On the probability of major-axis precession in triaxial ellipsoidal potentials

Orbits in triaxial ellipsoidal potentials precess about either the major or minor axis of the ellipsoid. In standard perturbation theory it can be shown that a circular orbit will precess about the minor axis if its angular momentum vector lies in a region bounded by two great circles which pass through the intermediate axis and which are inclined with minimum separation $i_T$ from the minor axis. We test the accuracy of the standard formula for $i_T$ by performing orbit integrations to determine $i_S$, the simulated turnover angle corresponding to $i_T$. We reach two principal conclusions: (i) $i_S$ is usually greater than $i_T$, by as much as 12 degrees even for moderate triaxialities, $A/1.2<B<C/0.8$. This reduces the expected frequency of polar rings. (ii) $i_S$ is not a single, well-defined number but can vary by a few degrees depending upon the initial phase of the orbit. This means that there is a reasonable probability for capture of gas onto orbits which precess about both axes. Interactions can then lead to substantial loss of angular momentum and subsequent infall to the galactic centre.