An astronomer is observing a nearby star system in an attempt to determine whether it is made up of one or two stars. If the star system is 4.1 light-years away and she is observing with blue light (400 nm) using a telescope with a 7.4 meter circular primary mirror, how far apart must the stars be (perpendicular to the line of sight) in order for her to tell them apart

x = 2.6 10⁹ m

Explanation:

In order to differentiate the two stars, you must apply the Rayleigh criterion, which establishes that the objects are separated if the maximum diffraction of one coincides with the first minute of diffraction of the other.

Based on this, the diffraction for slits with m = 1

                sin θ = λ / a

as the angles are very small

                sin θ = θ

               θ = λ / a

if instead of slits we use circular openings, the problem must be solved in polar coordinates, giving

                θ = 1.22 λ /D

let's calculate

                θ = 1.22 400 10⁻⁹ / 7.4

                θ = 6.59 10⁻⁸ rad

this is the tiny angle to be able to see the stars separately, if we use the definition of radians

               θ = x / R

where R is the distance to the system

             x =  θ R

let's reduce the distance to the SI system

             R = 4.1 light-years (9.461 10¹⁵ m / 1 light-year)

             R = 3.88 10¹⁶ m

let's calculate

             x = 6.59 10⁻⁸ 3.88 10¹⁶

             x = 2.6 10⁹ m

this is the minimum distance between the two stars to be able to differentiate them

the components of a vector are the parts of the vectors that are to each other.

b perpendicular