Invariant manifolds and the response of spiral arms in barred galaxies

The unstable invariant manifolds of the short-period family of periodic orbits around the unstable Lagrangian points $L_1$ and $L_2$ of a barred galaxy define loci in the configuration space which take the form of a trailing spiral pattern. In the present paper we investigate this association in the case of the self-consistent models of Kaufmann & Contopoulos (1996) which provide an approximation of real barred-spiral galaxies. We also examine the relation of `response' models of barred-spiral galaxies with the theory of the invariant manifolds. Our main results are the following: The invariant manifolds yield the correct form of the imposed spiral pattern provided that their calculation is done with the spiral potential term turned on. We provide a theoretical model explaining the form of the invariant manifolds that supports the spiral structure. The azimuthal displacement of the Lagrangian points with respect to the bar's major axis is a crucial parameter in this modeling. When this is taken into account, the manifolds necessarily develop in a spiral-like domain of the configuration space, delimited from below by the boundary of a banana-like non-permitted domain, and from above either by rotational KAM tori or by cantori forming a stickiness zone. We construct `spiral response' models on the basis of the theory of the invariant manifolds and examine the connection of the latter to the `response' models (Patsis 2006) used to fit real barred-spiral galaxies, explaining how are the manifolds related to a number of morphological features seen in such models.