Approximating stellar orbits: Improving on epicycle theory

Orbits that are already slightly eccentric, such as those occupied by many old stars in the Galactic disk, are not well approximated by Lindblad's epicycle theory. Here alternative approximations for hat orbits in axisymmetric stellar systems are derived and compared with results from numeric integrations. All of these approximations are more accurate than Lindblad's classical theory. I also present approximate, but canonical, maps from ordinary phase-space coordinates to a set: of action-angle variables. Unfortunately, the most accurate orbit approximation leads to nonanalytical R(t). However, from this approximation simple and yet very accurate estimates can be derived for the peri- and apocenters, frequencies, and action integrals of galactic orbits, even for high eccentricities, Moreover, further approximating this approximation allows for an analytical R(t) and still an accurate approximation to galactic orbits, even with high eccentricities.