Uniform-disc binary-star visibility ($V(Y)=\left|\frac{2J_1(x)}{x}\cos (\pi \beta Y/\lambda )\right|$)

Two identical uniform circular discs of angular diameter $\alpha$ and equal flux have $g_{\star }(Y)=2J_1(x)/x$, $x=\pi \alpha Y/\lambda$, with limiting value one at $Y=0$. Multiplication by $\cos (\pi \beta Y/\lambda )$ gives their signed binary coherence. Its first single-disc envelope zero is $Y=1.21967\lambda /\alpha$, while the binary modulation has a separate scale $\lambda /\beta$. Limb darkening and other nonuniform profiles change the envelope, not the elementary Fourier shift factor.