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The amount of light collected for a period of time is proportional to the area of the collecting aperture. We use our eyes -pupils have diameters of approximately 5 mm-, binoculars and telescopes. How much more light can we collect with a 50mm pair of binoculars compared with our eyes? And with an 8-inch telescope? If we could have access to a 10-meter class professional telescope, how large would the gain be?

50^2/5^2= 100 times more light

8in = 203.2 mm

203.2^2/5^2= 1652 times more light

10000^2/5^2= 4000000 times more light

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  • $\begingroup$ This is basically right in terms of total light from one small area of the sky that fits within the telescope's field of view. But remember when things (extended objects like nearby galaxies and nebulae) are magnified, the brightness is spread out to a larger area on the retina, so brightness per unit area has your aperture square term, but then divided by magnification squared. $\endgroup$
    – uhoh
    Feb 1, 2018 at 14:49

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You have the idea right! The amount of light collected is indeed proportional to the area of the collecting surface.

Now, if you want to take it a bit further, try to think about the following:

  • Binoculars have one lens for each eye, does this mean you get twice the amount of light in the end?

  • Reflecting telescopes almost always have a secondary mirror which results in part of the light being masked and not collected by the primary mirror. Now imagine you have an 8-inch telescope with a 2-inch secondary mirror, can you calculate again the light collection ratio compared to the naked eye?

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  • $\begingroup$ The amount of light that can be collected is proportional to the entrance aperture area. However, unless you account for lens systems aberrations, vignetting, etc., the amount of light that is presented to your Official Mark-I Eyeball will be less than that. $\endgroup$
    – Carl Witthoft
    Feb 1, 2018 at 14:02

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