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Chapter 32: Problem 31

Find the minimum telescope aperture that could resolve an object with angular diameter 0.35 arcsecond, observed at 500 -nm wavelength. (Note: 1 arcsec \(=1 / 3600^{\circ} .\) )

Short Answer

Expert verified
The minimum telescope aperture that could resolve an object with an angular diameter of 0.35 arcsecond, observed at a 500-nm wavelength, is 0.36 meters.

Step by step solution

01

Understand the Rayleigh criterion

The Rayleigh criterion for the resolution limit of a diffraction-limited system like a telescope can be expressed as \(θ = 1.22 \times \frac{λ}{d}\), where \(θ\) is the angular resolution, \(λ\) is the wavelength of light, and \(d\) is the aperture diameter of the telescope.
02

Convert the angular diameter to radians

The angular diameter given is 0.35 arcsecond, but our formula for the Rayleigh criterion requires that the angular diameter be in radians. Therefore, we must convert the angular diameter from arcseconds to radians. 1 arcsecond equals \(1/3600\) degrees, and there are \(\pi/180\) radians in a degree. Therefore, we convert 0.35 arcseconds to radians by multiplying \(0.35\) by \(1/3600\) and by \(\pi/180\), which gives us \(1.69684 \times 10^{-6}\) radians.
03

Solve for the aperture diameter

Now we can use the Rayleigh criterion to solve for \(d\). Rearranging the formula to isolate \(d\), we get \(d = 1.22 \times \frac{λ}{θ}\). Substituting \(λ = 500\) nm or \(500 \times 10^{-9}\) meters and \(θ = 1.69684 \times 10^{-6}\) radians, we get \(d = 1.22 \times \frac{500 \times 10^{-9}}{1.69684 \times 10^{-6}} = 0.3597\) meters.

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