deletions | additions       diff --git a/subsection_Faint_Images_of_the__.tex b/subsection_Faint_Images_of_the__.tex  index bcf2b87..7fadc68 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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$\sigma = \frac{2k_{B} T_{sys}}{A_{e} \times ( {(\Deltaν_{RF} \times \tau)}^{\frac{1}{2}}) }$   where $T_{sys}$ is the temperature of the interferometer system, $\Deltaν_{RF}$ is the receiver radio-frequency bandwidth, $\tau$ is the duration of the signal received from the interferometers. total integration.  The collecting area of circular parabolic radio telescopes is reduced to an effective area because the receiver is on the reflector axis, and together with its supporting legs, the receiver partially blocks the path of radiation falling onto the reflector. The angular resolution of a diffraction-limited telescope is given by $\theta \approx \frac{\lambda}{D}$ radians (where D is the diameter of the radio telescope dish and $\lambda$ is the wavelength of light). Large diameters are required to obtain sub-arc second resolution at radio wavelengths. For example, for 1 arcsecond resolution at 16 Hz need 50 km diameter. 
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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.