deletions | additions       diff --git a/subsubsection_Survey_Speed_label_sec__.tex b/subsubsection_Survey_Speed_label_sec__.tex  index e276961..a9001e0 100644  --- a/subsubsection_Survey_Speed_label_sec__.tex  +++ b/subsubsection_Survey_Speed_label_sec__.tex 
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In the original TIP, the assumed primary beam width of $\theta_P=15^\prime$ at 3~GHz implies an overall survey speed (SS) of $16.55$~deg$^2$/hour at S-band for a depth of 100~$\mu$Jy/beam. For the VLASS target per-epoch depth of 120~$\mu$Jy/beam this gives $23.83$~deg$^2$/hour. The value of $\theta_P=15^\prime$ at 3~GHz came from the Observational Status Summary (OSS) at that time, which stated "An approximate formula for the full width at half power in arcminutes is: $\theta_{PB} = 45/\nu_{\rm GHz}$."\footnote{\url{https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/fov} --- this may be revised with corrected values by the time you read this.} It turns out this was quite approximate. In fact, for the old system in VLA Memo 134 (Napier \& Rots 1982) the formula was given as $\theta_{P} = 44.255^\prime/\nu_{\rm GHz}$ which implies that at least for the old VLA that $\theta_P=14.75^\prime$ at 3~GHz, although it did not have any S-band receivers.   For the upgraded Jansky VLA, new wide-band feed horns were installed, including the new S-band system. In EVLA Memo 195 (Perley 2016) the measured sizes and polynomial fits to the primary beam over S-band are given, see Fig.~\ref{fig:sbandbeam}. What is apparent from this is that the primary beams are narrower than those assumed previously and varying over the band, with $\theta_{P} = 40.6$--$42.9^\prime/\nu_{\rm GHz}$. At band center, these values give $\theta_P=13.97^\prime$ at 3~GHz, which is 7\% narrower than that assumed in the original TIP and proposal. If we do the complete calculation using the ECT time on source for 120~$\mu$Jy/beam of $5.37$s this gives $20.59$~deg$^2$/hour. Thus, the required survey speed to reach the desired depth should be 14\% lower than that assumed above, and the survey will take 16\% longer to carry out. Equivalently, using the original survey speed of $SS=23.83\,deg^2$/hour will yield a full-band image depth of 129~$\mu$Jy/beam per epoch, and $74.5\,\mu$Jy/beam for 3 epochs combined. However, there is another compensatory factor. The mosaic imaging process weights the data explicitly by assigned weights (normally related to the rms noise) and implicitly by the beam area at each frequency (as the effective integration time for a channel is proportional to the beam area). This effect was pointed out by Condon (2014) and can simply be calculated through an effective MFS mean beam over frequency channels $k$  \begin{equation}  \bar{\Omega}_B = { \Sum_k w_k \Omega_{Bk} \over \Sum_k w_k }  \end{equation}  where for uniform weights $w_k=$const. and uniform frequency coverage over the band we can approximate by the integral  \begin{eqnarray}  \bar{\Omega}_B & = & {1\over \nu_{max}-\nu_{min}}\,\int_{\nu_{min}}^{\nu_{max}} d\nu\,\Omega_B(\nu)  \qquad \Omega_B(\nu) = \Omega_B(\nu_0)\,\left( {\nu_0 \over \nu}\right)^2 \\  & = & {\nu^2_0 \over \nu_{min}\,\nu_{max}}\,\Omega_B(\nu_0)  \end{eqnarray}  assuming the beam FWHM scales inversely by frequency. For full S-band $\nu_{min}=$2GHz, $\nu_{max}=$4GHz, and $\nu_{0}=$3GHz then   \begin{equation}\label{eq:ombeff}  \bar{\Omega}_B= {9\over8}\,\Omega_B(\nu_0=3{\rm GHz}) \qquad \bar{\theta}_P = \sqrt{9\over8}\,\theta_P(3{\rm GHz})  \end{equation}