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It is useful to define /eps/ such that Vo = Vc(1+eps). Then equation [1] can be
expanded to leading order as
R(theta) = Ro(1 + 2 eps - 2 eps cos(theta))
which checks out because:
-- R(theta) = Ro when theta=0
-- The new orbit is (2 eps) higher on average, which agrees with what we
get from KE + virial arguments.
========
To finish the task, we need to know dt/d(theta). It is tempting to write
R(theta)/Vo by dimensional analysis, but alas there is more to physics than
dimensional analysis. [I tried and failed to do this problem in my head;
this is the point where I dropped the ball.] The correct argument, using
conservation of momentum, throws another factor of R(theta)/Ro in there.
At the end of the task we get
t(new)
------- = 1 + 3 eps + higher-order terms
t(old) + terms that vanish when averaged over a whole orbit
That's the same "3" that several people have obtained by other means.