The Mathematical Application Of Solar Eclipse. Interpreting Solar Eclipse With Geometry

Challenges for the solar eclipse

First of all, the solar eclipse is necessary that the sun, the earth and the moon located in the right configuration. Compared with the sun, the size of the moon is quite small. This means that the only possible way to form the solar eclipse when the moon is located on the position in its orbit which takes it closest to the earth. And then in this way, the moon could shed a large shadow and be able to cover the entire sun.

In addition, total eclipses are also very short in duration since the moon needs to be positioned nearly perfectly between the earth and the sun.

In addition, as the moon moves and Earth spins this shadow races across the planet's surface at some 1,400 miles an hour, creating a relatively narrow line called the path of totality. Only sky watchers within this line—typically 10,000 miles long and just 100 miles wide—experience a total solar eclipse. Under such circumstances, only a few places on the Earth could experience the full solar eclipse, which turns out that the records on the documents and other resources are relatively few compare to the lunar eclipse.

Also, the sun is important to people. Therefore the ancient people were not able or dare to study or record the eclipse which leads to a situation that although the possibility of solar eclipse was similar to the lunar eclipse, the solar eclipse was considered rare than the lunar eclipse since the records were lost.

Saros Cycle

A Saros cycle is around 6,585.3211 days, or 18 years, 11 days, and 8 hours. In a saros cycle after the eclipse, the moon, sun and earth will have the same relative geometry, a straight line, and an almost identical eclipse will occur. The moon will have the same phase, the same distance from the earth. Additionally, since saros is nearly 18 years long, the earth will be nearly the same distance from the sun, and nearly the same Angle to the sun. Based on the date of eclipse, a saros later can predict almost the same eclipse. Each total solar eclipse track looked alike to the previous one, but moved 120 degrees to the west. [5]

The recurrence of similar solar and lunar eclipses, that is, when the eclipses have the same food type, similar time and pass through the same area (referring to the ground month and the shadow of the sun), they may have the same weather conditions, and the rainstorm floods generated are generally the same or similar, that is, the effects are similar. For example, the eclipses of the sun and the moon in 1662 and 1756 were of the same type, at similar times, and passed through the same regions. In these two years, the Yellow River was flooded, and the flood area, magnitude and disaster situation were basically the same. The 1853 and 1937 solar and lunar eclipses were very similar, and the weather conditions were generally the same in comparison with the two years, in which there were very similar floods in almost the same area in Henan.

In 1864, flood disaster occurred in the lower reaches of the Yellow River, Hunan and southern Liaoning, and the middle and lower reaches of the Yangtze river were earlier. In 1958, the middle reaches of the Yellow River, northern Sichuan, southern Liaoning and other places flooded, the lower reaches of the Yangtze river dry early. For another example, the recent solar and lunar eclipses in 1996 were similar to those in 1931. In 1931, data 131 recorded that in the middle of August, 1931, the Yangtze river basin had a major flood. According to the statistics of Hubei, Hunan, Jiangxi, Anhui, Jiangsu and other five provinces, 186 counties were affected by the flood in this year, the affected population was 28.5 million, 145,000 people died, the loss of property at that time silver 1.35 billion yuan; On July 18, more than 2 million people were evacuated, 810,000 rooms collapsed and more than 2.8 million were damaged, resulting in a direct economic loss of 40 billion yuan. If all the above examples show that the lunar eclipse on the same day has similar effects.

The relationship between rainstorm flood and solar eclipse saros cycle is obvious. The time, frequency and passing area of the solar and lunar eclipses in 1991 and 1973 were very similar. In 1973, a large flood occurred in the central Jilin province, and the situation in 1991 and 1973 was almost identical. And 18 years apart, for a saros cycle. In 1942, the second Songhua river in Jilin province was the year of flood. After a saros cycle, the solar and lunar eclipse in 1960 was the same as that in 1942.

For example, in 1953, the upstream area of Songhua river was flooded, and the maximum flood peak flow of Wudaogou hydrological station in the upstream area of Songhua river in Jilin province was 7120m3/ m2.It was the highest flood since 1888, and 18 years later (a saros cycle), the lunar and solar eclipses were much the same, resulting in flooding. In the rainstorm flood and solar eclipse saros cycle have obvious corresponding relationship, for no lunar eclipse year flood and saros cycle is more corresponding. In the data of nearly 100 years (1897-1998), the phenomenon of no lunar eclipse occurs alternately every 7 years. An 11-year lunar eclipse year must be followed by a 7-year lunar eclipse year, and a 7-year lunar eclipse year must be followed by an 11-year lunar eclipse year. Together, they are 18 years apart, the same as the saros cycle of the solar eclipse.

This indicates that the lunar eclipse also has a similar cycle to the solar eclipse saros cycle. In the event of a flood in a lunar year, the eclipse is the dominant factor, so the correlation between the saros cycle and the flood is more precise. Take the Yangtze river as an example. In 1951, there was a lunar eclipse year, and the Yangtze river was flooded. The year 2000 was the year without lunar eclipse and the year 1998 was the year without lunar eclipse.

