Influence of the warp

The break in the $z$ symmetry transforms the family of circular orbits in the unperturbed model into a set of families $pk$ (main families) with $p_z(0)\neq 0$. The significance of the indice $k$ will be explained further. Except at bifurcations, the $pk$ families are stable.

The shape of the corresponding orbits can be approximated with the following parametric function:

$$R(H) = R_0(H) + A_R(H) \cos(2\phi - \pi),$$ (7)

$$z(H) = z_0(H) + A_z(H) \cos\phi,$$ (8)

parametrised by the azimuthal angle $\phi$. For increasing $H$, $A_R$ increases from 0 to $1\,$kpc, while $A_z$ increases from 0 to $8\,$kpc. This latter variation corresponds almost exactly to the amplitude of the warp as a function of radius ( $A_z \cong wR^2$). In other words, the main orbit family follows the density maximum at any radius.

Another influence of the warp is to move the bifurcations toward slightly lower energies. The corresponding transverse bifurcations arise at $H=-0.0633$, $-0.0509$, $-0.0398$, $-0.0298$, and $-0.0176$. Moreover, a new bifurcation takes place within the limits of the diagram at $H=-0.0108$, where the subfamily starting at $H=-0.0176$ rejoins the main family. The other families would also rejoin the main family at higher energies, yet they would reach radii larger than $y(0)=50\,\rm kpc$, which was chosen as a limit for our study of a warped disk.

Figure 3: $H$-$p_z(0)$ phase space ($w=0.005$).

The subfamilies coming from the corresponding transverse bifurcations are given the symbol ``$sk\pm$'' where $k$ gives the frequency ratio $\nu / \Omega$ at the corresponding bifurcation and the sign is the one of the difference of $p_z(0)$ between the subfamily and the main family. These families evolve in the same way as family $pk$, i.e. they follow the warped disk, so may be significantly populated in a real disk. Note that the families with an odd $k$ conserve the bi-symmetry of the potential, which is not the case when $k$ is even. Fig. 4 displays a set of three orbits extracted from different subfamilies, $s4+$, $s5+$ and $s6-$. The right panels show the elevation $z$ of the orbits as a function of the azimuth $\phi$ in comparison to the elevation of the corresponding main family (dashed lines). The more the energy grows, the more the subfamilies follow well the main orbit.

Figure 4: Projections over the $y=0$, $z=0$ planes and the evolution of $z$ as a function of azimuthal angle $\phi$ of orbits of the families $s4+$ ($H=-0.03$), $s5+$ ($H=-0.02$) and $s6-$ ($H=-0.015$). The dashed line represents the potential minimum of the warped disk at a fixed radius which is similar to the main family.

Although in the absence of the warp all subfamilies are stable, this is no longer the case with a weak warp. This point will be discussed further below.