Invariant manifolds and the response of spiral arms in barred galaxies

Abstract

The unstable invariant manifolds of the short-period family of periodic orbits around the unstable Lagrangian points L1 and L2 of a barred galaxy define loci in the configuration space which tak e the form of a trailing spiral pattern. In previous works we have explored the association of such a pattern to the observed spiral patt ern in N-body models of barred-spiral galaxies and found it to be quite relevant. Our aims in the present paper are: a) to investigat e this association in the case of the self-consistent models of Kaufmann & Contopoulos (1996) which provide an approximation of real barred-spiral galaxies. b) to examine the dynamical role played by each of the non-axisymmetric components of the potential, i.e. the bar and the spiral perturbation, and their consequences on the form of the invariant manifolds, and c) to examine the relation of ‘r esponse’ models of barred-spiral galaxies with the theory o f the invariant manifolds. Our method relies on calculating the invariant manifolds for values of the Jacobi constant close to its value for L1 and L2. Our main results are the following: a) The invariant manifolds yield the correct form of the imposed spiral pattern provided that their calculation is done with the spiral potential term tur ned 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-l ike domain of the configuration space, delimited from below by the bound ary of a banana-like non-permitted domain, and from above either by rotational KAM tori or by cantori forming a stickiness zone. On the contrary, if the whole non-axisymmetric perturbation is artificially ‘aligned’ with the bar (i.e. there is no azimuthal shift of th e Lagrangian manifolds), the manifolds support a ring rather than a spiral structure. b) We construct ‘spiral response’ models on the b asis of the theory of the invariant manifolds and examine the connection of the latter to the ‘response’ models (Patsis 2006) used to fi t real barred-spiral galaxies, explaining how are the manif olds related to a number of morphological features seen in such models.