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Convert the distances to SI units: $r=1.5\times 10^{11}\mathrm{m}$ and $D=1.4\times 10^9\mathrm{m}$. The angular solar diameter isTwo receivers separated by a transverse baseline $b$ see a spread of relative wave phases of order $2\pi b\Theta /\lambda$. Averaging over incoherent source directions therefore reduces interferometric visibility when $b$ reaches the transverse optical coherence length scale $\lambda /\Theta$. At $\lambda =500\mathrm{nm}$,If “coherence length” specifically means the first zero of uniform-disk stellar visibility, a uniformly bright circular solar disk gives $g^{(1)}(b)=2J_1(x)/x$, with $x=\pi b\Theta /\lambda$. The first zero then lies at $\boxed{b_0=1.21967\lambda /\Theta \simeq 65\mu \mathrm{m}}$. This more specific number assumes a uniform disk and a first-zero criterion; the characteristic estimate is approximately $50$–$60\mu \mathrm{m}$. Solar brightness variation across the disk changes the exact contrast curve.

Each mutually incoherent patch of an extended star produces a displaced double-slit interference pattern. The detector adds their light intensities. For two apertures separated by a projected interferometric baseline $D$, a source direction offset by a small angle ${\vartheta }_x$ changes their phase difference by $2\pi D{\vartheta }_x/\lambda$. The normalized sum over the source's angular brightness distribution is its first-order optical coherence, whose magnitude is the interferometric visibility for equally bright channels.

For a uniform circular stellar disk of full angular diameter $\alpha$, the normalized angular Fourier transform gives the uniform-disk stellar visibilityHere $J_1$ is the order-one Bessel function of the first kind. This follows by integrating $e^{i(2\pi D/\lambda ){\vartheta }_x}$ over the disk and dividing by its area. In particular, its power series gives $2J_1(x)/x=1-x^2/8+O(x^4)$.

For an ordinary slit separation $d=1\mathrm{mm}$, using the paper's $\alpha =22.6\times 10^{-8}\mathrm{rad}$ and observing wavelength $\lambda =570\mathrm{nm}$,Even $d=10\mathrm{mm}$ gives a visibility loss of only $1.94\times 10^{-5}$. Thus millimetre separation leaves the interference fringes almost indistinguishable from those of a point source; it gives very little sensitivity to the diameter.

The movable mirrors in the Michelson stellar interferometer collect light at a large adjustable separation $D$ and redirect it into closely spaced output beams. They increase the sampling interferometric baseline independently of the fixed output slit spacing $d$: $D$ sets sensitivity to source structure, whereas the output optics set the screen's interference fringe spacing. The first Bessel function zero is $x_1=3.83170597$, henceThe interference fringes disappear at this baseline even though each beam separately remains bright.

The value $3.08\mathrm{m}$ assumes a uniform disk. The original question supplies a diameter without a surface brightness law; real limb darkening changes the precise visibility zero. A one-dimensional uniformly bright strip of the same full angular width instead has a sinc function visibility and first zero $D=\lambda /\alpha =2.522\mathrm{m}$. That is a different source model, not the circular-disk result. The robust conclusion without a surface brightness model is that a baseline of a few metres, rather than millimetres, is required.

The transverse optical coherence length summarizes the separation over which equal-time spatial optical coherence remains appreciable. It depends on a contrast criterion and on the source brightness profile. For a distant incoherent source of angular diameter $\Theta$, its characteristic scale is $\lambda /\Theta$. A uniform circular source gives uniform-disk stellar visibility $2J_1(x)/x$, with $x=\pi b\Theta /\lambda$ and baseline $b$. Its first zero is $b=1.21967\lambda /\Theta$. A uniformly bright one-dimensional strip instead has its first zero at $\lambda /\Theta$; the numerical factor is a source-model choice, not a universal constant.