Chapter 1 Part 1. Motions

The Motions of the Planets

The geocentric, or Ptolemaic, model of the Solar System has a nice concept - the Earth is in the center. This, unfortunately, runs into trouble quickly as it tries to deal with the actual motions of the planets in the sky. If you observe an outer planet (Mars, Jupiter or Saturn) over a period of time and plot its position on a star map, you will see a very interesting motion. The backward (east to west) part of the motion is called retrograde (backwards) motion. This retrograde motion of planets required the geocentric model to become loaded with epicycles, off-center equants, and orbits going around nothing. One might question why anything would orbit around nothing. The whole model represents the planets visible to the ancient observers.

The much simpler heliocentric model of Copernicus eliminates the tangle of epicycles while explaining the retrograde motion in a natural way. The simplicity is obvious; the epicycles and off-centering are gone. Look at Figure 1.5. Earth and Mars are shown in a series of positions separated by approximately equal time intervals. The numbers associate Mars and Earth at a given time. The line of sight from Earth to Mars (what we see) is shown. Here's a graphic frvaom Cornell that shows how it works.

Earth takes about 365.25 days to orbit the Sun while Mars takes 687 (Earth) days to do the same. This means that Earth is moving faster than Mars and will pass it occasionally (about every 26 months). Retrograde motion occurs while Earth passes between Mars and the Sun moving faster than Mars. Mars actually doesn't change its motion at all - retrograde motion is an illusion caused by Earth's motion.

Note that Earth passes between Mars and the Sun at roughly 780 day intervals; this interval is known as the Synodic Period of Mars. Early observers could apply this to the Copernican model to find the actual orbit period of any planet they could see even though they could not accurately observe it directly. Observing the synodic period is, on the other hand, fairly easy. Plot the position of Mars on the sky till you find it in a position directly opposite the Sun. Record that date, then watch Mars till it happens again; this will be about 780 days after the prior occurrence. Now you know the Synodic Period of Mars. You would also know that Earth's orbital period is approximately 365.25 days and that Mars is farther from the Sun than Earth and takes longer than Earth to go around the Sun.

Here's how you do it. Let's use S for the Synodic Period of Mars (observable). Use 365.25 for the orbital period of Earth (also observable). Use M for the orbital period of Mars (in Earth days) which is not directly observable. In one day Earth moves 1/365.25 of its orbit while Mars moves 1/M of its orbit. We don't know what M is, but we do know that Earth will lap Mars (catch up to it again) after S days, meaning that every day Earth gains 1/S of a circle on Mars.

Now use the equation 1/S = 1/365.25 - 1/M. Since we know S, we can put that in
1/780 = 1/365.25 - 1/M.
and rearrange the terms to get 1/M = 1/365.25 - 1/780. Do this yourself and see.

An inner planet (Venus or Mercury). Orbits faster than Earth, so the Synodic Period is the the time for them to lap the Earth. Simply rearrange the equation for this, getting
1/S = 1/V - 1/365.25
We are using Venus (v) in this case. Its observable Synodic Period is roughly 584 days. Rearrange the equation to:
1/V = 1/584 + 1/365.25
and get the orbital period of Venus (225 days).

Occam's Razor

Also note a description of Occam's Razor. It means that, if you must choose between several explanations for something, choose the simplest one. It is a heuristic and is not guaranteed. Heuristics can fail. Lots of things about the Universe are not simple. But - old Occam will improve your chances of making the right choice.

Notice that the use of Occam's Razor would make the choice between the geocentric model and the Copernican model quite easy.