## 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 ( ) of a telescope of effective collecting surface is linked to the observing wavelength () through

| (12)

This yields

with | (13)

We will use this relation twice.

- On one hand, we have to use the solid angle of the primary beam for and . Noting that this beam area corresponds to the main beam area of each telescope, this yields

  | (14)

- On the other hand, we have to use the solid angle of the synthesized beam for the conversion factor that we have to apply to the deconvolved product ( ). After calibration (including the the impact of the atmospheric decorrelation), imaging, and deconvolution (including a potential phase self-calibration), an interferometer mimick the observation by a perfect telescope of angular resolution equal to the synthesized beam. In this case, the beam area is equal to the main beam solid angle (i.e., the interferometric beam efficiency is equal to unity or ), and to the foward beam solid angle (i.e., ). We thus yield

  | (15)

  Note that we don't use the decorrelation efficiency in the later equation. This is also due to the fact that after the data reduction, the deconvolved product should appear as if it was observed by a perfect antenna whose response is exactly a Gaussian of angular size .

Combining Eq. , , and , we yield the usual

| (16)

which can be rewritten as

| (17)

where is the rms noise brightness, the half primary beam width, and and the half beamwidth along the major and minor axes of the synthesized beam.