Eclipse math can be totally mind-boggling

It’s not likely that many people will see the total phase of Friday's solar eclipse. Totality will be visible only from the middle of the South Pacific and only for, at the very most, 42 seconds.

Only from an aircraft or the deck of a ship positioned precisely within the very narrow (17-mile-wide, 27-kilometer-wide) path of totality would observers get a brief glimpse of a completely obscured sun, one of nature’s great spectacles.

It is a popular misconception that the phenomenon of a total eclipse of the sun is a rare occurrence. Quite the contrary, about once every 18 months, on average, a total solar eclipse is visible from some place on Earth’s surface. That’s two totalities for every three years.

However, seeing a total eclipse of the sun from a specific location is another story altogether.

Shadowy details
On the average, the length of the moon’s shadow at new moon is 232,100 miles (373,530 kilometers), and the distance to the nearest point of Earth’s surface is 234,900 miles (378,030 kilometers).

This means that when the moon passes directly in front of the sun, its shadow will usually miss Earth by  2,800 miles (4,500 kilometers), and the eclipse will merely be annular, with a dazzling ring of sunlight still visible around the moon’s silhouette.

Of course we all know that total eclipses do occur, because the new moon’s distance can vary between 217,730 miles (350,400 kilometers) and 247,930 miles (399,000 kilometers) from Earth’s surface, on account of the moon’s elliptical orbit.

INTERACTIVE What causes a solar eclipse?

As it turns out, the April 8 eclipse is one of those unusual hybrids where the eclipse is total over only a part of its path and annular throughout the rest. Near and at the ends of the path, the distance to the moon is too great (owing to the curvature of the earth) for its dark cone of shadow (called the umbra) to touch Earth’s surface. It’s only near the middle of the eclipse track that the tip of the umbra barely scrapes Earth, changing the character of the eclipse from annular into a total. Then, as the track approaches the Central American coast, the umbra moves off the earth’s surface and the eclipse switches back to annular.

Of all solar eclipses, about 35 percent are partial; 32 percent annular; 28 percent total; but only 5 percent are hybrids.

So now let’s return to our original question: How often a total eclipse can be seen from a specific point on the Earth’s surface?

The science of prediction
Predicting the details of a solar eclipse requires not only a fairly good idea of the motions of the sun and moon, but also an accurate distance to the moon and accurate geographical coordinates. Rough determinations of eclipse circumstances became possible after the work of Claudius Ptolemy (around A.D. 150), and diagrams of the eclipsed sun have been found in medieval manuscripts and in the first books printed about astronomy.

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 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 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 Earth’s surface is once in 375 years — a value that is very close to the figure arrived at by Russell, Dugan and Stewart.

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