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.