deletions | additions       diff --git a/Comparison to Toy Models.tex b/Comparison to Toy Models.tex  index 9a6c77d..4546963 100644  --- a/Comparison to Toy Models.tex  +++ b/Comparison to Toy Models.tex 
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2. ``Dumbell" 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.  3. M31 model - This model is based on the position and velocities of the 13 M31 satellites belonging to the corotating co-rotating  plane identified in \cite{Ibata_2013}. \citet{Ibata_2013}.  The three dimensional positions of the satellites are taken from \cite{McConnachie_2012} \citet{McConnachie_2012}  and the line-of-sight velocities are compiled from \cite{McConnachie_2012} \citet{McConnachie_2012}  and \cite{2013ApJ...768..172C} \citet{2013ApJ...768..172C}. Note that we only consider the 13 satellites that seem to be exhibiting coherent rotation; the two satellites aligned with the planar structure but with couter-aligned line-of-sight velocities are (for the purposes of this model) considered to be part of the isotropic background. We estimate proper motions of the 13 satellites on the assumption that they are in circular motion  3. Isotropic model - In this case, each host may again have 2-5 satellites. Each satellite is independently assigned a random position and velocity.