Question

A refracting telescope with a lens whose diameter is 26.0cm is used to view a binary star system that emits light in the visible region.

What is the minimum angular separation of the two stars for them to be barely resolved?

If the binary star is a distance of 4.00

Answer 1

Apply sin(angle) = 1.22(wavelength)/D

The wavelength range for visible light is from 400nm to 700nm according to most books (see note below)

Apply the formula for each extreme...

sin(angle) = 1.22(400 X 10-9)/.26

angle = 1.08 X 10^-4 degrees (That is 1.87 X 10^-6 rad)

And

sin(angle) = 1.22(700 X 10-9)/.26

angle = 1.88 X 10^-4 degrees (That is 3.28 X 10^-6 rad)

Since they are really close, that means it starts to be separated at 1.08 X 10^-4 degrees (1.86 X 10^-6 rad) but can be seen better at 1.88 X 10^-4 degrees (3.28 X 10^-6 rad)

To find the separation, multiply the radian measure by 4 X 10²⁰

That is either (1.86 X 10^-6)(4 X 10²⁰) = 7.44 X 10¹⁴ km or...

(3.28 X 10^-6)(4 X 10²⁰) = 1.312 X 10¹⁵ km

(NOTE: This solution is subject to interpretation which is why I gave you a couple of answers. Additionally, it also is subject to the 400 to 700 nm light range. My book uses those values, but your book may use something different. I have seen books use 350 nm to 750 nm so I really need to know what your book wants)

If the answers are off, they are not wrong, just not what your book wants. Please ask and I can help further. I used to teach Physics and can help further if needed. Problems like this are really specific to the text you are learning from. I know, I taught from many of them. Let me know if you have any question.