The name of interferometric “extended” source sensitivity can be
misleading. Indeed, the notion of noise for an extended source depends on
the source shape. For instance, a flat source of infinite extent would be
measured through the visibility exactly located the phase center, and all
other visibilities would only show noise. In this thought experiment the
noise level on the constant brightness would thus be the noise of a single
visibility! More generally, if we take a Gaussian source of a given FWHM,
only the visibilities located inside a radius of a few time
would contribute the the flux measurement of that
Gaussian, and the other visibilities would just be noise.
So the only well defined “extended” source sensitivity is the sensitivity for a source that exactly fills the Gaussian synthesized beam. This is what we will derive in this section.