Keywords: astro-noema-sensitivity-estimator

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## We yield 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 is expressed in unit of Jy/Beam that 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.

After calibration (including the calibration of the atmospheric decorrelation), imaging, and deconvolution (including a potential phase self-calibration), an interferometer mimick the observation by a telescope of angular resolution equal to the synthesized beam. However, the notion of effective collecting surface is ambiguous in this case. In order to generalize Eq. 5 to the final product of an interferometer, we use the fact that the solid angle resolution of a telescope of effective collecting surface is by definition linked to the observing wavelength ( ) through

| (11)

We can thus generalize Eq. 5 as

| (12)

On one hand, we have to use the solid angle of the primary beam for and . This yields

| (13)

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 ( )

| (14)

Note that we don't use the decorrelation efficiency in the later equation. This is 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. 10, 13, and 14, we yield

| (15)

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.

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