Chapter 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_.

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_.

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.

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_.

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:--

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)_.

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.

IV. THE SUN.