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Shadow Painting the Globe.

Stay long enough in one spot and you'll be in the path of a total eclipse of the Sun.

In the course of online discussion about the August 11th total solar eclipse, Govert Schilling of Utrecht, the Netherlands, posted the following message on an electronic mailing list last December 24th:

Suppose I would take a globe and paint the path of totality of every eclipse black. How many eclipses would I need before every part of the earth is black? In other words, how long would it take for every single spot on the earth to experience at least one total solar eclipse? (Or, put differently, are there places on the surface of earth where no total solar eclipse has occurred for, say, the past 3000 and the coming 3000 years?)

A fascinating quandary, indeed, and I accepted the challenge. Because the problem is very difficult, maybe even impossible, to solve theoretically, I chose a brute-force solution, consisting of calculating every solar-eclipse path until the globe is "fully painted."

I had considered plotting successive eclipse tracks on a map, but this seemed problematic. A very small region escaping from all total eclipses could easily remain unnoticed. Moreover, the northern and southern limits of totality, when plotted on a computer screen or drawn on paper, are not infinitely thin lines. Their thickness could cover places just outside, but very close to, the actual path where an eclipse is total.

I preferred a completely mathematical approach, considering a large number of points distributed as uniformly as possible over the Earth's surface. I chose sites at every integer degree of geographic longitude (0[degree sign] to 359[degree sign]) on each 1[degree sign] parallel of latitude (from 89[degree sign] north to 89[degree sign] south). I did not need to worry about the two poles, because a separate calculation had already shown that there was a total solar eclipse visible at the North Pole on July 6, 1815, and that another will be visible from the South Pole on January 16, 2094.

Now I had a spherical grid of 179 by 360, or 64,440 points. I wanted to construct a character string consisting of 64,440 zeros, one for each of the points on the globe. A zero would change into a one when the location it represented experienced totality. However, the computer language I used, QuickBASIC, does not accept such long strings; the maximum allowed length is 32,767 characters. Fortunately, half of 64,440 is just a little smaller than this maximum, so I performed the calculation for each hemisphere separately.

A requirement to calculate anything about a given solar eclipse (such as local circumstances or points on the northern and southern limits of totality) is having the so-called Besselian elements. Since these values had already been accurately calculated for all eclipses from 2000 B.C. to A.D. 3400, I could use these data for the computer program I wrote specially to solve the painted-globe problem.

First, I considered all total and annular-total eclipses from A.D. 1901 to 2200. For each eclipse, the circumstances at each of the 32,400 sites were calculated. After three centuries, 11,849 points (37 percent) remained zeros.

I continued the calculation by considering the earlier period from A.D. 1401 to 1900, but only for those points that had not yet been visited by a total eclipse. This allowed the calculation to become more and more rapid, as fewer zeros remained in the string. After this second run, 2,618 empty points lingered. I pressed further. After having considered all the years from 800 B.C. to A.D. 2200, just four empty points endured. Extending to A.D. 2800 finally covered those stragglers; now the long character string consisted completely of ones. The total calculation for the 32,400 points in the Northern Hemisphere had taken 90 minutes, very satisfying on my 166-megahertz Pentium processor.

Nevertheless, I was still only half done. I repeated the calculation for the Southern Hemisphere, again for the years from 800 B.C. to A.D. 2800. Five locations remained vacant. At this point, it appeared that some spots on the Earth's surface may not be under the Moon's shadow for 36 centuries, though this must be exceedingly rare. It is known that for a given location on the globe the mean frequency is about one total solar eclipse every 375 years; however, the actual distribution is very irregular. A small region in New Guinea experienced two total eclipses within 18 months in 1983-84!

The remaining five empty points in the Southern Hemisphere were finally covered by including 1400 to 800 B.C. Thus each of the 64,440 points was covered by at least one total solar eclipse between 1400 B.C. and A.D. 2800.

Of course, this grid of 64,440 points distributed over the globe does not represent all places on Earth's surface. Because the points along each parallel of latitude were separated by 1[degree sign] in longitude, they were much closer to each other at high latitudes than near the equator. Indeed, at the equator 1[degree sign] of longitude corresponds to 111 kilometers; this distance is reduced to only 2 km at 89[degree sign] latitude. Consequently, some remaining places were up to 79 km from the grid points considered.

