(a) As we discussed in class, in the geocentric model of the Solar System the epicycles of the three superior planets are not independent, but instead are arranged so that they

rotate in phase with one another and with the Sun's motion about the Earth. Explain why this was necessary, in terms of the observations available to Ptolemy. (b) How did Copernicus explain these same observations within his heliocentric model? (c) Both Ptolemy and Copernicus knew well the synodic periods of the planets (i.e., the intervals between successive oppositions of each planet, when it appeared directly opposite the Sun in the sky.) Show that in Ptolemy's model the period of a superior planet's motion around its deferent, PD and the period of its motion around its epicycle, PE, both measured relative to a fixed direction, must be given by PD = P₂ and PE = Pe where Pp is the true orbital period of the planet around the Sun and Pe is the true orbital period of the Earth around the Sun. [HINT: You might want to think about the planet's motion relative to the distant stars, and about your answer to part (a) above. You can also look at this in terms of the planet's synodic period, using the equation given in the Class Notes.] (d) For an inferior planet, on the other hand, show that PD Pe and PE = Pp. [Here again, it may be useful to consider the planet's synodic period, which in this case is twice the period between conjunctions with the Sun.] So both the Ptolemaic and Copernican models involve the same underlying periods for the planet's motions, but interpret them very differently.