matthew Passion" by Bach and "The History" by Herodotus.

In "The History" by Herodotus, there was a description on the solar eclipse predicted by the Greek philosopher Thales. After Assyria collapsed in 612 B.C., a battle began between Media and Lydia. In 585 B.C., the sixth year of the battle, a solar eclipse occurred during the war, which led to the achievement of peace. Thales had predicted this eclipse to the Ionian people. Some say he knew the solar eclipse Saros cycle that the Babylonians had recognized, and others say he calculated the smallest date that corresponds to an integral multiple of the synodic month and the nodical month (lowest common multiple).

I would like to take this opportunity to consider what made it possible to predict solar eclipses two thousand and several hundred years ago. Thales' ancestors were Phoenicians who were believed to have acquired information at various sites through trading earlier than others. Thales studied Babylonian astronomy and mathematics as well as Egyptian geometry. There is an anecdote about his falling down the well while looking up at the stars in the sky and being laughed at by his servant, and another anecdote about money making. When he was told that philosophy was not useful for making money, he bought up olive oil presses to make a large profit. Then, after demonstrating that money making was in fact easy for philosophers, he said "philosophers are not interested in money making." This anecdote reminds me that Dr. Kiyoshi Ito, who is famous for Stochastic Processes, a mathematical basis in money derivatives, used only saved money in an ordinary deposit.

In ancient times, predictions of solar eclipses were probably necessary for those in power. The Saros cycle, indicating that a solar eclipse occurs within a period of about 18 years, was well known in the Assyrian era.

I read an explanation for lunar and solar eclipses in the textbooks during my elementary school days. These texts said that a solar eclipse is an astronomical event that occurs when the moon moves between the earth and the sun and obscures the sun. An explanation for the waxing and waning of the moon in the same figure, however, said that a new moon cannot be observed. From the explanatory figure, it seemed to me that a solar eclipse occurred every month. I continued to wonder about this until my high school days, when Mr. Kojima, an earth sciences teacher, introduced the Saros cycle and gave a very clear explanation.

"A circle that runs along the surface containing the center of a sphere so as to cut it is called a great circle. For example, the equator is a great circle of the earth. Assuming a great circle generated by extending the equator to an imaginary celestial sphere, this is the celestial equator. The earth orbits the sun, while the sun, viewed from the earth, moves in constellations on the celestial sphere over one year. This is also a great circle called the ecliptic. Since the geographic axis is at a 23.4 degree incline, both the ecliptic and celestial equators are also at 23.4 degree inclines. The ecliptic is also a great circle generated when the orbital plane of the earth around the sun is extended to cross the celestial sphere.

The moon orbits the earth, and its orbital plane is at about a five degree incline with respect to the orbital plane of the earth around the sun. Though this value is much smaller than the inclination angle of the geographic axis (23.4 degrees), it is ten times the apparent spread of the moon (at an angle of 0.5 degrees). This is why the moon approaches the sun on the celestial sphere but often passes it without generating a solar eclipse. A great circle generated by extending the orbital plane of the moon to the celestial sphere is called moon's path. As you already know, both the celestial equator and moon's path are inclined by five degrees.

Figure Relationships among the Sun , Earth , and Moon 
The orbital plane of the moon is at about a five degree incline with reference to that of the earth. As shown in the right figure, a solar eclipse does not occur even if it is a new moon. In the middle figure, however, a solar eclipse occurs when it is a new moon, but does not occur when it is a crescent moon or half moon.

Do you know why the moon in the winter rises high and looks pale and chilly? The incline of the geographic axis causes the height of the sun to be lowered in the winter. The sun in the winter is located to the south of the celestial equator on the celestial sphere. But we often watch the moon at night. Since the moon lies opposite to the sun, its height is increased.

This is the reason a new moon does not always lead to a solar eclipse. In order for solar eclipses to occur, the sun and the moon need to be located near the intersection of the ecliptic and moon's path at the same time.

