5 Periodic orbits

Much of the orbital structure in a system can be assessed from the study of its periodic orbits, which sometimes are considered as the skeleton of the system. While periodic orbits have been widely investigated in 2D and 3D within bars, only few papers (Athanassoula et al. 1983; Contopoulos & Grosbol 1989; Sellwood & Wilkinson 1993) discuss them in discs surrounding bars.

Figure 4: Distinct looped periodic orbits in the axisymmetric potential  $\Phi _{\circ }$ of Eq. (6) which close after one rotation in the rotating frame and are symmetric with respect to the x- and/or y-axes. The orbits all have $(H-H_{\hbox{\tiny OLR}})/v_{\circ}^2=0.244$ and are not drawn at the same relative scale. For cross-identification, the line thickness of the orbits is the same as for the portion of the characteristic curves in Fig. 5a to which they belong. Orbits in the lower and upper halfs of the diagram give rise to respectively stable and unstable orbits at low eccentricities when the bar is added.

A good approach to investigate periodic orbits in a 2D rotating barred potential is to start with the axisymmetric limit. In this case, the only orbits which close whatever the value of $\Omega_{\rm P}$ are the circular orbits. All other bound orbits can be thought as a libration motion around these orbits and look like a rosette which never closes, except for some exceptional potentials like a point mass with $\Omega_{\rm P}=0$, or at resonances, where the radial and azimuthal frequencies $\omega_{\rm R}$ and $\omega_{\phi}$ satisfy the relation:

$$n_{\phi}\omega_{\rm R}=n_{\rm R}(\omega_{\phi}-\Omega_{\rm P}),$$ (8)

with integer values of $n_{\rm R}\geq 0$ and $n_{\phi}$. In the rotating frame, an $n_{\rm R}/n_{\phi}$ resonant orbit closes after $n_{\rm R}$ radial oscillations and $\vert n_{\phi}\vert$ orbital periods. Outside corotation, $n_{\phi}$ is negative and one may speak of outer $n_{\rm R}/\vert n_{\phi}\vert$ resonances. This paper discusses only outer resonances and the minus sign in their labelling will be omitted. While in the axisymmetric case the orientation of the resonant orbits is arbitrary, the virtual introduction of a bar will retain only those orbits which are reflection symmetric with respect to (at least one of) the bar principal axes, say the x- and y-axes. Figure 4 displays several orbits of this kind with $\vert n_{\phi}\vert=1$. There exists an infinity of resonant orbits which accumulate at corotation as $n_{\rm R}\rightarrow \infty$. Those with $n_{\rm R}>6$or closing after more than one rotation will not be considered in this paper. For even $n_{\rm R}$, there exists two distinct resonant orbits for each value of the Hamiltonian, depending on whether the x-axis coincides with orbital apocentre or pericentre. For odd $n_{\rm R}$, there are twice as many solutions because the orbits are no longer reflection symmetric with respect to both axes.

Figure 5: a) Characteristic diagram in the rotating axisymmetric potential (Eq. (6)) for the circular orbit (circ.) and the lowest order outer $n_{\rm R}/1$ resonant orbits which are symmetric with respect to the coordinate axes. The curves give the x-coordinate normalised by the OLR radius of the orbits when they cross the x-axis with $\dot{y}>0$ as a function of the Hamiltonian value relative to $H_{\hbox{\tiny OLR}}$, the H-value of the circular orbit at the OLR, and normalised by $v_{\circ }^2$, assuming that the potential rotates clockwise in the inertial frame. The short-dashed line gives the zero velocity curve (ZVC) and the vertical dotted line the H-value of the orbits sketched in Fig. 4. The grey 3/1and 5/1 resonance curves are shown only in the lower frame, which is a magnification of the rectangular box in the upper frame. b) Corresponding characteristic diagram in the barred potential of Eq. (5) with a bar strength F=0.10. The major axis of the bar coincides with the y-axis. The full and dotted parts of the curves stand for stable and unstable orbits respectively. The orbit labels refer to the nomenclature of Contopoulos & Grosbol (1989). The symmetric and asymmetric 3/1 and 5/1 resonance curves are plotted with grey lines in both the upper and lower frames, but labelled only in the lower one. c) Same as former diagram, but with F=0.20.

The characteristic curves of these resonant orbits in the H-x plane (Fig. 5a) all intersect the circular orbit curve (COC) at their point of lowest H, corresponding to a bifurcation. For even resonances, four branches emanate from the bifurcation, two from the COC and two from the resonance curve. The resonance branches above (towards larger x) and below the COC represent orbits with respectively apocentre and pericentre on the x-axis. The lower branch always passes through the zero velocity curve (ZVC), where the orbit becomes cuspy on the x-axis and then develops loops at higher value of the Hamiltonian. For such loop orbits, the x-coordinate of the characteristic curves does not trace the pericentre but the place where the orbit self-intersect and $\dot{x}\neq 0$. For odd resonances, the bifurcation has six branches: the two from the COC, two for the resonant orbits with radial extrema on the x-axis and which have properties similar to the former even resonance branches, and two for the resonant orbits with those extrema on the y-axis. The two latter branches have opposite $\dot{x}$ but degenerate into the same curve in the H-x characteristic diagram. All periodic orbits in the symmetry plane of an axisymmetric potential are stable.

Figures 5b and 5c show how the characteristic curves are modified when the bar component with major axis on the y-axis is added to the potential. The changes mainly occur at the bifurcations of the axisymmetric case. The bifurcations of the even resonances become gaps, with the right (low H) COC branch deviating into the lower resonance branch, and the upper resonance branch into the left COC branch, giving rise to a sequence of continuous orbit families. In the terminology introduced by Contopoulos & Grosbol (1989), the outermost of these families is called x₁(1), and the other families are divided into an upper x₁^*(i) and a lower x₁(i)sub-family at or near the point of minimum H, where the stability of the orbits appears to reverse. The six-branch bifurcations of the odd resonances (see the 1/1 and 3/1 resonances in the figures) split into two pitchfork bifurcations, one involving the resonant orbits symmetric with respect to the bar minor axis, which are stable near the bifurcation, and the other the resonant orbits non-symmetric relative to this axis, which are unstable near the bifurcation and qualified as asymmetric. As a by-product of this splitting, the segment of the x₁(i) characteristic curve between the two new bifurcations becomes unstable.

Figure 6: Orbits of the x1(1) (left) and x1(2) (right) families in the model with bar strength F=0.10, with the thick line representing the orbit of lowest apocentre and of lowest Hamiltonian respectively. The shaded ellipse sketches the bar. The full and dotted lines represent stable and unstable orbits respectively. The nearly circular x1(1) orbits extend out to infinity. Note the unstable x1(1) orbit at the 1/1 resonance.

At low eccentricity (i.e. small H), only those orbits with a pericentre on the bar minor axis seem to be stable. This could be the influence of the Lagrangian points $L_{\rm 4/5}$, which are stable fix points in the models. At high eccentricity, all orbit families undergo a change of stability and the characteristic curves in Fig. 5 are all interrupted at the point where the minor axis loop of the orbits touches the centre R=0. The curves actually go beyond this point, but the resonance number changes. For instance, the 2/1 orbits become 2/3 orbits. When increasing the bar strength (from Figs. 5b to 5c), the resonance gaps between successive x₁(i) curves and the separation between the pitchfork bifurcation pairs increases. The characteristic curves also become more twisted and the fraction of stable orbits decreases. In particular, almost all x₁(3) orbits are unstable for F=0.20. It should be noted that other authors (Athanassoula et al. 1983; Sellwood & Wilkinson 1993) report more complicated characteristic curves for the x₁(2) and higher order orbit families near the ZVC in their models, probably as a consequence of an m=4 Fourier component in the potential.

Some orbits of the above families are plotted in D2000, but obviously missing all eccentric x₁(1) and x₁(2) orbits with loops on the bar minor axis. A more complete set of orbits from these two families are given in Fig. 6. These orbits are the most important even $n_{\rm R}$ periodic orbits because they are usually associated with the largest invariant curve islands in surface of section maps. The orbits drawn with thick lines are examples of the perpendicularly oriented orbits that replace the circular orbit near the OLR radius and which have been proposed as a possible explanation for either the Hyades and Sirius streams (Kalnajs 1991) or the Hercules-LSR bimodality (D2000) in the observed u-v distribution. As we shall see in the next sections, the stable eccentric x₁(1) orbits also play an important role in shaping the local velocity distribution. Regular orbits trapped around them are indeed unlikely to be heavily populated by stars, but represent forbidden phase space regions for chaotic orbits.

