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Appendix A: Orbital structure

To show how the dynamical parameters of our galactic model influence the orbital structure of the system, we present for each case, color-coded grids of initial conditions , equivalent to surfaces of section, which allow us to visualize what types of orbits occupy specific areas in the phase-space.

Figure A.1a depicts the phase plane of the PH model when M_n = 0. One can observe that most of the phase space is covered by 2:1 resonant orbits, while there is also a weak chaotic layer that separates the areas of regularity. The outermost thick curve is the ZVC. In Fig. A.1b, we present a grid on the phase plane when M_n = 500, i.e., a model with a more massive central nucleus. It is evident that there are many differences with respect to Fig. A.1a, being the most visible: (i) the growth of the region occupied by chaotic orbits; (ii) an increase in the allowed radial velocity Ṙ of the stars near the center of the galaxy; and (iii) the absence of several families of resonant orbits (i.e., 1:1, 4:3, and 6:3 resonant orbits). In Fig. A.1c, we can see the structure of the phase plane of the OH model when M_n = 0. In this case, we observe that the phase plane is flooded with box orbits due to the absence of the central nucleus. On the other hand, in Fig. A.1d, where we have an OH model with a massive nucleus (M_n = 500), the portion of box orbits is confined considerably as a vast unified chaotic sea emerges surrounding several islands of secondary resonances.

	Fig. A.1 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: Mn = 0 and b) upper right: Mn = 500 and for the OH model when c) lower left: Mn = 0 and d) lower right: Mn = 500.
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The grid structure of the (R,Ṙ) phase plane for the PH model when c_n = 0.05 is presented in Fig. A.2a. We see that when the concentration of the spherical nucleus is very high, a solid chaotic sea exists at the outer parts of the phase plane, resulting in the complete absence of secondary resonances. On the other hand, when c_n = 0.50, we observe in Fig. A.2b that the area on the phase plane occupied by chaotic orbits has shrunk, thus leaving space for several resonant families (i.e., 1:1, 4:3 and other resonances) to increase their rates. Furthermore, the 12:7 chain of islands emerges inside the box domain. A similar comparison between lower and higher concentrated nucleus for the OH models is made in Fig. A.2c and d. We show that when c_n = 0.05 (Fig. A.2c) all the different resonant families are present and are surrounded by a unified chaotic sea. In the case where c_n = 0.50 (Fig. A.2d), the structure of the phase plane remains almost the same and the most prominent difference lies in the increasing rates of all of the regular families, which of course entails a reduction of the chaotic region.

	Fig. A.2 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: cn = 0.05 and b) upper right: cn = 0.50 and for the OH model when c) lower left: cn = 0.05 and d) lower right: cn = 0.50.
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A grid of initial conditions for the PH model when the disk has the minimum possible value (M_d = 4500) is given in Fig. A.3a. We observe that almost all of the resonant families are present forming well-defined sets of islands. It is interesting to note, the presence of a set of three islands corresponding to the so-called other resonances. In fact, this is the 5:3 resonance which, however, appears itself only in some isolated cases, so we do not feel it is necessary to include it in the list containing all the main resonant families under investigation. Figure A.3b shows a similar grid when M_d = 9000 (i.e., the maximum possible value of the mass of the disk). It is evident that the structure of the phase plane has several differences with respect to the Fig. A.3a which are: (i) the reduction of the region occupied by chaotic orbits; (ii) the approximately 40% increase in the allowed radial velocity Ṙ of the stars near the central region of the galaxy; and (iii) the appearance of a second smaller area near the center occupied by 2:1 banana-type orbit. In Fig. A.3c and d, we present two similar grids of initial conditions for the same values of M_d as in Fig. A.3a and b, respectively, but applied to the OH models this time. Once more, as the disk becomes more massive the extent of the chaotic sea decreases, thus amplifying the rates of all the regular families.

