Formulas and examples for resolution gain according to Köhler and Leinhos with consideration of transmission
To a former contribution of me [1] I have calculated some examples. I attach them in PDF format as well as in Libre Office Calc and MS Excel format (without macros) as ZIP-file. In table sheets you can change my values for own calculations. The examples provide guidance for choosing binoculars in twilight by considering the effects of changes in magnification, aperture, transmission and eye pupil diameter.
According to listed documents in the Merlitz paper [4] "binocular efficiency E is commonly defined as
E = R / r
where R stands for the range at which a target is detected with the binocular, and r the corresponding range achievable with the unaided eye."
It should be noted that:
1. Neglecting eye pupil diameter and applying the simple formula E ~ D * sqrt (T) for residual light (night) results in impossibly high values for resolution gain E.
2. The empirical formulas are based on the detection of Landolt rings [2], thus on the basis of black and white contrasts - not on the basis of self-luminous objects like stars. For astronomers these formulas are of limited relevance: For example, one can already see from the formula for residual light (night) that the magnification is only relevant for the exit pupil and thus the maximum effective eye pupil diameter. Every astronomer knows how differently the magnification works for the same objective lens diameter. In simplification, I have read: Magnification beats aperture.
3. The older contribution of me I attach [1], thereby please point 1. consider! The attachments of this post already contain the correction.
4. Further assumptions, restrictions and limits of the formulas of Köhler and Leinhos please read yourself, briefly described by Dr. Merlitz, see [4].
5. The exit pupil diameter and thus the maximum effective diameter of the eye pupils of binoculars in 8x40 binoculars is 5 mm, of 7x40 ~ 5.7 mm at night in example calculations.
6. I have calculated an eye pupil diameter of 6 mm at night for binoculars with larger exit pupils, i.e. for older observers. It is noticeable that with a constant eye pupil of 6 mm and increasing unusable aperture, the resolution gain continues to increase. The formulas are empirical approximations with limits of applicability. I chose the examples deliberately and thoughtfully to see this also.
Whoever finds errors is welcome to correct them. I do not have the writings of Köhler and Leinhos, perhaps someone can at least find or copy relevant excerpts from [3].
Best whishes an have fun with own calculations and findings. Jessie
[1] former post of me with important links for understanding: https://www.birdforum.net/threads/w...pil-size-as-we-age.402814/page-4#post-4118500
[2] Landolt ring, Landolt C: https://en.wikipedia.org/wiki/Landolt_C
[3] H. Köhler, R. Leinhos, Untersuchungen zu den Gesetzen des Fernrohrsehens, Optica Acta: International Journal of Optics
[4] H. Merlitz: Performance of binoculars: Berek’s model of target detection
Edit: Typo in attachments corrected.
To a former contribution of me [1] I have calculated some examples. I attach them in PDF format as well as in Libre Office Calc and MS Excel format (without macros) as ZIP-file. In table sheets you can change my values for own calculations. The examples provide guidance for choosing binoculars in twilight by considering the effects of changes in magnification, aperture, transmission and eye pupil diameter.
According to listed documents in the Merlitz paper [4] "binocular efficiency E is commonly defined as
E = R / r
where R stands for the range at which a target is detected with the binocular, and r the corresponding range achievable with the unaided eye."
It should be noted that:
1. Neglecting eye pupil diameter and applying the simple formula E ~ D * sqrt (T) for residual light (night) results in impossibly high values for resolution gain E.
2. The empirical formulas are based on the detection of Landolt rings [2], thus on the basis of black and white contrasts - not on the basis of self-luminous objects like stars. For astronomers these formulas are of limited relevance: For example, one can already see from the formula for residual light (night) that the magnification is only relevant for the exit pupil and thus the maximum effective eye pupil diameter. Every astronomer knows how differently the magnification works for the same objective lens diameter. In simplification, I have read: Magnification beats aperture.
3. The older contribution of me I attach [1], thereby please point 1. consider! The attachments of this post already contain the correction.
4. Further assumptions, restrictions and limits of the formulas of Köhler and Leinhos please read yourself, briefly described by Dr. Merlitz, see [4].
5. The exit pupil diameter and thus the maximum effective diameter of the eye pupils of binoculars in 8x40 binoculars is 5 mm, of 7x40 ~ 5.7 mm at night in example calculations.
6. I have calculated an eye pupil diameter of 6 mm at night for binoculars with larger exit pupils, i.e. for older observers. It is noticeable that with a constant eye pupil of 6 mm and increasing unusable aperture, the resolution gain continues to increase. The formulas are empirical approximations with limits of applicability. I chose the examples deliberately and thoughtfully to see this also.
Whoever finds errors is welcome to correct them. I do not have the writings of Köhler and Leinhos, perhaps someone can at least find or copy relevant excerpts from [3].
Best whishes an have fun with own calculations and findings. Jessie
[1] former post of me with important links for understanding: https://www.birdforum.net/threads/w...pil-size-as-we-age.402814/page-4#post-4118500
[2] Landolt ring, Landolt C: https://en.wikipedia.org/wiki/Landolt_C
[3] H. Köhler, R. Leinhos, Untersuchungen zu den Gesetzen des Fernrohrsehens, Optica Acta: International Journal of Optics
[4] H. Merlitz: Performance of binoculars: Berek’s model of target detection
Edit: Typo in attachments corrected.
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