The aperture (or more formally
entrance pupil) of a telescope is larger than the human eye pupil, so collects more light, concentrating it at the
exit pupil where the observer's own pupil is (usually) placed. The result is increased
illuminance – stars are effectively brightened. At the same time, magnification darkens the background sky (i.e. reduces its
luminance). Therefore stars normally invisible to the naked eye become visible in the telescope. Further increasing the magnification makes the sky look even darker in the eyepiece, but there is a limit to how far this can be taken. One reason is that as magnification increases, the exit pupil gets smaller, resulting in a poorer image – an effect that can be seen by looking through a small pinhole in daylight. Another reason is that star images are not perfect points of light; atmospheric turbulence creates a blurring effect referred to as
seeing. A third reason is that if magnification can be pushed sufficiently high, the sky background will become effectively black, and cannot be darkened any further. This happens at a background surface brightness of approximately 25 mag arcsec−2, where only
'dark light' (neural noise) is perceived. Point sources such as a star will be easier to detect than a diffuse object such as a galaxy or comet. Various authors have stated the limiting magnitude of a telescope with entrance pupil D centimetres to be of the form m = 5 \log D + N with suggested values for the constant N ranging from 6.8 to 8.7. Crumey obtained a formula for N as a function of the sky surface brightness, telescope magnification, observer's eye pupil diameter and other parameters including the personal factor F discussed above. Choosing parameter values thought typical of normal dark-site observations (e.g. eye pupil 0.7cm and F = 2) he found N = 7.69. Crumey obtained his formula as an approximation to one he derived in photometric units from his general model of human
contrast threshold. As an illustration, he calculated limiting magnitude as a function of sky brightness for a 100mm telescope at magnifications ranging from ×25 to ×200 (with other parameters given typical real-world values). Crumey found that a maximum of 12.7 mag could be achieved if magnification was sufficiently high and the sky sufficiently dark, so that the background in the eyepiece was effectively black. That limit corresponds to N = 7.7 in the formula above. More generally, for situations where it is possible to raise a telescope's magnification high enough to make the sky background effectively black, the limiting magnitude is approximated by m = 5 \log D + 8 - 2.5 \log (p^2 F / T) where D and F are as stated above, p is the observer's pupil diameter in centimetres, and T is the telescope transmittance (e.g. 0.75 for a typical reflector). Telescopic limiting magnitudes were investigated empirically by
I.S. Bowen at
Mount Wilson Observatory in 1947, and Crumey was able to use Bowen's data as a test of the theoretical model. Bowen did not record parameters such as his eye pupil diameter, naked-eye magnitude limit, or the extent of light loss in his telescopes; but because he made observations at a range of magnifications using three telescopes (with apertures 0.33 inch, 6 inch and 60 inch), Crumey was able to construct a system of simultaneous equations from which the remaining parameters could be deduced. Because Crumey used astronomical-unit approximations, and plotted on log axes, the limit "curve" for each telescope consisted of three straight sections, corresponding to exit pupil larger than eye pupil, exit pupil smaller, and sky background effectively black. Bowen's anomalous limit at highest magnification with the 60-inch telescope was due to poor seeing. As well as vindicating the theoretical model, Crumey was able to show from this analysis that the sky brightness at the time of Bowen's observations was approximately 21.27 mag arcsec−2, highlighting the rapid growth of light pollution at Mount Wilson in the second half of the twentieth century. ==Large observatories==