parallel to the ecliptic. Thus, if E C (Fig. 63) represent the ecliptic, K its pole, s the situation of the star, M that of the moon, and s m' an arc passing through s and parallel to the arc E C, we have in place of m M, m' M = m M-m m', and in place of S m, s m'. The hourly variation of S'm must also be reduced to the arc s m'. 458. The reduction of the difference of longitude of the moon and star, to the parallel to the ecliptic, passing through the star, is effected by multiplying this difference by the cosine of the latitude of the star. For, let A B (Fig. 65) be an arc of the ecliptic, and A' B' the corresponding arc of a circle parallel to it; then, since similar arcs of circles are proportional to their radii, we have, BC: B'C'::AB: A'B' == but, Hence, = AB.B'C' =AB cos B B'. BC The reduction of the relative hourly motion in longitude to the parallel in question, is obviously effected in the same manner. CHAPTER XVII. OF THE PLANETS AND THE PHENOMENA OCCASIONED BY THEIR MOTIONS IN SPACE. Apparent Motions of the Planets with respect to the Sun. 459. The apparent motion of an inferior planet, with reference to the sun, is materially different from that of a superior planet. The inferior planets always accompany the sun, being seen alternately on the east and west side of him, and never receding from him beyond a certain distance, while the superior planets are seen at every variety of angular distance. This difference of apparent motion arises from the difference of situation of the orbits of an inferior and superior planet, with respect to the orbit of the earth, the one lying within and the other without the earth's orbit. Let CA CB (Fig. 66) represent the orbit of either one of the inferior planets, Venus for example, and P K T the orbit of the earth; which we will suppose to be circles and to lie in the same plane; and let M L N represent the sphere of the heavens to which all bodies are referred. Suppose, for the present, that the earth is stationary in the position P, and through P draw the lines P A, P B, tangent to the orbit of Venus, and prolong them on till they intersect the heavens at a and b. When Venus is at C, (the earth being at P,) she will be in superior conjunction, and when at C' in inferior conjunction. Now, by inspecting the figure, it will be seen that in passing from C to C', she will be seen in the heavens on the east side of the sun, and in passing from C' to C on the west side of the sun; also, that in passing from C to A she will recede from the sun in the heavens, from A to C' approach him, from C' to B recede from him again, and from B to C approach him again. a and b will be her positions in the heavens at the times of her greatest eastern and western elongations. When Venus is to the east of the sun, she is seen in the evening, and called the Evening Star; and when to the west, she is seen in the morning, and called the Morning Star. 460. We have in the foregoing investigation supposed the earth to be stationary, a supposition which is contrary to the fact; but it is plain that the only effect of the earth's motion in the case under consideration, as it is slower than that of the planet, is to cause the points A, C, B to advance in the orbit, without altering the nature of the apparent motion of the planet with respect to the sun. The orbits of the earth and planet are also ellipses of small eccentricity, and are slightly inclined to each other, instead of being circles and lying in the same plane: on this account, as the greatest elongations will occur in various parts of the orbits, they will differ in value. The greatest elongation of Venus varies from 45° to 47° 12'. Its mean value is about 46°. 461. Owing to the circumstance of the orbit of Mercury being within the orbit of Venus, the greatest elongation of this planet is less than that of Venus. It varies between the limits 16° 12', and 28° 48'; and is, at a mean, 22° 30'. 462. Next, suppose P K T (Fig. 66) to be the orbit of a superior planet, and CA C' B that of the earth; and as the velocity of the earth is much greater than that of the planet, let us, for the present, regard the planet as stationary in the position P, while the earth describes the circle CAC. When the earth is at C, the planet, being at P, is in conjunction with the sun. When the earth is at A, SA P the elongation of the planet, is 90°. When it arrives at C' the planet is in opposition, or 180° distant from the sun. And when it reaches B, the elongation is again 90°. At intermediate points the elongation will have intermediate values. If, now, we restore to the planet its orbitual motion, we shall manifestly be conducted to the same results relative to the change of elongation, as the only effect of such motion will be to throw the points A, C', B forward in the orbit. It appears, then, that in the course of a synodic revolution a superior planet will be seen at all angular distances from the sun, both on the east and west side of him. From conjunction to opposition, that is, while the earth is passing from C to C', the planet will be to the right, or to the west of the sun; and will therefore be below the horizon at sunset, and rise some time in the course of the night. But, from opposition to conjunction, or while the earth is moving from C' to C, it will be to the east of the sun, and therefore above the horizon at sunset. 