As everyone will know, there is still no simple formula that describes the ability of binoculars to lighten civil twilight and part of nautical twilight. And therefore, that can indicate the luminous power of the instrument, or at least the nominal one.
The twilight factor indicates the ability to read the detail in very low light conditions, which correspond to the brightness of the astronomical twilight (the darkest one), or technically from about 0.3 to 0.003 cd/m^2 or even, from the moonlight to about the night.
In this case, therefore, the Zeiss engineers used an inconsistent term to name this factor. A term that has often confused users, since in practice the twilight factor has nothing to do with twilight or even with twilight binoculars.
In fact, for example, all 10x35 and 7x50 binoculars will have the same twilight factor 18.7 = √(10x35) = √350 = √(7x50).
But will a 10x35 binocular really have the same ability to "lighten the twilight" that has a 7x50?
Obviously not! Thus, the twilight factor will be completely unsuitable for describing the light power of binoculars.
So I tried to reason on the question, and gathering all the knowledge and experiences, I first hypothesized and then found that in fact both the enlargement and the pupillary surface of the binoculars, both contribute in proportional way to increase the ability to lighten the observations in the twilight (as described above) and in very shady situations of equal environmental brightness.
The magnification multiplies the amount of light in the eye, in proportion to the apparent superficial increase of the observed object, in the same way as the larger surface of the exit pupil, increases the luminous flux directed towards the eye. In this way, both values concur together to increase the visibility of the objects observed, and the most suitable factor to represent this capacity, can be expressed with a formula that equals the magnification and the pupillary surface, like their product, providing so also a nominal value, numeric and indicative of the "light power" of the binoculars.
For example: 8x56 = 8x(π((56/8)/2)^2) = 8x((56/8)^2x0.7854) = 8x(38.484...) = approximately 308 (where 0.7854 is an abbreviation of Pi).
Thus, 7x50 = 7x(π((50/7)/2)^2) = 7x((50/7)^2x0.7854) = about 280 and 10x35 = 10x((35/10)^2x0.7854) = about 96. This indicates consistently the true difference of the luminous power in the twilight, between a 7x50 and a 10x35 (considering them of equal optical quality and total transmittance).
Logically, the resulting value is considered as nominal because it uses the nominal values of the format and does not count the transmittance or the various optical qualities involved. But basically this formula can also be simplified a lot and adapted to the simplest calculators, excluding the Greek Pi from the calculations and replacing the exit pupil with the value of the aperture, directly using the format data.
Also in this mode, the calculation will always find proportionate and consistent results between the various binoculars exposed to the formula:
For example: 8x56 = 56^2/8 = approximately 390, but also 7x50 = 50^2/7 = approximately 360 and 10x35 = 35^2/10 = approximately 120. Where, 360/120 = approximately 280/96.
Over time I have been able to do various experiments in various circumstances of ambient light and so I could see that the results of this formula are actually consistent with reality and can be very useful for cataloging the differences in brightness between the various formats.
For example, each 10x28 binocular has a nominal light output of 78 pln. Every 10x36, of 130 pln and every 10x70, of 490 pln.
Similarly, each 8x25 binocular has a nominal light output of 78 pln. Every 8x32, of 128 pln and every 8x56, of 390 pln.
Thus, up to 80 pln, fall the less bright or low power binoculars. Between 120 and 180 pln, those of medium power and between 340 and 500 pln, the brighter ones, often considered even the true crepuscular binoculars.
The twilight factor indicates the ability to read the detail in very low light conditions, which correspond to the brightness of the astronomical twilight (the darkest one), or technically from about 0.3 to 0.003 cd/m^2 or even, from the moonlight to about the night.
In this case, therefore, the Zeiss engineers used an inconsistent term to name this factor. A term that has often confused users, since in practice the twilight factor has nothing to do with twilight or even with twilight binoculars.
In fact, for example, all 10x35 and 7x50 binoculars will have the same twilight factor 18.7 = √(10x35) = √350 = √(7x50).
But will a 10x35 binocular really have the same ability to "lighten the twilight" that has a 7x50?
Obviously not! Thus, the twilight factor will be completely unsuitable for describing the light power of binoculars.
So I tried to reason on the question, and gathering all the knowledge and experiences, I first hypothesized and then found that in fact both the enlargement and the pupillary surface of the binoculars, both contribute in proportional way to increase the ability to lighten the observations in the twilight (as described above) and in very shady situations of equal environmental brightness.
The magnification multiplies the amount of light in the eye, in proportion to the apparent superficial increase of the observed object, in the same way as the larger surface of the exit pupil, increases the luminous flux directed towards the eye. In this way, both values concur together to increase the visibility of the objects observed, and the most suitable factor to represent this capacity, can be expressed with a formula that equals the magnification and the pupillary surface, like their product, providing so also a nominal value, numeric and indicative of the "light power" of the binoculars.
For example: 8x56 = 8x(π((56/8)/2)^2) = 8x((56/8)^2x0.7854) = 8x(38.484...) = approximately 308 (where 0.7854 is an abbreviation of Pi).
Thus, 7x50 = 7x(π((50/7)/2)^2) = 7x((50/7)^2x0.7854) = about 280 and 10x35 = 10x((35/10)^2x0.7854) = about 96. This indicates consistently the true difference of the luminous power in the twilight, between a 7x50 and a 10x35 (considering them of equal optical quality and total transmittance).
Logically, the resulting value is considered as nominal because it uses the nominal values of the format and does not count the transmittance or the various optical qualities involved. But basically this formula can also be simplified a lot and adapted to the simplest calculators, excluding the Greek Pi from the calculations and replacing the exit pupil with the value of the aperture, directly using the format data.
Also in this mode, the calculation will always find proportionate and consistent results between the various binoculars exposed to the formula:
Nominal luminous power (pln) = Aperture^2/Magnification
For example: 8x56 = 56^2/8 = approximately 390, but also 7x50 = 50^2/7 = approximately 360 and 10x35 = 35^2/10 = approximately 120. Where, 360/120 = approximately 280/96.
Over time I have been able to do various experiments in various circumstances of ambient light and so I could see that the results of this formula are actually consistent with reality and can be very useful for cataloging the differences in brightness between the various formats.
For example, each 10x28 binocular has a nominal light output of 78 pln. Every 10x36, of 130 pln and every 10x70, of 490 pln.
Similarly, each 8x25 binocular has a nominal light output of 78 pln. Every 8x32, of 128 pln and every 8x56, of 390 pln.
Thus, up to 80 pln, fall the less bright or low power binoculars. Between 120 and 180 pln, those of medium power and between 340 and 500 pln, the brighter ones, often considered even the true crepuscular binoculars.