I had always wanted a nice animation depicting periodic solutions of the restricted three-body problem, and was always a skosh underwhelmed by the animations I’ve found. I then decided, on a quiet afternoon, to see if the new (to me) capabilities of Mathematica were up to the task, and I was not at all disappointed:

(click on the image to see a bigger version of the animation)

This is one of my favorite Arenstorf orbits (see also these slides); I went with the route taken in Hairer/Nørsett/Wanner and used the Bulirsch-Stoer extrapolation method for numerically solving the underlying differential equation (Method -> "Extrapolation" in NDSolve[]). (Note that for this animation, I used the Arenstorf orbit depicted in the Bulirsch-Stoer paper instead of the slightly more complicated path depicted in Hairer/Nørsett/Wanner; I can be persuaded to produce an animation for that version, though… ;) )

It wasn’t much trouble to depict the Earth, the Moon, and a spaceship. For ease of visualization, I (grudgingly) decided to exaggerate the sizes of the Moon and the spaceship; if I went with the actual scale relative to the Earth, you’d probably have a hard time seeing those. Mathematica‘s Texture[] function, with texture maps obtained from here, as well as ExampleData[{"Geometry3D", "SpaceShuttle"}] for the spaceship, were useful in this regard.

The one thing I did have trouble with is depicting the starry backdrop of outer space in Graphics3D[], so I elected to skip it. Still, the animation looks quite picturesque, no?

Added 2/7/2012:

I finally decided to do the “four loop” version depicted in Hairer/Nørsett/Wanner as well:

(click on the image to see a bigger version of the animation)

Just wanted to say HI. I found your blog a few days ago on Technorati and have been reading it over the past few days.