A Description of the Solar Eclipse Calculation

The first part of the Surya Siddhanta gives a rather lengthy description of how to calculate the Hindu day count. Given a particular date the Hindu day count is the number of days elapsed between the end of creation and the particular date given. Creation ended 1,955,880,000 years before midnight February 18, 3102 B.C. at Ujjayni, India. The Hindu day count  is very similar to the Julian Day used by modern astronomers; the Julian Day is the number of days elapsed since noon January 1, 4713 B.C. at Greenwich, England. The Hindu calculation is overly complicated because it is based on a lunar calendar system and in order to determine the number of days one must first determine the number of lunar months. Learn more about the Hindu Lunar Calendar.

For a particular day, once the day count has been determined, the Surya Siddhanta describes how to determine the celestial longitude (measured along the ecliptic) of the particular body (Sun, Moon, or planet) in question. The longitude is calculated by first calculating the mean longitude for the particular body; this is the longitude that the body would have if it were moving around the celestial sphere at a constant rate. (It has the same meaning in modern calculations of positions of the Sun, Moon, and planets.) Then a correction is calculated based on the assumption that the body is sometimes ahead and sometimes behind the mean longitude. The correction amounts to assuming that the body moves on a smaller circle and the center of this circle is at the mean longitude. At times the body is at the leading edge of the smaller circle, and thus ahead of the mean longitude; later, as the body moves around the smaller circle it is near the trailing edge of the circle and thus behind the mean longitude. This is exactly the same as described by Ptolemy (150 A.D.) in his Almagest; the smaller circle is called an epicycle by Ptolemy. The Hindus apparently did not believe the body actually traveled around the epicycle however, instead they believe there was some sort of force which at times drew the body ahead in its motion, and at other times pulled it back in its motion. 

In any case the true longitude of the body is then computed by adding the correction to the mean longitude. 

A solar eclipse can only occur on a day when there is a New Moon. Since every descent ancient astronomer would already have algorithms for determining the dates of the New Moon we do not need do discuss how that is done here. In particular if you have a lunar calendar the New Moon is always the first day of each month; simple! 

To determine whether a solar eclipse occurs one must not only know the day of the New Moon but one must also know the time at which the New Moon occurs; that is the time of that day when the Sun and Moon have the same true longitude. The Hindu method is quite simple; the true longitudes of the Sun and Moon are determined for the time of the beginning of the New Moon day; they are then re-determined for the beginning of the next day. By taking differences one determines the rate at which the Sun and Moon are moving in longitude that day, and then it is a simple matter of interpolation to determine the time of day at which they had identical true longitudes. At this point the time of the New Moon has been established as well as the true longitudes of the Sun and Moon at that time. 

The path that the Sun takes as it moves around the celestial sphere is called the ecliptic. The path of the Moon, as it moves around the celestial sphere, is sometimes north of the ecliptic, and sometimes south of the ecliptic. The point at which the Moon crosses the ecliptic moving north is called the ascending node and the point at which the Moon crosses the ecliptic moving south is called the descending node. 

An eclipse can only occur if the Sun and Moon are both close to the nodes of the orbit of the Moon, and it must also be a New Moon. Sometimes the New Moon occurs when the Moon is away from the nodes, and thus so far from the ecliptic that it passes by the Sun without passing in front of it. Therefore in order to determine if a solar eclipse occurs on a particular New Moon one needs determine the positions of the ascending and descending nodes of the orbit of the Moon. 

The ascending and descending nodes are treated by the Hindu astronomers as though they are celestial bodies. They are given names, Rahu and Ketu and their positions, true longitudes, are calculated by the same algorithm that the positions of the other bodies are calculate. Rahu and Ketu are called the shadow planets, as they are not real planets but they move as though they were. Learn more about Rahu and Ketu . I have also prepared an animation to help you better understand the orbit of the Moon, nodes, and how eclipses come about.

Knowing the true longitudes of the Sun and Moon at the time of the New Moon (they are identical of course) and the true longitudes of Rahu and Ketu allows one to calculate the celestial latitude of the Moon at this time. It is determined by knowing the difference in longitude between the nodes and the Moon and the amount by which the plane of the orbit of the Moon is tilted with respect to the plane of the path of the Sun. The Surya Siddhanta gives prescriptions for performing this calculation. 

The Surya Siddhanta gives numbers for the distances to the Sun and the Moon. With these and the celestial latitude of the Moon (in ecliptic coordinates) the Surya Siddhanta then tells how to calculate the location of the shadow of the Moon relative to the Earth. From this one can determine if the shadow hits the Earth and if so where. Knowing the distances to the Sun and Moon, and the sizes of the Sun and Moon, the Surya Siddhanta then describes how to calculate the angular size of the Sun and Moon at the time of the eclipse. This determines whether the eclipse will be a total eclipse or an annular eclipse. 

Comments on the Calculations

The times of solar eclipses predicted by the Surya Siddhanta algorithms compare well with modern calculations considering the simplicity of the methods. They are often within 30 minutes. This is not an acceptable error, however,  if you are using these calculations to determine the location on Earth from which one will see the total eclipse. The Earth rotates by 7.5 degrees in 30 minutes which means, if there is a 30 minute error, the shadow of the Moon on the Earth will be 7.5 degrees in latitude different from where the Surya Siddhanta predicts it will be at the moment of the New Moon. At the equator this amounts to about 500 miles. The accuracy is sufficient however to determine the approximate time of day of an eclipse and one would be amply warned to start looking for the eclipse.

The calculation of the celestial latitude of the Moon at the time of the New Moon is close enough to allow one to get a good idea as to whether an eclipse will occur or not and also in which hemisphere of the Earth it might be total.

I have also written a Brief History of the Surya Siddhanta. 

I have written a Javascript program to carry out the calculations for the circumstances of solar eclipses according to the prescriptions of the Surya Siddhanta. It is found on my  Circumstances of a Solar Eclipse page.