deletions | additions       diff --git a/Comparison.tex b/Comparison.tex  index a3f5f87..3f9e5c2 100644  --- a/Comparison.tex  +++ b/Comparison.tex 
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\centering  \begin{tabular}{lllllllll}  \hline \hline  Model & $\chi ^2$, $\alpha \leq 45^o$ &DOF, $\alpha \leq 45^o$& reduced $\chi ^2$, $\alpha \leq 45^o$& 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 
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In Figure \ref{fig:disks} we examine the kinematic SDSS data in comparison with our ``disk model." We plot the fraction of satellite pairs that are corotating as a function of the opening angle $\alpha$. The blue line corresponds to a statistical sample comprised only of satellite disks, the green line to a sample comprised 50\% of satellite disks and 50\% of isotropic satellites, and the red line 10\% of disks and 90\% of isotropic satellites. The dashed green line corresponds to a sample with 50\% M31 model and 50\% isotropic composition. The black line is a purely isotropic sample.  We find strong disagreement between the disk models with $\geq 50\%$ isotropic contribution  and the teal line denoting the SDSS data. Significantly, the presence of inclined, rotated and out-of-phase planes in the toy models results in a significant signal in the $\sim 20^{\circ} \, \lt \alpha \, \lt \sim 60^{\circ}$ regime which is not seen in the data. The data does agree reasonably well with models where 90\% of the hosts have satellites distributed isotropically in phase space; however, the 90\% isotropic model fails to reproduce the sharpness of the decline in corotation at $\alpha \sim 10^{\circ}$. The M31 sample is similar to the 50\% disk/50\% isotropic sample with the posiitoins of the satellites tailored to match the positions of M31 satellites. Our results strongly exclude the possibility of coherently rotating disks, the objection could be raised that the velocity selection criteria used to select the SDSS systems systematically removes all inclined, rotated and out-of-phase systems; we will address this possible objection in a later subsection.