The space coverage of the orbit families can be determined from x-yplots like in Fig. 6. For each position in real space, there will be an (infinite) set of periodic orbits passing through. The velocity trace in planar velocity space of the above described orbits, as well as the curves delineating some of the main resonances in the underlying axisymmetric potential, are indicated in Figs. 7 and 8 for various azimuthal angles $\varphi$ and a realistic range of galactocentric distances relative to the OLR, and for two different bar strengths. Here the angle $\varphi$ is measured from the bar major axis and increases towards the direction opposite to galactic rotation, i.e. coincides with the traditional in-plane inclination angle of the bar. All space positions are reached by many of the considered orbit families, and sometimes several traces are produced by orbits from the same family: for instance, at $\varphi=90^{\circ}$, there is a large range of R with three traces from x₁(1) orbits, mainly due to the loops on the x-axis of the high-H orbits. The traces of the 1/1, 1/1 asym, x₁(1), x₁^*(2), x₁(2) and x₁^*(3) orbits with non-nearly vanishing |u|-velocity all fall very close to the associated resonance curves, indicating that the axisymmetric approximation used to compute these curves works well for $n_{\rm R}\leq 4$. Not all resonant periodic orbits, i.e. those with traces on the resonance curves, are unstable. In particular, orbits on the 2/1 (OLR) and 1/1 resonance curves are stable x₁(1) and 1/1 orbits at $\varphi=90^{\circ}$ and unstable x₁^*(2) and 1/1 asym orbits at $\varphi=0^{\circ}$, except for a x₁^*(2) and a x₁(1) orbit with u=0for $R/R_{\hbox{\tiny OLR}}\geq 1.1$. Hence resonance regions of phase space are not necessary unstable and depleted as asserted in D2000 for the OLR.

There are sometimes several orbits from the same family plotted very close to each other, like for example the three unstable x₁(2) orbits at $R/R_{\hbox{\tiny OLR}}=0.9$, $\varphi =30^{\circ}$ and $(v/v_{\circ},u/v_{\circ})\approx (-0.85,-0.80)$ for F=0.20. This happens when the sequence of orbits within the family reverses its progression in the x-y plane towards a given direction and very close to the current space position, causing an accumulation of orbits near this position with different local velocities. One may also note that the continuous transitions between some orbit families can cause periodic orbits from different families to have almost identical traces, as for example the x₁^*(2) and x₁(2) near the centre of the frame $R/R_{\hbox{\tiny OLR}}=1.1$ and $\varphi=90^{\circ}$ in Fig. 8.

Figure 7: Liapunov divergence timescale of the orbits in the u-v plane as a function of position in real space, for a bar strength F=0.10. The horizontal and vertical axes of each frame are $v/v_{\circ }$ and $u/v_{\circ }$respectively, with v positive in the direction of rotation and u towards the anti-centre, and the origin at the circular orbit of the axisymmetric part of the potential $\Phi _{\circ }$. The timescales are in units of local circular period  $t_{\Omega }$ in $\Phi _{\circ }$ and greyscale coded. The dark and white regions respectively represent regular and chaotic orbits. The oval and polygonal symbols indicate the positions of the periodic orbits, with a different symbol for each orbit family. Full and empty symbols respectively stand for stable and unstable orbits. The full lines open towards the right (increasing v) are the H12 and H45 contours. The dash-dotted, solid, dashed and dotted lines open towards the left respectively give the locations of the outer 1/1, 2/1, 4/1 and 6/1 resonant orbits in $\Phi _{\circ }$.

Figure 8: As Fig. 7, but for a bar strength F=0.20.

The rather large number of stable simple periodic orbits through each space position and the numerous stellar streams observed in the Solar neighbourhood may suggest that at least some of them are related to periodic orbits, as anticipated in Kalnajs (1991). For instance, beside the idea of the x₁(1) and low-eccentricity x₁(2) induced streams near the OLR radius, the $\varphi =30^{\circ}$ frames in Fig. 7 betray interesting stable eccentric x₁(2) and 5/1 asym orbits at $R/R_{\hbox{\tiny OLR}}=1.2$ and $R/R_{\hbox{\tiny OLR}}=1.1$ respectively and with $v/v_{\circ}\approx -0.6$ and $u/v_{\circ}\approx 0.05-0.15$, which fall very close to the velocity of the young Arcturus stream (see Table 1).