Question

why is total solar eclipse is not frequent?

Answer 1

Since the distance to the moon varies, the width of the path of totality differs from one eclipse to another.  This width will change even during a single eclipse, because different parts of the Earth lie at different distances from the moon and also because of geometrical effects as the shadow falls at an oblique angle onto the Earth's surface.

In calculating a solar eclipse, one of the first steps is to determine the shadow's relation to the "fundamental plane," which passes through the Earth's center and is perpendicular to the moon-sun line. The path of the axis of the shadow across this plane is virtually a straight line. It is from this special geometry, that the intersection of the moon's dark shadow cone with the rotating spheroid of our Earth must be worked out, using lengthy procedures in trigonometry. To say the least, these factors can make the calculations quite involved (although today's high-speed PCs can effortlessly crunch the numbers, making the task much easier). 

In their classical textbook "Astronomy" (Boston, 1926), authors H.N. Russell, R.S. Dugan and J.Q. Stewart noted that:

"Since the track of a solar eclipse is a very narrow path over the earth's surface, averaging only 60 or 70 miles in width, we find that in the long run a total eclipse happens at any given station only once in about 360 years."

More recently, Jean Meeus of Belgium, whose special interest is spherical and mathematical astronomy, recalculated this figure statistically on an HP-85 microcomputer and found that the mean frequency for a total eclipse of the sun for any given point on the Earth's surface is once in 375 years. A value that is very close to the figure that Russell, Dugan and Stewart arrived at. [Amazing Solar Eclipse Photos]

This NASA chart shows the shadow path of the "ring of fire" annular solar eclipse of April 28-29, 2014. The chart, prepared by NASA eclipse expert Fred Espenak, also lists times for the maximum eclipse, as well as start and stop times.Credit: NASA Goddard Space Flight Center/Fred Espenak

Without retracing these computations, there is perhaps another way to check the validity of these answers. In the table below, is a listing of 25 cities. Twenty-three are in North America, plus two others: Honolulu, on the Hawaiian Island of Oahu, and Hamilton, the capital of Bermuda. Using two computer programs designed to scan through the centuries for eclipses, I first searched for the date of the most recent total solar eclipse that was visible from each city, then searched for the date when the next total eclipse for that city would take place.

The average number of years between eclipses turned out to be nearly 534 years. Considering our relatively small survey of 25 cities, this is reasonably close to the once-in-almost four-century rule.

A botched opportunity

All of us who enjoy solar eclipses should be indebted to those astronomers who pioneered doing these extensive calculations; otherwise we would not know exactly where to position ourselves for the big event. Prussian astronomer Friedrich Bessel introduced a group of mathematical formulas in 1824 (now called "Besselian Elements") that greatly simplified the calculation of the position of the sun, moon and Earth. [Related: The World's 1st Televised Solar Eclipse]

It is too bad that Bessel's procedures were not available in the late 18th century, when Samuel Williams, a professor at Harvard, led an expedition to Penobscot Bay, Maine, to observe the total solar eclipse of Oct. 27, 1780. As it turned out, this eclipse took place during the Revolutionary War and Penobscot Bay lay behind enemy lines. Fortunately, the British granted the expedition safe passage, citing the i