deletions | additions       diff --git a/Comparison.tex b/Comparison.tex  index 50252e0..0a53348 100644  --- a/Comparison.tex  +++ b/Comparison.tex 
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\section{Comparison to Toy Models}  \label{sec:models}  \begin{table*}  \centering  \begin{tabular}{lllllllll}  \hline \hline  Model & $\chi ^2$, $\alpha \leq 45^o$ &DOF, $\alpha \leq 45^o$& reduced $\chi ^2$, $\alpha \leq 45^o$& $p$, $\alpha \leq 45^o$ & $\chi ^2$ & 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*}  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: