Presentation #101.06 in the session Galaxy Dynamics 1: Spirals and Bars.
Highly eccentric orbits in the gravitational potentials of typical galaxy disks have different apsidal precession frequencies than near circular (epicyclic) orbits of equal semi-major axis. As a result, at a given pattern speed, resonant orbits with a significant eccentricity have a mean radius different than the Lindblad resonance of the corresponding near circular orbit. Thus, the location of the ILR curve in a radius-frequency plot, depends on a second dimension, eccentricity, and is in fact a region not a curve in such a plot. Since eccentric orbits range over a finite range of radius, individual resonant eccentric orbits are better represented by a line segment over that range in a radius frequency plot, rather than their mean radius. At most orbital frequencies, stable orbits do not extend over the full range of eccentricities. The area of the extended, stable, eccentric-orbit, resonance (EER) region depends sensitively on the structure of the disk potential. An example potential consisting of an exponential disk plus a central point-mass illustrates this as the mass ratio of the two components varies. More generally, such examples suggest how central concentration affects the EER region, and determines the range of eccentricities for stable resonant orbits in galaxy disks. These results have important ramifications for bars or spiral waves built around a backbone of resonant orbits.