deletions | additions       diff --git a/Comparison to Toy Models.tex b/Comparison to Toy Models.tex  index 6994724..34eaf4c 100644  --- a/Comparison to Toy Models.tex  +++ b/Comparison to Toy Models.tex 
...
In this section we make comparisons of the dynamical SDSS data to simple toy models. We begin by detailing how each toy model is constructed. In the disk and isotropic models, hosts are permitted to have more than two satellites; in these cases, hosts are four times more likely to have n as n+1 satellites.   1. 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 so as to be corotating in circular motion. Fiducially, all All  satellites are assigned a 3D velocity of 100 km/s. 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. 
...
\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 $90^{\circ}$, 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 on-sky opening  angle between the satellites, where the vertex of the angle is at the location of the host. $\alpha$  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 $\alpha$s less than $90^{\circ}$, 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 $\sim 20^{\circ} \, \lt \alpha \, \lt \sim 60^{\circ}$ 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 this is not distinguishable from the case where all satellites are are distributed isotropically and no disks are present at all. Notably, the 90\% isotropic model fails to reproduce the high degree of corotation at small $\alpha$, which was the reason for this investigation in the first place. 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.