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What is the three-body problem? (Advanced)

*How to obtain a mathematical model of the motion of the Moon around the Earth, considering solar perturbations. *

The problem you are referring to is called the "three-body problem", and is one of the most difficult in celestial mechanics. Since the end of the eighteen century, we know that this problem is not integrable, i.e., you cannot find an analytical solution. What you can do is to find an approximate solution, by expanding in series of orbital elements the so called "disturbing function", or, more commonly, integrate numerically the equations of motions. There are no really good ways to do this without mathematics and/or programming.

Both methods are not exactly straightforward, so, if celestial mechanics is a new subject for you, I would suggest to start deriving the solution for the orbital elements of a simpler problem, the so-called two-body problem (i.e., like the case of the Earth orbiting the Sun, without anyother planets). Chapters 4 and 6 in Danby (1992, Fundamental of Celestial Mechanics, Ed. Willmann-Bell) are a very good starting point for the two-body problem. If you are already familiar with the two-body problem, then you may want to read chapter 8 (and eventually 9) in Danby, about the three-body and n-body problem. Another reference is Murray and Dermott 1999, Solar System Dynamics, Ed. Cambridge. Chapters 6 (the disturbing function), 7, and 8 would be a useful source of information. Keep in mind that this is a book that requires a substantial background in mathematics.

Well, I hope I have been of some help, please let me know if you have further questions.

*This page was last updated on July 18, 2015.*

### About the Author

Valerio is currently a Professor at the São Paulo State University in Brazil (UNESP), and he mostly works with asteroid dynamics. He went to college in Italy at the University "La Sapienza", took his Ph. D. in Qstronomy at Cornell University, and then went to Brazil in 2004 for various pos-docs that then "evolved" into his current permanent position.

More information on Valerio and his research is available at: www.feg.unesp.br/~vcarruba/