# Epicycles

## The Earth/Moon system - modelling an ellipse

The Moon orbits the Earth in an ellipse with an eccentricity of about 5.5%. This is quite small so in order to see the effects more clearly, we shall start with an ellipse with an eccentricity of 30%

Our first approximation is a deferent (ie a large circle) whose radius is equal to the semi-major axis of the ellipse on which the simulated Moon rotates in the correct (anti-clockwise) direction with a frequency of one. On running the program you will find that the simulated Moon leads and lags the real Moon by up to 340.

The second approximation is to add an offset to the deferent - ie make it rotate about a different centre. If yopu set the offset fraction to 30% (ie the same as the eccenricity of the ellipse) you will see a marked imporvement in the model which now differs by only 170. If you turn on the trails you will see just how close the agreement between the two curves is. It is worth noting here that adding an offset to the deferent is exactly equivalent to adding an epicycle which rotates in the reverse direction with a frequency of 1. When counting epicycles, therefore, we shall count the offset as an extra epicycle.

Comparing the shapes of the two orbits by using the trails feature is rather misleading because the most important feature of an elliptical orbit is not the very slight distortion in shape from a circle, it is the profound effect it has on the speed of the object as it goes round the orbit. In fact, neither Ptolemy nor Copernicus were interested in finding the best approximation to the actual orbit because, not being able to measure the radial distance to the Moon, they had no idea what the actual orbit looked like. All they were interested in was 'saving the appearances' - ie minimising the angular discrepancy between the position of the simulated Moon from the observed position of the breal one.

We can improve this fit even more by increasing the offset to 45%. Now the maximum discrepancy is a little over 80

As you watch the two Moons orbiting round the Earth, you will notice that the real Moon seems to rotate round the simulated Moon in an anti-clockwise direction twice every orbit. What this means is that if we add another epicycle (epicycle 1) whose frequency is twice that of the deferent and whose radius is equal to the extra amount we have added to the offset (ie 15%) we can reduce the maximum discrepancy to around 30. (This epicycle must rotate in the corect direction and be in phase so leave the two check boxes unchecked.)

Now if you turn your attention to the panel at the bottom, this discrepancy is shown in greater detail. The real Moon is shown fixed while the simulated Moon oscillates round it, sometimes leading, sometimes lagging behind. The vertical lines are at degree intervals and the discs are half a dgeree in diameter - ie the actual angular size of the Moon. You will observe that during one complete cycle, the simulated Moon makes two oscillations back and forth which are of unequal sizes. We can reduce tha maximum discrepancy a bit further by making these two oscillation the same size. This happens when the epicycle has a radius of approximately 20% of the deferent.

Now that we have an approximately sinusoidal discrepancy, it is clear a second epicycle of appropriate radius, frequency and phase will greatly reduce it. The best that can be done is with a radius of 4%, a frequency of 1, reversed and in antiphase. This reduces the maximum discrepancy to less than a degree! And this is with an ellipse with a starting eccentricity of 30%! See what you can achieve with a starting eccentricity of 15%