calculated numerically. Our modelling of the vertical tracer will use a plane at a height $z$ of 100pc.

Halo Potential
--------------

We make use of the flattened, but axi-symmetric, pseudo-isothermal halo model proposed by @Sackett1994A, where we assume that the equatorial plane of the halo matches that of the galaxy. In this model the density is stratified in concentric ellipsoids, as given by $$\rho_\textrm{halo}(R,z) = \frac{\rho_{0,\textrm{halo}}\,R_c^2}{R_c^2 + R^2 + z^2/q^2}.$$ The ellipsoids formed by this density distribution have axis ratio $q\equiv c/a$, with core radius $R_c$.

The potential due to this density distribution is given by @Sackett1990A as $$\begin{aligned}
\Phi_\textrm{halo}(R,z) &=\ \  2 \pi G q \,\rho_{0,\textrm{halo}} \,R_c^2 \int_0^{1/q} \left[\frac{1}{x^2(1-q^2) + 1}\right]&\nonumber \\ &\times  \log\left\{1+\frac{x^2}{R_c^2} \left[\frac{R^2}{x^2(1-q^2) +1}+z^2\right]\right\}\,dx.& \end{aligned}$$

@Sackett1994A provides the solution to the forces associated with this halo in spherical coordinates: $$\begin{aligned}
F_R(R,z) &=& \frac{-v_H^2 R \gamma}{h \arctan \gamma} \left[ \frac{\mu^2}{\mu^2-1} \left(\frac{\arctan \gamma\mu}{\gamma\mu} - \frac{\arctan\gamma}{\gamma}\right)\right.\label{eqn:halor} \nonumber \\ 
&&\left.- \frac{\nu^2}{\nu^2-1} \left(\frac{\arctan \gamma\nu}{\gamma \nu} - \frac{\arctan \gamma}{\gamma} \right) \right] \,,\\
F_z(R,z) &=& \frac{-v_H^2 z \gamma}{h \arctan \gamma}\left(\frac{\arctan \gamma\mu}{\gamma \mu} - \frac{\arctan \gamma \nu}{\gamma \nu}\right)\,,\label{eqn:haloz}\end{aligned}$$ where $$\gamma \equiv \frac{\sqrt{1-q^2}}{q}\,\, ,\,\, \nu\equiv \sqrt{\frac{2c}{b-h}}\,\,,\,\, \mu\equiv\sqrt{\frac{2c}{b+h}},$$ and $$\begin{aligned}
h&\equiv&\sqrt{b^2-4ac}  \, ,\\
a &\equiv& (1-q^2) R_c^2 \, ,\\
b &\equiv& z^2 + R^2 + (1-q^2) R_c^2 