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Chapter 25: Problem 73

About how close to each other are two objects on the Moon that can just barely be resolved by the \(5.08-\mathrm{m}-\) (200-in.)-diameter Mount Palomar reflecting telescope? (Use a wavelength of \(520 \mathrm{nm}\).)

Short Answer

Expert verified
Answer: The smallest distance between two objects on the Moon's surface that can just barely be resolved by the 5.08-meter diameter Mount Palomar reflecting telescope, using a wavelength of 520 nm, is approximately 0.0693 km (about 69.3 meters).

Step by step solution

01

Find the angular resolution

First, we need to find the angular resolution of the telescope using the formula: Angular resolution = 1.22 * (wavelength / aperture) Here, the aperture is 5.08 meters, and the wavelength is 520 nm (which we need to convert to meters). To convert nanometers to meters, divide by \(10^9\): 520 nm = \((520 * 10^{-9})\) m Now we can find the angular resolution: Angular resolution = 1.22 * (\((520 * 10^{-9})\) m / 5.08 m)
02

Calculate the angular resolution

Performing the calculation, we get: Angular resolution = 1.22 * (\((520 * 10^{-9})\) m / 5.08 m) = \(1.249 * 10^{-7}\) radians Now we have the angular resolution in radians. To find the distance between two objects on the Moon's surface, we will need to convert the angular resolution to degrees.
03

Convert angular resolution to degrees

To convert radians to degrees, multiply by \(\frac{180}{\pi}\): Angular resolution_degrees = \(\frac{180}{\pi} * 1.249 * 10^{-7}\) radians = \(7.16 * 10^{-6}\) degrees Now we have the angular resolution in degrees.
04

Calculate the linear distance on the Moon's surface

Next, we need to find the smallest distance between two objects on the Moon's surface that can just barely be resolved by the telescope. To do this, we can use the small-angle approximation formula: Distance_between_objects = (Angular_resolution_degrees / 360) * (2 * π * Moon_radius) The average radius of the Moon is approximately 1,737.1 km. Plugging in the values, we get: Distance_between_objects = \((7.16 * 10^{-6}\) degrees / 360) * (2 * π * 1737.1 km)
05

Find the distance between two objects on the Moon's surface

Performing the calculation, we get: Distance_between_objects = \((7.16 * 10^{-6}\) degrees / 360) * (2 * π * 1737.1 km) ≈ 0.0693 km So, two objects on the Moon's surface can just barely be resolved by the \(5.08-\mathrm{m}\)-diameter Mount Palomar reflecting telescope if they are about 0.0693 km (about 69.3 meters) apart.

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