Equal-brightness binary-star visibility ($g(Y)=g_{\star }(Y)\cos (\pi \beta Y/\lambda )$)

Let identical spatially incoherent stellar brightness profiles of equal integrated flux be displaced by projected angles $\pm \beta /2$ along an interferometer baseline. The Fourier shift property makes their normalized signed coherence $g(Y)=g_{\star }(Y)\cos (\pi \beta Y/\lambda )$. The observed visibility is its absolute value. The slow single-star envelope depends on each star's brightness profile, while the oscillatory factor records projected separation. Unequal flux prevents complete cancellation at the binary minima.

For two identical equally bright stars, zeros of the binary factor occur at $Y_j=(j+1/2)\lambda /\beta$, where $\beta$ is separation projected along the baseline. Their repeated spacing is $\Delta Y=\lambda /\beta$. Multiple samples separate these rapid zeros from zeros of a resolved stellar envelope. Baseline rotation is needed to recover an unknown two-dimensional separation direction; the first zero alone does not determine both diameter and separation.

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.