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The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity of solving the effect of three bodies which all pull on each other while moving, a total of six interactions. Mathematician Richard Arenstorf while at NASA solved a special case of this problem, by simplifying the interactions to four, because, the effect of the spacecraft's gravity upon the motion of the vastly more massive Earth and Moon is practically non-existent. Arenstorf found a stable orbit for a spacecraft orbiting between the Earth and Moon, shaped like an '8'


Was Arenstorf's solution purely analytical, or did he use numerical mechanisms? Is the '8' shape an optimal path, meaning the route on which the spacecraft would expand the least amount of energy? If yes, how was this requirement included in the derivation in mathematical form?

If anyone has a clean derivation for this problem, that would be great, or any links to books, other papers, etc.

Note: Apparently there was an earlier related mathoverflow question on this as well:


Arenstorf's technical report is here


The most clean, succint description of the equations I could find were here



Restricted 3 Body Problem mentioned in this book (chapter 8)


Some vpython (visual python) code can simulate Apollo 13 trip,


the result is the 8-orbit as seen here

enter image description here

Some explanation on the math, and superposition of gravities is also mentioned in this link.


I have no idea yet how this would relate to Arenstorf's method, I believe in this code Earth and Moon are not moving, but I thought it'd be nice to share.


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Arenstorf actually found his orbits through numerics. There's a short discussion in Hairer/Norsett/Wanner with references. I'll post it if nobody beats me to it... –  J. M. Jul 31 '11 at 11:53
he used numerics.. interesting. thanks. Hairer/Norsett/Wanner discussion would be much appreciated. –  BB_ML Jul 31 '11 at 14:22
I'm far away from my notes, so it might take me a while to write something. :( I was about to suggest looking at Fehlberg's "Runge-kutta type formulas of high-order accuracy and their application to the numerical integration of the restricted problem of three bodies" but there seems to be no digital copy of that available. Additionally, there's related work (in German) by Filippi that should interest you as well. –  J. M. Jul 31 '11 at 14:41
This was crossposted to MO. In the future, please wait some time before posting your question in multiple fora, and when you do, provide links to the other posts - as you can imagine, it would be frustrating for someone to put time into answering your question here, only to see hear from you that you'd already gotten the solution elsewhere. –  Zev Chonoles Jul 31 '11 at 15:45
See this as well... –  J. M. Jul 31 '11 at 15:49

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