Two stars are 3.7 1011

mapart and are equally distant from the earth. A telescope has

anobjective lens with a diameter of 1.03m and just detects these

stars as separate objects. Assume thatlight of wavelength 550 nm is

being observed. Also assume thatdiffraction effects, rather than

atmospheric turbulence, limit theresolving power of the telescope.

Find the maximum distance thatthese stars could be from the

earth.

L = 5.68 10¹⁷ m

Explanation:

The resolution of the telescope is given by diffraction

a sin θ = m λ

Where a is the separation of the linear slits, λ the wavelength, m an integer that determines the order of diffraction

In this case, suppose that the premieres meet Rayleigh's criteria, which establishes that the central maximum of the diffraction of an object coincides with the first minimum of diffraction of the second object. In this case m = 1

sin θ = λ / a

In the case of circular openings, polar coordinates must be used, so the equation is

sin θ = 1.22 λ / D

Where D is the diameter of the lens or tightness. Since the distances are very large and the small angles we can approximate the sine to the radian angle value

θ = 1.22 λ / D

Let's use trigonometry to find the angle. We have the separation of the premieres y = 3.7 10¹¹ m and

tan θ = y / L

θ = y / L

Let's replace

y / L = 1.22 λ / D

L = y D / 1.22 λ

calculate

L = 3.7 10¹¹ 1.03 / (1.22 550 10⁻⁹)

L = 5.68 10¹⁷ m