deletions | additions       diff --git a/Comparison to Toy Models.tex b/Comparison to Toy Models.tex  index b55b5a2..f5673ef 100644  --- a/Comparison to Toy Models.tex  +++ b/Comparison to Toy Models.tex 
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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 subjected to a random rotation and oberved along the z direction, i.e. the z direction is taken to be the line of sight and the xy plane is taken to be the plane of the sky.  \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 model procceeds as in the disk model. \item M31 model - This model is based on the position and velocities of the 13 M31 satellites belonging to the co-rotating plane identified in \citet{Ibata_2013}. The three dimensional positions of the satellites are taken from \citet{McConnachie_2012} and the line-of-sight velocities are compiled from \citet{McConnachie_2012} and \citet{2013ApJ...768..172C}. Note that we only consider the 13 satellites exhibiting coherent rotation; the two satellites aligned with the planar structure but with couter-aligned line-of-sight velocities are considered part of the isotropic background. We assign proper motions to the satellites to place them in circular orbits around the host.  
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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\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 $90^{\circ}$, 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 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 and the black line to a purely isotropic sample; these cases will be discussed in greater detail in a subsequent subsection.  We find strong disagreement between the disk models with $\geq 50\%$ 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 $\alpha \gt \sim  10^{\circ}$. Our results seem 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. \subsubsection{COP model}  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 COP  model. Due to the strong constraints of the model, the pure dumbbell COP  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 Instead, we plot only  the inclusion of 50\%  isotropic satellite systems; indeed the data agrees and 90\% isotropic cases, both of which are in  quite well good agreement  with the inclusion of 90\% isotropic systems, and unlike SDSS data. This agreement is particularly notable at small $\alpha$, where  the pure disk scenario, this scenario sharp dropoff in corotating fraction at $\alpha \sim 10^{\circ}$  is distinguishable from captured by  the purely isotropic case. model.  Whether or not the ``dumbbell model" COP model  is physical will be discussed at length in \ref{sec:Discuss}; we conclude this subsection with the tentative claim that the ``dumbbell model" is the better fit to the kinematic data. \begin{table}[h]  \begin{tabular}{lll}