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Chapter 34: Problem 43

Calculate and compare the angular resolutions of the Hubble Space Telescope (aperture diameter \(2.40 \mathrm{~m}\), wavelength \(450 . \mathrm{nm}\); illustrated in the text), the Keck Telescope (aperture diameter \(10.0 \mathrm{~m}\), wavelength \(450 . \mathrm{nm}\) ), and the Arecibo radio telescope (aperture diameter \(305 \mathrm{~m}\), wavelength \(0.210 \mathrm{~m}\) ). Assume that the resolution of each instrument is diffraction limited.

Short Answer

Expert verified
Answer: The order of image quality from highest to lowest is: Keck Telescope, Hubble Space Telescope, and Arecibo Radio Telescope.

Step by step solution

01

Hubble Space Telescope Resolution

To calculate the angular resolution of the Hubble Space Telescope, we will use the given values for the aperture diameter (D) as \(2.40 \mathrm{m}\) and the wavelength (λ) as \(450 \mathrm{nm}\). Plug these values into the formula and convert the wavelength to meters for consistency: $$ \theta_{Hubble} = 1.22\frac{450 \times 10^{-9} m}{2.4 m} $$
02

Keck Telescope Resolution

To calculate the angular resolution of the Keck Telescope, we will use the given values for the aperture diameter (D) as \(10.0 \mathrm{m}\) and the wavelength (λ) as \(450 \mathrm{nm}\). Plug these values into the formula and convert the wavelength to meters for consistency: $$ \theta_{Keck} = 1.22\frac{450 \times 10^{-9} m}{10.0 m} $$
03

Arecibo Radio Telescope Resolution

To calculate the angular resolution of the Arecibo Radio Telescope, we will use the given values for the aperture diameter (D) as \(305 \mathrm{m}\) and the wavelength (λ) as \(0.210 \mathrm{m}\). Plug these values into the formula: $$ \theta_{Arecibo} = 1.22\frac{0.210 m}{305 m} $$
04

Calculate The Resolutions

Now, calculate the angular resolutions for each telescope: $$ \theta_{Hubble} = 2.275 \times 10^{-7} radians \\ \theta_{Keck} = 5.46 \times 10^{-8} radians \\ \theta_{Arecibo} = 8.524 \times 10^{-4} radians $$
05

Compare The Resolutions

Now that we have the angular resolutions for each telescope, we can compare them. Lower angular resolution means better image quality. From the calculated values, we can see that the order of image quality (from highest to lowest) is: Keck Telescope, Hubble Space Telescope, and Arecibo Radio Telescope.

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