deletions | additions       diff --git a/subsection_The_Liu_Quasar_Pair__.tex b/subsection_The_Liu_Quasar_Pair__.tex  index dcd0515..86337e1 100644  --- a/subsection_The_Liu_Quasar_Pair__.tex  +++ b/subsection_The_Liu_Quasar_Pair__.tex 
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  \subsection{The Liu Quasar Pair Catalogue}  The quasar pair catalogue sample used in this project is obtained from \cite{Liu_2011}. This catalogue has 1286 AGN pairs in the redshift range $0.02 < z <0.33$. some of the criteria of the sample AGN includes angular separation, transverse proper separation, line of sight (LOS) velocity offset. \cite{Liu_2011} AGN that make up the parent sample are mostly optically-selected from the spectroscopic sample of the SDSS DR7. The SDSS DR7 consists of 105 783 objects. Further selection criteria are applied to refine the parent sample to ensure the study of genuinely interacting AGN pairs rather than chance superpositions. By using line of sight (LOS) velocity differences (or offset) where LOS velocity is the velocity measured along the line of sight of an observer, and transverse separation - the projected distance within the LOS direction - $r_{p}$. \citet{Liu_2011} Liu et al. (2011)  claims to significantly increase the probability that genuine AGN pairs are selected. The velocity offset – removes chance superpositions. ${100h_{70}}^{-1} = 1$ if we assume the Hubble (H) constant is equal= 70 km/s/Mpc. The Hublle constant (H) accounts for /is a measure of the expansion rate of the Universe ( more details in \cite{Huchra_1992}). \citet{Liu_2011} Liu et al. (2011)  further observes that most pairs with separation greater than 100 kpc are found to be closely separated but not interacting with one another. Selecting those with separation less than 100 kpc helps remove these points that are not a part of our sample space. The separation between merging AGN is a few galactic diameters for the average galaxy. \cite{Liu_2011} further Justifies the use a LOS velocity offset $ \Delta v < 600 km/s$ as a method to further refine the desired sample space. Plotting the distribution of $ \Delta v$ of AGN pairs (Figure 2 provided in \cite{Liu_2011}) shows us that that most AGN pairs in the catalogue are found within the $300 ≲ \Delta v < 600 km /s$ interval. Even so, \citet{Liu_2011} Liu et al. (2011)  further states that $ \Delta v$ range within which the pair is still interacting depends on the environment – there could be a chance that some data point were erroneously cut from the sample; however, the combined LOS velocity offset, projected separation and visual inspection (discussed below) aim to minimize this. Visual inspection of SDSS images for spectroscopically selected images further removes 34 AGN pairs from the parent sample. This means that strongly interacting AGN pairs can have $\Delta v$ very large.  \citet{Liu_2011} Liu et al. (2011)  notes that corrections are to be made to the sample due to SDSS’ limitations, since the sample is biased against very low separation pairs due to the hardware limitations of the fiber width (pairs with angular separation ≤ 55” lead to the spectroscopic data of the pair not being captured unless one of the AGN is captured on an overlapping plate). The correction is done by supplementing the observed pairs (with angular separation ≤ 55” ) with modified ([C-1]Np) pairs. The Np pairs are randomly drawn from the sample pairs with separation smaller than 55”. details of this correction are contained in \cite{Liu_2011}.