Basic Formulas


Focal Length of Eyepieces


The magnification of the telescope is given by the focal length f of the eyepiece. Magnification V is simply (with F as focal length of the telescope):




therefore e.g. for f=4mm, F=1000mm a magnification of 250.


Figure 1: Graphic for some important definitions


Which magnifications are useful?


The smallest useful magnification is given, if the exit pupil is as big as the eye’s pupil, if adapted to darkness. This is about 7 to 8mm, depending on the observers age. If a smaller magnification is chosen, the exit pupil will increase and the amount of light outside the eye’s pupil will be lost. Therefore it is possible to use these smaller magnifications, but the image will be identical to that of a smaller telescope – according to the light lost.

The exit pupil A is calculated by the aperture ration of the telescope (lens resp. mirror diameter D divided by the focal length F) and the focal length of the eyepiece. The formula is given by:




For the smallest magnification one will get:




The lowest useful magnification is therefore dependent only on the lens resp. mirror diameter! The according maximum focal length of the eyepiece is:




That is for a system with an aperture ratio of F/D=6 an eyepiece with 48mm focal length.


For the useful maximum magnification I would like to make a short excursion.


The maximum theoretical resolution of a telescope is limited by diffraction at the edge of the objective (lens or primary mirror). Without any derivation the common value in use for this resolution is given by




That means, that an instrument with an objective diameter of 140mm is able to resolve two lines if their distance is bigger than 1“ (1 arc second). 280mm will lead to a resolving power of 0.5”, 70mm to 2”.

At the end of the telescope another optical element is placed – the human eye. The human eye, caused by the distance of light sensitive cells in the retina, is able to resolve an angular distance of about 1 arc minute. From this it follows that the maximum magnification VRes, at which in principle the maximum resolving power of the telescope is used, is given by




That is for an objective of 200mm diameter just a magnification of 86! Any magnification which is going ahead this limit will lead to a “diffuse” image! But there are a couple of reasons to use much higher magnifications:


  1. At low illumination levels, and this is the case in most of astronomical observations, the eye’s resolution is going down to 2 or 3 arc minutes. According to this the magnification can be increased unless the impression of an diffuse image occurs.
  2. It is very exhausting to view details at the edge of the eye’s resolution power. Magnification beyond the theoretical limit leads to more relaxed viewing.
  3. Planets are very bright at low magnifications. Using higher magnification reduces the area brightness and contrasts become more obvious.


For this the practical limit for the maximum magnification is not given by D [mm]* 0,43, but




For refracting telescopes you will find up to D [mm] * 2,5.


The according minimum focal length of the eyepiece is given by




This is e.g. for a system with focal length F=1000mm and objective diameter of 100mm an eyepiece focal length of 5mm.


Field of View


You have to differentiate between “apparent field of view (AV)“ and “real field of view (RV)“.

The apparent field of view is characteristic for an eyepiece and does not depend on the telescope used. It is the viewing angle when looking into the eyepiece. Depending on the eyepiece design apparent field of views differ from about 30° (Huygens) to about 82° (Nagler type). To give a vivid example, if looking into a Huygens eyepiece you will have the impression of looking into a tunnel, whereas the view into a wide angle eyepiece will you make feel as if sitting in the first row of a cinema.

The real field of view is the area on the sky, given in degrees, which you will see if looking through the telescope. It is dependent on the apparent field of view, but also on the magnification as defined above. The formula is




From this it is clear that for the same magnification you will see with a wide angle eyepiece (70° AV) an area with a diameter 2.3 time bigger than with an orthoscopic eyepiece (30° AV), if both eyepieces have the same focal length. I.e. the area visible with the wide angle is increased by a factor of 5.4! For the observation of large objects wide angle eyepieces are therefore recommended, the impression when viewing through the telescope is really impressive.

Real field of views are varying from about 3° (31mm Nagler type with telescope of focal length 800mm) down to a few arc minutes.


Comment for advanced readers:


Formula (9) is commonly used (so by us), but however it is not correct at all. The formula neglects any distortion effects, and distortion lead to the fact that magnification is not a constant over the complete field of view. In case of pincushion distortion the magnification increases towards the edge of the field of view. As we use formula (9) for our measurements of the apparent field of view, our results for the AV are somewhat less (about 0° to 3°, depending on the distortion level) than the “real” AV according to its definition. The advantage is, that if you use our AV measurement results and calculate your RV by formula (9), you will get a correct result. The disadvantage is, that there are inherently differences between the manufacturers AFoV and ours.


The maximum apparent field of view is limited by physics for a given diameter of the field lens. It can be calculated from the focal length of the eyepiece and the diameter of the field lens resp. field stop:




For attention: If yoou use your pocket calculator in setting „rad“, or if you use Excel for the calculation, you have to multiply with a factor 180/π (arctan in radian!).


The field lens diameter is clearly limited by the eyepieces insertion diameter. Eyepieces are available with diameters of 24.5mm, 1 ¼” and 2”. Depending on the focal length of the eyepiece the maximum possible apparent field of view is given in the following table. The values are theoretical, as the sleeve has an inner diameter which is less than the insertion diameter. Therefore the reachable field of view is some degrees less than given here.


Insertion Diameter

Focal Length of Eyepiece (mm)

Maximum Apparent Field of View (°)
















31,8mm (1 ¼“)















50,8mm (2“)
















Table 1: Maximum Apparent Field of View for Eyepieces with different Insertion Diameters and Focal Lengths.

Bigger apparent field of views are not possible without distortion! From this table it becomes clear where to use the big 2“-eyepieces: at long focal lengths, if the required apparent field of view is not reachable with smaller eyepieces. However, at e.g. 15mm focal length one can achieve exactly the same apparent field of view either with a 1 ¼“ eyepiece than with an 2“ eyepiece.