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 Ocean 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) 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 the 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 the Earth's surface 234,900 miles (378,030 kilometers).

This means that when the Moon passes directly in front of the Sun, its shadow will usually miss the Earth by some 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 the Earth's surface, on account of the Moon's elliptical orbit.

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 the Earth's surface. It's only near the middle of the eclipse track that the tip of the umbra barely scrapes the 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 150 A.D.), 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 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."

Eclipse Terms

Partial eclipse: The Moon covers only part of the Sun.

Total eclipse: The Moon covers the entire disk of the Sun along a narrow path across the Earth.

Annular eclipse: The Moon is too far from Earth to completely cover the Sun. A thin ring of the Sun's disk surrounds the Moon.

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.

Double check

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.

But it should first be stressed that the nearly four-century wait is merely a statistical average. Indeed, over a much shorter span of time, the paths of different eclipses can sometimes criss-cross over a specific place, so in some cases the wait might not be so long at all. In fact, a forty-mile stretch of the Atlantic coast of Angola, just north of Lobito, experienced a total solar eclipse on June 21, 2001 and was treated to another on Dec. 4, 2002, after less than 18 months!

On the other hand, as Mr. Meeus recently discovered, some spots on the Earth's surface may not see a total solar eclipse for 36 centuries (" . . . though this must be exceedingly rare," he notes).

On our listing of 25 selected cities, how close would we come to the computed mean-frequency of nearly 400-years between total eclipses?

Here is the list:

Location

Most recent totality

Next scheduled totality

Years between eclipses

Anchorage, AK

1943, February 4

2399, August 2*

456.5

Atlanta, GA

1778, June 24

2078, May 11

299.9

Boston, MA

1959, October 2

2079, May 1

119.6

Calgary, Alberta

1869, August 7

2044, August 23

175.0

Chicago, IL

1806, June 6*

2205, July 17

399.1

Dallas, TX

1623, October 23

2024, April 8

401.5

Denver, CO

1878, July 29

2045, August 12

167.0

Halifax, Nova Scotia

1970, March 7

2079, May 1

109.1

Hamilton, Bermuda

1532, August 30**

2352, February 16

819.5

Honolulu, HI

1850, August 7

2252, December 31

402.4

Houston, TX

1259, October 17**

2200, April 14

940.5

Las Vegas, NV

1724, May 22

2207, November 20

483.5

Los Angeles, CA

1724, May 22

3290, April 1

1,565.9

Mexico City, Mexico

1991, July 11

2261, December 22

270.4

Miami, FL

1752, May 13*

2352, February 16

599.8

Montreal, Quebec

1932, August 31*

2024, April 8*

91.6

New Orleans, LA

1900, May 28

2078, May 11

178.0

New York, NY

1925, January 24*

2079, May 1

154.3

Phoenix, AZ

1806, June 16

2205, July 17

399.1

St. Louis, MO

1442, July 7**

2017, August 21

575.1

San Francisco, CA

1424, June 26**

2252, December 31

828.5

Seattle, WA

1860, July 18

2645, May 17

784.8

Toronto, Ontario

1142, August 22**

2144, October 26

1,002.2

Washington, DC

1451, June 28**

2200, April 14

748.8

Winnipeg, Manitoba

1979, February 26

3356, September 16

1,377.6

A single asterisk (*) denotes that either the northern or southern limit of the Moon's umbral shadow only grazes a specific city; only part of that metropolitan area will see a total eclipse while the other part sees a partial eclipse. A double asterisk (**) indicates a date when the now-defunct Julian Calendar was in effect.

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 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.

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 interest of science above political differences.

And yet in the end, it was all for naught.

Williams apparently made a fatal error in his computations and inadvertently positioned his men at Islesboro - outside the path of totality - likely finding this out with a heavy heart when the waning crescent of sunlight slid completely around the dark edge of the Moon and started thickening!

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Joe Rao serves as an instructor and guest lecturer at New York's Hayden Planetarium. He writes about astronomy for The New York Times and other publications, and he is also an on-camera meteorologist for News 12 Westchester, New York.