# 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 and at first glance the orbit looks completely circular. Our first approximation is therefore a deferent (ie a large circle). On running the program you will find that the simulated Moon leads and lags the real Moon by up to 6º. This is shown in detail in the panel at the bottom. 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.

If you turn the trails on you will see immediately that the main difference between the circular orbit of the simulated moon and the elliptical orbit of the real moon is not its shape, but the fact that the ellipse appears to be off centre. The second approximation is to add an offset to the deferent - ie make it rotate about a different centre. (This device is known as an 'eccentric deferent' but I prefer to use the word 'offset' so as to avoid confusion with the 'eccentricity' of the elliptical orbit we are trying to reproduce.)

If you set the offset fraction to 5.5% (ie the same as the eccenricity of the ellipse) you will see a marked imporvement in the model which now differs by only 3º. If you turn on the trails you will see just how close the agreement between the two curves is.

Comparing the shapes of the two orbits by using the trails feature is rather misleading though, 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 11%. Now, even though there is an obvious deiscrepancy between the simulated and the real orbits, the maximum angular discrepancy is reduced to a mere 0.14º. This is about the limit of Ptolemy's observations and I am sure that he would have been very happy with this result. The problem with the Moon is that, because of the influence of the Sun's gravity as well as the Earth's, its orbit is far more complex than a simple ellipse and no amount of tinkering with epicycles and the like will achieve a satisfactory fit.

It is useful to experiment with this mode using a larger eccentricity so that you can see more clearly what is going on. Set the eccentricity to 9% and the offset to 9% also. (The eccentricity of Mars' orbit is 9%) You will find that the angular discrepancy is as much as 5º. Now click on the 'Equant' checkbox. Now the simulated 'Moon' rotates uniformly about a point equidistant from the centre of the deferent opposite to the 'Earth'. (This is shown in violet.) Immediately the angular discrepancy reduces to a fraction of a degree. (Mathematically speaking, the use of an equant is a second order approximation to an elliptical one - and a very good one it is too.)

When Copernicus was devising his heliocentric theory, he was very unhappy about Ptolemy's use of the equant as it seemed to him to flout the fundamental principle of uniform motion in a circle. He therefore set about removing the equants and replacing them with epicycles. Lets see how this works.

Set the eccentricity to 20% and the offset fraction to 40%. Uncheck the Equant box. Note the angular discrepancy. In one orbit, the simulated moon leads by 2½º, then lags by 1½º, then leads by 1½º and finally lags by 2½º.

Now reduce the offset by a few percentage points and add an epicycle of frequency 2 and radius equal to the amount by which you have just reduced the offset. You will find that the angular discrepancy reduces considerably. By experiment, find out which combination of offset and epicycle radius reduces the discrepancy to its lowest level. I don't think you will beat the equant!

Another thing you might like to try is to see how close you can get the simulated moon to follow the real one. Set the epicycle radius to zero and adjust the offset until the simulated moon seems to roll round the real Moon in an anti-clockwise direction twice every orbit. Now add an epicycle of frequency 2 and of sufficient radius to bring the two moons into line. You should be able to reduce the angular discrepancy to about a degree and the radial discrepancy to less than 4%. Not too bad really but not good enough for Kepler!