The two stars in a binary star system have an angular separation of $1.0 \times 10 ^ { - 5 } \mathrm { rad }$. If the wavelength of the light from the system is $500 \mathrm {~nm}$, what is the smallest diameter aperture telescope that could just resolve this system to show that it contains two stars?
A) $6.1 \mathrm {~cm}$
B) $5.0 \mathrm {~cm}$
C) $0.61 \mathrm {~cm}$
D) $0.50 \mathrm {~cm}$

Question

Quizplus · Accepted Answer

The smallest diameter aperture telescope that could just resolve this system can be calculated using the Rayleigh criterion formula: $D = 1.22 \frac{\lambda}{\theta}$, where $D$ is the diameter of the aperture, $\lambda$ is the wavelength of light, and $\theta$ is the angular separation of the stars. Substituting $\lambda = 500 \times 10^{-9} \, \text{m}$ (or $500 \, \text{nm}$) and $\theta = 1.0 \times 10^{-5} \, \text{rad}$, we get $D = 1.22 \times \frac{500 \times 10^{-9}}{1.0 \times 10^{-5}} = 1.22 \times 50 \times 10^{-3} \, \text{m} = 61 \times 10^{-3} \, \text{m} = 6.1 \, \text{cm}$.

The Two Stars in a Binary Star System Have an Angular