Letters on Astronomy / in which the Elements of the Science are Familiarly Explained in Connection with Biographical Sketches of the Most Eminent Astronomers

by Denison Olmsted

It is found, by calculation, that the length of the moon's shadow, on an average, is just about sufficient to reach to the earth; but the moon is sometimes further from the earth than at others, and when she is nearer than usual, the shadow reaches considerably beyond the surface of the earth. Also, the moon, as well as the earth, is at different distances from the sun at different times, and its shadow is longest when it is furthest from the sun. Now, when both these circumstances conspire, that is, when the moon is in her perigee and along with the earth in her aphelion, her shadow extends nearly fifteen thousand miles beyond the centre of the earth, and covers a space on the surface one hundred and seventy miles broad. The earth's shadow is nearly a million of miles in length, and consequently more than three and a half times as long as the distance of the earth from the moon; and it is also, at the distance of the moon, three times as broad as the moon itself.

An eclipse of the sun can take place only at new moon, when the sun and moon meet in the same part of the heavens, for then only can the moon come between us and the sun; and an eclipse of the moon can occur only when the sun and moon are in opposite parts of the heavens, or at full moon; for then only can the moon fall into the shadow of the earth.

Fig. 42.

The nature of eclipses will be clearly understood from the following representation. The diagram, Fig. 42, exhibits the relative position of the sun, the earth, and the moon, both in a solar and in a lunar eclipse. Here, the moon is first represented, while revolving round the earth, as passing between the earth and the sun, and casting its shadow on the earth. As the moon is here supposed to be at her average distance from the earth, the shadow but just reaches the earth's surface. Were the moon (as is sometimes the case) nearer the earth her shadow would not terminate in a point, as is represented in the figure, but at a greater or less distance nearer the base of the cone, so as to cover a considerable space, which, as I have already mentioned, sometimes extends to one hundred and seventy miles in breadth, but is commonly much less than this. On the other side of the earth, the moon is represented as traversing the earth's shadow, as is the case in a lunar eclipse. As the moon is sometimes nearer the earth and sometimes further off, it is evident that it will traverse the shadow at a broader or a narrower part, accordingly. The figure, however, represents the moon as passing the shadow further from the earth than is ever actually the case, since the distance from the earth is never so much as one third of the whole length of the shadow.

It is evident from the figure, that if a spectator were situated where the moon's shadow strikes the earth, the moon would cut off from him the view of the sun, or the sun would be totally eclipsed. Or, if he were within a certain distance of the shadow on either side, the moon would be partly between him and the sun, and would intercept from him more or less of the sun's light, according as he was nearer to the shadow or further from it. If he were at c or d, he would just see the moon entering upon the sun's disk; if he were nearer the shadow than either of these points, he would have a portion of this light cut off from his view, and more, in proportion as he drew nearer the shadow; and the moment he entered the shadow, he would lose sight of the sun. To all places between a or b and the shadow, the sun would cast a partial shadow of the moon, growing deeper and deeper, as it approached the true shadow. This partial shadow is called the moon's penumbra. In like manner, as the moon approaches the earth's shadow, in a lunar eclipse, as soon as she arrives at a, the earth begins to intercept from her a portion of the sun's light, or she falls in the earth's penumbra. She continues to lose more and more of the sun's light, as she draws near to the shadow, and hence her disk becomes gradually obscured, until it enters the shadow, when the sun's light is entirely lost.

As the sun and earth are both situated in the plane of the ecliptic, if the moon also revolved around the earth in this plane, we should have a solar eclipse at every new moon, and a lunar eclipse at every full moon; for, in the former case, the moon would come directly between us and the sun, and in the latter case, the earth would come directly between the sun and the moon. But the moon is inclined to the ecliptic about five degrees, and the centre of the moon may be all this distance from the centre of the sun at new moon, and the same distance from the centre of the earth's shadow at full moon. It is true, the moon extends across her path, one half her breadth lying on each side of it, and the sun likewise reaches from the ecliptic a distance equal to half his breadth. But these luminaries together make but little more than a degree, and consequently, their two semidiameters would occupy only about half a degree of the five degrees from one orbit to the other where they are furthest apart. Also, the earth's shadow, where the moon crosses it, extends from the ecliptic less than three fourths of a degree, so that the semidiameter of the moon and of the earth's shadow would together reach but little way across the space that may, in certain cases, separate the two luminaries from each other when they are in opposition. Thus, suppose we could take hold of the circle in the figure that represents the moon's orbit, (Fig. 42, page 197,) and lift the moon up five degrees above the plane of the paper, it is evident that the moon, as seen from the earth, would appear in the heavens five degrees above the sun, and of course would cut off none of his light; and it is also plain that the moon, at the full, would pass the shadow of the earth five degrees below it, and would suffer no eclipse. But in the course of the sun's apparent revolution round the earth once a year he is successively in every part of the ecliptic; consequently, the conjunctions and oppositions of the sun and moon may occur at any part of the ecliptic, and of course at the two points where the moon's orbit crosses the ecliptic,—that is, at the nodes; for the sun must necessarily come to each of these nodes once a year. If, then, the moon overtakes the sun just as she is crossing his path, she will hide more or less of his disk from us. Since, also, the earth's shadow is always directly opposite to the sun, if the sun is at one of the nodes, the shadow must extend in the direction of the other node, so as to lie directly across the moon's path; and if the moon overtakes it there, she will pass through it, and be eclipsed. Thus, in Fig. 43, let BN represent the sun's path, and AN, the moon's,—N being the place of the node; then it is evident, that if the two luminaries at new moon be so far from the node, that the distances between their centres is greater than their semidiameters, no eclipse can happen; but if that distance is less than this sum, as at E, F, then an eclipse will take place; but if the position be as at C, D, the two bodies will just touch one another. If A denotes the earth's shadow, instead of the sun, the same illustration will apply to an eclipse of the moon.

Fig. 43.

Since bodies are defined to be in conjunction when they are in the same part of the heavens, and to be in opposition when they are in opposite parts of the heavens, it may not appear how the sun and moon can be in conjunction, as at A and B, when they are still at some distance from each other. But it must be recollected that bodies are in conjunction when they have the same longitude, in which case they are situated in the same great circle perpendicular to the ecliptic,—that is, in the same secondary to the ecliptic. One of these bodies may be much further from the ecliptic than the other; still, if the same secondary to the ecliptic passes through them both, they will be in conjunction or opposition.