Part 18

128. The Sidereal and Synodical Periods of an Inferior Planet.--The time it takes a planet to make a complete revolution around the sun is called the sidereal period of the planet; and the time it takes it to pass from one aspect around to the same aspect again, its synodical period.

The synodical period of an inferior planet is longer than its sidereal period. This will be evident from an examination of Fig. 147. S is the position of the sun, E that of the earth, and P that of the planet at inferior conjunction. Before the planet can be in inferior conjunction again, it must pass entirely around its orbit, and overtake the earth, which has in the mean time passed on in its orbit to E'.

While the earth is passing from E to E', the planet passes entirely around its orbit, and from P to P' in addition. Now the arc PP' is just equal to the arc EE': hence the planet has to pass over the same arc that the earth does, and 360° more. In other words, the planet has to gain 360° on the earth.

The synodical period of the planet is found by direct observation.

129. The Length of the Sidereal Period.--The length of the sidereal period of an inferior planet may be found by the following computation:--

Let a denote the synodical period of the planet, Let b denote the sidereal period of the earth, Let x denote the sidereal period of the planet. Then 360°/b = the daily motion of the earth, And 360°/x = the daily motion of the planet, And 360°/x - 360°/b = the daily gain of the planet: Also 360°/a = the daily gain of the planet: Hence 360°/x - 360°/b = 360°/a. Dividing by 360°, we have 1/x - 1/b = 1/a; Clearing of fractions, we have ab - ax = bx: Transposing and collecting, we have (a + b)x = ab:

Therefore x = ab/a+b.

130. The Relative Distance of an Inferior Planet.--By the relative distance of a planet, we mean its distance from the sun compared with the earth's distance from the sun. The relative distance of an inferior planet may be found by the following method:--

Let V, in Fig. 148, represent the position of Venus at its greatest elongation from the sun, S the position of the sun, and E that of the earth. The line EV will evidently be tangent to a circle described about the sun with a radius equal to the distance of Venus from the sun at the time of this greatest elongation. Draw the radius SV and the line SE. Since SV is a radius, the angle at V is a right angle. The angle at E is known by measurement, and the angle at S is equal to 90°- the angle E. In the right-angled triangle EVS, we then know the three angles, and we wish to find the ratio of the side SV to the side SE.

The ratio of these lines may be found by trigonometrical computation as follows:--

VS : ES = sin SEV : 1.

Substitute the value of the sine of SEV, and we have

VS : ES = .723 : 1.

Hence the relative distances of Venus and of the earth from the sun are .723 and 1.

Superior Planets.

131. The Superior Planets.--The superior planets are those which lie beyond the earth. They are Mars, the Asteroids, Jupiter, Saturn, Uranus, and Neptune.

132. Apparent Motion of a Superior Planet.--In order to deduce the apparent motion of a superior planet from the real motions of the earth and planet, let S (Fig. 149) be the place of the sun; 1, 2, 3, etc., the orbit of the earth; a, b, c, etc., the orbit of Mars; and CGL a part of the starry firmament. Let the orbit of the earth be divided into twelve equal parts, each described in one month; and let ab, bc, cd, etc., be the spaces described by Mars in the same time. Suppose the earth to be at the point 1 when Mars is at the point a, Mars will then appear in the heavens in the direction of 1 a. When the earth is at 3, and Mars at c, he will appear in the heavens at C. When the earth arrives at 4, Mars will arrive at d, and will appear in the heavens at D. While the earth moves from 4 to 5 and from 5 to 6, Mars will appear to have advanced among the stars from D to E and from E to F, in the direction from west to east. During the motion of the earth from 6 to 7 and from 7 to 8, Mars will appear to go backward from F to G and from G to H, in the direction from east to west. During the motion of the earth from 8 to 9 and from 9 to 10, Mars will appear to advance from H to I and from I to K, in the direction from west to east, and the motion will continue in the same direction until near the succeeding opposition.

The apparent motion of a superior planet projected on the heavens is thus seen to be similar to that of an inferior planet, except that, in the latter case, the retrogression takes place near inferior conjunction, and in the former it takes place near opposition.

133. Aspects of a Superior Planet.--The four aspects of a superior planet are shown in Fig. 150, in which S is the position of the sun, E that of the earth, and P that of the planet.

When the planet is on the opposite side of the earth to the sun, as at P, it is said to be in opposition. The sun and the planet will then appear in opposite parts of the heavens, the sun appearing at C, and the planet at A.

When the planet is on the opposite side of the sun to the earth, as at P'', it is said to be in superior conjunction. It will then appear in the same part of the heavens as the sun, both appearing at C.

When the planet is at P' and P''', so that a line drawn from the earth through the planet will make a right angle with a line drawn from the earth to the sun, it is said to be in quadrature. At P' it is in its western quadrature, and at P''' in its eastern quadrature.

134. Phases of a Superior Planet.--Mars is the only one of the superior planets that has appreciable phases. At quadrature, as will appear from Fig. 151, Mars does not present quite the same side to the earth as to the sun: hence, near these parts of its orbit, the planet appears slightly gibbous. Elsewhere in its orbit, the planet appears full.

All the other superior planets are so far away from the sun and earth, that the sides which they turn towards the sun and the earth in every part of their orbit are so nearly the same, that no change in the form of their disks can be detected.

135. The Synodical Period of a Superior Planet.--During a synodical period of a superior planet the earth must gain one revolution, or 360°, on the planet, as will be evident from an examination of Fig. 152, in which S represents the sun, E the earth, and P the planet at opposition. Before the planet can be in opposition again, the earth must make a complete revolution, and overtake the planet, which has in the mean time passed on from P to P'.

In the case of most of the superior planets the synodical period is shorter than the sidereal period; but in the case of Mars it is longer, since Mars makes more than a complete revolution before the earth overtakes it.

The synodical period of a superior planet is found by direct observation.

136. The Sidereal Period of a Superior Planet.--The sidereal period of a superior planet is found by a method of computation similar to that for finding the sidereal period of an inferior planet:--