Suppose you wanted to be able to see small details on the Moon using an optical telescope. What is the smallest diameter of the objective lens required to resolve a 8.45 m object on the Moon? (Hints: Use the Rayleigh Criterion for circular apertures. The angle is very small, so it is safe to use the θ = sin(θ) = tan(θ) approximation, where θ is the angle measured in radians. Assume that the telescope works best in the middle of the visible spectrum, i.e. at λ = 550 nm. The Earth-Moon distance is 3.847×105 km.)

The Rayleigh Criterion says if two objects are separated by the angle [tex]\theta=1.22\lambda/D[/tex], the objects are resolvable. In this, [tex]\lambda[/tex] is the wavelength and [tex]D[/tex] is the diameter of the aperture of the lens.

We don't know [tex]\theta[/tex] or [tex]D[/tex], but the criterion also says that the resolution angle can be equal to the resolving power [tex]x[/tex] divided by the distance to the object [tex]d[/tex]. The resolving power is the distance that permits differentiates two objects from each other or the size of the object that can be resolved.

[tex]\theta=1.22\frac{\lambda}{D} =\frac{x}{d}[/tex]

Then:

[tex]1.22\frac{\lambda}{D} =\frac{x}{d}[/tex]

Solving for D

[tex]1.22\frac{\lambda}{D} =\frac{x}{d}\\1.22\lambda=\frac{xD}{d}\\D=1.22\frac{\lambda d}{x} \\D=1.22\frac{(550\times10^{-9})(3.847\times10^8)}{8.45} \\D=1.22\frac{211.585}{8.45} \\ D=1.22(25.0396)\\D=30.5m[/tex]

Parrain Parrain · Answer 2 · 2022-04-05T16:19:33+02:00

Based on the Earth-Moon distance, and the length of the object, the smallest diameter required of the objective lens is 30.55 m.

What is the diameter required?

Resolving power of telescope = (Length of object / Earth-Moon distance)

Resolving power = (1.22 x λ) / diameter

Solving for d gives:

(1.22 x λ) / diameter = (8.45 / 3.847 x 10⁸)

(1.22 x 550 x 10⁻⁹) / diameter = (8.45 / 3.847 x 10⁸)

Diameter = (1.22 x 550 x 10⁻⁹ x 3.847 x 10⁸) / 8.45

= 30.55 m

Find out more on the Rayleigh Criterion of Circular Apertures at https://brainly.com/question/25755075.