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Register a new userRegular motions in double bars. II. Survey of trajectories and 23 models
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We show that stable double-frequency orbits form the backbone of double bars, because they trap around themselves regular orbits, as stable closed periodic orbits do in single bars, and in both cases the trapped orbits occupy similar volume of phase-space. We perform a global search for such stable double-frequency orbits in a model of double bars by constructing maps of trajectories with initial conditions well sampled over the available phase-space. We use the width of a ring sufficient to enclose a given map as the indicator of how tightly the trajectory is trapped around a double-frequency orbit. We construct histograms of these ring widths in order to determine the fraction of phase-space occupied by ordered motions. We build 22 further models of double bars, and we construct histograms showing the fraction of the phase-space occupied by regular orbits in each model. Our models indicate that resonant coupling between the bars may not be the dominant factor reducing chaos in the system.
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Bars in galaxies are mainly supported by particles trapped around stable periodic orbits. These orbits represent oscillatory motion with only one frequency, which is the bar driving frequency, and miss free oscillations. We show that a similar situation takes place in double bars: particles get trapped around parent orbits, which in this case represent oscillatory motion with two frequencies of driving by the two bars, and which also lack free oscillations. Thus the parent orbits, which constitute the backbone of an oscillating potential of two independently rotating bars, are the double-frequency orbits. These orbits do not close in any reference frame, but they map onto loops, first introduced by Maciejewski & Sparke (1997). Trajectories trapped around the parent double-frequency orbit map onto a set of points confined within a ring surrounding the loop.
The backbone of double bars is made out of double-frequency orbits, and loops, their maps, indicate the bars extent, morphology and dynamics.
The intrinsic photometric properties of inner and outer stellar bars within 17 double-barred galaxies are thoroughly studied through a photometric analysis consisting of: i) two-dimensional multi-component photometric decompositions, and ii) three-dimensional statistical deprojections for measuring the thickening of bars, thus retrieving their 3D shape. The results are compared with previous measurements obtained with the widely used analysis of integrated light. Large-scale bars in single- and double-barred systems show similar sizes, and inner bars may be longer than outer bars in different galaxies. We find two distinct groups of inner bars attending to their in-plane length and ellipticity, resulting in a bimodal behaviour for the inner/outer bar length ratio. Such bimodality is related neither to the properties of the host galaxy nor the dominant bulge, and it does not show a counterpart in the dimension off the disc plane. The group of long inner bars lays at the lower end of the outer bar length vs. ellipticity correlation, whereas the short inner bars are out of that relation. We suggest that this behaviour could be due to either a different nature of the inner discs from which the inner bars are dynamically formed, or a different assembly stage for the inner bars. This last possibility would imply that the dynamical assembly of inner bars is a slow process taking several Gyr to happen. We have also explored whether all large-scale bars are prone to develop an inner bar at some stage of their lives, possibility we cannot fully confirm or discard.
The dynamics in three-dimensional billiards leads, using a Poincare section, to a four-dimensional map which is challenging to visualize. By means of the recently introduced 3D phase-space slices an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincare recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps we find that: (i) Trapping takes place close to regular structures outside the Arnold web. (ii) Trapping is not due to a generalized island-around-island hierarchy. (iii) The dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.
The method to study oscillating potentials of double bars, based on invariant loops, is introduced here in a new way, intended to be more intelligible. Using this method, I show how the orbital structure of a double-barred galaxy (nested bars) changes with the variation of nuclear bars pattern speed. Not all pattern speeds are allowed when the inner bar rotates in the same direction as the outer bar. Below certain minimum pattern speed orbital support for the inner bar abruptly disappears, while high values of this speed lead to loops that are increasingly round. For values between these two extremes, loops supporting the inner bar extend further out as its pattern speed decreases, and they become more eccentric and pulsate more. These findings do not apply to counter-rotating inner bars.
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