= Solution
<Temporal coherence> compares the field at different times at one location: a narrow spectrum maintains a predictable <phase> relation over a longer delay. <Spatial coherence> compares the field at different locations at the same time: different angular source components illuminate a baseline with different <phase differences>, which wash out <interference> when averaged. These are properties of correlations, not claims that all thermal source points have a common <phase>.
Assume a distant spatially incoherent, quasimonochromatic source, small angles, identical slit transmission, matched polarizations and equal mean <intensity> at the two slits. For an angular component $\theta$, the relative input <phase> is $-kd\theta$. Let $\varphi$ be the additional observation-plane <phase>. Incoherent angular components add <intensities>, giving
$$
I_{\rm obs}(\varphi)=2F(0)+2\operatorname{Re}[e^{i\varphi}F(d)],
\qquad F(d)=\int I(\theta)e^{-ikd\theta}\,d\theta.
$$
Its extrema are $2F(0)\pm2|F(d)|$, hence
$$
\boxed{V(d)=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}
=\frac{|F(d)|}{F(0)}.}
$$
For symmetric $I(\theta)$, $F(d)$ is a real cosine transform. \b[Physical visibility is its normalized absolute value]; the printed proportionality omits that qualification. A negative signed coherence reverses fringe <phase> rather than giving negative contrast. This is the small-angle <Van Cittert-Zernike theorem>. <Temporal coherence> analogously depends on the <Fourier transform> of the spectrum.
A <Michelson stellar interferometer> uses separated collecting mirrors to obtain a large adjustable projected baseline without requiring a monolithic objective of that diameter. Each collector supplies <light> to a common combiner. Optical path compensation keeps the delay within the <temporal coherence> range; varying mirror separation samples <spatial coherence> and resolves stellar angular structure finer than a small collector alone.
Let $g_\star(Y)$ be the normalized signed coherence of one star, and take the baseline along the projected binary separation. The <equal-brightness binary-star visibility> for identical profiles centred at $\pm\beta/2$ give
$$
\boxed{g(Y)=g_\star(Y)\cos\!\left(\frac{\pi\beta Y}{\lambda}\right),\qquad
V(Y)=|g(Y)|.}
$$
For \b[uniform circular discs of <angular diameter> $\alpha$],
$$
\boxed{g_\star(Y)=\frac{2J_1(x)}x,\qquad
x=\frac{\pi\alpha Y}{\lambda},\qquad g_\star(0)=1.}
$$
For <binary separation from visibility-zero spacing>, the rapid binary visibility zeros satisfy $Y_j=(j+\tfrac12)\lambda/\beta$, so their repeated spacing is $\lambda/\beta$. The slower single-star envelope first vanishes at $Y=1.21967\lambda/\alpha$. The <stellar angular diameter from a visibility envelope> and the binary separation follow by fitting both scales; multiple baselines distinguish envelope zeros from binary zeros.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2014/paper-1-oe2014-binary-visibility.png]
{title=Original normalized binary-star visibility sketch for two uniform circular discs, with projected separation eight times their diameter; binary zeros and the stellar-diameter envelope are labelled}
Angular width alone does not specify a unique brightness profile. If the intended one-dimensional model is a uniform angular strip of full width $\alpha$, replace the disc factor by $\operatorname{sinc}(\pi\alpha Y/\lambda)$ with $\operatorname{sinc}u=\sin u/u$; its first envelope zero is $\lambda/\alpha$. <Limb darkening> changes the disc envelope too. Rotating the baseline is necessary to recover an unknown separation direction; a single orientation measures only its projection.
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