Approximating Stellar Orbits: Improving on Epicycle Theory

Already slightly eccentric orbits, 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 flat orbits in axisymmetric stellar systems are derived and compared to 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 phasespace coordinates to a set of action-angle variables. Unfortunately, the most accurate orbit approximation leads to non-analytical R(t). However, from this approximation simple and yet very accurate estimates can be derived for the periand apo-centers, frequencies, and actions 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. Subject headings: stellar dynamics, celestial mechanics – galaxies: kinematics and dynamics – methods: analytical e-mail: [email protected]