Practical application of KAM theory to galactic dynamics: II. Application to weakly chaotic orbits in barred galaxies

Abstract

Owing to the pioneering work of Contopoulos, a strongly barred galaxy is known to have irregular orbits in the vicinity of the bar. By definition, irregular orbits can not be represented by action-angle tori everywhere in phase space. This thwarts perturbation theory and complicates our understanding of their role in galaxy structure and evolution. This paper provides a qualitative introduction to a new method based on KAM theory for investigating the morphology of regular and irregular orbits based on direct computation of tori described in Paper 1 and applies it to a galaxy disc bar. Using this method, we find that much of the phase space inside of the bar radius becomes chaotic for strong bars, excepting a small region in phase space between the ILR and corotation resonances for orbits of moderate ellipticity. This helps explain the preponderance of moderately eccentric bar-supporting orbits as the bar strength increases. This also suggests that bar strength may be limited by chaos! The chaos results from stochastic layers that form around primary resonances owing to separatrix splitting. Most investigations of orbit regularity are performed using numerical computation of Lyapunov exponents or related indices. We show that Lyapunov exponents poorly diagnose the degree of stochasticity in this problem; the island structure in the stochastic sheaths allow orbit to change morphology while presenting anomalously small Lyapunov exponent values (i.e. weak chaos). For example, a weakly chaotic orbit may appear to change its morphology spontaneously, while appearing regular except during the change itself. The numerical KAM approach sensitively detects these dynamics and provides a model Hamiltonian for further investigation. It may underpredict the number of broken tori for strong perturbations.