The power
pattern of a short
dipole antenna is
$$P \propto \sin^2 \theta\rlap{\quad
\href{AntennaTheory.html#DipolePattern}{\rm {(3A1)}}}$$
The power emitted by a short ($l \ll \lambda$) dipole driven by a
current $I = I_0 e^{-i \omega t}$ is
$$\langle P \rangle = {\pi^2 \over 3 c} \biggl({I_0 l \over
\lambda}\biggr)^2 \rlap{\quad\href{AntennaTheory.html#DipolePower}{\rm
{(3A2)}}}$$
Radiation
resistance is defined by
$$R \equiv {2 \langle P \rangle \over
I_0^2}\rlap{\quad
\href{AntennaTheory.html#RadiationResistance}{\rm {(3A3)}}}$$
Energy conservation implies the
average power gain of a lossless antenna is
$$ \langle G \rangle = 1\rlap{\quad
\href{AntennaTheory.html#AverageGain}{\rm {(3A4)}}}$$
The effective
area of an antenna is defined by
$$ A_{\rm e} \equiv P_\nu / S_{\rm
(matched)}\rlap{\quad
\href{AntennaTheory.html#EffectiveArea}{\rm {(3A5)}}}$$
The average effective area of any
lossless antenna is $$\langle A_{\rm e} \rangle =
{\lambda^2 \over 4 \pi}\rlap{\quad
\href{AntennaTheory.html#AverageArea}{\rm {(3A6)}}}$$
Reciprocity
implies
$$G(\theta, \phi) \propto
A_{\rm e}(\theta, \phi)\rlap{\quad
\href{AntennaTheory.html#Reciprocity}{\rm {(3A7)}}}$$
Reciprocity and energy conservation imply:
$$A_{\rm e}(\theta, \phi) = {\lambda^2
G(\theta, \phi) \over 4 \pi}\rlap{\quad
\href{AntennaTheory.html#GainArea}{\rm {(3A8)}}}$$
Antenna
temperature is defined by
$$T_{\rm A} \equiv {P_\nu \over k}\rlap{\quad
\href{AntennaTheory.html#AntennaTemp}{\rm {(3A9)}}}$$
The antenna temperature produced by an
unpolarized point source of flux density $S$ is
$$T_{\rm A} = {A_{\rm e} S \over 2 k}\rlap{\quad
\href{AntennaTheory.html#AntSens}{\rm {(3A10)}}}$$
An effective collecting area $A_{\rm e} \approx 2761 {\rm ~m}^2$
corresponds to a point-source sensitivity of $1 {\rm ~K~Jy}^{-1}$.
Beam
solid
angle is defined by
$$\Omega_{\rm A} \equiv \int_{4 \pi} P_{\rm n} (\theta, \phi) d
\Omega\rlap{\quad
\href{AntennaTheory.html#BeamSolidAngle}{\rm {(3A11)}}}$$
where $P_{\rm n} (\theta, \phi)$ is the power pattern normalized to one
at the peak.
For a uniform compact source of brightness temperature $T_{\rm B}$
covering solid angle $\Omega_{\rm s}$,
$$T_{\rm A} \approx T_{\rm B} {\Omega_{\rm s} \over \Omega_{\rm A}}$$
The main
beam
solid
angle is
defined by the integral over the main beam to the first zero only:
$$\Omega_{\rm MB} \equiv \int_{\rm MB} P_{\rm n} (\theta, \phi) d
\Omega\rlap{\quad
\href{AntennaTheory.html#MainBeamSolidAngle}{\rm {(3A12)}}}$$
and is used in the definition of main
beam
efficiency:
$$\eta_{\rm B} \equiv {\Omega_{\rm MB} \over \Omega_{\rm A}}\rlap{\quad
\href{AntennaTheory.html#MainBeamEfficiency}{\rm {(3A13)}}}$$