Morphologies introduced by bistability in barred-spiral galactic potentials

MNRAS 448, 3081–3092 (2015) doi:10.1093/mnras/stv206 Morphologies introduced by bistability in barred-spiral galactic potentials L. Tsigaridi1,2 and P. A. Patsis1‹ 1 Research 2 Section Center for Astronomy, Academy of Athens, Soranou Efessiou 4, GR-115 27 Athens, Greece of Astrophysics, Astronomy and Mechanics, Department of Physics, University of Athens, GR-15783 Zografos, Athens, Greece Accepted 2015 January 28. Received 2015 January 23; in original form 2014 November 24 We investigate the orbital dynamics of a barred-spiral model when the system is rotating slowly and corotation is located beyond the end of the spiral arms. In the characteristic of the central family of periodic orbits, we find a ‘bistable region’. In the response model, we observe a ring surrounding the bar and spiral arms starting tangential to the ring. This is a morphology resembling barred-spiral systems with inner rings. However, the dynamics associated with this structure in the case we study is different from that of a typical bar ending close to corotation. The ring of our model is round, or rather elongated perpendicular to the bar. It is associated with a folding (an ‘S’-shaped feature) of the characteristic of the central family, which is typical in bistable bifurcations. Along the ‘S’ part of the characteristic, we have a change in the orientation of the periodic orbits from an x1-type to an x2-type morphology. The orbits populated in the response model change rather abruptly their orientation when reaching the lowest energy of the ‘S’. The spirals of the model follow a standard ‘precessing ellipses flow’ and the orbits building them have energies beyond the ‘S’ region. The bar is structured mainly by sticky orbits from regions around the stability islands of the central family. This leads to the appearance of X features in the bars on the galactic plane. Such a bar morphology appears in the unsharp-masked images of some moderately inclined galaxies. Key words: chaos – galaxies: kinematics and dynamics – galaxies: spiral – galaxies: structure. 1 I N T RO D U C T I O N In dynamical systems, a ‘bistability situation’ usually refers to cases where a system has two stable equilibrium states. In a bifurcation diagram, the curve of steady state displays an ‘S’ shape as a certain parameter of the system varies. The ‘S’ is delimited by two saddle-node bifurcation points. Between them, we have two stable and one unstable steady states (see e.g. Angeli, Ferrell & Sontag 2003; Lynch 2007; Strogatz 2014). The corresponding situation in Hamiltonian Galactic Dynamics is depicted in the characteristic of a family of periodic orbits as two successive tangent bifurcations (see e.g. Contopoulos 2004) facing opposite directions. These two bifurcations share the same unstable branch. In other words, the characteristic folds twice as the varying parameter, usually the Jacobi constant EJ , increases. Foldings of the characteristic have been encountered by Skokos, Patsis & Athanassoula (2002a) and Skokos, Patsis & Athanassoula (2002b), in 3D Ferrers bar potentials. However, the way they affect the face-on morphology of a model has not been examined in those papers. Nevertheless, it was clear that the foldings of the characteristics affect to a larger degree slowly rotating models. In this paper, we present the implications of  E-mail: the presence of such a folding of the characteristic of the main family of periodic orbits for the dynamics of a slowly rotating barred-spiral potential. Slowly rotating models of disc galaxies have been proposed in the past to describe the dynamics of normal (non-barred) spiral galaxies with open spiral arms. In these models (stellar and gaseous), the symmetric, strong spiral structure extends inside corotation (Contopoulos & Grosbøl 1986, 1988; Patsis, Contopoulos & Grosbøl 1991; Kranz, Slyz & Rix 2003; Martos et al. 2004; Junqueira et al. 2013). Contrarily, in barred galaxy models, corotation is usually placed close to the end of the bar (Contopoulos 1980). Recently, Font et al. (2014) presented a list with 32 barred galaxies in which they estimated the ratio of the corotation to the bar radius, Rc /Rb , to be between 0.94 ± 0.08 < Rc /Rb < 2.1 ± 0.5. Model bars ending well inside corotation have been found in N-body simulations (Combes & Elmegreen 1993) as well as in response models of barred potentials derived from near-infrared observations (Rautiainen, Salo & Laurikainen 2008). In all these studies, slowly rotating bars are associated with late-type barred-spiral galaxies. It is generally believed that bars in barred galaxies are supported by the x1 family of elliptical, stable, periodic orbits, which extends along the major axis of the bar (Contopoulos & Papayannopoulos 1980), or, in the three-dimensional case, by the corresponding families of the  C 2015 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society ABSTRACT 3082 L. Tsigaridi and P. A. Patsis MNRAS 448, 3081–3092 (2015) The structure of the paper is the following: in Section 2, we briefly present again our potential (for more details, see Paper I). The results of our study are described in Section 3 and refer to the building of the response features, which are the ring, the bar and the spirals. These results are discussed in Section 4 and in Section 5 we enumerate our conclusions. 2 S U M M A RY O F M O D E L P RO P E RT I E S The model has been extensively described in Paper I. It is a twodimensional model of the general form  mc (r) cos (mϕ) + ms (r) sin (mϕ). (r, ϕ) = 0 (r) + m=2,4,6 (1) The components 0 , mc , andms of the equation above are given as polynomials of the form n an rn , n = 0, . . . , 8. The radial variation of the perturbation force normalized over the radial axisymmetric one is given in fig. 1 in Paper I. The equations of motion are derived from the Hamiltonian H ≡  1 1 2 ẋ + ẏ 2 + (x, y) − 2p (x 2 + y 2 ) = EJ , 2 2 (2) where (x, y) are the coordinates in a Cartesian frame of reference rotating with angular velocity p . (x, y) is the potential (1) in Cartesian coordinates with the bar aligned approximately with the y-axis, EJ is the numerical value of the Jacobi constant, hereafter called the energy, and dots denote time derivatives. 3 S L OW LY ROTAT I N G M O D E L S By varying p between 10 < p < 30 km s−1 kpc−1 , we obtained always a kind of barred-spiral response. In this range of pattern speeds, the corotation radius of the models, Rc , takes values between 12  Rc  4.3 kpc, respectively. Nevertheless, while the pitch angle of the response spirals varied considerably in models with different pattern speeds (it was larger for lower pattern speeds), the radius of the response bar varied only between 2.85 < Rb < 2.95 kpc. For p > 30 km s−1 kpc−1 , the size of the response bar was clearly decreasing. For example, for p = 35 km s−1 kpc−1 , we estimated it to be about 2.45 kpc. The changes that are introd (...truncated)