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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 2body problem reduces to the equation of
the \onebody 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
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 Fall '09
 dion

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