	Fig. A.3 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: Md = 4500 and b) upper right: Md = 9000 and for the OH model when c) lower left: Md = 4500 and d) lower right: Md = 9000.
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In Fig. A.4a, we present the orbital structure of a grid of initial conditions for the PH model when b = 4. We show that almost all resonant families are present, forming different sets of islands. Here we have to point out the existence of an additional resonant family, that of the 5:3 resonant orbits, which correspond to the so-called “other resonances” and produce the set of the well-defined purple triple islands at the phase plane. Figure A.4b shows a similar grid of initial conditions when the core radius of the disk-halo possess its maximum possible value (b = 8). One may distinguish several differences in the structure of the phase plane with respect to that shown in Fig. A.4a. The main differences are the following: (i) the area occupied by chaotic orbits has been reduced significantly and it is confined to the very outer parts of the phase plane; (ii) the bifurcated 10:5 family has been adsorbed by the main 2:1 family; (iii) the presence of higher resonances such as the 12:7 family is much stronger; and (iv) the 4:3 resonance is depopulated and the corresponding islands are so tiny that they appear as isolated points in the grid. Similar grids of initial conditions for the same values of b as in Fig. A.4a and b, respectively, but for the OH models, are shown in Fig. A.4c and d. We observe that at highest value of b (Fig. A.4d) the amount of initial conditions corresponding to chaotic orbits is significantly greater with respect to Fig. A.4c, thus limiting the extent of all the different areas of stability. Furthermore, higher resonant orbits (i.e., the 8:5 family) appear only in models with large values of b.

	Fig. A.4 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: b = 4 and b) upper right: b = 8 and for the OH model when c) lower left: b = 4 and d) lower right: b = 8.
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The orbital structure of a grid of initial conditions for the PH model when α = 2.5 is presented in Fig. A.5a. We observe that the vast majority of the phase plane is covered by resonant 2:1 banana-type orbits. Specifically, there are two regions corresponding to 2:1 resonant orbits. It should be emphasized that the 1:1 resonant family, which is a basic family, is absent in PH galaxy models with sufficient small values of the scale length of the disk. A similar grid of initial conditions when α = 5 is given in Fig. A.5b. This grid has many similarities with respect to that shown in Fig. A.5a, but it also has several important differences. The most visible differences are the growth of the area corresponding to chaotic orbits, the appearance of the main 1:1 family, the absence of the second small region occupied by 2:1 resonant orbits, and the disappearance of secondary resonances such as the 4:3 and the 12:7. In Fig. A.5c and d we present grids of initial conditions for the same values of α as in Fig. A.5a and b, respectively, but for the OH galaxy models. It is evident, that in the case of the highest value of α, that is in Fig. A.5d, the portion of the chaotic and the 1:1 resonant orbits is considerably larger, while at the same time, all the other regions of stability have been reduced (i.e., regions corresponding to box, 2:1, and 3:2 resonant orbits). Moreover, we should point out that in both cases, several higher, secondary resonant orbits (i.e., 11:7, 13:8, 14:9) emerge mainly inside the area of the box orbits.

	Fig. A.5 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: α = 2.5 and b) upper right: α = 5 and for the OH model when c) lower left: α = 2.5 and d) lower right: α = 5.
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Figure A.6a depicts the orbital structure of a grid of initial conditions for the prolate dark matter halo (PH) model when h = 0.1. As in all previous cases, more than half the phase plane is covered by 2:1 banana-type orbits. A unified chaotic layer exists at the outer parts of the phase plane and surrounds most of the different stability islands. However, the islands produced by the 4:3 resonant orbits are so small that they appear as lonely points in the gird. Things are quite similar in Fig. A.6b where h = 1. We see that the overall grid structure is maintained and the observed differences are minor. In fact, the most noticeable differences are the following: (i) the extent of the chaotic layer is smaller, giving space to box orbits; (ii) the portion of the 4:3 resonant orbits looks more prominent; and (iii) the extent of the 12:7 resonant orbits has been reduced. Similar grids of initial conditions for the same values of h exist, as in Fig. A.6a and b, but for the OH galaxy models are shown in Fig. A.6c and d, respectively. Once more, we note that the change of the value of the scale height of the disk does not cause significant influence on the structure of the phase plane. All it does is to shrink the area occupied by chaotic orbits thus, allowing mainly the 1:1 resonant orbits to enlarge their rate.

