Manifold spirals in barred galaxies with multiple pattern speeds

¹ Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efessiou 4, 115 21 Athens, Greece
e-mail: cefthim@academyofathens.gr, mharsoul@academyofathens.gr, gcontop@academyofathens.gr
² Department of Mathematics, University of Padova, Via Trieste, 63, 35121 Padova, Italy

Received: 8 October 2019
Accepted: 22 November 2019

Abstract

In the manifold theory of spiral structure in barred galaxies, the usual assumption is that the spirals rotate with the same pattern speed as the bar. Here, we generalize the manifold theory under the assumption that the spirals rotate with a different pattern speed than the bar. More generally, we consider the case in which one or more modes, represented by the potentials V₂, V₃, etc., coexist in the galactic disk in addition to the bar’s mode V_bar, but the modes rotate with pattern speeds, Ω₂, Ω₃, etc., which are incommensurable between themselves and with Ω_bar. Through a perturbative treatment (assuming that V₂, V₃, etc. are small with respect to V_bar), we then show that the unstable Lagrangian points L₁ and L₂ of the pure bar model (V_bar, Ω_bar) are continued in the full model as periodic orbits, in the case of one extra pattern speed, or as epicyclic “Lissajous-like” unstable orbits, in the case of more than one extra pattern speeds. We use GL₁ and GL₂ to denote the continued orbits around the points L₁ and L₂. Furthermore, we show that the orbits GL₁ and GL₂ are simply unstable. As a result, these orbits admit invariant manifolds, which can be regarded as the generalization of the manifolds of the L₁ and L₂ points in the single pattern speed case. As an example, we computed the generalized orbits GL₁, GL₂, and their manifolds in a Milky-Way-like model in which bar and spiral pattern speeds were assumed to be different. We find that the manifolds produce a time-varying morphology consisting of segments of spirals or “pseudorings”. These structures are repeated after a period equal to half the relative period of the imposed spirals with respect to the bar. Along one period, the manifold-induced time-varying structures are found to continuously support at least some part of the imposed spirals, except at short intervals around specific times at which the relative phase of the imposed spirals with respect to the bar is equal to ±π/2. The connection of these effects to the phenomenon of recurrent spirals is discussed.