Source and observed flux density

The integral of the brightness over the source extension yields the total source flux density

$\displaystyle \boxed{%
\ensuremath{F_\ensuremath{\mathrm{sou}}}(\ensuremath{\n...
...},\phi_{}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}},
}$ (63)

where

$F_\ensuremath{\mathrm{sou}}$ [ $\mathrm{W\,m^{-2}\,Hz^{-1}}$] flux density of source,
$B_{}$ [ $\mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$] brightness as a function of position over source,
$d\Omega$ [ $\mathrm{sr}$] infinitesimal solid angle of sky $(= \sin \theta\,d\theta\,d\phi)$.

When the source is observed with an antenna of power pattern ( $P_\ensuremath{\mathrm{ant}}$), the observed flux density in direction ( $\theta_{0},\phi_{0}$) is

$\displaystyle \ensuremath{F_\ensuremath{\mathrm{obs}}}(\ensuremath{\theta_{0},\...
...{}-\phi_{0}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}},$ (64)

where

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

This integral is a correlation. We now introduce the mirror symmetric of the antenna pattern to yield a convolution

$\displaystyle \ensuremath{F_\ensuremath{\mathrm{obs}}}(\ensuremath{\theta_{0},\...
...{0}-\phi_{}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}},$ (65)

where$\displaystyle \quad %
\ensuremath{\tilde{P}_\ensuremath{\mathrm{ant}}}(\ensure...
...rm{ant}}}(\ensuremath{\theta_{}-\theta_{0},\phi_{}-\phi_{0}},\ensuremath{\nu}).$ (66)

The distinction of these two patterns is only important for non-axisymmetrical cases.