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ay45c4-page9 - It is convenient to reduce the system to...

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Unformatted text preview: It is convenient to reduce the system to simpler form by working in terms of the separation vector of the two bodies: r = r1 r2 = vector from body 2 to body 1. r1 − r2 = r M1 r1 r2 CoM M2 Then consider equation (1) divided by M1 minus equation (2) divided by M2: r r = (1 r2) = Gj(rM1 +rM32) (r1 r2) j 1 or 2  r = GrM r 3 where M = M1 + M2 is the total mass of the system. This equation says that body 1 moves as if it was of very small mass being attracted by mass (M1 + M2) at body 2; the separation vector is accelerated by the total mass of the system. The fact that the 2-body problem reduces to the equation of the \one-body problem" (test mass in the central force eld of a ctitious mass M ) is an amazing and nonobvious result! With the relations r = r1 r2 M1r1 + M2r2 = 0 we can nd how to calculate r1 and r2 from the solutions for r: r1 = M2 r M r 2 = M1 r : M 68 ...
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