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\end{tabular}  \caption{$\chi ^2$ and reduced $\chi ^2$ for various statistical samples of models described in text. Disk models (i.e. M31-like models, including the model based on M31 iteself) are strongly disfavored relative to our COP model or the simple isotropic case.}  \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} $0^{\circ}  \lt \alpha \lt 180^{\circ}$. 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. We  will examine the statistical samples in more detail in the remainder of this section. \subsubsection{Disk Model and M31 model}