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The History of Solar Eclipse Estimations

Many people in the world had found the rule of solar eclipse long time ago. The records estimated in ancient times such as Chaldean and Chinese had built an important basis of solving the mystery phenomenon of the disappearing sun and moon. Within the ancient people’s discoveries that help the experiment in modern astronomy, scientists use saros more as the most accurate eclipse estimation.

At first, Chinese used lunar months to create a calendar named Taichu. Taichu Calendar is a calendar that was created in the Han dynasty, using the 24 solar terms to count time. By using this calendar, ancient Chinese people created a theory named “Sanchong” to estimate when would solar eclipse happen. The Sanchong theory find that solar eclipse would happen each 135 lunar months, also similar to 146.5 nodical months, fitting the rule that modern astronomer estimates. [5] However, this rule is not accurate enough to estimate the solar eclipse’s happening time, since the Moon and the Sun can line up straightly in order to let the Moon cover lights radiated from the Sun.

The other estimation, saros cycle, works better to estimate the happening time of eclipses.

The first finding of solar eclipse estimation was believed first recorded by the Chaldeans. Chaldeans used a lot of time in order to estimate the happening time of solar eclipse. Later it was spread to other places and was known by famous astronomers such as Pliny the Elder, Hipparchus, and Ptolemy. [6] In 1691 Edmond Halley named the cycle of solar eclipse as “Saros”. The word means “repetition” in latin, and accurately refers to what the cycle shown. [7]

According to the records of the Chaldeans, each saros cycle was about 6585.32 days. This means that there must be a solar eclipse seen by people after the last eclipse happened over nearly 18 years. The Chaldeans learnt the regularity of solar eclipse through long-time observations, and actually fit in the results of what the modern astronomers’ researches. Comparing saros and modern researches, the saros cycle is as long as 242 nodical month and 223 lunar month. The two times was so similar that everyone can be surprised that how much intelligence ancient people had.

Scientific significance of solar eclipse

As a great natural phenomenon, solar eclipse is of great significance in many scientific researches.

Normally, as the most efficient energy source for solar system, Sun emits light which is too powerful and blocks light reflected from other stars beside, nor corona, the outermost layer of Sun, can be observed with enough details. [9] However, as the Moon blocks most sunlight during the solar eclipse, scientists can obtain more exact data of the celestial bodies then.

During the solar eclipse in 1868, physicist Jules Janssen and astronomer Joseph Norman Lockyer observed and examined the light coming from the more inner layer of the Sun than corona, which cannot be observed in usual days. [10] After analyzing the characteristics of the light, they independently discovered a new element which Norman named “helium”. [11]

The famous Theory of Relativity by Albert Einstein also found its strong proof during the solar eclipse on May 29, 1919, when the light from the star behind the Sun was detected( see in Fig5) This enabled scientists to determine whether the starlight is deflected by the Sun’s gravity as stated in Relativity Theory by comparing the calculated positions of the star before the Sun reached the position in the sky and during the eclipse. [12]

Solar eclipse also helps historians in dating important historical events. Since solar eclipse is a tremendous and predicable phenomenon, researchers can compare its descriptions in written records and the scientifically calculated time and site where the eclipse happened to narrow down the when and where of certain past events.

The eclipse was recorded in the writings of the Greek historian Herodotus before the Persian invasion of Greece, which had been estimated to occur around 480 BCE. A modern astronomer found that the area would have seen the eclipse on Feb. 17, 478 BCE, which helps approaching to the timing of the invasion.

Reference list

1. https://momath.org/wp-content/uploads/2017/08/EclipseMathFacts.pdf
2. momath.org, Eclipse Math Facts
3. https://eclipse2017.nasa.gov/eclipse-who-what-where-when-and-how
4. https://tasks.illustrativemathematics.org/content-standards/tasks/1145
5. https://eclipse2017.nasa.gov/what-saros-cycle
6. http://www.zhlzw.com/Mind/201008/248840.html
7. Naturalis Historia II.10[56]
8. 'saros'. Encarta Dictionary. Microsoft. Archived from the original on June 8, 2009.
9. Aschwanden, M. J. (2004). Physics of the Solar Corona. An Introduction. Praxis Publishing. ISBN 978-3-540-22321-4. https://www.space.com/33797-total-solar-eclipse-2017-guide.html
10. Kochhar, R. K. (1991). 'French astronomers in India during the 17th – 19th centuries'. Journal of the British Astronomical Association. 101 (2): 95–100. Bibcode:1991JBAA..101...95K
11. Thomson, William (August 3, 1871). 'Inaugural Address of Sir William Thomson'. Nature. 4 (92): 261–278 [268]. Bibcode:1871Natur...4..261.. doi:10.1038/004261a0. PMC 2070380. Frankland and Lockyer find the yellow prominences to give a very decided bright line not far from D, but hitherto not identified with any terrestrial flame. It seems to indicate a new substance, which they propose to call Helium
12. “Testing General Relativity.” NASA, NASA,eclipse2017.nasa.gov/testing-general- relativity .
13. Redd, Nola Taylor. “Here's What Scientists Have Learned From Total Solar Eclipses.” Space.com, Space Created with Sketch. Space, 17 May 2017, https://www.space.com/36785-solar-eclipse-science-throughout-history.html.