Estimating the limiting magnitude of binoculars

This is an attempt to create a number that allows the comparison of binoculars of different apertures and magnifications.

The ability to see a faint star, or other source of light such as a faint galaxy or star cluster, depends primarily on the light-collecting aperture D.

Of premium importance is the light-collecting aperture (of diameter D). Assuming no losses, the light gathered compared to the eye pupil of diameter E, is (D/E)². This would in itself increase the limiting magnitude by 5 log(D/E). (NB log is the decimal logarithm!)

The other important factor is the magnification m. With higher magnifications, the light from the sky background is "spread out" more, and appears darker. The exit pupil X=D/m The brightness of the background, as seen in the binoculars, is reduced by X/E², or in magnitudes/square arcsecond -5 log(X/E).

The diagram shows the data for limiting magnitude from Blackwell (see here). You can see that in the range from 18.8 to 23.8 magnitudes/square arcsecond, where averted vision is normally preferred, the gain in limiting magnitude is 0.4 magnitudes per magnitude of background reduction, and this is linear within this range. It covers dark (21 magnitudes/square arcsecond) or moderately light polluted skies, as seen in binoculars of commonly used magnifications. With the equation in the previous paragraph, this gives the gain in limiting magnitude as 2 log(E/X), or - 2 log(D/E)+ 2 log(m).

These factors together give that the increase of limiting magnitude is 5 log(D/E) - 2 log(D/E) + 2 log(m) or:

3 log(D/E) + 2 log(m)

This is valid only as long as the exit pupil X is less than or equal to the eye pupil. If it is larger, the background brightness will appear without reduction, but the effective aperture is now mE, less than the full aperture mX=D. The total gain in magnitudes is now 5 log(m), independent of D!

*=exit pupil is larger than eye pupil

It should also be noted that the limiting magnitude gain cannot be fully utilized with the binoculars hand-held due to the unavoidable shaking of the image.

Discussion

The number given here is a rough guide to the gain in limiting magnitude for stars or small diffuse objects, seen against a dark sky with averted vision. It may be of some usefulness in the selection of binoculars, if you consider its limitations. In the example above, you see that the higher "power" or magnification of the 20x60 outweighs the larger aperture of the 11x70, thanks to the darker background it shows. However, this is only true if the better dark adaptation can be taken advantage of. The optical losses within the binoculars have not been considered, but (with background darkening allowed for) 15% light losses will lower the limiting magnitude gain by about 0.1 magnitudes.

There have been other attempts - Roy L Bishop of Royal Astronomical Society has suggested a "Visibility Factor" as

You can find an article about this by Sky & Telescope: Power and Aperture in Binoculars

The "Visibility Factor" is beautifully simple - expressed in magnitudes gain would be: 2.5 log(D/E) + 2.5 log(m). Compared to my formula, it slightly overestimates the effect of magnification and underestimates the importance of aperture.