# 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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