Question

Estimate the linear separation (in kilometers) of two objects at a distance of 2.6 × 108 km that can just be resolved by an observer on Earth (a) using the naked eye and (b) using a telescope with a 5.0-m diameter mirror. Use the following data: diameter of pupil = 5.0 mm; wavelength of light = 550 nm.

Answer 1

Answer:

34892 km

34.892 km

Explanation:

L = Distance of two objects = $2.6\times 10^8\ km$

$\lambda$ = Wavelength = 550 nm

When d = Diameter of pupil = 5 mm

Linear separation

$\frac{D}{L}=\frac{1.22\lambda}{d}\\\Rightarrow D=\frac{1.22\lambda L}{d}\\\Rightarrow D=\frac{1.22\times 550\times 10^{-9}\times 2.6\times 10^8\times 10^3}{5\times 10^{-3}}\\\Rightarrow D=34892000\ m=34892\ km$

Linear separation = 34892 km

When d = 5 m

$\frac{D}{L}=\frac{1.22\lambda}{d}\\\Rightarrow D=\frac{1.22\lambda L}{d}\\\Rightarrow D=\frac{1.22\times 550\times 10^{-9}\times 2.6\times 10^8\times 10^3}{5}\\\Rightarrow D=34892\ m=34.892\ km$

Linear separation = 34.892 km