The Heavens Above: A Popular Handbook of Astronomy

by J. A. Gillet; W. J. Rolfe

III. INFERIOR AND SUPERIOR PLANETS.

Inferior Planets.

124. The Inferior Planets.—The inferior planets are those which lie between the earth and the sun, and whose orbits are included by that of the earth. They are Mercury and Venus.

Fig. 144.

125. Aspects of an Inferior Planet.—The four chief aspects of an inferior planet as seen from the earth are shown in Fig. 144, in which S represents the sun, P the planet, and E the earth.

When the planet is between the earth and the sun, as at P, it is said to be in inferior conjunction.

When it is in the same direction as the sun, but beyond it, as at P'', it is said to be in superior conjunction.

When the planet is at such a point in its orbit that a line drawn from the earth to it would be tangent to the orbit, as at P' and P''', it is said to be at its greatest elongation.

Fig. 145.

126. Apparent Motion of an Inferior Planet.—When the planet is at P, if it could be seen at all, it would appear in the heavens at A. As it moves from P to P', it will appear to move in the heavens from A to B. Then, as it moves from P' to P'', it will appear to move back again from B to A. While it moves from P'' to P''', it will appear to move from A to C; and, while moving from P''' to P, it will appear to move back again from C to A. Thus the planet will appear to oscillate to and fro across the sun from B to C, never getting farther from the sun than B on the west, or C on the east: hence, when at these points, it is said to be at its greatest western and eastern elongations. This oscillating motion of an inferior planet across the sun, combined with the sun's motion among the stars, causes the planet to describe a path among the stars similar to that shown in Fig. 145.

Fig. 146.

127. Phases of an Inferior Planet.—An inferior planet, when viewed with a telescope, is found to present a succession of phases similar to those of the moon. The reason of this is evident from Fig. 146. As an inferior planet passes around the sun, it presents sometimes more and sometimes less of its bright hemisphere to the earth. When the earth is at T, and Venus at superior conjunction, the planet turns the whole of its bright hemisphere towards the earth, and appears full; it then becomes gibbous, half, and crescent. When it comes into inferior conjunction, it turns its dark hemisphere towards the earth: it then becomes crescent, half, gibbous, and full again.

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.

Fig. 147.

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:—

Fig. 148.

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.

Fig. 149.

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.

Fig. 150.

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.

Fig. 151.

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'.

Fig. 152.

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:—

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 will 360°/b = daily motion of the earth,

And 360°/x = daily motion of the planet;

Also 360°/b - 360°/x = daily gain of the earth.

But 360°/a = daily gain of the earth:

Hence 360°/b - 360°/x = 360°/a

1/b - 1/x = 1/a

ax - ab = bx

(a-b)x = ab

x = ab/(a-b).

Fig. 153.

137. The Relative Distance of a Superior Planet.—Let S, e, and m, in Fig. 153, represent the relative positions of the sun, the earth, and Mars, when the latter planet is in opposition. Let E and M represent the relative positions of the earth and Mars the day after opposition. At the first observation Mars will be seen in the direction emA, and at the second observation in the direction EMA.

But the fixed stars are so distant, that if a line, eA, were drawn to a fixed star at the first observation, and a line, EB, drawn from the earth to the same fixed star at the second observation, these two lines would be sensibly parallel; that is, the fixed star would be seen in the direction of the line eA at the first observation, and in the direction of the line EB, parallel to eA, at the second observation. But if Mars were seen in the direction of the fixed star at the first observation, it would appear back, or west, of that star at the second observation by the angular distance BEA; that is, the planet would have retrograded that angular distance. Now, this retrogression of Mars during one day, at the time of opposition, can be measured directly by observation. This measurement gives us the value of the angle BEA; but we know the rate at which both the earth and Mars are moving in their orbits, and from this we can easily find the angular distance passed over by each in one day. This gives us the angles ESA and MSA. We can now find the relative length of the lines MS and ES (which represent the distances of Mars and of the earth from the sun), both by construction and by trigonometrical computation.

Since EB and eA are parallel, the angle EAS is equal to BEA.

SEA = 180° - (ESA + EAS)

ESM = ESA - MSA

EMS = 180° - (SEA + ESM).

We have then

MS : ES = sin SEA : sin EMS.

Substituting the values of the sines, and reducing the ratio to its lowest terms, we have

MS : ES = 1.524 : 1.

Thus we find that the relative distances of Mars and the earth from the sun are 1.524 and 1. By the simple observation of its greatest elongation, we are able to determine the relative distances of an inferior planet and the earth from the sun; and, by the equally simple observation of the daily retrogression of a superior planet, we can find the relative distances of such a planet and the earth from the sun.