Actually, Jan Kyster is incorrect. The brightness of the image is only partially determined by the diameter of the objective lens. It is possible to put enough magnification on a larger objective lens to make the image dimmer than a lower power view through a smaller objective lens, so objective diameter alone is not responsible for image brightness. The other factors are light transmission efficiency through the system (clarity of glass, number of optical elements, dirt and foreign matter in the system) and the magnification.
The exit pupil is the diameter of the total field of view that your eye is looking at though the eyepiece. The image will appear brighter until it reaches the diameter of a fully dark-adapted eye pupil, about 7mm for a young adult and decreasing to about 5mm for an older adult. Any image of over 7mm exit pupil will not look any brighter to your eye than a 7mm or whatever is the diameter of your pupil at the time.
The way you calculate the exit pupil is to take the diameter of the objective lens (the first lens in the system at the top of the periscope) and divide that by the magnification. For instance, have you ever wondered why 50x7 binoculars are such a popular size? 50mm/7x=just a hair over a 7mm exit pupil. Personally I own a 50x10 pair of binoculars because for me a 5mm exit pupil is just as bright and I can use the extra magnification.
So exit pupil equals diameter of objective lens/magnification. Brightness at the same exit pupil will be greater for a larger diameter objective lens because there is more light condensed into that little diameter.
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