Compare the light-gathering power of a telescope with the 9-centimeter diameter mirror to that of the human eye. (Take the diameter of the pupil of...

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The light-gathering power of a telescope is the quantity of light which a telescope can gather over a time period and at a given light intensity.

When two telescopes (or a telescope and a human eye) are at a places with the same light intensity, then the quantity of light gathered over a period of time depends only on the area of the opening. For a mirror telescope it is the area of a mirror, for an eye it is the area of a pupil.

The area of a circle with the diameter `d` is `(pi d^2)/4.` Therefore, the ratio of areas is equal to the square of the ratio of diameters. In our case it is:

`(9/0.5)^2=18^2=` 324 (times). This is how many times bigger the area of the opening of the telescope is compared to the human eye, which directly relates to how much more light the telescope can gather in comparison to the eye.

That said, the largest mirrors in existing telescopes are about 10 m in diameter.