Recovering galactic orbits by perturbation theory

Abstract

With the aim of finding useful representations for the orbit structure in galaxy potentials, we have applied three Hamiltonian perturbations methods to a perturbed isochrone sphere. We find that axisymmetric galaxy potentials can be treated successfully as perturbed spherical systems, for axial ratios at least as flat as 0.6 in density. In these potentials the orbital tori can be constructed by perturbation theory and a corresponding approximate invariant ('third integral') can be found. We test these tori by comparing with numerically integrated orbits, and by calculating the component of the Hamiltonian flow normal to them. We show that they give an excellent description of the orbits in the meridional plane of the system, but the methods are also applicable to orbits with azimuthal motion. The methods we investigate are: (i) a first-order averaging method, which is less successful in this case, for reasons we explain; (ii) a resonant method, which gives a good description of both loops and boxes in the meridional plane; (iii) a higher order method based on the Kolmogorov-Arnold-Moser theory. This is restricted to orbit families already present in the unperturbed system - in this case loops - but improves accuracy at higher orders.