Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section A / 3

The angular separation is $\Delta \theta =10^{11}/(2\times 10^{17})=5.00\times 10^{-7}\mathrm{rad}$. Each star's angular diameter is below $5\times 10^{-9}\mathrm{rad}$, so treating it as a point source is justified at these diffraction scales.

At300nm the Airy disk first-zero radius is $1.22\lambda /D=1.525\times 10^{-7}\mathrm{rad}$. The separation is3.28 times this radius: two well-separated bright peaks occur. The stars are mutually incoherent, so their point spread functions add as intensities, not amplitudes:The value at u=0 is defined by continuity. Faint diffraction rings remain on either side of each principal peak. ^[ref]

At1000nm the first-zero radius is $5.083\times 10^{-7}\mathrm{rad}$, slightly larger than the $5.00\times 10^{-7}\mathrm{rad}$ separation. The images therefore overlap strongly and just fail the conventional Rayleigh criterion, rather than forming two clean isolated peaks.

The summed Airy profile can still have a shallow central dip and two local maxima: Rayleigh's criterion is a resolution convention, not the exact condition for a profile to become single-peaked. The two-panel original plot in part(a) shows the correct intensity sums at both wavelengths. ^[ref]