1-D Effective Potential

The potential, , for a Plummer sphere is shown. An orbit with particular angular momentum has an effective potential, _eff. The circular orbit, radius R_g, is the minimum energy orbit with this angular momentum. A radial impulse adds energy (but no AM) to yield an orbit that oscillates in radius.

In the limit of small amplitude oscillations, the form of the minimum in _eff is parabolic, giving SHM motion in radius.

The x-y orbital trajectory of the system shown on the left. Clearly, the frequency of the radial oscillation is significantly higher than the orbital frequency, so the orbit isn't closed and regresses quickly.