Past exam of the optics and electrodynamics course of the University of Cambridge / 2019 / Section A / A3 / Transverse coherence estimate

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.