Pointing tolerance of a circular receiving aperture ($\sin {\theta }_{1/2}\simeq 0.5145\lambda /D$)

For a uniformly illuminated circular aperture of diameter D, normalized received power is $[2J_1(x)/x{]}^2$ with $x=\pi D\sin \theta /\lambda$. The first null has $\sin \theta =1.21967\lambda /D$ and the half-power point has $\sin \theta \simeq 0.5145\lambda /D$. A pointing tolerance must name the tolerated power loss; the first-null angle is not a good low-loss tolerance. These estimates require an ideal aperture pattern and are only approximate for a dish a few wavelengths across.