Telescope Calculations

Introductory Notes | Calculations | References

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.

Overview of Formulae

Note: For definitions in a small glossary, see page Quick & Dirty Astronomy Glossary.

 

Introductory Notes

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...).

 

Calculations

Aperture

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..

Examples

  • Heritage 76: 76 mm
  • Heritage 100P: 100 mm
  • Heritage 114P: 114 mm
  • Heritage P130: 130 mm
    
  • Newton 6": 150 mm
  • Dobson 8": 200 mm
  • Dobson 10": 254 mm
    
  • ETX 90/EC: 90 mm
  • Skymax-102: 102 mm
  • Skymax-127: 127 mm
    
  • C5: 125 mm
  • C8: 2032 mm
  • PS 72/432: 72 mm
  • TLAPO 1027: 102 mm

Focal Ratio

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

  • Focal Ratio (f) = (Focal length of the telescope) / (Aperture of the telescope)

Examples

  • Heritage 76: 300 mm / 76 mm = 3.95 (about 4)
  • Heritage 100P: 400 mm / 100 mm = 4
  • Heritage 114P: 500 mm / 114 mm = 4.4
  • Heritage P130: 650 mm / 130mm = 5
  • Newton 6": 750 mm / 150 mm = 5
    
  • Dobson 8": 1200 mm / 200 mm = 6
  • Dobson 10": 1270 mm / 254mm = 5
  • ETX 90/EC: 1250 mm / 90mm = 13.89
  • Skymax-102: 1300 mm / 102 mm = 12.75
  • Skymax-127: 1500 mm / 127 mm = 11.81
    
  • C5: 1250 / 125 mm = 10
  • C8: 2032 mm / 203 mm = 10
  • PS 72/432: 432 mm / 72 mm = 6
  • TLAPO 1027: 714 mm / 102 mm = 7

Light Gathering Power

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

  • Light gathering power = (aperture in mm)² / 49

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

Examples

  • Heritage 76: 76 mm² / 49 = 118
  • Heritage 100P: 100 mm² / 49 = 204
  • Heritage 114P: 114 mm² / 49 = 265
  • Heritage P130: 130 mm² / 49 = 345
  • Newton 6": 150 mm² / 49 = 459
    
  • Dobson 8": 200 mm² / 49 = 816
  • Dobson 10": 254 mm² / 49 = 1317
  • ETX 90/EC: 90 mm² / 49 = 165
  • Skymax-102: 102 mm² / 49 = 212
  • Skymax-102: 127 mm² / 49 = 329
    
  • C5: 125 mm² / 49 = 319
  • C8: 203 mm² / 49 = 841
  • PS 72/432: 72 mm² / 49 = 106
  • TLAPO 1027: 102 mm² / 49 = 212

Magnification (Visual Power)

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

  • Magnification = (Focal length of the telescope) / (Focal length of the eyepiece)

Examples (10mm Eyepiece)

  • Heritage 76: 300 mm / 10 mm = 30 x
  • Heritage 100P: 400 mm / 10 mm = 40 x
  • Heritage 114P: 500 mm / 10 mm = 50 x
  • Heritage P130: 650 mm/10 mm = 65 x
  • Newton 6": 750 mm / 10 mm = 75 x
    
  • Dobson 8": 1200 mm / 10 mm = 120 x
  • Dobson 10": 1270 mm / 10 mm = 127 x
  • ETX 90/EC: 1250 mm / 10 mm = 125 x
  • Skymax-102: 1300 mm / 10 = 130 x
  • Skymax-127: 1500 mm / 10 = 150 x

Maximum Practical Visual Power (Maximum Useful Magnification)/Minimum Usable Focal Length of Eyepieces

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

  • Maximum Practical Visual Power = (Aperture in mm) * X = Aperture * X

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:

  • Beneficial Visual Power = Aperture / 0.7 = (Primary Mirror Diameter) / 0.7

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 small-scale deep-sky 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...):

  • Minimum usable focal length of eyepiece = (Focal length of the telescope) / (Maximum usable magnification)

Examples


Maximum Usable Focal Length of Eyepieces/Minimum Useful Visual Power (Minimum Useful Magnification, Normal Magnification)

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

  • Maximum Usable Focal Length of Eyepiece = (Focal Ratio) * (Exit Pupil)

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

  • Focal Length of Eyepiece = (Focal Ratio) * 6.5 mm

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

  • Focal Length of Eyepiece = (Focal Ratio) * 7 mm

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

  • Minimum Useful Visual Power (Magnification) = (Focal Length of Telescope) / (Maximum Usable Focal Length of Eyepiece) = (Focal Length of Telescope) / ((Focal Ratio) * (Exit Pupil)) = (Aperture of telescope) / (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).

