Time-dependent secular evolution in galaxies

Lynden-Bell & Kalnajs (1972) presented a useful formula for computing the long-range torque between spiral arms and the disk at large. The derivation uses second-order perturbation theory and assumes that the perturbation slowly grows over a very long time: the time-asymptotic limit. This formula has been widely used to predict the angular momentum transport between spiral arms and stellar bars between disks and dark-matter halos. However, this paper shows that the LBK time-asymptotic limit is not appropriate because the characteristic evolution time for galaxies is too close to the relevant dynamical times. We demonstrate that transients, not present in the time-asymptotic formula, can play a major role in the evolution for realistic astronomical time scales. A generalisation for arbitrary time dependence is presented and illustrated by the bar--halo and satellite--halo interaction. The natural time dependence in bar-driven halo evolution causes quantitative differences in the overall torque and qualitative differences in the physical- and phase-space location of angular momentum transfer. The time-dependent theory predicts that four principal resonances dominate the torque at different times and accurately predicts the results of an N-body simulation. In addition, we show that the Inner Lindblad Resonance (ILR) is responsible for the peak angular momentum exchange but, due to the time dependence, the changes occur over a broad range of energies, radii and frequencies. We describe the implication of these findings for the satellite--halo interaction using a simple model and end with a discussion of possible impact on other aspects secular galaxy evolution.