463. To find the length of the synodic revolution of a planet. Let us first take an inferior planet, Venus for instance. Suppose we assume, at a given instant, the sun, Venus, and the earth to be in the same right line; then, after any elapsed time (a day for instance), Venus will have described an angle m, and the earth an angle M around the sun. Now, m is greater than M; therefore at the end of a day, the separation of Venus from the earth, (measuring the separation by an angle formed by two lines drawn from Venus and the earth to the sun) will be m-M; at the end of two days (the mean daily motions continuing the same), the angle of separation will be 2 (m-M); at the end of three days, 3 (m—M); at the end of s days, s (m— M). When the angle of separation then amounts to 360°, that is, when = s (m-M) 360°, the sun, Venus, and the earth must be again in the same right line, and in that case, In which expression s denotes the mean duration of a synodic revolution, m and M being taken to denote the mean daily motions. We may obtain from equation (128) another equation, in which the synodic revolution is expressed in terms of the sidereal periods of the earth and planet. and M Let P and p denote the sidereal periods in question; then since 1d. M° P: 360°, : = and m = ; substituting P Equations (128), (129), although investigated for an inferior planet, will answer equally well for a superior planet, provided we regard m as standing for the mean daily motion of the earth, M for that of the planet, p for the sidereal period of the earth, and P for that of the planet. For, the earth holds towards a superior planet the place of an inferior planet, and a synodic revolution of the earth to an observer on the planet, will obviously be a synodic revolution of the planet to an observer on the earth. 464. Equation (128) shows that the length of a mean synodic revolution depends altogether upon the amount of the difference of the mean daily motions of the earth and planet, and is the greater the less is this difference. It follows therefore that the synodic revolution is the longest for the planets nearest the earth. It appears by equation (129), that the length of a synodic revolution is, for an inferior planet, greater than the sidereal period of the planet, and for a superior planet, greater than the sidereal period of the earth. The actual lengths of the synodic revolutions of the different planets are given in Table V. 465. The mean synodic revolution of a planet being known, and also the time of one conjunction or opposition, we may easily ascertain its mean elongation at any given time, and thus approximately the time of its rising, setting, and meridian passage. Stations and Retrogradations of the Planets. 466. The apparent motions of the planets in the heavens, as has already been stated (Art. 8), are not, like those of the sun and moon, continually from west to east, or direct, but are sometimes also from east to west, or retrograde. The retrograde motion takes place over arcs of but a small number of degrees; and in changing the direction of their motions, the planets are for several days stationary in the heavens. These phenomena are called the Stations and Retrogradations of the planets. We now propose to inquire theoretically into the particulars of the motions in question, and to show how the phenomena just mentioned result from the motions of the planets in connection with the motion of the earth. Let CA C' B (Fig. 66), represent the orbit of an inferior planet, and P K T the orbit of the earth; both considered as circles, and as situated in the same plane. If the earth were continually stationary in some point P of its orbit, it is plain that while the planet was moving from B the position of greatest western elongation, to A the position of greatest eastern elongation, it would advance in the heavens from b to a; that, while it was moving from A to B, that is, from greatest eastern to greatest western elongation, it would retrograde in the heavens from a to b; and that, in passing the points A and B, as it would be moving directly towards or from the earth, it would for a time appear stationary in the heavens in the positions a and b. But the earth is in fact in motion, and the actual apparent motion of the planet is in consequence materially different from this. Let A, A' (Fig. 67), be the positions of the planet and earth at the time of the greatest eastern elongation, C', P their positions at inferior conjunction, and B, B' their positions at the greatest western elongation. At the time of the greatest eastern elongation while the planet describes a certain distance A D on the line of the centres of the earth and planet, the earth moves forward in its or |
