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\begin{tabular}{lllll}  \hline \hline  Model & $\chi ^2$, $\alpha \leq 45^o$ & reduced $\chi ^2$, $\alpha \leq 45^o$& $\chi ^2$ & reduced $\chi ^2$\\ \hline  Pure disk & 139.16 & 69.58  & 582.56 & 36.41  \\ Disk, 50\% isotropic & 58.02 & 29.01  & 180.90 & 11.31  \\ Disk, 90\% isotropic & 5.77 & 1.92  & 25.57 & 1.50  \\ \hline M31, 50\% isotropic & 9.50 & 3.16  & 40.14 & 2.36  \\ \hline COP, 50\% isotropic & 11.77 & 5.88  & 19.23 & \\ 1.20\\  COP, 90\% isotropic & 0.96 & 0.48  & 8.71 & 0.54  \\ \hline 100\% isotropic & 3.96 & 0.99  & 11.66 & 0.65  \label{tab:chis}  \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, $0^{\circ} \lt \alpha \lt 180^{\circ}$. 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.  We will examine the statistical samples in more detail in the remainder of this section. \subsubsection{Disk Model and M31 model}