Heliocentric to Geocentric graphing

We will make a simple transformation from a heliocentric coordinate system to a geocentric coordinate system. This exercise makes approximations that the heliocentric orbits are circular with the sun directly at the center of every orbit. As we studied in class and read in Crowe's book on the Ptolemaic system the ancients knew that the earth was not the dead center of every celestial body's deferent. Eccentricities were part of the required parameters to fit their quasi-geocentric models. However, we don't need to go into the complexities of the actual Ptolemaic models to see the main features of their geocentric models.

1) On a sheet of graph paper, for example here,  place the sun at the center of the (X, Y) coordinates (0,0).

2) Draw two circles of radii 4 cm and 6 cm. The inner circle is for the "earth" and the outer circle for a fictitious planet p.

3) We will assume that when the earth completes a 360 degrees orbit the planet only goes through 180 degrees, that is it takes two orbits of earth for the planet to complete one orbit.

4) We want to determine the motion of the planet as seen from earth. This means we need to know how far away in the X and Y directions the planet is from the earth.

5) We can determine this by finding the positions of the earth and the planet in the heliocentric system and subtracting the planet's position from the earth's position with respect to the sun. The table below should be filled in and extended as needed to assure a complete orbit for the planet.

 theta earth theta planet Xe Ye Xp Yp Xp - Xe Yp - Ye 0 0 20 10 40 20 theta_e theta_e/2 . . . . 720 360

6) The last two columns tell you where the planet is with respect to the earth. Take another sheet of graph paper and place the earth at (0, 0). Plot the last two columns from the table on this graph. Draw a smooth curve through your data points.  Don't simply connect dot to dot. This is the apparent orbit of the planet with respect to the earth. It is the geocentric orbit.

7) If you need more data points add as many as you need to see what a smooth orbit should look like.

8) The report consists of the table of values above and the geocentric orbit plot. Include one page of narrative noting any interesting features you see in the plot and including a short description of why the ancient felt justified in assuming a stationary earth.

NOTE: This same table and plot works well in a spread sheet. You will need to use trigonometric functions to calculate the coordinates.