Problem 57

  The largest refracting telescope in the world is at the Yerkes Observatory in Williams Bay, Wisconsin. The objective of the telescope has a diameter of 1.02 m. Two objects are $3.75 \times 10^4 m$ from the telescope. With light of wavelength 565 nm, how close can the objects be to each other so that they are just resolved by the telescope?

SOLUTION:
For minimum angular separation, we use Rayleigh's criterion

$$\theta_{min} \approx 1.22 \frac{\lambda}{D}$$

$$\theta_{min} \approx 6.76 \times 10^{-7} rad ,$$

where $\lambda$ is the wavelength of light and D is the aperature diameter (of the telescope). If y is the separation and L the distance to the stars, then, because $\theta_{min}$ is very small,

$$y = L\theta_{min} = 0.0254 m .$$