Copernicus versus Ptolemy

The planet Mars is observed to move slowly eastward relative to the fixed stars from night to night. Occasionally, however, it appears to double back on its path, moving westward for a few months. This is called a retrograde loop.

Consider the heliocentric model of Copernicus. Assume, by way of approximation, that Mars and Earth describe circular orbits, that these orbits are coplanar, and that Mars is in opposition at time t=0, i.e. the Sun, Earth and Mars lie along a line, in that order, at t=0.

Take , , and .

1. What is ?
2. Carefully (and roughly to scale, if possible) draw a diagram of Copernicus's heliocentric system. By marking on it several sequential positions of Mars and Earth, show that Mars, as viewed from Earth, executes a retrograde loop against the background of the fixed stars in a time interval about t=0.
3. Estimate approximately how long the retrograde loop lasts, as viewed from Earth.
4. Calculate the time that elapses between successive retrograde loops.
5. Calculate the angular separation between the locations of successive retrograde loops, as viewed from Earth. What would be the angular separation if Mars described a different orbit with ? Comment.

Ptolemy devised a geocentric explanation of retrograde loops involving epicycles.

6. Carefully (and roughly to scale, if possible) draw a diagram of Ptolemy's geocentric system. By marking on it several sequential positions of Mars and Earth, show how the geocentric system leads to retrograde loops.
7. What features of real planetary motion (as opposed to the idealised motion considered here) did Ptolemy's model fail to explain economically?

In reality, the orbit of Mars is inclined with respect to the plane of the Earth's orbit (the ecliptic) by an angle . This is shown in Fig. 1.

 
Figure 1:

8. Write down and .
9. What is the relative position of Mars as seen from Earth?
10. Project this relative position onto the y-z plane of the sky for , , , , , 0, , , , , . Graph the result neatly on ruled axes, appropriately scaled. You should get a retrograde loop.