= Solution
The angular separation is $\Delta\theta=10^{11}/(2\times10^{17})=5.00\times10^{-7}\,\mathrm{rad}$. Each star's angular diameter is below $5\times10^{-9}\,\mathrm{rad}$, so treating it as a point source is justified at these diffraction scales.
At300nm the <Airy disk> first-zero radius is $1.22\lambda/D=1.525\times10^{-7}\,\mathrm{rad}$. The separation is3.28 times this radius: \b[two well-separated bright peaks] occur. The stars are mutually incoherent, so their <point spread functions> add as intensities, not amplitudes:
$$
I(\theta)=I_\star\left[A(\theta-\Delta\theta/2)+A(\theta+\Delta\theta/2)\right],\quad
A(u)=\left[\frac{2J_1(\pi Du/\lambda)}{\pi Du/\lambda}\right]^2.
$$
The value at u=0 is defined by continuity. Faint diffraction rings remain on either side of each principal peak. <University Physics Volume 3 by Samuel J. Ling and others>[§4.5, circular-aperture diffraction and Rayleigh angular scale]{ref}
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-1-pa2010-star-images.png]
{title=Two-star profiles}
{description=Summed Airy intensities for equal incoherent point sources at the two specified wavelengths.}
{source=paper-1-pa2010-star-images.py}
Back to article page
