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The Restricted Three-Body Problem

# The Restricted Three-Body Problem - S Widnall 16.07...

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S. Widnall 16.07 Dynamics Fall 2008 Version 1.0 Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem The Three-Body Problem In Lecture 15-17, we presented the solution to the two-body problem for mutual gravitational attraction. We were able to obtain this solution in closed form. For bodies of comparable mass, the solution showed that the bodies orbited about their center of mass, with orbital periods that could be obtained analytically by transforming the two-body problem into a ”Kepler” problem with modified gravitational forces. For a body of negligible mass, such as a satellite orbiting the earth, we were able to place the earth at rest at the center of our coordinates, and obtain a simple expression for the orbit of the satellite. No such solution is available for the three-body problem. The general solution to the three-body problem must be obtained numerically, and even with these tools, the solution is chaotic: that is small changes in initial conditional lead to widely diverging behavior. For analysis of space missions, such a spacecraft moving between planets, or from earth to the moon, we may apply the model known as the restricted three-body problem, in which the mass of a small spacecraft is a ff ected by the gravitational forces from two bodies, but the large bodies do not feel the influence of the spacecraft. The restricted three-body problem is most easily analyized/understood in a coordinate system rotating with the two primary bodies. Since the solution of the two-body problem is that of the two primary masses rotating at constant angular velocity about their center of mass, in a coordinate system rotating with angular velocity , these two bodies are stationary. The third body moves in their gravitational fields and experiences the additional acceleration terms associated with the rotation of the coordinate system. Consider a two-body system of mass M 1 and M 2 in an inertial references frame. We add a small mass m 3 which does not a ff ect the large masses. 1

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We consider planar x,y motion only. The equations of motion are x ¨ 1 = GM 2 ( x 1 x 2 ) (1) (( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ) (3 / 2) x ¨ 2 = GM 1 ( x 2 x 1 ) (2) (( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 ) (3 / 2) x ¨ 3 = GM 1 ( x 3 x 1 ) GM 2 ( x 3 x 2 ) (3) (( x 3 x 1 ) 2 + ( y 3 y 1 ) 2 ) (3 / 2) (( x 3 x 2 ) 2 + ( y 3 y 2 ) 2 ) ( 3 / 2) y ¨ 1 = GM 2 ( y 1 y 2 ) (4) (( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ) (3 / 2) GM 1 ( y 2 y 1 ) y ¨ 2 = (( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 ) (3 / 2) (5) ¨ = GM 1 ( y 3 y 1 ) GM 2 ( y 3 y 2 ) (6) y 3 (( x 3 x 1 ) 2 + ( y 3 y 1 ) 2 ) (3 / 2) (( x 3 x 2 ) 2 + ( y 3 y 2 ) 2 ) ( 3 / 2) The solution to the motion of M 1 and M 2 is a Kepler problem, where the two bodies move in circular orbits about their center of mass with a frequency G ( M 1 + M 2 ) = R 3 , (7) where R is the distance between them. R = r 1 + r 2 ( r 1 is negative). We can use these equations to solve numerically for the motion of the small mass m 3 in the presence of the gravitational field of M 1 and M 2 ; in general no analytic solution is possible.
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The Restricted Three-Body Problem - S Widnall 16.07...

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