	Fig. A.6 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: h = 0.1 and b) upper right: h = 1 and for the OH model when c) lower left: h = 0.1 and d) lower right: h = 1.
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In Fig. A.7a, we present the orbital structure of a grid of initial conditions for the prolate dark matter halo (PH) model when β = 0.1. Undoubtedly, this is a very interesting phase plane that is very different from what we have seen so far. We observe the existence of a vast chaotic sea that embraces many islands of stability formed by different regular families of orbits. The meridional 2:1 banana-type and the linear 1:1 resonant orbits form two different islands on the grid, while box orbits have an anemic presence. Moreover, we distinguish several purple regions corresponding to other types of resonances. Here we have to point out that is the first time we have encountered such an intense presence of that type of orbits. Our numerical calculations suggest that these regions are produced by two different types of orbits. The set of the triple islands inside the region of the 10:5 orbits correspond to the 10:3 resonance, while the double set of islands inside the large 2:1 region and also above the box orbits is produced by the 8:4 resonance. The grid shown in Fig. A.7b corresponds to the case where β = 0.9. We see that everything is now back to normal and the previous complicated structure has vanished. Again, we can distinguish many sets of islands formed by miscellaneous resonances. The small islands inside the region of box orbits correspond to the 13:8 resonance, the set right above the box orbits corresponds to the 8:5 resonance, while the set of the double islands embedded in the chaotic sea corresponds to the 3:2 resonant family. The orbital structure of the grid in the case of the spherical dark halo (β = 1) is presented in Fig. A.7c. Considering the previous analysis of the “other resonances” shows that the orbital structure has remained almost unperturbed. Finally, Fig. A.7d, shows the case of a highly flattened halo where β = 1.9. Here, the increase of several types of orbits (i.e., 1:1, 3:2, 4:3, chaotic) spurred both box and 2:1 orbits to reduce their rates and to be limited to the center of the phase plane.

	Fig. A.7 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: β = 0.1 and b) upper right: β = 0.9 and for the OH model when c) lower left: β = 1 and d) lower right: β = 1.9.
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The orbital structure of a grid of initial conditions for the prolate dark matter halo (PH) model when c_h = 7 is shown in Fig. A.8a. One may observe that the vast majority of the phase plane is covered by initial conditions corresponding to 2:1 banana-type orbits. In fact, there are two distinct regions formed by these orbits. At the outer parts of the phase plane, a chaotic layer is present that contains small stability islands. In particular, we can distinguish a tiny region formed by the 1:1 resonant orbits, while the points on the grid corresponding to the 4:3 resonant orbits are hardly visible. Figure A.8b shows a similar grid of initial conditions for a prolate galaxy model when c_h = 22. It is evident that the orbital structure has many significant differences with respect to that presented in Fig. A.8a. The most noticeable differences are: (i) the amount of chaos has increased considerably leading to a vast chaotic sea; (ii) box orbits have been greatly depopulated and now are confined only to the center of the grid; (iii) all the bifurcated resonances (i.e., 6:3 and 12:5) have disappeared, while new resonant families such as the 3:2 family appear; and (iv) the total area of the phase plane is reduced. In Fig. A.8c, we present another grid of initial conditions for an oblate dark halo galaxy model when c_h = 7. In this case, all the expected types of orbits are present forming well-defined regions in the phase plane. The purple dots correspond to resonant orbits of higher multiplicity (i.e., 11:7 and 13:8 resonant orbits). A similar oblate dark halo grid when c_h = 22 is shown in Fig. A.8d. We show that the extent of chaos is significantly larger than that observed in Fig. A.8c. Moreover, the higher resonant 8:5 family is completely absent. With a closer look at the overall orbital structure, we realize that the grid of Fig. A.8d is very similar to that shown in Fig. A.8b. Those two grids correspond to the same value of scale length of the halo c_h = 22, but to different shapes of the halo (prolate and oblate, respectively). Therefore, we may conclude that in high values of c_h, or in other words, when the halo is much less concentrated, the orbital structure is the same, regardless of the particular shape of the halo (prolate or oblate).

