The Sun: Ruler, Fire, Light, and Life of the Planetary System

As regards the least possible number of eclipses, it is obvious that, as there must be two eclipse-seasons in the year, and at least one eclipse in each, we cannot have less than two eclipses in the course of a year. In this case each eclipse is solar and central.

As regards intermediate cases, we need make no special inquiry. Many combinations are possible. The most common case is that in which there are four eclipses-two solar and two lunar. Further, it may be noticed that, whatever the number of eclipses, from two to seven inclusive, there must always be two solar eclipses at least in each year.

And now we may turn from the particular mode of con sidering eclipses, which we have thus far followed. There is another by which we might have arrived at similar results almost as readily. We might, instead of viewing the Earth and Moon in imagination from the Sun, have traced the course of the Sun and Moon around the heavens. Both methods of dealing with eclipses are employed by astronomers, the method used in the preceding pages corresponding to what is termed the method of projection, the other to the method of direct calculation from the celestial ordinates of the Sun and Moon. For our present purpose, however, one method is all that need be considered.*

And now, before closing this essay, I will consider in the usual manner the nature of the Moon's shadow-cone in solar eclipses, and of the Earth's shadow-cone in lunar eclipses.

Eclipses of both sorts may be regarded as illustrated together in fig. 105. Here E is the Earth, and the Moon is shown in two In the Popular Science Review for July 1868 I have exhibited t line of reasoning by which the results deduced above can be obtained by considering the apparent motions of the Moon and Sun around the celestial vault. By the artifice of regarding these motions as taking place on a sphere which can be viewed from without, and shifted or rotated so as to illustrate the various relations dealt with, the whole subject may be very conveniently discussed. The student of astronomy does well to examine all such questions by as many independent methods as possible; but in the present treatise there is not space for a complete investigation of the theory of eclipses on the second plan.

places, at м, directly between the Earth and Sun, and at the point opposite м, in the heart of the Earth's shadow-cone. The true geometrical shadows of the Earth and Moon are shown black, the true geometrical penumbræ are shaded. It must be

FIG. 105.

understood, however, that the vertical dimensions have had to be exaggerated; the angle at c ought properly to contain. but about half a degree. Such an angle could not be conveniently employed in illustrating our subject.

ce and ce' produced touch the Sun's globe; so also do the boundaries of the Moon's black shadow. The boundaries me and me' touch the Sun's globe after crossing; so also do the boundaries of the Moon's penumbra.

The distance EC is variable, being as great as 870,300 miles when the Earth is in aphelion, and as small as 843,300 miles when the Earth is in perihelion. The Moon's orbit round the Earth has a mean radius of 238,770 miles. Thus the Earth's shadow extends about three and a half times as far from the Earth as the Moon's orbit.

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The end of the Moon's shadow is represented on a larger scale (and without the penumbra) in figs. 106 and 107. In fig. 106 the shadow's extreme point c does not reach the Earth; in fig 107 it passes far beyond the Earth. The two

figures represent the extreme possible range of the shadowpoint either way. In fig. 106 there is shown beyond c a shaded anti-cone. From any point within this the Sun will be annularly eclipsed. Thus the section at a a' includes that part of the Earth whence at the moment an annular eclipse of the Sun is visible. On the other hand, the section u u' in fig. 107 includes the part of the Earth whence the Sun is totally eclipsed. It is important to notice that the greatest possible width of u u' is about 173 miles.†

Now in fig. 105 the points m m m' m' may be supposed to lie on the Moon's orbit seen in plano. If we suppose this orbit not to lie in the plane of the paper, but tilted at an angle of 5° (or rather to an angle as much larger than 5° as the shadow-angle at c is increased beyond its true value of half a minute), then by conceiving the whole figure turned about CM, until the Moon's orbit is seen sideways, this orbit, according to the direction in which the tilt existed, would exhibit a shape resembling one of those shown in fig. 98 (only more open on account of the exaggeration of the tilt. And, further, if we could watch from such a standpoint that is (obviously) a standpoint towards which or from which the Earth was travelling during a period of about 346 days, we should see the Moon's orbit passing in

It is obvious that the Sun will seem to be annularly eclipsed from any point within this anti-cone; for lines drawn from any such point to the circle on the Moon in which the shadow-cone begins will form a cone (right or oblique) which, beyond the Moon, will be wholly within the extension of the shadow-cone's geometrical surface. Hence a portion of the Sun's globe must lie outside and around this inner cone, This portion will be visible, therefore, from the vertex of this inner cone (the point within c a a') as a ring of light, whose boundaries will be concentric or eccentric according as the inner cone is right or obliquein other words, according as its vertex lies or not on the axis of the shadow-cone produced.

The extreme limits of central solar eclipses result when, first, the Sun's diameter has its greatest value, 32′ 36′′-4, and the Moon's its least, 29′ 21′′-9, in which case a ring of light 1' 37"-2 wide remains; and, secondly, when the Sun's diameter has its least and the Moon's its greatest value, in which case the Moon's disc overlaps the Sun's by 59"-6, and the Sun continues for several minutes totally eclipsed.

orderly succession though all such phases as are exhibited in fig. 98.

Now, clearly, since for a lunar eclipse to occur the Moon must enter the cone m' c m' opposite m, while for the occurrence of a solar eclipse the Moon must enter this cone opposite m', lunar eclipses must on the whole be less numerous than total ones, for the cone is appreciably narrower opposite m than opposite m'. It is, however, also obvious, that when the Moon is in the Earth's shadow she is eclipsed as viewed from a whole hemisphere of the Earth, whereas when the Moon casts a shadow on the Earth the Sun is only eclipsed as viewed from parts of the Earth which are traversed by that shadow. The extent of such regions falls very far short* of half the Earth's surface. Hence solar eclipses are less frequent at any given station than lunar ones.

But it is worthy of notice that, if penumbral lunar eclipses are included, more lurar eclipses than solar ones occur in any long period of time. For, clearly, the section of the penumbral cone opposite m is greater than that of the cone mc m' opposite m', since both cones enclose the Sun, but the vertex of the former is nearer to the Sun than that of the latter, and therefore the vertical angle of the former cone is the greater.

It is convenient to notice in conclusion, that in every period of 21,600 lunations there are on the average 4,072 solar eclipses and 2,614 lunar eclipses, not counting penumbral ones. If penumbral lunar eclipses are included, the number rises to 4,231. In all there are (on the average) 6,686 lunar and solar eclipses in the course of every 21,600 lunations, the total rising to 8,303 when penumbral lunar eclipses are added.

*The extent of the region actually in shadow at a given moment will vary in different eclipses, and at different hours during the same eclipse. It will be least of all when the Moon's real shadow has its greatest possible extent (i. e. when the Sun is in aphelion, the Moon in perihelion, and both as near the zenith as they can be compatibly with those conditions). At such a time the edge of the penumbra forms a circle (approximately) having a radius equal to the Moon's diameter diminished by about 86 miles, or a radius of about 2,078 miles. The extent of the Earth's surface then in shadow is easily shown to be about one 37th part of the whole surface of the Earth.

TABLE I.

Principal Elements of the Sun.

Equatorial horizontal parallax at mean distance from the

Earth