Question

An astronaut in a space shuttle claims she can just barely resolve two point sources on Earth’s surface, 160 km below. Calculate their (a) angular and (b) linear separation, assuming ideal conditions. Take λ = 540 nm and the pupil diameter of the astronaut’s eye to be 5.0 mm.

Short answer

The angular separation of the astronaut is $0.0076°$ and linear separation of the astronaut is 1.21 km.

Step-by-step solution

Rayleigh criterion

Rayleigh’s criterion suggests that if one object's central diffraction maxima coincide with the other object's first diffraction minima, then those two objects are said to be just resolved. The minimum angular separation of the just resolved objects is determined using,

${\mathbf{\theta }}_{\mathbf{R}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{22}\frac{\mathbf{\lambda }}{\mathbf{d}}$

where $\mathbf{\lambda }$ is the wavelength of light, and d is the diameter of the aperture.

Given Data

The separation between two points on earth: 160 km

The wavelength of light used: 540 nm

Diameter of astronaut’s eye: 5.0 mm

Determine the angular and linear separation.

In this case, two points on the earth 160 km from the astronaut is just resolvable. The angular separation between them is

$$\begin{gathered} {\theta }_R=1.22\left(\frac{540\times {10}^{-9}m}{5.0\times {10}^{-3}m}\right) \\ =1.32\times {10}^{-4}rad \\ =\left(7.55\times {10}^{-3}\right)° \end{gathered}$$

We know that linear separation is related to angular separation by

$$\begin{gathered} s=r\theta \\ =\left(160km\right)\left(7.55\times {10}^{-3}°\right) \\ =1.21km \end{gathered}$$

Hence the angular separation is $0.0076°$and linear separation is 1.21 km.