Calculations with Optical Instruments


Telescope, Binocular

Basically, a binocular consists of two parallel, identical telescopes, one for each eye. Here you can calculate the exit pupil, focal length, magnification and limiting magnitude.

Focal length objective:
Focal length eyepiece:

M = fob / fok

The two focal length must have the same units, commonly those are millimeters.

Exit pupil (mm):
Magnification M:
Objective diameter D (mm):

EP = D / M

At a binocular, often magnification and objective diameter are indicated, e.g. as 8x42. The larger the exit pupil, the more light-gathering it is and the easier it is to look through. 7 mm is the approximate maximum size of the human pupil, higher values don't give any more gain in light. Small objective diameter and large magnification create a small exit pupil, which can make it hard for beginners to see anything at all. Values below 1 mm are difficult.

Twilight Factor for Binoculars and Spotting Scopes:

Objective diameter D (mm):
Magnification M:
Twilight factor T:

T = √ M * D

The twilight factor has common values between 5 and 25. The higher the value, the better is the resolution in twilight.

Practical maximum magnification:
Objective diameter D (mm):

Mmax = D * 2

At higher magnifications, the image becomes blurred and doesn't reveal further details.


Limiting magnitude m (mag):
Objective diameter D: (mm)
Minimum magnification Mmin:

m = 7.5 + 5 * log10 ( D/10 )
Mmin = D / 7

The limiting magnitude is an estimation of the brightness, up to which objects are visible with a certain objective diameter. Not considered here are important factors like seeing and equipment quality. There are different formulas for the estimation of the limiting magnitude, this calculator uses a rather simple one.
The minimum magnification results from the exit pupil, weaker magnifications are possible, but don't reach the limiting magnitude.

Ratio of apparent magnitudes:
Magnitudes a, b (mag):
Ratio R

R = 5 100  b-a

The brightness of celestial bodies is measured in mag. Very bright starts have 0 mag or less, the faintest visible to the naked eye have 6 mag. The scale is logarithmic, 5 mag difference means 100 times more or less brighter.
↔ swaps the values of a and b.

Indication of size:
mm = cm = Inch, ″

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