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Chapter 4: Problem 68

Can an astronaut orbiting Earth in a satellite at a distance of \(180 \mathrm{km}\) from the surface distinguish two skyscrapers that are \(20 \mathrm{m}\) apart? Assume that the pupils of the astronaut's eyes have a diameter of \(5.0 \mathrm{mm}\) and that most of the light is centered around \(500 \mathrm{nm}\).

Short Answer

Expert verified
The astronaut cannot distinguish the two skyscrapers that are 20 m apart. Using the Rayleigh Criterion, we calculated the angular resolution of the astronaut's eyes to be \(1.22 \times 10^{-4} \, rad\). We also found the actual angular separation between the skyscrapers to be \(1.11 \times 10^{-4} \, rad\). Since the actual angular separation is less than the angular resolution, the astronaut cannot distinguish the two skyscrapers.

Step by step solution

01

Determine angular resolution using Rayleigh Criterion

To find the angular resolution, we can use the Rayleigh Criterion formula, which states that the minimum angular resolution θ is given by: \(\theta = \dfrac{1.22 \cdot \lambda}{D}\), where λ is the wavelength of the light and D is the diameter of the aperture (in this case, the diameter of the pupil). For our problem, λ = 500 nm (given) and D = 5.0 mm (given).
02

Convert units to meters

To perform calculations, we need to convert the given values to meters. λ = 500 nm = \(5.0 \times 10^{-7}\) m and D = 5.0 mm = 0.005 m.
03

Calculate angular resolution

Using the Rayleigh Criterion formula from Step 1 and values from Step 2, we can now calculate the angular resolution θ: \(\theta = \dfrac{1.22 \cdot 5.0 \times 10^{-7} m}{0.005 m}\) \(\theta = 1.22 \times 10^{-4} \, rad\)
04

Calculate the actual angular separation between the skyscrapers

To check if the astronaut can distinguish the skyscrapers, we need to compare the angular resolution with the actual angular separation between the skyscrapers. To find that separation, we can use the formula \(\alpha = \dfrac{d}{h}\), where α is the angular separation, d is the distance between the skyscrapers (20 m), and h is the height from the Earth's surface to the astronaut (180 km). First, we need to convert 180 km to meters: 180 km = 180,000 m. Now, we can plug the values into the formula: \(\alpha = \dfrac{20 m}{180,000 m}\) \(\alpha = 1.11 \times 10^{-4} \, rad\)
05

Compare the angular resolution with the actual angular separation

Now that we have both the angular resolution θ and the actual angular separation α, we can compare them to determine if the astronaut can distinguish the skyscrapers. If the actual angular separation is greater than or equal to the angular resolution, then they can be distinguished. In this case, \(\alpha = 1.11 \times 10^{-4} \, rad\) and \(\theta = 1.22 \times 10^{-4} \, rad\). Since the actual angular separation (α) is less than the angular resolution (θ), the astronaut cannot distinguish the two skyscrapers that are 20 m apart.

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