\, ,\\
c &\equiv& z^2.\end{aligned}$$ Note that to calculate $q>1$ one requires the use of complex numbers, although these return to real numbers at the end of the calculation. Only using real numbers will force a fit to be constrained to $q<1$. The calculation has a singularity at $q=1$, which we automatically substitute with $q=0.999$ where required.

The asymptotic halo velocity $v_H$ is defined as $$v_H^2 = \frac{4\pi G\,\rho_{0,\textrm{halo}} \,R_c^2\, q \arccos q}{\sqrt{1-q^2}}\label{eqn:dmhalovelocity}$$

Fitting Strategy
----------------

Using Equation \[eqn:vresidual\], we calculate the observed circular rotation curve due to the dark matter halo $v_\textrm{halo}^2$ at the mid-plane of the galaxy over the full range of $R$. In a similar way we calculate the observed vertical force gradient $dF_{z,\textrm{halo}}/dz(R,z)$ due to the dark matter halo at a height $z$ of 100pc, using Equation \[eqn:zresidual\]. We inspect the results to determine in which range of radii $R$ they are sufficiently reliable. In contrast to @OBrien2010D we fit both the vertical and radial tracers simultaneously. Since the rotation curve decomposition is less sensitive to noise than the vertical force gradient decomposition, we can often fit a larger radial range for the rotation curve decomposition than the vertical force gradient decomposition. We will fit the 