The frequency of total and annular solar eclipses for a given place

The following text was first published in the Journal of the British Astronomical Association, Vol. 92, No. 3, pages 124-126 (April 1982).

A classical question concerning solar eclipses is the following one: How often can a total or an annular eclipse of the Sun be expected at a given point on the Earth's surface? In their classical textbook Astronomy (Boston, 1926), H.N. Russell, R. S. Dugan and J. Q. Stewart write (vol. I, page 227):

"Solar eclipses that are total somewhere or other on the earth's surface are not very rare, averaging one for about every year and a half. But at any given place the case is very different. 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."

These authors, however, give no details about how this mean frequency of one total eclipse every 360 years has been found. The theoretical calculation is not easy, by reason of the large variations of the length and width of the path from one eclipse to another. Even for a given eclipse, the width can vary widely along the path. For the total eclipse of 1981 July 31, for instance, the width was 57 kilometers at the beginning of the path, reached a maximum value of 108 km near the middle, and then decreased to 51 km towards the end.

In order to recompute the answer given by Russell, Dugan and Stewart, and to find the mean frequency for annular eclipses too, we attacked the problem statistically. For every solar eclipse in the period A.D. 1700 to 2299, the local circumstances at 408 'standard points' on the Earth's surface were calculated. The following standard points have been chosen : the points at latitudes +80°, +70°, +60°, etc., to -80° on the 24 meridians of longitudes +180°, +165°, +150°, etc., to -165°.

The calculation, made on an HP-85 microcomputer, proceeded automatically. After calculating the Besselian elements of an eclipse, the machine examined for each of the 408 standard points whether or not there was a total or annular eclipse there, after which the next eclipse was calculated, etc. A total or annular eclipse at a standard point was considered to be visible if, and only if, at the time of maximum eclipse, the geometric altitude of the center of the Sun's disk was positive.

A part of the computer outprint looks as shown in the box on the next page. The second column gives the longitude (measured positively westward from the meridian of Greenwich), and the third column the latitude.

As mentioned above, the investigation was made over a time-period of six centuries, this value having been chosen in order to avoid any possible effect with a period of six centuries. It is well known, for instance, that the mean frequency of total lunar eclipses varies with time, the period being 586 years. [See Chapter 16 and, for solar eclipses, see Chapter 10].

For the period 1700-2299, we found 665 total and 1208 annular eclipses at standard points, distributed over the various latitudes as indicated in Table 13. A, columns (2) and (3). These values are plotted in Figure 13.a — see the two solid lines.

There is an evident latitude effect in the distribution of these total and annular eclipses. For instance, we see at once that total eclipses are less frequent in the zone of southern latitudes 30° to 80° than in the northern hemisphere. Annular eclipses are more frequent at latitudes 50°S to 80°S than near the equator.

These distributions are explained by the combination of the following effects:

(i) in the equatorial regions, total eclipses are more frequent, and annular eclipses are less frequent, than at higher latitudes because the equatorial regions are generally closer to the Moon, whose disk thus appears larger;

(ii) on the other hand, the lunar shadow moves in approximately the same direction as the rotating surface of the Earth. Therefore, the mean duration of solar eclipses at a given place will be longer, and their mean frequency less, than it would be for a non-rotating Earth. This is connected with the fact that the probability for the Sun to be eclipsed (expressed, for instance, in minutes per century) remains independent of the speed of rotation of the Earth. As a consequence, near the equator a solar eclipse will be somewhat rarer;

(iii) in the summer months, the Sun is for a longer time above the horizon, increasing the frequency of visible eclipses there. In the northern hemisphere, this occurs around the time when the Earth is near the aphelion of its orbit, resulting in a smaller-than-average solar disk, thus favouring the occurrence of a total eclipse, and disfavouring that of an annular one; the opposite holds for the southern hemisphere.

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