Historically, the study of orbits in potentials has focused on periodic orbits. In systems like disk galaxies small perturbations to closed periodic orbits (e.g., the epicyclic and vertical perturbations of circular orbits) provided a good analytic description of most orbits. Self-consistent distribution functions are thought to be “parented” by stable periodic orbits (Arnold 1978). Early works (e.g., Contopoulos & Papayannopoulos 1980) identified and characterized the stability properties of the periodic orbit families in rapidly rotating bars. The most important periodic families in two-dimensional bars were identified as the prograde x1 family, which is elongated along the major axis of the bar, and the prograde stable x2 and unstable x3 families, which are elongated perpendicular to the bar (primarily found at small radii). The retrograde x4 (stable) orbit family is also elongated perpendicular to the bar at small radii, but becomes rounder as it extends to larger radii (for detailed description of orbit families and how they are identified see Contopoulos & Grosbol 1989; Sellwood & Wilkinson 1993; Binney & Tremaine 2008; Sellwood 2014b). In the frame of reference rotating with the bar, all of these families are characterized by a 1:2 resonance between the tangential oscillation frequency (Ωϕ) and the radial or epicyclic frequency (ΩR). Indeed, studies of orbits in 2D N-body bars largely confirmed the picture arising from the study of periodic orbits and showed that many regular orbits elongated along the bar were parented by x1 orbits, a small fraction were parented by retrograde x4 orbits (Sparke & Sellwood 1987), and none were parented by prograde x2 orbits. The realization that bars can also undergo buckling instabilities (Combes & Sanders 1981; Raha et al. 1991), which makes them develop substantial vertical thickness and peanut-shaped morphologies, led to the study of periodic orbits in three-dimensional bars (Pfenniger 1984; Martinet & de Zeeuw 1988; Pfenniger & Friedli 1991; Skokos et al. 2002a, 2002b). It was shown that the appearance of specific morphological features in images of bars, such as the X-shape and peanut features seen in edge-on bars and the boxy/rectangular isophotes and “ansae” of face-on bars, could be explained by orbits trapped around specific periodic orbit families (Patsis et al. 2002, 2003, 2010). The introduction of a third dimension did not drastically change the picture of the nature of periodic orbits, and it was found that 3D bars are composed primarily of vertical bifurcations (resonances) of the x1 family and a few additional families (e.g., Pfenniger & Friedli 1991; Skokos et al. 2002a, 2002b). Most studies of periodic orbits in analytic potentials consider prograde x2 orbits (but not retrograde x4 orbits) to be another fundamental building block of bars (Skokos et al. 2002a; Binney & Tremaine 2008). In our study we do not find any orbits parented by the periodic prograde x2 orbit in our initial bar model, but we do find orbits that are parented by the periodic retrograde x4 orbit—a result that is consistent with previous studies (Sparke & Sellwood 1987; Pfenniger & Friedli 1991; Voglis et al. 2007).
Citation: Sellwood 2014b
Citation function: Background; Background