Let d1 be the diameter of the pupil and d2 be the diameter of the telescope. The collecting area is proportional to the square of the diameter. The ratio of areas is d^2
This is the ratio of luminosities that we can see with the naked eye and with the telescope. We can express this ratio as a magnitude difference m - m2 = -2.5 log10(4.0 X 104)
This means that the faintest objects we can see with the telescope are 11.5 mag fainter than the faintest objects we can see with the naked eye. If the naked eye can see down to 6 mag, the telescope-aided eye can see down to 17.5 mag. This illustrates the great improvement in light-gathering power with the telescope.
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