Sensitivity Model

The ‘system equivalent flux density’ (SEFD) for a single dish is given by:

\[SEFD_{dish} = \frac{2kT_{sys}}{\eta_A A}\]

where:

  • \(k\) is the Boltzmann constant so that \(kT_{sys}\) measures the power received from background emission and all other sources of unwanted signal within the system, that is \(T_{sys} = T_{spl} + T_{sky} + T_{rcv} + T_{cmb} + ...\)

  • \(\eta_A\) is the dish efficiency

  • \(A\) is the geometric dish area.

The SEFD for an interferometer array made up of two types of dish is given by:

\[SEFD_{\mathrm{array}} = \frac{1}{\sqrt{ \frac{n_{\mathrm{SKA}}(n_{\mathrm{SKA}} - 1)}{SEFD_{\mathrm{SKA}}^2} + \frac{2 n_{\mathrm{SKA}} n_{\mathrm{MeerKAT}}}{SEFD_{\mathrm{SKA}} SEFD_{\mathrm{MeerKAT}}} + \frac{n_{\mathrm{MeerKAT}}(n_{\mathrm{MeerKAT}} - 1)}{SEFD_{\mathrm{MeerKAT}}^2} }}\]

where \(n_{\mathrm{SKA}}\) is the number of SKA antennas, \(n_{\mathrm{MeerKAT}}\) is the number of MeerKAT antennas, \(SEFD_{\mathrm{SKA}}\) is the SEFD computed for an individual SKA antenna, and \(SEFD_{\mathrm{MeerKAT}}\) is the SEFD computed for an individual MeerKAT antenna.

We define the telescope sensitivity here as the minimum detectable Stokes I flux (1 \(\sigma\)). This is equal to the noise on the background power, obtained using the radiometer equation \(\sigma = SEFD / \sqrt{2 B t}\), corrected for atmospheric absorption:

\[\Delta S_{min} \exp (-\tau_{atm}) = \frac{SEFD_{array}}{\eta_s \sqrt{2Bt}} Jy\]

where:

  • \(\Delta S_{min}\) is the source flux density above the atmosphere

  • \(\eta_s\) is the efficiency factor of the interferometer

  • \(B\) is bandwidth

  • \(t\) is integration time

  • \(\tau_{atm}\) is the optical depth of the atmosphere towards the target

See Implementation for more details.