Yielding the interferometric extended source sensitivity

The point source sensitivity is well adapted to unresolved sources because it directly delivers the estimation of the flux of these sources. For extended sources, the point source sensitivity that is expressed in unit of Jy/Beam, is difficult to understand because it depends on the synthesized beam resolution in a non-trivial way. When a source is resolved (extended compared to the expected synthesized beam), it is much easier to think in temperature brightness. We thus convert back to a brightness temperature scale, but we now do it at the synthesized beam resolution.

In order to generalize Eq. [*] to the final product of an interferometer, we use the fact (see Sect. [*]) that the beam area ( $\Omega_\ensuremath{\mathrm{ant}}$) of a telescope of effective collecting surface $A_\ensuremath{\mathrm{eff}}$ is linked to the observing wavelength ($\lambda$) through

$\displaystyle \ensuremath{\Omega_\ensuremath{\mathrm{ant}}} \, \ensuremath{A_\ensuremath{\mathrm{eff}}}= \ensuremath{\lambda}^2.$ (12)

This yields

$\displaystyle \ensuremath{F_\ensuremath{\mathrm{}}}= \ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{}}}\ensuremath{T_\ensuremath{\mathrm{A}}^\star}$   with$\displaystyle \quad
\ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm...
...thrm{ant}}}\,\ensuremath{F_\ensuremath{\mathrm{eff}}}}{\ensuremath{\lambda}^2}.$ (13)

We will use this relation twice.

Combining Eq. [*], [*], and [*], we yield the usual

$\displaystyle \boxed{%
\ensuremath{\sigma_\ensuremath{\mathrm{K}}} = \frac{\en...
...remath{\theta_\ensuremath{\mathrm{min}}}}{4\,\ln{2}\,\ensuremath{\lambda}^2},
}$ (16)

which can be rewritten as

$\displaystyle \boxed{%
\ensuremath{\sigma_\ensuremath{\mathrm{K}}}
= \frac{\en...
...ant}}}-1)\,\ensuremath{d\nu}\,\ensuremath{\Delta t_\ensuremath{\mathrm{}}}}},
}$ (17)

where $\sigma_\ensuremath{\mathrm{K}}$ is the rms noise brightness, $\theta_\ensuremath{\mathrm{prim}}$ the half primary beam width, and $\theta_\ensuremath{\mathrm{maj}}$ and $\theta_\ensuremath{\mathrm{min}}$ the half beamwidth along the major and minor axes of the synthesized beam.