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.