Warped disk model

In order to compute periodic orbits, we have based our model on a superposition of three simple Miyamoto-Nagai potentials (Nagai & Miyamoto [1975]). The three components can be viewed as representing respectively a bulge, a visible disk and a gas disk containing a large amount of dark matter. In cylindrical coordinates, the total potential can be written as :

$$\Phi_0(R,\phi,z) = -\sum_{i=1}^{3} \frac{GM_i} {\sqrt{R^2+\left(a_i+\sqrt{z^2 + b_i^2}\right)^2}}\ .$$ (1)

Observations suggest that warps seen on edge take the shape of an integral sign, i.e., the disk deformation is proportional to the cosine azimuthal angle $\phi$ and increases with radius. Analytically, the warp is well represented by a term $\Delta z$ where :

$$\Delta z = w R^2 \cos\phi,$$ (2)

and $w$ is an adjustable parameter characterizing the warp's amplitude.

Replacing $z$ by $z-\Delta z$ in the potential (1) we get an analytical warped disk potential model :

$$\Phi_{w}(R,\phi,z) = -\sum_{i=1}^{3} \frac{GM_i} {\sqrt{R^2+(a_i+d_i)^2}},$$ (3)

with

$$d_i=\sqrt{\left(z-w R^2 \cos\phi\right)^2 + b_i^2}\ .$$ (4)

Note that the potential is bi-symmetric with respect to the $y=0$ plane, $\Phi(x,y,z) = \Phi(-x,y,-z)$.