The orbital plane of the moon, however, is gradually shifting like the movement of a gyroscope. Thus the intersection goes round the celestial sphere in 18.61 years. This is much faster than the 26 thousand years comprised by the movement cycle of the equatorial plane of the earth (precession). As a result, a cycle with which the moon crosses the ecliptic (called one nodical month) is 27.2122 days, while a cycle with which the sun crosses the moon's path (called one eclipse year) is 346.62 days, a little shorter than one year.

On the other hand, a cycle from a new moon to the next one (called one synodic month) is 29.5306 days, which is longer than one nodical month. Since solar eclipses require positioning near the intersection and a new moon, a cycle from one solar eclipse to the next one is calculated as the least common multiple of one eclipse year and one synodic month, which is called the Saros cycle. At this time, 19 eclipse years are approximately equal to 223 synodic months, so the duration of the Saros cycle is 18 years and 11.33 days. Since the portion after the decimal point represents one-third of a day, solar eclipses in accordance with the Saros cycle occur at longitudes shifting by 120 degrees. A cycle in which solar eclipses occur at the same place is three times the Saros cycle. That is, 54 years and 34 days."

On receiving such a logical explanation, my longstanding question was answered. The Saros cycle predicts solar eclipses with identical conditions in terms of location and degrees, and comprises many series. The current solar eclipse and the annular eclipse witnessed in the Hachijo islands belong to different series.

Mr. Kojima also taught us why the tides occur, and why the rotation of the oscillating plane of the Foucault pendulum depends on latitudes. The reasons he gave us were very profound, and they deeply impressed every student in the classroom. Anyway, I was surprised to learn that solar eclipses were predicted with high accuracy only through observation and arithmetic in the Babylonian era, when the Copernican system had not been established yet.

The Pythagorean Theorem was already popular in the Babylonian era. When I talked about the method of least squares during my class on experiments in physics, I thought about the wisdom that people had more than two thousand years ago as I examined documents with cuneiform writing on them with my students. The concept of Hilbert Space, on which quantum mechanics is based, also originated in the Babylonian era.

At an Open Campus event held from August 1 (Saturday) to August 3 (Monday), my students will present the results of their studies through experiments and poster displays, and they will introduce the literature concerning Babylonian arithmetic and astronomy. I hope you will ask these presenters many questions.
http://www.astro.phys.waseda.ac.jp/index.html

Bibliographies

Herodotus, The Persian Wars, Loeb Classical Library (1920)
Herodotus "The History", Translated by Chiaki Matsudaira, Iwanami Shoten (1971)
Anthology of early Greek natural philosophy, Edited by Yoshinobu Kusakabe, Chikumashobo, (2004)
B.L. van der Waerden "Science Awakening", Translated by Tamotsu Murata and Katsuzo Sato, Misuzu Shobo (1984)

Additionally, a lot of helpful descriptions are provided online. Please search for desired sites using keywords such as: eclipse, Saros, and cycle. The "Assyrian Dictionary" (22 volumes, Chicago University) is also available to the public.

Tsuneaki Daishido
Professor on the Faculty of Education and Integrated Arts and Sciences, Waseda University

Specialty: Astrophysics and radio astronomy
1964 Graduated from the municipal Toyama Senior High School in Tokyo
1969 Graduated from the Department of Physics, Graduate School of Science and Engineering, Waseda University
1975 Completed Graduate Studies at the School of Science (Astronomy), the University of Tokyo and acquired a Doctor of Science Degree
1975-1977 Fellowship researcher of the Japan Society for the Promotion of Science (National Astronomical Observatory of Japan)
1977 Assistant Professor at the School of Education, Waseda University
1980 Associate Professor
1985 Professor

Research fields include construction of a 64-element radio interferometer at the top of the No. 15 building and the large-scale radio interferometer in Nasu, and observation of a radio transient object. This object was detected by Waseda University earlier than anyone in the world, but it has not been identified yet. We do not know even whether the object came from inside or outside the Galaxy

External activities: Member of the Administration Committee at the National Astronomical Observatory of Japan, Member of the Evaluation committee at the National Institute of Information and Communications Technology, Lecture on "Array antenna" (at the Department of Electronics and Electrical Engineering, Keio University), and Editor of "Parity" (Maruzen)