So close to a perfect circle

Consider the motion of a point mass under the influence of an attractive central force. The potential for the force is proportional to the inverse distance from the central point O to the point mass. For every value of the angular momentum $$L$$ of the object around O we have an equilibrium point in the radial direction, which corresponds to a circular orbit. We now consider a small perturbation of a circular orbit into a low eccentricity ellipse with the same angular momentum. If we look at the radial distance from O to the object on the elliptical orbit, we see that the distance changes from some $$r_{min}$$ to $$r_{max}$$ with some frequency $$f$$. If at $$L_1=1~\mbox{kg}\cdot\mbox{m}^2/\mbox{s}$$ we get $$f_1=f(L_1)=1~\mbox{Hz}$$, find the frequency $$f_2$$ for $$L_2=10~\mbox{kg}\cdot\mbox{m}^2/\mbox{s}$$ in Hz.

Bonus thought: The value of the frequency at $$L_3=0$$ (without angular momentum) is defined as the bare frequency. Is this a finite value?

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