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\begin{table*}
\centering
\begin{tabular}{lllllllll}
\hline \hline
Model & $\chi ^2$, $\alpha \leq 45^o$ &DOF, $\alpha \leq 45^o$& reduced $\chi ^2$, $\alpha \leq 45^oreplace_contentamp; $p$, $\alpha \leq 45^o$ & $\chi ^2$ & DOF & reduced $\chi ^2$ & $p$\\ \hline
Pure disk & 139.16 & 3 & 69.58 & $\sim 0$ & 582.56 & 17 & 36.41 & $\sim 0$ \\
Disk, 50\% isotropic & 58.02 & 3 & 29.01 & $\sim 0$ & 180.90 & 17 & 11.31 & $\sim 0$ \\
Disk, 90\% isotropic & 5.77 & 3 & 1.92 & 0.123 & 25.57 & 17 & 1.50 & 0.083 \\ \hline
M31, 50\% isotropic & 9.50 & 4 & 3.16 & 0.050 & 40.14 & 18 & 2.36 & 0.002 \\ \hline
COP, 50\% isotropic & 11.77 & 2 & 5.88 & 0.002 & 19.23 & 16 & 1.20 & 0.257 \\
COP, 90\% isotropic & 0.96 & 2 & 0.48 & 0.618 & 8.71 & 16 & 0.54 & 0.924 \\ \hline
100\% isotropic & 3.96 & 4 & 0.99 & 0.411 & 11.66 & 18 & 0.65 & 0.864
\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 disfavored relative to our COP model or the simple isotropic case.}
\end{table*}
\section{Comparison to Toy Models}
\label{sec:models}
In order to gain insight as to whether the data presented in Figures \ref{fig:zoom} and \ref{fig:full} does indeed argue for the existence of coherently rotating satellite structures, we compare the SDSS data to mock observations of simple, idealized "toy" models of satellite systems. These models are not intended to give detailed descriptions of satellite phase space distributions or physical behavior. Rather, we treat them as simple test cases to which the data can be compared. We begin by detailing how each toy model is constructed: