Calculating Limiting Magnitude
Limiting magnitude can be calculated by using a telescope.
As a first approximation, the gain in magnitudes of a telescope is, where D1 is the diameter of the telescope's primary light gathering component, and D0 is the diameter of the eye's dark adapted pupil. Both quantities must be measured in the same units. D0 varies from person to person but is typically 6-7mm (~1/4").
A 10-inch (D1=254mm) telescope therefore would provide a gain of about 8 magnitudes beyond what could be observed without it. Thus, if one is at a site where the naked eye limiting magnitude (NELM) is 5, the telescope will allow one to see stars as faint as about magnitude 13.
In reality a telescope allows one to see much fainter stars because at higher powers the background is darkened and contrast increased. A typical 10-inch scope at high power (250x or more) will easily reach magnitude 15. See the Telescope Limiting Magnitude Calculator. Derived from this site the formula is
where
D = objective or main mirror diameter in mm
P = power or magnification
t = transmission factor, usually 0.85-0.9.
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