mnsfs logo The 3-Body Problem
Newton, Lagrange and Universal Gravitation

Isaac Newton tells us that every atom in the universe exerts a tiny gravitational pull on every other one. Fortunately, we don't need to go into that level of detail every time we want to calculate an orbit. Reasonable calculation relies on several simplifications.

For example, since planets are very nearly spheres, one can calculate its gravitational effect as if all of the mass of the planet were concentrated smack dab in its center -- that is, as if the planet were converted to a black hole of the same mass.

Sometimes this is an oversimplification. Our Earth is not a perfect sphere -- it bulges at the Equator -- and as our Earth and Moon orbit around each other this bulge causes little wiggles and precessions which can build up into effects that are significant in just a couple thousand years.

Astronomers, and spacecraft navigators, often calculate an orbit considering only the most important forces first and then go back and add fudge factors to account for perterbations caused by lesser influences. For example, if one wanted to compute Venus' orbit it would make sense to consider just Venus and the Sun first. The result might be accurate enough -- if not, one could add corrections to account for the gravitational attraction of Jupiter, the Earth, and so on.

Lagrange wanted to find an elegant way to calculate orbits for an arbitrary number of celestial bodies. A Universe with one body is trivial -- it just sits there. Two is also easy -- they both orbit in elipses around their common center of mass. But move to a Universe with three bodies and the math blows up. Three bodies of approximately the same mass and approximately equidistant from each other will dance with each other only chaotically with each other until one of the masses gets flung off into the void, or until a collision takes place.

Lagrange Points diagram The special cases Lagrange discovered are only stable if the three masses are huge, small, and tiny with respect to each other. Fortunately for us this describes a system of the Earth, Moon, and a space colony ... until the time when we can build colonies that rival the size of the Moon.


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Last rev: September 06, 2004 at 12:04 AM (a Monday) -- counter


 


 

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