On this page, I present some simple formulas for telescopes and the calculation results for telescopes that I own, owned, or find interesting. In addition, I provide a few useful links. I do not describe the terms here, see page Quick & Dirty Astronomy Glossary for descriptions.
Note: For definitions in a small glossary, see page Quick & Dirty Astronomy Glossary.
As a telescope owner, you may have some requirements, but these can be satisfied by the different types of telescopes only to a certain degree:
The following calculations enable telescope users to determine some characteristics of their telescopes and eyepieces and thus, to better judge what these can do and what not. Regrettably, some "astronomy jargon" is needed here. Therefore, I try to explain some of the used terms in a small glossary, often using Wikipedia articles. Further definitions can be found on the Internet (I provide a few links...).
The term aperture refers to the diameter of the opening of a telescope. For mirror telescopes this is either the diameter of the primary mirror or a value that takes care of obstructions that limit the light receiving area, for refractors it is the lens diameter..




The focal ratio of a telescope is given by the ratio of the focal length of the telescope and its aperture:




The light gathering power of a telescope is expressed in multiples of the light gathering power of the human eye:

(The maximum aperture of the naked eye is about 7 mm)



The magnification of a telescope is calculated from the ratio of the focal length of the telescope and the focal length of the eyepiece:



The maximum practical visual power / useful magnification is more or less determined by the aperture of the telescope:

X amounts to:
Note: Stoyan (Deep Sky Reiseführer) speaks of the "beneficial" visual power, at which the airy disk is still not resolved and at which the magnitude limit of the telescope is reached. It is calculated as:

This corresponds more or less to a factor X of 1.5 (exact: 1.43) in the first formula. Since this leads to very similar data, I do not list the results obtained by this formula.
Note: For smallscale deepsky objects Stoyan (Deep Sky Guide) proposes to go far beyond the beneficial visual power up to the maximum visual power, which is twice as high as the beneficial visual power (and corresponds to a factor X of 3). Depending on the telescope type and the seeing (air turbulence), this is, however, not always possible. Smaller telescopes reach their maximum visual power easier because it is lower than that of large telescopes and thus, the seeing has less influence.
The minimum (practically) usable focal length of eyepieces is calculated from the maximum useful magnification and the focal length of the telescope (seems to be my own idea...):

The maximum usable focal length of eyepiece is determined by the exit pupil and the focal ratio of the telescope:

For an exit pupil of 6.5 mm, we get the formula:

For an exit pupil of 7 mm (often used in examples), we get the formula:

The minimum useful visual power (minimum useful magnification, normal magnification) is determined by the size of the exit pupil:

If the magnification is too low, parts of the light that leaves the eyepiece cannot be utilized by the human eye (the exit pupil is too large).
Telescope  Focal Length  Focal Ratio  Exit Pupil = 6.5 cm  Exit Pupil = 7 cm  
MUFLEP  MUM  MUFLEP  MUM  
Heritage 76  300  3.95  26 mm  11.68  27.65 mm  10.85 
Heritage 100P  400  4  26 mm  15.38  29 mm  14.29 
Heritage 114P  500  4.4  28.5 mm  17.54  30.7 mm  16.29 
Heritage P130  650  5  32.5 mm  20.00  35 mm  18.57 
Newton 6"  750  5  32.5 mm  23.00  35 mm  21.43 
Dobson 8"  1200  6  39 mm  30.77  42 mm  28.57 
Dobson 10  1270  5  32.5 mm  39.08  35 mm  36.29 
ETX 90/EC  1250  13.89  90.3 mm  13.84  97.23 mm  12.86 
Skymax102  1300  12.75  82.8 mm  15.70  89.2 mm  14.57 
Skymax127  1500  11.81  76.8 mm  19.53  82.7 mm  18.14 
The apparent field of view determines the angle that is shown by an eyepiece as a section of the sky. It depends on the type of the eyepiece an is usually given by the manufacturer of the eyepiece. See the glossary for more information.
Note: SkyWatcher lists 42° as a suitable value for most amateur eyepieces. Obviously, these are Kellnertype eyepieces (SkyWatcher delivers these together with its budget telescopes).
The true field of view determines the size of objects that can be observed in a telescope (Example: The moon corresponds to a field of view of about 0.5°)



The apparent field of view of an eyepiece is not always known or one does not trust the manufacturer's specifications. If it is possible to measure the field stop of an eyepiece or it is in the technical data of the eyepiece, then the true field of view can be determined using the following rule of thumb:

The maximum true field of view for a telescope port (1.25", 2", 3", ...) is obtained by entering the maximum field aperture for that port into the above formula. Since the receptacle has a certain thickness and the lenses are screwed with rings, the maximum field aperture is always smaller than the port dimension (which is, after all, also the outer dimension). For 1.25" eyepieces, I found values between 27 and 29 mm (29 mm seems to be the absolute maximum), for 2" eyepieces a value of 46 mm.
The 1.25" mount is "maxed out" with a 25 mm eyepiece with 70° viewing angle or a 32 mm Plössl eyepiece with 52° viewing angle with respect to the maximum field of view, the 2" mount with a 40 mm eyepiece with 70° angle of view, or a 56 mm Plössl eyepiece with 52° angle of view. The 70° eyepieces show the objects larger, the 52° eyepieces show a brighter (larger exit pupil), smaller image.
The true field of view F of an eyepiece may not always be known and can be determined using a stopwatch (thanks to Jörg Meyer!):
Locate a star of known declination d close to the celestial equator and place it at the eastern edge of the field of view in the eyepiece (motor off!). Measure the time t that the star needs to move through the field of view and enter it into the following equation:

