The restricted three-body problem consists of two masses in circular orbits about each other and a third body of much smaller mass whose effect on the two larger bodies can be neglected.
(a) Define an effective potential V(x, y) for this problem where the x axis is the line of the two larger masses. Sketch the function V(x, 0) and show that there are two “valleys” (points of stable equilibrium) corresponding to the two masses. Also show that there are three “hills” (three points of unstable equilibrium).
(b) Using a computer program, calculate some orbits for the restricted three-body problem. Many orbits will end with ejection of the smaller mass. Start by assuming a position and a vector velocity for the small mass.
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