Page:EB1911 - Volume 08.djvu/921

 Recurrence of Remarkable Eclipses.

From the property of the Saros it follows that eclipses remarkable for their duration, or other circumstances depending on the relative positions of the sun and moon, occur at intervals of one saros (18 y. 11 d.). Of interest in this connexion is the recurrence of total eclipses remarkable for their duration. The absolute maximum duration of a total eclipse is about 7′ 30″; but no actual eclipse can be expected to reach this duration. Those which will come nearest to the maximum during the next 500 years belong to the series numbered 4 and 6 and in the list which precedes. These occurring in the years 1937, 1955, &c., will ultimately fall little more than 20″ below the maximum. But the series 4, though not now remarkable in this respect, will become so in the future, reaching in the eclipse of June 25, 2150, a duration of about 7′ 15″ and on July 5, 2168, a duration of 7′ 28″, the longest in human history. The first of these will pass over the Pacific Ocean; the second over the southern part of the Indian Ocean near Madras.

All the national annual Ephemerides contain elements of the eclipses of the sun occurring during the year. Those of England, America and France also give maps showing the path of the central line, if any, over the earth’s surface; the lines of eclipse beginning and ending at sunrise, &c., and the outlines of the shadow from hour to hour. By the aid of the latter the time at which an eclipse begins or ends at any point can be determined by inspection or measurement within a few minutes.

V. Methods of computing Eclipses of the Sun.

The complete computation of the circumstances of an eclipse ab initio requires three distinct processes. The geocentric positions of the sun and moon have first to be computed from the tables of the motions of those bodies. The second step is to compute certain elements of the eclipse from these geocentric positions. The third step is from these elements to compute the circumstances of the eclipse for the earth generally or for any given place on its surface. The national Astronomical Ephemerides, or “Nautical Almanacs,” give in full the geocentric positions of the sun and moon from at least the early part of the 19th century to an epoch three years in advance of the date of publication. It is therefore unnecessary to undertake the first part of the computation except for dates outside the limits of the published ephemerides, and for many years to come even this computation will be unnecessary, because tables giving the elements of eclipses from the earliest historic periods up to the 22nd century have been published by T. Ritter von Oppolzer and by Simon Newcomb. We shall therefore confine ourselves to a statement of the eclipse problem and of the principles on which such tables rest.

Two systems of eclipse elements are now adopted in the ephemerides and tables; the one, that of F. W. Bessel, is used in the English, American and French ephemerides, the other—P. A. Hansen’s—in the German and in the eclipse tables of T. Ritter von Oppolzer. The two have in common certain geometric constructions. The fundamental axis of reference in both systems is the line passing through the centres of the sun and moon; this is the common axis of the shadow cones, which envelop simultaneously the sun and moon as shown in figs. 1, 2, 3. The surface of one of these cones, that of the umbra, is tangent to both bodies externally. This cone comes to a point at a distance from the moon nearly equal to that of the earth. Within it the sun is wholly hidden by the moon. Outside the umbral cone is that of the penumbra, within which the sun is partially hidden by the moon. The geometric condition that the two bodies shall appear in contact, or that the eclipse shall begin or end at a certain moment, is that the surface of one of these cones shall pass through the place of the observer at that moment. Let a plane, which we call the fundamental plane, pass through the centre of the earth perpendicular to the shadow axis. On this plane the centre of the earth is taken as an origin of rectangular co-ordinates. The axis of Z is perpendicular to the plane, and therefore parallel to the shadow axis; that of Y and X lie in the plane. In these fundamental constructions the two methods coincide. They differ in the direction of the axis of Y and X in the fundamental plane. In Bessel’s method, which we shall first describe, the intersection of the plane of the earth’s equator with the fundamental plane is taken as the axis of X. The axis of Y is perpendicular to it, the positive direction being towards the north. The Besselian elements of an eclipse are then:—x, y, the co-ordinates of the shadow axis on the fundamental plane; d, the declination of that point in which the shadow axis intersects the celestial sphere;, the Greenwich hour angle of this point; , the radius of the circle, in which the penumbral or outer cone intersects the fundamental plane; and  ′, the radius of the circle, in which the inner or umbral cone intersects this plane, taken positively when the vertex of the cone does not reach the plane, so that the axis must be produced, and negatively when the vertex is beyond the plane.

Hansen’s method differs from that of Bessel in that the ecliptic is taken as the fundamental plane instead of the equator. The axis of X on the fundamental plane is parallel to the plane of the ecliptic; that of Y perpendicular to it. The other elements are nearly the same in the two theories. As to their relative advantages, it may be remarked that Hansen’s co-ordinates follow most simply from the data of the tables, and are necessarily used in eclipse tables, but that the subsequent computation is simpler by Bessel’s method.

Several problems are involved in the complete computation