CHAPTER XIII. ECLIPSES OF THE SUN AND MOON.-OCCULTATIONS. 222. Eclipses occur only at New and Full Moon. As an eclipse of the sun is caused by the moon passing between the sun and earth (214), it can only occur when the moon is in conjunction with the sun, that is, at the time of new moon. An eclipse of the moon is caused by the interposition of the earth, between the sun and moon, which prevents in whole or in part, the illumination of the latter by the former. It must, therefore, occur when the moon is in opposition, that is, at the time of full moon. If the moon's orbit coincided with the plane of the ecliptic, there would necessarily be an eclipse of the sun at every new moon, since the moon would in that case pass directly between the sun and earth; and an eclipse of the moon at every full moon, as the earth would then be directly between the sun and moon. But as the orbit is inclined to the ecliptic, an eclipse can only occur when the moon at the time of new and full moon, is at, or near one of its nodes. In other cases the moon is too far north or south of the ecliptic, to cause an eclipse of the sun, or to be itself eclipsed. ECLIPSES OF THE MOON. 223. Earth's shadow and penumbra. The magnitude of the sun being far greater than that of the earth, and both being globular bodies, the shadow of the earth must evidently be of a conical form. Let AB and hg, Fig. 37, be sections of the sun and earth by a plane passing through their centres S and E; and let AC and BC, and also, AH and BK, be tangents common to the two sections. Then, will gCh be a section of the earth's conical shadow or umbra, as it is frequently called, and EC will be the axis of the shadow. If the plane CEhK, be supposed to revolve round the axis EC, the tangent hK will describe the convex surface of the frustum of a cone, within the whole of which, the light of the sun must be more or less obstructed by the earth. That part of the frustum, which is included between the umbra and convex surface, that is, the part, of which HgChK is a section, is called the earth's penumbra. An 224. Beginning or end of an eclipse of the moon. eclipse of the moon is regarded as beginning or ending at the instant her edge touches the earth's shadow. Thus, if mn be a part of the moon's orbit, the eclipse begins when the moon is at a, and ends when she is at e. Prior, however, to the beginning of an eclipse, while the moon is passing from the edge of the penumbra to the edge of the shadow, she must evidently suffer a gradual but increasing, diminution of her light. This circumstance, renders it difficult, if not impracticable, to observe with accuracy the instant at which the eclipse begins. On account of the gradual increase of the moon's light in passing from the shadow, the same difficulty occurs at the end. Sometimes the moon, at full moon, though too far north or south of the ecliptic to come in contact with the shadow, may still be sufficiently near, to pass through the penumbra. In this case the moon suffers a dimunition of light without being eclipsed. 225. Length of the earth's shadow. The length of the earth's shadow exceeds three times the distance of the moon from the earth. Let us assume the moon to be at one of her nodes at the instant she is in opposition at c. Then, will the centres of the sun, earth and moon be in the same straight line SC, parallax of the point C. Hence, (93. G). Hence, since (95, 96 and 100), always exceeds 53′ and - is always less than 17', it follows that EC must be more than three times Ec, the moon's distance from the earth. 226. Semi-diameter of the earth's shadow. The apparent semi-diameter of the earth's shadow, at the distance of the moon is called the semi-diameter of the shadow. Thus, the angle bEc is the semi-diameter of the shadow. The point c is called the centre of the shadow. The semi-diameter of the earth's shadow, is equal to the sum of the moon's and sun's horizontal parallaxes, less the sun's apparent semi-diameter. semi-diameter of the earth's shadow. Ebg-ECg and (225), ECg= — «', '), or S + - 8. 9", and 16' 1" which are their mean values, very nearly, we have the mean value = Cor. Since S 41' 9", we have 2S 82' 18". The diameter of the shadow is, therefore, more than twice the moon's apparent diameter, and consequently, the moon may be entirely enveloped in the shadow. Scholium. In obtaining the above expression for the semidiameter of the shadow, the shadow is assumed to be limited by those rays of the sun which are tangents to the sun and earth. It is, however, found that the observed duration of an eclipse always exceeds the duration computed on this supposition. This is accounted for, by assuming that most of those rays which pass near the surface of the earth, are absorbed by the lower strata of the atmosphere. The extent of the obstruction to the passage of the light being thus increased, the diameter of the shadow and consequently the duration of the eclipse, must also be increased. In consequence of the difficulty in ascertaining the exact time of beginning and end of the eclipse (224), astronomers have differed as to the amount of the correction that should be made. According to the observations and computations of Dr. Mäedler, a German astronomer, who has recently given particular attention to the subject, the computed semidiameter should be increased by about a 50th part*. Hence, 227. Moon visible when entirely enveloped in earth's shadow. Another effect of the action of the earth's atmosphere is perceptible in eclipses of the moon. Those rays from the sun which enter the atmosphere and are so far from the surface as not to be absorbed, have their directions changed, and leave the atmosphere with a greater inclination to the axis of the shadow, as represented by the dotted line in the figure. In this way a sufficient quantity of light is generally thrown on the moon to render her visible, even when in the middle of the shadow. She appears, while in the shadow, with a dull red or copper-coloured light. 228. Moon's angular distance, from the centre of the earth's shadow, at the beginning or end of a lunar eclipse. This distance is equal to the sun of the semidiameters of the earth's shadow and moon. For, as the eclipse begins when the moon's centre is at a, the angle aEc expresses her distance from the centre of Astr. Nach. Nos. 256, 286 and 338. the shadow at that time. But aEc is equal to the sum of bEc and aEb; or, angular distance, aEc S+d. = When the moon is first entirely in the shadow, or when she begins to emerge from it, her angular distance from the centre of the shadow will evidently be, S. Cor. When at, or near, the time of full moon, the moon's angular distance from the centre of the shadow does not become less than S, there evidently cannot be an eclipse; and when it does become less, there must be an eclipse. 229. Lunar Ecliptic Limits. Referring the points and orbit to the celestial sphere, let c', Fig. 34, be the place of the centre of the earth's shadow in the ecliptic and M' the place of the moon's centre in her orbit NF, when the angular distance c'M' is perpendicular to the orbit and is equal to S. Then it is evident, that, according as the distance of the centre of the shadow from the node N, or of the sun from the opposite node, is greater or less than Nc', the least distance of the centres of the moon and shadow must be greater or less than S+. Hence, it follows (228. cor.), that there can never be an eclipse of the moon when the distance of the sun from the nearest node is greater than the greatest value of Nc', and that there must always be one when this distance is less than the least value of Nc'. The greatest and least values of Nc' are, therefore, called the lunar ecliptic limits. Similar quantities for eclipses of the sun are called solar ecliptic limits. Now, it is known both from observations and from investigations in physical astronomy, that at the time of the syzygies the inclination of the moon's orbit has always nearly its greatest value of 5° 17'. Taking, therefore, this value of c'NM' and the greatest and least value of S+, which, including the correction of S, are about 63′ 17′′ and 53' 8", the right angled spherical triangle c'M'N gives for |