The exit pupil determines, how bright the image of a certain object, for example, the moon, will appear in the eye piece. For the same exit pupil it will appear with the same brightness, irrespective of the telescope, its aperture, and its magnification.
If the exit pupil of an eyepiece is too small, objects appear too dim (below 1 mm for deep sky objects, below 0.7 mm for planets, below 0.5 mm for the moon and bright double stars), if it is larger than that of the human eye (>7 mm), only part of the light hits the human eye. For galaxies, choose an exit pupil of 23 mm, not at all the maximum magnification (from the Internet).
The exit pupil can be determined in two ways, both of which lead to the same formula:

Thus, depending on your point of view, the exit pupil of an eyepiece can be calculated either from the magnification or the focal ratio of a telescope or binocular.


The diameter of the Airy disk, which is the effective aperture diameter of an optical system, determines its resolving power. Two points can be separated reliably according to the Rayleigh criterion if the maxima of their images are separated by at least the radius of the Airy disk. The diameter also indicates the minimum size with which stars are imaged in the telescope.
The diameter of the Airy disk in µm can be calculated as follows:

Example (Vaonis Vespera): A focal ratio of F/4 and a wavelength of 0.55 µm (550 nm) lead to a diameter of 5.37 µm >> ideal sensor pixel size is 2.68 µm.
The diameter of the airy disc in angular units (seconds) is:

Example (Vaonis Vespera): An aperture of 50 mm and a wavelength of 0.00055 mm (550 nm) lead to a diameter of 5.54" >> is above a FWHM of 5".
*) The numerical value corresponds to the Rayleigh criterion.
Resolution (or resolving power) is defined as the ability to separate two closely spaced objects (e.g., binary stars in astronomy). There are two empirically found criteria for this ability:

The Rayleigh criterion is a heuristic condition for the distance between two light sources in order to recognize them as separate. This minimum distance is equal to the distance of the first minimum from the center of the diffraction pattern. In practice, rules of thumb are used for both criteria. That is why I leave out the complex mathematics here. The two rules of thumb are:

The resolution depends solely on the aperture of the telescope. Obviously, the Dawes criterion is considered "more practical" (see the article by Stefan Gotthold), and the manufacturers also provide this value (probably also because it looks better).
Example Table: Resolution Values Given by the Manufacturers and the "Rule of Thumb" Formulae
Telescope  Focal Length *  Aperture  Rayleigh  Dawes  Manufacturer 
Refractor  400...700  60  2.3"  1.93"  
SW Heritage 76  300  76  1.82"  1.53"  1.51" 
Refractor  500...900  80  1.725"  1.45"  
Meade ETX 90EC  1250  90  1.53"  1.29"  1.3" 
SW Heritage 100P  400  100  1.38"  1.16"  1.15" 
SW Skymax102  1300  102  1.35"  1.14"  1.15" 
SW Skymax127  1500  127  1.08"  0.91"  0.91" 
SW Explorer 150PDS  750  150  0.92"  0.77"  0.77" 
GSO GSD 680  1200  200  0.69"  0.58"  0.58" 
Meade Lightbridge 10"  1270  254  0.54"  0.46"  0.45" 
*) Not needed here
The above resolution formulae apply to fine structures such as double stars. In his book Moonhopper, Lambert Spix provides the following empirically determined formulae for the moon, which correspond to much higher resolutions, i.e. enable capturing much smaller structures:

According to Spix, these "theoretic" values have to be doubled in most nights due to atmospheric turbulence. Structures below 0.16" (300 m) are practically invisible even with large telescopes (larger than 10").
Example Table: Resolution Values and Doubled Resolution Values Calculated According to the "Empirical Formulae"
Theoretical 
Doubled 

Telescope  Focal Length *  Aperture  Crater  Groove  Crater  Groove 
Refractor  400...700  60  0.65"  0.38"  1.30"  0.76" 
SW Heritage 76  300  76  0.51"  0.30"  1.03"  0.61" 
Refractor  500...900  80  0.49"  0.29"  0.98"  0.58" 
Meade ETX 90EC  1250  90  0.43"  0.26"  0.87"  0.51" 
SW Heritage 100P  400  100  0.39"  0.23"  0.78"  0.46" 
SW Skymax102  1300  102  0.38"  0.23"  0.76"  0.45" 
SW Skymax127  1500  127  0.31"  0.18"  0.61"  0.36" 
SW Explorer 150PDS  750  150  0.26"  0.15"  0.52"  0.31" 
GSO GSD 680  1200  200  0.20"  0.12"  0.39"  0.23" 
Meade Lightbridge 10"  1270  254  0.15"  0.09"  0.31"  0.18" 
*) Not needed here; 0.52" roughly corresponds to 1 km on the moon.
The rules of thumb are based on a light wavelength of 550 nm. Derivations and exact formulas can be found here.






The field of view (in arcminutes, or ') is simply:

D is the sensor dimension (width, height, diameter) in millimeters; the effective focal length is the telescope focal length including the effect of focal reducers or extenders.
Examples
03.02.2021 