Actions of highly eccentric orbits

ABSTRACT The challenge presented by computing actions for eccentric orbits in axisymmetric potentials is discussed. In the limit of vanishing angular momentum about the potential’s symmetry axis, there is a clean distinction between box and loop orbits. We show that this distinction persists into the regime of non-zero angular momentum. In the case of a Stäckel potential, there is a critical value $I_{3\rm crit}(E)$ of the third integral $I_3$ below which $I_3$ does not contribute to the centrifugal barrier. An orbit is of box or loop type according as its value of $I_3$ is smaller or greater than $I_{3\rm crit}$. We give algorithms for determining $I_{3\rm crit}(E)$ and the critical action $J_{z\rm crit}$ below which orbits in any given potential are boxes. It is hard to compute the actions and especially the frequencies of orbits that have $J_z\simeq J_{z\rm crit}$ using the Stäckel Fudge. A modification of the Fudge that alleviates the problem is described.