Answer:
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
