An Astronomy StackExchange (ASE) user asked:

Why does the radii of the outer planets’ epicycles have to be aligned with the Earth-Sun radius in the Ptolemaic model?

Always loving creating animated models of planetary theories, I decided to create some in order to answer that question in a better way than possible on ASE.

First, some astronomical background. In the Ptolemaic model, the Sun, the Moon, and the planets revolve around the Earth. This has since been demonstrated to be erroneous, but for a while, astronomers didn’t know better, and it took Copernicus and the observations of Galileo to confirm that the Earth is a planet like any other, and that all planets revolve around the Sun—only the Moon actually revolves around the Earth. But back to Ptolemy’s model: to be precise, planets move on small circles called epicycles, which themselves move on a larger circle, the deferent. Ptolemy figured, following a long demonstration and calculations based on his observations (in the Almagest; if you read French, I have translated it in that language), that the center of the deferent is not the Earth, but is slightly offset from it, from an amount that differs from one planet to another. Also, the center of uniform motion (later called the “equant”) is not coincident with the center of the deferent, but is twice as far away from the Earth as the deferent’s center is, in the same direction.

The question on ASE began with:

An opposition is an alignment Sun–Earth–planet, in that order. When it happens, the planet rises at sunset and sets at sunrise, being visible all night (technically, atmospheric refraction extends this to a few nights before and after opposition as well, but that’s another story).

Retrograde motion is the apparent “backing up” of a planet along its path in the sky from one night to the next. We now understand that it is caused by the Earth “catching up” with the planet in their respective orbits around the Sun—a little like when we pass cars on the road: they seem to “back up” relatively to us, but we all know they still move forward.

In order to answer the question, let’s first see what would happen if the radius of the epicycle, defined as an imaginary line between the epicycle center and the planet) was not parallel to the Earth–Sun line at opposition. In the animations below (to scale except for body sizes), the Earth is the blue dot, the deferent’s center is the grey dot, and the equant is the red dot. The planet (purple) moves on its epicycle (small grey circle), which itself moves on the deferent (large grey circle). In Ptolemy’s system, at our scale, the Sun would be really close to the Earth, so let’s replace it with a lime-colored line pointing in its direction away from Earth.

In order to keep things as realistic as possible, let’s use the same parameters as Ptolemy determined for Saturn, with the difference that, in Animation 1 (below, left) the anomaly (the planet’s position on its epicycle) is shifted so that the epicycle radius (the line from the epicycle to the planet; in orange) is always at a 90° angle with the Earth–Sun line. All other parameters are as determined by Ptolemy. In Animation 2 (left), all of Ptolemy’s original parameters have been kept, and it can be seen that the epicycle radius is parallel to the Earth–Sun line.

The animation window measures 840 pixels by 440 pixels, so make sure your screen is large enough! Also, this is best seen on a computer, not a mobile device.

You’ll notice that Saturn is already in retrograde motion when Animation 1 starts. However, opposition in this animation happens just after retrograde motion has ended. Things remain the same for the next opposition, which also begins just after Saturn has resumed its direct motion. In Animation 2, however, it can be seen that retrograde motion starts before opposition; opposition happens exactly in the middle of the retrograde period. Both animation pause at the first opposition, to resume after you click the respective Continue button which will then appear, and end at the second opposition.

I hope these animations help you understand why oppositions can happen at the middle of retrograde motion, as observed in reality, only if the radius of the epicycle is parallel to the Earth–Sun line. One may wonder how dogma and religion were strong in ancient times, to make astronomers refuse to see that the Sun is in the middle of the Solar System.

© 2024 EcliptiQc / Pierre Paquette
Last Updated 2024-07-17 at 00 h 55 UTC