Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

Abstract

In the presence of a strong m = 2 component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L 1 or L 2 . The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine-Gordon equation. Numerical examples are given in an N-body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar-spiral pattern.

Invariant manifolds and the response of spiral arms in barred galaxies

“…(8). Beyond this distance, the manifolds exhibit a typical behavior called a "bridge" in our previous works (Voglis et al 2006a;Tsoutsis et al 2008). Namely, the manifold W U PL1 forms an inner spur consisting of a number of lobes connecting segments of the manifold from A to their continuation, which starts in the neighborhood of the other periodic orbit, PL2.…”

“…We should stress that an analysis of the Floquet matrix of the PL1 or PL2 families yields that only the directions of the unstable invariant manifolds W In summary, the theory of the invariant manifolds, viewed as either the loci on which lies the continuous flow of a swarm of orbits (Romero-Gomez et al 2006, 2007, or the loci of apocentric positions of these orbits (Voglis et al 2006a;Tsoutsis et al 2008), predicts the formation by the manifolds of a trailing spiral pattern beyond corotation. Naturally, the central question that should be posed now is whether (and up to what extent) the spiral arms formed self-consistently in real galaxies can be associated with the spiral patterns formed by the invariant manifolds W U PL1,2 .…”

“…To understand how the chaotic orbits may establish a longtime flow supporting the spiral structure, Voglis et al (2006a) examined a particular subset of points of the unstable manifold W U PL1 , namely the locus of all apocentric positions of the orbits on W U PL1 . This is defined by taking the intersection of W U PL1 with the so-called surface of section of the apocentric positions on a hypersurface of constant E J , defined as the set of phase space points satisfying p r = 0,ṗ r < 0.…”

“…Naturally, the central question that should be posed now is whether (and up to what extent) the spiral arms formed self-consistently in real galaxies can be associated with the spiral patterns formed by the invariant manifolds W U PL1,2 . In our previous works (Voglis et al 2006a;Tsoutsis et al 2008), we examined this question by considering the spiral arms formed in an N-Body model of a barred galaxy and found such an association to be quite relevant.…”

Analytical forms of chaotic spiral arms

“…The manifold theory was proposed in 2006 (Voglis et al 2006b;Romero-Gomez et al 2006) and was explored in detail in a number of subsequent papers (Romero-Gomez et al 2007;Tsoutsis et al 2008Tsoutsis et al , 2009Athanassoula et al 2009a,b;Harsoula et al 2011;Athanassoula 2012). The theory predicts a number of morphological correlations between the spiral arms and the bar strength and/or the pattern speed (see Pérez-Villegas et al (2015) for comparison of these features with observations as well as Dobbs & Baba (2014) for a review).…”