The (wrong) graph has been drawn with 12 cycles, based on approximately 12 lunar months per year. But the result is obviously not to scale. If you try to draw it to scale, the truth jumps out at you; it can't have this form.
Yeah, this fact makes it awfully hard to draw realistically. Just try to draw a circle (Earth's orbit around the sun) and then a convex curve that crosses it from time to time. Pretty hard not to make it concave.
Given the Earth-Moon distance of about 380000 km and Earth-Sun distance of about 150000000 km, the lunar orbit deviates from Earth's orbit about 0.25% at most; if the Earth-Moon distance [1] were one meter, the maximum deviation would be ~2.5 millimeters. So, yeah, "awfully hard" just about describes the situation.
[1] The average distance - Earth's orbit is not a circle, after all.
Does the fact that the earths orbit is not a circle lead to an instability? So the deviation will change over time - whether by converging on a circle or spinning away. [totally naive question]
The Moon is slowly moving away due to tidal interaction. If the Earth and Moon were tidally locked (both presenting the same face to each other at all times) then the orbit would remain stable. If the Earth would be spinning slower than the Moon, then the Moon would be slowly pushed inward.
An interesting consequence is that in a billion year's time, after Earth's oceans would have evaporated, this tidal interaction would greatly reduce, "fixing" the Earth-Moon system in place.
As caf said, and as Johannes Kepler first correctly deduced, stable periodic orbits in a two-body gravitationally bound system are ellipses with an eccentricity (deviation from a perfect circle) dependent on the system. A perfectly circular orbit (eccentricity 0) is of course theoretically possible, but would require careful fine-tuning. Earth's orbit is rather close to a circle, as is Venus's, but Mercury, Mars, and especially Pluto have notably eccentric orbits.
The case of a system of more than two bodies is of course trickier. One cannot analytically solve the orbits even in a three-body system, not to mention a real planetary system. The orbits are chaotic and everything perturbs everything else in complex ways. This does not mean that the orbits are intrinsically unstable or that their evolution cannot be numerically predicted, however. For reasons presented below, for most practical purposes the orbit of any planet in the Solar System can be approximated as an ellipse.
There's a selection effect present: in areas of the protoplanetary disc where a nearby growing planet caused major disturbations, another planet couldn't form in the first place, and those planetesimals that did form, would eventually collide and merge with their more massive neighbor, or in extreme cases, be ejected from the Solar System altogether. This is the "cleaning the neighborhood" effect that is now used in the definition of a planet.
Also, if two planets have orbital periods close to a simple integer ratio, such as 3:2, the resonance magnifies their mutual perturbations over time. Unsurprisingly, there are no such simple ratios in the Solar System.
Every contemporary planetary body has thus been selected for having an orbit stable over a timescale of at least billions of years. The asteroid belt consists of remnants of the original disc: a fifth rocky planet could have formed there if not for the effect of Jupiter, preventing the coalescence of anything larger than Ceres. The belt was also originally much denser than currently; over the eons Jupiter has pruned every rock unfortunate enough to have an orbit just a bit too unstable.
Based on observations of other planetary systems, we now know that a giant planet could form or otherwise end up in a very eccentric orbit and thus prevent other stable orbits in a large part of its planetary system. This was surprising, given the nearly-circular orbits of the giants in our system. In retrospect, it is clear why we observe our system to be so well-behaved - otherwise, Earth wouldn't have formed and we would not be here making these observations in the first place! This is an instance of the so-called Weak Anthropic Principle.
Why wouldn't it? When it's going around the earth, it's going to be going in the opposite direction from the earth's orbit part of the time. The question is which motion is bigger.
In the ptolemeic system, the planets were thought to move with epicycles, or "wheels within wheels". This is why astrolabes had gears attached to gears.
Then copernicus came along and blew that all away with his heliocentric model, which was then given mathematical foundation by Johannes Kepler and his laws of planetary motion.
I would really like to meet these mathematicians who described it that way. I suspect he asked two and one of them was busy at that moment and not really paying much attention to his question.
I thought it would look like a sine wave on a circle...Well but that totally unexpected.
Do check the guys other writings they are really very interesting.
I understood the argument that the speed of the Earth's orbit dominated the Moon's, but I was having a really tough time visualizing it. The wiki article (http://en.wikipedia.org/wiki/Orbit_of_the_Moon#Path_of_Earth...) has a diagram that explains it (http://en.wikipedia.org/wiki/File:Moon_trajectory1.svg)