The capture and escape of stars

The shape of galaxies depends on their orbital populations. These populations change through capture into and escape from resonance. Capture problems fall into distinct cases depending upon the shape of the potential well. To visualise the effective potential well for orbital capture, a diagrammatic approach to the resonant perturbation theory of Born is presented. These diagrams we call equiaction sections. To illustrate their use, we present examples drawn from both galactic and Solar System dynamics. The probability of capture for generic shapes of the potential well is calculated. A number of predictions are made. First, there are barred galaxies that possess two outer rings of gas and stars (type R ′ 1R ′ 2). We show how to relate changes in the pattern speed and amplitude of the bar to the strength of the two rings. Secondly, under certain conditions, small disturbances can lead to dramatic changes in orbital shape. This can be exploited as a mechanism to pump counter-rotating stars and gas into the nuclei of disk galaxies. Tidal resonant forcing of highly inclined orbits around a central mass causes a substantial increase in the likelihood of collision. Thirdly, the angular momentum of a potential well is changed by the passage of stars across or capture into the well. This can lead to the creation of holes, notches and high velocity tails in the stellar distribution function, whose form we explicitly calculate.