I repeated the whole calculation for other points, so that the separation in both longitude and latitude was now 11/42[degree sign]. After again considering the period from 1400 B.C. to A.D. 2800, there remained two empty points: neighboring spots in the southern Pacific Ocean at latitude 62[degree sign] 30[cent] south, and west longitudes 160[degree sign] 30[cent] and 161[degree sign]. These two stubborn places were finally covered by extending the calculation to A.D. 3158.

In the end, all 258,482 points I considered on the globe (including the two poles) were covered at least once by total solar eclipse tracks during 46 centuries - a far cry from the 50,000 to 500,000 years guessed by one of the readers of Schilling's message. Another reader even thought that it would take millions of years to paint the globe completely.

Nevertheless, there's still plenty of room between those quarter-of-a- million points. But from the foregoing we may assume with confidence that these regions are likewise covered at least once by the Moon's umbra in a period of not more than 4,000 or 5,000 years.

Hey, Not So Fast . . .

Readers familiar with eclipse calculation may wonder how I accounted for the quantity called "Delta T" in this project.

Delta T (DT) is a very important quantity in the calculation of solar eclipses and of occultations of stars by the Moon. It is the difference, at a given instant, between Universal Time and the uniform time scale known as Dynamical Time. While the latter is defined by atomic clocks, Universal Time is based on the rotation of the Earth around its axis. As this rotation gradually slows, DT increases at an ever faster rate the farther we are away from A.D. 1900, into the future as well as into the past (see the February issue, page 53).

We need accurate knowledge of DT for the calculation of exact times of an eclipse, but it is also of utmost importance for determining exactly where the path of totality falls on Earth's surface and the magnitude of the eclipse (how much of the Sun's disk is covered by the Moon) at a given place. Differences arise because the Earth rotates during the DT interval and 1[degree sign] of longitude corresponds to 4 minutes of time. Therefore, if the adopted value for DT is, say, 4 minutes too large, the calculated positions of the centerline and limits will be 1[degree sign] of longitude too far to the east. Latitude is not affected.

The situation is all the more complicated because DT is not increasing regularly. It undergoes unpredictable fluctuations. For instance, from 1968 to 1980 DT increased by almost exactly one second per year, but from December 1997 to December 1998 the increase was only 0.5 second. Consequently, it's impossible to predict the exact time a solar eclipse in A.D. 2100 begins at a given place even if we make use of a perfectly accurate lunar ephemeris.

The value of DT is known with sufficient accuracy from about A.D. 1620 to 2000. For years outside this period, I used formulas recently issued by Jean Chapront of the Bureau des Longitudes, Paris. The reader may object that some of the eclipses I considered "total" at certain points were or will be, in fact, partial because the adopted value for DT was "wrong." I don't dispute this. On the other hand, some other eclipses, which according to my calculations were only partial, certainly were or will be total.

Since I found at least one total eclipse during 1400 B.C. to A.D. 3158 for all of those 258,482 points, we may be sure that they will have a total eclipse at least once during about 40 or 50 centuries for any value we adopt for DT. To validate this assumption, I repeated the whole calculation using zero for DT for all years. Such a value is absolutely wrong for epochs in the distant past and future. Starting from 1400 B.C., I found that by A.D. 2870 all except one of the considered 258,482 places on the Earth's surface are painted by a total eclipse. Going back to 1638 B.C. covered it.

We may conclude that after about 45 centuries the whole Earth is painted by tracks of total solar eclipses. However, it takes only 20 centuries to paint more than 99 percent of the globe, and 30 centuries to coat more than 99.9 percent. So the second half of those 45 centuries is needed to ascertain that the last few remaining patches not yet touched by the lunar shadow are finally covered too.

Jean Meeus, a pioneer of astronomical calculations with personal computers, is a frequent contributor to Sky & Telescope.
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Title Annotation:solar eclipse
Author:Meeus, Jean
Publication:Sky & Telescope
Date:Aug 1, 1999
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