= Beam solid angle
{title2=$\Omega_A$}
The <beam solid angle> of an <antenna (radio)> is the integral of its normalized radiated-power pattern over the complete sphere. If $U(\theta,\phi)$ is radiated <power> per unit <solid angle> and $U_{\max}$ is its maximum, then
$$
\Omega_A=\int_{4\pi}\frac{U(\theta,\phi)}{U_{\max}}\,d\Omega,
\qquad d\Omega=\sin\theta\,d\theta\,d\phi.
$$
It is measured in <steradians> and includes sidelobes and backlobes, not just a chosen main-beam boundary. Since total radiated <power> is $P_{\rm rad}=\int U\,d\Omega$, peak <antenna directivity> satisfies
$$
D_0=\frac{4\pi U_{\max}}{P_{\rm rad}}=\frac{4\pi}{\Omega_A}.
$$
The definition relates concentration of radiation to the entire pattern, rather than to a single width measurement.<Antenna Models for Electromagnetic Compatibility Analyses by C. W. Wang and T. Keech>[Chapter 2, antenna parameters and directivity in terms of normalized radiation intensity]{ref}
For a narrow, axisymmetric Gaussian power pattern $U/U_{\max}=e^{-\theta^2/(2\sigma^2)}$, replace $\sin\theta$ by $\theta$ near the axis and extend the negligible tail to infinity. The resulting <Gaussian beam solid angle> is $\Omega_A\simeq2\pi\sigma^2$. Its full half-power width is $\theta_{\rm HP}=2\sqrt{2\ln2}\,\sigma$, so $\Omega_A\simeq\pi\theta_{\rm HP}^2/(4\ln2)$. These approximations require a narrow beam with no significant additional lobes; a broad pattern or strong sidelobes requires the full integral.
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