Examples (Maximum Usable Focal Length of Eyepieces/Minimum Usable Visual Power (Minimum Usable Magnification))

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
Skymax-102 1300 12.75 82.8 mm 15.70 89.2 mm 14.57
Skymax-127 1500 11.81 76.8 mm 19.53 82.7 mm 18.14

Field of View

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: Sky-Watcher lists 42° as a suitable value for most amateur eyepieces. Obviously, these are Kellner-type eyepieces (Sky-Watcher 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°)

  • True field of view = (Apparent field of view) / Magnification = (Apparent field of view) * (Focal length of the eyepiece) / (Focal length of the telescope)

Examples (10 mm Eyepiece, 42°)

  • Heritage 76: 42° / 30 = (42° * 10 mm) / 300 mm = 1.4°
  • Heritage 100P: 42° / 40 = (42° * 10 mm) / 400 mm = 1.05°
  • Heritage 114P: 42° / 50 = (42° * 10 mm) / 500 mm = 0.84°
  • Heritage P130: 42° / 65 = (42° * 10 mm) / 650 mm = 0.65°
  • Newton 6 ": 42° / 120 = (42° * 10 mm) / 750 mm = 0.56°
    
  • Dobson 8": 42° / 120 = (42° * 10 mm) / 1200 mm = 0.35°
  • Dobson 10": 42° / 127 = (42° * 10 mm) / 1270 mm = 0.33°
  • ETX 90/EC: 42° / 125 = (42° * 10 mm) / 1250 mm = 0.34°
  • Skymax-102: 42° / 130 = (42° * 10 mm) / 1300 mm = 0.32°
  • Skymax-127: 42° / 150 = (42° * 10 mm) / 1500 mm = 0.28°

Field of View Calculated from the Field Stop

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:

  • 57.3 *Field stop [mm] / Telescope focal length [mm]

Which Eyepieces Have the Maximum True Field of View?

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.


Measuring the Field of View

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:

  • G = (t * 15 * cos d) / 60 (arc minutes)

Exit Pupil

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 2-3 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:

  • Exit pupil = (Aperture of the telescope) / Magnification
  • Exit pupil = (Focal length of the eyepiece) / (Focal ratio)
  • =>
  • Exit pupil = (Aperture of the telescope) * (Focal length of the eyepiece) / (Focal length of the telescope)

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.

Examples (10 mm Eyepiece)

  • Heritage 76: 76 mm / 30 = (76 mm * 10 mm) / 300 mm = 2.53 mm
  • Heritage 100P: 100 mm / 40 = (100 mm * 10 mm) / 400 mm = 2.5 mm
  • Heritage 114P: 114 mm / 50 = (114 mm * 10 mm) / 500 mm = 2.28 mm
  • Heritage P130: 130 mm / 65 = (130 mm * 10 mm) / 650 mm = 2 mm
  • Newton 6": 150 mm / 750 = (150 mm * 10 mm) / 750 mm = 2 mm
    
  • Dobson 8": 200 mm / 120 = (200 mm * 10 mm) / 1200 mm = 1.67 mm
  • Dobson 10": 254 mm / 127 = (254 mm * 10 mm) / 1270 mm = 2 mm
  • ETX 90/EC: 90 mm / 125 = (90 mm * 10 mm) / 1250 mm = 0.72 mm
  • Skymax-102: 102 mm / 130 = (102 mm * 10 mm) / 1300 mm = 0.78 mm

Airy Disk

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:

