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The three

The three - The three-body problem The equations of motion...

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The three-body problem The equations of motion of two bodies moving under their mutual gravitational attraction are soluble algorithmically. However, as soon as a third body is added to the scenario, the equations become generally insoluble algorithmically and we are fonsoluble algorithmically and we are forced to numeric solutions. We shall look at the case of a third body of negligible mass moving in the gravitational field due to two larger bodies: this would be the case for, say, a space probe or a small satellite moving under the influence of the Earth and Sun. We take "universal" axes X,Y,Z, in which one large body, mass m 0 is the centre of coordinate system x,y,z, centred on m 0 and parallel to X,Y,Z (see diagram). At a distance r 1 from m 0 is another large body, mass m 1 . A small body of mass m, negligible compared to m 0 and m 1 , is at distance r from m 0 and [ro] from m 1 . Now the equation of motion of m can be broken up to three components. In the X direction we have: m d 2 X = G m m 0 (X 0 - X) + Gm m 1 (X 1 - X) __ _____ ______ ____ ______ dt 2 r

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