Angular Separation of Stars: Min Resolved w/ Diffraction Effects

[grouper]

Homework Statement

What is the minimum angular separation an eye could resolve when viewing two stars, considering only diffraction effects?

Homework Equations

θ=(1.22*λ)/D

The Attempt at a Solution

I tried estimating with λ=550 nm and D=5.0 mm (pupil diameter) which appeared in another problem about viewing stars and got 1.34e-4 rad, but this was incorrect. Our book states the best eye resolution is around 5e-4 rad so I tried that as well, but it wasn't correct either. This problem must want something more concrete than an estimation but I'm not sure where to go with it.

 

[CanIExplore]

They're asking for a minimum here. According to rayleigh equation which you've written there, what criterion would minimize the angle theta?

 

[grouper]

I suppose either a smaller wavelength or a larger diameter; do you think I should be using a different λ for my estimation? If I use λ=400 nm (keeping D=5.0 mm), that yields θ=9.76e-5 rad, but that's not correct either. I got the 5.0 mm diameter from another problem, so I'm not sure adjusting the diameter will get a correct answer either.

 

[grouper]

I figured out the problem; they were asking for arcs. Wasn't indicated anywhere but I eventually gave up on this problem since it's due tonight and when it showed the correct answer it read "0.46' of arc". Would've been nice if they said that in the problem, especially considering all the other problems in our book deal in radians. Oh well.

 

[asteriatic]

I would approach this problem by first understanding the concept of diffraction and its effects on resolving power. Diffraction is the bending of light waves around obstacles, and in the case of viewing stars, it refers to the spreading out of light from a single point source. This spreading out of light results in a decrease in the resolution of the image formed by the eye.

To calculate the minimum angular separation that the eye can resolve, we can use the Rayleigh criterion, which states that two point sources can be resolved if the peak of one source is separated from the peak of the other source by at least one diffraction spot. This spot is defined as the first minimum of the diffraction pattern, where the intensity of the light is significantly reduced.

Using the equation provided, θ=(1.22*λ)/D, we can calculate the minimum angular separation by plugging in the values for λ (wavelength of light) and D (pupil diameter). However, it is important to note that this equation assumes perfect conditions, such as a perfect eye with no aberrations or defects. In reality, the resolution of the eye can be affected by various factors, such as the quality of the eye's optics and the individual's visual acuity.

In the case of this problem, if we use the values of λ=550 nm and D=5.0 mm, we get a minimum angular separation of 1.34e-4 rad, which is close to the estimation given in the problem. However, as mentioned before, this value may not be entirely accurate due to the limitations of the equation and the assumptions made.

In conclusion, the minimum angular separation an eye can resolve when viewing two stars, considering only diffraction effects, can be calculated using the Rayleigh criterion. However, this value may vary depending on the individual's eye and other factors. It is important to keep in mind that this is just a theoretical calculation and may not accurately reflect real-world situations.