  • D [µm] = 2.43932 * λ [µm] * (Focal ratio of the telescope) = 2.44 * 0.55 * (Focal ratio of the telescope) (λ = wavelength in µm; rule of thumb*)
  • D [µm] = 1,3441626 * (Focal ratio of the telescope) (wavelength = 550 nm; simple formula) = 1.344 * (Focal ratio of the telescope) (even simpler rule of thumb)

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:

  • D ["] = 360 * 3600 / pi * arcsin (1.21966 * λ [mm] / (Aperture of the telescope [mm])) (λ = wavelength in mm, e.g. 550 nm = 0,55 µm = 0,00055 mm)
  • D ["] = 360 * 3600 / pi * 1.21966 * λ [mm] / (Aperture of the telescope [mm]) (λ = wavelength in mm, e.g. 550 nm = 0,55 µm = 0,00055 mm; linear approximation)
  • D ["] = 276.73 / (Aperture of the telescope [mm]) (wavelength = 550 nm; rule of thumb) = 277 / (Aperture of the telescope [mm]) (wavelength = 550 nm; even simpler rule of thumb*)

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

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:

  • Rayleigh criterion: diffraction discs do not touch each other
  • Dawes criterion: diffraction discs produce an 8-shaped image

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:

  • Rule of thumb (Rayleigh criterion): Resolution of the telescope (") α = 138 / aperture (objective/mirror diameter in mm)
  • Rule of thumb (Dawes criterion): Resolution of the telescope (") α = 116 / aperture (objective/mirror diameter in mm)

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"  
S-W 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"
S-W Heritage 100P 400 100 1.38" 1.16" 1.15"
S-W Skymax-102 1300 102 1.35" 1.14" 1.15"
S-W Skymax-127 1500 127 1.08" 0.91" 0.91"
S-W 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

Resolution for the Moon

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:

  • Dark dots against a bright background (craters): resolution of the telescope (") α = 39 / aperture (objective/mirror diameter in mm)
  • Black lines against a light background (grooves): resolution of the telescope (") α = 23 / aperture (objective/mirror diameter in mm)

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"
S-W 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"
S-W Heritage 100P 400 100 0.39" 0.23" 0.78" 0.46"
S-W Skymax-102 1300 102 0.38" 0.23" 0.76" 0.45"
S-W Skymax-127 1500 127 0.31" 0.18" 0.61" 0.36"
S-W 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.


Rules of Thumb for Fitting Telescope and Camera Sensor

The rules of thumb are based on a light wavelength of 550 nm. Derivations and exact formulas can be found here.

Resolving Power (after Rayleigh)

  • Resolving power [rad] = 0.00067 / Aperture [mm]
  • Resolving power ["] = 138.4 / Aperture [mm]

Pixel Size and Telescope Focal Length (Based on Resolving Power after Rayleigh)

  • Pixel size [µm] = Focal ratio * 0.3355
  • Focal length [mm] = Pixel size [µm] * Aperture [mm] / 0.3355

Pixel Size and Telescope Focal Length (Based on Seeing)

  • Pixel size [µm] = Telescope focal length [mm] * FWHM ["] / 412.5
  • Telescope focal length [mm] = Pixel size [µm] / (FWHM ["] * 412.5

Airy Disk

  • D [µm] = 2.44 * 0.55 * Focal ratio = 1.344 * Focal ratio
  • D ["] = 276.73 / Aperture [mm]

Image Scale

  • Image scale ["/pixel] = 206.265 * Pixel size [µm] / Focal length [mm]

Optimum Aperture Ratio, Optimum Focal Length

  • fo (BW) = Pixel size [μm] * 3.57
  • fo (Color) = Pixel size [μm] * 5.00 (according to a posting by Gerd Düring the B&W value of 3,57 is also valid for color)
  • Optimum focal length = fo* Aperture
  • Optimum focal length = Focal length * fo * Focal ratio = Focal length * fo / (f value of the telescope)

Field of View

The field of view (in arc-minutes, or ') is simply:

  • FOV [']= 3436 * D [mm] / (Effective focal length [mm])
  • FOV [°]= 3436 * D [mm] / ((Effective focal length [mm]) * 60) = 57.3 * D [mm] / (Effective focal length [mm])

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


References

 

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03.02.2021