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\Section{Comparison to Toy Models}
In this section we make comparisons of the dynamical SDSS data to simple toy models
1. Disk model
2. ``Dumbell" model
3. Isotropic model
In each case, we assign a number of satellites per host and rotate and rotate each host to a random viewing angle. A statistical sample consists of N $\sim 10^5$ hosts, and most statistical samples will consist of a mixing of the isotropic model with one of the other two models.
\Subsection{Kinematic Comparisons}
In Figure \ref{fig:disks} we examine the kinematic SDSS data in comparison with our ``disk model." The left panel shows the fraction of satellite pairs that are corotating\footnote{Satellite pairs are definited as ``corotating" if they have opposite-signed velocity offsets relative to the hosts and are their associated $\alpha$ is greater than 90\degree, or they have same-signed velocity offsets relative to the hosts and their associated $\alpha$ is less than 90\degree ; otherwise, they are deemed counter-rotating} as a function of the on-sky angle between the satellites, where the vertex of the angle is at the location of the host. The right panel gives the same information in a cumulative sense: The SDSS data is plotted as a solid teal line. Other lines indicate increasing levels of contamination from hosts whose satellite populations are distributed isotropically. The statistical sample comprised only of the disk model is plotted as a dashed blue line. We will only consider only $\alphas$ less than 90 \degrees, as this is the regime we might expect satellites to evolve relatively independently.
We find strong disagreement between the non-trivial disk models 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 $\sim 40 \degree \, \lt \theta \, \lt \sim 80 \degree$ 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 this is not distinguishable from the case where all satellites are are distributed isotropically and no disks are present at all. While this result seems to 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.
Figure \ref{fig:bells} shows the same information as Figure \ref{fig:disks} with the disk model being replaced by the ``dumbbell model." It is quite apparent that the dumbbell model describes the data better than the disk model. Due to the strong constraints of the model, the pure dumbbell corotation fraction model is undefined over much of the regime (satellite pairs separated by e.g. 60 degrees simply do not exist no matter how the model is rotated), therefore we do not plot it. This can be rectified by the inclusion of isotropic satellite systems; indeed the data agrees quite well with the inclusion of 90\% isotropic systems, and unlike the pure disk scenario, this scenario is distinguishable from the purely isotropic case. Whether or not the ``dumbbell model" is physical will be discussed at length in \ref{sec:Discuss}; we conclude this subsection with the claim that the ``dumbbell model" is the better fit to the kinematic data.
\Subsection{Astrometric Comparison}
\Subsection{Velocity Modeling and Cuts}
