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\end{table}  In Table \ref{tab:chis} we give the $\chi ^2$ values describing the goodness of fit for several statistical samples of modelled systems. We first quote the $\chi ^2$ considering only $\alpha \lt 45^{\circ}$, corresponding to systems with satellites in close to oppositely aligned configurations. We then give the $\chi ^2$ for the full domain, $0^{\circ} \lt \alpha \lt 180^{\circ}$. The $\chi ^2$ are displayed this way in order to illustrate how the picture changes when full domain of $\alpha$ is considered, as in Figures \ref{fig:full} and Figure \ref{fig:zoom}.   We also give the reduced $\chi ^2$ values for each model. These values are calculated by removing two degrees of freedom for the COP model, and a single degree of freedom from the disk model, since these models are progressive more restrictive as compared to the isotropic case. The COP and isotropic models produce lower $\chi^2$ values than the disk models.  We will examine the statistical samples in more detail in the remainder of this section. \subsubsection{Disk Model and M31 model}