How this calculator works
Magnification exchanges field width and exit pupil size. A beam larger than the observer pupil loses some aperture at the eye; a smaller exit pupil dims extended objects approximately with its area. The reported stellar limit is an explicitly ideal aperture-only comparison against a user-supplied naked-eye limit, not an observing forecast. It omits sky background, transmission, seeing and observer response. Resolution uses a 550 nm Rayleigh criterion.
Formula
M = Fscope/Feye; exit pupil = D/M; f-ratio = Fscope/D; true field ≈ apparent field/M. Relative extended-source brightness = min[1,(exit/eye pupil)²]. Ideal magnitude gain = 5 log10(Deffective/eye pupil).
Worked example
A 100 mm aperture, 1000 mm focal length and 20 mm eyepiece give 50×, f/10 and a 2 mm exit pupil. For a 5 mm eye pupil, ideal extended-source brightness is 16% of naked-eye surface brightness.
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Frequently asked questions
Does more magnification always show more detail?
No. Aperture, seeing and optical quality set the detail available; excessive magnification spreads that light over a larger image.
Why is surface brightness capped at 100%?
An ideal passive telescope cannot increase the surface brightness of an extended object beyond the naked-eye view at a matched pupil.
How reliable is the stellar limiting magnitude?
It is a simple light-gathering comparison. Actual limits depend strongly on sky conditions, magnification, losses and the observer.