Observed brightnesses

The brightness observed in direction ( $\theta_{0},\phi_{0}$) is the flux density observed in this direction divided by the typical beam area

$\displaystyle \boxed{%
\ensuremath{B_{\ensuremath{\mathrm{obs}}}}(\ensuremath{...
...}-\phi_{}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}},
}$ (67)

where

$B_{obs}$( $\theta_{0},\phi_{0}$) [ $\mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$] observed brightness,
$B_{}$ [ $\mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$] actual brightness as a function of position over source,
$\Omega_\ensuremath{\mathrm{}}$ [sr] typical beam area,
$P_\ensuremath{\mathrm{ant}}$ [ $\mathrm{dimensionless}$] normalized antenna power pattern,
$d\Omega$ [ $\mathrm{sr}$] infinitesimal solid angle of sky $(= \sin \theta\,d\theta\,d\phi)$.

The typical beam area is an ambiguous notion. Indeed, there are three different solid angles that can be used in this formula: The antenna beam solid angle ( $\Omega_\ensuremath{\mathrm{ant}}$), the forward beam solid angle ( $\Omega_\ensuremath{\mathrm{fb}}$), or the main beam solid angle ( $\Omega_\ensuremath{\mathrm{mb}}$)

$\displaystyle \ensuremath{B_{\ensuremath{\mathrm{ant}}}}(\ensuremath{\theta_{0},\phi_{0}},\ensuremath{\nu})$ $\displaystyle = \frac{1}{\ensuremath{\Omega_\ensuremath{\mathrm{ant}}}}\,\ensur...
...{0}-\phi_{}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}},$ (68)
$\displaystyle \ensuremath{B_{\ensuremath{\mathrm{fb}}}}(\ensuremath{\theta_{0},\phi_{0}},\ensuremath{\nu})$ $\displaystyle = \frac{1}{\ensuremath{\Omega_\ensuremath{\mathrm{fb}}}}\,\ensure...
...{0}-\phi_{}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}},$ (69)
and$\displaystyle \quad
\ensuremath{B_{\ensuremath{\mathrm{mb}}}}(\ensuremath{\theta_{0},\phi_{0}},\ensuremath{\nu})$ $\displaystyle = \frac{1}{\ensuremath{\Omega_\ensuremath{\mathrm{mb}}}}\,\ensure...
...{0}-\phi_{}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}}.$ (70)

It is trivial to show that

$\displaystyle \boxed{%
\ensuremath{B_{\ensuremath{\mathrm{fb}}}} = \frac{1}{\e...
...{B_\ensuremath{\mathrm{eff}}}}\,\ensuremath{B_{\ensuremath{\mathrm{fb}}}}. %
}$ (71)

At IRAM, the observed brightness obtained after a single-dish calibration are by default computed using the forward beam solid angle.