Extending Newton's Apsidal Theorem

For the Kepler potential $\Phi_{0} = \mu/r$, the orbit is closed and the apsidal angle is a full circle $\Delta \Theta = 2 \pi$. For the potential $\Phi (r) = \mu/r^{2-n}$, the orbit is open [Bertrand's theorem] and the trajectory has the general shape of a rosette. Newton found an expression for the Apsidal precession for small eccentricities $\Delta \Theta = 2 \pi/\sqrt{n}$. We extend this result for arbitrary orbital parameters; we introduce a description in terms of an effective angular momentum and a Keplerian potential $\Phi_{\rm eff} = - \rho/r$. We find an exact expression for the Apsidal precession. For $|n|<0.3$, we find $\Delta \Theta = 2\pi/n^{\alpha}$ and find expressions for the exponent $\alpha$ that are correct with 1\%. We discuss possible applications to the orbits of stars in elliptical galaxies.