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\item Disk model - In this model, 2-5 satellites are placed randomly between 0 and 200 kpc from the origin on the xy plane and then randomly given a z coordinate randomly between -10 and 10 kpc. All satellites are assigned a 3D velocity of 100 km/s such that each satellite is in circular motion about the host. The model is randomly rotated and viewed along the z axis.  \subsection{Kinematic Comparisons}  \begin{table*}  \centering  \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 & & 582.56 & \\   Disk, 50\% isotropic & 58.02 & & 180.90 & \\  Disk, 90\% isotropic & 5.77 & & 25.57 & \\ \hline  M31, 50\% isotropic & 9.50 & & 40.14 & \\ \hline  COP, 50\% isotropic & 11.77 & & 19.23 & \\  COP, 90\% isotropic & 0.96 & & 8.71 & \\ \hline  100\% isotropic & 3.96 & & 11.66 &   \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}  \item Corotating oppositely-aligned pairs (COP) model - Here, each host is restricted to exactly two satellites. Once the first satellite is randomly placed on the xy plane, the placing of the second satellite is restricted such that the angle between the position vectors of the two satellites is greater than $170^{\circ}$. From there, each satellite is assigned a z cooridate between -10 and 10 kpc and the system is subject to random rotation and viewed along the z axis.