deletions | additions       diff --git a/subsection_Faint_Images_of_the__.tex b/subsection_Faint_Images_of_the__.tex  index d608941..bcf2b87 100644  --- a/subsection_Faint_Images_of_the__.tex  +++ b/subsection_Faint_Images_of_the__.tex 
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\subsection{Faint Images of the Radio Sky (FIRST)}  The Faint Images of the Radio Sky at Twenty-cm (FIRST) is presently the most sensitive large-area ($\gg 1 square degree$)survey degree$) survey  at radio wavelengths \cite{Becker_1995}. It is a radio snapshot survey performed at the NRAO Very Large Array (VLA) facility. FIRST covers approximately ${10 000 square degree}$ with a resolution of approximately 5 arcseconds. Coverage is shown in Figure 6 and Figure 7. FIRST produces 3-minute snapshots covering a hexagonal grid of the sky, using $2 \times 7$ 3-MHz frequency channels centered at 1365 and 1435 MHz The survey catalogue contains around one million sources, and it is estimated that nearly $15\%$ of these sources have optical counterparts. The FIRST survey area has been selected to correspond with that of the SDSS. (Sky Survey). FIRST provides a database that is uniform in angular resolution and flux density sensitivity and it offers the opportunity to produce the largest unbiased survey for statistical analysis. FIRST’s design enables the search for radio variability of sources on timescales of minutes to years \cite{Thyagarajan_2011} \subsubsection{Trade-off between area and depth in FIRST survey} 
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where $B_{v}(T)$ is the Planck distribution brightness. The associated sensitivity is then proportional to the ratio of the system sensitivity and the solid angle subtended by the receiver's diffraction pattern.  Given that the sensitivity limit of FIRST and our knowledge of radio source populations, we know this survey is dominated by the FR-I and FR-II sources described earlier. That is, radio jets that results from collimated electrons spiraling around magnetic field lines at relativistic speeds following acceleration by a centered SMBH. Despite the challenges in selecting an unbiased sample, the Liu et al. catalogue is the largest sample of quasar pairs complied to date.