	Fig. A.8 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: ch = 7 and b) upper right: ch = 22 and for the OH model when c) lower left: ch = 7 and d) lower right: ch = 22.
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A grid of initial conditions showing the orbital structure of the prolate dark matter halo model when L_z = 1 is presented in Fig. A.9a. We observe the existence of a dense and unified chaotic sea, while the majority of the regular domains are located near the central region of the phase plane, although there are important stability islands surrounding them. Moreover, we should notice the complete absence of 4:3 or higher resonant orbits. Figure A.9b shows a similar grid of initial conditions corresponding to the L_z = 50 prolate halo model. It is evident that the structure of this phase plane differs greatly from the previous one. The most significant differences are: (i) the entire phase plane is covered by regular orbits therefore chaotic motion, if any, is negligible; (ii) numerous types of miscellaneous resonant orbits belonging to the “other resonances” class are spread all over the phase plane (3:2, 5:3, 4:5, 8:5, 5:7, 6:7, 10:7, 11:7, 13:8, 14:9 resonant orbits, mentioning most of them); (iii) the allowed radial velocity Ṙ of stars passing near the center of the galaxy is almost decreased by half; and (iv) the permissible area on the (R,Ṙ) plane is reduced. In Fig. A.9c, we present another grid of initial conditions for the L_z = 1 oblate halo model. A vast chaotic sea is observed surrounding all the different stability islands. Furthermore, we should note, a lack of higher resonant orbits, while the 8:5 resonant orbits appear as extreme isolated points on the grid. Things are very different in Fig. A.9d where the grid of the L_z = 50 oblate halo model is depicted. The main differences with respect to the structure of the grid shown in Fig. A.9c are very similar to those described earlier in the prolate dark halo case. Again, the 8:5 resonant orbits are hardly visible. However, we should point out that in this case, the number of the “other resonances” orbits (4:5, 6:5, 7:5, 6:7, 8:7, 10:7, 8:9, 12:11, 13:11, mentioning the most important of them) is lower.

	Fig. A.9 Orbital structure of the (R,Ṙ) phase plane for the PH model when a) upper left: Lz = 1 and b) upper right: Lz = 50 and for the OH model when c) lower left: Lz = 1 and d) lower right: Lz = 50.
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	Fig. A.10 Orbital structure of the (R,Ṙ) phase plane for a) left: the PH model and b) right: the OH model when E = 700.
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The orbital structure of a grid of initial conditions of the prolate dark matter halo model when E = 700 (the maximum energy level studied) is presented in Fig. A.10a. It is clearly seen that the vast majority (about 60%) of the phase plane is covered by initial conditions corresponding to the 2:1 banana-type orbits. As in many previous prolate halo grids, the 2:1 resonant orbits form two separate islands of stability on the phase plane. In Fig. A.10a, we note a complete lack of 4:3 and 12:7 boxlet orbits, while we can identify the presence of “other resonant” orbits. Inside the main 2:1 region, there is a thin ribbon of initial conditions produced by the 8:4 resonance, which is a bifurcated subfamily of the main 2:1 family (like the 6:3 and the 10:5 families of orbits). Moreover, just outside the box orbits we see a set of five islands of initial conditions corresponding to

the 12:5 resonance. In Fig. A.10b, a similar grid of initial conditions for the same value of the energy (E = 700), but for the oblate case, is given. We observe that all the orbit families are present forming distinct well-defined stability islands, which are embraced by a unified chaotic sea. We should mention, that there is a strong presence of resonant orbits of higher multiplicity inside the area occupied by box orbits. Our numerical calculations indicate that these initial conditions correspond either to 10:7 or 13:8 resonant orbits.

© ESO, 2014