Question

Two stars 12 light-years away are barely resolved by a 220cm (mirror diameter) telescope.

How far apart are the stars in meters? Assume ?=550nm and that the resolution is limited by diffraction.

Express your answer using two significant figures.

Answer 1

If the stars are just resolved, then they are separated by the diffraction limit of the telescope, which is set by the Rayleigh criterion:

theta=1.22*lambda/d, where lambda=550nm and d=2.2 m (as the givens). So you can find theta:

In radians:
theta=1.22*(550*10^-9)/2.2
=3.05 * 10^-7 radians = 0.065 arcsec

then you know that
tan(theta/2)=d/(2*R), where R =12 ly and d is the distance between the two stars. But since theta is small, so is theta/2 so this equation simplifies to

theta/2=d/(2*R) which becomes (since 2 cancels
theta=d/R so then
d=R*theta= (12 ly)*(3.05*10^-7)=3.66*10^-4 ly