The sensitivity of a single baseline between two antennas \(i,j\) can, according to Eq. 9-14 in Wrobel & Walker (1999), be written as $$ \Delta S_{ij} [\mathrm{Jy/beam}]= \frac{1}{\eta}\sqrt{\frac{SEFD_iSEFD_j}{2\Delta\nu\tau n_p}} $$ where \(SEFD_i\) is the System Equivalent Flux Density of antenna \(i\), \(\eta\) is the system efficiency (taken to be \(0.7\)), \(\Delta\nu\) the observing bandwidth [Hz], \(\tau\) the time on source [seconds], and \(n_p\) the number of polarisations averaged together [1 or 2]. Given \(n\) antennas where each baseline has equal weight (natural weighting), the standard deviation (sensitivity) of the combined image is the propagated standard error of the weighted mean: $$ \Delta S = \left(\sum_{i=1}^n \sum_{j=i+1}^n \Delta S_{ij}^{-2}\right)^{-1/2}=\frac{SEFD^*}{\eta\sqrt{2\Delta\nu\tau n_p}} \quad \mathrm{where}\\ SEFD^* = \left(\sum_{i=1}^n \sum_{j=i+1}^n \frac{1}{SEFD_iSEFD_j}\right)^{-1/2} $$ which sums over all baselines without double counting, i.e. \(n(n-1)/2\) terms.

Additional notes

• Data losses due to e.g. additional RFI can be accounted for by reducing \(\tau\) or \(\Delta\nu\).
• Other weighting schemes, such as Briggs robust 0.5, normally result in 10% higher image noise (but improved resolution).