The Hubble Space Telescope has an aperture of 2.4 m and focuses visible light (380–750 nm). The Arecibo radio telescope in Puerto Rico is 305 m (1000 ft) in diameter (it is built in a mountain valley) and focuses radio waves of wavelength 75 cm. (a) Under optimal viewing conditions, what is the smallest crater that each of these telescopes could resolve on our moon? (b) If the Hubble Space Telescope were to be converted to surveillance use, what is the highest orbit above the surface of the earth it could have and still be able to resolve the license plate (not the letters, just the plate) of a car on the ground? Assume optimal viewing conditions, so that the resolution is diffraction limited.

Step by step solution

01

Rayleigh’s Criterion

According to the Rayleigh criterion for the diffraction limit to resolution, two images are only resolvable when the center of one's diffraction pattern is directly over the first minimum of the other's diffraction pattern.

02

The smallest crater that each of the telescopes could resolve

1. Using the expression;

In the case of Hubble;

In the case of Arecibo;

Hence, under optimal viewing conditions, the smallest crater by each telescope is

03

Highest orbit above the surface of the earth

1. Taking the size of the plate as 0.3m, the condition can be expressed over R;
   

Hence, the highest orbit is 1500km.

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